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1 2 3
1
2
3
4
5
6
AN ELEMENTARY TREATISE
ON TBB
DYNAMICS OF A SYSTEM OF RIGID
BODIES.
E
AN ELEMENTARY TREATISE
ON THE
DYNAMICS OF A SYSTEM OF
RIGID BODIES.
/"
BY
EDWARD JOHN ROUTH, M.A., F.R.S., F.R.A.S., F.G.S., &c.
LATE FEIiliDW AND LATE ABSIBTANX TDIOB OF 8T FETEB'S COLLEOE, CAUBBIBOE^
LATE EXAMINEB IN THE UNIVEBSITY OF LONDON.
TUmH EDITION, REVISED AND ENLAB
Uonljon :
MACMILLAN AND CO
1877.
[All Righta reserved.]
PBINTBD BY a J, ci^y, a.^.
AT THB 0MVBBSITY PEE88.
QA
HI
PREFACE.
In this edition I have made many additions to every part
of the subject. I have been led to do this, because there are
so many important applications which it did not seem proper
to pass over without some notice. I have found how difficult
it is not to render a book formidable to the student by its size
and yet to supply some information at least on all the chief
points of a great subject. I believe the reader will not find
any portion treated at greater length than is necessary to render
the argument intelligible.
As in the former editions, each chapter has been made as
far as possible complete in itself, so that all that relates to any
one part of the subject may be found in the same place. This
arrangement will be found convenient for those who are already
acquainted with the subject, as it will enable them to direct
their attention to those parts in which they may feel most in-
terested. It will also enable the student to select his own order
of reading the subject. The student who is just beginning
Dynamics may not wish to be delayed by a long chapter of
preliminary analysis before he enters on the real subject of the
book. He may therefore begin at D'Alembert's Principle and
R. D. b
VI
PREFACE.
only read those parts of Chapter I. to which reference is made.
Other readers may also wish to pass on as soon as possible to
the great principles of Angular Momentum and Vis Viva.
Though a different order will be found advisable for different
persons, I have ventured to indicate a list of Articles to which
those who are just beginning the subject should first turn their
attention.
It will be observed that a chapter has been devoted to the
discussion of Motion in Two Dimensions. This course has been
adopted because it seemed expedient to separate the difHculties
of Dynamics from those of Solid Geometry.
I have attempted to give a slight historical notice whenever
I felt it could be briefly done. This course, if not carried too
far, will I believe be found to add greatly to the interest of the
subject. But the success of this attempt is far from complete.
In the earlier portions of the subject I had the* guidance of
Montuela, and further on there was Prof. Cay ley '3 Report to the
British Association. With the help of these the task became
comparatively easy; but in some other portions the number of
Memoirs which have been written is so vast, that anything but
the slightest notice has been rendered impossible. A useful
theorem is many times discovered, and probably each time with
some variations. It is thus often difficult to ascertain who is
the first author. I have therefore found it necessary to correct
some of the references given in the second edition, and to add
references where there were none before.
Throughout each chapter there will be found numerous ex-
amples, many very easy and others which are intended for the
more advanced student. In order to obtain as great a variety
of problems as possible, I have added a further collection at
the end of each chapter, taken from the Examination Papers
which have been set in the University and in the Colleges.
PREFACE.
VII
Some of these ure such excellent illustrations of dyuamical
piinciples that they will certainly be of the greatest assistance
to the student.
I cannot conclude without expressing how much I am in-
debted to Mr Webb, of St John's College, for the great assist-
ance he has given me in correcting the proofs of the first eight
chapters, and for the suggestions he has made to me. Most of
the examples in these chapters have also been very kindly
verified by him. Several others also of my friends have greatly
assisted me by correcting some proof-sheeta for me, particularly
Mr Edwards, of Sidney Sussex College, who has read the proofs
of the last three chapters.
Some portions of this edition have been written several years
ago, and the printing has extended over two years. This course,
though open to many objections, was rendered unavoidable by
the pressure of other engagements. I have theretore found it
necessary to add a few Notes, chiefly historical, at the end of
the treatise.
EDWARD J. ROUTH.
Fetebhovse,
Aptil 24, 1877.
Page 77.
Lino 80.
For h + h read h+h'.
„ 103,
II
23.
For in that case read in these oases.
,, 113.
>»
18.
For 0O8 2tf read oostf.
„ 181.
i>
8.
For ,-«'*, tan -i?««d-«'* tan -1?.
2 -At* n 2-/[*» fi.
« 141.
For V read V thronghont the page.
,^ 146.
II
1.
Omit the word necessary.
„ 255.
II
2.
For is read is parallel to.
„ 264.
M
38.
For 1 read 1 .
„. 287.
II
14,
For single read simple.
„ 297.
II
21.
For ^h read ^2h.
„ 299.
II
80.
For by A read by 2A.
M 808.
M
34.
For read -0.
>» >i
II
35.
For of three read of the axes referred to three.
II II
II
86.
For Art. 235 read Note to Art. 235.
„ 304.
»l
17.
For f" read <^",
„ 813.
II
17.
For t read t, t^.
„ 330.
II
16.
For - read +.
»i »•
II
17.
For X read f(x).
„ 331.
11
8.
For - read +.
CONTENTS.
CHAPTER I.
ON MOMENTS OF INERTIA.
ABTS.
1—11. On finding Moments of Inertia by integration
12—18. Other methods of finding Moments c' Inertia
19—38. The Ellipsoids of Inertia ....
84 — 47. On Equimomental Bodies and on Inversion
48 — 66. On Principal Axes
PAOES
1-9
9—15
15-23
23—32
82—46
CHAPTER II.
d'alembert's principle, &c.
66 — 88. D'Alembert's Principle and the Bqnations of Motion .
84 — 87. Impulsive forces
47—60
60—64
CHAPTER III.
MOTION ABOUT A FIXED AXIS.
88—91. Ecinations of Motion 65—67
92—97. The Pendulum 67—76
98 — 108. Length of the seeoads' pendulum 76 — 82
109. Oscillation of a watch-balance 82—84
110—118. Pressures on the fixed axis 84—93
119. The Centre of Percussion 93—94
120—122. The Ballistic Pendulum 94—99
CHAPTER IV.
MOTION IN TWO DIMENSIONS.
123—138. General Methods and Examples 100—120
139—141. The Stress at any point of a rod 120—123
142—161. On Friction 123—131
152—168. Cn Impulsive Forces 131- -146
169—172. On Initial Motions 146—160
173—181. On Belative Motion and Moving Axes .... 151—168
Examples 159—163
CONTENTS.
I<
ADTS.
182—196.
197—217.
218—226.
227—238.
239—242.
243—263.
261—277.
278—283.
284—289.
290—294.
295—304.
305-316.
CHAPTER V.
MOTION IN THREE DIMENSIONS.
Translation and Rotation .
Composition of Botations .
Motion referred to Fixed Axes
Euler's Equations
Expressions for Angular M)meutum
Moving Axes and Kelative Motion
Motion relative to the Earth
PAORS
164—170
170—181
181—187
187—194
194-197
197—213
213-226
CHAPTER VI.
ON MOMENTU.M.
On Momentum, with examples 227 — 233
Sudden changes of motion 233 — 239
The Invariable Plane 239 — 242
Impulsive forces in three dimensions 242 — 247
Impact of rough elastic bodies 247—256
Examples 255—257
CHAPTER VII.
VIS VIVA.
318—333. The Force-Fu ction and Work . . . .
334 — 35% Conservation of Vis Viva and energy .
353—367. Carnot's, Gauss', and Clausius' Theorems .
358—365. Newton's Principle of Similitude
366—390. Lagrange's and Sir W. K. Hamilton's Equations
391 — 398. Principles of Least Aotion and Varying Action .
399 — 409. Solution of the general equations of motion
Examples . .
258—267
267—279
279—283
283—287
288—304
306—312
313—321
322—324
CONTENTS.
Xi
CHAPTER VIII.
SMALL OSCILLATIONS.
ARTS.
410—415. OsoillationB with one degree of freedom
416 426. First method of forming the equations of motion
427 — 430. Second method of forming the equations of motion
431 — 437. Oscillations with two or more degrees of freedom
43g — 443. Composition of oscillations and transference of Energy
444 — 461. Lagrange's Method of forming the equations of motion
462 — 469. The energy test of Stability, with an extension to certain
cases of motion
470 — 484. Oscillations about Steady Motion with application to the
Governor and Laplace's three particles, and some
general theorems on Stability
485—489. Tlie Calculus of Fmite Differences
490 —495. The Cavendish Experiment
496 — 507. Oscillations of the second order
Examples
PAGES
325—331
831—841
341-845
845—354
854—356
356—369
369—376
375—886
886—889
389—394
395—401
401—403
CHAPTER IX.
MOTION OF A BODY UNDER THE ACTION OF NO FORCES.
608—510. Solution of Euler's Equations 404—407
611 — 522. Poinsot's and Mac Cullagh's construction for the motion . 407 — 417
623 — 536. The Cones described by the invariable and instantaneous
axes 417—426
637-540. Motion of the Principal Axes . . . . . 427—429
541—544. Motion when .4 ^ B 430—432
545-552. Motion when G« = J»Z'. . . . ' . . . . 432—437
653—666. Correlated and Contra-related Bodies 437 — 441
Examples 442—443
CHAPTER X.
557-
672-
-571.
-588.
689—698.
599.
MOTION UNDER ANY FORCES.
Motion of a Top 444—467
Motion of a sphere on perfectly rough surfaces of various
forms and on an imperfectly rough inclined plane.
Billiards 457-473
Motion of a Solid Body on a plane which is perfectly
rough, imperfectly rough, or smooth .... 473 — 485
Motion of a Rod 485—487
Examples 487-489
Xll
CONTENTS.
CHAPTER XL
PRECESSION AND NUTATION, &0.
ABTB.
COO— 609. On the Potential .....
610—624. Motion of the Earth about its centre of gravity
625—634. Motion of the Moon about its centre of gravity
PAOKS
490—498
499—618
614—522
CHAPTEE XII.
MOTION OF A STRING OR CHAIN.
636—640. The Equations of Motion .
641-644. On Steady Motion ....
645 — 660. On Initial and Impulsive Motions
661—662. Small Oscillations of a loose chain
663 —672. Small Oscillations and energy of a tight string
623—528
628—532
632—635
636—546
647-666
NOTES.
On D'Alembert's Principle 557
On Euler's Geometrical Equations 658
On the Impact of Bodies 669
On Sir W. B. Hamilton's Equations 560
On the Principle of Least Action 560
On Sphero-ConioB 662
Miscellaneous Notes 664
The student, to whom this subject is entirely new, is advised to read first the
following Articles :— 1— 24, 36, 48—51, 66—68, 71, 73—93, 98—102, 110—116, 119
—120, 123—150, 152—163, 156—163, 169, 171—191, 197—208, 218—220, 227—
832, 235, 239—241, 243—246, 278—281, 284—285, 290—293, 295—298, 318-328,
834—348, 350, 366—369, 874—376, 410—412, 416—419, 424, 427—430, 444—445,
449-461, 462—464, 490—495, 608—609, 511-519, 522, 587, 641—544.
I i
CHAPTER I.
ON FINDING MOMENTS OF INERTIA BY INTEGRA.TION.
1. In the subsequent pages of this wofk it will be found
that certain integrals continually recur. It is therefore convenient
to collect these into a preliminary chapter for reference. Though
the bearing of these on Dynamics may not be obvious beforehand,
yet the student may be assured that it is as useful to be able to
write down moments of inertia with facility as it is to be able
to quote the centres of gravity of the elementary bodies.
In addition however to these necessary propositions there are
many others which are useful as giving a more complete view of
the arrangement of the axes of inertia in a body. These also
have been included in this chapter though they are not of the
same importance as the former.
2. All the integrals used in Dynamics as well as those used
in Statics and some, other branches of Mixed Mathematics are
included in the one form
jijoifi/hydxdydz,
where (a, /8, 7) have particular values. In Statics two of these
three exponents are usually zero, and the third is either unity
or zero, according as we wish to find the numerator or denomi-
nator of a coordinate of the centre of gravity. In Dynamics
of the three exponents one is zero, and the sum of the other two
is usually equal to 2. The integral in all its generality has not
yet been fully discussed, probably because only certain cases have
any real utility. In the case in which the body considered is
a homogeneous ellipsoid the value of the general integral has
been found in gamma functions by Lejeune Dirichlet in Vol. iv.
of Liouville's Journal. His results were afterwards extended by
Liouville in the same volume to the case of a heterogeneous
ellipsoid in which the strata of uniform density are similar
ellipsoids.
In this treatise, it is intended to restrict ourselves to the con-
sideration of moments and products of inertia, as being the only
cases of the integral .vhich are iseful in Dynamics.
R. D. 1
i
.
2
MOMENTS OF INERTIA.
i. ]
!M
3. If the mass of every particle of a material system bo
multiplied by the square of its distance from a straight line, the
sum of the products so formed is called the moment of inertia of
the system about that line.
If M be the mass of a system and k be such a quanti^^^y that
MJ(? is its moment of inertia about a given straight line, then k
is called the radius of gyration of the system about that line.
The term " moment of inertia " was introduced by Euler, and
has now got into general use wherever Rigid Dynamics is studied.
It will be convenient for us to use the following additional terms.
If the mass of every particle of a material system be multi-
plied by the square of its distance from a given plane or from a
given point, the sum of the products so formed is called the
moment of inertia of the system with reference to that plane or
that point.
If two straight lines Ox, Oy be taken as axes, and if the mass
of every particle of the system be multiplied by its two co-
ordinates X, y, the sum of the products so formed is called the
product of inertia of the system about those two axes.
This might, perhaps more conveniently, be called the product
of inertia of the system with reference to the two co-ordinate
planes xz, yz. .
4. Let a body be referred to any rectangular axes Ox, Oy,
Oz meeting in a point 0, and let x, y, z be the co-ordinates of any
particle w, then according to these definitions the moments of
inertia about the axes of x, y, z respectively will be
A = Xm{f-\-z\ B = tm{e-\-a^), a = tm{x' + y^).
The moments of inertia with regard to the planes yz, zx, xy,
respectively, will be
A'^^Xma?, B' = Xmy\ C' = t^.nz'.
The products of inertia with regard to the axes yz, zx, xy,
will be
D = Xmyz, J? = %mzx, F= Xmxy.
Lastly, the moment of inertia with regard to the origin will be
/T = Sm (ir ' + y' + z^) = tmr\
ivhere r is the distance of the particle m from the origin.
5. The following propositions may be established without
difficulty, and will serve as illustrations of the preceding defi-
nitions.
(1) The three moments of inertia A, B, G about three
rectangular axes are such that the sum of any two of them is
greater than the third.
BY INTEGRATION.
3
(2) The sum of the moments of inertia about any three
rectangular axes meeting at a given point is always the same ;
and is equal to twice the moment of inertia with respect to that
point.
For A ;-5 + C=2Sm(x' + j/* + 2') = 22»ir', and is therefore independent of the
directions of the axes.
(3) The sum of the moments of inertia of a system with
reference to any plane through a given point and its normal at
that point is constant and equal to the moment of inertia of the
system with reference to that point.
Take the given point as origin and the plane as the plane of xy, then
C"+ G='2.m,r^, which is independent of the direction of the axes.
Hence we infer that
A' = \{B-\rC-A), B'==l{C^A-B), and C'=\iA+B-C).
(4) Any product of inertia as D cannot numerically be
so gi'eat as \A.
(5) If A, B, F be the moments and product of inertia of a
lamina about two rectangular axes in its plane, then AB is greater
than F^.
If t be any quantity we have At^-\-2Fl + D='S.m{yl-Vxy=a. positive quantity.
Hence the roots of the quadratic A0-v2Fl + B=(i are imaginary, and therefore
AB>F^.
(6) Prove that for any body
{A + B-C){B + C-A) > ^E',
{A-\-B- C){B + C-A){G+A-B) = SDEF.
(7) Prove that the moment of inertia of the surface of a
hemisphere of radius a and mass M about the diameter perpen-
dicular to the base is Jlffa'.
For, complete the ..phere, then by (2) the moment of inertia about any diameter
is two-thirds of tho moment of inertia with respect to the point.
6. It is clear that the process of finding moments and products
of inertia is merely that of integration. We may illustrate this
by the following example.
To find the mor.ient of inertia of a uniform triangular plate
about an axis in its plane passing through one angular pc'nt.
Let ABC be the triangle, Ai/ the axis about which the
moment is required. Draw Ax perpendicular to At/ and produce
BC to meet Ay in D. The given triangle ABC may be regarded
as the difference of the triangles ABU, AC I). Let us then first
find the moment of inertia of ABD. Let PQP'Q' be an ele-
mentary area whoso sides PQ, FQ' arc parallel to tho base AJJ,
1—2
MOMENTS OF INERTIA.
and let PQ cut Ax in M. Let /3 be the distance of the angular
point B from the axis Ay, AM= x and AD = l.
I
i
m
\ii
V
B — X
Then the elementary area PQP'Q' is clearly r dx, and
B —x
its moment of inertia about Ay is fil _ dx .x\ fi being the
mass per unit of area. Hence the moment of inertia of the
triangle ABD
Similarly if 7 be the distance of the angular point C from the
axis Ay, the moment of inertia of the triangle A CD is nl~^'
Hence the moment of inertia of the given triangle ABC is
H'Tai^^ "y*)' Now^^)S and ^Zy are the areas of the triangles
ABD, ACD. Hence if M be the mass of the triangle ABG, the
moment of inertia of the triangle about the axis Ay is
Ex. If each element of the mass of the triangle be multiplied by the nth power
of its distance from the straight line through the angle A, then it may be proved
in the same way that the sum of the products is
2M_ /3''+l->y"^-l
(w + i){n + 2) /3-7 •
7. When the body is a lamina the moment of inertia about an
axis perpendicidar to its plane is equal to if>e sum of the moments
BY INTEGRATION. 6
of inertia about any two rectangular axes in its plane drawn from
the point where the former axis meets the plane.
For let the axis of 2 be taken as the normal to the plane, then,
if A, B, C be the moments of inertia about the axes, we have
A = ^my\ B^Xmx", G^Xmix' + f),
and therefore G=A + B,
We may apply this theorem to the case of the triangle. Let
fi', 7' be the distances of the points B, C from the axis Ax. Then
the moment of inertia of the triangle about a normal to the plane
of the triangle through the point A is
= f(^* + ^7 + 7^ + /3'^ + /3V + 7'^;.
8. The following moments of inertia occur so frequently that
they have been collected together for reference. The reader is
advised to commit to memory the following table :
The moment of inertia of
(1) A rectangle whose sides are 2a and 26
about an axis through its centre in its plane per- ")
pendicular to the side 2a J
about an axis through its centre perpendicu- ) _
lar to its plane j ~
(2) An ellipse semi-axes a and h
about the major axio t = mass ^-j
mass
a*
mass -K- ,
o
a' + y
minor axis h = mass
a
4'
about an axis perpendicular to its plane) _ a'+h*
through the centre ] ~ "^^^^ 4 •
In the particular case of a circle of radius a, the moment of
2
inertia about a diameter is mass j- , and about a perpendicular to
2
its plane through the centre mass -^ .
(3) An ellipsoid semi-axes a, h, c
about the axis a = mass — - — .
In the particular case of a sphere of radius a the moment of
, 2
inertia about a diameter = mass ■= a^
5
»; 1
6
MOMENTS OF INERTIA.
'.
:! I
hi
1 i
f ■ i
Hi '
I
I' !
(4) A right solid whose sides are 2a, 26, 2c
about an axis through its centre perpendicular ] _ 6' + c*
to the plane containing the sides b and o j ~ ^^^^ 3" •
These results may be all included in one rule, which the
author has long used as an assistance to the memory.
Moment of inertia ) (s"«^ of squares of perpendicular
about an axis [= mass semi-axes) ^
of symmetry J 3, 4 or 5
The denominator is to be 3, 4 or 5, according as the body is
rectangular, elliptical or ellipsoidal.
Thus, if we wanted the moment of inertia of a circle of radius
a about a diameter, we notice that the perpendicular semi-axis in
its plane is the radius a, and the semi-axis perpendicular to its
plane is zero, the moment of inertia required is therefore M -^ ,
if M be the mass. If we wanted the moment about a perpendi-
cular to its plane through the centre, we notice that the perpen-
dicular semi-axes are each equal to a and the moment required is
therefore
M
a" fa'
= 1/
or
2*
9. As the process for determining these moments of inertia is very nearly tho
same for all these cases, it will be sufficient to consider only two instances.
To determine the moment of inertia of an ellipse about the minor axis.
Let the equation to the eUipse hoy=- ^a'-' - «■'. Take any elementary area PQ
parallel to tho axis of ift then clearly the moment of inertia is
4/t / x^i/dx = ip,- J x'' t^a'-x^dx,
where n is tho mass of a unit of a 'ea.
B
To integrate this, put ;(;=a sin <p, then tho integral becomes
w n
I Pa 9 _i -9^11 J /*!! 1 - cos 4a , ira*
a' / •' cos^ (p sma <pd<f)=a* I _— ^ d0 1^^ ^„ ;
'',1 *^0 o 10
. •. the moment of inertia =u7r«6 :r=masB-7 .
4 4
hi
BY INTEGRATION.
To determine the moment of inertia of an ellipsoid about a prlneipal diameter.
Let tho equation to tho elUpsoid be -« + tj + -3= !• Take any elementary area
PNQ parallel to the plane of yz. Its area is evidently rPN . Qff. Now PN is tho
C
value of z when ^=0, and QN the value of y when z =0, as obtained from the equa-
tion to the ellipsoid; .♦. PA'=- V*'-«*. QiV=- Va'-«';
irhc
.'. the area of the elements —? (a^ - x*).
Let n be the mass of a unit of volume, then the whole moment of inertia
4 . J^ + ca
= mass
6Hc»
Ex. I. The moment of inertia of an arc of a circle whose radius is a and which
subtends an angle 2a at the centre
(c) about an axis through its centre perpendicular to its plane =3f a',
[b) about an axis through its middle point perpendicular to its plane
(c) about the diameter which bisects the arc =M (l — 5—^ | ^ .
Ex. 2. The moment of inertia of the part of the area of a parabola out off by
3
any ordinate at a distance x from the vertex is M^x^ about the tangent at the
vertex, and 3/^ about the principal diameter, where y is the ordinate correspond-
5
ing to X'.
MOMENTS OF INERTIA.
Ex. 3. The momoat of inertia of the area of the lemnisoate r* = a* cos i$ about a
line through the origin iu its plane and perpendicular to its axis ia M —.
8ir+8
48
o».
Ex. 4. A lamina ia bounded by four rectangular hyperbolas, two of them have
the axes of co-ordinates for asymptotes, and the other two have the axes for principal
diameters. Prove that the sum of the moments of inertia of the lamina about the
co-ordinate axes is
|(<i»-«"')(/3'-n,
where aa, /3/3' are the semi-major axes of the Lyperbolas.
Take the equations xy=u, x^-y*=v, then the two moments of inertia are
A = jJx*J du dv aaH B= fh/*J du dv, where -^ is the Jacobian of uv with regard to
xy. This gives at once 4 + B= J ijdudv, where the limit* are clearly m = | to
^,r=/3«tor=/3'».
Ex. 5. A lamina is bounded on two sides by two similar ellipses, the ratio of
the axes in each being m, and on the other two sides by two similar hyperbolas, the
ratio of the axes iu each being n. These fotir curves have their principal diameters
along the co-ordinate axes. FroviQ that the product of inertia about the co-ordinate
fa' - o') CS' - fl''>
axes is ^ — . , , ,, ' , where aa', BB' are the semi-major axes of the curves.
4 (m' + »')
10. Many moments of inertia may be deduced from those
given in Art. 8 by tiie method of differentiation. Thus the
moment of inertia of a solid ellipsoid of uniform density p about
the axis of a is known to be k trdbcp — = — • Let the ellipsoid
increase indefinitely little in size, then the moment of inertia of
the enclosed shell is
, (4 , 6' + &
a i^irabcp — ^
This differentiation can be effected as soon as the law according
to which the ellipsoid alters is given. Suppose the bounding
ellipsoids to be similar, and let the ratio of the axes in each be
= », - = 7. Then
4 p^ + 0"
moment of inertia of solid ellipsoid = n'fppg, e ^^
.*. moment of inertia of shell = ^ "irppq (p' + <f) a*da.
In vhe same way
4
mass of solid ellipsoid = ^ irppqa^ ;
.*. mass of shell = iirpjjqd^da.
a* COB 20 about a
OTHER METHODS.
Hence the moment of inertia of an indefinitely thin ellipsoidal
6* + c'
shell of mass J/" bounded by similar ellipsoids is M — « — .
By reference to Art. 8, it will be seen that this is the same as
the moment of inertia of the circumscribing right solid of equal
mass. These two bodies therefore have equal moments of inertia
about their axes of symmetry at the centre of gravity.
11. The moments of inertia of a heterogeneous body whoso
boundary is a surface of uniform density may sometimes be found
by the method of diiferentiation. Suppose the moment of inertia
of a homogeneous body of density D, bounded by any surface of
uniform density, to be known. Let this when expressed in terms
of some parameter a be ^ (a) D. Then the moment of inertia of a
stratum of density D will be <f>' (a) Dda. Replacing D by the
variable density p, the moment of inertia required will be
1/30' (a) da.
Ex. 1. Shew that the moment of inertia of a heterogeneous ellipsoid about tho
major axis, the strata of uniform density being similar concentric ellipsoids, and
the density along the major axis varying as the distance from the centre, is
il/^Cft' + c').
Ex. 2. The moment of inertia of a heterogeneous ellipse about the minor axis,
the strata of uniform density being confocal ellipses and the density along the minor
axis varymg as the distance from the centre, is 7^7 „ ■< . » — z — 5 •
•' " '20 20^+0*- 3ac'
Other methods of finding moments of inertia.
12. The moments of inertia given in the table in Art. 8 are
only a few of those in continual use. The moments of inertia of an
ellipse, for example, about its principal axes are there given, but
we shall also frequently want, its moments of inertia about other
axes. It is of course possible to find these in each separate case
by integration. But this is a tedious process, and it may be often
avoided by the use of the two following propositions.
The moments of inertia of a body about certain axes through
its centre of gravity, which we may take as axes of reference, are
regarded as given in the table. In order to find the moment of
inertia of that body about any other axis we shall investigate,
(1) A method of comparing the required moment of inertia
with that about a parallel axis through the centre of gravity.
10
MOMENTS OP INERTIA.
(2) A method of determining the moment of inertia about
this parallel axis in terms of the given moments of inertia about
the axes of reference.
13. Piiop. I. Given the moments and products of inertia
about all axes through the centre of grav'ty of a body, to deduce
the moments and products about all other parallel axes.
The moment of inertia of a body or system of bodies about
any axis is equal to the moment of inertia about a parallel axis
through the centre of gravity plus the moment of inertia of the
whole mass collected at the centre of gravity about the original
axis.
The product of inertia about any two axes is equal to the
product of inertia about two parallel axes through the centre of
gravity plus the product of inertia of the whole moss collected at
the centre of gravity about the original axes.
Firstly, take the axis about which the moment of inertia is
required as the axis of z. Let m be the mass of any particle of
the body, which generally will be any small element. Let x, y, z
be the co-ordinates of m, S, y, i those of the centre of gravity
G of the whole system of bodies, x', y', z those of m referred to
a system of parallel axes through the centre of gravity.
Then since
\m^
Xmy'
Xm
\mz
- are the co-ordinates of the
%m ' lim ' S»i
centre of gravity of the system referred to the centre of gravity
as the origin, it follows that Swa?' = (,„ 2my' = 0, Sw/ = 0.
The moment of inertia of the system about the axis of z is
V
= Sw («' + 3/*),
= Sm (i" -I- ^ + 2m (»" + y\ -f- 2a; . Swa/ + 2y . ^my.
Now Sw (S' + p") is the moment of inertia of a mass 2m
collected at the centre of gravity, and 2m (aj"+y'*) is the moment
of inertia of the system about an axis through (?, also Sma;' = 0,
2my = ; whence the proposition is proved.
Secondly, take the axes of x, y as the axes about which the
product of inertia is required. The product required is
= 2ma;^ = tm {x + x') (^ + y'),
= xy . 2m + 2m x'y + xZmy + y'Xmx'
= xy%m + 'Sitnx'y.
Now xy . 2m is the product of inertia of a mass 2m collected
at G and Xmxy is the product of the whole system about axes
through G ; whence the proposition is proved.
OTHER METHODS.
11
Lot there bo two parallel axes A and 7? at tlLstancos a and b
from the centre of gravity of tho body. Then, if M bo the mass
of tho material system,
moment of inertia] ,- , _ jmomcnt of inertia ,,.,
about A ) \ about B
Hence when the moment of inertia of a body about one axis
is known, that about any other parallol axis may be found. It is
obvious that a similar proposition holds with regard to tho pro-
ducts of inertia.
14. The preceding proposition m ly bo generalised as follows.
Let any system be in motion, and let x, y, z be tho co-ordinates
at time t of any particle of mass m, thoii -y- , -j:^ 7/7 ^"^^ *^^^
d^x cCv cPz
velocities, and ;^ > j^ > j^ ^^^ accelerations of tho particle
resolved parallel to the axes. Suppose
tr -%} A f dx d^x dy d^y dz d*z\
V=Xm<f>{x. j^. ^,. y, -^^, J, z, ^^, ^,j
to be a given function depending on the structure and motion of
the system, the summation extending throughout the system.
Also let <f> be an algebraic function of the first or second order.
Thus <l> may consist of such terms as
Aa? + Bx^^ + c(^^\Eyz + Fx +
where A, B, C, &c. are some constants. Then the following
general principle will hold.
"The value of V for any system of co-ordinates is equal to
the value of V obtained for a parallel system of co-ordinates with
the centre of gravity for origm plus the value of V for the whole
mass collected at the centre of gravity with reference to the first
system of co-ordinates."
For let X, y, z, be the co-ordinates of the centre of gravity,
(lOS d'JT {LSR
and let a; = » + x\ &c. •*• 77: = ;^ + "jT > ^^'
Now since j> is an algebraic function of the second order of
X, -r. , -^ ; y, &c. it is evident that on making the above sub-
stitution and expanding, the process of squaring &c. Avill lead to
three sets of terms, those containing only x, -7- , -1-5 , &c., those
containing the products of x, x &c., and lastly those containing
i, ■ '
(i^ i
12
MOMENTS OP INEBTU.
dx
only a?', , , &c. The first of these will on the whole make up
ff> (x, r: > &c.] , and the last <j) (x, -,- , &c. j .
Hence we have
where A, B, C, &c. are some constants.
/_ dx'\ . _ dx'
Now the term Xm ( a; -17 ) is the same as x%m —tt , and this
fw y*
vanishes. For since %mx' = 0, it follows that Xm -y- = 0. Simi-
an
larly all the other terms in the second line vanish.
Hence the value of V is reduced to two terms. But the first
of these is the value of V at the origin for the whole mass col-
lected at the centre of gravity, and the second of these the value
of V for the whole system referred to the centre of gravity as
origin. Hence the proposition is proved.
The proposition would obviously be true if -7-3 , -A , -7-3 ,
or any higher differential coefficients were also present in the
function V.
15. Prop. IL (Hven the moments and products of inertia
about three straight lines at right angles meeting in a point, to
deduce the moments and produces of inertia about all other axes
meeting in that point.
Take these three straight lines as the axes of co-ordinates.
Let A, B, G be the moments of inertia about the axes of x, 1/, z;
D, E, F the products of inertia about the axes of yz, zx, xy. Let
a, )3, 7 be the direction-cosines of any straight line through the
origin, then the moment of inertia / of the body about that line
will be given by the equation
/ = ^a' + 5/8' + CV" - 2Z)/97 - 2^7x - 2Fa^.
Let P be any point of the body at which a mass m is situated,
and let x, y, z be the co-ordinates of P. Let ON be the line
whose direction-cosines are a, /9, 7, draw PN perpendicular to ON".
Since ON'\b the projection of OP, it is clearly
= xa + yli + z%
also OP" = x^ ^y' + z\ and 1 = a" + /8« + 7'^
OTHER METHODS.
13
The moment of inertia / about 0N= Xm PN^
= 2m [x^ + y + s^ _ (ax + ^y + ^zf]
= till {{x' + y + 3") {%' + /3' + 7') - [ax + ^y + yzf]
= Xm Q/^ + z') a' + tm {z" + x^) /S^ + Im (x' + f) y""
— 2Xmyz . fiy — 2%mzx . y% — 2Xmxy . a/3
= Aa^ + B^+ CV" - 2D/37 - 2Ey2 - iFa^.
It may be shewn in exactly the same manner that if A'B'C
be the moments of inertia with regard to the planes yz, zx, xy,
then the moment of inertia with regard to the plane whose direc-
tion-cosines are a, /9, 7 is
/' = A'o? + 5'/3» + Cy + 2D^y + 2EyoL + 2 Fa^.
It should be remarked that this formula differs from the
moment about a straight line in the signs of the three last
terms.
16. When three straight lines at right angles and meeting in
a given point are such that if they be taken as axes of co-ordi-
nates the products Xmxy, Xmyz, Xmzx all vanish, these are said
to be Principal Axes at the given point.
The three planes through any two principal axes are called
the Principal Planes at the given point.
The moments of inertia about the principal axes at any point
are called the Principal Moments of Inertia at that point.
17. The fundamental formula in Art. 15 may bo much sim-
plified if the axes of co-ordinates can be chosen so as to bo
principal axes at the origin. In this case the expression takes
the simple form
I==Ai' + B^'+Cy\
A method will presently be given by which we can always
find these axes, but in some simpler cases wo may determine
; I
^ m
H^]
H!
■■ ' i
p
14
MOMENTS OF INERTIA.
their position by inspection. Let the body be symmetrical about
the plane of xy. Then for every element m on one side of the
plane whose co-ordinates are (x, y, z) there is another element of
equal mass on the other side whose co-ordinates are (a;, y, —z).
Hence for such a body ^irnxz = and Itmyz = 0. If the body be
a lamina in the plane of xy, then the z of every element is zero,
and we have again %mxz = 0, Xmyz = 0.
Eecurring to the table in Art. 8, we see that in every case the
axes, about which the moments of inertia are given, are principal
axes. Thus in the case of the ellipsoid, the three principal
sections are all planes of symmetry, and therefore, by what has
just been said, the principal diameters are principal axes of
inertia. In applying the fundamental formula of Art. 15 to any
body mentioned in the table, we may therefore always use the
modified form given in this article.
18. Letusnow consider how the two important propositions of Arts. 13 and 15
are to be applied in practice.
Ex. 1. Suppose we want the moment of inertia of an elliptic area of mass M
and semiaxes a and b about a diameter making an angle with the major axis. The
moments of inertia about the axes of a and b respectively are M -g and M -j .
Then by Art. 17 the moment of inertia about the diameter isM — cos* + M-r sin' 0.
4 4
If r be the length of the diameter this is known from the equation to the ellipse to
MaW
be the same as -^ — , , which is & very convenient form in practice.
Ex. 2. Suppose we want the moment of inertia of tha same ellipse about a
tangent. Let f> be the perpendicular from the centre on the tangent, then by Art.
13, the required moment is equal to the moment of inertia about a parallel axis
through the centre together with Mp^ = — — ^ +Mp*= -r!P\ since j)r=db.
Ex. 3. As an example of a different kind, let ns find the moment of inertia of an
ellipsoid of mass 3/ and semiaxes (a, 6, c) with regard to a diametral j)?ane whose direc-
iiou-cosines referred to the principal planes are (a, j9, 7). By Art. 8, the moments of
inertia with regard to the principal axes are M — = — , M — -= — , M ■ - .
555
Hence
by Art. 5, the moments of inertia with regard to the principal planes are M - ,
.M ■^, I/-5 . Hence the required moment of inertia is "^ (a^a? + b'^p,'^ + c'^'^^). If p
^ , M-= . Hence the required moment of inertia is ^^
5 5 5
be the perpendicular on the parallel tangent plane, wo know by solid geometry that
this IS the same sis M ■=■.
Ex. 4. The moment of inertia of a rectangle whose sides are 2a, 26 about a
diagonal is
2M aW
, 3 ((•■!. I //.'•
L
Arts. 13 and 15
to the ellipse to
ELLIPSOIDS OF INERTIA. 16
Ex. 6. If ki, k^ be the radii of gyration of an elliptic lamina about two conjugate
diameters, then j3 + jiri= ^ ^^2 "^ pj *
Ex. 6. The sum of the moments of inertia of an elliptic area about any two
tangents at right angles is always the same.
Ex. 7, If M be the mass of a right cone, a its altitude and b the radius of the
3
base, then the moment of inertia about the axis is ilf r-^'; ^bat about a straight
line through the vertex perpendicular to the axis ib M^(a^ + j\, that about a slant
side M ^ -^ — rj ; that about a perpendicular to the axis through the centre of
3
gravity i^ ^ ^ («" + ^S").
Ex. 8. If a be the altitude of a right cylinder, b the radius of the base, then the
moment of inertia about the axis is ilf ^ and that about a straight lino through the
I centre of gravity perpendicular to the axis is
M
«-)•
Ex. 9. The moment of inertia of a body of mass At about a straight line whoso
x-f _y-g _ z-h
\ equation is
= referred to any rectangular axes meeting at the
I m n
'^Im centre of gravity is
I AP + Bm^ + Cn^ - 2Dmn - 2Enl - 2Flm + M{f ^+g^ + h?-(fl + gm + kn)%
$ where {I, m, n) are the direction-cosines of the straight line.
Ex. 10. The moment of inertia of an elliptic disc whose equation is
ax^ + 2bxy + cy^ + 2dx + 2eij+l=0,
M -Ha
about a diameter parallel to the axis of x, is
, , where M is the mass and
4 ' (ac-by
II is the determinant oc - 6' + 2bed -ae^- ccP, usually called the discriminant.
Ex. 11. The moment of inertia of the elliptic disc whose equation in arcal co-
ordinates is <p {xyz) = about a diameter parallel to the side a is
-^(a) -2K\Ty-dz}'^'
I where A is the area, II the discriminant and K the bordered discriminant.
nes are M ■
a, 26 about a
The Ellipsoids of Inertia.
19. The expression which has been found in Art. !'.5 for the
moment of inertia / about a straight line whose direction-cosines
are (a, /i?, 7),
I = Aoi' + B^' + Cy^-2D/3y-2Eyx-2Fal3,
admits of a very useful geometrical interpretation.
I i
16
MOMENTS OF INERTIA.
\ >i
Let ca radius vector OQ move in any manner about the given
point 0, and be of such length that the moment of inertia about
OQ may be proportional to the inverse square of the length.
Then .if R represent the length of the radius vector whose direc-
tion-cosines are (a, ,8, y), we have / = -y^s- . where e is some
R'
constant introduced to keep the dimensions correct, and M is the
mass. Hence the polar equation to the locus of Q is
Me*
Aoi^ + B^'+ 6V - 2Z>/3y - 2EyoL - 2Fa^.
Transforming to Cartesian co-ordinates, we hav«
Me* = AX^ + BY'+CZ'-2DYZ- 2EZX- 2FXY,
which is the equation to a quadric. Thus to every point of a
material body there is a corresponding quadric which possesses
the property that the moment of inertia about any radius vector
is represented by the inverse square of that radius vector. The
convenience of thit, construction is, that the relations which exist
between the moments of inertia about straight lines meeting at
any given point may be discovered by help of the known proper-
ties of a quadric.
Since a moment of inertia is essentially positive, being by
definition the sum of a number of squares, it is clear that every
radius vector R must be real. Hence the quadric is always an
ellipsoid. It is called the momental ellipsoid, and was first used
by Cauchy, Exercices de Math. Vol. ii.
20. The momental ellipsoid is defined by a geometHcal pro-
perty, viz. that any radius vector is equal to some constar.c divided
by the square root of the moment of inertia about that radius
vector. Hence whatever co-ordinate axes are taken, we must
always arrive at the same ellipsoid. If therefore the momental
ellipsoid be referred to any set of rectangular axes, the coefficients
of X\ Y\ Z\ -2YZ, -2ZX, -2XY in its equation will still
represent the moments and products of inertia about the axes of
co-ordinates.
Since the discriminating cubic determines the lengths of the
axes of the ellipsoid, it also follows that its coefficients are un-
altered by a transf or nation of axes. But these coefficients are
A + B+0,
AB + BC + CA-D'-E^-F\
ABC - 2BEF - AD'' - BE' - CF\
Hence for all rectangular axes having the same origin, these are
invariable and all greater than zero.
ELLIPSOIDS OF IXEHTIA.
17
about the given
of inertia about
of the length.
tor wliose direc-
here
€ IS some
ct, and M is the
lis
2Fa/3.
;ry point of a
which possesses
ly radius vector
lus vector. The
ions which exist
lines meeting at
e known proper-
)sitive, being by
clear that every
"ic is always an
was first used
geometrical pro-
jonstar.c divided
tout that radius
iaken, we must
the momental
the coeflficie'^tv)
lation will still
•out the axes of
lengths of the
icients are un-
fefficients are
[rigin, these fire
21. It should be noticed that the constant e is arbitrary,
Ithouo-h when once chosen it cannot be altered. Thus we have a
series of similar and similarly situated ellipsoids, any one of
[which may be used as a momental ellipsoid.
When the body is a plane lamina, a section of the ellipsoid
[corresponding to any point in the lamina by the plane of the
[lp,mina, is called a momental ellipse of that point.
22. If principal axes at any point of a body be taken as
(axes of co-ordinates, the equation to the momental ellipsoid takes
the simple form AX"" + BY^ + CZ^=Me\ where J/ is the mass
[and e* any constant. Let us now apply this to some simple cases.
Ex. 1. To find the momental ellipsoid at the centre of a material elliptic disc.
T2 ^2 /I- 4. ?i2
I Taking the same notation as before, we have A = 'M j, B = M j , C = M .
iPIence the ellipsoid is
4 4
a" + 6"
Z^=Mt*.
|Siuce 6 is any constant, this may be written
If When Z=0, this becomes an ellipse similar to the boundary of given disc. Hence
;; ywe infer that the momental ellipse at the centre of an elliptic area is any similar
'^d similarly situated elUpse. This also follows from Art. 18, Ex. 1.
^ Ex. 2, To find the momental ellipsoid at any point O of a material straight rod
A B of mass M and length 2a. Let the straight line OAB be the axis of x, O the
/•forigin, the middle point of AB, 00 =c. If the material line can be regarded as
■M ■ /a' \
^Jipndefinitely thin, .4=0, i'=ilf( — + c') = C, henci the momental ell'psoid is
3p + Z*=e"', where e' is any constant. The momenta! ellipsoid is therefore an
elongated spheroid, which becomes a right cylinder having ihe straight line for axis,
vheu the rod becomes indefinitely thiiu
Ek. ' The momental ellipsoid at the centre of a material ellipsoid is
(62 + c") A'2 + (fl' 4 a^) P + (a2 + 6«) Z^ = e*,
vhere c is any constant. It should be noticed that the longest and shortest axes of
he momental ellipsoid coincide in direction with the longest and shortest axes
Respectively of the material ellipsoid.
23. By a consideration of some simple properties of ellipsoids,
[ho following propositions are evident :
I. Of the moments of inertia of a body about axes meeting at
given point, the moment of inertia about one of the principal
iixes is greatest and about another least.
For, in the momental ellipsoid, the moment of inertia about
|iny radius vector from the centre is least when that radius vector
R. D. 2
f,l
'«
kb'
\i
■■ii
. !
18
MOMENTS OF INERTIA.
is greatest and vice versd. And it is evident that the greatest and
least radii vectores are two of the principal diameters.
It follows by Art. 5 that of the moments of inertia with
regard to all planes passing through a given point, that with
regard to one principal plane is greatest and with regard to
another is least.
II. If the three principal moments at any point be equal
to each other, the ellipsoid becomes a sphere. Every diameter is
then a principal diameter, and the radii vectores are all equal.
Hence every straight line through is a principal axis at 0, and
the moments of inertia about them are all equal.
For example, the perpendiculars from the centre of gravity of
a cube on the three faces are principal axes ; for, the body being
referred to them as axes, we clearly have Xmxy = 0, %myz = 0,
Sm^ic = 0. Also the three moments of inertia about them are by
symmetry equal. Hence every axis through the centre of gravity
of a cube is a principal axis, and the moments of inertia about
them are all equal.
Next suppose the body to be a regular solid. Consider two
planes drawn through the centre of gravity each parallel to a faco
of the solid. The relations of these two planes to the solid are
in all respects the same. Hence also the m omental ellipsoid at
the centre of gravity must be similarly situated with regard to
each of these planes, and the same is true for planes parallel to all
the faces. Hence the ellipsoid must be a sphere and the moment
of inertia will be the same about every axis.
24. At every point of a matenal system there ai^ always three
principal axes at right angles to each other.
Construct the momental ellipsoid at the given point. Then it
has been shown that the products of inertia about the axes are
half the coefficients of — XY, — YZ, — ZX in the equation to the
momental ellipsoid referred to these straight lines as axes of co-
ordinates. Now if an ellipsoid be referred to its principal dia-
meters as axes, these coefficients vanish. Hence the principal dia-
meters of the ellipsoid are the principal axes of the system. But
every ellipsoid has at least three principal diameters, hence every
material system has at least three principal axes.
25. Ex. 1. The principal axes at the centre of gravity being the axes of refer-
ence, prove that the momental ellipsoid at the point (p, q, r) is
~2qrYZ-2 rp ZX - 2pq X !'=£*,
when referred to its centre as origin.
ELLIPSOIDS OF INERTIA.
10
the greatest and
;ers.
of inertia with
loint, that with
with regard to
oint be equal
very diameter is
;s are all equal,
il axis at 0, and
tre of gravity of
the body being
y = 0, '%myz = 0,
tout them are by
;entre of gravity
of inertia about
I. Consider two
parallel to a faco
to the solid are
mtal ellipsoid at
. with regard to
les parallel to all
and the moment
aj'e always three
joint. Then it
ut the axes are
equation to the
s as axes of co-
principal dia-
le principal dia-
e system. But
rs, hence every
ig the axes of refer-
"')
Ex. 2. Show that the cubic equation to find the three principal moments of
inertia at any point {p, q, r) may be written in the form of a determinant
I-A
M
n
rp
ri
J-R
M
■ r' - r"
rp
qr
qr
I-C
M
ri^ - o'
= 0.
If (I, m, n) bo proportional to the direction-cosines of the axis corresponding to
' any one of the values of I, their values may be found from the ec^uationa
\I-{A + Mq'> + Mr'^)]l^Mpqm-i-Mrpn=iO, j
Mpql + { / - (Z? + A/r" + Mp'^) ]m + Mqrn= 0,
Mrpl + Mqi-m+ {I- (C + Mp^ + M>f)ln:=0.
Ex. 3. If 5-0 be the equation to the momental ellipsoid at the centre of
[gravity referred to any rectangular axes written in the form given in Art. li),
I then the momental ellipsoid at the point P whose co-ordinates are (p, q, r) is
S+3I {p^ + 2" + !•«) (Z* +Y^ + Z^)-M(pX+qY + rZf = 0.
I Hence show (1) that the conjugate planes of the straight line OP in the momental
I ellipsoids at and P are parallel and (2) that the sections perpendicular to OP
^ have their axes paralleL
26. The reciprocal surface of the momental ellipsoid is
^nother ellipsoid, which has also been employed to represent, geo-
iinetrically, the positions of the principal axes and the moment of
"linertia about any line.
We shall requue the following elementary proposition. The reciprocal surface
of the ellipsoid -j + |j + ij = 1 is the ellipsoid a^x^ + IV + c'^" = e*.
Let ON be the perpendicular from the origin on the tangent plane at any
[point P of the first ellipsoid, and let I, m, n be the direction-cosines of ON, then
\0N'=aH'' + b^m'^+cH^ Produce OiVto Q so that 0Q=^, then Q is a point on
t e*
I the reciprocal surface. Let 0Q=R; .: =a*l^ + h''m^ + chi^. Changing this to
[rectangular co-ordinates, we get e*=a'x'^ + b^y^ + ch\
To each point of a material body there corresponds a series of
[similar momental ellipsoids. If we reciprocate these we got
lanother series of similar ellipsoids coaxial with the first, and
[such that the moment of inertia of the body about the perpen-
jdiculars on the tangent planes to any one ellipsoid are propor-
[tional to the squares of those perpendiculars. It is, however, con-
Ivenient to call that particular ellipsoid the ellipsoid of gyration
I which makes the moment of inertia about a perpendicular on a
I tangent plane equal to the product of the mass into the square
so
MOMENTS OF INERTIA.
lljf
f
of that perpendicular. If Mho the mass of the body and A, B,
the principal moments, the equation to the ellipsoid of gyration is
A"^ B'^ G~ M'
It is clear that the constant on the right-hand side must be
-jTj., for when Y and Z are put equal to zero, ^Y' must by
A
definition be -r>.
M
27. Conversely, the series of momontal ellipsoids at any point
of a body may be regarded as the reciprocals, with different
constants, of the ellipsoid of gyration at that point. They are
all of an opposite shape to the ellipsoid of gyration, having their
longest axes in the direction of the shortest axis and their shortest
axes in the direction of the longest axis of the ellipsoid of gy-
ration. The momental ellipsoids however resemble the general
shape of the body more nearly than the ellipsoid of gyration.
They are protuberant where the body is protuberant and com-
pressed where the body is compressed. The exact reverse of this
is the case in the ellipsoid of gyration. See Art. 22, Ex. 3.
28. Ex. 1. To find the ellipsoid of gyration at the centre of a material elliptic
disc. Taking the values of A, B, C given in Art. 22, Ex. 1, we see that the
Z2 1
~~V
ellipsoid of gyration is -— + -^ +
Ex. 2. The ellipsoid of gyration at any point of a material rod AB is
■jp + r~2T— -J + r"a — i - ^> tfl'^ing tl'e same notation as in Art. 22, Ex. 2. This is
a very flat ellipsoid which when the rod is indefinitely thin becomes a circular area
whose centre is at 0, whose radius is /^^a^+c'^ and whose plane is perpendicular
to the rod.
Ex. 3. It may be shown that the general equation to the ellipsoid of gyration
referred to any set of rectangular axes meeting at the given point of the body is
= 0,
A
-F
-E
MX
-F
B
-D
MY
-E
-D
C
MZ
MX
MY
MZ
M
or when expanded
(UC - D^)X^ + {CA -E^)Y^-\-{AB - F-^)Z'i + 2{AB ■¥EF)YZ
+ 2(BE+FI))ZX+2{CF+DE)XY
=^(AB0-AD^-BE^-CF'-2DEF).
The right-hand side, when multiplied by M, is the discriminant obtained by
leaving out the last row and the last column, and the coefficients of X\ Y'^, Z",
2ZX, 2XY, 2YZ are the minors of this discriminant.
ELLIPSOIDS OF INERTIA.
31
n
29. The use of the ellipsoid -whose equation referred to the
principal axes at the centre of gravity is
has been suggested by Legendre in his Fonctions Elliptiques.
This ellipsoid is to be regarded as a homogeneous solid of such
density that its mass is equal to that of the body. By Art. 8,
Ex. 3, it possesses the property that its moments of inertia
with regard to its principal axes, and therefore by Art. 15 its
moments of inertia with regard to all planes and axes, are the
same as those of the body. Wo may call this ellipsoid the equi-
momental ellipsoid or Legendre' s ellipsoid.
Ex. If a plane move so that the moment of inertia with regard to it is always
proportional to the square of the perpendicular from the centre of gravity on tBa
plane, then this plane envelopes an ellipsoid similtir to Lbgendre's ellipsoid.
30. There is another ellipsoid which is sometimes useJ^ By Art. 15 the
moment of inertia with reference to a plane whose direction-cosines are (a, ^, 7) is
/' = 2;Ha;'. a« + i,ni7/'>./3» + 2wi32. 7" + 22771^/2. j37-l-22»i2a5. 70 + 22ma!y. o^.
Hence, as in Art. 19, we may construct the ellipsoid
Smx'. Z4+2mj/». r»+2mz'. Z=' + 22ni?/2. YZ + 21.mzx . ZX+2,l.mxy .XY=Mi*.
Then the moment of inertia with regard to any plane through the centre of the
ellipsoid is represented by the inverse 9quar3 of the radius vector porpeudfoular te
that plane.
If we compare the equation of the momenta! ellipsoid with that of this ellipsoid^
we see that one may be obtained from the other by subtracting the same quantity
from each of the coefficients of X^, Y\ Z^, Hence the two ellipsoids have their
circular sections coincident in direction.
This ellipsoid may also be used to find the moments of inertia about any
straight line through the origin. For we may deduce from Art. 5 that the moment
of inertia about any radius vector is represented by the difference between the
inverse square of that radius vector and the sum of *he inverse squares of the
semi-axes. This ellipsoid is a reciprocal of Legendrp'd ellipsoid. All these ellipsoids
have their principal diameters coincident in direction, and any one of them may be
used to determine the directions of the principal axes at any point.
31. When the body considered is a lamina, the section of the ellipsoid of
gyration at any point of the lamina by the plane of the lamina is called the ellipse
;of gyration. If the plane of the lamina be the plane of xy, we have 2m3'=0*,
The section of the fourth ellipsoid is then clearly the same as a momenta! ellipse at
[ the point. If any momenta! ellipse be turned round its centre through a right
angle it evidently becomes similar and similarly situated to the ellipse of gyration.
So that, in the case of a lamina, any ons of these ellipses may be easily ohanr,ed
I into the others.
32. A straight line passes tJirough a fixed point O a^id moves
about it in such a manner that the moment of inertia about the line
is always the same and equal to a given quantity I. 2 find the
equation to the cone generated by the straight line.
22
MOMENTS OF INERTIA.
fl
Itimii
m
I
Let the principal axes at bo taken as the axes of co-ordi-
nates, and let (a, ff, 7) be the direction-cosines of the straight lino
in any position. Then by Art. 17 we have Aa' + B^ + G^f = I.
Hence the equation to the locus is
(^-/)a«-f-(5-/)/3'+(a-/)7' = 0,
or, transforming to Cartesian co-ordinates,
{A-I)x' + {B-I)y'+{C-I)z''=^0.
It appears from this equation that the principal diameters of
the cone are the principal axes of the body at the given point.
The given quantity I must be less than the greatest and
greater than the least of the moments A, B, C. Let A, B, C be
arranged in descending order of magnitude ; then if / be less
than B, the cone has its concavity turned towards the axis C, if /
be greater than B the concavity is turned towards the axis A, if
7= B the cone becomes two planes which are coincident with the
central circular sections of the momental ellipsoid at the point 0.
The geometrical peculiarity of this cone is that its circular
sections in all cases are coincident in direction with the circular
sections of the momental ellipsoid at the vertex.
This cone is called an equimomental cone at the point at which
its vertex is situated.
83. The properties of products of inertia of a body about different sets of axa
are not so useful as to require a complete discussion. The following theorems will
serve as exercises.
Ex. 1. If any point be given and any plane drawn through it, then two
straight lines at right angles Ox, Oy can always be found such that the product of
inertia about these lines is zero.
These are the axes of the section of the momental ellipsoid at the point
formed by the given plane.
Ex. 2. If two other straight lines at right angles Oac', O1J be taken in the same
plane making an angle measured in the positive direction with Ox, Oy rc^ectively,
then the product of inertia F about Ox', Oy' is given by the equation *
F'=\6va2e(A-Ji),
where A , B are the moments of inertia about Ox, Oy.
Ex. 3. If I be the moment of inertia about any line in this plane making an
angle 9 with Ox, then
I=Acoi?e + Bim-9.
For the section of the momental ellipsoid by the plane is the ellipse whose
equation is Ax- + By- = M(*, whence the property follows at once.
tL
EQUIMOMENTIAL BODIES.
23
point at which
i at the point
aken in the same
Ex. 4. Let (Nm") (^V"') be the direction-cosinca of two straight linos Ox', 0;/
I at right angles passing through the origin and referred to the principal axes at
I us axes of oo-ordinates. Then the product of inertia ahout these linos is
F = XX'Smx' + nix' ^my' ■{ w'Sjms'.
For let (x'yV) be the co-ordinates of any point {xi/z) referred to Ox', Oij' and a
I third line Oz' as nevr axes of co-ordinates. Then
«'=Xa!+/ty-t-M, and y'=:\'x + n'ij + v'z.
Hence, since F'='S.nu!y', the theorem follows by simple multiplication.
Since XV + /*/»' + ""' = 0, we have
-f" =4 XX' 4 5/*/*' + (?"»''.
Ex. 5. If (X/i»') he the dircction-cosinos of an axis Ox', then the direction-
cosines (XVi/') of another axis Oy' at right angles such that the product of inertia
I about Ox', Oy' is zero, are given by the equations
. X' ^ m' _ ^ "' _
(B-C)iu> {C-A)v\ {A-Ji)\fi'
For by (4) the equations to find X'/tV are
A\\'+BniJif + Ci'i>'=0,)
\\'+Hfx' + i>y' = 0,)
whence the theorem follows by cross multiplication.
By (1) this is equivalent to the geometrical theorem. Given a radius vector
Ox' of an ellipsoid, find another radius vector Oy' such that Ox', Oy' are principal
diameters of the section xfOy'.
Ex. 6. Let (Imn) be the direction-cosines of any given straight line Oz', and let
jy, E' be the products of inertia about Oz', Oy'; 0/, Ox', where Ox', Oy' are any
two straight lines at right angles. Then as Ox', Oy' turn round Oz', Z)''^ + E''^ ia
constant, and
D'a + E'^ ={A- B)^ (Im)' + {B- C)' {mn}^ + {C-A)^ (nl,^.
For by (4), - U=Al\^-Bmit->r Cnv, - E'=Al\'-i- Bm/j! + Cnv' ;
.'. jy^ + E"=AH^(X' + \''')+2ABlmQi^i.+\'iJi.') + &c.
But V + X'« = l-Zi'=m2+n»,)
\fi+\'ix'=-lm, )
whence by substitution the theorem follows at once.
Ex. 7. If A', B' be the moments of inertia about Ox', Oy', then as Ox', Oy'
turn round Oz', the value of A'B' -F'^ is constant, and
A'B'-F'^=^BCl^-[- CA ;u2 + A £n\
plane making an
le ellipse whose
On Equimomental Bodies.
84. Two bodies or systems of bodies are said to be equi-
momental when their moments of inertia about all straight lines
arc C(iual each to each.
I'
r
ii
> K '
R
IT B- ^
I
( !
2t
MOMENTS OF IVKRTIA.
35. If two systcmH have tho same centre of gravity, the same
mass, the same principal axes and principal moments at tho centre
of gi'avity, it follows from tho two fimdamcntal propositions of
Arts. 13 and 15 that their moments of inertia about all straijjlit
lines are equal, each to each.
That the converse theorem is also true may bo shown thus.
We know by Art. 13 that of all straight lines having a given
direction in a body, tiiat straight line has the least moment of
inertia which passes through the centre of gravity. It is clear that
these least moments of inertia could not be equal in two bodies
for all directions unless they had a common centre of gravity.
Of all straight lines through the centre of gravity those which
have the greatest and least moments of inertia are two of tho
principal axes, hence these and therefore also the third principal
axis must be coincident in direction if the two bodies are equi-
momental. The principal moments of inertia must then be equal,
because all moments arc equal. Lastly, by Art. 1.3, tho two
systems could not have equal moments about two parallel axes,
each to each, unless their masses were ccpial.
It is easy to see that two equimomental systems must have
the same momental ellipsoid, and therefore the same principal
axes at every point.
3C. To find the moments and products of inertia of a triangle
about an]/ axes ivhatever.
If /3 and 7 be the distances of the angular points B, C, of a
triangle ABC from any straight line AX through the angle A, in
the plane of the triangle, it is known that tho moment of inertia
M
of the triangle about AX\» y (/3* + ^7 + y^, where M is the mass
of the triangle.
Let three equal particles, the mass of each being -^ , be placed
o
at the middle points of the three sides. Then it is eu ../ seen,
that the moment of inertia of the three particles about AX is
•which is the same as that of the triangle. The three particles
treated as one system, and the triangle, have the same centre of
gravity. Let this point be called 0. Draw any straight line OX'
through the common centre of gravity parallel to AX, then it
is evident that the moments of inertia of the two systems about
OX' are also equal.
Since this equality exists for all straight lines through in
the plane of the triangle, it will be true for two straight lines 0X\
+
(I)'- &
H
EQUIMOMRNTAL BODIES.
25
be shown thus.
M is the mass
irough in
y at right angles, ami therefore also for a straight lino OZ*
1 perpendicular to the plane of the triangle.
One of the principal axes at of the triangle, and of the
[syHtem of three particles, is normal to the plane, and therefore the
same for the two systems. The principal axes at in the plane,
arc those two straight lines about which the moments of inertia
J are greatest and least, and therefore by what precedes these axes
[are the same for the two systems. If at any point two systems
jliave the same principal axes and principal moments, they have
jalso the same moments of inertia about all axes through that
f)()int, and the same products of inertia about any two straight
iiies meeting in that point. And if this point be the centre of
Igravity of both systems, the same thing will also be true for any
jother point.
If then a particle whose mass is one-third that of the triangle
[be placed at the middle point of each side, the moment of inertia
3f the triangle about any straight line, is the same as that of the
Isystem of particles, and the product of inertia about any two
ietraight lines meeting one another, is the same as that of tho
Isystem of particles about tho same straight lines.
§ 37. Three points D, E, F can always be found such that the
products and moments of inertia of three equal particles placed
lit D, E, F, may be the same as the products and moments of
Inertia of any plane area. For let be the centre of gravity of
the area, Ox, Oy the principal axes at in the plane of the area,
and il7a' and M^^ be the moments of inertia about these axes.
Let {xy)y {xy), {x"y") be the co-ordinates of D, E, F, m the
Imass of a particle, so that M= Sm.
Then we must have m (a;*+ x' + x"^) = Jl/yS*,
xy + xy + ic'y = 0.
Also, since the two systems must have the same centre of
jravity, ic + u;' + a;" = 0, y + y + y" = 0.
Eliminating x'y, x"y" from these equations, we get
diich is the equation to a momental ellipse. It easily follows,
i,hat D may be taken any where on this ellipse, and E and F are
it the opposite extremities of that chord which is bisected in some
)oint iV by the produced radius DO, so that 0N= \0D.
38. A momental ellipsoid at the centre of gravity of any
triangle may be found as follows.
Vit-"^
2G
MOMENTS OF INERTIA.
m
MJ! -i
, \'
i> t
Let an ellipse be inscribed in the triangle touching two of the
bides AB, BO in their middle points F, I). Then, by Garnet's
Theorem, it touches the third side CA in its middle point E.
Since DF is parallel to CA the tangent at F, the straight line
joining F to the middle point iV of DF passes through the centre,
and therefore the centre of the conic is the centre of gravity of
the triangle.
This conic may be shown to be a momenta! ellipse of the
triangle at 0. To prove this, let us find the moment of inertia of
the triangle about OF. Let OE=r, and let the semi-conjugate
diameter be r, and w the angle between r and r'. Now ON=^r,
and hence from the equation to the ellipse FN^ = ^r'^,
therefore moment ofi
inertia about OF
= P/.
1 '!
sm 0), =
M A'"
irV '
where A' is the area of the ellipse, so that the moments of inertia
of the system about OF, OF, OD are proportional inversely to
OF^, OF^, OD^. If we take a momental ellipse of the right
dimensions, it will cut the inscribed conic in F, F, and D, and
therefore also at the opposite ends of these diameters. But two
conies cannot cut each other in six points unless they are identical.
Hence this conic is a momental ellipse at of the triangle.
A normal at to the plane of the triangle is a principal axis
of the triangle (Art. 17). Hence a momental ellipsoid of the
triangle has the inscribed conic for one principal section. If a
and b be the lengths of the axes of thi;^ conic, c that of the axis
of the ellipsoid which is perpendicular to the plane of the lamina,
we have by Arts. 7 and 19
1-11
If the triangle be an equilateral triangle, the momental ellip-
soid becomes a spheroid, and every axis through the centre of
gravity in the plane of the triangle is a principal axis.
Since any similar and similarly situated ellipse is also a
momental ellipse, we might take the ellipse circumscribing the
triangle, and having its centre at the centre of gravity, as the
momental ellipse of the triangle.
39. Ex. 1. A momental ellipse at an angular point of a triangular area touches
the opposite side at its middle point and bisects tlie adjacent sides.
Ex. 2. The principal radii of gyration at the centre of gravity of a triangle
are i;u roots of the equation
where A is the area of the triangle.
llci
EQUIMOMENTAL BODIES.
27
liing two of the
en, by Carnot's
liddle point E.
he straight lino
ough the centre,
re of gravity of
1 ellipse of the
ent of inertia of
serai-conjugate
Now ON=y,
lr\
M ^
2 • ttV* '
nents of inertia
lal inversely to
ie of the right
F, and D, and
eters. But two
ey are identical,
triangle.
a principal axis
3llipsoid of the
section. If a
hat of the axis
of the lamina,
nomental ellip-
the centre of
CIS.
pse is also a
imscribing the
gravity, as the
gular area touches
s.
dty of a triangle
Ex. 3. The direction of the principal axes at the centre of graviiy of a tri-
[ angle may be constructed thus. Draw at the middle point D of any side BQ
I lengths DII= — , BH'— — along the perpendicular, where p is the perpendicular
! P P
[from A on BC and P, k"^ are the principal radii of gyration found by the last ex-
f ample. Then OH, OH' are the directions of the principal axes at 0, whose
I moments of inertia are respectively il/A" and Mk'-.
Ex. 4. The directions of the principal axes and the principal moments at the
I centre of gravity may also be constructed thus. Draw at the middle point D of
BO
I any side BC a perpendicular DK =
2J3'
Describe a circle on OK as diameter
i and join D to the middle point of OK cutting the circle in R and S, then OR, OS
are the directions of the principal axes, and the moments of inertia about them are
[respectively M
DS"'
2
and M
DB^
Ex. 6. Let four particles each one-sixth of the mass of the area of a parallelo-
I gi-am be placed at the middle points of the sides and a fifth particle one-third of the
I same mass be placed at the centre of gravity, then these five particles and the area
I of the parallelogram are equimomental systems.
Ex. 6. Let four particles each one-twelfth of the mass of the area of a quadri-
: lateral be placed at each corner and let a negative mass also one-twelfth be placed
at the intersection of the diagonals and a sixth particle three-quarters of the same
mass be placed at the centre of gravity, then these six particles and the area of the
quadrilateral are equimomental systems.
Ex. 7. Let three particles each one-sixth of the mass of an elliptic area be placed
one at one extremity of the major axis and the other two at the extremities of the
ordinate which bisects the semi-axis major, and let a fourth particle whose mass is
one-half that of the area be placed at the centre of gravity. Then the moments
and products of inertia of the system of four particles and of the elliptic area are
the same for all axes whatever.
Ex. 8. Any sphere of radius a and mass M is equimomental to a system of
m four particles each of mass ^ ( - ) placed so that their distances from the centre
make equal angles with each other and are each eqiial to r and a fifth particle equal
to the remainder of the mass of the sphere placed at the centre.
40. To find the moments and products of inertia of a tetra-
hedron about any axes whatever.
Let ABGD be the tetrahedron. Through one angular point
D draw any plane and let it be taken as the plane of xy. Let D
I be the area of the base ABC', a, /8, 7 the distances of its angular
I points from the plane of xy, and p the length of the pcrpendiculor
^" from D on the base ABC.
Let PQR be any section parallel to the base ABC and of
thickness du, where u is the perpendicular from D on PQR. The
moment of inertia of the triangle PQR with respect to the plane
28
MOMENTS OF INERTIA.
of xy is the same as that of three equal particles, each one-third
its mass, placed at the middle points of its sides. The vohime of
W
the element PQR = -^ pdu. The ordinates of the middle points
of the sides AB, BG, CA are respectively ^—^^ 9~^> '^~Y^ '
Hence, by similar triangles, the ordinates of the middle points of
PQ, e;j.iJP are also ^i', ''-±^''-, ^^
The moment of inertia of the triangle PQR with regard to the
plane xi/ is therefore
Integrating from u = to u=p, we have the moment of
inertia of the tetrahedron vfith. regard to the plane xi/
where Fis the volume.
If particles each one-twentieth of the mass of the tetrahedron
were placed at each of the angular points and the rest of the
mass, viz. four-fifths, were collected at the centre of gravity, the
moment of inertia of these five particles with regard to the plane
of xi/ would be
which is the same as that of the tetrahedron.
The centre of gTavity of these five particles is the centre of
gi'avity of the tetrahedron, and they together make up the mass
of the tetrahedron. Hence, by Art. 13, the moments of inertia of
the two systems with regard to any plane through the centre of
gravity are the same, and by the same article thir, equality will
exist for all planes whatever. It follows by Art. 5, that the mo-
ments of inertia about any straight line are also equal. The two
systems are therefore equimomental.*
41, If the distance of every point in a given figure in space
from some fixed plane be increased in a fixed ratio, the figure
thus altered is called the projection of the given figure. By pro-
" This result was proposed as a Problem in the Mathematical Tripos in an
interval of the publication of the preceding and following results, thus anticipating
the author by a few days.
<8
EQUIMOMENTAL BODIES.
29
3, each one-third
The volume of
B middle points
^+7 7+a
» 2 ' 2 •
middle points of
th regard to the
I)}-
the moment of
the tetrahedron
the rest of the
of gravity, the
ird to the plane
the centre of
:e up the mass
its of inertia of
h the centre of
. equality will
, that the mo-
ual. The t-vo
figure in space
tio, the figure
fure. By pro-
ical Tripos in an
tlius anticipating
jecting a figure from three planes as base planes at right angles
in succession, the figure may be often much simplified. Thus an
ellipsoid can always be projected into a sphere, and any tetra-
hedron into a regular tetrahedron.
It is clear that if the base plane from which the figure is
projected be moved parallel to itself into a position distant D
from its former position, no change of form is produced in the
projected figure. If n be the fixed ratio of projection the pro-
jected figure has merely been moved through a space nl) perpen-
dicular to the base plane. We may therefore suppose the base
plane to pass through any given point which may be convenient.
42. If two bodies are equimomental, their projections are also
equimomental.
Let the origin be the common centre of gravity, then the
two bodies are such that 2w = 2m' ; Xmx = 0, Xm'x' = 0, &c.,
Xmx^ = ^m'x'^, ^myz = 'liVii/'z', &c., unaccented letters referring
to one body and accented letters to the other. Let both the
[bodies be projected from the plane of xi/ in the fixed ratio 1 : n.
Then any point whose co-ordinates are (x, y, z) is transferred to
\{x, y, nz) and {x', y', z) to {x, y, nz). Also the elements of mass
wi, vn become nm and nm. It is evident that the above equalities
are not affected by these changes, and that therefore the projected
bodies are equimomental.
The projection of a momental ellipse of a plane area is a
moinental ellipse of the projection.
Let the figure be projected from the axis of x as base line,
pso that any point {x, y) is transferred to {x, y') where y' = ny,
^nd any element of area m becomes m' where m' =■ nm. Then
■''?' Ill
I l,mx^ = - Xm'af, tmxy = -g Xm'xy\ Xmf/ = -3 Smy".
,/si '* n n
lie momental ellipses of the primitive and the projection are
l^mfX' - 2XmxyXY+ Xmx^Y^^Me\
XmyX" - 2Xm'xy'X' F + Xm'x' Y" = M'e\
[■o project the former we put X' = X, Y'=nY, its equation then
fbecomes identical with the latter by virtue of the above equalities
Jif we put e = en.
I 43. Ex. 1. A momental ellipse of the area of a square at its centre of gravity
;|is easily seen to be the inscribed circle. By projecting these first with one Ride aa
-J base line, and secondly with a diagonal as base, the square becomes successively a
rectangle and then a parallelogram. Hence a momental ellipse at the centre of
-gravity of a parallelogram is the inscribed conic touching at the middle points of
• the sides.
30
MOriEXTS OF INERTIA.
w
.\ i
Ex. 2. By jjrojectiug an equilateral triangle into any triangle, we may infer the
results of some of the previous articles, but the method will be best exx)lained by its
application to a tetrahedron.
Ex. 3. Since any ellipsoid may be obtained by projecting a sphere, we infer by
Ai't. 39, Ex. 8, that any solid ellipsoid of mass M is equimomental to a system of
four particles each of mass -y~ ^ placed on a similar ellipsoid whose linear dimen-
sions are n times as great as those of the material ellipsoid, so that the eccentric
lines of the particles make equal angles with each other and a fifth particle equal to
the remainder of the mass of the sphere placed at the centre of gravity.
If this material ellipsoid be the Legendre's ellipsoid of any given body, we
see that any body whatever is equimomental to a system of five particles placed as
above described on an ellipsoid similar to the Legendre's ellipsoid of the body.
Ex. 4. Show that a solid oblique cone on an elliptic base of mass M is equimo-
mental to a system of three particles each ^ - 21/ placed on the circumfeience of the
3
base so that the differences of their eccentric angles are equal, a fourth particle — M
placed at the middle point of the straight line joining the vertex to the centre of
gi'avity of the base, and a fifth particle to make up the mass of the cone placed at
the centre of gravity of the volume.
44. To find the equimomental ellipsoid of any tetrahedron.
The moments of inertia of a regular tetrahedron with regard
to all planes through the centre of gravity are equal by Art. 23.
If r be the radius of the inscribed sphere, the moment with
regard to a plane parallel to one face is easily seen by Art. 40
3^.2 " _
to be M -^ . If then we describe a sphere of radius p = ^S r,
o
with its centre at the centre of gravity, and its mass equal to
that of the tetrahedron ; this sphere and the tetrahedron will be
equimomental. Since the centre of gravity of any face projects
ir.tr. the centre of gravity of the projected face, we infer that
the ellipsoid to which any tetrahedron is equimomental, is similar
and similarly situated to that inscribed in the tetrahedron and
touching each face in its centre of gravity, but has its linear
dimensions greater in the ratio 1 : J3. It may also be easily
seen that the sphere whose radius is p = ,^3^ touches each edge
of the regular tetrahedron at its middle point. Hence we infer
that the equimomental ellipsoid of any tetrahedron touches each
edge at its middle point and has its centre at the centre of gravity
of the volume.
These results may also be deduced from Art. 25, Ex. 2, with-
out the use of projections.
1
EQUIMOMEXTAL BODIES.
31
.0, we may infer the
best explained by its
sphere, we infer by
ental to a system of
ivhose linear dimeu-
10 that the eccentric
fth particle equal to
gravity.
my given body, we
! particles placed as
lid of the body.
mass M is equimo-
ircumfeience of the
g
ourth particle — M
rtex to the centre of
: the cone placed at
'etrahedron.
:on with regard
ual by Art. 23.
moment with
een bv Art. 40
adiu3 p = JSr,
mass equal to
hedron will be
y face projects
we infer that
ntal, is similar
trahedron and
has its linear
also be easily
hes each edge
ence we infer
1 touches each
ntre of gravity
>, Ex. 2, with-
in 45, Ex 1. If E^! be the sum of the squares of the edges o' a tetrahedron, 2?" the
Buia of the squares of the areas of the faces and V the volume, show that the semi-
Hxesof the ellipsoid inscribed in t} J tetrahedron, touching each face in the centre of
Wavity and having its centre at the centre of gravity of the tetrahedron, are the
%oots of
ITS P2 72
2^3
P^+2^3.P^
%nd if the roots be i-p^:kp.^J=p.^, then the moments of inertia with regard to the
'W
^F,^
W
rincipal iJlanes of the tetrahedron are M -^ , M -g" , M
I 4p
i Ex. 2. If a perpendicular EP be di-awn at the centre of gravity E of any face = —
Sphere p is the perpendicular from the opposite corner of the tetrahedron on that
,|acp, then /» is a point on the principal plane con-espondiug to the root p of the
jpubic.
I 46. To explain hoiu the theory of invers.n can he applied to
Mnd moments of inertia.
f Let a radius vector drawn from some fixed origin to any point P of a
fguro be produced to P' where the rectangle OP .OP' = k^ where k is some given
uantity. Then as P travels all over the given figure, P' traces out another
Vhich is called the inverse of the given figure.
" Let {x, y, z) be the coordinates of P, (as', ij z') those of P'; r, r' the radii vectores,
dv, dv' corresponding polar elements of volume; /), p', dm, dm' their respective
densities and masses. Let du be the solid angle subtended at by either dv or dv'.
Then
dv'=r'^dudi
a/
-67
r^du
"i^ov! dm=pdv, dm'= p'dv'. If then we
and sincp — = - wo have sd'^ dv'= ( " ) a;^ dv.
take p'—l-j p we have Ix'- dm'='^x^ dm, with similar equalities in the case of all
|he other moments and products of inertia.
Hence we infer, that if a homogeneous body be inverted with regard to a point
?, and the density of the new body vary inversely as the tenth power of the distance
^rom 0, then these two bodies have the same moments of inertia about all straight
iiues through 0.
Ex. The density of a solid sphere varies inversely as the tenth power of the
listance from an external point 0. Prove that its moments of inertia about any
i^traight line through is the same as if the sphere were homogeneous and equal in
iensity to that of the heterogeneous sphere at a point where the tangent from
icets the sphere. Prove that if the density had varied inversely as the sixth power
sf the distance from 0, the masses of the two spheres would have been equal. What
|s the condition thev should have a common centre of gravity ?
47. The theory of equimomental particles is of considerable
;, use in finding the centre of pressure of any area vertically im-
linicrscd in a homogeneous fluid under the action of gravity. It
|may be proved fi-om hydrostatical principles that if the axis of
1
S2
MOMENTS OF INERTIA.
'!i
; V
I
I
1
4
'
tc be in the effective surface, and the axis of y vertically down-
wards, the co-ordinates of the centre of pressure are
Product of inertia about the axes
X =
Y=
moment of area about Ox
Moment of inertia about Ox
moment of area about Ow
We see therefore that two equimomental areas have the same
centre of pressure. The preceding proposition may be used with
considerable effect.
Ex, Prove that the centre of pressure of any triangle wholly immersed is tlio
centre of gravity of three weights placed at the middle points of the sides and each
proportional to the depth of the point at which it is placed.
On the positions of the Principal Axes of a system.
48. Prop. A straight line "being given it is required to find at
what point in its length it is a principal aods of the system, and if
any such point eocist to find the other two principal axes at that
point.
Take the straight line as axis of z, and any point in it as
origin. Let C be the point at which it is a principal axis, and let
Cx', Gy' be the other two principal axes.
Let C0 = h, ^ = angle between Ca;' and Ox. Then
x' = X cos 6-\-y sin ^
2/' = — a; sin ^ + ^ cos 6
z' = z — h
Hence 1.mxz = cos OXmxz + sin OXmyz ) _ ^
— h (cos OXmx + sin d'^my) ) ~
%rtiyz = — sin 6%mxz + cos OXmyz ]_ ^
— A (— sin 6%mx + cos &%my) J "
sin 20
0)
(2)
Imx'y = T.m {y^ — a?) ^— + 'tmxy cos 20 = (3)
The last equation shows that
tan2g = /fry.. (4)
2,m [x —y) ^ '
2F
B~A'
accordi?ig to the previous notation.
PRINCIPAL AXES.
ss
y vertically down-
3 are
xes
las have the same
may be used with
wholly immersed is the
s of the sides and each
of a system.
required to find at
' the system, and if
xcipal axes at that
ny point in it as
ncipal axis, and let
Then
= (1)
= (2)
= (3)
(4)
The equations (1) and (2) must be satisfied by the same value
h. Eliminating h we get y.mxz Sm?/ = Xmyz %nx as the con-
[ition that the axis of z should be a principal axis at some point
its length. Substituting in (1) we have
%mx
, _ ^myz __ zmxz
Xmy
(5)
The equation (5) expresses the condition that the axis of z
lould be a principal axis at some point in its length ; and
le value of h gives the position of this point. The positions
the other two principal axes may then be found by equa-
|on (4).
If Sma;« = and Xm^/z — 0, the equations (1) and (2) arc
)th satisfied by h = 0. These are therofore the sufficient and
jcessary conditions that the axis of z should be a principal axis
the origin.
Tf the system be a plane lamina and the axis of z be a normal
the plane at any point, we have z = 0. Hence the conditions
%mxz = and Xmyz = are satisfied. Therefore one of the
l^incipal axes at any point of a lamina is a normal to the plane
ftt that point.
In the case of a surface of revolution bounded by planes per-
pendicular to the axis, the axis is a principal axis at any point of
its length.
Again equation (4) enables us, when one principal axis is
given, to find the other two. If ^ = a be the first value of 6, all
tjje others are included in = a + n ; hence all these values give
^ly the same axes over again.
49. Since (4) does not contain h, it appears that if the axis of
»,be a principal axis at more than one point, the principal axes at
l^ose points are parallel. Again, in that case (5) must be satis-
fed by mjre than one value of h. But since h enters only in the
"fst power, this cannot be unless
Xmx = 0, Swy = 0,
Xmxz = 0, Xmyz = ;
that the axis must pass through the centre of gravity and be a
Kncipal axis at the origin, and therefore (since the origin is arbi-
■iry) a principal axis at every point in its length.
If the principal axes at the centre of gravity be taken as the
bs of x, y, z, (1) and (2) are satisfied for all values of h. Hence.
a straight line be a principal axis at the centre of gravity, it is
principal axis at every point in its length.
R. D. 3
34
MOMENTS OF INERTIA.
h >
^
50. Let the system be projected on a plane perpendicular to
the given straight line, so that the ratios of the elements of mass
to each other are unaltered. The given straight line, which has
been taken as the axis of z, cuts this plane in 0, and will be a
principal axis of the projection at 0, because the projected system
being a plane lamina, the conditions '^mxz = 0, z.myz = are
both satisfied. Since z does not appear in equation (4), it follows
that if the given straight line be a principal axis at some point G
in its length, the other two principal axes at C will be parallel to
the principal axes of the projected system at 0. These last may
often be conveniently found by the next proposition.
51. Ex. 1. The principal axes of a right-angled triangle at the right angle are,
one perpendicular to the plane and two others inclined to its sides at the angles
j^tan~i -J — y^, where a and 6 are the sides of the triangle adjacent to the right
angle.
Take the formula tan2^ = -^--— , Art. 48, then hy Axt. 8, A = M ^, B = M ^ ,
F = M
ah
12'
Ex. 2, The principal axes of a quadrant of an ellipse at the centre are, one
perpendicular to the plane and two others inclined to the principal diameters at the
angles ^ tan"* - ^_^ , where a and h are the semi-axes of the ellipse.
Ex. 3. The principal axes of a cube at any point P are, the straight line
joining P to the centre of gravity of the cube, and any two straight lines at P
perpendicular to PO, and perpendicular to each other.
Ex. 4. Prove that the locus of a point P at which one of the principal axes is
parallel to a given straight line is a rectangular hyperbola in the plane of which the
centre of gravity of the body lies, and one of the asymptotes is parallel to the given
straight line. But if the given straight line be parallel to one of the principal axes
at the centre of gravity, the locus of P is that principal axis or the perpendicular
principal plane.
Take the origin at the centre of gravity, and one axis of co-ordinates parallel
to the given straight line.
Ex. 5. An edge of a tetrahedron will be a principal axis at some point in its
length, only when it is perpendicular to the opposite edge. [Jullien.]
Conversely if this condition be satisfied, tLd edge will be a principal axis at a
2
point C such that 0C=^ ON where N is the middle point of the edge and is the
foot of the perpendicular distance between it and the opposite edge.
52. Prop. Giveti the positions of the principal axes Ox, Oy,
Oz at the centre of gravity O, and the momen:,s of inertia about
them, to find the positions of the principal axes at any point P in
the plane ofxy, and the moments of inertia about those axes.
PRINCIPAL AXES.
35
it to the right
iinates parallel
me point in its
Let the mass of the body be M, and let A, B be the moments
of inertia about the axes Ox, Oy, of which we shall suppose A the
greater.
Take two points 8 and // in the axis of greatest moment, one
on each side of the origin so that
08=^ on
-w
M
These points may be called the foci of inertia for that principal
plane.
Because these points ai'e in one of the principal axes at the
centre of gravity, the principal axes at ^ and fT are parallel to the
axes of co-ordinates, and the moments of inertia about those in the
plane of xy are respectively A and B + M . Oti^ = A, and these
being equal, any straight line through S or // in the plane of xy
is a principal axis at that point, and the moment of inertia about
it is equal to A.
If P be any point in the plane of xy, then one of the principal
axes at P will be perpendicular to the plane xy. For if ^, §- be
the co-ordinates of P, the conditions that this line is a principal
axis are
Swi (a? — w) s = I
which are obviously satisfied because the centre of gravity is the
origin, and the principal axes the axes of co-ordinates.
The other two principal axes may be found thus. If two
straight lines meeting at a point P be such that the moments of
inertia about them are equal, then provided they are in p princi-
pal plane the principal axes at P bisect the angles between these
two straight lines. For if with centre P we describe the momental
ellipse, then the axes of this ellipse bisect the angles between any
two equal radii vectores.
Join SP and HP; the moments of inertia about 8P, HP are
each equal to A. Hence, if PG and PT are the internal and
3-2
36
MOMENTS OF INERTIA.
VV i
external bisectors of the angle 8PH; PO, PT are the principal
axes at P. If therefore with S and H as foci we describe anjj
ellipse or hyperbola, the tangent and normal at any point are the
pnncipal axes at that point.
63. Take any straight line MN through the origin, making
an angle 6 with the axis oi x. Draw SM, IIN perpend icular.s on
MN. The moment of inertia about it is
= ^ cos'^ + ^sin"^
==A-{A-B) sin'
= A-M.{08ainey
= A-M.8]\P.
Through P draw PT parallel to MN, and let 8Y and HZ be the
perpendiculars froni S and H on it. The moment of inertia about
PT is then
= moment about MN+ M. MY^
= A + M{MY- 8M) {MY+ 8M)
= A + M.8Y.IIZ.
In the same way it may be proved that the moment of inertia
about r^ line PO passing between H and 8 is less than A by the
mass into the product of the perpendiculars from 8 and H on PG.
If therefore with S andU as foci we describe any ellipse or
hyperbola, the moments of inertia about any tangent to either of
these curves is constant.
It follows from this that the moments of inertia about the
• • 1 . D 1.7,. ..(8P±HPV
prmcipal axes at P are equal to ij + iM ( ^ J .
For if a and h be the axes of the ellipse we have a^ — b^ = 08^
A-B
M
and hence
A^M.SY.HZ=A + Mb^ = B + Ma^ = B + M
f8P + HP\'
\ 2
and the hyperbola may be treated in a similar manner.
54. This reasoning may be extended to points lying in any
given plane passing through the centre of gravity of the body.
Let Ox, Oy be the axes in the given plane such that the product
of inertia about them is zero (Art. 33). Construct the points 8
and // as before, so that 08^ and OIP are each equal to the
difference of the moments of inertia about Ox and Oy divided by
the mass. Draw Sy' a parallel through 8 to the axis of ?/, the
PRINCIPAL AXES.
37
ia about the
product of inertia about Sx, Si/' is equal to that about Ox, Oy
together with the product of inertia of the whole mass collected
at 0. Both these are zero, hence the section of the momental
ellipsoid at 8 is a circle, and the moment of inertia about every
straight line through S in the plane xOy is the same and e(jual
to that about Ox. We can then sliow that the moments of
inertia about PH and PS are equal ; so that PG, Pl\ the internal
and external bisectors of the angle SPH are the principal dia-
meters of the section of the momental ellipsoid at P by the given
plane. And it also follows that the moments of inertia about the
tangents to a conic whoso foci are S and H are the same.
55. Ex. 1. To ilnd tho foci of inertia of an elliptic area. The moments of
iuertia about the major and minor axes are M -r and 21/ -, . Hence tho minor axis
4 4
iii the axis of greatest moment. Tho foci of inertia therefore lie in the minor axis
at a distance from the centre = ^ ija^ - b', i.e. half tho distance of tho geometrical
foci from the centre.
Ex. 2. Two particles each of mass m are placed at the extremities of the minor
axis of an elliptic area of mass M, Prove that the principal axes at ai?y point of
the circumference of tho ellipse will bo the tangent and normal to the ellipse, pro-
., , m 5 e'^
Ex. 3. At the points which have been called foci of inertia two of the principal
moments are equal. Show that it is not in general true that a point exists such
that the moments of inertia about all axes through it are the same, and find the con-
ditions that there may be such a point.
Eefer the body to the principal axes at tLo centre of gravity. Let P be the point
required, {x, y, z) its co-ordinates. Since the momental ellipsoid at P is to be a
sphere, the products of inertia about all rectangular axes meeting at P are zero.
Hence, by Art. 13, xy = 0, yz=0, zx=0. It follows that two of the three x, y, z
must be zero, so that the point must be on one of the principal axes at the centre
of gravity. Let this be called tho axis of z. Since the moments of inertia about
three axes at P parallel to the co-ordinate axes are A + 3/ z", B + Mz"^ and C, we see
that these cannot be equal unless A = B and each is less than C. There are then
two points on the axis of unequal moment which are equimomental for all axes.
[Poisson and Binet.]
56. Given the positions of the pnmipal axes at the centre of
gravity and the moments of inertia about them, to find the
positions of the principal axes*, and the principal moments at a>;y
other point P.
Let the body be referred to its principal axes at the centre of
gravity 0, let A, B, C be its principal moments, the mass of the
* Some of the following theorems were given by Sir William Thomson and
Mr Towusend, in two articles which appeared at the same time in the Mathematical
Journal, 1846. Their demonstrations are different from those given in this treatise.
i
38
MOMENTS OF INERTIA.
1!
I
M
m
i*
body hemrr taken as unity. Construct a quadric confocal with
the ellipsoid of gyration, and let the squares of its semi-axes be
a'-s A +\, h'^ B + \, 0^= G + \. Let us find the moment of
inertia with regard to any tangent plane.
Let (a, )9, 7) be the dii'cction angles of the perpendicular to
any tangent plane. The moment of inertia, with regard to a
parallel plane through 0, is
^ (Jcos''a + i?cos'/3+ Ccos'7).
The moment of inertia, with regard to the tangent pl.ano, is
formed by adding the square of the perpendicular distance be-
tween the planes, viz.
we get
{A + X) cos'a + {B + \) cos'yS + (C+X) cos" 7,
moment of inertia with re-] A + B + C
gard to a tangent planej 2
B+G-A
+ \
Thus the moments of inertia with regard to all tangent planes to
any one quadric confocal with the ellipsoid of gyration are the
same.
These planes are all principal planes at the point of contact.
For draw any plane through the point of contact P, then in the
case in which the confocal is an ellipsoid, the tangent plane
parallel to this plane is more remote from the origin than this
plane. Therefore, the moment of inertia with regard to any plane
through P is less than the moment of inertia with regard to a
tangent plane to the confocal ellipsoid through P. That is, the
tangent plane to the ellipsoid is the principal plane of greatest
moment. In the same way the tangent plane to the confocal
hyperboloid of two sheets through P is the principal plane of
least moment. It follows that the tangent plane to the confocal
hyperboloid of one sheet is the principal plane of mean moment.
Through a given point P, three confocals can be drawn, the
normals to these confocals are, by Art. 16, the principal axes at P.
By Art. 5, Ex. 3, the principal axis of least moment is normal
to the confocal ellipsoid and of greatest moment normal to the
confocal hyperboloid of two sheets.
57. The moment of inertia with regard to the point P is, by
Art. 14, s,— + OP^. Hence, by Art. 5, Ex. 3, the moments
PRINCIPAL AXES.
39
)cncHcular to
of inertia about the normals to the tliroo confocals through P
whose parameters are \p \, \ arc respectively
0P'-\, 0P'-\, 0P'-\.
58. If wo describe any other confocal and draw a tangent
cone to it whoso vertex is P, the axes of this cone are known to
be the normals to the three confocals through P. This gives
another construction for the principal axes at P.
If this confocal diminish without limit, imtil it becomes a
focal conic, then the priiicipal diameters of the systom at P are
the principal diameters of a cone whose vertex is P and base
a focal conic of the ellipsoid of gyration at the centre of gravity.
.59. If we wish to use only one quadric, we may consider the
confocal ellipsoid through P. Wo know* that the normals to the
♦ These propositions are to bo found in books on Soliil Geometry, they may also
bo proved as follows.
I,et the confocal ellipsoid pass near P and approach it indefinitely. The base
of the enveloping cone is ultimately the Indicatrix ; and as the cone becomes ulti-
mately a tangent plane, one of its axes is ultimately a perpendicular to the plane of
the Indicatrix. Now in any cone two of its axes are parallel to the principal diame-
ters of any section perpendicular to the third axis. Hence the axes of the envelop-
ing cone are the normal to the surface and parallels to the prim i pal diameters of
the Indicatrix. But all parallel sections of an ellipsoid are similiir and similarly
situated, hence the principal diameters of the Indicatrix are parallel to the princi-
pal diameters of the diametral section parallel to the tangent plane at P.
To find the principal moments, we may reason as follows. Let a tangent plane
to the ellipsoid be drawn perpendicular to any radius vector OQ of the diametral
section of OP, then the point of contact T, OQ and OP will lie iu one plane when
(
e
•::(!
^1
40
MOMENTS OF INERTIA.
other two confocals are tangents to the lines of curvature on the
ellipsoid, and are also parallel to the principal diameters of che
diametral section made by a plane parallel to the tangent plane
at P. And if D^D^ be these princijtal semi-diameters, we know
that
\ =\ -Z>„
Hence, if through any point P we describe the quadric
X'
y
A + \^ B^-\ C \
1,
the axes of co-ordinates being the principal axes at the centre of
gravity, then the principal axes at P are the normal to this
tjuadric, and parallels to the axes of the diametral section made
by a plane parallel to the tangent plane at P. And if these axes
be 2Z>, and ^D^, the principal moments at Pare
OF'-K OP'-\ + D,\ OP'-X + D^\
Ex, If two bodies have the same centre tf gravity, the same principal axes at
the centre of gravity and the differences of i,l.3ir principal moments equal, each to
each, then these bodies have the same priuclpcl axes at all points.
60. The axes of co-ordmates being tJie principal axes at the
centre of gravity it is required to express the condition that any
given straight line may he a principal axis o/t some point in its
length and to find that point.
Let the equations to the given straight line be
^-f^y-9^z-h
I m n
(1).
OQ is an axis of the section. For draw through T a section parallel to the diame-
tral section, and let 0' be its centre, and let O'Y' be a perpendicular from 0' on the
tangent plane, which touches at T. Then OQ, <yY' and OP are in one plane.
Now consider the section whose centre is 0' ; O'Y' is the perpendicular on the tan-
gent to an ellipse whose point of contact is T. Hence O'Y', OT do not coincide
unless O'Y' be the direction of the axis of the ellipse. But this section is similar
to the diametral section to which it was drawn parallel. Hence OQ is an axis of
the diametral section.
Let PR be a straight line drawn through P parallel to OQ! to meet in R the
tangent plane which touches in T. Then RP, RT are two tangents at right angles
to the ellipse PQT. Hence
OB? = sum of the squares of the semi-axes of the ellipse
because OP, OQ are conjugate diameters.
The moment of inertia about PR, a perpendicular to a tangent plane, has been
proved above to bo OR^ - X, hence the moment of inertia about a parallel through P
to the axis OQ is OP'^ + OQ" - \.
PRINCIPAL AXES.
41
then it must be a normal to the quadric
*' +^+ ^'
.(2)
at the point at which the straight line is a principal axis.
Hence comparing the equation to the normal to (2) with (1),
wo have
X
A + \
fil,
y
:bTx = ^^' ai:x = '*^
(3),
these six equations must be satisfied by the same values of x, y, z,
\ and fi.
Substituting for x, y, z from (3) in (1), we get
I m n
eliminating fi from these last equations we have
9 J^ ^ f
J?
j_
I'
m
m
n
n
I
A-B B-C~ C-A~^
(4).
This ctearly amounts to only one equation, and is the required
condition that the straight line should be a principal axis at some
point in its length.
Substituting for x, y, z from (3) in (2), we have
\{V+-7a^^n'')=\- [AV + Bm'' + On') ,
which gives one value only to A,. The values of \ and /* having
been found, equations (3) will determine x, y, z, the co-ordinates
uf the point at which the straight line is a principal axis.
The geometrical meaning of this condition may be found by
the following considerations, which were given by Mr Townsend
in the Mathematical Journal. The normal and tangent plane at
every point of a quadric will meet any principal plane in a point
and a straight line, which are pole and polar Avith regard to the
focal conic in that plane. Hence to find whether any assumed
straight line is a principal axis or not, draw any plane perpen-
dicular to the straight line and produce both the straight lino
and the plane to meet any principal plane at the centre of gravity.
If the line of intersection of the plane be parallel to the polar
line of the point of intersection of the straight line with respect
to the focal conic, the axis will be a principal axis, if otherwise it
will not be so. And the point at which the assumed straight line
is a principal axis may be found by drawing a plane through the
^i
i
I
I, '
1 I
'J !
'i
42
MOMENTS OF INERTIA;
polar line perpendicular to the straight line. The point of inter-
section is the required point.
The analytical condition (4) exactly expresses the fact that the
polar line is parallel to the intersection of the plane.
61. Ex. 1. Given a plane ^ + ?^+--l=0, there is always some point in it
f !J h
nt which it is a principal plane. Also this point is its intersection with the straight
line fx-A=gy- B=hz - C.
Ex. 2. Let two points P, Q he so situated that a principal axis at P intersects a
principal axis at Q. Then if two planes be drawn at P and Q perpendicular to
these principal axes, their intersection will he a principal axis at the point where
it is cut by the plane containing the principal axes at P and Q. [Mr Townsend.]
i''or let the principal axes at P, Q meet any principal plane at the centre of
giiivity in p, q, and let the perpendicular planes cut the same principal plane in
ZiV, MN. Also let the perpendicular planes intersect each other in RX. Then
RN is perpendicular to the plane containing the points P, Q, p, q. Also since the
polars of p and q are LN, MN, it follows that pq is the polar of the point N. Hence
the straight line UN satisties the criterion of the last Article.
Ex. 3. If P be any point in a principal plane at the centre of gravity, then
every axis which passes through P, and is a principal axis at some point, lies in one
of two perpendicular planes. One of these planes is the principal plane at the
centre of gravity, and the other is a plane perpendicular to the polar line of P with
regard to the focal conic. Also the locus of all the points Q at which QP is a prin-
cipal axis is a circle passing through P and having its centre in the principal plane.
[Mr Townsend.]
Ex. 4. The edge of regression of the developable surface which is the envelope
of the normal planes of any line of curvature drawn on a confocal quadric is a
curve such that all its tangents are principal axes at some point in each.
62. To find the locus of the points at which two principal
moments of inertia are equal to each other.
The principal moments at any point P are
I,= OF'-\, I^=OP'-\ + D^\ I,= OP'-X + D,\
If we equate /, and I^ we have D^ = 0, and the point F must
lie on the elliptic focal conic of the ellipsoid of gyration.
If we equate /^ and I.^ wo have D^ = D^, so that P is an um-
bilicus of any ellipsoid confocal with the ellipsoid of gyration. The
locus of these umbilici is the hyperbolic focal conic.
In the first of these cases we have \ = — C, and D^ is the semi-
diameter of the focal conic conjugate to QP. Hence i>/+ OP^ —
sum of squares of semi-axes = -k — 4- i? — C The three prin-
cipal moments are therefore I^ = I^= OF' + C, I^ = A+B — C, uud
the axis of unequal moment is a tangent to the focal conic.
The second case may be treated in the same way by using
a confocal hyperboloid, we therefore have I^= 1^= 0P'+ B,
PRINCIPAL AXES.
43
is the envelope
oint P must
I^ = A + C—B, and the axis of unequal moment is a tangent
to the focal conic.
63. To find the curves on any confocal quadric at which a
principal moment of inertia is equal to a given quantity I.
Firstly. The moment of inertia about a normal to a confocal
quadric is 0P* — \. If this be constant, we have OP constant,
and therefore the required curve is the intersection of that
quadric with any concentric sphere. Such a curve is a sphero-conic.
Secondly. Let us consider those points at which the moment
of inertia about a tangent is constant.
Construct any two confocals whose semi-major axes are a and
a. Draw any two tangent planes to these which cut each other
at right angles. The moment of inertia about their intersection
is the sum of the moments of inertia with regard to the two
planes, and is therefore
= B+G-A + a^+a\
Thus the moments of inertia about the intersections of perpendicular
tangent planes to the same confocals are the same.
Let a, a', a" be the semi-major axes of the three confocals
which meet at any point P, then since confocals cut at right
angles, the moment of inertia about the intersection of the con-
focals a', a" is
I, = B+C-A + a:''-Va"\
The intersection of these two confocals is a line of curvature
on either. Hence the moments of inertia about the tangents to any
line of curvature are equal to one another; and these tangents are
principal axes at the point of contact.
On the quadric a draw a tangent PT making any angles (f>
and T: — <f> with the tangents to the lines of curvature at the
z
point of contact P. If T^, /, be the moments about the tangents
to these lines of curvature, the moment of inertia about the
tangent PT
= /j cos'^ <f> + ^a sin" (j>
= B-^ C-A+ {a"^ + a') cos" <f> +■ (a" + a") sui" <f>.
J-nt along a geodesic on the quadric a, a'^siu'c^ + a'^cos'^ is
constant. Hence the moments of inertia about the tangents to any
geodesic on the quadric are equal to each other.
64. Ex. 1. If a straight line touch any two confocals v.hose semi-major axes
are a, a', the moment of inertia about it is ^ + C - il + o' + a'^.
44
MOMENTS OF INERTIA.
Ex. 2. When a' body is referred to its principal axes at the centre of grafity,
Bhovv how to find the coordinates of the point P at which the three principal
moments are equal to three given quantities IJ^ly [Jullien's Problem.]
The] elliptic co-ordinates of P are evidently a" = i (Jj + 13-/1 -5- C+^) &c. ;
and the co-ordinates (x, y, z) may then be[found by Dr Salmon's formula,
a:-a-'a'
&o.
{A-B){A-0
Ex. 3. Let two planes at right angles touch two confocals whose semi-major
axes are a, a'; and let a, a' be the values of a, a', when the confocals touch the intersec-
tion of the planes; then a* + a'='=a^ + a''', and the product of inertia with regard to
the two planes is aV* - a^'a'^'.
65. The locus of all those points at which one of the prin-
cipal moments of inertia of the body is constant is called an equi-
momental surface.
To find the equation to such a surface we have only to put I^
constant, this gives \ = r^ — I. Substituting in the equation to
the subsidiary quadric, the equation to the surface becomes
cc
y
+
= 1.
Through any point P on an e(|ui-momcntal surface describe
the confocal quadric such that the principal axis is a tangent
to a line of curvature on the quadric. By Art. 63 one of the
intersections of the equi-momental surface and this quadric is the
line of curvature. Hence the principal axis at P about which
the moment of inertia is / is a tangent to the equi-momental
surface.
Again, construct the confocal quadric through P such that
the principal axis Is a normal at P, then one of the intersections
of the raomental surface and this quadric is the sphero-conic
through P. The normal to the quadric, being the principal axis,
has just been showr to be a tangent to the surftxce. Hence the
tangent plane to the equi-momental surface, is tlic plane which
contains the normal to the quadric and the tangent to the sphero-
conic.
To draw a perpendicular from the centre on this tangent
plane, we may follow Euclid's rule. Take PP' a tangent to the
sphero-conic, (h'op a perpendicnhir from on PP, this is the
radius vector OP, because PP is a tangent to the sphere. At P
in the tangent plane draw a perpendicular to PP, this is the
normal PQ to the (]uadric. From drop a pei-pendicular OQ on
this normal, then Oi} is a normal to the tangent plane. Hence
this construction,
If 1^ he any point on an equi-momental surface whose para-
meter is I and OQ a perpendicular from the centre on the tangent
PRINCIPAL AXES.
45
plane, then PQ is the jmncipal axis at P about which the moment
of inertia is the constant quantity I.
The equi-momental becomes Fresnel's wa,ve surfaca when
/ is greater than the greatest principal moment of inert;a at the
centre of gravity, 'llie general form of the surface is too Avell
known to need a minute discussion here. It consists of two
sheets, which become a concentric sphere and a spheroid when two
of the principal moments at the centre of gravity are equal.
When the principal moments are unequal, there are two singu-
larities in the surface.
(1) The two sheets meet at a point P in the plane of the
greatest and least moments. Ax P there is a tangent cone to
the surface. Draw any tangent plane to this cone, and let OQ
be a perpendicular from the centre of gravity on t^iis tangent
plane. Then PQ is a principal axis at P. Thu? iheic pre an
infinite number of principal axes at P because an infinite number
of tangent planes can be drawn to the cone. But at any given
point there cannot be more than three principal axes unless two
of the principal axes be equal, and then the locus of the principal
axes is a plane. Hence the point P is sitimted on a focal conic,
and the locus of all the lines PQ is a normal plane to the conic.
The point Q lies on a sphere whose diameter is OP, hence the
locus of ^ is a circle,
(2) The two sheets have a common tangent plane which
touches the surface along the curve. This curve is a circle whose
plane is perpendicular to the plane of greatest and least moments.
Let OP be a perpendicular from on the plane of the circle,
then P' is a point on the circle. If R be any other point on the
circle the principal axis at R is RP'. Thus there is a circular
ring of points at each of which the principal axis passes through
the same point and the moments of inertia about these principal
axes are all equal.
The equation to the equi-momental surface may also be used
for the purpose of finding the three principal moments at any
point whose co-ordinates {x, y, z) are given. If we clear the equation
of fractions, we have a cubic to determine I whose roots ore the
three principal moments.
Thus let it be required to find the locus of all those points
in a body at which any symmetrical function of the three prin-
cipal moments is equal to a given quantity. We may express
this symmetrical function in terms of the coefficients by the usual
rules, and the equation to the locus is found.
Ex. 1. If au equi-momental surface cut a quadric confocal with the ellipsoid
of gyration at the centre of gravity, *lien the iutersections are a spbero conic and a
line of curvature. But if tlip qualric l)c an rllipsoid, both these cannot be real.
46
MOMENTS OF INERTIA.
V ,',;
I' *■
U
n
I
For if the surfEce cut the ellipsoid in both, let P be a point on the line of
curvature, and 1" a point on the Bphero-conic, then by Art, 59, OP" + D^^ = OP"^,
which is less than ^ + X. But OP'^ + Di« + D,* = 4 -f- B + C + 3\, therefore D^^>B +
C+2\, which is >A + 2\. Hence i>j>than the greatest radius vector of the ellip-
soid, which is impossible.
Ex. 2. Find the Ic-us of all those points in a body at which
(1) the sum cl the principal moments is equi." to a given quantity I.
(2) the bnm of the products of the principal moments taken two and two
together, is equal to P,
(3) the product of the principal moments is equal to P,
The results are
(1) a sphere whose radius is a/ —ir\f " > ^^' 13.
(2) the surface
(x? + f+zy+{A + B + C){x^ + if + z^))_ ,
+ Ax' + By^ + Cz' + AB + BC+CA i '
(3) the surface A'B'C - A 'yV - B'z V - C'x'y^ - 2a; Vz" = P,
where A'— A +y^ + z'', with similar expressions for B*, C.
CHAPTER II.
d'alembert's principle, &c.
66. The principles, by which the motion of a single particle
iinder the action of given forces can be determined, will be found
discussed in any treatise on Dynamics of a Particle. These prin-
ciples are called the thioe laws of motion. It is shown that if
(x, y, z) be the co-ordinates of the particle at any time t referred
to three rectangular axes fixed in space, m its mass ; X, Y, Z the
forces resolved parallel to the axes, the motion may be found by
solving the simultaneous equations,
dt
dt
dt
If we regard a rigid body as a, collection of material particles
connected by invariable relations, we might write down the equa-
tions of tha several particles in accordance with the principles just
stated. The forces on each particle are however no longer known,
some of them being due to the mutual actions of the particles.
We assume (1) that the action between two particles is along
the line which joins them, (2) that the action and reaction be-
tween any two are equal and opposite. Suppose there are n
particles, then there will be 3w equations, and, as shown in any
treatise on Statics, 3?i — 6 unknown reactions. To find the
motion it will be necessary to eliminate these unknown quanti-
ties. We may expect to find six resulting equations, and these
will be shown, a little further on, to ^e sufficien+ to determine the
motion of the body.
When there are several rigid bodies which mutually act and
re-act on each other the problem becomes still more complicated.
But it is unnecessary for us to consider in detail, either this or the
preceding case, for D'Alembert has proposed a method by which
all the necessary equations may be obtained without writing down
the equations of motion of the several particles, and without
making any assumption as to the nature of the mutual actions
except the following, which may be regarded as a natural conse-
quence of the laws of motion.
The internal actions and reactions of any system of rigid bodies
in rdotion are in eqidlihrium amongst themselves.
48
D ALEMBERT S PRINCIPLE.
I' ) '
1
67. To explain D'Alemhert's Principle.
In the application of this principle it will be convenient to
use the term effective force, which may be defined as follows.
When a particle is moving as part of a rigid body, it is acted
on by the external impressed forces and also by the molecular
reactions of the other particles. If we considered this particle to
be separated from the rest of the body, and all these forces re-
moved, there is some one force which, under the same initial
conditions, would make it move in the same way as before. This
force is called the effective force on the particle. It is evidently
the resultant of the impressed and molecular forces on the par-
ticle.
Let m be the mass of the particle, {x, y, z) its co-ordinates
referred to any flxed rectangular axes at the time t. The accele-
d'x d^y , d?z
rations of the particle are ^^, ^'^ and ~ . Let / be the resul-
tant of these, then, as explained in Dynamics of a Particle, the
effective force is measured by mf.
Let F be the resultant of the impressed forces, R the resultant
of the molecular forces on the particle. Then mf is the resultant
of F and R. Hence if mf be reversed, the three F, R, and mf are
in equilibrium.
We may apply the same reasoning to every particle of each
body of the system. We thus have a group of forces similar to R,
a group similar to F and a group similar to mf these three groups
will form a system of forces in equilibrium. Now by D'Alembert's
principle the group R will itself form a system of forces in equili-
brium. Whence it follows that the group F will be in equilibrium
with the group mf Hence
If forces equal to the effective forces hut acting in exactly oppo-
site directions were applied at each point of the system these woxdd
he in equilibrium luith the impressed forces.
68. By this principle the solution of a dynamical problem is
reduced to a problem in Statics. The process would be as fol-
lows. We first choose some quantities by means of which the
position of the system in space may be fixed. We then express
the effective forces on each element in terms of these quantities.
These reversed will be in equilibrium with the given impressed
forces. Lastly, the equations of motion for each body may be
formed, as is usually done in Statics, by resolving in three direc-
tions and taking moments about three straight lines.
(19. Before the publication of D'Alembert's principle a vast number of Dynami-
cal problems had been solved. These may be found scattered through the early
volumes of the Momoir.« of St Pctersburf?, Berlin and Pnris, in the works of John
D ALEMBERT S PRINCIPLE.
4t»
Bernoulli and the Opuscules of Euler. They require for the most part the dctormi-
ration of the motions of several bodies with or without weight which push or pull
each other by menus of threads or levers to which they are fastened or along which
they can glide, and which having a ccrtai:. impulse given them at first are then left
to themselves or are compelled to move in given lines or surfaces.
The postulate of Huyghens, "that if any weights are put in motion by the force
of gravity they cannot move so tliat the centre of gravity of them all shall rise
liigher than the place from which it descended," was generally one of the principles
of the solution : but other principles were always needed in addition to those, and
it required the exercise of ingenuity and skill to detect the most suitable in each
case. Such problems were for some tune a -sort of trial of strength among mathe-
maticians. The Trait6 de Di/namique published by D'Alembert in 1743, put an end
to this kind of challenge by supplying a direct and general method of resolving or
at least throwing into equations any imaginable problem. The mechanical diffi-
culties were in this way reduced to difficulties of Pure Mathematics. See Montucla,
Vol. III. page 616, or Whewell's version of the same in his History of the Inductive
Sciences.
D'Alembert uses the following words :—" Soient A, /?, C, &c. les corps qui com-
posent le systeme, et supposons qu'on leur ait imprime les mouvemens a, b, c, 4c.
qu'ils soient forces, h cause de leur action mntuelle, de changer dans les mouvemens
n, b, c, &c. II est clair qu'on peut regarder le mouvement a imprime au corps A
comme compost du mouvement a, qu'il a pris, et d'un autre mouvement a ; qu'on
peut de meme regarder les mouvemens 6, c, <fec. comme composfis dos mouvemens
b, j9; c, 7; &c., d'ou il s'ensuit que le mouvement des corps A, B, C, &c. entr'eux
auroit 6t6 le meme, si au lieu de leur donuer les impulsions a, f>, c, on leur eftt
doun6 li-la-fois les doubles impulsions a, o; b, ^; &c. Or par la supposition les
corps A, B, G, &c. ont pris d'eux-mgmes les mouvemens a, b, c, &c. done les mou-
vemens a, /3, 7, &c. doivent etre tels qu'ils ne d6rangent rien dans les mouvemens
a, b, c, &c. c'est-3,-dire que si les corps n'avoient rei;u que les mouvemens a, «, 7,
&c. ces mouvemens auroient dft se detruire mutuellement, et le systeme demeurer
en repos. De Ik xesulte le principe suivant pour trouver le mouvement de plusieurs
corps qui agissent les uns sur les autres. Dccomposez les mouvemens o, 6, c&c. im-
primes a chaque corps, chacun en deux autres a, a; b, /3; c, 7; etc. qui soient tels
que si Ton n'eftt imprimS aux corps que les mouvemens a, b, c, &c, ils eussent pu
conserver les mouvemens sans se nuire rCciproquement ; et que si on ne leur efit
imprim6 que les mouvemens o, ^, 7, &c. le systeme ffit demeur^ en repos ; il est
clair que a, b, c, &c. seront les mouvemens que ces corps prendront en vertu de leur
action. Ce qu'il falloit trouver."
70. As an e-xample of D'Alembert's principle let us consider
the following problem.
A heavy body is capable of motion by two hinges about a hori-
zontal axis, which axis is made to rotate with a uniform angular
velocity w about a vertical axis intersecting it in a point 0. It is
required to find the conditions that the body may be inclined at a
constant angle to the vertical.
Let the horizontal axis which is fixed in the body be taken as
axis of y, and let two other axes also fixed in the body be taken
origin 0. Let 6 be the angle
as a set of rectangular axes with
R. D.
4
r :- '
I i
I
h.;..
ou
d'alembert's rniNCiPLE.
the plane of yz makes with a vertical piano through Oj.
object is to find the relation between d and a>.
Our
By hj'pothesis each particle P describes a horizontal circle
whose centre C is in the vertical through 0. If r be the radius
CP of this circle, and m the mass, the effective force on the
particle is mai'i and is directed along the radius. When reversed
this will act in the direction CP.
The impressed forces on the body are its weight which may be
supposed to act at the centre of gravity and the actions at the
hinges. To avoid these last, we shall take moments about the
axis Oj/. Then the moment of the reversed effective forces toge-
ther with the moment of the weight will be zero.
Let M be the mass of the body, x, jj, z the co-ordinates of the
centre of gravity, | its distance from the vertical plane through Oy.
The moment of the weight is Mg^. The resolved part of the
effective force parallel to O.v has no moment about Oi/. The
moment of the resolved part perpendicular to the vertical plane
through Oy is mal^p if p be the distance of the particle from that
plane. The equation of moments gives if CO = u
Mg^ -{■ Zmoi'pu = 0.
By jDrojecting the co-ordinates on CO and OP we have -
ti = — X m\ 6 + z cos 6,
p= X cos 6 + z ?,m. d,
f = X cos 6 -{-z sin 0.
Substituting we get
Mg [x cos ^ + i sin 6) = o)" [\ sin 26%m (x^ — z') — cos ^Otmxz],
When the shape and structure of the body are known, the
integrals Sm (x^ — z^) and Sm scz can be found by the methods of
DALEMBERTH PUINCIPLK.
51
the preceding chapter or by direct integration. Tliis equation
will then give the required relation between 6 and w.
It may be noticed that the only manner in which the form of
the body enters into the equation is through its moments and
products of inertia. If we change the body into any equi-mo-
mental one, the equation comiecting and w will be imaltered.
So far as this problem is concerned, a body may be said to bo
given Dynamically when its mass, centre of gravity, principal
axes, and principal moments at the centre of gravity are given.
This remark will be found to be of general application.
Ex. 1. If the body be pnshoil along the axis of y and made to rotate abont the
vertical with tlio Barae angular velocity as before, show that uo effect is produced on
the inclination of the body to the vertical.
Ex. 2. If the body bo a heavy disc capable of turning about a horizontal axis
Oy in its own plane, show that the piano of the disc will be vertical unless w' > "—
whera h is the distance of the centre of gravity of the disc from Oij and k the radius
of gyration about Oy.
Ex. 3. If the body bo a circular disc capable of turning about a horizontal axis
perpendicular to its plane and intersecting the disc iu its cu-cumference, show that
if the tangent to the disc at the hinge make an argle with the vertical, the angular
velocity w must be a./ — '-
sin d
Ex. 4. Two equal balls A and B ore attached to the extremities of two equal
thin rods Aa, Bb. The ends a and h are attached by hinges to a fixed point O and
the whole is set in rotation about a vertical through as in the Governor of the
Steam Engine. If the mass of the rods be neglected show that the time of rotation
is equal to the time of oscillation of a pendulum whose length is the vertical distance
of either sphere below the hinges at 0.
Ex. 5. If in the last example m be the mass of either thin rod and M that of
either spliere, I the length of a rod, r the radius of a sphere, h the depth of either
centre below the hinge, then the length of the pendulum is
l + r M{l-\-r) + \ml
71. To apply D'Alemherfs principle to obtain the equations
of motion of a system of riyid bodies.
Let (.r, y, z) be the co-ordinates of the particle m at the time
t referred to any set of rectangular axes fixed in space. Then
— t. V^i and -TT, will be the accelerations of the particle. Let
dt' ' dt' * dt"
X, Y, Z be the impressed accelerating forces on the same particle
resolved parallel to the axes. By D'Alembert's principle the
forces
-^^-T^^ '"(^'-§)- -^-%
4—2
I' /
,) )
;(;.
52
d'alembert's principle.
together with similar forces on every particle will be in equi-
librium. Hence by the principles of Statics we have the equation
= zmX,
',m
dC
and two similar equations for y and z', these are obtained by
resolving parallel to the axes. Also we have
;»i
(^1^-40=^""*^-^^'
and two similar equations for zx and xy ; these arc obtained by
taking moments about the axes.
These equations may be written in the more convenient forma
d ^ dx ^ -.r
(A).
,(B).
d ^ dz V '7
J-, 2m -J- « 2»iZ
dt dt
In a precisely similar manner by taking the expressions for
the accelerations in polar co-ordinates we should have obtained
another but equivalent set of equations of motion.
72. Let us consider the meaning of these equations without
reference to axes of co-ordinates. The effective forces are to be
equivalent to the impressed forces. But as shown in Statics any
system of forces and therefore each of these is equivalent to a
single force and a single couple at some point taken as origin.
These resultaiic forces and couples must therefore be equivalent,
each to each.
If we multiply the mass m of any particle P by its velocity v
we have the momentum mv of the particle. Let us represent this
in direction and magnitude by a straight line PP'. Then, just as
in Statics, this momentum is equivalent to an equal and parallel
linear momentum at which we may represent by OM, and a
couple whose moment is mvp, where p is the perpendicular dis-
tance between OM and PP'. The plane of this couple is the
D ALEMBKRT S rUINCIPLfc:.
53
plane containing 0^F and PP", and it may as usual be represented
in direction and magnitude by an axis ON perpendicular to its
plane. This couple is sometimes called an angular momentum.
Let 0^f', ON' bo the positions of these two lines after an
interval of time dt. Then MM', NN' will represent in direction
and magnitude the linear momentum and the angidar or couple
momentum added on in the time dt. Hence the elective force
on any particle vi is equivalent to a single linear eti'ective force
acting at represented by - ,.-, fvnd a single effective couple
NN'
represented by — '^ .
Let OV, on be two straight linos drawn through the origit.
to represent in direction and magnitude the resultant linear
momentum and resultant couple momentum of the whole system
at any time t. Let OV, OH' be the positions of these lines at
the time t-{-dt. Then OF is the resultant of the group 03f cor-
responding to all the particles of the system, and V the resultant
VV
of the group OM'. Hence - ,.— represents the whole linear ef-
fective force of the system at the time t. By similar reasoning
HH'
--rr represents the resultant effective couple of the system. Thus
it appears that the points Faud //trace out two curves in space
whose properties are analogous to those of the hodograph in
Dynamics of a particle. From this reasoning it follows, that if
Vx be the resolved part of the momentum of a system in the
direction of any straight line Ox, and H„ the moment of the
momentum about that straight line, then — , * and —jf are re-
spectively the resolved part along, and the moment about that
straight line, of the effective force of the whole system.
Let us now refer the whole system to Cartesian co-ordinates
flOC Cm ?J fl 2
as in Art, 71. We see that m -i- , m Sr . ''^ r are the resolved
dt dt dt
Hence OF is the
parts of the momentum of the particle m.
resultant of Sm
dx
%m
dx\
dy , ^ dz ., f dy
J . ^„^ -;- , and im-rr. Also m\x-'i —y .
dt dt dt \ dt "^ dt J
is the moment of the momentum of the particle m about the
axis of z. Hence OH is the resultant of
H^'ir
dx'\
diJ'
y — ^
\,m
( dz di/\
VTt-'dtJ'
\,m
dx
'di
X
dz\
dtl'
Now D'Alembert's principle asserts that the whole effective
forces of a system are together equivalent to the impressed forces.
fc
! l-MSli:
54
d'alembert's principle.
ri
I il
Hence whatever co-ordinates may be lised, if X and L be the
resolved parts and moment of the impressed moving forces re-
spectively along and about any fixed straight line which we shall
call the axis of x, the equations of motion are
dK
dt
~ = X,
dJI^
dt
= L.
The first of these corresponds to equations (A), the second to
equations (B) of Art. 71.
We may notice the following cases.
(1) If no impressed forces act on the system, the two lines
OV, OH are absolutely fixed in direction and magnitude through-
out the motion.
(2) If all the impressed forces pass through a fixed point,
let this point be chosen as the origin, then though OF may be
variable, OZ/is fixed in position and magnitude.
(3) If all the impressed forces be equivalent to a system
of couples, then though OH may be variable, V is fixed in
position and magnitude*.
73. The equations of motion of Art. 71 are the general equa-
tions of motion of any dynamical system. They are, however,
extremely inconvenient in their present form. When the system
considered is a rigid body and not merely a finite number of
separate particles, the 2's are all '^icfinite integrals. There are
also an infinite number of xb, ?/'s and ^'s all connected together
by an infinite number of geometrical equations. It will be neces-
sary, as suggested in Art. 68, to find some quantities which may
determine the position of the body in space and express the
effective forces in terms of these quantities. These are called the
co-ordinates of the bodi/f. It is most important in theoretical
dynamics to choose these co-ordinates properly. They should be
(1) such that a knowledge of them in terms of the time determines
the motion of the body in a convenient manner, and (2) such that
the dynamical equations when expressed in terms of them may
be as little complicated as possible.
74. Let us first enquire how many co-ordinates are necessary
to fix the position of a body.
The position of a body in spa?e is given when we know the
co-ordinates of some point in it and the angles which two straight
lines fixed in the body make with the axes of co-ordinates. There
• In a memoir on the differential coefficients and determinants of lines, Mr Cohen
lias discussed some of the properties of those resultant lines. rhU. Trans. 1862.
t Sir W. Hamilton uses the phrase "marks of position," but subsequent writers
have adopted the term co-ordinates. Sec Caylry's licport to the Brit. Assoc, 1857.
D ALEMBERTS PRINCIPLE.
55
re necessary
are three geometrical relations existing between these six angles,
so that the position of a body may be made to depend on sia;
independent variables, viz. three co-ordinates and three angles.
These might be taken as the co-ordinates of the body. By the
term "co-ordinates of a body" is meant any quantities which de-
termine the position of the body in space.
It is evident that we may express the co-ordinates (x, i/, z) of
any particle m of a body in terms of the co-ordinates of that body
and quantities which are known and remain constant during the
motion. First, let us suppose the system to consist only of a
single body, then if we substitute those expressions for x, y, z in
the equations (A) and (B) of Art. 71, we shall have six equations
to determine the six co-ordinates of the body in terms of the
time. Thus the motion will be found. If the system consist of
several bodies, we shall, by considering each separately, have six
equations for each body. If there be any unknown reactions be-
tween the bodies, these will ])e included in A'^ F, Z. For each
reaction there will bo a corresponding geometrical relation con-
necting the motion of those bodies. Thus on the Avhole we shall
have sufficient equations to determine the motion of the system.
When the motion is in two dimensions these six co-ordinates
become three. These ai-e the tv/o co-ordinates of the fixed point
in the body, and the angle some straight line fixed in the body
*makes with a straight line fixed in space.
75. Let us next consider how the equations of motion formed
by resolution can be simplified by a proper choice of co-ordinates.
We must find tlio resolved part of the momentum and the re-
solved part of the effective forces of a system in any direction.
Let the given direction be taken as the axis of x. Let {x, y, z)
he the co-ordinates of any particle whose mass is m. The re-
dx
solved part of its momentum in the given direction is tn -jr .
Hence the resolved part of the momentum of the whole system is
dx -
2)H -y- • Let (x, y, z) be the co-ordinates of the centre of gravity
of the system and M the whole mass. Then Mx == ^mx ;
•• '^^di=^"'dt'
Hence the resolved pai't of the momentum of a system in any
direction is equal to the whole mass multiplied into the resolved part
of the velocity of the centre of gravity.
That is, the linear momentum of a system is the same as if the
whole mass tuere collected into its centre of gravity.
'nf
,'m.m s
56
d'alembert's principle.
!.|;
fM
m 1^
In the same way, the resolved part of the effective forces of a
system in any direction is eqwal to the whole mass multiplied into
the resolved part of the acceleration of the centre of gravity.
It appears from this proposition that it will be convenient to
take the co-ordinates of the centre of gravity of each rigid body in
the system as three of the co-ord'inates of that body. We can then
express in a simple form the resolved part of the effective forces
in any direction.
76. Lastly, let us consider how the equations of motion formdti
by taking moments can be simplified by a proper choice of the
three remaiining co-ordinates. We must find the moment of the
momentum and the moment of the effective forces about any
btraifjht line.
Let the given straight line be taken as the axis of x, then
following the same notation as before, the moment of the mo-
mentum about the axis of x is
S"'(4:-i)
If, then, wo
Now this is an expression of the second degree,
substitute y = y-Vy, ^ = s -f- s', we get by Art. 14
wlrere M is the mass of the system or body under consideration.
The second term of this expression is the moment about the
axis of X of the momentum of a mass M moving witli the centre
of gravity.
The first term' is the moment about a straight line para'llel to
the axis of x, not of the actual momenta of all the several parti-
cles but of their momenta relatively to that of the centre of gravity.
In the case of any particular body it therefore depends only on the
motion of the body relatively to its centre of gravity. In finding
its value we shall suppose the centre of gravity reduced to rest by
applying to every particle of the system a velocity equal and oppo-
site to that of the centre of gravity. Hence Ave infer that
The moment of the momentum of a system about any straight
line is equal to the moment of the- momentum of the ivhole mass
supposed collected at its centre of gravity and moving with it,
together with the moment of the momentum of the system relaVive to
Us centre of gravity about a straight line drawn parallel to the given
straight line through the centre of gravity.
In the same way, this proposition will bo also true if for the
"momentum" of the svstom avc substitute " efteetive force."
j! I
Sssstfs
D ALEMBERT S PRINCIPLE.
B7
By taking the axis Ox through the centre of gravity, we see
that the moment of the relative momenta about any straight line
through the centre of gravity is equal to that of the actual
momenta.
77. It appears from the preceding article that it will be con-
venient to refer the angular motion of a body to a system of
co-ordinate axes meeting at the centre of gravity. A general
expression for the moment of the effective forces about any straight
line through the centre of gravity cannot be conveniently investi-
gated at this stage. Different expressions will be found advanta-
geous under different circumstances. There are three cases to
which attention should be particularly directed: (1) when the
body is turning about an axis fixed in the body and fixed in
space ; (2) when the motion is in two dimensions, and (3) Euler's
expression when the body is turning about a fixed point. These
will be found at the beginnings of the third and fourth chapters
and in the fifth chapter respectively.
78. The quantity Sm [^-^
dy dx
i.—y-r.j expresses the moment of
the momentum about the axis of z. It is then called the angtilar
momentum of the system about the axis of z. There is anothei'
interpretation which can be given t© it. If we transform to polar
co-ordinates, we have
dy dx dd
dt ^ dt dt
Now \r^d6 is the elementary area described round the origin
in the time dt by the projection of the particle on the plane of xy.
If twice this polar area be multiplied by the mass of the particle,
it is called the area conserved by the particle in the time dt round
the axis of z. Hence
.^ / dy div\
^'''V'dt-yit)
is called the area conserved by the system' in a unit of time, or
more simply the area conserved.
79. We may now enunciate two important propositions, which
follow at once from' the preceding results. It will, however, be
more useful to deduce them' from first principles.
(1) The motion of the centre of gravity of a system acted on hy
any forces is the same as if all the mass were collected at the centre
of gravity and all' the forces were applied at that 'point parallel to
their former directions.
(2) The motion of a body, acted on by any forces, about its
centre of gravity is the same as if the centre of gravity ivere fixed
and the same forces acted on the body.
1 !^
Its
'^/
^ !
1 15
f
H
flp^
'!
n
I !
58
DALEMBERTS PRINCIPLE.
Tak' '^g any one of the equations (A) we have
at
If X, y, z bo the co-ordinates of the centre of gravity, then
xXtn = Hmx ;
dt
S?/i = ^mX,
and the other equations may be treated in a similar manner.
But these are the equations which give the motion of a mass
2?^ acterl on by forces XmX, &c. Hence the first principle is
proved.
Taking any one of equations (B) we have
tm [x-^r -y -j^^j = Sm {xY - yX).
Let x=x + x' , y =y jf.y^ 2=2 + z', then by Art. 14 this equa-
tion becomes
"tm (.
2,/
„' A'
^'^-i/-:u^\ +
, d'x
It
X
d^y - d^x
df
y -^) tm = Xm{x Y- yX).
Now the axes of co-ordinates are quite arbitrary, let them be
so chosen that the centre of gravity is passing through the origin
at the moment under consideration. Then ^ = 0, ^ = 0, but
~ , ~ are not necessarily zero. The equation then becomes
This equation does not contain the co-ordinates of the centre
of gravity and holds at every separate instant of the motion and
therefore is always true. But this and the two similar equations
obtained from the other two equations of (B) are exactly the equa-
tions of moments we should have had if we had regarded the
centre of gravity as a fixed point and taken it as the origin of
moments.
80. These two important propositions are called respectively
the principles of th'^ conservation of the motions of translation and
rotation. The first was given by Newton in the fourth corollary
to the third law of motion, and was afterwards generalized by
D'Alembert and Montucla. The second is more recent and seems
to have been discovered about the same time by Euler, Bernoulli
and the Chevalier d'Arcy.
81. By the first principle the problem of finding the motion
of the centre of gravity of a system, however complex the system
d'alembeut's principle.
59
may be, is reduced to the problem of finding the motion of a
single particle. By the second the problem of finding the angular
motion of a free body in space is reduced to that of determining
the motion of that body about a fixed point.
In using the first principle it should be noticed that the im-
pressed forces are to be applied at the centre of gravity parallel
to their former directions. Thus, if a rigid body be moving
under the influence of a central force, the motion of the centre of
gravity is not generally the same as if the whole mass were col-
lected at the centre of gravity and it were then acted on by the
same centr^-l force. What the principle asserts is, that, if the
attraction of the central force on each element of the body be
found, the motion of the centre of gravity is the same as if these
forces were applied at the centre of gravity parallel to their
original directions.
If the impressed forces act always parallel to a fixed straight
line, or if they tend to fixed centres and vary as the distance from
those centres, the magnitude and direction of their resultant are
the same whether we suppose the body collected into its centre of
gravity or not. But in most cases care must be taken to find the
resultant of the impressed forces as they really act on the body
before it has been collected into its centre of gravity.
82. From this proposition we may infer the independence of
the motions of translation and rotation. The motion of the centre
of gravity is the same as if the whole mass were collected at that
point, and is therefore quite independent of the rotation. The
motion round the centre of gravity is the same as if that point
were fixed, and is therefore independent of the mo-tion ot that
point.
83. We may now collect together for reference the results of
the preceding articles.
Let u, V, w be the velocities of the centre of gravity of any
rigid body of mass M resolved parallel to any tliree fixed rect-
angular axes, let h^, h^, k^ be the three moments of tlic momentum
relative to the centre of gravity about three recLangular axes
fixed in direction and meeting at the centre of gravity. Then the
effective forces of the body are equivalent to the three effective
forces M-r- y M -j-., ^^-r acting at the centre of gravity parallel
to the directions into which the velocities have been resolved, ajid
to the three effective couples —i^ , ~ , -~ about the axes- meet-
ing at the centre of gravity about which the moments were taken.
The effective forces of all the other bodies of the system may be
expressed in a similar manner.
M
! f
i»:
I
60
d'alembert's principle.
Then all these effective forces and couples, being reversed, will
V-^ in equilibrium with the impressed forces. The equations of
equilibrium may then be found by resolving in such directions and
taking moments cbout such straight lines as may be most con-
venient. Instead of reversing the effective forces it is usually
found more con\euient to write the impressed and effective forces
on opposite sides of the equations.
Application of UAlemhert's Principle to impulsive forces.
84. If a force F act on a particle of mass m always in the
same direction,, the equation of motion is
where v is the velocity of the particle at the time t. Let T be the
interval during which the force acts, and let v, v' be the velocities
at the beginning and end of that interval. Then
m
(u'-u)=J
Fdt.
Now suppose the force F to increase without limit while the
interval T decreases without limit. Then the integral may have
a finite limit. Let this limit be P. Then the equation becomes
m {v —v) = P.-
The velocity in the interval T has increased or decreased from
V to V. Supposing the velocity to have remained finite, let V be
its greatest value during this interval. Then the space described
is less than VT. But in the limit this vanishes. Hence the
particle has not moved during the action of the force F. It has
not had time to move but its velocity is suddenly changed from
V to v.
"We may consider that a proper measure has been found for a
force when from that measure we can deduce all the effects of the
force. In the case of finite forces we have to determine both the
change of place and the change in the velocity of the particle. It
is therefore necessary to divide the whole time of action into
elementary times and determine the effect of the force during
each of these. But in the case of infinite forces which act for an
indefinitely short time, the change of place is zero, and the change
, of velocity is the only element to be determined. It is therefore
more convenient to collect the whole force expended into one
measure. Such a force is called an impulse. It may be defined
as the limit of a force which is infinitely great, but acts only
during an infinitely short time. There are of course no such
' '^haa^M^sti^'^'-it
d'alembert's principle. 61
forces in nature, but there are forces which are very great, and
act only during a very short time. The blow of a hammer is
a force of this kind. They may be treated as if they were im-
pulses, and the results will be more or less correct according to
the magnitude of the force and the shortness of the time of
action. They may also be treated as if they were finite forces,
and the displacement of the body during the time of action of the
force may be found.
The quantity P may be taken as the measure of the force.
An impulsive force is measured by the whole momentum gener-
ated by the impulse.
85. In deter. idning the effect of an impulse on a "body, the
effect of all finite forces which act on the body at the same time may
he omitted.
For let a finite force / act on a body at the same time as an
impulsive force F. Then as before we have
rX rT
Fdt fdt .
m m m m
But in the limit fT vanishes. Similarly the force / may be
omitted in the equation of moments.
86. To obtain the general equations of motion of a system
acted on by any number of impulses at once.
Let u, V, w, u\ V, vo be the velocities of a particle of mass m
parallel to the axes just before and just after the action of the
impulses. Let X\ Y', Z' be the resolved parts of the impulse on
m parallel to the axes.
Taking the same notation as before, we have the equation
or integrating
tm{u'-'u) = tm\ Xdt = XX' (1).
Jo
Similarly we have the equations
2w (v' - v) =XY' (2),
Xm{w'-w) = tZ' (3).
Again the equation
2m (x -^ - y~j = tm (x Y- yX)
I
I
t)2 d'alemrert's principle.
becomes on integration
or taken between limits,
Xm[x{v'-v)-i/{u'-u)] = %{xY'-yX') (4),
and the other two equations become
Xm\i/{iv'—iv) — z {v —v)\ =S iyZ' — zY') (5),
Xm[z{u'-u)-x{w-iv)]='Z{zX'-xZ') (6).
In all the followins: investijjations it will be found convenient
to use accen+'^d letters to denote the tates ' mocion after impact
■which correspond to those denoted Ir < m- . IjC letters unaccented
before the action of the impulse. Sii. ; !;•. Ganges in direction
and magnitude of the velocities of tl - ixt^l particles of the
bodies are the only objects of investigati-m, it v i ' be more conve-
nient to express the equations of motion in terms of these veloci-
ties, and to avoid the introduction of such symbols ^^ -jii ~^> -f-
87. In applying D'Alembert's Principle to impulsive forces the
only change which must be made is in the mode of measuring the
effective foices. If (u, v, w), {u\ v, w') be the resolved parts of the
velocity of any particle, just before and just after the impulse, and
if 7nbe its mass, the effective forces will be measured by m{u'—^l),
m (v — v), and m [lo' — V)). The quantity mf ai Art. 67 is to be
regarded as the measure of the impulsive force which, if the parti-
cle were separated from the rest of the body, would produce these
changes of momentum.
In this caae, if we follow the notation of Arts. 75 and 76, the
resolved part of the effective force in the direction of the axis of z
dz
is the difference of the values of Sm -r just before and just after
the action of the impulses, and this is the same as the difference
dz
of the values of M -j- at the same instants. In the same way the
moment of the effective forces about the axis of z will be the
difference of the values of
.1^ / dii dx^
just before and just , ♦'ter the action of the impulses.
We may therefore extend the general proposition of Art. 83 to
impulsive forces in the following manner.
Let (u, V, tu), {u', v', w) be the velocities of the centre of gravity
of any rigid body of mans il/ just before and just after the action
D ALEMBERT S PRINCIPLE.
03
W.
(5),
(6).
;onvctiient
tcr impact
inaccented
I direction
;les of the
lore conve-
liese veloci-
x dy dz^
Tt'tt' dt'
le forces the
jasuring the
parts of the
impulse, and
^y m{it—u),
67 is to be
if the parti-
•oduce these
and 76, the
the axis of z
id just after
be difference
lame way the
will be the
of Art. 83 to
Ure of gravity
ter the action
of the impulses resolved parallel to any three fixed rectangular
axes. Let (/i,, h,^, h^, {h', //./, h^') be the three moments of the
momentum relative to tlic centre of gravity about three rect-
angular axes fixed in direction and meeting at the centre of
gravity, the moments being taken just before and just after the
impulses. Then the effective forces of the body are equivalent to
the three effective forces M{u —%C), M(v' — v), M{w' — w) acting
at the centre of gravity parallel to the rectangular axes together
with the three effective couples (/*,'— A,), {k^ — Ik^), {hj — /<g) about
those axes.
These effective forces and couples being reversed will be in
equilibrium with the impressed forces. The C(iuations of equili-
brium may then be formed according to the rules of Statics.
Ex. 1. Two particles moving in the same plane are projected in parallel bnt
opposite directions with velocities inversely proportional to their masses. Find the
motion of their centre of gravity.
Ex. 2. A person is placed on a perfectly smooth table, show how ho may get
off.
Ex. 3. Explain how a person sitting on a chair, is able to move the chair
across the room by a series of jerks, without touching the ground with his feet.
Ex. 4. A person is placed at one end of a perfectly rough board which rests on
a smooth table. Supposing he walks to the other end of the board, determine how
much the board has moved. If he stepped off the board, show how to determine its
subsequent motion.
Ex. 5. The motion of the centre of gravity of a shell shot from a gun in vacuo
is a poi'abola, and its motion is unaffected by the biu-sting of the shell.
Ex. 6. A rod revolving uniformly in a horizontal plane round a pivot at its ex-
tremity suddenly snaps in two : determine the motion of each part.
Ex. 7. A cube slides down a perfectly smooth inclined plane with four of its
edges horizontal. The middle point of the lowest edge comes in contact with
a small fixed obstacle and is reduced to rest. Determine if the cube is also reduced
to rest, and show that the resultant impulsive action along the edge will not in
general act along the inclined plane.
Ex. 8. Two persons A and B are situated on a perfectly smooth horizontal
plane at a distance a from each other. A throws a ball to B which reaches B after
a time t.
ma
Show that A will begin to slide along the plane with a velocity =-- where
jM t
M is his own mass and 7)i that of the ball. If the plane were perfectly rough
explain in general terms the nature of the pressures between ^'s feet and the
plane which would prevent him from sliding. Would these pressures have a single
resultant ?
Ex. 9. A cannon rests on an imperfectly rough horizontal plane and is fired
with such a charge that the relative velocity of the ball and cannon at the moment
when the ball leaves the cannon is V. If M be the mass of the cannon, m that of
1^
.
;! ' *
64 DAIEMBERTS PRINCIPLE.
the ball and n the coefficient of friction, show that the cannon will recoil a distance
\ ir ',' ) o on the plane.
Ex. 10. A spherical cavity of radius a is cut out of a cubical mass so that the
centre of gravity of the remaining mass is in the vertical through the centre of the
cavity. The cubical mass rests on a perfectly smooth horizontal plane, but the
interior of the cavity is perfectly rough. A sphere of mass m, and radius b, rolls
down the side of the cavity starting from rest with its centre on a level with the
centre of the cavity. Show that when the sphere next comes to rest, the cubical
mass has moved through a space
2in {a - b)
M+itt
where M is the mass of the remaining
portion of the cube,
i'ough or smooth ?
Will the result be the same if the cavity were imperfectly
Ex. 11. Two railway engines drawing the same train are connected by a loose
chain and come several times in succession into collision with each other; the
leading engine being a little top-heavy and the buffers of both rather low. The
fore-wheels of the first engine are observed to jump up and down. What dynamical
explanation can be given of this rocking motion ? At what level should the buffers
be placed that it may not occur? Camb. Tramac. Vol. vii.
Ex. 12. Sir C. Lyell in his account of the earthquake in Calabria in 1783,
mentions two obelisks each of which was constructed of three great stones laid on
top of each other. After the earthquake, the pedestal of each obelisk was found to
be in its original place, but the separate stones above were turned partially round
and removed several inches from their position without falling. The shock which
agitated the building was therefore described as having been horizontal and vorti-
cose. Show that such a displacement would be produced by a fiimple rectilinear
shock, if the resultant blow on each stone did not pass through its centre of gravity.
See Mallet's Dynamics of Earthquakes.
I i
!! I
a distance
riM ri HI I i
BO that the
mtre of the
ae, but the
.dius b, rolls
(rel with the
, the cubical
lie remaining
3 imperfectly
;ed by a loose
■h other; the
ler low. The
hat dynamical
uld the buffers
labria in 1793.
stones laid on
jk was found to
partially round
:he shock which
ontal 0nd vorti-
mple rectilinear
entre of gravity.
CHAPTER III.
MOTION ABOUT A FIXED AXIS.
88. A rigid body can turn freely about an axis fixed in the
body and in space, to find the moment of the effective forces about
the a^s of rotation.
Let any plane passing through the axis and fixed in space be
taken as a plane of reference, and let 6 be the angle which any
other plane through the axis and fixed in the body makes with
the first plane. Let m be the mass of any element of the body,
7* its distance from the axis, let <^ be the angle a plane through the
axis and the element m makes with the plane of reference.
The velocity of the particle m is r -^ in a direction perpendi-
cular to the plane containing the axis and the particle. The
moment of the momentum of this particle about the axis is
clearly mr^ -^
particles is 2 \^mr^ ,J
Hence the moment of the momenta of all the
Since the particles of the body are rigidly
connected with each other, it is obvious that -r- is the same for
d4> .
dt
dd
dt
every particle, and equal to ^ . Hence the moment of the mo-
d0
menta of all the particles of the body about the axis is "Zmr^ ->- ,
i.e. the moment of inertia of the body about the axis multiplied
into the angular velocity,
d?<^
The accelerations of the particle m are r
de
and-r(^
perpendicular to, and along the directions in which r is measured,
J'lA
the moment of the moving forces of m about the axis is mr^ -X ,
hence the moment of the moving forces of all the particles of the
body about the axis is 2 [mr^
d^
dt'
By the same reasoning as
i !
ii
R. D.
i!
w- i
GO
MOTION ABOUT A FIXED AXIS.
before this is ('(jual to Xiiir'* , .^ , i.e. the moment of inertia of the
body about the axis into the angular acceleration.
89. To determine the motion of a body about a fixed axis
under the action of any forces.
By D'Alembert's principle the effective forces when reversed
will be in equilibrium with the impressed forces. To avoid intro-
ducing the unknown reactions at the axis, let us take moments
about the axis.
First, let the forces be impulsive. Let w, w be the angular
velocities of the body just before and just after the action of the
forces. Then, following the notation of the last article,
G)'. 2wr'— ft) . S;«r' = L,
where L is the moment of the impressed forces about the axis ;
moment of forces about axis
O) — ft)
moment of inertia about axis *
This equation will determine the change in the angular velo-
city produced by the action of the forces.
Secondly, let the forces be finite. Then taking moments about
the axis, we have
d^d ^ J J
d'd
de
moment of forces about axis
de
(It
moment of inertia about axis '
This equation when integrated will give the values of and
at any time. Two undetermined constants will make their
appearance in the course of the solution. These are to be deter-
dff
mined from the given initial values of 6 and -j- . Thus the whole
motion can be found.
90. It appears from this proposition that the motion of a
rigid body about a fixed axis depends on ( 1) the moment of the
forces about that axis and (2) the moment of inertia of the body
about the axis. Let Mk^ be this moment of inertia, so that k is
the radius of gyration of the body. Then if the whole mass of
the body were collected into a particle and attached to the fixed
axis by a rod without inertia, whose length is the radius of gyra-
tion k, and if this system be acted on by forces having the same
moment as before, and be set in motion with the same initial
i' )
mmm0amm»
tia of the
fixed axis
a reversed
void intro-
) moments
he angular
;tion of the
the axis ;
mgular velo-
)ments about
ues of 6 and
make their
to be deter-
lus the whole
motion of a
.loment of the
a of the body
,, so that k is
'hole mass of
. to the fixed
adius of gyra-
ting the same
same
GENERAL PRINCIPLES.
e7
dt
values of and '-^ , then the whole subsequent angular or gyra-
tory motion of the rod will be the same as that of the body. We
may say briefly, that a body turning about a fixed axis is dyna-
mically given, when we know its mass and radius of gyration.
91. Ex. A prrfecthj rough circular horizontal hoard iit capable, of revolving
freely round a vertical axis through it» centre. A vian ichose weight is equal to that
of the hoard walks on and round it at the edgt : when he has completed the circuit
what will be his position in space f
Let a bo the radius of the board, Mk' its moment of inertia about the vertical
axis. Let w be the angular velocity of tlip board, u' that of tlie man about the
vortical axis at any time. And let F be the action between the feet of the man and
tlie board.
The equation of motion of the board is by Art. 89,
Mk^'^=-Fa
at
(1).
The equation of motion of the man is by Art. 79,
du' „
Ma
dt
.(2).
Eliminating F and integrating, we get
the constant being zero, because the man and the board start from rest. Let 0,
e' be the angles described by the board and man round tlie vertical axis. Then
w=— , w'=— , and h'e + a^' = 0. Hence, when e'-tf = 2ir, we have ^' = 7^^— -2t.
at at k^ + a^
Tills gives the angle in space described by the man.
It k^=- we have «'=„ t.
Let V be the mean relative velocity with which the man walks along the board.
Then w'-w=-
of the board.
«= -
Va
k'+a;'
2 V
o a
This gives the mean angular velocity
On the Pendulum.
92. A body moves about a fixed horizontal axis acted on by
rjravity only, to determine the motion.
Take the vertical plane through the xis as the plane of refer-
ence, and the plane through the axis and the centre of gravity as
the plane fixed in the body. Then the equation of motion is
d^d _ moment of forces , .
df moment of inertia
Mgh sin
«
r
ii
' ii
08
MOTION ABOUT A FIXED AXIS.
wliere A is the distance of the centre of gravity from the axis and
Mk^ is the moment of inertia of the body about an axis through
the centre of gravity parallel to the fixed axis. Hence
(2).
The equation (2) cannot be integrated in finite terms, but if
the oscillations be small, we may reject the cubes and higher
powers of 6 and the equation will become
Hence the time of a complete oscillation is Itt sJ — r— • If
h and k be measured in feet and g = 3218, this formula gives the
time in seconds.
The equation of motion of a particle of any mass suspended
by a string / is
'J^f+f.sin^ = (3),
which may be deduced from equation (2) by putting k = and
// = I. Hence the angular motions of the string and the body
under the same initial conditions will be identical if
1 =
J^ + h'
h
(4).
This length is called the lenr/th of the simple equivaleiH
pendulum.
Through G, the centre ot gravity of the body, draw a perpen-
dicular to the axis of revolution cutting it in C Then C is called
the centre of suspension. Produce GG to so that CO = l. Then
is called the centre of oscillation. If the whole mass of the
body were collected at the centre of oscillation and suspended by
a thread to the centre of suspension, its angular motion and time
of oscillation vvould be the same as that of the body under the
same initial .cumstances.
The equation (4) may be put under another form. Since
CG = h and OG = 1 —h, we have
(r (7. 6rO= (rad.)" of gyration about (7, "
CG . CO = (rad.)^ of gyration about C,
OG. 0C== (rad.)' of gyration about 0.
Any of these equations show that if be made the centre of
suspension, the axis being parallel to the axis about which k was
taken, then C will be the centre of oscillation. Thus the centres
THE PENDULUM.
69
axis and
s through
(2).
ms, but, if
nd higher
1}^. If
gh
a gives the
suspended
(3).
^ fc = and
'd the body
(4).
equivalent
w a perpen-
[1 is called
= 1 Then
mass of the
uspended by
:>n and time
y under the
nrm. Since
[he centre of
(which k was
Is the centres
of oscillation and suspension are convertible and the time of oscilla-
tion about each is the same.
If the time of oscillation be given, I is given and the equa-
tion (4) will give two values of h. Let these values be h^, h^.
Let two cylinders bo described \\\i\\ that straight line as axis
about which the radius of gyration k was taken, and let the
radii of these cylinders be /«,, K^ Then the times of oscillation of
the body about any generating lines of these cylinders are the
//*
same, and are approximately equal to 27rA/ - .
With the same axis describe a third cylinder whoso radius
IS
k. T\\Qnl=1k +
{h -kY
h
, hence I is always greater than 2k,
and decreases continually as h decreases and approaches the value
h. Thus the length of the equivalent pendulum continually de-
creases as the axis of suspension approaches from without to the
circumference of this third cylinder. When the axis of suspension
is a generating line of the cylinder the length of the equivalent
pendulum is 2,k. When the axis of suspension is within the
cylinder and approaching the centre of gravity the length of the
equivalent pendulum continually increases and becomes infinite
when the axis passes through the centre of gravity.
The time of oscillation is therefore least when the axis is a
generating line of the circular cylinder whose radius is k. But the
time about the axis thus found is not an absolute minimum. It
is a minimum for all axes drawn parallel to a given straight line
in ihe body. To find the axis about which the time is absolutely
a minimum we must find the axis about which A; is a minimum.
Now it is proved in Art. 23, that of all axes through G tlie
* Til 9 position of the centre of oscillation of a body was first correctly deter-
mined by Huygbens in his Ilorologiim OscillatoHum jiiiblished at Paris in 1073.
The most important of the theorems given in the text were discovered by him. As
D'Alembert's principle was not known at that time, Hnygliens had to discover some
principle for himself. The liypothesis was, that when several weights are put in
motion by the force of gravity, in whatever manner they act on each other tlu^r
centre of gravity cannot be made to mount to a height greater than that from which
it had descended. Huygbens considers that he assumes here only that a heavy body
cannot of itself move upwards. The next step in the argument was, that at any
instant the velocities of the particles are such that, if thoy were separated from
each other and properly guided, the centre of gravity could be made to mount to a
second position as high as its first position. For if not, consider the imrticlos to
start from their last positions, to describe the same paths reversed, and then again
to be Jbined together into a pendulum ; the centre of gravity would rise to its first
position ; but if this be higher than the second position, the liypothesis would bo
contradicted. This principle gives the same equation which the modern principltj
of Via Viva would give, and the rest of the solution is not of much intercut.
i!
1
70
MOTION ABOUT A FIXED AXIS.
axis about which the moment of inertia is least or greatest is one
of the principal axes. Hence the axis about which the time of
oscillation is a minimum is parallel to that principal axis through
O about which the moment of inertia is least. And if Mk^ be the
moment of inertia about that axis, the axis of suspension is at a
distance k measured in any direction from the principal axis.
; .■
:
93. Ex. 1. Find the time of tlie small oscillations of a cube (1) when one
side is fixed, (2) when a diagonal of one of its faces is fixed; the axis in both
cases being horizontal.
Result. If 2a be a side of the cube, the length of the simple ecjuivalent pendu-
lum is in the first case ^ - a, and in the second case - a.
Ex. 2. An elliptic lamina i" such that when it swings about one latus rectum
as a horizontal axis, the other latus rectum passes thi'ough the centre of oscillation,
prove that the eccentricity is J.
Ex. 3. A circular arc oscillates about an axis through its middle point perpen-
diculai" to the plane of the arc. Prove that the length of the simple equivalent
pendulum is independent of the length of the arc, and is equal to twice the radius.
Ex. 4. The density of a rod varies as the distance from one end, find the axis
perjoendicular to it about which tlie time of oscillation is a minimum.
Jiesult. The axis passes through either of the two points whose distance from the
centre of gravity is -^r a, where a is the length of the rod.
Ex. 5. Find what axis in the area of an ellipse must be fixed that the time of
a small oscillation may be a minimum.
Result, The axis must be parallel to the major axis, and bisect the semi-minor
axis.
Ex. 6. A uniform stick hangs freely by one end, tlie other end being close to the
ground. An angular velocity in a vertical plane is then communicated to the stick,
and when it has risen through an angle of 90", the end by which it was hanging is
loosed. What must be the initial angular velocity so that on falling to the ground
it may pitch in an upright position ?
Jiesult. The reriuired angular velocity w is given by
2a ^ 2a
j(2n + l)
2\
(2^^+1)^+1
where n is any integer, and 2a is the length of the rod.
Ex. 7. Two bodies can move freely and independently under the action of
gravity about the same horizontal axis ; their masses are m, m', and the distances of
their centres of gravity from the axis are h, h'. If the lengths of their simple equi-
valent pendrJums be L, L', prove that when fastened together the length of tli o
. , . J , ... , mhL + m'h'L'
equivalent pendulum will be - ,- .--- •
mn + mh.'
THE PENDULUM.
71
Ex. 8. Wlien it is required to regulate a clock without stopping the pendulum,
it is usual to add or subtract some small weight from a platform Attached to the
pendulum. Show that in order to make a given alteration in the going of the clock
by tlie addition of the least possible weight, the platform must be placed at a dis-
tance from the point of suspension equal to half the sir. pie equivalent pendulum.
Show also that a sliglit error in the position of the platform will not affect tho
weight required to be added.
Ex. 9. A circular table oentre is supported by three legs AA\ BB', C(7' which
rest on a perfeunly rough horizontal floor, and a heavy particle P is placed on tho
table. Suddenly one leg CC gives way, show that tho tabic and the particle will
immediately separate if pc be greater than k^ ; where p and c arc the distances of P
and respectively from the MweAB joining the tops of the legs, and k is the radius
of gyration of the table and legs about the line A'B' joining the points where tho
legs rest on the floor.
The condition of separation is that the initial normal acceleration of the point
of the table at P should be greater than the normal acceleration of the particle
itself.
Ex. 10. A string without weight is placed round a fixed ellipse whose plane ia
vertical, and the two ends are fastened together. The length of the string is greater
than the perimeter of the ellipse. A heavy particle can slide freely on the string
and performs small oscillations under the action of gravity. Prove that the simple
equivalent pendulum is the radius of curvature of the confocal ellipse passing
through the position of equilibrium of the particle.
94. In a clock whicli is regulated by a pendulum, it is neces-
sary that tlie time of oscillation should be invariable. As all
substances expand and contract with every alteration of tempera-
ture, it is clear that the distance of the centre of gravity of the
pendulum from the axis and the moment of inertia about that
axis will be continually altering. The length of the simple equi-
valent pendulum does not however depend on either of these
elements simply, but on their ratio. If then we can construct a
pendulum such that the expansion or contraction of its different
parts does not alter this ratio, the time of oscillation will be un-
affected by any changes of temperature. For an account of the
various methods of accomplishing this which have been suggested,
we refer the reader to any treatise* on clocks. We shall here only
notice for the sake of illustration one simple construction, whicli
has been generally used. It was invented by George Graham
about the year 1715.
Some heavy fluid, such as mercury, is enclosed in a cast-iron cylindrical jar
into the top of whicli an iron rod is screwed. This rod is then suspended in tho
usual manner from a fixed point. The downward expansion of the iron on any
increase of temperature tends to lower the centre of oscillation, but tho upward ex-
pansion of the mercui'y tends on the contrary to raise it. It is re(iuirod to doter-
* Rcid oil Clocks; Denison's treatise on Clocls and Clockmnldnn in Wcalc's
Si'rics, 1W)7; Caiitain Kator's treatise on .l/rr/jft;;/''? in Laidnor's Ciiflopiritiuy 1880.
i
I
t ■;
72
MOTION ABOUT A FIXED AXIS.
mine the condition that the position of the centre of oscillation may on tlie whole
be unaltered.
Let Mk'^ be the moment o^ inci-tia of the iron jar and rod about the axis of sus-
pen iioE, c the distiUiofl ii; tlaalr common centre of gravity from that axis. Let I be
the len<3th of the iKniinam from the point of suspension to the bottom of the jar,
a the internal ratlius of the jar. Let nM be the mass of the mercury, k the
height it occupies in the jar.
The moment of inertia of the cylinder of mercury about a straight line through
its centre of gravity perpendicular to its axis is by Art. 18, Ex. 8, nM ( To + t ) •
Hence the moment of inertia of the whole body about the axis of suspension is
Mn\^ +
aud the moment of the whole mass collected at its centre of gravity is
Mn(l-^+Mc.
The length L of the simple equivalent pendulum is the ratio of these two, aud on
reduction we have
X=
n (^-lh + P +
:->"
^¥T)
(!)•
+ c
Let the linear expansion of the substance which forms the rod and jar be denoted
by a and that of mercury by /3 for each degree of the thermometer. If the thermo-
meter used be Fahrenheit's, we have a =-0000065668, /3 = -00003336, accordiug to
some experiments of Dulong and Petit. Thus we see that o and /3 are so small that
their squares may be neglected. In calculating the height of the mercury it must
be remembered that the jar expands laterally, and thus the relative vertical expan-
sion of the mercury is 3j8 - 2a, which we shall represent by y.
If then the temperature of every part be increased t", we have a, I, Jc, c, all
increased in the ratio l + a< : 1, while h is increased in the ratio 1-H7i; : 1. Sir>(.j L
is to be unaltered, we have
rdL dL
\da dL
, dL, dL
dk dc
)'
But Lis & homogeneous function of one dimension, hence
dL dL, dL, dL dL ^ ,
da, dl dk dc dh
The condition becomes therefore by substitution
a
0-7
hdL
Ldh'
Let A, Bhe the n" "aerator and denominator of the expression for L given by
equation (1). Then 'ikng the logarithmic differential
I dh ~ .i' ^ B B V""T~ + 2'^*
THE pry;jULUM.
7Z
he whole
il'-Dii the required condii . . is
da of 8UH-
Let I be
)f the jar,
iry, h the
lie through
ion is
two, aud on
(1).
ai be denoted
; the thermo-
accordiug to
80 small that
•cnry it must
jrtical expau-
|«, I, h c, all
1. Si^HJ L
Lr L given by
3(/3-«)
>
This calculation is of more theoretical than practical importance, for the nume-
rical values of a and /3 depend a good deal on the purity of the metals and on the
mode in which they have been worked. The adjustment must therefore be finally
made by experimt 'it.
In the iuvestiijation we have supposed a and /3 to be absolutely constant, but
this is only a itv near approximation. Thus a change of 80" Fah. would alter /3
by less than a t ftieth of its value.
When the adjustment is made the compensation is not strictly correct, for the
iron jar and mercury have been suppo.ed to be of uniform temperatiu'e. Now the
different materials of which the pendulum is composed absorb heat at different
rates and therefore while the temperature is changing there will be some slight
error in the clock.
95. Another cause of error in a clock pendulum is the buoy-
ancy of the air. This produces an upward force acting at the
centre of gravity of the volume of the pendulum equal to the
weight of the air displaced. A very slight modification of the
fundamental investigation in Art. 92 will enable us to take this
into nccoiint. Let V be the volume of the pendulum, D the
density of the air ; I\, h^, the distances of the centres of gravity of
the ma;-s and volume respectively from the axis of suspension,
Mkj' the moment of inertia of the mass about the axis of suspen-
sion. Let us also suppose the pendulum to be symmetrical about
a plane through the axis and either centre of gravity,
equation of motion is then
J. lie
iric' -^"^ = - 3Ii/h, sin 6 + VDg\ sin 6
(!)•
By the same reasoning as before we infer that if I be the
length of the equivalent pendulum
M' , , VD
(2).
The density of the air is continually changing, the changes being
indicated by variations in the height of the barometer. Let h be
VD
the value of h^ — h^ —j.j for any standard density D. Suppose the
actual density to be Z> + 8D and let l+ol be the corresponding
length of the seconds pendulum, then we have by differentiation
Ic'Bl , V8D , ,, ,
-rp = Aj "Tf" » ^^^ therefore
I " h M D '
:i
/!
;
11
■
t '
; t
! li
i \ l\
7^ MOTION ABOUT A FIXED AXIS.
If T be the time of oscillation, we have
8T im
V (J
and .•
T 2 1'
96. Ex. 1. II the centres of gravity of the mass and vohime were very nearly
coincident and th., weight of the air displaced were T-^^nr <*^ ^^ weight of the
penduhira, show that a rise of one inch in the barometer would canae «n error in
the seconds pendulum of nearly 2 sec. per day.
Ex. 2. If we affix to the peudiUum rod produced upwards a body of the same
volume as the pendulum bob but of very small weight, so that the centre of gravity
of she volume lies in the axis of suspension, shovv that the correction for buoyancy
vanishes. This method was suggested in 1871 by the Astronomer Eoyal, but ho
remarks that this construction woiUd probably be inconvenient in practice.
Ex. 3. If a barometer be attached to the pendulum show that the rise or fall of
the mercury as the density of the air changed could be so arranged as to keep the
time of vibration unaltered. This method was suggested first by Dr Robinson of
Armagh in 1831 in the fifth volume of the memoirs of the Astronomical Society,
and afterwards by Mr Deiiison in the Astronomical Notices for Jan. 1873. In the
Armagh Places of Stars published in 1859, Dr Robinson describes the difficulties
he found in practice before he was satisfied with the working of the clock.
The theory of this construction is that in differentiating equation (2) we are to
suppose k^ and h^ variable and I constant. This gives — ^ — = 8 (^^^i) - 8 {h^VD).
V
Let r be the rise of the barometer in the glass tube, ?•' the fall in the cistern, then
r' — nvr, whore jftis a known fraction depending on the dimensions of the barorriter.
Let a and h be the depths of the mercmy in the tube and cistern below the axis of
suspension, 1c the diameter of the tube, p the density of the mercury. Since irc'^pr
is the quantity of mercury added to the top of tne mercury in the tube and taken
away from tiie istern, we have
8(iWi=) = Tcv|(a-|y-^6 + 0'|,
These are acrurate if the barometer be merely ". bent tube so that the cylinders
transferred are similar as well as equal; in this case m — l. If the area of the
cistern be grt it ;r than that of the tube wo have here neglected the difl'erence of the
moments of inertia of the two cylindp .r about asf - through their centre of gravity.
As r is seldom more than one inch, wo may write tln.'e
d{Mk^) = 7,v'',>r{a;^--b-^),
d(Mhi) = Trc^pr{a-h).
Since D is very small, we may neglect the variations of Vh.^ when multiplied
by D. Thus we have
ST) ..r^IIpa + h-l
D " rm., " r '''
THE PENDULUM.
75
where H=b-a is the height of the barometer. If the temperature of the au- bo
luialtercil wo have
5/) SIf
-'j ami )• (1 + m) = dU. The required condition is therefore
irc-Ifp JI a + b~ I ,
It in clearly necessary that n + b>l. The jar of merenry in Graham's mercurial
l)eudiiliim might be used as the cistern of the barometer, as Mr Denisou remarks.
The height of the barometer being 30 inches this would hardly be effective unless
the pendulum was longer than the seconds pendulum, which is about 39 inches.
Prof. Bankine read a paper to the British Association in 18/33 in which ho
proposed to use a clock with a ccnti-ifugal or revolving pendulum, part of whicli
should consist of a siphon barometer. The rising and falling of the barometer would
affect the rate of going of the clock and thence the mean height of the mercurial
column dm'ing any long period would register itself.
Ex. 4. If the pendulum be supposed to drag a quantity of air with it which
bears a constant ratio to the density D of the surrounding air and adds yD to the
moment of inertia of the pendulum without increasing the moving power, show that
the change produced in the simple equivalent pendulum by a change of density SD
is given by 51=7 ,..- . Show that this might be included in Dr Eobinson's mode
of correcting for buoyancy.
97. In many experimental investigations it is necessary to
determine the moment of inertia of the body experimented on
about some axis. If the body be of regular shape and be so far
homogeneous that the errors thus produced are of the order to be
neglected, we can determine the moment of inertia by calculation.
But sometimes this cannot be done. If we can make the body
oscillate under gravity about any axis parallel to the given axis
placed in a horizontal position, we can determine by equation (4)
of Art. 92 the radius of gyration about a parallel axis through the
centre of gravity. This requires however that the distances of the
centre of gravity from the axes should be very accurately found.
Sometimes it is more convenient to attach the body to a pendulum
of known mass whose radius of gyration about a fixed horizontal
axis has been previously found by observing the time of oscilla-
tion. Then by a new determination of the time of oscillation, the
moment of inertia of the compound body, and therefore of the
given body, may be found, the m.asses being known.
If the body be a lamina, Vire may thus find the radii of gyra-
tion about three axes passing through the centre of gravity. By
measuring three lengths along these axes inversely proportional to
these radii of gyration, we have three points on a momental ellipse
at the centre of gravity. The ellipse may then be easily con-
structed. The directions of its principal diameters are the princi-
pal axes, and the reciprocals of their lengths '.epresent on the same
scale as before the principal radii of gyration.
n
!
II h
70
JIOTION ABOUT A FIXED AXIS.
If the body be a solid, six observed radii of gyration will deter-
mine the principal axes and moments at the centre of gravity.
But in most eases some of the other circumstances of the par-
ticular problem under consideration will simplify the process.
On the length of the Seconds Pendulum.
98. The oscillations of a rigid body may be used to determine
the numerical value of the accelerating force of gravity. Let t be
the half time of a small oscillation of a body made in vacuo about
a horizontal axis, h the distance of the centre of gravity from the
axis, k the radius of gyration about a parallel axis through the
centre of gravity, j ' . n we have by Art. 92,
k'' + h' = \hT' (1),
where \ = '-^ so that X is the length of the simple pendulum
TT
whose complete time of oscillation is two seconds.
We might apply this formula to any regular body for which
A;* and h could be found by calculation. Experiments have thus
been m^'^^e with a rectangular bar, drawn as a wire and suspended
from one end. In this case , which is- the length of the
h
simple equivalent pendulum is easily seen to be two-thirds of the
length of the rod. The preceding formula then gives X or g as
soon as the time of oscillation has been observed. By inverting
the rod and taking the mean of the results in each position any
error arising from want of un'formity in density oi figure may
be partially obviated. It has, lowever, been found impracticable
to obtain a rod sufficiently uniform to give results in accordance
with each other.
99. If we make a body oscillate about two parallel axes in
succession not at the same distance from the centre of gravity, we
get two equations similar to (1), viz.
k'+h" = U'T"] ^*'^-
Between these two we may now eliminate k"^, thus
''-J^^hr'-hW' (3).
This equation gives X. Since k"^ has disappeared, the form and
structure of the body is now a matter of no importance. Let a
body be constructed with two apertures into which knife edges
I.ENOTII OF THE SECONDS PENDULUM.
77
ieter-
avity.
! par-
?rmiue
at T be
) about
Din tbe
gh the
.•(1).
Qdulum
r which
„ve thus
spended
I of the
s of the
or g as
Qvorting
ion any
re may
icticable
ordance
axes in
Lvity, we
(2).
,.(3).
)rm and
Let a
Ife edges
can be fixed. By means of these resting either on a horizontal
phine or in two triangular apertures to prevent shpping, the body
can be made to oscillate through small arcs. The perpendicular
distances h, h' of the centre of gravity from the axes must then be
measured with great care. The formula will then give \.
100. In Capt. Kater's method the body has a .sliding weight
in the form of a ring which can be moved up and down by means
of a screw. The body itself has the form of a bar and the
apertures are so placed that the centre of gravity lies between
them. The ring weight is then moved until the two times of
oscillation are exactly equal. The equation (3) then becomes
/<+//
= T
.(4).
which determines \. The advantage of this construction is that
the position of the centre of gravity, which is very difficult to find
by experiment, is not required. AH we want is ^ + h , the exact
distance between the knife edges. The disadvantage is that the
ring weight has to be moved until two times of oscillation, each of
which it is difficult to observe, are made equah
101. The equation (3) can be written in the form
We now see that if the body be so constructed that the times of
oscillation about the two axes of suspension are very nearly equal
r^ — r'^ will be small, and therefore it will be sufficient in the last
term to substitute for h and li their approximate values. The
position of the centre of gravity is of course to be found as accu-
rately as possible, but any small error in its position is of no very
great consequence, for these errors are multiplied by the small
quantity t'^ — t '^ The advantage of this construction over Kater's
is that the ring weight may be dispensed with and yet the only
element which must be measured with extreme accuracy is h -\ h ,
the distance between the knife edges.
102. Tn order to measure the distance between the knife
edges, Captain Kater first compared the different standards of
length then in use, in terms of each of which he expressed the
length of his pendulum. Since then a much more complete com-
parison of these and other standards has been made under the
direction of the Commission appointed for that purpose in 1843.
FML Trans. 1857.
Having settled his unit of length, Captain Kater proceeded to
measure the distance between the knife edges by means of micro-
78
MOTION ABOUT A FIXED AXIS.
j
I:
r ;
i!
scopes. Two different metliods were used, which liowever cannot
be ()*Bcribed here. As an illustration of the extreme care neces-
sary in these measurements, the following fact may be mentioned.
Though the images of the knife edges were always perfectly sharp
and well defined, their distance when seen on a black ground was
•000572 of an inch less than when seen on a white ground. This
difference appeared to be the same whatever the relative illumi-
nation of the object and ground might be so long as the difference
of character was pr^erved. Three sets of measurements were
taken, two at the beginuing of the experiments, and the third after
some time. The object of these last was to ascertain if the knife
edges had suffered from u^k?. The mean results of these three dif-
fered by less than a ten-thousandth of an inch from each other,
the distance to be measured being 3944085 inches.
103. The time of a single vibration cannot be observed di-
rectly, because this would requii*© the fraction of a second of time
as shown by the clock to be estinoiated either by the eye or ear.
The difficulty may be overcome by observing the time, say of a
thousand vibrations, and thus the error of the time of a single vi-
bration is divided by a thousand. The labour of so much counting
may however be avoided by the use of "the method of coinci-
dences." The pendulum is placed in front of a clock pendulum
whose time of vibration is slightly differept. Certain marks made
on the two pendulums are observed by % telescope at the lowest
point of their arcs of vibration. The field of view is limited by
a diaphragm to a narrow aperture across which the marks are
seen to pass. At each succeeding vibration one pendulum follows
the other more closely, and at last its mark is completely covered
by the other during their passage across the field of view of the
telescope. After a few vibrations it appears again preceding the
other. In the interval from one disappearance to the next, one
pendulum has made, as nearly as possible, one complete oscillation
more than the other. In this manner 530 half-vibrations of a
clock pendulum, each equal to a second, Avere found to orrespond
to 532 of Captain Kater's pendulum. The advantage of this
method of observation is such, that an error of one second in noting
the interval between two coincidences would occasion an error of
only 0*63 in the number of vibrations in 24 hours. The ratio of
the times of vibration of the pendulum and the clock pendulum
may thus be calculated with extreme accuracy. The rate of going
of the clock must then be found by astronomical means.
104. The time of vibration thus obtained will require several
corrections which are called "reductions." For instance, if the
oscillation be not so small that we can put sin ^ = ^ in Art. 92, we
must make a reduction to infinitely small arcs. The general
method of effecting this will be considered in the chapter on Small
Pi
LENGTH OP THE SECONDS PENDULUM.
cannot
neces-
tioned.
' sharp
nd was
ThiH
illumi-
ference
s were
rd after
le knife
ree dif-
1 other,
:ved di-
of time
? or ear.
say of a
ingle vi-
3ounting
f coinci-
sndulum
•ks made
le lowest
nited by
rks are
follows
covered
iw of the
ling the
ext, one
cillation
Ions of a
espond
of this
n noting
error of
ratio of
sndulum
of going
Irr
le several
if the
\i. 92, Ave
general
)n Small
Oscillations. Another reduction is necessary if wo wish to reduce
the result to what it would have been at the level of the sea.
The attrnction of the intervening laud may be allowed for by
Dr Young's rule {Phil. Trans, 1819). We may thus obtain the
force of gravity at the level of the sea, supposint^- all the land
above this level were cut off and the sea constrained to keep its
present level. As the level of the sea is altereil by the attraction
of the land, further corrections are still necessary if we wish to re-
duce the result to the surface of that spheroid which most nearly
represents the earth. See Cainh. Phil. Trans. Vol. X.
M. Baily gives as the length of the pendulum vibrating in half
time a mean solar second in the open air in this latitude 39'13.S
inches, and the length of a similar pendulum vibrating sidereal
seconds 38'919 inches.
105. The obsen'ations must be made in the air. To correct for this we have to
make a reduction to a vacuum. This reduction consists of throe parts: (1) The
correction for buoyancy, (2) Du Buat's correction for the air dragged along by the
pendulum, (3) The resistanc' of the air.
Let V be the volume of the pendulum which may be found by measuring the
dimensions of the body. As the "rtiductiou to a vacuum " is only a correction, any
small unavoidable errors in calculating the dimensions will produce an effect only
of the second order on the value of X. Let p be the density of the air when tho
body is oscillating about one knife edge, p' the density when oscillating about tho
other. If the observation be made within an hour or two hours, we may put p = p'.
The effect of buoyancy is allowed for by supposing a force Vp<j to act upwards at tho
centre of gravity of the volume of the body. If the body be made as nearly as pos-
sible symmetrical about the two knife edges this centre of gravity will be half way
between the knife edges.
Du Buat discovered by experiment that a pendulum drags with it to and fro a
certain mass of air which increases the inertia of the body without adding to the
moving force of gravity. This result has been confirmed by theory. The mass
dragged bears to the mass of air displaced by the body a ratio which depends on the
external shape of the body. Let us represent it by fi Vp. If the body be symmetri-
cal about the knife edges, so that the external shape is the same whichever edge is
made the axis of suspension, n will be the same for each oscillation. Since this
mass is to be collected at the centre of gravity of the volume, we must add to the
fc* of equation (1) in Art. 92, and therefore also in Ai-t. 98, the term^wFp ( — - j .
Taking these two corrections the equation (1) of Ai-t. 98 will now become
m
m \ 2 J \ m 2 J'
where m is the mass of the pendulum. Similarly for the oscillation abont the other
knife edge,
m \ 2 J \ VI 2 J
We mu«t eliminate k^ as before. If the obsei-vations about the two knife
,.«:^
r^'^.
IMAGE EVALUATION
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80
MOTION ABOUT A FIXED AXIS.
ii
edges succeed each other r.t a short interval we may put p=p', and then Dn Buat's
correction will disappear. This is of course a very great advantage. We then have*
h + h'
,n-%,j-;,._,,,(,.^).
the last term heing very small because t and r' are nearly equal.
de
The resistance of the air will be some function of the angular velocity ^ of the
pendulum. Since -^ is very small we may expand this function and take only the
first power. Supposing Maclaurin's theorem not to fail, and that no coefficient of a
higher power than the first is very great, this gives a resistance proportional to
(10
di'
The equation of motion will therefore take the form
iir .
where — is the time of a complete oscillation in a vacuum and the term on the
n
right-hand side in that due to the resistance of the air. The discussion of this
equation will be found in the chapter on Small Oscillations.
106. In constructing a reversible pendulum to measure the
force of gravity, the following are points of importance.
1. The axes of suspension, or knife edges, must not be at the
same distance from the centre of gravity of the mass. They
should be parallel to each other.
2. The times of oscillation about the two knife edges should
be nearly equal.
3. The external form of the body must be symmetrical, and
the same about the two axes of suspension.
4. The pendulum must be of such a regular shape that the
dimensions of all the parts can be readily calculated.
These conditions are satisfied if the pendulum be of a rect-
angular shape with two cylinders placed one at each end. The
external forms of these cylinders are to be equal and similar, but
one is to be solid and the other hollow, and such that by calcula-
tion of moments of inertia the distance between the knife edges is
to be as nearly as possible equal to the length of the simple equi-
valent pendulum.
5. The pendulum should be made, as far as possible, of one
metal, so that as the temperature changes it may be always similar
to itself. In this case since the times of oscillations of similar
bodies vary as the square root of their linear dimensions, it is
easy to reduce the observed time of oscillation to a standard tem-
* This formula was mentioned to the author as the one used in the late experi-
ments by Capt. Heaviside to determine the length of tlie seconds pendulum.
LENGTH OF THE SECONDS PENDULUM.
81
=.hr^-h-,
perature. The knife edges however must be made of some strong
substance not likely to be easily injured.
107. Ex. 1. If the knife edges be not perfectly sharp, let r be the difference of
their radii of curvature, show that
h?-h'^ + {h^h')r
\
very nearly when the pendulum vibrates in vacuo. It appears that the correction
vanishes if the knife edges be only equally sharp. By interchanging the knife edges
we have the same equation with the sign of r changed. By making a few observa-
tions we may thus determine r A proposition similar to this has been ascribed to
Laplace by Dr Young.
Ex. 2. A heavy spherical ball is suspended successively by a very fine wire
from two points of support A and B whose vertical distance b has been carefully
measured, thus forming two pendulums. The lowest point of the ball is, on each
suspension, made to be as exactly as possible on the same level, which level is
approximately at depths a and a' below A and B respectively. If r be the radius of
the ball, wliich is small compared with a or a', and I, V the lengths of the simple
,.a
very nearly. By coimt-
l — V 2
equivalent pendulum, prove that —7— = 1 - - , , , ,
b 5(a-r)(a-r)
ing the number of oscillations performed in a given time by each pendulum, show
I
how to find ratio -, .
V
Thence show how to find g and point out which lengths must
be most carefully measured and which need only be approximately found, so as to
render this method effective. This method is mentioned in Cirant's iirtory of
Physical Astronomy, page 155, as having been used by Bessel.
108. The length of the seconds pendulum has been used as a
national standard of length. By an Act of Parliament passed in
1824, it was declared that the distance between the centres of the
two points in the gold studs in the straight brass rod then in the
custody of the clerk of the House of Commons, whereon the words
and figures "standard yard, 1760" were engraved, shall be the
original and genuine standard of length called a yard, the brass
being at the temperature of 62° Fah. And as it was expedient that
the said standard yard if injured should be restored of the same
length by reference to some invariable natural standard, it was
enacted, that the new standard yard should be of such length that
the pendulum, vibrating seconds of mean time in the latitude of
London in a vacuum at the level of the sea, should be 39"1393
inches.
On Oct. 16, 1834, occurred the fire at the Houses of Parlia-
ment, in which the standards were destroyed. The bar of 1760
was recovered, but one of its gold pins bearing a point was melted
out and the bar was otherwise injured.
In 1838 a commission was appointed to report to the govei-n-
ment on the course best to be pursued under the peculiar circum-
stances of the case.
R. D.
6
y
I Hill f
82
MOTION ABOUT A FIXED AXIS.
In 1841 the commission reported that they .were of opinion
that the definition by which the standard yard is declared to be
a certain brass rod is the best which it is possible to adopt. With
resp'ict to the provision for restoration they did not recommend
a reference to the length of the seconds pendulum. " Since the
passing of the act of 1824 it has been ascertained that several
elements of reduction of the pendulum experiments therein re-
ferred to are doubtful or erroneous: thus it was shown by Dr
Young, Phil. Trans. 1819, that the reduction to the level of the
sea was doubtful ; by Bessel, Astron. Nachr. No. 128, and by
Sabine, Phil. Trans. 1829, that the reduction for the weight of air
was erroneous ; by Baily, Phil. Trans. 1832, that the specific
gravity of the pendulum was erroneously estimated and that the
faults of the agate planes introduced some elements of doubt ; by
Kater, Phil. Trans. 1830, and by Baily, Astron. Soc. Memoirs,
Vol. IX., that very sensible errors were introduced in the operation
of comparing the length of the pendulum with Shuckburgh's scale
used as a representative of the legal standard. It is evident,
therefore, that the course prescribed by the act would not neces-
sarily reproduce the length of the original yard."
The commission stated that there were several measures
which had been formerly accurately compared with the original
standard yard, and by the use of these the length of the original
yard could be determined without sensible error.
In 1843 another commission was appointed to compare all the
existing measures and construct from them a new Parliamentary
standard. Unexpected difticulties occurred in the course of the
comparison, which cannot be described here. A full account of
th.e proceedings of the commission will be found in a paper
contributed by Sir G. Airy to the Royal Society in 1857.
Oscillation of a Watch Balance.
109. A rod B'CB can turn freely about its centre of gravity
C which is fixed, and is acted on by a very fine spiral spring CPB.
The spring has one end fixed in position in such a manner that
the tangent at C is also fixed, and has the other end B attached
to the rod so that the tangent at B makes a constant angle with
the rod. The rod being turned through any angle, it is required
to find the time of oscillation. This is the construction used
in watches, just as the pendulum is used in clocks, to regulate
the motion.
Let Cx be the position of the rod when in equilibrium, and
let be the angle the rod makes with Cx at any time t, MF the
moment of inertia of the rod about G. Let p be the radius of
/I
OSCILLATION OF A WATCE BALANCE.
83
curvature at any point P of the spring, p^ the value of p when in
equilibrium. Let (a?, i/) be the co-ordinates of P referred to G as
origin and Cx as axis of x. Let us consider the forces which act
on the rod and the portion BP of the spring. The forces on the
rod are X, Y the resolved parts of the reaction at C parallel to the
axes of co-ordinates, and the reversed eflfective forces which are
J2i3
equivalent to a couple 3Ik^ -j^ . The forces on the sj»ring are, the
reversed effective forces which are so small that they may be
neglected, and the resultant action across the section of the spring
at P. This resultant action is produced by the tensions of the
innumerable fibres which make up the spring, and these are
equivalent to a force at P and a couple. When an elastic spring
is bent so that its curvature is changed, it is proved both by
experiment and theory that this couple is proportional to the
change of curvature at P. We may therefore represent it by
eI ], where E depends only on the material of which the
spring is made and on the form of its section.
Taking moments about P to avoid introducing the unknown
force at P, we have
^^^f=
E
Cv.)--^
+ Yx.
This equation is true whatever point P may be chosen. Con-
sidering the left side constant at any moment and (a;, y) variable,
this becomes the intrinsic equation to the form of the spring.
Let BP = s, multiply this equation by ds and integrate along
the whole length I of the spiral spring, we have
ds
i
Now — is the angle between two consecutive normals, hence
P
ds
is the angle between the extreme normals.
Now at A
the normal to the spring is fixed throughout the motion, therefore
6—2
J I
lA
i'^'
84
MOTION ABOUT A FIXED AXIS.
K 1 is the angle between the normals at B in the t'tvo
positions in which 6 - 6 and ^ = 0. But since the normal at B
makes a constant angle with the rod, this angle is the angle 6
which the rod makes with its position of equilibrium. Also if
x, y be the co-ordinates of the centre of gravity of the spring at
the time t, we have \xds = xl, \yds = yl. Hence the equation of
motion becomes
Let us suppose that in the position of equilibrium there is no
pressure on the axis (7, then X and Y will, throughout the motion,
be small ^^aantities of the order 6. Let us also suppose that the
fulcrum is placed over the centre of gravity of the spring when
at rest. Then if the number of spiral turns of the spring be
numerous and if each turn be nearly circular, the centre of gravity
will never deviate far from C. So that the terms Yx and Xy are
each the product of two small quantities, and are therefore at least
of the second order. Neglecting these terms we have
^^ de " I ^•
Hence the time of oscillation is 27r
/Mm
It appears that to a first approximation the time of oscillation
is independent of the form of the spring in equilibrium, and
depends only on its length and on the form of its section.
This brief discussion of the motion of a watch balance is taken
from a memoir presented to the Academy of Sciences. The
reader is referred to an article in Liouville's Journal, 1860, for a
further investigation of the conditions necessary for isochronism
and for a determination of the best forms for the spring.
Pressures on the fixed cuds.
110. A body moves about a fi,xed axis under the action of any
forces, to find the pressures on the axis.
First. Suppose the body and the forces to be symmetrical
about the plane through the centre of gravity perpendicular to
the axis. Then it is evident that the pressures on the axis are
reducible to a single force at G the centre of suspension.
Let F, G be the actions of the point of support on the body
resolved along and perpendicular to CO, where is the centre
oJ
tl
re
li
fo
h(
pi
b(
tb
ID
bi
V{
PRESSUKES ON THE FIXED AXIS.
85
II
of gravity. Let X, Y be the sum of the resolved parts of
the impressed forces in the same directions, and L their moment
round C.
Let CO = h and d = angle which CO makes with any straight
line fixed in space.
Taking moments about C, we have
d?d _ L
de~M{k^^K') ^ ^•
The motion of the centre of gravity is the same as if all the
forces acted at that point. Now it describes a circle round C\
hence, taking the tangential and normal resolutions, we have
^de = -w- (2)'
-^Kdt) — M" (^)-
Equation (1) gives the values of -^ and -j-, and then the
pressures may be found by equations (2) and (3).
If the only force acting on the body be that of gravity, let
be measured from the vertical. If the body start from rest in
that position which makes CO horizontal, we have
X=Mg cos 6, Y^ — Mg smd, L = — 3fghsiud;
d'e_ gh . ^
•*• dt' — F+T"''''
integrating, we have
but when G = — , -j: vanishes, therefore (7=0; substituting these
values (2) and (3), we get
y M
86 MOTION ABOUT A FIXED AXIS.
-F=Mg COS 0.-j^j^,
where B is the angle which CO makes with the vertical.
Let "^ be the angle the direction of the pressure at G makes
with the line GO, the angle being measured from GO downwards
to the left, then
cot -^ = ( 1 + 3 p) cot 0,
which is a convenient formula to determine the direction of the
pressure*.
111. Secondly. Suppose either the body or the forces not to
be symmetrical.
Let the fixed axis be taken as the axis of z with any origin
and plane of xz. These we shall afterwards so choose as to sim-
plify our process as much as possible. Let x, y, i be the co-ordi-
nates of the centre of gravity at the time t.
Let 0) be the angular velocity of th«.
acceleration, so that /= -— .
y, f the angular
Now every element m of the body describes a circle about the
axis, hence its accelerations along and perpendicular to the radius
vector r from the axis are — wV and fr. Let be the angle
* Let il/.iZ be the resultant of F and G, and let a=g~^ and &=g /' g,
cos' i^ sin'^iA' 1
then - J^ - + — pX= _. Construct an ellipse with C for centre and axes equal
to a and h measured along and perpendicular to CO. Then the resultant pressure
varies as the diameter along which it acts. And the direction may be found thus ;
let the auxiliary circle cut the vertical in F, and let the perpendicular from F on
CO cut the ellipse in R. Then CR is the direction of the pressure.
PRESSURES ON THE FIXED AXIS. 87
which r makes with the plane of xz at any time, then from the
resolution of forces it is clear that
— = - ft)V cos 6 —fr sin ^ = — tJ'x —fiji
similarly -^ = — m^y +fx.
These equations may also be obtained by differentiating the
equations a? = r cos ^, y = r aind twice, remembering that r is
constant.
Conceive the body to be fixed to the axis at two points, distant
a and a from the origin, and let the reactions of the points on
the body resolved parallel to the axes be respectively F, Q, H\
The equations of motion of Art. 71 then give
tmX+ F+F'=^tm^ = Xm{- o>'x -ft/)
= -cB'J/5-/ify (1),
tmY+G+G'='Zmj^,=Xm{-a>'y+fx)
^-tJ'My+fMx (2),
2mZ+ir+ir' = Sw^ = (3).
Taking moments about the axes, we have
tm{2/^-z7)-Ga-G'a' = 'Zm(y^-zj^^
= (o^Xmyz —fXmxz (4):
by merely introducing z into the results in (2), •
tm{zX-xZ) + Fa + F'a==Xm(z~-x^^
= — a>*Xmxz —fXmyz (5),
%m{xY-.yX) =tm{x^^-y^)
= Mk'\f (6).
f 1
i i.
'I
III
%
88 MOTION ABOUT A FIXED AXIS.
Equation (6) serves to determine / and w, and equations (1),
(2), (4), (5) then determine F, G, F', 0'\ //and W are indeter-
minate, but their sum is given by equation (3).
Looking at these equations, we see that they n'ould be greatly
simplified in two cases.
First, if the axis of ^ be a principal axis at the origin,
"^inxz = 0, "Zmyz = 0,
and the calculation of the right-hand sides of equations (4) and
(5) would only be so much superfluous labour. Hence, in at-
tempting a problem of this kind, we should, when possible, so
choose the origin that the axis of revolution is a principal axis
of the body at that point.
Secondly, except the determination of / and to by integrating
equation (G), the whole process is merely an algebraic substitution
of / and to in the remaining equations. Hence our results will
btill be correct if we choose the plane of wz to contain the centre
of gravity at the moment under consideration ; this will make
^ = 0, and thus equations (1) and (2) will be simplified.
112. If the forces which act on the body be impulsive, the
equations will require some alterations.
Let to, to' be the angular velocities of the body just before and
just after the action of the impulses. In the case in which the
body and forces are symmetrical, the equations (1), (2), (3) of
Art. 110 become respectively
^^'-^^ilJl/ZTF) "^^'
Y-\- G " "^ "
7, («.•-<.) = ^^ (2),
= ^-^ • (3).
where all the letters have the same meaning as before, except
that F, G, X, Y are now impulsive instead of finite forces.
Let us next consider the case in which the forces on the body
are not symmetrical. Let u, v, w, u', v , w' be the velocities
resolved parallel to the axes of any element m whose co-ordinates
are x, y, z. Then u = — yto, u' = — yto', v = xto, v' = xto', and
w, V) are both zero.
The several equations of Art. Ill will then be replaced by the
following:
2 A^ ^-FaF = Sw {\i -u) = - tmy {to' - to)
= -3/^(0,' -a,) (1),
PRESSURES ON THE FIXED AXIS. 89
2 r + O' + 6^' = 2m (v - v) = Swa; (o)' - 0))
= Mu:((o'-(o) (2),
2Z+//+//' = (3),
XQ/Z-zY) -Ga- G'a' = Xm [y (w'-w)-z{v -?;)}
= — Xmxz . (o>' — w) (4),
2 {zX- xZ) +fli + Fd = tm {z (u' -u)-x{w'-w)}
= — Imi/z . («()' — <u) (5),
2 (x Y-yX) ^ 2m (x' + y) . (o)' - w) (6).
These six equations are sufficient, to determine &>', F, F\
O, G' and the sum H+ H' oi the two pressures along the axis.
These equations admit of simplification when the origin can
be so chosen that the axis of rotation is a principal axis at that
point. In this case the right-hand sides of equations (4) and (5)
vanish. Also if the plane of xz be chosen to pass through the
centre of gravity of the body, we have ^ = 0, and the right-hand
side of equation (1) vanishes.
113. Ex. A door is suspended by tieo hinges from a fixed axis making an angle
a with the vertical. Find the motion and pressures on the hinges.
Since the fixed axis is evidently a principal axis at the middle point, -Tfe shall
take this point for origin. Also we shall take the plane of xz so that it contains the
centre of gravity of the door at the moment imder consideration.
The only force acting on the door is gravity, which may he supposed to act at
the centre of gravity. We must first resolve thie parallel to the axes. Let if> be
■/ 1 fj
the angle the plane of the door makes with a vertical plane through the axis of
suspension. If we draw a plane ZON such that its trace ON on the plane of XOY
makes an angle ^ with the axis of x, this will be the vortical plane through the
'/■■
ii
90 MOTION ABOUT A FIXED AXIS.
nxis; and if wo draw V in tiiis plane making ZO V=a, OV will be vertical. Hence
the resolved parts of gravity are
X =<7 sin a cos 0, r=(7HinoHin0, Z=:-*7C0flo.
Since the resolved parts of the effective forces are the same as if the whole mass
were collected at the centre of gravity, the six equations of motion are
3/j/ sin a cos + /•+/"= -u^Mx (1),
Mfj Bina Bin <i> + + 0'=fMx (2),
-iMircoso + //+//'=0 (S),
-Oa+0'a=0 (4),
Afg COB ax + Fa -F'a-0 (5),
booause the fixed axis is a principal axis at the origin,
- Mfj Bin a Bin if,. x = Mk''.-^^ (6).
Integrating the last equation, we have
C + 2r/ sin o cos ^ = fc '*w'.
Suppose the door to be initially placed at rest, with its plane making an angle /3
with the vertical plane through the axis; then when 0=/3, u=0; hence
k'^w^ = 2r/i sin o (cos - cos /3) )
and k'^/= ~u sinasin<t>.x )'
By substitution in the first four equations F, F', 0, 0', may bo found.
114. It should be noticed that these equations do not depend
on the form of the body, but only on its moments and products
of inertia. We may therefore replace the body by any equi-
momental body that may be convenient for our purpose.
This consideration will often enable us to reduce the compli-
cated forms of Art. Ill to the simpler ones given in Art. 110.
For though the body may not be symmetrical about a plane
through its centre of gravity perpendicular to the axis of sus-
pension, yet if the momental ellipsoid at the centre of gravity be
symmetrical about this plane we may treat the body as if it were
really symmetrical. Such a body may be said to be Dynamically
Symmetrical. If at the same time the forces be symmetrical
about the same plane, and this will always be the case if the axis
of suspension be horizontal and gravity be the only force
acting, we know that the pressures on the axis must certainly
reduce to a single pressure, which may be fouod by Art. 110.
115. Ex. 1. A uniform heavy lamina in the form of a sector of a circle is
suspended by a horizontal axis parallel to the radius which bisects the arc, and
oscillates under the action of gi'avity. Show that the pressures on the axis are
equivalent to a single force, and find its magnitude.
Ex. 2. An equilateral triangle oscillates about any horizontal axis situated in
its own plane, show that the pressures are equivalent to a single force and find its
magnitude.
PRESSURES ON THE FIXED AXIS.
91
116. If a body be set in rotation about any axis which is
a principal axis at some point in its length, and if there be
no impressed forces acting on the body, it follows at once from
these conditions that the pressures on the axis are equivalent
to a single resultant force acting at 0. Hence if be fixed in
space, the body will continue to rotate about that axis as if it also
were fixed in space. Such an axis is called a permanent axis of
rotation at the point 0.
If the body be entirely free and yet turning about an axis
of rotation which does not alter its position in space, we may
suppose any point we please ii* the axis to be fixed. In this case
the axis must be a principal axis at every point of its length.
It must therefore by Art. 49 pass through the centre of gravity.
The existence of principal axes was first established by Scgner
in the work Specimen Theorim Turhinum. His course of in-
vestigation is the opposite of that pursued in this treatise. He
defines a principal axis to be such that when a body revolves
round it the forces arising from the rotation have no tendency
to alter the position of the axis. From this dynamical definition
he deduces the geometrical properties of these axes. The reader
may consult Prof. Cayley's report to the British Association on the
special problems of Dynamics, 18U2, and Bossut, Histoire de
MatMmatiqiie, Tome ii.
117. Suppose the body to start from rest and to be acted on
by a couple, let us discover the necessary conditions that the
pressures on the fixed axis may be reduced to a single resultant
pressure. Supposing such a single resultant pressure to exist, we
can take as origin that point of the axis at which it is intersected
by the single resultant. Then the moments of the two pressures
on the axis of rotation about the co-ordinate axes will vanish.
Hence since © = the equations (4), (5), and (6) of Art. 112 become
L = -fXmxz, M=-fXmi/z, N=Mky,
where we have written L, My iVfor the three moments 'Zm{yZ—z Y),
&c. of the impressed forces about the co-ordinate axes.
The plane of the couple whose resolved parts about the axes
are L, M, N, is known by Statics to be
LX + 3fY + NZ=0,
or in our case,
-tmxzX-XmyzY+Mk"Z=0 (1).
Let the momental ellipsoid at the fixed point be constructed,
and let its equation be
AX' + BY'+CZ'-2DYZ- 2EZX- 2FXY= e*.
The equation to the diametral plane of the axis of Z is
^EX-DY+ CZ^O (2).
i
ii
n u
I
92
MOTION ABOUT A FIXED AXIS.
Comparing (1) and (2) we see that the plane of the resultant
couple must be the diametral plane of the axis of revolution.
Since the pressures on the axis are equivalent to a single
resultant force acting at some point of the axis, we may suppose
this point alone to be fixed and the axis of rotation to be other-
wise free. If then a body at rest with one point fixed be acted on
by any couple, it will begin to rotate about the diametral line of
the plane of the couple with regard to the momental ellipsoid at
the fixed point.
Thus the body will begin to rotate about a perpendicular to
the plane of the couple only when the plane of the couple is
parallel to a principal plane of the body at the fixed point.
If the acting couple be an impulsive couple, the equations of
motion, by Art. 112, will be the same as those obtained above when
(o is put zero and <a' written for f. Hence the same conclusion
will follow.
The body will not in generil continue to rotate about the dia-
metral line.
118. Ex. 1. If a body at rest have one point fixed and be acted on by any
couple whose axis is a radius vector OP of the ellipsoid of gyration at 0, the body
will begin to turn about a perpendicular from on the tangent plane at P.
Ex. 2. A soUd ellipsoid is fixed at its centre, and is acted on by a couple in a
plane whose direction-cosines referred to the principal diameters are (I, m,n). Prove
I
that the direction-cosines of the initial axis of rotation are proportional to
ly^ + c^'
m
o''' -t- a"
and
u? + b^'
Ex. 3. Any plane section being taken of the momental ellipsoid of a body at a
fixed point, the body may be made to rotate about either of the principal diameters
of this section by the application of a couple of the proper magnitude whose axis is
the other principal diameter.
For assume the body to be turning uniformly about the axis of 2. Then the
couples which must act on the body to produce tliis motion are L=w^^myz,
Af = -w'Snuez, iV=0. Then by taking the axis of x such that Sma»=0 we see that
the axis of the ' juple must be the axis of x and the magnitude of the couple will
be L=w^2myz,
Ex. 4. A body having one point fixed in space is made to rotate about any
proposed straight line by the application of the proper couple. The position of the
axis of rotation when the magnitude of the couple is a maximum, has been called
an axis of maximum reluctance. Show that there are six axes of maximum
reluctance, two in each principal plane, ea«h two bisecting the angles between the
principal axes in the plane in which they are.
Let the axes of reference be the principal axes of the body at the fixed point,
let (/, wi, h) be the direction-cosines of the axis of rotation, (X, /*, v) those of the axis
THE CENTRE OF PERCUSSION.
'J3
resultant
on.
a single
suppose
)e other-
acted on
il line of
ipsoid at
icular to
jouple is
ations of
)ve when
)nclusion
the dia-
l on Ly any
), the body
P.
couple in a
, »). Prove
I
to
l»« + ca'
I body at a
I diameters
lose axis is
Then the
re see that
jouple will
about any
tion of the
jeen called
maximum
etween the
xed point,
of the axis
of the couple 0. Then by the last question and tho fifth and sixth examples of
Art. 33, we have
(B-C)^ ~{Cr-A)'iil'~ jA^BJhii '
We haTe then to make a maximum by variation of {Imn) subject to the con-
dition f + »i* + n* = l. The positions of these axes were first investigated by
Mr Walton in the Quarterly Journal of Mathematics, 1865.
The Centre of Percussion.
119. When the fixed axis is given and the body can be so
struck that there is no impulsive pressure on the axis, any point
in the line of action of the force is called a centre of percussion.
When the line of action of the blow is given, the axis about
which the body begins to turn is called the axis of spontaneous
rotation. It obviously coincides with the position of the fixed
axis in the first case.
Prop. A body is capable of turning freely about a fixed
axis. To determine the conditions that there shall be a centre of
percussion and to find its position.
Take the fixed axis as the axis of z, and let the plane of xs
pass through the centre of gravity of the body. Let X, Y, Z be
the resolved parts of the impulse, and let ^, t], ^ be the co-ordi-
nates of any point in its line of action. Let Mk'^ be the moment
of inertia of the body about the fixed axis. Then since ^ = 0, the
equations of motion are, by Art. 61,
X=0
Y=Mx{(o'-m)
Z =
(1).
'nZ~^Y=- (o)' - co) tmxz
KX- ^Z= - (ft)' - o)) tmyz\ (2).
^Y-'riX={a>'-<o).MTc^ )
The impulsive pressures on the fixed axis arc omitted because by
hypothesis they do not exist.
From these equations we may deduce the following conditions.
I. From (1) we see that X = 0, Z= 0, and therefore the force
must act perpendicular to the plane containing the axis and tho
centre of gravity.
IL Substituting from (1) in the first two equations of (2) we
%mxz
!'
'\
have Xmyz — and f =
At
X
Since the origin may be taken
94
MOTION ABOUT A FIXED AXIS.
,1'.
anjrwhere in the axis of rotation, let it be so chosen that 'tmxz= 0.
Then the axis of z must be a principal axis at the point where a
plane passing through the line of action of the blow perpendicular
to the axis cuts the axis. So that there can be no centre of
percussion unless the axis be a principal axis at some point in its
length.
III. Substituting from (1) in the last equation of (2) we have
I = -- . By Art. 92 this is the equation to determme the centre
of oscillation of the body about the fixed axis treated as an axis of
suspension. Hence the perpendicular distance between the line
of action of the impulse and the fixed axis must be equal to the
distance of the centre of oscillation from the axis.
If the fixed axis be parallel to a principal axis at the centre of
gravity, the line of action of the blow will pass through the centre
of oscillation.
The Ballistic Pendulum.
120. It is a matter of considerable importance in the Theory of
Gunnery to determine the velocity of a bullet as it issues from the
mouth of a gun. By means of it we obtain a complete test of any
theory we have reason to fonn concerning the motion of the bullet
in the gun ; or we may find by experiment the separate effects
produced by varying the length of the gun, the charge of powder,
or the weight of the ball. By determining the velocity of a bullet
at different distances from the gun we may discover the laws
which govern the resistance of the air.
It was to determine this initial velocity that Mr Robins about
1743 invented the Ballistic Pendulum. Before his time but little
progress had been made in the true theory of military projectiles.
His New Principles of Gunnery was soon translated into several
languages, and Euler added to his translation of it into German an
extensive commentary ; the work of Eulor's being again trans-
lated into English in 1784. The experiments of Robins were all
conducted with musket balls of about an ounce weight, but they
were afterwards continued during several years by Dr Hutton,
who used cannon balls of from one to nearly three pounds in weight.
These last experiments are still regarded as some of the most
trustworthy on smooth-bore guns.
There are two methods of applying the ballistic pendulum,
both of wliich were used by Robins. In the first method, the gun
is attached to a very heavy pendulum ; when the gun is fired the
recoil causes the pendulum to turn round its axis and to oscillate
THE BALLISTIC PENDULUM.
95
of any
(lulum,
he gun
red the
scillatc
through an arc which can be measured. The velocity of the
bullet can be deduced from the magnitude of this arc. In the
second method, the bullet is fired into a heavy pendulum. The
velocity of the bullet is itself too great to be measured directly,
but the angular velocity communicated to the pendulum may bo
made as small as we please by increasing its bulk. The arc of
oscillation being measured, the velocity of the bullet can be found
by calculation.
The initial velocity of small bullets may also be determined by
the use of some rotational apparatus. Two circular discs of paper
are attached perpendicularly to the straight line joining their
centres, and are made to rotate about this straight line with a
great but known angular velocity. Instead of two discs, a cylinder
of paper might be used. The bullet being fired through at least
two of the moving surfaces, its velocity can be calculated when
the situations of the two small holes made by the bullet have
been observed. This was originally an Italian invention, but it
was much improved and used by Olinthus Gregory in the early
part of this century.
121. A rifle is attached in a horizontal position to a large
block of wood which can turn freely about a horizontal axis. The
rifle being fired, the recoil causes the pendulum to turn round its
axis, until brought to rest by the action of gravity. A piece of
tape is attached to the pendulum, and is drawn out of a reel
during the backward motion of the pendiduni, and thus serves to
measure the amount of the angle of recoil. It is required to find
the velocity of the bullet.
The initial velocity of the bullet is so much greater than that
of the pendulum that we may suppose the ball to have left the
rifle before the pendulum has sensibly moved from its initial posi-
tion. The initial momentum of the bullet may be taken as a
measure of the impulse communicated to the pendulum.
Let h be the distance of the centre of gravity from the axis of
suspension ; / the distance from the axis of the rifle to the axis of
suspension; c the distance from the axis of suspension to the
point of attachment of the tape, m the mass of the bullet; ilf that
of the pendulum and rifle, and n the ratio of ilf to m; b the
chord of the arc of the recoil which is measured by the tape. Let
k' be the radius of gyration of the rifle and pendulum about the
axis of suspension, v the initial velocity of the bullet.
The explosion of the gunpowder generates an equal impulsive
action on the bullet and on the rifle. Since the initial velocity of
the bullet i" v, this aci'ion is measured by mv. The initial angular
velocity generated in tlie pendulum by this impulse is by Art. 89
I
il
96
MOTION ABOUT A FIXED AXIS.
i '■■
Is '«
0) = J.-' . The subsequent motion is given (Art. 92) by the
equation
d*e ah . a
•••©'=^+1^'^"^^^
when ^ = we have -r = w, and if a be the angle of recoil, when
at
= a, -T.=0. Hence w' = -R^ (1 — cos a). Eliminating a> we have
V = -Tr- . 2 sin ^ \gh. But the chord of the arc of the recoil is
6 = 2c sin 2 ;
nhk' i—r
:. v= — T. va/i.
cf
The magnitude of k' may be foimd experimentally by ob-
serving the time of a small oscillation ^ the pendulum and rifle.
If '£ be a half-time we have r= tt a/ -r- (^^t. 92.)
This is the formula given by Poisson in the second volume of
his Mecaniqiie. The reader will find in the Philosophical Maga-
zine for June 18.54, an account of some experiments conducted by
Dr S. Haughton from which, by the use of this formula, the initial
velocities of rifle bullets were calculated.
Tlio formula must however be regarded as only a first approximation, for the
recoil of the pendulum when the gan is fired without a baU has been altogether
neglected. In Dr Haughton's experiments the (barge of powder was compijiratively
small, and this assumption was nearly correct. But in some of Dr Button's experi-
ments, where comparatively large charges of powder were used, the recoil without a
ball was found to be very considerable.
To allow for this Dr Button, following Mr Eobins, assumed that the effect of
the charge of powder on the recoil of the gun is the same either with or without a
ball. If p be the momentum generated by the powder, the whole momentum gene-
rated in the pendulum will be mv+p instead of mv. Proceeding as before, we find
If we now repeat the experiment, with an equal charge without a ball, we have
p _ „ "<c J^^ where &o is the chord measured by the tape. Subtracting one result
from the other, we have
m cj ^^
THE BALLISTIC PENDULUM.
97
by the
il, when
we have
recoil is
r by ob-
and rifle.
volume of
al Maga-
lucted by
,he initial
ion, for the
altogether
mpiiratively
.on's experi-
without a
Thus Dr Hntton^s formula differs from Foisson's in this refspoct, that the chord of
vibration is first found for any charge without a ball and then for an equal charge
with a ball : the difference of these chords is regarded as the chord which is due to
the recoil of the ball.
When the magnitude of the charge of powder is small, the two methods of using
the ballistic pendiUum give nearly the same result. With large charges Dr Hutton
found that the difference was very considerable, a less velocity being indicated by
the method of observing the recoil than by that of firing the ball into the pendulum.
He therefore inferred that the effect of the charge of powder on the recoil of the gun
is not the same when it is fired without a ball as when it is fired with one.
We may in some measure understand the reason of this discrepancy if we con-
sider separately the effects of the inflamed powder while the ball is in the gun and
after it has left the barrel. Supposing, merely as an approximation, that the gas
urging the ball forward is of uniform density ; its centre of gravity, at the moment
when the ball is leaving the gun, will be at the middle point of the barrel and mov-
ing relatively to the gun with half the relative velocity of the ball. If /x be the mass
of the powder, the angular velocity w' communicated to the pendulum will be given
approximately by Mk'^u = { m + g J w/. After the ball has left the gun, the inflamed
powder escapes from the mouth and continues to exert some pressure tending to
increase the recoil. The determination of this motion is a problem in Hyd /-
namios which has not yet been properly solved and which cannot be discussed here.
We may, however, suppose that ilobins' principle applies more nearly to this part
of the motion than to the whole. If so, the momentum generated by the issuing
gas, considered as an impulse, is nearly the same for a given charge and a given
gun, whatever the magnitude of the ball may have been.
If p' be the momentum thus generated we have
If Vo and \ be the values of v and 6 when the gun is fired without a ball, we have
M{b-h,)h' ,~
„-^(r„-r)=-— ^- Jgh.
2m,
Since Vq is greater than v, this equation would show that, for considerable charges,
Dr Hinton's formula will give too small a value for <-. The value of v^ is however
very imperfectly known.
•I
v
the effect of
without a
ntum geue-
ore, we find
11, we have
g one result
122. A gun is placed in front of a heavy pendidum, which
can turn freely about a horizontal axis. The ball strikes the pen-
dulum hm'izontally at a distance ifrom the axis of suspension. It
penetrates into the wood a short distance and communicates a
momentum to the pendulum. The chord of the arc being measured
as before by apiece of tape, find the velocity of the bidlet
The time, which the bullet takes to penetrate, is so short that
we may suppose it completed before the pendulum has sensibly
moved from its initial position. If we follow the same notation
as before, the moment of inertia of the pendulum and ball about
R. D. 7
Jin
ii
!
'1
98
MOTION ON A FIXED AXIS.
V-
the axis of suspension will be Mk'' + mC, and the distance of the
centre of gravity will be -in^ — • Following the same reasoning,
we find
_b^g (Mk" + mt^)^ (Mh + mi)^
ci m
If the gun be placed as nearly as possible opposite the centre
of gravity of the pendulum, we may put h = i in the small terms,
and since M is large compared with m the formula takes the
simple form
Jl/+m hh ,—
v =
m
pen-
where I is the distance of the centre of oscillation of the
dulum and ball from the axis of suspension.
The inconvenience of this construction as compared with the
former is that the balls remain in the pendulum during the time
of making one whole set of experiments. The weight, and the
positions of the centres of gravity and oscillation, will be changed
by the addition of each ball which is lodged in the wood. Even
then the changes produced in the pendulum itself by each blow
are omitted. A great improvement was made by the French in
conducting their experim'^nts at Metz in 1839, and at L'Orient
in 1842. Instead of a mass of wood, requiring frequent renewals,
as in the English pendulum, a permanent r^cepteur was substi-
tuted. This receiver is shaped within as a truncated cone, which
is sufficiently long to prevent the shot from passing entirely
through the sand with which it is filled. The front is covered
with a thin sheet of lead to prevent the sand from being shaken
out. This sheet is marked by a horizontal and by a vertical
line, the intersection corresponding to the axial line of the cone,
so that the actual position of the shot when entering the re-
ceiver can be readily determined by these lines.
Ex. 1. Show that after each bullet has been fired into a ballistic pendulum
constructed on the English plan, h must be increased by ^ (t - h) and Ihy -rfii-l)
nearly in order to prepare the formula for the next shot.
Ex. 2. Dr Haughton found that, for rifles fired with a constant charge, the
initial velocity of the bullet varies as the square root of the mass of the bullet in-
versely and as the square root of the length of the gun directly. Show from this,
that the force developed by the explosion of the powder diminished by the friction
of the barrel is constant as the ball traverses the rifle.
Dr Hutton found that in smooth bores the velocity increases in a ratio some-
what less than the square root of the length of the gun, but greater than the cube
root of the length. Show that this might he expected from the decreased friction in
a smooth bore as compared with a rifle
i
THE BALLISTIC PENDULUM.
99
Ex. 8. If the Telocity of a bullet issuing from the mouth of a gun 80 inches
long be 1000 feet per second, show that the time the bullet tool: to traverse the gun
was about -^ of a second.
Ex. 4. It has been found by experiment that if a bullet be fired into a large
fixed block of wood, the penetration of the bullet into the wood varies nearly as the
square of the velocity, though as the velocity is very much increased the depth of
penetration falls short of that given by this rule. Assuming this rule, show that
the resistance to penetration is constant and that the time of penetration is the
ratio of twice the space to the initial velocity of the bullet. In an experiment of
Dr Button's a ball fired with a velocity of 1500 feet per second was found to pene-
trate about 14 inches into a block of sound dry elm : show that the time of penetra-
tion was TTT of a second.
1
7—2
n .
i
iiit
11' •
CHAPTER IV.
MOTION IN TWO DIMENSIONS.
On the Equations of Motion.
123. The position of a body in space of two dimensions
may be determined by the co-ordinates of its centre of gravity,
and the angle some straight line fixed in the body makes with
some straight line fixed in space. These three have been called the
co-ordinates of the body, and it is our object to determine them
in terms of the time.
It will be necessary to express the effective forces of the body
in terms of these co-ordinates. The resolved parts of these
effective forces parallel to the axes have been already found in
Art. 79, all that is now necessary is to find their moment about
the centre of gravity. If (a?', y) be the co-ordinates of any
particle of mass m referred to rectangular axes meeting at the
centre of gravity and parallel to axes fixed in space, this moment
has been shown in Art. 72 to be equal to -r^ , where
dt'
h = t
m
V dt y dt)'
Let be the "angular co-ordinate" of the body, i.e. the angle
some straight line fixed in the body makes with some straight line
fixed in space. Let (r', <f>') be the polar co-ordinates of any par-
ticle m referred to the centre of gravity of the body as origin.
Then r' is constant throughout the motion, and -—- is the same
rid
for every particle of the body and equal to t- . Thus the an-
gular momentum h, exactly as in Art. 88, is
THE EQUATIONS OF MOTION.
101
where M1^ is the moment of inertia of the body about its centre
of gravity.
The angle is the angle some straight line fixed in the body
makes with a straight line fixed in space. Whatever straight
dd
lines are chosen -^ is the same. If this be not obvious, it may
be shown thus. Let QA, O'A' be any two straight lines fixed in
the body inclined at an angle a to each other. Let OB, OB be
two straight lines fixed in space inclined at an angle /9 to each
other. Let AOB=0, A'0'B' = ff, then ^ + /3 = l^+a. Since
a and y9 are independent of the time, -j1 = -ji • -By this propo-
sition we learn that the angular velocities of a body in two di-
mensions are the same about all points.
The general method of proceeding will be as follows.
Let {x, y) be the co-ordinates of the centre of gravity of
any body of the system referred to rectangular axes fixed in space,
M the mass of the body. Then the effective forces of the body
are together equivalent to two forces measured by JZ-rr, M-z^
at dv
acting at the centre of gravity and parallel to the axes of co-
ordinates, together with a couple measured by Mh^ -^ tending to
turn the body about its centre of gravity in the direction in which
6 is measured. By D'Alembert's principle the effective forces of all
the bodies, if reversed, will be in equilibrium with the impressed
forces. The dynamical equations may then be formed according
to the ordinary rules of Statics.
For example, if we took moments about a point T"hose co-
ordinates are (p, q) we should have an equation of the form
M
((-.)f-(y-.)'i^}+3f^g=A
where L is the moment of the impressed forces and the other
letters have the same meaning as before. In this equation (p, q)
may be the co-ordinates of any point whatever, whether fixed
or moving. Just as in a statical problem, the solution of the
equations may frequently be much simplified by a proper choice
of the point about which to take moments. Thus if we wished
to avoid the introduction into our equations of some unknown
reaction, we might take moments about the point of application
or use the principle of virtual velocities. So again in resolving
!'
ii
;i
102
MOTION IN TWO DIMENSIONS.
d'x
\l
I
our forces wo might replace the Cartesian expressions M -rp ,
M -j^ by the polar forms
dt*
M
ff-'(f)'}
„„a./14(4*)
r dt\ dtt
lor the resolved parts parallel and perpendicular to the radius
vector. If V be the velocity ol the centre of gravity, p the radius
of curvature of its path, we may sometimes also use with advantage
the forms M-yr and M— for the resolved parts of the effective
dt p ^
forces along the tangent and radius of curvature of the path of the
centre of gravity.
124. As we shall have so frequently to use the equation
formed by taking moments, it is important to consider other forms
into which it may be put. Let the point about which wj are
to take moments be fixed in space, so that it may be chosen as
the origin of co-ordinates. Then the moment of the effective
forces on the body M is
ih(4r-4:)+^4:}=
The attention of the reader is directed to the meaning of the
several parts of this expression. We see that, as explained in
Art. 72, the moment of the effective forces is the differential
coefficient of the moment of the momentum about the same point.
The moment of the momentum by Art. 76 is the same as the moment
about the centre of gravity together with the moment of the whole
mass collected at the centre of gravity, and moving with the velocity
of the centre of gravity. The moment round the centre of gravity
is by the first Article either of Chap. iii. or Chap. IV. equal to
Mk^ -r: and the moment of the collected mass is Jf [a; -^ — y -^ J ,
where {x, y) are the co-ordinates of the centre of gravity. Hence
in space of two dimensions we have for any body of mass M
angular momentum round
the origin
^(4?-4)+^^l-
If we prefer to use polar co-ordinates, we can put this into
another form. Let (r, ^) be the polar co-ordinates of the centre of
gravity, then,
angular momentum round ") _ , , , d^ ^, g dd
the origin ) dt dt'
If V be the velocity of the centre of gravity, and p the per-
pendicular from the origin on the tangent to its direction of
THE EQUATIONS OF MOTION.
103
B radius
le radius
Ivantage
eflfective
:h of the
squation
er forms
, wj are
losen as
effective
ig of the
ained in
ferential
ae point,
moment
le whole
velocity
gravity
equal to
-yi:&
Hence
M
d0
dt'
his into
jentre of
the per-
ction of
motion, the moment of momentum of the mass collected at tho
centre of gravity is Mvp, so that wo also have
angular momentum round) _ , - . -, , d$
the origin [ ~ r ^t'
]
dt
It is clear from Art. 7G that ihis is the instantaneous angular
momentum of the body about the origin, whether it is fixed or
moveable, though in the latter case its dift'erential coefficient with
regard to t is not the moment of the effective forces.
Since the instantaneous centre of rotation may be regarded as
a fixed point, when we have to deal only with the coordinates and
with their first differential coefficients with regard to the time, we .
have
angular momentum round the
instantaneous centre
= i/(r' + A;')
dt'
If Mk'^ be the moment of inertia about the instantaneous
centre, this last moment may be written MTc* -^ .
In taking moments about any point whether it be the centre
of gravity or not, it should be noticed that the Mk* in all these
formulae is the moment of inertia with regard to the centre of
gravity, and not with regard to the point about which ^ "^ arc
taking moments. It is only when we are taking moments ubout
the instantaneous centre or about a fixed point that we can use
the moment of inertia about that point instead of the moment
of inertia about the centre of gravity, and in that case our expres-
sion for the angular momentum includes the angular momentum
of the mass collected at the centre of gravity.
125. Suppose we form the equations of motion of each
body by resolving parallel to the axes of co-ordinates and by
taking moments about the centre of gravity. Wc shall get
three equations for each body of the form
M -jw = i^cos ^ + jR cos -Jr -H . . .
M
d^
df
= JP sin ^ + JB sin -^^ + . . .
Mk
!^d'e _
de
= Fp. +Mq
iV,
where F is any one of the impressed forces acting on the body,
whose resolved parts are J^cos (f>, Fsin ^, and whose moment
about the centre of gravity is Fp, and B is any one of the re-
actions. These we shall call the Dynamical equations of t!ie body.
: I
L
i:n
i
104
MOTION IN TWO DIMENSIONS.
1
t!
, -;
I \
Bc'widt'8 these there will be certain geometrical cquatiuuM
expressing the connections of the system. As every such forced
connection is accompanied by a reaction and every reaction bv
some forced connection, the number of geometrical equations will
be the same as the number of unknown reactions in the system.
Having obtained the proper number of equations of motion
we proceed to their solution. Two general methods have been
proposed.
First Method. Ditferuntiate the geometrical equations twice
with respect to t, and substitute for v , , -jtj , ^ , , from the
dynamical ecjuations. We sliall then have a sufficient number
of equations to determine the reactions. This method will oe of
great advantage whenever the geometrical equations are of the
form
Ax + Bi/+ Ce = D (2),
where A, B, C, D ave constants. Suppose also that the dynamical
equations are such that when written in the form (1) they contain
only the reactions and constants on the right-hand side without
any x, y, or 6. Then, when we substitute in the equation
obtained by differentiating (1), we have an equation containing
only the reactions and constants. This being true for all the
geometrical relations, it is evident that all the reactions will be
constant throughout the motion and their values may be found.
Hence when these values are substituted in the dynamical equa-
tions (1), their right-hand members will all be constants and the
values of x, y, and 6 may be found by an easy integration.
If however the geometrical equations are not of the form (2),
this method of solution will usually fail. For suppose any geo-
metrical equation took the form
x'-¥f = c\
containing squares instead of first powers, then its second dif-
ferential equation will be
d'x . d^y /dx^
X
df'^'^df ^
+
(ly-.
72 _ 7 2
and though we can substitute t'or-~ , -^, we cannot, in general,
eliminate the terms
dxY
di
™^ ('fX
THE EQUATIONS OF MOTION.
105
12C. The reactions iu a ilynamical problem are in many
cases producetl by tho pressures of some smooth fixed obstacles
which are touched by tho moving bodies. Such obstacles can only
push, and therefore if the equation showed that such a reaction
changes sign at any instant, it is clear that the body will leave the
obstacle at that instant. This will occasionally introduce discon-
tinuity into our eciuations. At first tho system moves under
certain constraints, and our equations are found on that suppo-
sition. At some instant which may be determined by the vanish-
ing of some reaction, ore of the bodies leaves its constraints and
the equations of motion have to be changed by the omission of
this reaction. Similar remarks apply if the reactions be produced
by the pressure of one body against another.
It is important to notice that when this first method of solu-
tion applies, the reactions are constant throughout the motion, so
that this kind of discontinuity can never occur. If a moving
body be in contact with another, they will either separate at the
beginning of the motion or will always continue in contact.
127. Suppose that in a dynamical system we have two bodies
which press on each other with a reaction R; let us consider
how we should form the corresponding geometrical equation.
We have clearly to express the fact that the velocities of tiie
points of contact of the two bodies resolved along the dire^v
tion of R are equal. The following proposition will be oftei;
useful. Let a body be turning about a point G with an angular
velocity -j7 = f^ in a direction opposite to the hands of a watch,
and let G be moving in the direction GA with a velocity V. It
is required to find the velocity of any point P resolved in any
direction PQ, making an angle (f) with GA. In the time dt the
whole body, and therefore also the point P, is moved through a
space Vdt parallel to GA, and during the same time P is moved
pei-pendicular to CP through a space w . GP . dt Resolving
parallel to PQ, the whole displacement of P
= {Vco!i<f>^(o.GPsmGPN)dt.
ill
:
, 1
• i '
1(
i .M
-, "I
W
\M
f
106
MOTION IN TWO DIMENSIONS.
If ON'=p be the perpendicular from G on PQ, we see that the
velocity of P parallel to PQ is = F cos — top.
It should be noticed that this is independent of the position of
P on the straight line PQ. It follows that the velocities of all
points in any straight line PQ resolved along PQ are the same.
In practice, therefore, we only use that point in the direction
of PQ which is most convenient, and this is generally the foot of
the perpendicular from the centre of gravity.
If (x, y, 6), {x, y\ 0) be the co-ordinates of the two bodies,
q, q' the perpendiculars from the points (ar, y), [x, y') on the direc-
tion of any reaction B, y^ the angle the direction of R makes with
the axis of x, the required geometrical equation will be
dx
di
, + ^
d0 di
X
cosylr + -f sinylr + -j-q= -^cosf
dt
dt^ dt
j^lyl
^3 . , d& ,
If the bodies be perfectly rough and roll on each other
without sliding, there will be two reactions at the point of contact,
one normal and the other tangential to the common surface of the
touching bodies. For each of these we shall have an equation
similar to that just found. But if there be any sliding friction
this reasoning will not apply. This case will be considered a little
further on.
128. Second Method of Solution. Suppose in a dynamical
system two bodies of masses M, M' are pressing on each other
with a reaction R. Let the equations of motion of M be those
marked (1) in Art. 125, and let those of M' be obtained from
these by accenting all the letters except R, i/r and t, and writing
— R for Ry ifr and t being of course unaltered. Let us multiply
the equation of JW by 2 -3- , 2 ^ , 2 -5- respectively, and those of
M' by corresponding quantities. Adding all these six equations,
we get
^^ \dt de ^ dt dt^" dt dej ^ **'•
cir<( J ^-^ ' , du d&
+ &C.
+ 2B(cost|
. , dy d&
-2«H^^'+»°^f +?'f)-
The coefficient of R will vanish by virtue of the geometrical
equation obtained in the last Article. And this reasoning will
apply to all the reactions between each two of the moving bodies.
/
THE EQUATIONS OF MOTION.
107
that the
►sition of
es of all
le same,
direction
e foot of
bodies,
he direc-
kes with
eh other
f contact,
,ce of the
equation
If friction
d a little
ynamical
Lch other
be those
led from
writing
multiply
those of
uations.
)metrical
ling will
: bodies.
Suppose the body M to press against some external fixed
obstacle, then in this case B acts only on the body M, and its
coefficient will be restricted to the part included iu the first
bracket. But the velocity of the point of contact resolved along
the direction of R must vanish, and therefore the coefficient of
R is again zero.
Let A be the point of application of the impressed force F,
and let -4 be the velocity of .4 resolved along the direction of action
oiF.
Then we see that the coefficient of 2F is -4- .
at
df
It also
follows from the definition of -j that Fdf is what is called in
Statics the virtual moment of the force F.
"We have thus a general method of obtaining an equation free
from the unknown reactions of perfectly smooth or perfectly
rough bodies. The rule is, Multiply the equations having
dx
It'
X
d'y iw«^
M-r^, M-jY, Mk^-j^, &c. on their left-hand sides by
df
di
df
-jr, -J- , &c., and add together all the resulting equations for all
cLZ dt
the bodies. The coefficients of all the unknown reactions will be
found to be zero by virtue of the geometrical equations.
The left-hand side of the equation thus obtained is clearly
a perfect differential. Integrating we get
«m-$M^'h'^-'-^hf-
where C is the constant of integration.
In practice it is usual to omit all the intermediate steps and
Avrite down the resulting equation in the following manner:
where U is the integral of the virtual moment of the forces.
This is called the equation of Vis Viva. Another proof will
be given in the chapter under that heading.
129. The left-hand side of this equation is called the vis viva
of the whole system. Taking any one body M, we may say thau
^dy\\
,dtj
visviva of Jf =ilf-|l-^j +
-^m
If the whole mass were collected into its centre of gravity and
were to move with the velocity of the centre of gravity, k would be
) 1;
M
108
MOTION IN TWO DIMENSIONS.
m
zero, and the vis viva would be reduced to the two first terms.
These terms are therefore together called the vis viva of transla-
tion, and the last term is called the vis viva of rotation.
If V be the velocity of the centre of gravity, we may write this
equation
vis viva of M= Mv^ + il/A;' (-r-J •
If we wish to use polar co-ordinates, we have
v.vivaor^=..g)V.(t)V..(|)] .
where [r, ^) are the polar co-ordinates of the centre of gravity.
If p be the distance of the centre of gravity from the instanta-
dd
neous centre of rotation of the body, p -tt is clearly the velocity
at
of the centre of gravity, and therefore
vis viva of J/ = ilf (p^ + A;") ^^y .
The right-hand side of the equation of vis viva, after division
by 2, is called sometimes the force function of the forces and
sometimes the luork of the forces. It may always be obtained
by writing down the virtual moment of the forces according to
the rules of Statics, integrating the result and adding a constant.
Frequently it is convenient to avoid introducing the unknown
constant C by taking the integral between limits. We then
subtract from the left side the initial vis viva, and from the right
side the initial value of the force function.
130. If there is only one way in which the system can move,
that motion will be determined by the equation of vis viva. But
if there be more than one possible motion, we must find another
integral of the equations of the second order. What should be
done will depend on the special case under consideration. The
discovery of the proper treatment of the equations is often a
matter of great difficulty. The difficulty will be increased, if
in forming the equations care has not been taken that they
should have the simplest possible forms.
131. In many cases a great simplification of the equations
will be effected by a proper choice of the direction in which to
resolve the forces, or of the points about which we take moments.
First we should search if there be any directioi. in which the
resolved part of the impressed forces vanishes. By resolving in
this direction wc get an ccjuation which can bo immediately
fl !
THE EQUATIONS OF MOTION.
109
integrated. Suppose the axis of x to be taken in this direction ;
lot M, M', &c. be the masses of the several bodies, x, al, &c. the
abscissae of their centres of gravity, then by Art. 123 we have
''w*''w^
which by integration gives
^S + ^'f +
..=0,
= c,
where is some constant to be found from the initial conditions.
This equation may also be again integrated if required.
This result might have been derived from the general princi-
ples of the conservation of the translation of the centre of gravity
laid down in Art. 79. For since there is no impressed force
parallel to the axis of x, the velocity of the centre of gravity of the
whole system resolved in that direction is constant.
132. Next we should search if there be any point about
which the moment of the impressed forces vanishes. By taking
moments about that point we again have an equation which
admits of immediate integration. Suppose this point to be taken
as origin, and the letters to have their usual meaning, then by
the first article of this chapter we have
w^^-
yf,)^MU
^1 =
tlie % referring to summation for all the bodies of the system
Integi'ating as in Art. 124 we have
{^(^l-/|)+^43=^-
where C is some constant to be determined by the initial condi-
tions of the question.
This equation expresses that if the impressed forces have no
moment about any point, the angular momentum about that
point is constant throughout the motion. This result follows at
once from the reasoning in Chap, li.
133. A homogeneous sphere rolls directly down a perfectly rough inclined plane
under the action of gravity. Find the motio7i.
Let a he tlie inclination of the plane to the horizon, a the radius of the sphere,
mJc'^ its moment of inertia about a horizontal diameter.
Let be that point of the inclined plane which was initially touched by ti.ri
sphere, and N the point of contact at the time t. Then it is obviously convenient
to choose for origin and OIV for the axis of ,«■.
I
If
\\ \
:'
■, 1
i
I
I
110
MOTION IN TWO DIMENSIONS.
The forces which act on the sphere are first the reaction R perpendicular to ON,
secondly, F the friction acting at N along NO and mg acting vertically at C the
centre.
The effective forces ^ro m -^ , m -A acting at C parallel to the axes of x and y
d'0
and a couple mk* -z-^ tending to turn the sphere round C in the direction NA,
'^
Here 6 is the angle any fixed straight line in the hody makes with a fixed straight
line in space. We shall take the fixed straight line in the body to be the radius
CA, and the fixed straight line in space the normal to the inclined plane. Then
is the angle turned through by the sphere.
Resolving along and perpendicular to the inclined plane we have
m-^=mgBxaa-F (1),
m ■^= -mg COB a + R (2).
Taking moments about N to avoid the reactions, we have
""^d^"*"^ ^~'"^**'"*" ^ '"
Since there are two unknown reactions F and R, we shall require two geome-
trical relations. Because there is no slipping at N, we have
x=ae (4).
Also because there is no jumping y=a (5).
Both these equations are of the form described in the first method. Differcn-
tiating (4) we get j^ =<* ^ • Joining this to (3) we have
diJ = a-«T*«^'^°" (^^-
2
Since the sphere is homogeneous, Ji^=^a^, and we have
5
d^x 5 .
If the sphere had been sliding down a smooth plane, the equation of motion
would have been
d'x
EXAMPLES.
Ill
80 that two-sevenths of gravity is used in tnming the sphere, and five-sevenths in
urging the sphere downwards. Supposing the sphere to start from rest we have
clearly
1 6
«=2
and the whole motion is determined.
jflfsino.e',
In the above solutions, only a few of the equations of motion have been used,
and if only the motion had been required it would have been unnecessary to write
down any equations except (3) and (4). If the reactions also be required, we must
use the remaining equations. From (1) we have
From (2) and (5) we have
F-=mg Bin a.
R=mg COB a.
It is usual to delay the substitution of the value of k^ in the equations until the
end of the investigation, for this value is often very complicated. But there is
another advantage. It serves as a verification of the signs in our original eqiiations,
for if equation (6) had been
we should have expected some error to exist in the so! ition. For it seems clear
that the acceleration could not be made infinite by any alteration of the internal
structure of the sphere.
Ex. If the plane were imperfectly rough with a coefficient of friction /i less
than f sin a, show that the angular velocity of the sphere after a time t from rest
would be
5/1 g cos a
•i
l1
11
■1l '1 '
n
i
t.
134. A homogeneous sphere vols down another perfectly rough fixed sphere.
Find the motion.
Let a and b be the radii of the moving and fixed spheres, respectively, C and
the two centres.
Let OB be the vertical radius of the fixed sphere, and = / BOO. Let F and B
be the friction and the normal reaction at N. Then resolving tangentially and
normally to the path of C we have
nm
Vi
»1
i '
.
I!
'I
112 MOTION IN TWO DIMENSIONS.
(a + 6)^=flrsm^-- (1),
(« + ^)(ty=^°°«^-^ <2).
Let ^ be that point of the moving sphere which originally coincided with £.
Then if bo the angle which any fixed line, as CA, in the body makes with any fixed
line in space, as the vertical, we have by taking moments about C
dt^~mk'^ ^ ''
It should be observed that we cannot take as the angle ACO because, though
CA is ^xed in the body, CO is not fixed in space.
The geometrical equation is clearly
a(e-<f>) = b<p (4).
No other is wanted, since in forming equations (1) and (2) the constancy of tho
distance CO has been already supposed.
The form of equation (4) shows that we can apply the first method. We thus
obtain
and wo are finally led to the equation
By multiplying by 2 ~ and integrating we get after determining the constant
fdd>Y 10 o ,,
the rolling body being supposed to start fron rest at a point indefinitely near B,
This result might also have been deduced from the equation of vis viva. Tho
vis viva of the sphere is m | v" + P ( -37 ) | and r = (a + 6) -^ . The force function is
m lgdy=mffy if y be the vertical space descended by the centre. We thus have
<" + ^)'' (5)' + *"©' = ^^^" + ^^ (1-COS0),
which is easily seen to lead, by help of (4), to the same result.
To Und whero the body leaves the sphere we must put R=0. This gives by (2)
{a + h)(-^\=gcos<p; .•. y (/(l-co3^)=gicos0; .•.cos^ = r^. It may be re-
marked that this result is independent of the magnitudes of the spheres.
Ex. 1. If the spheres had been smooth the upper sphere would have left the
lower sphere when cos = |.
Ex. 2. A rod rests with one extremity on a smooth horizontal plane and tho
other on a smooth vertical wall at an inclination a to the horizon. If it then slips
down, show that it will leave the wall when its inclination is sin"^ (I sin o).
Del. We thus
he constant
EXAMPLES.
113
Ex, 8. A beam is rotating on a smooth horizontal plane about one extremity,
which is fixed, under the action v,f no forces except the resistance of the atmo'^phere.
Supposing the retarding effect of the resistance on a small element of the beam of
length a to be Aa (vel.)', then the angular velocity at the time t is given by
* ^ "*"* ■ [Queens' Coll.]
u O ^Mk*
t.
Ex. 4. An inclined plane of mass M is capable of moving freely on a smooth
horizontal plane. A perieotly rough sphere of mass m is placed on its inclined face
and rolls down under the action of gravity. If a! be the horizontal space advanced
by the inclined plane, x the part of the plane rolled over by the sphere, prove that
(Jf + m) x'= ma; cos a,
\x - cos ox' = \g sin o«',
where a is the inclination of the plane to the hprizon.
Ex. 5. Two equal perfectly rough spheres are placed in unstable equilibrium,
one on the top of the other ; the lower sphere resting on a perfectly smooth table.
The slightest disturbance being given to the system, shew that the spheres will
continue to touch each other at the same points and if be the inclination to the
vertical of the straight line joining the centres,
(P + oHo" sin" e) (^)''=2^a (1 - cos 2(9).
Ex. 6. Two unequal perfectly smooth spheres are placed in unstable equilibrium
one on the top of the other ; the lower sphere resting on a perfectly smooth tablp
A very slight disturbance being given to the system, shew that the spheres will
separate when the straight line joining the centres makes an angle with the verti-
caJ, given by the equation -^ cos' ^ - 3 cos A + 2 = 0, where M is the mass of the
m + m
lower and m of the upper sphere.
Ex. 7. A sphere of mass M and radius a is constrained to roll on a perfectly
rough curve of any form and initially the velocity of its centre of gr.-vvity is V. If
the initial velocity were changed to V, shew that the normal reaction would be
friction would be unaltered, p being the
y>3_ yn
increatdd by M — and that the
p-a
radius of curvature of the curve at the point of contact.
135. A rod OA can turn about a hinge at 0, lohile the end A rests on a smooth
toedge which can slide along a smooth horizontal plane through 0. Find the motion.
Let a=the inclination of the wedge, il!f=its mass and «= OC.
;: '
\ have left the
R. D
^njaatti
^!
I
;i
U
114 MOTION IN TWO DIMENSIONS.
Let l=ihe length of the beam, m=itB mass aud 0=A OC.
Let iZ- the reaction at 4. Then we have
the dynamical equations,
d'x _ iZ sin o ,, ,
dt^-~M~ ^^'•
,„ M . cos (a-O)- mg ^ cos 6
d£^~ mk^ _ ■ ^"
and the geometrical equation,
x = -. — .siu(o-e) (3).
sma '
It is obvious we must apply the second method of solution. Hence
...dxd^x „ ...dedW , „de „„\ . dx . , ..d0\
^'^didt^+^'''^dlW'=-'''^^''''^dt+^^n''dt-^^'^'^''-^^di\'
The coefficient of R is seen to vanish by differentiating ec[uation (3). Inte-
grating we have
This result might have been written down at once by the principle of vis viva.
For the vis viva of the wedge is clearly M{-r-\ and that of the rod Mk^ \dt/'
The virtual moment of the forces is - mgdy where y is the altitude above OC of the
centre of gravity of the rod OA, hence twice the force function is C-2mgy. Since
y=^l sin 0, this reduces to the result already written down.
Substituting from (3) we have
\M^^^,-coBHa-e) + mh-'\(~^y=C-mglBm0 (4).
If the beam start from rest when 0=p, then C=mgl sin /3.
This equation cannot be integrated any further. We cannot therefore find in
terms of t. But the angular velocity of the 'jeam, and therefore the velocity of the
wedge, is given by the above equation.
136. Two rods A B, BC are hinged together at B and can freely slide on a
amooth horizontal plane. The extremity A of the rod AB is attached by another
hinge to a fixed point on the table. An elastic string AC, whose unstretched length
is equal to A B or BC, joins A to the extremity C of the rod BC. Initially the two
rods and the string form an equilateral triangle and the system is started with an
angular velocity CI round A . Find the greatest length of the elastic string during the
motion. Find also the angular velocities of the rods when they are at right angles^
and the least value of Q that this may be possible.
Let the length of either rod be 2a, mP the moment of inertia of cither about its
centre of gravity, so that k^=~ . Let D and E be the middle points of the rods,
and let (r, 6) be the polar co-ordinates of E referred to A as origin.
The only forces on the system are the reaction of the hinge at A and the tension
of the elastic string A C. If we search for any direction in which the sum of the
resolved parts of these vanishes, we can find none, since tho direction of the
EXAMPLES.
115
(1).
(2),
.(3).
di\'
(3). Inte-
of vis viva.
e OC of the
mgy. Since
'©••
(4).
)re find in
locity of the
/ slide on a
by another
died length
ally the two
ted with an
g during the
•ight angles^
ler about its
of the rods,
the tension
sum of the
ion of the
reaction is at present unknown. But since the lines of action of both these forces
pass through A, their moments about A vanish, and therefore, by Art. 132, the
angular momentum about A is constant throughout the motion aud equal to its
initial value. Let u, <a' be the angular velocities ot AB, BC at auy instant (. The
angular momentum of BC about A is by Art. 124 m (r* -j- + Pw'). The angular
momentum of A£ is by the same article m(k'' + a^) w, since AB ia turning about A
as a fixed point. The initial values of these are respectively m(Sa^Sl + k^O), and
m{k'+a^)Q, since w, w' and -j- are each initially equal to 0, and r is initially
equal to the perpendicular from A on the opposite side of the equilateral triangle
formed by the system. Hence
m{ifl + a^u + mk^w'+mr'^=m{2k^ + ia')Q (1).
We may obtain another equ'-.Lion by the use of the principle of vis viva. The
vis viva of the rod BC is by Art. 129 m |(^^ +r» {j\ +Jc>u'A . The vis viva of
AB is by the same article m (P + a") w" since it is turning round 4 as a fixed point.
The initial values of these are respectively m (Sa" + k^) fl* and m (/fc" + a") QK If T be
the tension of the string, p its length at time /, the force function of the tension is
X{- T)dp. According to the rule given in Statics to calculate virtual moments,
the minus sign is given to the tension because it acts to diminish p; and the limits
are 2a to p because the string has stretched from its initial length 2a to p. By
Hooke's law T=E ^ ^ "" y so that, by integration, the force .unction= -E-~
2^ , , ^j 6- . -" -- 4„ •
The reaction at A does not appear by Art. 128. The equation of vis viva is
therefore
m(;!^ + a')wHmj^|^y + >-2(^y + JfcV2J=wi(2i!;' + 4a'')n'-£^^'^ (2).
There are only two possible independent motions of the rods. We can turn A B
about A and BC about B, all other motions, not compounded of these, are incon-
sistent with the geometrical conditions of the question. Two dynamical equations
8—2
116
MOTION IN TWO DIMENSIONS.
are snffioient to dotemaine these, and these we have jnst obtained. All the other
equations which may be wanted must be derived from geometrical considerations.
We must now express the geometrical conditions of the question. Let ^ bo tho
supplement of the angle ABC, then
rf*
r'=5o' + 4o'cos^
Since ^ is the relative angular velocity of the rods BC, AB,
dt
= w - w
dr
di'
- 2a' Bin (w' - «)
Let \f> be the angle EAB, then
sin f = sin
.(8).
.(4),
.(6).
(C),
d^p de ,
and smce 37 = 37 - w, we have
dt dt
<iOByp\^^-uj = (^cos4>+-^Bin^</,yu'-io) (7).
Also from the triangle ABC
p!i + 2a''=2r«
From these eight equations we can eliminate w, w', r, -n> p> ^ and -77 .
(tt 0ib
(8).
Wo shall
d<l>
then have a differential equation of the first order to solve, containing ^ and -~
Hi'
It is required to find the greatest length of the elastic string during the motion.
At the moment when p is a maximum, -i^=0 and tho whole system is therefore
moving as if it wt.e a rigid body. We therefore have for a single moment w, u' and
■^ all equal to each other and ^7=0. The two first equations become, when we
have substituted for I? its value — ,
(6a«+3r2)w=14««0
3J?
I-
(Sa" + 3r«) w2 = 14a« 0* - ^ (/) - 2af
Eliminating w and substituting for r from (8) we have the cubic
(3p»+16a«){p-2a) = 2i^'. (p + 2a).
which has one positive root greater than 2a.
It is also required to find the motion at the instant when the rods are at right
angles. At this moment <l>=^ and hence by (3) r = a V^. by (5) -3^ = - -j= a{u'-u),
do 1
^y (7) -J- = H (w' + 4«). Substituting in equations (1) and (2) we get
at 5
17
EXAMPLES.
117
From those two equations we may easily find u and (■/. It is easily seen that the
10 E
values of u, u' will not be real unless 0'>t;^ (\/2 - 1)*.
7 ma ^ '
We moy often save ourselves the trouble of some elimination if wo form tho
equations derived from the principles of angular momentum and vis viva in a
slightly different manner. Tho rod BC is turning round P with an angular velocity
w', while at the same time B is moving perpendicularly to AB with a velocity 2au.
The velocity of E is therefore the resultant of aw' perpendicular to BC and 2au per-
pendioiUar to AH, both velocities, of course, being applied to the point E. When
we wish our results to be expressed in terms of u, u' we may use these velocities to
express the motion of E instead of the polar co-ordinates (r, 0).
Thus in applying the principle of angular momentum, we have to take tho
moment of the velocity of E about A . Since tho velocity 2au is perpendicular to
AB, the length of the perpendicular from A on its direction isAB together with the
projection of BE on A B, v^hich is 2a + a cos if>. Since the velocity att/ is perpen-
dicular to BE, the length cf tho perpendicular from A on its line of action is BE
together with the projectioa of ^B on BE, which is a + 2a cos ^. Hence the angu-
lar momentum of the rod BC about A is, by Art. 124,
ink"- w'+ 2inau (2a + a cos <f>) + mau' {a + 2o cos 0).
The principle of augula'" momentum for the two rods gives therefore
m (P + Sa" + 2a' cos 0) w + wi (i» + a" + 2a' cos 0) w' = m (2/fca + 4a') 0.
Tho right-hand side of this equation, being the initial value of the angular momen-
tum, is derived from the left-hand side by putting cos 0= - 4 and w = w'=0.
In applying the principle of vis viva, we require the velocity of E. Begarding
it as the resultant of 2au and au' we see that, if v be this velocity,
«' = (2aw)' -f- (aw')'' + 2 . 2aw . ow' cos </>.
The initial value being found, as before, by putting cos 0= - J, «=«'=0, the princi-
ple of vis viva gives, by Art. 129,
,(/)-2a)a
m (A" + 5a') oP + m (A' + a') w" + ima^ uu' cos = m (2i' + 4a«) 0' - £ '
2a
The force function is foimd in the same manner as before. If we join to this equa-
tion (4) given above, and substitute p=4acos --, we have just three equations to find
u, w', and if>. If these quantities are all that are required, as in the two cases con-
sidered above, this form of solution has the advantage of brevity. When p is a
maximum, we put w=w', when the rods are at right angles, we put cos 0=0. The
equations then lead to the results already given.
137. The boh of a heavy pendulum contains a spherical cavity tohich is filled
with water. To determine the motion.
Let be the point of suspension, the centre of gravity of the solid part of the
pendulum, MK^ its moment of inertia about and let 00=h. Let C be the centre
of the sphere" of water, a its radius and OC=c. Let m be the mass of the water.
If we suppose the water to be a perfect fluid, the action between it and the case
must, by the definition of a fluid, be normal to the spherical boundary. There will
therefore be no force tending to turn the fluid round its centre of gravity. As the
pendulum oscillates to and fro, the centre of the sphere will partake of its motion,
but there will be no rotation of the water.
i
j
I f
i %
118
' II '
i
MOTION IN TWO DIMENSIONS.
The ofTtictivo forccB of the water are by Art. 128 equivalent to the efTcctive force o!
the whole mass collected at ItH centre of grovity together with a couple niA' -
where w in the angnlar velocity of the water, and m/t' itn moment of inertia ahout a
diameter. But u has jUHt been proved zero, hence thiw couple may be omitted. It
followH that in all problems of this kind wliere the body does not turn, or turns with
uniform angular velocity, we may collect the body into a .lingle particle placed at
its centre of gravity.
The pendulum and the collected fluid now form a rigid body turning about n
fixed axis, hence if be the angle CO a fixed line in the body makes with the
vertical, the equation of motion by Art. 88 is
d^0
(MK^ + «ic«) ^ + (Mh + mc) g sin = 0,
where in finding the moment of gravity, 0, and C Lave been supposed to lie in a
straight line.
The length L' of the simple equivalent pendulum is, by Art. 92,
~ Mh + mc
Let ml' bo the moment of inert' \ of the sphere of water about a diameter.
Then if the water were to become solid and to bo rigidly connected with the case,
the length L of the simple equivalent pendulum would bo, by similar reasoning.
L =
Mh+mn
It appears that L'<L, so that the time of oscillation is less than when the
whole is solid.
138. If we refer to tlie equations of motion of a body given
in Art. 125, we see that tlic motion depends on (1) the mass of
the body, (2) the position of the centre of gravity, (3) the external
forces, (4) the moments of inertia of the body about straight lines
through tlie centre of gravity, (5) the geometrical equations. Two
bodies, however different they may really be, vhich have these
characteristics the same, will move in the same inauner, i.e. their
centres of gravity will describe the same path, and their angular
motions about their centres of gravity will be the same. It is
often convenient to use this proposition to change the given body
into some other whose motion can be more simply found.
For example, if a sphere have an eccentric spherical cavity
filled with fluid of the same density as that of the solid sphere,
the motion of the sphere is independent of the position of the
cavity, so that, if it be more convenient, we may put the cavity at
the centre. To prove this, we may notice that since the sphere of
fluid does not rotate, or rotates with uniform angular velocity, the
motion is unaltered by collecting the fluid into a particle placed
at its centre. This being done, the first, second, third, and fifth
characteristics are clearly independent of the position of the cavity.
As for the fourth characteristic, let a be the radius of the sphcro.
EXAMPLES.
110
ive furoe of
I to lie in a
I when tlio
b that of tho cavity, j the distance of its centre from the centro
of the sphere, D the density, tlien the moment of inertia of tho
solid part of the sphere is |7ra'. ]f a'- ;^7rfe* .(](?>'' + c'). Tho
moment of inertia of the fluid coliocttMl into its ctMitro is ^ttJ'.c*.
When wo add those togothor o disappoars, so that tho whole
moment of inertia is independent of tho position of the cavity.
The motion of a uniform triangular area moving under tho
action of gravity is another example. If we replace the area by
throe wires forming its perimeter hut without weight, the geome-
trical conditions of the motion will in general be inialterod, and if
we also place at the middle points of these wires three weights,
each one-third of the mass of the triangle, this body will have
all its characteristics the same as that of the real triangle, and
may replace it in any problem.
When a string connecting two parts of a dynamical system
passes over a rough pulley, it was formerly the custom to con-
sider tho inertia of rotation of the pulley by replacing it by
another pulley of the same size but without mass and loaded
with a particle at its circumference. If a be the radius of the
pulley, k its radius of gyration about the centre, m its mas.s, the
mass of the particle is -jj^». so that in a cylindrical pulley the
mass of the particle is half that of the pulley. This mass must
then be added on to the other particles attached to the string.
For example, if two heavy masses j\f, M' be connected by a string
passing over a cylindrical pulley of mass m, which can turn freely
about its axis, the ecpiation of motion is
h
{M^M'^f^'%^(U-M'),
wliere v is the velocity. Here the inertia of tlio pulley is taken
account of by simply adding - - to the mass moved. If the pulley
be moveable in space as well as free to rotate, its inertia of trans-
lation is as usual taken account of by collecting the whole mass
into its centre of gravity. As this representation of the inertia of
rotation is not often used now, the demonstration of the above
remarks, if any be needed, is left to the reader.
Ex. 1. A rod AB whose centre of gravity is at t'ae middle point C oi AB has its
extremities A and B constrained to move along t'lro straight lines Ox, Oy inclined
at right angles and is acted on hy any forces. Shew that the motion is the same as
if the whole mass were collected into its centre of gravity and all the forces reduced
A"
in the ratio 1 + — j : 1 where 2a ia the length A B and it is the radius of gyration
about the centro of gravity.
ti.
n
120
MOTION m TWO DIMENSIONS.
I: -h
r I'ij
i ^ir
K(
Ex. 2. A cironlar disc whose centre of gravity is in its centre rolls on a perfectly
rough curre under the action of any forces, she^ that the motion of the centre is
the same as if tha curve were smooth and all the forces were reduced in the ratio
1 + — : 1, where a is the radius of the disc and k is the radius of gyration about the
a*
centre. But the normal pressures on the curve in the two cases are not the same.
In any position of the disc they differ by X yzJa ^^^^^ ^ is the force on the disc
resolved along the normal to the rough curve.
On the stress at any point of a rod.
139. Suppose a rod OA to be in equilibrium under the action
of any forces, it is required to determine the action across any
section of the rod at P. This action may be conceived to be the
resultant of the tensions positive or negative of the innumerable
fibres which form the material of the rod. All these we know by
Statics may be compounded into a single force B and a couple O
acting at any point Q we may please to choose. Since each por-
tion of the rod is in equilibrium, these must also be the resultants
of all the external forces which act on the rod on one side of the
section at P. If the section be indefinitely small it is usual to
take Q in the plane of the section, and these two, the force R
and the couple G, will together measure the stress''^ at the
section.
If the rod be bent by the action of the forces, the fibres on
one side will all be stretched and on the other compressed. The
rod will begin to break as soon as these fibres have been sufiici-
ently stretched or compressed. Let us compare the tendencies of
the force B and the couple G to break the rod. Let A be the
area of the section of the rod, then a force F pulling the rod will
cause a resultant force R= F, and will produce a tension in the
F
fibres which when referred to a unit of area is equal to -j . The
same force F acting on the rod at an arm from P whose length
is p, will cause a couple O = Fp, which must be balanced by the
couple formed by the tensions. Let 2a be the mean breadth of
the rod, then the mean tension referred to a unit of area produced
F v
by is of the order 7 . - . Now if the section of the rod be very
small - will be large. It appears therefore that the couple, when
it exists, will generally have much more effect in breaking the
* Sir W. Thomson has appropriated the word strain to the alterations of volume
and figme produced in an clastic body by the forces applied to it, and the word
Ktress to the elastic pressures.
ON STRESS.
121
a perfectly
e centre is
[u the ratio
1 about the
t the same.
on the disc
he action
jross any
to be the
umerable
know by
couple G
jach por-
■esultants
ie of the
usual to
force R
* at the
fibres on
bd. The
n suffici-
encies of
be the
rod will
n in the
r. The
e length
by the
eadtli of
jroduced
be very
le, when
:ing the
of volume
the word
rod than the force. This couple is therefore often taken to
measure the whole effect of the forces to break the rod. The
" tendency to break" at any point P of a rod. OA of very small
section is measured by the moment about P of all the forces which
act on either of the sides OP or PA of the rod.
The resolved part of the force B perpendicular to the rod is
called the shear. This is therefore equal to all the forces which
act on either of the sides OP or PA resolved perpendicular to the
rod.
If the rod be in motion the same reasoning will, by D'Alem-
bert's principle, be applicable ; provided we include the reversed
effective forces among the forces which act on the rod.
In most cases the rod will be so little bent that in finding
the moment of the impressed forces we may neglect the effects
of curvature.
If the section of the rod be not very small, this measure of
the "tendency to break" becomes inapplicable. It then becomes
necessary to consider both the force and the couple. This does
not come within the limits of the present treatise, and the reader
is referred to works on Elastic Solids.
In the case of a string the couple vanishes and the force acts
along a tangent to the string. The stress at any point is there-
fore simply measured by the tension.
140. A rod OA, of length 2a, and mass m, which can turn freely about one
extremity 0, falls under the action of gravity in a vertical plane. Find the " tendency
to break" at any point P.
Let du be any element of the rod distant u from P and on the side of P nearer
the end A of the rod, and let OP=x. Let be the angle the rod makes with the
vertical at the time t. The effective forces on du are
hVi
m
du, ^ .d^0 , du, , ,fdey
-(x+«)^ and -m^^ («: + «) (^^j
du
respectively perpendicular and along the rod. The impressed force is wi — g acting
^a
vertically downwai'ds. The effective forces being reversed the tendency to break
at P is equal to the moment about P of all the forces whiclr act on the part PA of
the rod. If this be called L, we have
the limits being from u^ to «= 2a - x. Also taking moments about 0, the equation
of motion is
W> d^d
Hence we easily find
m. -^ ^/a - "*5"* ^"^ ^''
innmnB ,„
^ I
122
MOTION IN TWO DIMENSIONS.
The meaning of the minus sign is that the forces tend to bend PA round P in the
opposite direction to that in which has been measured.
To find where the rod supposed equally strong throughout is most likely to
break, we must make L a maximum. This gives -7- =0 and therefore x=-rr. The
' ax A
point required is at a distance from the fixed end equal to one-third of the length of
the rod. This point, it should be noticed, is independent of the initial conditions.
To find the shear at P we must resolve perpendicularly to the rod. If the result
be called 1", we have
du . . r du , . d-$
the limits being the same as before. This gives
mg sin 9
Y=
lOa"
(2a - x) (2a - 3x),
I I
which vanishes when the tendency to break is a maximum, and is a maximum at a
distance from the fixed end equal to two- thirds of the length of the rod.
To find the tension at P we must resolve along the rod. If the result be called
X, we have
^ r du ^ r du , , (deY
^= -> 27*^"°' ^ + >2-a (^ + «) \dt) '
If the rod start from rest at an inclination o to the vertical, we find, by integrating
the equation of motion, ( -jr ) = k^ (cos a - cos 6), Hence
X=^^(2a-x){- 4a cos ^ -1- 3 (cos a - cos 9) (2a f a;)}.
From these equations we may deduce the following results. (1) The magnitudes
of the stress couple and of the shear are independent of the initial conditions.
(2) The magnitude of either tho couple or the shear at any given point of the rod
varies as the sine of the inclination of the rod to the vertical. (3) The ratio of the
magnitudes of the stress couples at any two given points of the rod is always the
same, and the same proposition is also true of the shear. (4) The tension depends
on the initial conditions and unless the rod start from rest in the horizontal position,
the ratio of the tensions at any two given points varies witl- the position of tho rod.
141. A rigid hoop complctehj cracked at one point rolls on a perfectly rough
horizontal plane and is acted on by no forces but gravity. Prove that the wrench
couple at the point of tlic hoop most remote from the crack tcill he a maximum ivhen-
ever, the crack being lower than the centre, the inclination of the diameter through
2
the crack to the horizon is tan~^~ . [The Math. Tripos, 1864.]
TT
Let u be the angular velocity of tho hoop, a its radius. The velocity of any
point P of the hoop is the resultant of a velocity aw parallel to the horizontal pliiue
and an equal velocity aw along a tangent to the hoop. Tlio first is co^istant in
direction and magnitude and therefore gives nothing to tho acceleration of P. The
latter is constant in magnitude but variable in direction and gives aw' as the
acceleration which is directed along a radius of the hoop. Lot A be tue cracked
point, /i tho other end of tho diameter, V tho centre,, tho inclination of ACfi to
ON STRESS.
123
tho horizon. Let PP" be any element on the upper half of the circle, BCP=(t>.
Then the wrench couple, or tendency to break, at B is proportional to
/ [ - aw' a sin + gr {a COS ^ - a cos {<fi + fl)}] ad<f> = - 2a''w^ + ga^ (cos ^jr + 2 sin 0).
•'0
2 *
This is a maximum when tan — -,
Ex. 1. A semicircular wire ^ J5 of radius a is rotating on a smooth horizontal
plane about one extremity A with a constant angular velocity w. If a<j) be the arc
between the fixed point A and tho point where the tendency to break is greatest,
prove that tan <f> = ir-</>. If the extremity £ be suddenly fixed and the extremity
A let go, prove that the tendency to break is greatest at a point P where
^ tan PBA = PBA.
Ex. 2. Two of the angles of a heavy square lamina, a side of which is a, are
connected with two points equally distant from the centre of a rod of length 2a, so
that the square can rotate about the rod. The weight of the square is equal to the
weight of the rod, and the rod when supported by its extremities in a horizontal
position is on the point of breaking. The rod is then held by its extremities in
a vertical position, and an angular velocity w is then impressed on the square.
Shew that it will break if « >
V'i-
[Coll. Exam.]
Ex. 3. A wire in the form of the portion of the curve r=a (1 + cos 0) cut off by
the initial line rotates about the origin with angular velocity w. Prove that the
TT 12 v/2 „
tendency to break at the point tf=^ is measured by m — ^ — w^a'.
[St John's CoU.]
I'
y of any
tal plane
taut in
P. The
as the
cracked
AC/i to
On Friction hetiveen Imperfectly Bough Bodies.
142. When one body rolls on another under pressure, the two
bodies yield slightly, and are therefore in contact along a small
area. At every point of this area there is a mutual action be-
tween the bodies. The elements just behind the geometrical
point of contact are on the point of separation and may tend to
adhere to each other, those in front may tend to resist com-
pression. The whole of the actions across all the elements are
equivalent to (1) a component R, normal to the common tan-
gent plane, and usually called the reaction; (2) a component #
in the tangent plane usually called the friction ; (3) a couple L
about an axis lying in the tangent plane and which we shall call
the couple of rolling friction ; (4) if the bodies have any relative
angular velocity about their common normal, a couple N about
this normal as axis which may be called the couple of twisting
friction.
124
^•OTION IN TWO DIMENSIONS.
143. These two coupres are found by experiment to be in
most cases very small and are generally neglected. But in certain
cases where the friction forces are also small, it may be necessary
to take account of them.
144. When one body presses against another over any small
area, the force of friction acts in such a direction and with such a
magnitude that it is just sufficient to prevent sliding. Both the
magnitude and direction of friction may. therefore, be unknown
beforehand, and their determination will be part of the problem
under consideration. It is found by experiment that no more
than a certain amount of friction can be called into play, and
when more is required to keep the bodies from sliding on each
other, sliding will begin. T'lis amount is called limiting friction.
The magnitude of this limit is found to bear a ratio to the normal
pressure which is rery nearly constant for the same two bodies.
Though all experimenters have not entirely agreed with each other
as to the accuracy of this result, yet it has been found generally that,
if the relative motion of the two bodies be the same at all points of
the area of contact, this ratio is nearly independent of the extent of
the area and of the relative velocity. If, however, the bodies have
remained in contact for some time under pressure in a position
of equilibrium, it is found that, for the more compressible bodies,
the ratio is a little greater than after motion has begun. This
ratio has been called the coefficient of friction of the materials of
the two bodies. Its constancy is generally assumed by mathema-
ticians. When the friction which can be called into play is insuf-
ficient to prevent sliding, the bodies slide on each other. In this
c-,ise the magnitude of the friction is equal to its limiting value,
and the direction of the friction is opposite to that of relative
motion.
145. If the bodies be perfectly rough, the coefficient of friction
is infinite, and there is no limit to the amount of friction which'
can be called into play. There can, therefore, be no sliding be-
tween the bodies.
146. Discontinuity of motion will often occur when a body
moves under the action of friction. Suppose the body rolls on a
rough surface, the friction called into play just prevents sliding,
and is possibly variable in magnitude and direction. By writing
down and solving the equations of motion we can find the ratio of
the friction F to the normal pressure R. If this ratio be always
less than the coefficient fi of friction, enough friction can always
be called into play to make the body roll on the rough surface.
In this case we have obtained the true motion. But if at any
instant the ratio -^ thus found should be greater than the co-
to be in
a certain
lecessary
Qy small
h such a
Both the
inknown
problem
no more
lay, and
on each
friction.
e normal
bodies.
ich other
ally that,
points of
extent of
lies have
position
e bodies,
n. This
terials of
athema-
is insuf-
In this
ig value,
relative
friction
n which
ing be-
a body
lis on a
sliding,
writing
ratio of
} always
always
surface.
at any
the co-
IMPERFECT FRICTION.
121
i
efficient of friction, the point of contact will begin to slide at that
moment. Jn this case the equations do not represent the true
motion. To correct them we must replace the unknown friction
F by fiR, and remove the geometrical equation which expresses
the fact that there is no slipping between the bodies. The ecjua-
tions must now be again solved on this new supposition. It is of
course possible that another change may take place. If at any
instant the velocities of the points of contact become equal to
each other, all the possible friction may not be called into play.
At that instant the friction ceases to be equal to fiR and becomes
again unknown in magnitude and direction.
Discontinuity may also arise in other ways. When, for example,
one body is sliding over another, the friction is opposite to the
direction of relative motion, and numerically equal to .he normal
reaction multiplied by the c efficient of friction. If then, during
the course of the motion the direction of the normal reaction
should change sign, while the direction of motion remains un-
altered ; or if the direction of motion should change sign Awhile
the normal reaction should remain unaltered, the sign of the
coefficient of friction must be changed. This may modify the
dynamical equations and alter the subsequent motion. The same
cause of discontinuity operates when a body moves in a resisting
medium, when the law of resistance is an even function of the
velocity, or any function which does not change sign when the
direction of motion is changed.
In some cases the motion may be rendered indeterminate by
the introduction of friction. Thus, we have seen in Art. Ill, that
when a body swings on two hinges, the pressures on the hinges
resolved in the direction of the straight line joining them cannot
be found. The sum of these components can be found, but not
either of them. But there was no indeterminateness in the
motion. If however these hinges were imperfectly rough, there
would be two friction couples, one at each hinge, acting on the
body. The common axis of these couples would be the straight
line joining the hinges. The magnitude of each would be- equal
to the pressure resolved along its axis multiplied by a constant
depending on the roughness of the hinge. If the hinges were
unequally rough, the magnitude of the resultant couple would
depend on the distribution of the pressure on the two hinges. In
such a case the motion of the body would be indeterminate.
147. A homogeneous sphere is placed at rest on a rough inclivnd plane, the
copffieient offrictio.t being /i, determine whether the sphere tvill slide or roll.
Let F be the friction required to make the sphere roll. The problem then
F
becomes the same as that discussed in Art. 133. We have, therefore, - =^ tan a,
whore a is the inclination of the plane to the horizon.
126
MOTION IN T'VO DIMENSIONS.
m
If then I tan a be not greater than /tc, the solution given in the article referred
to is the correct one. But it /k^ tan a the sphere will begin to slide on the
inclineJ plane. The subsequent motion will be given by the equations
m j-j =nig sina-fxR
0= -mg cos a + R
vw -, ji- + vik^ ^ = mga f in o
dfi
dt^
whence we have, rememberiug that k^ = i a',
d-x
^.i= 9 (sin a -n cos a)
d^0 . g
^, = |M-cosa
Since the snhere starts from rest, we have by integration
aj^Jf^t" (sino-/xcosa) \
e=^fji.^t'^ cos a j
The Velocity of the point of the sphere in contact with the plane is
dx 10 ^ , . . .
But since, by hypothesis, n is less than f tan a, this velocity can never vanish.
The friction therefore will never change to rolling friction. The motion has thus
been completely determined.
148. A homogeneous sphere is rotating about a horizontal diameter, aiid is
gently placed on a rough horizontal plane, the coefficient of friction being /i. Deter-
mine the subsequent motion.
Since the velociiy of the point of contact with the horizontal plane is not zero,
the sphere will evidently begin to slide, and the motion of its centre will be along a
straight line perpendicular to the initial axis of rotation. Let this straight line be
taken as the axis of x, and let 6 be the angle between the vertical and that radiiis of
the sphere which was initially vertical. Let a be the radius of the sphere, mJc^ its
moment of inertia about a diameter, and Q the initial angular velocity. Let R be
the normal reaction of the plane. Then the equations of motion are clearly
d^x
m^^ = y.R
Q=mg-R \
m:
(1),
whence we have
d^
dt^''
■M
d^e , g
(2).
,d9
Integrating, and remembering that the initial value of — is 0, we have
x = \ixgt'^ )
* a
.(S).
referred
) on the
ir vanish,
has thus
', and is
. Deter-
not zero,
along a
line be
radius of
mk^ its
Jet R be
... (1).
(2).
IMPERFECT FRICTION.
127
But it is evident that these equations cannot represent the whole motion, for
they would make j- , the velocity of the centre of the sphere, increase continually.
This is quite contrary to experience. The velocity of the point of the sphere in
contact with the plane is
dx do .-. ,
This vanishes at a time
ti=;
.(4).
■(6).
(8).
At this instant the friction suddenly changes its character. It now becomes
only of sufficient magnitude to keep the point of contact of the sphere at rest. Let
F be the friction required to effect this. The equations of motion will then be
dH _ -\
= m(/-R
and the geometrical equation will be a; = a9.
Differentiating this twice, and substituting from the dynamical equations, we
get F{a'+k^) = 0, and therefore F=0. That is, no friction is required to keep the
point of contact of the sphere at rest, and therefore noi:e will be called into play.
The sphere will therefore move uniformly with the velocity which it had at the
UX
time t^. Substituting the value of t^ in the expression for ^ obtained from equa-
(It
tions (3) we find that this velocity is f aO. It appears therefore that the sphere
will move with a uniformly increasing velocity for a time f — and will then move
uniformly with a velocity f aQ. It may be remarked that this velocity is independ-
ent of /x.
If the plane be perfectly rough, n is infinite, and the time t^ vanishes. The
sphere therefore immediately begins to move with a uniform velocity =f aO.
149. In this investigation the couple of rolling friction has been neglected. Its
effect would be to diminish the angular velocity. The velocity of the lowest point
of the sphere would then tend to be no longer zero, and thus a small sliding friction
will be required to keep that point at rest. Suppose the moment of the friction-
couple to be measured by Jmg, where / is a constant. Introducing this into the
equations (5) the third is changed into
mP - , = -Fa -frng,
the others re:aainiug imaltered. Solving these as before we find
We see from this that F is negative and retards the sphere. The effect of the
couple is to call into play a friction-force which gradually reduces the sphere to
rest.
As tho sphere moves in the air we may wish to determine the effect of its resist-
ances. The chief part of this resistance may be pretty accurately represented by a
t V
li
I?
■J i
\m
mm
128
MOTION IN TWO DIMENSIONS.
force m/3 - acting at the centre in the direction opposite to motion, v being the
velocity of the sphere and /3 a constant whose magnitude depends on the density of
the air. Besides this there will be also a small friction between the sphere and air
whose magnitude is not known so accurately. Let us suppose it to be represented
by a couple whose moment is myv^ where 7 is a constant of small magnitude. The
equations of motion can be solved without difficulty, and we find
tan-x..v/to_tan- 7^^= .^^^f^,,
\' fg V fg a' + k*
where V is the velocity of the sphere at the epoch from which t is measured.
150. In order to determine by experiment the magnitude of
rolling friction, let a cylinder of mass M and radius r be placed on
a rough horizontal plane. Let two weights whose masses are P
and P + phe suspended by a fine thread passing over the cylinder
and hanging down through a slit in the horizontal plane. Let F
be the force of friction, L the couple at the point of contact A of
the cylinder with the horizontal plane. Imagine p to be at first
zero, and to be gradually increased until the cylinder just moves.
When the cylinder is on the point of motion, we have by resolving
horizontally F= aud by taking moments L =pgr. Now in the
experiments of Coulomb and Morin p was found to vary as the
novmal pressure directly, and as r inversely. When p was great
enough to set the cylinder in motion. Coulomb found that the
acceleration of the cylinder was nearly constant, and thence we
may conclude that the rolling friction was independent of the
velocity. M. Morin found that it was not independent of the
length of the cylinder.
The laws which govern the couple of rolling friction are similar
to those which govern the force of friction. The magnitude is
just sufficient to prevent rolling. But no more than a certain
amount can be called into play, and this is called the limiting
rolling couple. The moment of this couple bears a constant ratio
to the magnitude of the normal pressure. This ratio is called the
coefficient of rolling friction. It depends on the materials in con-
tact, it is independent of the curvatures of the bodies, and, in
some cases, of the angular velocity.
No experiments seem to have been made on bodi-,;^ which
touch at one point only and have their curvatures in all direc-
tions unequal. But since the magnitude of the couple is indepen-
dent of the curvature, it seems reasonable to assume that the
axis of the rolling couple, when there is no twisting couple, is the
instantaneous axis of rotation.
In order to test these laws of friction let us compare the
results of the following problem with experiment.
> being the
) density of
ere and air
represented
tude. The
ired.
nitude of
placed on
ses are P
! cylinder
!. Let F
tact A of
)e at first
st moves.
resolving
)w in the
ry as the
was great
that the
hence we
it of the
at of the
re similar
fnitude is
a certain
3 limiting
tant ratio
called the
Is in con-
3, and, in
i-^ which
all direc-
I indepen-
that the
pie, is the
ipare the
IMPERFECT FRICTION. 129
151. A carnage on n pairs of wheels is dragged on a level horizontal plane by a
horizontal force 2P tcith uniform motion. Find the magnitude of P.
Let the rndii of the wheels be respectively r,, r^, &c., their weights w^, u\, &o.,
and the radii of the axles p^, p^, &o. Let 2Wbe the whole weight of the carriage,
2Qp 2Qjj, &c. the pressures on the several axles, so that W= ZQ. Let the pressures
between the wheels and axles be Ri, i?a, &c. and the pressures on the ground Ii\,
R\, &c. Let C be the common centre of any wheel and axlo, P their point of con-
tact, and A the point of contact of the wheel with the ground. Let the angle
ACP = 6 supposed positive when P is behind AC. Let p. be the coeflacient of the
force of sliding friction at P and / the coefficient of the couple of rolling friction
at A. The equations of equilibrium for any wheel, foimd by resolving vertically
and taking moments about A, are
R'=q + ic (1),
p.R{rQos0-p)-Rr&iue=fR' (2).
The friction force at A does not appear because we have not resolved horizontally.
The equations of equilibrium of the carriage, foimd by resolving vertically and hori-
zontally, are
Rco^0 + p.Rmne = Q (3),
"L (R sin. e-iiR COS d)-\-P = Q (4).
The effective forces have been omitted because the carriage is supposed to move
uniformly, so that the M-r.ol the carriage and the mh"^ — of the wheel are both
Up (It
zero.
The first three of these equations give by eliminating R and R'
u(costf--)-sintf ,, .
costf+/isintf r\ QJ ^
This gives the value of 9. In most wheels - and ^ are both small as well as /. In
° r Q
such a case /* cos tf - sin is a small quantity. If therefore ja=tane we have d=e
very nearly.
The third and fourth of these equations give by eliminating R
p^^ HCOBO-Bine
p. sin + cos
(/tsinff + cos^r r )
by equation (5). If - be small, it will be sufficient to substitute for 6 in the first
term its approximate value e. This give"?
i.=ZJ,m.5«^/«ll"i (6).
Here we have neglected terms of the order y-jQ'
If all the wheels are equal and similar we have, since 2Q= TV,
P=sinejt7+/^ (7).
Thus the force required to drag a carriage of given weight with any constant velocity
is very nearly independent of the number of wheels.
R. D. 9
! P\
W:
\m
130
MOTION IN TWO DIMENSIONS.
In a gig tho wheels are usually larger than in a four-wheel carriage, and there-
fore the force of traction is usually less. In a four-wheel carriage the two fore
wheels must be small in order to pass under the carriage when turning. This will
cause the term sin e - Qj in the expression for P containing the radius r^ of the
fore wheel to be large. To diminish the effect of this term, tho load should bo eo
adjusted that its centre of gravity is nearly over tho axle of the large wheels, tho
pressure Qj in tho numerator of this term will then be small.
A variety of experiments were made by a French engineer, M. Morin, at Metz in
the years 1837 and 1838, and afterwards at Coiu-bevoie in 1839 and 1841, with a
view to determine with the utmost exactness the force necessary to drag carriages
of different kinds over the ordinary roads. These experiments were undertaken by
order of the French Minister of War, and afterwards under the directions of the
Minister of Public Works. The eiiect of each element was determined separately,
thus the same carriage was loaded with different weights to determine the effect of
pressure and dragged on the same road in tho same state of moistvire. Then the
weight being the same, wheels of different radii but the same breadth were used, and
80 on.
The general results were that for carriages on equal wheels, the resistance varied
as the pressure directly and the diameter of the wheels inversely, and was independ-
ent of the number of v jieels. On wet soils the resistance increased as the breadth
of the tire was decreased, but on solid roads the resistance was independent of the
breadth of the tire. For velocities which varied from a foot pace to a gallop, the
resistance on wet soils did not increase sensibly with the velocity, but on solid roads
it did increase with the velocity if there wore many inequalities on the road. As
an approximate result it was found that the resistance might be expressed by a
formula of the kind a + bV, where a and b are two constants depending on the nature
of the road and the stiffness of the carriage, and V is the velocity.
M. Morin's analytical determination of the value of P does not altogether agree
with that given here, but it so happens that this does not materially affect the
comparison between theory and observation. See his Notions Fondamentales de
Mecanique, Paris 1855. It is easy to see that M. Morin's experiments tend to con-
firm the laws of rolling friction stated in a previous article.
Ex. 1. A homogeneous sphere is projected without rotation directly up an
imperfectly rough plane, the inclination of which to the horizon is a, and the
coefficient of friction fi. Show that the whole time during which the sphere
ascends the plane is the same as if the plane were smooth, and that tLe iiuie
2 tan a
during which the sphere slides is to the time during which it rolls as 1 :
7 fi
Ex, 2. A homogeneous sphere of mass M is placed on an imperfectly rough
table, the coefficient of friction of which is fi. A particle of mass m is attached to
the extremity of a horizontal diameter. Show that the sphere will begin to roll or
slide according as u is greater or less than „ .,„ ,„., s .
this value, show that the sphere will begin to roll.
If fjL be equal to
Ex. 3. A rod AB has two small rings at its extremities which slide on two
rough horizontal rods Ox, Oy at right angles. The rod is started with an angular
velocity when very nearly coincident with Ox, show that if the coefficient of fric-
IMPULSIVE FORCES.
131
and there.
le two foro
This will
B Ti of the
ihoultl be Fo
wheels, the
, at Metz in
1841, with n
rag carriages
clertaken by
itions of the
1 separately,
the effect of
). Then the
ere used, and
stance varied
ras independ-
i the breadth
odent of the
a gallop, the
on solid roads
he road. As
pressed by a
on the nature
ogether agree
illy affect the
amentales de
tend to con-
rectly up an
a, and the
the sphere
hat iliii vlme
2 tan a
rfectly rough
s attached to
!gin to roll or
be equal to
slide on two
h an angular
icient of fric-
tion is less than ^2, the motion of the rod is given by fl = ■ ^ log f 1 + ' ^ , J until
2
tan 6=-, and that when the rod reaches Oy, its angular velocity is w, where
ill
What is the motion if /u' > 2 ?
(2-/i«)(4-/x'0*
On Impulsive Forces.
152. In the case in which the impres.sed forces are impulsive
the general principle emuiciated in Art. 123 of this chapter re-
quires but slight modification.
Let {u, v), {ii, v) be the velocities of the centre of gravity of
any body of the system resolved parallel to any rectangular axes
respectively just before and just after the action of the impulses.
Let CO and &>' be the angular velocities of the body about the centre
of gravity at the same instants. And let MJc' be the moment of
inertia of the body about the centre of gravity. Then the effective
forces on the body are equivalent io two forces measured by
M{u' — u) and M{v' — v) acting at the centre of gravity parallel
to the axes of co-ordinates together with a couple measured
by M¥{w'-(o).
The resultant effective forces of all the bodies of the system
may be found by the same rule. By D'Alembert's principle
these will be in equilibrium with the impressed forces. The
equations of motion may then be found by resolving in such
directions and taking moments about such points as may be found
most convenient.
In many cases it will be found that by the use of Virtual
Velocities the elimination of the unknown reactions may be
effected without difficulty.
153. A string is wound round the circumference of a circular reel, and the free
end is attached to a fixed point. The reel is then lifted up and let fall so that at the
moment when the string becomes tight it is vertical, and a tangent to the reel. The
whole vwtion being supposed to take place in one plnne, determine the effect of the
impulse.
The reel in the first instance falls vertically without rotation. Let v be the
velocity of the centre at the moment when the string becomes tight ; v', u the
velocity of the centre and the angular velocity just after the impulse. Let T be the
impulsive tension, mk^ the moment of inertia of the reel about its centre of gravity,
a its radias.
In order to avoid introducing the imknown tension into the equations of motion,
let us take moments about the point of contact of the string with the reel ; we then
have
m(v'-v)a + mKW=0 (1).
y— 2
^1
132
MOTION IS TWO DIMENSrONfl.
JuHt after the impact tlio part of the reel in contact with the string has no
velocity.
Henco v'-aw'^O (2).
av
Solving these we have w' =
a-+k-'
If the reel be a homogeneous cylinder
2v
k^ = -x , and in this case wo have w'= ^ , f = r,
impulsive tension, we have resolving vertically
7/i(v'~v)= - T.
If it bo required to find the
(3).
Hence
To find the subsequent motion. The centre of the reel bcyins to descend verti-
cally, and there is no horizontal force on it. Hence it will continue to descend in
a vertical straight lino, and throughout all the subsequent motion the string is
vertical. The motion may therefore be easily investigated as in Art. 18.S. If we
put o = 7i I and let F=tho finite tension of the string, it may be shown that F=one-
2
third of tlie weight, and that the reel descends with a uniform acceleration = 5 f;.
o
The initial velocity of the reel has been found in this article =v', so that the space
1 2
descended in a time t after the impact is =v't + - . -yt^.
Ex. 1. An inelastic sphere of radius a sliding on a smooth horizontal plane
impinges on a fixed rough point at a height c above the plane, show that if the
velocity of the sphere bo a/ — / --^j— > it will just roll over the point.
Ex. 2. A rectangular parallelepiped of mass 3j», having a square base ABCD,
rests on a horizontal plane and is moveable about CD as a hinge. The height of
the solid is 3a and the side of the base a. A particle m moving with a horizontal
velocity v strikes du'ectly the middle of that vertical face which stands on ^ S and
lodges there without penetrating. Show that the solid will not upset unless
^^ fja. [King's Coll.]
v'>
9
154. Four equal rods each of length 2a and mass m are freely jointed so as to
form a rhombus. The system falls from rest with a diagonal vertical under tlie action
of gravity and strikes against a fixed horizontal inelastic plane. Find the subse-
quent motion.
Let AB, BC, CD, DA be the rods and let .4C be the vertical diagonal impinging
on the horizontal plane at A. Let V be the velocity of every point of the rhombus
just before impact and let a bo the angle any rod makes with the vertical.
Let u, V be the horizontal and vertical velocities of the centre of gravity and u
the angular velocity of either of the upper rods just after impact. Then the
effective forces on either rod are equivalent to the force m [v - V) acting vertically
and mu horizontally at the centre of gravity and a couple mk^u tending to increase
the angle a. Let R be tho impulse at C, the direction of which by the rule of
symmetry is horizontal. To avoid introducing the reaction at £ into onr equations,
let us take moments for the rod BC about B and we have
mJc^u + m (y - V)a sin a - mua cos a = - ]i.2a cos a (1).
ug has no
(2).
UB cyliniler
to find the
.(3).
jscend verti-
deHcend in
he string ia
IbH. If we
that F=one-
2
eration = Kf/'
liat the space
izontal plane
ow that if the
at.
hase ABCD,
lie height of
a horizontal
on ^B and
upset unless
inted so as to
der tlie action
ind the subse-
al impinging
the rhomhus
cal.
gravity and w
it. Then the
ing vertically
g to increase
ly the rule of
owe eqxiations,
(!)•
IMPULSIVE FORCES. 133
Each of the iower rods will begin to tarn round its extremity A aa a fixed point.
If w' bo its angular velocity juHt after impact, the moment of tlio momentum about
A just after impact will bo m(ifc- + a')w' and just before will be mrasino. The
difference of these two is tlio moment about A of the effective forces on either of
the lower rods. Wo may now tiiiio moments about A for the two rods AB, BO
together and wo have
m (P + a") w' - m Va sin a - mk^u + m (u - V)a sin a + mw . 3a cos a = TJ . 4a cos o . . . (2),
The geometrical equations may be found thus.
Since the two rods must make equal angles with the vertical during the whole
motion we have
w' = w (3).
Again, since the two rods are connected at // the velocities of the extremities of
the two rods must bo the same in direction and magnitude, liosolviug those hori-
zontally and vertically, we have
« + aw cos o = 2aw' cos o (4),
t>-acosina = 2aw'8iuo (5).
These five equations are sufficient to determine the initial motion.
Eliminating R between (1) and (2), substituting for m, v, u' in terms of u from
the geometrical equations, we find
_3 Tsino
"~2 ' all + aBiu^ia' ^"''
In this problem we might have avoidoil ! he intvoductiou of the unknown reaction
R by the use of Virtual Velocities. Sui^oso wo give tho system such a displace-
ment that tho incliuatiou of each rod to the vortical is increased by the same
quantity Sa. Then tho principle of Virtual Velocities gives
»iA'«5o - m [v - V)S (3a cos a) + mu5 (a sin o) + m (k" + a^) w'5a + m Vd {a cos a) = 0,
which reduces to
(2^-3 + rt^) w - Va sin a + 3 (v - F) a sin a + ua cos c = 0,
and the solution may be continued as before.
Ex. 1. Prove that the rhombus loses by the impact , — „ . ■- of its
1 + 3 sm* a
momentura.
Ex. 2. Show that the direction of the impulsive action at tho hinges B or D
makes with the horizon an angle whose tangent is — .
tan a
To find the subsequent motion. This may be found very easily by the method of
Vis Viva. But in order to illustrate as many modes of solution as iiossible, we
shall proceed in a different manner. The effective forces on either of the
upper rods will be represented by tho differential coefficients m-j-, m . , mk^^,
and the moment for either of tho lower rods will be m (P + a*) -z- . Let d be the
angle any rod makes with tho vertical at the time t. Taking moments in the same
way as before, we have •
mP -,+ ni -rr a sin ^ - ?» -7- a cos 6= - B .2a cos 6 + t)wa sin (1)',
dt at at " ^ "
m (k^ + a^)-, — mk^ -r -t- wi-p a sin ^ + m -.- . 3a cos = 7? . 4a cos + Imna sin ^. . . (2)'.
dt dt dt dt o \ /
i
\-\
134
MOTION IN TWO DIMENSIONS.
The geometrical equations are the same as those given above, with d written
for a.
Eliminating R and substituting for u, v, we get
do
multiplying both sides by u=-7- and integrating, we get
{2 (i« + a- ; i- 8a« sin2 6] w^ = C - Sffa cos $.
Initially when 0=a, u has the value given by equation (6). Hence we find that
the angular velocity w when the inclination of any rod to the vertical is is
given by
{l + 3sbxH)i^=^l . - ^^f.°-. +^(coBa-cosg).
' 4a^ 1 + d sm* a a
Ex, 1. A square is moving freely about a diagonal with angular velocity u,
when one of the angular po'nts not in that diagonal becomes fixed; determine the
impulsive pressure on the f-.cd point, and show that the instantaneous angular
velocity will be - . [Christ's Coll.]
Ex. 2. Three equal rods placed in a straight line are jointed by hinges to one
another; they move with a velocity v perpendicular to their lengths; if the middle
point of the middle one become suddenly fixed, show that the extremities of the
other two will meet in a time -^r— , a being the length of each rod. [Coll. Exam.]
yv
Ex. 3. The points ABCD are the angular points of a square; AB, CD are two
equal similar rods connected by the string BO. The point A receives an impulse
in the direction AD, show that the initial velocity of A is seven times that of the
poin*. D. [.Queens' CollJ
Ex, 4. A series of equal beams AB, BC, CD is connected by hinges; the
beams arc placed on a smooth horizontal plane, each at right angles to the two
adjacent, so as to form a figure resembling a set of steps, and an impulse is given
at the end A along A B : determine the impulsive action at any hinge. [Math.
Tripos.]
Result. If Xn be the impulsive action at the »"■ angular point, show that
^ir^i - 5^sn+a - 2X2,1+3 - and Zjn+j - 5Z8„+i - iX^^ = 0. Thence find Z„.
155. A free lamina of any form is turning in its own plane abotit an imtanta-
neous centre of rotation S and impinges on an obstacle at P, situated in the straight
line joining the centre of gravity G to S. To find the point P wlien tlie magnitude
of the blow is a maximum*.
First, let the obstacle P be a fixed point.
Let GP=x, and let It be the force of the blow, l^st SO=h, and let w, 0/ be the
angular velocities about the centre of gravity before and after the impact. Then hu
* Poinsot, Surla percussion des corps, Liouvilles Journal, 1857; translated in
the Annals of Philosophy, 185*^.
IMPULSIVE FORCES.
135
written
e find that
al is is
velocity w,
srmine the
as angular
nges to one
the middle
ities of the
11. Exam.]
CD are two
an impulse
that of the
linges; the
to the two
se is given
ge. [Math.
show that
m imtanta-
the straight
magnitude
w, (■/ he the
Then ha
ranslated in
is the linear velocity of Q just before the impact; let v' be its linear velocity just
after the impact. We have the equations
-Rx
(i).
(2).
w — w =
Ml
and supposing the point of impact to he reduced to rest,
v' + xu)'=0
Substituting for u/ and v' from (1) in equation (2), we get
»= + &■'
This is to be made a maximum. Equating to zero its differential coefficient
with respect to x, we get
x^ + 2hx-l?=0 •. (3);
One of these values of x is positive and the other negative. Both these corre-
spond to viaximum points of percussion, hut opposite in direction. Thus there is a
point P with which the body strikes in front and a point P' with which it strikes
in rear of its own translation in space more forcibly than with any other point.
Ex. 1. Show that the two points P, P' are equally distant from S, and if be
the centre of oscillation with regard to /S as a centre of suspension, SP^=SO . SO.
Ex. 2. If P be made a point of suspension, P' is the corresponding centre of
oscillation. Also PP' is harmonically divided in and 0.
Ex. 3. The magnitudes of the blows at P, P' are inversely proportional to their
distances from G.
Secondly, let the obstacle be a free particle of mass m.
Then, besides the equations (1), we have the equation of motion of the particle
m. Let V be its velocity after impact,
• • r — — •
m
The point of impact in the two bodies will have after impact the same velocity,
hence instead of equation (2) we have V'=v'+xu'.
Eliminating w', v', V, we get
(M + m) ^-i + mx"
This is to be made a maximum. Equating to zero its differential coefficient
with respect to a;, we find
;= - 7t± ^h^ + k^ {^'^m) <*^'
This point does not coincide with that found when the obstacle was fixed, unless
m is infinite. To find when it coincides with the centre of oscillation, we must put
k^=xh. This gives
M _x + h
~h~
m
, ov \il=x + h\iQ the length of the simple equivalent
pendulum,— ■=-, Since F'= — , it is evident that when i? is a maximum V
m h m
is a maximum. Hence the two points found by equation (4) might bo called the
centres of greatest communicated velocity.
^i
138
MOTION IN TWO DIMENSIONS.
There are otber singiilar points in a moving body whose positions may be found;
thus Vie might inquire at what point a body must impinge against a fixed obstacle,
that first the linear velocity of the centre of gravity might be a maximum, or
secondly, that the angular velocity might be a maximum. These points, respec-
tively, have been called by Poinsot the centres of maximum Reflexion and Conver-
sion. Beferriug to equations (1), we see that when v' is a maximum R is either a
maximum or a minimum, and hence it may be shown that the first point coincides
with the point of gieatest impact. When u' is a maximum, we have to make
a - ryr« = maximum.
Mir
Substituting for H, this gives «* - 2 t- oj - /;*= 0.
If be the
centre of oscillation, we have GO_=j . Let this length be represented by h'. Then
this equation becomes
x^-2h'x-k^=0 (5).
The roots of this equation are the same functions of h' and I- that those of
equation (3) are of k and k, except that the signs are opposite. Now <S and are
on opposite sides of 0, hence the positions of the two centres of maximum Con-
version bear to and the same relation that the positions ot the two centres of
maximum Beflexion do to S and G. If the point of suspension be changed from S
to 0, the positions of the centr«s of maximum Beflexion and Conversion are inter-
changed.
Ex. A free lamina of any form is turning in its own plane about an instanta-
neous centre of rotation S and impinges on a fixed obstacle P, situated in the
straight line joining the centre of gravity G to S. Find the position, of P, first, that
the centre of gravity may be reduced to rest, secondly,, that its. velocity after impact
may be the same as before but reversed in direction.
Result. In the first case, P coincides either with G or with the centre of oscil-
lation. In the second case the points x=GP are fovmd from the equation
where(S(?=A. [Poinsot.]
k^ k^ ^
156. Two bodies impinge on each other, to explain tlie nature
of the action which takes place between them.
When two spheres of any hard material irapipge on each
other, they appear to separate almost immediately, and a finite
change of velocity is generated in each by their mutual action.
Thi^ sudden change of velocity is the characteristic of an im-
pulsive force. Let the centres of gravity of the spheres be
moving before impact in the same straight line with velocities
u and V. Then after impact they will continue to move in the
same straight line, and let u, v be the velocities. Let m, i»i' be
the masses of the spheres, R the action between them, then we
have by Article 152,
R
u
u = —
V —v =
m
R
a).
IMPULSIVE FORCES.
137
(1).
These equations are not sufficient to determine the three quan-
tities td, V, R. To obtain a third equation we must consider what
takes place during the impact.
Each of the balls will be slightly compressed by the other, so
that they will no longer be perfect spheres. Each also will, in
general, tend to return to its original shape, so that there will be
a rebound. The period of impact may therefore be divided into
two parts. First, the period of compression, while the distance
between the centres of gravity of the two bodies is diminishing,
and secondly the period- of restitution, while the distance between
the centres of gravity is increasing. At the termination of this
second period the bodies separate.
The arrangement of the particles of a body being disturbed by
impact, we ought to determine the relative motions of the several
parts of the body. Thus we might regard each body as a collec-
tion of free particles connected by their mutual actions. These
particles being thus set in motion might continue always in motion
oscillating about some mean positions.
It is however usual to assume that the changes of shape and
structure are so small that the effect in altering the position of the
centre of gravity and the moments of inertia of the body may be
neglected, and that the whole time of impact is so short that the
motion of the body in that time may be neglected. If for any
bodies these assumptions are not true, the effects of their impact
must be deduced from the equations of the second order. We
may therefore assert that at the moment of greatest compression
the centres of gravity of the two spheres are moving with equal
velocity.
The ratio of the magnitude of the action between the bodies
during the period of restitution to that during compression is
found to be different for bodies of different materials. In some
cases this ratio is so small that the force during the period of re-
stitution may be neglected. The bodies are then said to be inelastic.
In this case we have just after the impact u—v. This gives
„ mm / K , / iniu + wi'i;
it= — . — r(M — t'), whence u =
wi + m
m-Vm
If the force of restitution cannot bo neglected, let 22 be the
whole action between the balls, R^ the action up to the moment
of greatest compression. The magnitude of R must be found by
experiment. This may bo done by determining the values of %i
and V , and thus determining R by means of equations (I). These
experiments were made in the first instance by Newton, and the
result is that -=7- is a constant ratio depending on the material of
the balls. Let this constant ratio be called I + e. The quantity
n
■ i
,1
133
MOTION IN TWO DIMENSIONS.
u
e is always less than unity, in the limiting case when e = 1 the
bodies are said to be perfectly elastic.
The value of e being supposed known the velocities after
impact may be easily found. The action E^ must be first calcu-
lated as if the bodies were inelastic, then the whole value of R
may be found by multiplying this result by 1 + e. This gives
i2 =
mm
-r{u-v){\+e),
m + m
whence u and v may be found by equations (1).
157. As an example, let us consider how the motion of the reel discussed in
Art. 153 would be affected if the string were elastic.
Since the point of the reel in contact with the string has no velocity at the
moment of greatest compression, the impulsive tension found in the article referred
to, measures the whole momentum communicated to the reel from the beginning of
the impact up to the moment of greatest compression. By what has been said in
the last article, the whole momentum communicated from the beginning to the
t^'rmination of the period of restitution will be foimd by multiplying the tension
found in Art. 153 by i + e, if c be the measm-e of the elasticity of the string. This
gives
The motion of a reel acted on by this known impulsive force is easily found.
Eesolving vertically we find
m{v' -v)= -\mv(l + e).
Taking moments about the centre of gravity
mWu' = \ mva (1 + c) ,
whence v' and w' may be found.
Ex. A uniform beam is balanced about a horizontal axis through its centre
of gravity, and a perfectly elastic ball is let fall from a height h on one extremity ;
determine the motion of the beam and ball.
Remit. Let M, m be the masses of beam and ball, 2a = length of beam, V, V
the velocities of ball at the moments just before an ' after impact, w' the angular
velocity of the beam. Then
6mV „. „ Sm-M
« =
(M+Zm)a'
V'=V.
Zm + M'
158. Hitherto we have only considered the impulsive action
normal to the common surface of the two bodies. If the bodies
are rough there will clearly be an impulsive friction called into
play. Since an impulse is only the integral of a very great force
acting for a very short time, we might suppose that impulsive
friction obeys the laws of ordinary friction. But these laws are
founded on experiment, and we cannot be sure that they are
correct in the extreme, case in which the forces are very great.
This point M. Morin undertook to determine by experiment at
the express request of Poisson. He found that the frictional
IMPULSIVE FORCES.
139
found.
impulse between two bodies which strike and slide bears to the
normal impulse the same ratio as in ordinary friction, and that
this ratio is independent of the relative velocity of the striking
bodies. M. Morin's experiment is described in the following
example.
159. A box AB which can be loaded with shot so as to be of any proposed
weight has two vertical beams AC, BD erected on its lid ; CD is joined by a cross
piece and supports a weight equal to mg attached to it by a string. The weight of
the loaded box is Mg. A string AEF passes horizontally from the box over a
smooth pulley E and supports a weight at F equal to (M+m)gn. The box can
slide on a horizontal plane whose coefficient of friction is n, and therefore having
been once set in motion, it moves in a straight line with a uniform velocity which
we will call V, Suddenly the string supporting m^r is cut, and this weight falls into
the box and immediately becomes fixed to ihe box. Show that an impulsive fric-
tion is called into play between the box un i the horizontal plane. Prove that if
the velocity of the box immedia';*i!y after the impiii^o is again equal to F, the coeffi-
cient of impulsive friction is equal to Cat of finite friction. Find also the whole
space passu! over by the box in any time which includes the impact.
160. When two inelastic bodies impinge on each other at
some point A, the points in contact at the beginning of the im-
pact have a relative velocity both along the common tangent
plane at A and also along the normal. Thus two reactions will be
called into play, a normal force and a friction, the ratio of these
two being /*, the coefficient of friction. As the impact proceeds
the relative normal velocity gets destroyed, and is zero at the
moment of greatest compression. Let R be the whole momentum
transferred normally from one body to the other in this very
short time. This force R is an unknown reaction, to determine it
we have the geometrical condition that just after impact the
normal velocities of the points in contact are equal. This condi-
tion must be expressed in the manner explained in Art. 127.
The relative sliding velocity at A is also diminished. If it
vanishes before the momeat of greatest compression, then during
the rest of the imj)act, only so much friction is called into play,
and in such a direction, as is necessary (if any be necessary) to
prevent the points in contact at A from sliding, provided that
this amount is less than the limiting friction. Let F be the
whole momentum transferred tangentially from one body to the
other. This reaction F is to be determined by the condition
that just after impact the tangential velocities of the points
in contact are equal. If, however, the sliding motion does not
vanish before the moment of greatest compression, then the whole
of the friction is called into play in the direction opposite to that
of relative sliding, and we have F—fiR. Generally we may dis-
tinguish these two cases in the following manner. In the first
case it is necessary that the values of F and R found by solving
140
MOTION IN TWO DIMENSIONS.
i
1 ■
tho equations of motion should be such that F<fili. In the
second case, the final relative velocity of the points in contact at
A must be in the same direction after impact as before. These
are however not sufficient conditions, for it is possible that,
in more complicated cases, the sliding may change, or tend to
change, its direction during the impact. See Art. 164.
161. If the impinging bodies be elastic, there may be both
a iiormal reaction and a friction during the period of restitution.
Sometimes we shall have to consider this stage of the motion as a
separate problem. The motion of the bodies at the moment of
greatest compression having been determined, these are to be
regarded as the initial conditions of a new state of motion under
different impulses. The friction called into play during restitu-
tion must follow the same laws as that during compression. Just
as before, two cases will present themselves, either there will be
sliding during the whole period of restitution or only during a
portion of it. These are to be treated in the manner already
explained.
162. There is one very important difference between the
periods of compression and restitution. During the compression
the normal reaction is unknown. The motion of the body just
before compression is given, and we have a geometrical equation
expressing the fact that the relative normal velocity of the points
in contact is zero at the termination of the period of compression.
From this geometrical equation we deduce the force of compres-
sion. The motion of the body just before restitution is thus
found, but the motion just after is the thing we want to deter-
mine. For this, we have no geometrical equation, but the force
of restitution bears a given ratio to the force of compression, and
is therefore knovn.
163. A spherical ball moving witlwut rotation on a smooth horizontal plane
impinges with velocity v against a rough vertical wall ivhose coefficient of friction
is iJL. The line of motion of the centre of gravity before incidence making an angle
a ivith the normal to the wall, determine the motion jutt after impact.
Let M, V be the velocities of the centre of the ball just before impact resolved
along and perpendicular to the wall in such directions that they are both positive.
Then u = F sin o, v=V cos a. Let u', v' be the velocities of the centre at any
iustf nt during the impact, resolved in the same directions, w' the angular velocity
at that instant. Let R he the normal blow from tho beginning of the impact up to
that instant, i^ tho fricLional blow. Then we have •
u'-
H
z=
F -
m
v'-
■V
=
R
m
w'
=
Fa
mk' J
IMPULSIVE FORCES.
141
;j
If the instant considcreil be any moment subsequent to that at which the tangential
velocity of the point of contact vanishes, we have u'-nu' = 0. This gives, since
A;* = 10", F=7»iy sina. Since F is independent of the moment considered, we see
that no friction is called into play after the taup'" .tial velocity of the point of con-
tact is reduced to zero.
At the moment of greatest compression the normal velocity of the point of
contact is zero, hence v' = and .•. 7J = )«rcosa. If Fhe<ixR, i.e. if f tana<j«,
these will ho the proper values of F and R, and by substituting in the equations the
motion of the sphere may be found.
But if Y tan a > /u, the sphere will separate slightly from the wall before sufficient
friction has been called into play to reduce the tangential velocity of the point of
contact to zero. In this ease we must replace F by fxR in the equations. At the
momei ^. of greatest compression we have as before v' = 0. This gives R = mv cos a.
By substituting in the equations the motion of the sphere may be found. The
initial velocity of the point of contact is easily seen to be n' -au' = v(Hm o-/«5coso).
If this were negative, the friction at the end of the impact would be acting in the
direction of relative motion, which is impossible. This solution is therefore correct
only if ^ tan a> fi.
If the sphere be imperfectly elastic, a normal force of restitution is called into
play equal to emv cos a. If then ^ mv sin ahe <fi(l + e)mv cos o, the friction neces-
sary to destroy the tangential velocity of the point of contact is less than the
limiting friction. In this case by writing i?=f7w« sin o, 11 = {1 + e) mv coa a in the
equations of motion, wo can find u', v' and w'. If ^viv sin o be >ix(l + e)mv cos o,
we must put R=:(l + e)mveoaa, F-ix{l+e)mvcosa, and the same equations will
now give u', v' and w'.
164. Two rough bodies of any form impinge on each other in
a given manner. It is required to find the motion just after
impact.
Let O, 0' be the centres of gravity of the two bodies, A the
142
MOTION IN TWO DIMENSIONS.
point of contact. Let V, V be tho resolved velocitie.s of G just
Defore impact, parallel to the tangent and normal respectively
at .4 ; u, V the resolved velocities at any time t after the com-
mencement of the impact, but before its termination. Then t is
indefinitely small. Let VL be the angular velocity of the body,
whose centre of gravity is G, just before impact, oi the angular
velocity after the interval t. Let M be the mass of the body,
k its radius of gyration about G. Let GN be a perpendicular
from G on tho tangent at A, and let AN = x, NG=y. Let
accented letters denote corresponding quantities for the other
body.
Let R be the whole momentum communicated to the body M
in the time t of the impact by the normal pressure, and let F be
the momentum communicated by the frictional pressure. We
shall suppose these to act on the body whose mass is M in the
directions NG, NA respectively. Then they must be supposed to
act in the opposite directions on the body whose mass is M'.
Since II represents the Avhole momeutum communicated to
the body M in the direction of the normal, the momentum com-
municated in the time dt is dR. As the bodies can only push
against each other, dR must be positive, and, by Art. 12(j, when
dR vanishes, the bodies separate. Thus the magnitude of R may
be taken to measure the progress of the impact. It is zero at the
beginning, gradually increases throughout, and is a maximum at
the termination of the impact. It will be found more convenient
to choose R rather than the time t as the independent variable.
The dynamical equations are by Art. 152
M{u-U)=-F
M{v-V)=R ' (1),
]\W {(o - n) = Fi/ + Rx .
M'{u'-U')=F
M'{v'-V')=-R [ (2).
The relative velocity of sliding of the points in contact is by
Art. 127
S = u — yoi — u — y'a>' (.3),
and the relative velocity of compression is by the same article
C=v'+a;'o)' — v — xm (4).
Substituting in these equations from the dynamical equations
we find
S=8,-aF-hR (5),
G just
)ectively
le com-
'hen t is
le body,
angular
le body,
ndicular
: y. Let
le other
body M
let i^ be
re. We
1/ in the
iposed to
r.
cated to
um com-
11 ly push
20, when
»f R may
ro at the
imum at
nvenient
riable.
...(1),
Lct is by
...(3),
tide
...(4).
quations
.(6).
IMPULSIVE FORCES. 143
C=C,-bF-a'R (G),
where 8,= U- yn- U' -y'Cl' (7),
C, = V'+x'il' - V-xil (8),
_ 1 1 j/ y" .Q.
'^~iM'^M''^MIc''^ilW' ^^^'
""^M^ M'^TW^MV' ^^ ^'
M¥ M'k' ^ ''
These may be called the constants of the impact. The first
two 8^, Cq represent the initial velocities of sliding and com-
pression. These we shall consider to be positive ; so that the
body If is sliding over the body M' at the beginning of the com-
pression. The other three constants a, a, h are independent of
the initial motion of the striking bodies. The constants a and a'
are essentially positive, while h may have either sign. It will be
found useful to notice that aa > V.
165. "When 6=0, the discussion of these equations, as in
Art. 163, does not present any difficulty, but in the general case
it is more easy to follow the changes in the forces, if we adopt a
graphical method. Let us draw two lengths AB, AF along the
normal and tangent at A in the directions NO, -4 -AT respectively,
to represent the magnitudes of R and F at any moment of the
impact. Then if we consider AR and AF to be the co-ordinates
of a point P, referred to AR, AF as, axes of R and F, the changes
in the position of P will indicate to the eye the changes that
take place in the forces during the progress of the impact. It will
be convenient to trace the two loci determined hy 8=0, C = 0.
By reference to (5) and (6) we see that they are both straight
lines. These we shall call the straight lines of no sliding and of
greatest compression. To trace these, we must find their inter-
cepts on the axes of F and R. Take
^C= ■^, A8=^, AG' = ^\
a a b
AS' = -^
b
then *Si^', CG' will be these straight lines. Since a and a' are
necessarily positive, while b has any sign, we see that their inter-
cepts on the axes of F and R respectively are positive, while their
intercepts on the axes of R and F must have the same sign.
Since aa' > 6*, the acute angle made by the line of no sliding with
the axis of F is greater than that made by the line of greatest
compression, i.e. the former line is steeper to the axis of i^than
the latter. It easily follows that the two straight lines cannot
iU,
t I
V
f
it
144
MOTION IN TWO DIMENSIONS.
intersect in the quadrant contained by HA produced and FA
produced.
106. In the beginning of the impact the bodies slide over
each other, hence, as explained in Art. 144, the whole limiting
friction is called into play. The point P therefore moves along a
\
R
.
>
\
c
\
s-
^^
\
T
A.
S
c
straight line AL, defined by the equation F=fili, where fi is the
coefficient of friction. The friction will continue to be limiting
until F reaches the straight line 88'. If R^ be the abscissa of
this point we find 72.= — " , . This gives the whole normal
^ '^ aji + h ^
blow, from the beginning of the impact, until friction can change
from sliding to rolling. If R^ is negative, the straight lines AL
and 88' will not intersect on the positive side of the axis of F.
In this case the friction will be limiting throughout the impact.
If R^ is positive the representative point P will reach 88 '. After
this onlj so much friction is called into play as will suffice to
prevent sliding, provided this amount is less than the limiting
friction. If the acute angle which 88' makes with the axis of ^
be less than tan"^ fi, the friction dF necessary to prevent sliding
will be less than the limiting friction /ic?Z?. Hence P must
travel along 88' in such a direction that the abscissa R con-
tinues to increase positively. In this case the friction will not
again become limiting during the impact.
But if the acute angle which 88' makes with the axis of i2 be
grbater than tan'^/t, the ratio of dF to dR will be numerically
greater than fx, and more friction is necessary to prevent sliding
than can be called into play. The friction will therefore continue
to be limiting, and P, after reaching 88', must travel along a
straight line, making the same angle with the axis of R that AL
does. But this angle must be measured on the opposite side of
the axis of R, for when the point P has crossed 88' the direction
IMPULSIVE FOllCES.
145
I ■:
incl FA
ide over
limiting
along a
3 fi is the
limiting
bscissa of
5 normal
n change
lines AL
ixis of F.
impact.
After
uffice to
limiting
axis of R
it sliding
P must
R con-
will not
3 of -R be
merically
it sliding
continue
along a
that AL
side of
direction
of relative sliding and therefore the direction of friction is
changed. In this case it is clear that the friction will continue
limiting throughout the impact.
When P passes tlie straight line CC, compression ceases and
restitution begins. But the passage is marked by no peculiarity
except this. If R^ be tlie abscissa of the point at which P cro.sse8
CC, the whole impact, for experimental reasons, is .supposed to
terminate when the aKscissa of P is R,^ = R^ (1 + e), e being the
measure of the elasticity of the two bodies.
It is obvious that a great variety of cases may occur according
to the relative positions of the three straiglit lines AL, SS' and
CC. But in all cases the progress of the impact may be traced
by the method just explained, which may be briefly stated thus.
The representative point P travels along AL, until it meets SS'.
It then proceeds either along SS', or along a .straight line
making the same angle with the axis of R as AL does, but on the
opposite side. The one along which it proceeds is the steeper to
the axis of F. It travels along this line in such a direction as to
make the abscissa R increase. The complete value of R for the
whole impact is for.ud by multiplying the abscissa of the point at
which P crosses CC by 1+e. The complete value of F is the
corresponding ordinate O'f P. Substituting these in the dyna-
mical equations (1) and (2), the motion just after impact may be
easily found.
If the bodies be smooth, the straight line AL coincides with
the axis of R. The representative point P must travel along the
axis of R and the complete value of R for the whole impact is
found by multiplying the abscissa of C by 1 + e.
167. It is not necessary that the friction should keep the
same direction during the impact. The friction must keep one
sign when P travels along AL. But when P reaches SS', its
direction of motion changes, and the friction dF called into play in
the time dt may have the same sign as before or the opposite.
But it is clear that the friction can change sign only once during
the impact.
It is possible that the friction may continue limiting through-
out the impact, so that the bodies slide on each other throughout.
The necessary conditions are that either the straight line SS'
must be less steep to the axis of F than AL, or the point P
must not reach the straight line SS' until its abscissa has be-
come greater than R.^. The condition for the first case is, that
h must be greater than fia. The abscissae of the intersections
S
of AL with SS' and CC are respectively R^ = — ^ and
R. D. 10
M
irmsmmsim
116
' bfi+a
MOTION IN TWO DIMENSIONS.
The condition for tlio second case is necessary,
that i?, must be positive, and R^ either negative or positively
greater than R^ (1 + e).
168. Ex. 1. Show that the reprosentative point P as it travels in the manner
directed in the text must cross the line of greatest compression, and that the
abscissa R of the point at which it crosses this straight line must be positive.
Ex. 2. Show that the conic whose equatidii referred to the axes of R and F if)
aF^ + 2bFR + a'lP=f, where e is some constant, is an ellipse, and that the straight
lines of no sliding and greatest compression are parallel to the conjugates of the
axes of P and R respectively. Show also that the intersection of the straight lines
of no sliding and greatest compression must lie in that angle formed by the conju-
gate diameters which contains or is contained by the first quadrant.
Ex. 3. Two bodies, each turning aboiit a fixed point, impinge on each other,
find the motion just after impact.
Let 0, G\ in the figure of Art. 164, be takon as the fixed points. Taking
moments about the fixed points, the results will be nearly the same as those given
in the case considered in the text.
Initial Motions.
169. Suppose a system of bodies to be in equilibrium and
that one of the supports suddenly gives way. It is required to
find the initial motion of the bodies and the initial values of the
reactions which exist between the several bodies.
The problem of finding the initial motion of a dynamical
system is the same as that of expanding the co-ordinates of the
moving particles in powers of the time t. Let (x, y, 6) be the
co-ordinates of any body of the system. For the sake of brevity
let us denote by accents differential coefficients with regard to the
time, and let the suffix zero denote initial values. Thus x^'
cPx
denotes the initial value of -^ • ^7 Taylor's theorem we have
a; = a + <'i2 + <"^+
the term x^ is omitted because we shall suppose the system to
start from rest.
First, let only the initial values of the reactions he required.
The dynamical equations will contain the co-ordinates, their second
differential coeflficients with regard to t, and the unknown re-
actions. There will be as many geometrical equations as re-
actions. From these we have to eliminate the second differential
.(1)
INITIAL MOTIONS.
147
iccssary,
)8itively
le manner
that the
ive.
and F is
le straight
bes of the
light lines
the conju-
ach other,
). Taking
iiose given
ium and
[uired to
s of the
i^namical
of the
he the
brevity
d to the
have
..(1) :
jTstem to
'equired.
second
own re-
as re-
'erential
coeflflciv^nts r.nd find the reactions. Tlio process will bo as follows,
which is really the same as the first method of solution described
in Art. 125.
Write down the geometrical equations, differentiate each twice
and then simplify the results by substituting for the co-ordinates
their initial values. Thus, if wt use Cartesian co-ordinates, let
<f> (x, y, 6) =0 be any geometrical relation, we have since a?,' = 0,
" rfv "" '" ~
dx
dd
The process of differentiating the equations may sometimes
be much simplified when the origin has been so chosen that the
initial values of some at least of the co-ordinates are zero. We
may then simplify the equations by neglecting the squares and
products of all such co-ordinates. For if we have a term a?, its
second differential coefficient is 2 [xx" -f x'^), and if the initial
value of 03 is zero, this vanishes.
The geometrical equations must be obtained by supposing the
bodies to have their displaced position, because we require to
differentiate them. But this is not the case with the dynamical
equations. These we may write down on the supposition that
each body is in its initial position. These equations may be
obtained according to the rules given in Art. 125. The forms
there given for the effective forces admit in this problem of some
simplifications. Thus since r^ = 0, ^^ = 0, the accelerations along
and perpendicular to the radius vector take the simple forms r^ '
and rt/>o". So again the acceleration — along the normal vanishes.
If, for example, we know the initial direction of motion of the
centre of gravity of any one of the bodies, we might conveniently
r ^solve along the normal to the path. This will supply an equa-
tion which contains only the impressed forces and such tensions
or reactions as may act on that body. If there be only one re-
action, this equation will suffice to determine its initial value.
We may also deduce from the equations the values of x^',
y"> ^o"> ^^^ ^^^"^ ^y substituting in equation (1) we have found
the initial motion up to terms depending on f.
170. Secondly, let the initial motion he required. How many
terms of the series (1) it may be necessary to retain will depend
on the nature of the problem. Suppose the radius of curvature
of the path described by the centre of gravity of one of the bodies
to be required. We have
xy -yx
10—2
tavm
148
MOTION IN TWO DIMENSIONS.
4
' • I
fl
and by differentiating equation (1)
in ^
«' =a:;'< + <"f^ + ajr,^ +
X
&c. =&c.;
results which may also he obtained by a direct use of Taylor's
theorem.
If then the body start from rest, the radius of curvature is
zero. But if cc^'y^' ~" ^o "Vo" = ^> "^^ ^^^^
p = 3
v,'*'o
+yn
^0 Vo ~^o Vo
To find these differential coefficients we may proceed thus.
Differentiate each dynamical equation twice and then reduce
it to its initial form by writing for so, y, 0, &c. their initial values,
and for x, y, ff nxo. Differentiate each geometrical equation
four times and then reduce each to its initial form. We shall
thus have sufficient equations to determine x^', x^", x^^, &c., B^,
R^, iJg", &e., where M is any one of the unknown reactions. It
will often be an advantage to eliminate the unknown reactions
from the equations before differentiation. We shall then have
only the unknown coefficients w^', x^", &c. entering into the equa-
tions.
If we know the direction of motion of one of the centres
of gravity under consideration, we can take the axis of « a tangent
V*
to its path. Then we have p = ^ , where x is of the second order,
y of the first order, of small quantities. We may therefore neg-
lect the squares of x and the cubes of y. This will greatly sim-
plify the equation?. If the body start from rest we have a?/ = 0,
and if x^' = 0, we may then use the formula
171. Ex. A circular disc is hung tip by three equal strings attached to three
points at equal distances in its circumference, and fastened to a peg vertically over
the centre of the disc. One of these strings is suddenly cut. Determine the initial
circumstances of motion.
INITIAL MOTIONS.
149
Let bo the peg, AB the circle seen by an eye in its plane. Let OA bo tbe
string which is cut and C be the nuddle point of the chord joining the points of the
K
a
Taylor's
vature is
ied thus.
I reduce
il values,
equation
^6 shall
, &c., i?„,
ions. It
reactions
len have
he equa-
} centres
tangent
ad order,
'ore neg-
atly sim-
e 4' = 0,
ed to three
ically over
the initial
m
circle to which the two other strings are attadied. Then the two tensions, each
equal to T, are throughout the motion equivalent to a resultant tension R along CO.
If 2a be the angle between the two strings, we have
i2=22'cosa.
Let I be the length of OC, /3 be the angle GOO, a be the radius of the disc. Let
(«, y) be the co-ordinates of the displaced position of the centre of gravity with
reference to the origin Or x being measured horizontally to the left and y vertically
downwards. Let d be the angle the displaced position of the disc makes with AB.
By drawing the disc in its displaced position it will be seen that the co-ordinates
of the displaced position of C7 are x - 2 sin /3 cos and y-lwt\.^m\.$. Hence since
the length OC remains constant and equal to I we have
a;S + y» _ 21 sin /3 (o! cos tf + y sin tf) = Z" cos' /3.
Suppose the initial tensions only to be required. It will be sufficient to differ-
entiate this twice. Since we may neglect the squares of small quantities, we
may omit a;", put cos 0=t,e,m.$ = e. The process of differentiation will not then be
very long, for it is easy to see beforehand what terms will disappear when we equate
the differential coefficients (x', y', ff) to zero, and put for (x, ^,^) their initial values
(0, I cos ft 0). We get
^o" cos /3 = sin /3 (V -^ I cos /35o") •
This equation may also be obtained by an artifice which is often useful. The
motion of Q is made up of the motion of C and the motion of Q relatively to C.
Since C begins to describe a circle from rest, its acceleration along CO is zero.
Again, the acceleration of relatively to C when resolved along CO is QC -r^ cos ft
The resolved acceleration of G is the sum of these two, but it is also
j/o" cos p - Xff' sin ft
Hence the equation follows at once.
In this case wo require the differential equations only in their initial form.
These are
ma!o"=i?osin/3
my(^"=mg - Rq cos /3
}■
mk%' = i?o I sin /3 cos /3)
where m is the mass of the body. Substituting in the geometrical equation we find
I?0 = Wflf. jij
cos
L
i V
i ■ I
l + Ti,sin''j9cos''/3
150
MOTION IN TWO DIMENSIONS.
The tension of any string, before the string OA was cut, may be found by the
[es of Statics, and is clearly T,
change of tension can be found.
rules of Statics, and is clearly T, = 5 — — , where 7 is the angle AOO. Hence the
' 3 cos y
172. Ex. 1. Two strings of equal length have each an extremity tied to a
weight G and their other extremities tied to two points ^, J5 in the same horizontal
line. If one be cut the tension of the other is instantaneously altered in the ratio
l:2cos''?. [St Pet. Coll.]
Ex. 2. An elliptic lamina is supported with its plane vertical and transverse
axis horizontal by two weightless pegs passing through the foci. If one pin be
released show that if the eccentricity of the ellipse be a/ ^ , the pressure on the
other pin will be initially unaltered. [Coll. Exam.]
Ex. 3. Three equal particles A, B, C repelling each other with any forces, are
tied together by three strings of unequal length, so as to form a triangle right-
angled at ^. If the string joining B and C be cut, prove that the instantaneous
changes of tension of the strings joining BA, CA are J TcosB and ^ jf cos C respec-
tively, where B and C are the angles opposite the strings joining CA, AB respec-
tively, and T is the repulsive force between B and C.
Ex. 4. Two uniform equal rods, each of mass m, are placed in the form of the
letter X on a smooth horizontal plane, the upper and lower extremities being con-
nected by equal strings ; show that whichever string be cut, the tension of the other
is the same function of the inclir.ation of the rods, and initially is | mg sin a, where
a is the initial inclination of the rods. [St Pet. Coll.]
Ex. 6. A horizontal rod of mass m and length 2a hangs by two parallel strings
of length 2a attached to its ends : an angular velocity w being suddenly communi-
cated to it about a vertical axis through its centre, show that the initial increase
of tension of either string equals —^ , and that the rod will rise through a space
&g
[Coll. Exam.]
Ex. 6. A particle is suspended by three equal strings of length a from three
points forming an equilateral triangle of side 26 in a horizontal plane. If one
string be cut the tension of each of the others is instantaneously changed in the
.. 3a''-46a r^ „ ,,
ratio 2 (ct"- ft") ' [Coll. Exam.]
Ex. 7. A sphere resting on a rough horizontal plane is divided into an infinite
number of solid lines and tied together again with a string ; the axis through which
the plane faces of the lines pass being vertical. Show that if the string be cut
the pressure on the plane is diminished instantaneously in the ratio 45t^ : 2048.
[Emm. Coll.]
RELATIVE MOTION.
151
a space
; 2048.
On Relative Motion or Moving Axes.
173. In many dynamical problems the relative motion of
the different bodies of the system is frequently all that is required.
In these cases it will be an advantage if we can determine this
without finding the absolute motion of each body in space. Let
us suppose that the motion relative to some one body (A) is
required. There are then two cases to be considered, (1) when
the body (A) has a motion of translation only, and (2) when it
has a motion of rotation only. The case in which the body (A)
has a motion both of translation and rotation may be regarded
as a combination of these two cases. Let us consider these in
order.
174. Let it be required to find the motion of any dynamical
system relative to some moving point C We may clearly reduce
C to rest by applying to every element of the system an accelera-
tion equal and opposite to that of C. It will also be necessary to
suppose that an initial velocity equal and opposite to that of C
has been applied to each element.
Let /be the acceleration of C at any time t If every particle
m of a body be acted on by the same accelerating force / parallel
to any given direction, it is clear that these are together equi-
valent to a force f%m acting at the centre of gravity. Hence to
reduce any point of a system to rest, it will be sufficient to
apply to the centre of gravity of each body in a direction opposite
to that of the acceleration of C a force measured by Mf, where
M is the mass of the body and/ the acceleration of C.
■The point G may now be taken as the origin of co-ordinates.
We may also take moments about it as if it were a point fixed in
space.
Let us consider the equation of moments a little more minutely.
Let (r, 6) be the polar co-ordinates of any element of a body
whose mass is m referred to (7 as origin. The accelerations of
the particle ^^^^ j^a — ''(751) ^^^ ~'JfV~^)' *^°°S ^^^ perpen-
dicular to the radius vector r. Taking moments about C, we get
[moment round G of the impressed forces
,8 dO\ _ plus the moment round G of the reversed
dt) effective forces of G supposed to act at the
centre of gravity.
If the point G be fixed in the body and move with it, -j-
will be the same for every element of the body, and, as in Art. 88,
2w
dt V
we have tm ;^ (*'" 77) = ^^^'
de
w '\
\ I I
11. i i
V i
i-il
i
152
MOTION IN TWO DIMENSIONS.
175. From the general equation of moments about a moving
point G we learn that we may use the equation
day _ moment of forces about C
dt moment of inertia about G
in the following cases.
First. If the point G be fixed both in the body and in space ;
or, if the point u being fixed in the body move in space with
uniform velocity ; for the acceleration of G is zero.
Secondly. If the point G be the centre of gravity ; for in that
case, though the acceleration of G is not zero, yet the moment
vanishes.
Thirdly. If the point G be the instantaneous centre of rota-
tion*, and the motion be a small oscillation or an initial motion
which starts from rest. At the time t the body is turning about G,
and the velocity of G is therefore zero. At the time t + dt, the
body is turning about some point G' very near to G. Let GG'— da,
then the velocity of G is oida-. Hence in the time dt the velocity
of G has increased from zero to oada; therefore its acceleration is
(o -j:. To obtain the accurate equation of moments about G we
dt
dc
must apply the eflfective force Xm . to -7- in the reversed direction
dt
at the centre of gravity.
dcr
But in small oscillations a> and -r- are
dt
both small quantities whose squares and products are to be
neglected, and in an initial motion &> is zero. Hence the moment
cf this force must be neglected, and the equation of motion will
be the same as if G had been a fixed point.
It is to be observed that we may take moments about any
point very near to the instantaneous centre of rotation, but it will
usually be most convenient to take moments about the centre in
its disturbed position. If there be any unknown reactions at the
centre of rotation, their moments will then be zero.
176. If the accurate equation of moments about the instan-
taneous centre be required, we may proceed thus. I-et L be the
moment of the impressed forces about the instantaneous centre,
* If a body be in motion in one plane it is known tbat the actual displacement
of every particle in the time dt is the same as if the body had been turned through
some angle udt about some fixed point O. This may be proved in the same way as
the corresponding proposition in Three Dimensions is proved in the next Chapter.
See Art. 183. The point C is called the instautauoous centre of rotation, and w is
called the instantaneous angular velocity. See also Salmon's Higher Plane Curves^
1852, Arts. 246 and 2C4.
BELATIVE MOTION.
163
the centre of gravity, r the distance between the cent: o of
gravity and the instantaneous centre G, M the mass of the body ;
then the moment of the impressed forces and the reversed
effective forces about C is
L-Mw^^.r cos GC'G:
at
If k be the radius of gyration about the centre of gravity, the
equation of motion becomes
writing for cos QC'C its value -j- .
dt
177. Ex. 1. Two heavy particles ■whos& masses ( •• m and ta' are connected by
an inextensible string, which is laid over the vertex of a double iTicUned plane whose
mass is M, and which is capable of moving freely on a smooth horizontal plane.
Find the force which must act on the wedge that the system may be in a state of
relative equilibrium.
Here it will be convenient to reduce the wedge to rest by applying to every
particle an acceleration / equal and opposite to that of the wedge. Supposing this
done the whole system is in equilibrium. If F be the required force, we have by
resolving horizontally (M + m+m')f=F.
Let a, a' be the inclinations of the sides of the wedge to the horizontal. The
particle m is acted on by mg vertically and mf horizontally. Hence the tension of
the string is m(i/6ina+/cosa). By considering the particle m', we find the
tension to be also m' {g sin a' -/cos a'). Equating these two we have
,_ m sin a - m sin a
^ ~ ml cos a' + m cos a
Hence F is found.
178. Ex 2. A cylindrical cavity whose section is any oval curve and wlwse
generating lines are horizontal is made in a cubical mass tohich can slide freely on a
smooth horizontal plane. The surface of the cavity is perfectly rough and a sphere is
placed in it at rest so that the vditcal plane through the centres of gravity of the
mass and the sphere is perpendicular to the generating lints of the cylinder. A
momentum B is communicated to the cube by a blow in this vertical plane. Find the
motion of the sphere relatively to the cube and the least value of the blow that the
sphere may not leave the surface of the cavity.
Simultaneously with the blow B there will be an impulsive friction between the
cube and the sphere. Let M, m be the masses of the cube and sphere, a the radius
of the sphere, k its radius of gyration about a diameter. Let Fq be the initial
velocity of the cube, Vq that of the centre of the sphere relatively to the cube, w„ the
initial angular velocity. Then by resolving horizontally for the whole system, and
taking moments for the sphere alone about the point of contact, we have
wi(i'o+Fo) + il/Fo = i?|
■(1),
'i t]
il
l\
!i1
!■ ; i
164 MOTION IN TWO DIMENSIONS.
and sinoe there is no eliding
ro-a«o=0 (2).
To find the Bubsequent motion, let {x, y) be the co-ordinates of the centre of the
sphere referred to rectangular axes attached to the cubical mass, x being horizontal
and y vertical, then the equation to the cylindrical cavity being given, y is a known
fimotiou of X. Let ^ be the angle the tangent to the cavity at the point of contact
of the sphere makes with the horizon, then tan^=-^. Lot V be the velocity of
the cubical mass, then, by Art. 131,
m{^ + V\-k-MV=B (3).
If Tg be the initial vis viva and y^ the initial value of y, we have by the equation
of vis viva
'"!(§+ ^y+(fy+^"W+^^''=^«-^'"^^2^"2'») (^)'
where u is the angular velocity of the sphere at the time t. If v be the velociiy of
the centre of the sphere relatively to the cube, we have since there is no sliding
v=au. Eliminating F and » from these equations, we have
(|)*.j(UtenV)(u^:)-^! = C'.-2«, (5).
where Cg= +'^ffyo'
(M+m) ^M + (M+m)^}
This equation gives the motion of the sphere relatively to the cube.
To find the pressure on the cube, let us reduce the cube to rest. Let R be the
pressure of the sphere on the cube, then the whole effective force on the cube is
JZ sin 1^ parallel to the axis of x. By Art. 174 we must therefore apply to every
particle an acceleration — ^r^-^ opposite to this effective force. The sphere will
then be acted on by ^^ 72 sin ^ in a horizontal direction in addition to the reaction
E, the friction and its own weight. Besolving the forces on its centre along a
normal to its path we have
vhere p is the radius of curvature of the path of the centie of the sphere. Elimi'
nating -y- by the help of the equation of vis viva, we have
<'-^.»+p-<i+i--iS.*)=^- p).
where mgF=p f 1 + -5 - -V f, ) "^m ^"^' ^ /' ^^^ ^ P ^°^^ ^^^ change sign, is
essentially a positive quantity.
At the point where the sphere leaves the surface of the cavity R vanishes.
Putting R=0, we have an equation to determine ^ at this point, (7 being a known
function of the initial conditions. If the sphere is to go all round the cylindrical
cavity, the values of cos ^ given by this equatio^j must be all imaginary or numeri-
RELATIVE MOTION.
155
•(6).
callj greater than unity. If the sphere is just to go all round, then R must be
positive tliroughout and must vanish at the point where it is least. In this case we
have R and -^ simultaneously zero.
Differentiating we have
dlogp
008^(1 + - - TT- — cos«^) = (-5 + 3-Trr— oos'^) sinf
.(8).
This equation, since p is given as a function of \j/ from the equation to the cylinder,
determines ^; C is then known from (7) when R is put equal to zero, and thence
the required value of B.
We may notice that the position of the point at which R is to be put zero is
independent of the initial conditions, and depends on the form of the cavity and the
ratio of the masses of the cube and sphere. This point cannot be at the highest
point of the cavity unless the radius of curvature of the cavity is at that point a
maximum or minimum. If the section of the cavity be a circle or an ellipse
having its major axis horizontal, then the equation to find y// is satisfied only when
yp=ir. In this case we find as the least value of the blow £ to be given to the cube
that the sphere may go all round
^=\M+(M+m)~\. J4(3f+m)/3+^3f+(3f+m)^)^'j.
where a and ^ are the semi-axes of the ellipse.
179. Next, let us consider the case in which we wish to refer
the motion to two straight lines Of, Or}, turning round a fixed
origin with angular velocity a.
Let Ox, Oy be any fixed axes and let the angle xO^=0. Let
f = OM, 7} = PM be the co-ordinates of any point P.
It is evident that the motion of P is made up of the motions
of the two points M, N by simple t'ddition. The resolved parts of
the velocity of M sue -^ and f a> along and perpendicular to OM.
• ii
1
,.'•
^•-*
^"-^d
>
r-l
X
The resolved parts of the velocity of N are in the same way -^
and 7}oi along and perpendicular to ON. By adding these with
their proper signs we have
156
MOTION IN TWO DIMENSIONS.
velocity of P I d^_
paraUelto 0^)~di~"^'^'
velocity oi P\_dri ,,
parallel to Orj) dt
In the same way by adding the accelerations of M and N we
have
acceleration of P ) <?'^ _i: * l^/«\
parallel io 0^]~lif~^'^ "^Tt ^^ *"''
By using these foi i Ijb i/is-'oad of -^ and -^ we may refer
the motion to the moving axes 0^, Or).
In a similar manner we may use polar co-ordinates. In this
case if (r, <f>) be the polar co-ordinates of P, we have
acceleration of PI
accelerp 'ion of <
parai ' i
along rad. vect.
acceleration of P)
M d'r fd<f>^ V
perp. to rad. vect
I
=Jlf*(f-)}
180. Ex. 1. Let the axes 0{, Orj be oblique and make an angle a with each
other, prove that if the velocity be represented by the two components u, v parallel
to the axes,
dt ^ ^
u=^ - «f cot a-uT] coseo a,
Cut
» = ^+ w>7 cot a + wf coseo a.
In this case PM is parallel to Ot/. The velocities o! M and N are the same as
before. Their resultant is, by the question, the same as the resultant of u and v.
By resolving in any two directions and equating the components we get two equa-
tions to find u and v. The best directions to resolve along are those perpendicular
to 0% and O17, for then v, is a'usent from one of the equations and v from the other.
Thus tt or V may bo found separately when the other is not wanted.
Ex. 2. If the acceleration be represented by the components X and F, prove
Jt = 3- — cim cot a - wv cosec a,
at
„ i/o ,
r= 7- + &w cot a + wu coseo o.
dt
These may be obtained in the same way by resolving velocities and accelerations
perpendicular to 0^ and O17.
I '^l
RELATIVE MOTION.
157
prove
181. Ex. A particle under the action of any forces moves on a smooth curve
which is constrained to turn with angular velocity u about a fixed axis. Find the
motion relative to the curve.
Let na suppose the motion to be in three dimensions. Take the axis of Z as
the fixed axis, and lot the axes of (, t) be fixed relatively to the curve. Then the
equations of motion are
7] dt
dt'
Z + Rn
.(1).
where X, Y, Z are the resolved parts of the impressed accelerating forces resolved
parallel to the axes, R is the pressure on the curve, and (Z, m, n) the direction-
cosines of the direction of R. Then since R acts perpendicularly to the cutvp
,d^ dn dt .
Suppose the moving curve to be projected orthogonally on the plane ' ! j, % let
a be the arc of the projection, and v'= — be the resolved part of the velocity ^ -allel
(tt
to the plane of projection. Then the equations may be written in the f ""m
dt^
= Z + Rn.
The two terms 2uv' ^ and - 2&«/ ^ may be regarded as the resolved parts of a
force 2uv' acting in a direction whose direction-cosines are
j,_dri
m' = -
da
»'=0.
These satisfy the equation I'-^+m' -p+n' t^=0«
Hence the force is perpendicular to the tangent to the curve, and also perpen-
dicular to the axis of rotation. Let R^ be the resultant of the reaction R and of the
force 2(iw'. Then Rf also acts perpendicularly to the tangent, let {I", m", n") be the
direction-cosines of its direction.
The equations of motion therefore become
dt^
da ,
dt'
•(2).
Hi
;!!
',h
I' \
158
MOTION IN TWO DIMENSIONS.
These are the equations of motion of a particle moving on a Jixfd curve, and
acted on in addition to the impressed forces by two extra forces, viz. (1) a force wV
tending directly from the axis, where r is the distance of the particle from the axis,
and (2) a force -j- r perpendicular to the plane containing the particle and the axis,
and tending opposite to the direction of rotation of the curve.
In any particular problem we may therefore treat the curve as fixed. Thus
suppose the curve to be turning round the axis with uniform angular velocity.
Then resolving along the tangent we have
dv_ dx ydj/ dz J ^
d»~ da ds da da '
where r is the distance of the particle from the axis. Let V be the initial value of
V, rg that of r. Then
r2- V^=2f(Xdx+ Ydy + Zdz) + u^r*-r^').
Let t'o be the velocity the particle would have had under the action of the same
forces if the curve had been fixed. Then
Hence
vj* -V^=2 f(X dx+Ydij + Z dz).
r'-VQS=w«(,.3-ry').
The pressure on the moving curve is not equal to the pressure on the fixed
curve. The pressure R on the moving curve is clearly the resultant of the pressure
J?' on the fixed curve, and a pressure 2e<yv' acting perpendicular both to the curve
and to the axis in the direction of motion of the curve.
Thus suppose the curve to be plane and revolving imiformly about an axis per-
pendicular to its plane, and that there are no impressed forces. We have, resolving
along the normal,
- = _ wV sin ^ + R',
P
where <p is the angle r makes with the tangent.
If p be the perpendicular drawn from the axis on the tangent, we have, there-
fore,
£=- + u^p + 2uv.
P
This example might also have been advantageously solved by cylindrical co-ordi-
nates. The fixed axis might be taken as axis of z and the projection on the plane
of asy referred to polar co-ordinates. This method of treating the question is left to
the student as an exercise.
Ex. If w be variable, we have in a similar manner
v'
„ »" _ - dv I
llr=- + u'p + 2uv + -j- Jr
p dt ^
■^-p\
RELATIVE MOTION.
159
EXAMPLES •.
1. A circular hoop, which is free to move on a smooth horizontal plane, carries
on it a small ring - th of its weight, the coefficient of friction between the two being
n
ft. Initially the hoop is at rest and the ring has an angnlar velocity u abont the
centre of the hoop. Show that the ring will be at rest on the hoop after a time
1 + n
fJLU
2. A heavy circular wire has its plane vortical and its lowest point at a height
h above a horizontal plane. A small ring is projected along the wire from its
highest point with an angular velocity about its centre equal to wn
x/!«'
the
instant that the wire is let go. Show that when the wire reaches the horizontal
plane, the particle will just have described n revolutions.
8. A heavy uniform sphere rolls on a rough plane and is acted on by a fixed
centre of force in the plane varying inversely as the square of the distance ; if the
sphere be projected along the plane from a given point in it, in a direction opposite
to that of the centre of force, find the roughness of the plane at any point, suppos*
ing the whole of it to be required.
4. Two equal uniform rods of length 2a, loosely jointed at one extremity, are
a. J 2
placed symmetrically upon a fixed smooth sphere of radius
and raised into a
horizontal position so that the hinge is in contact with the sphere. If thoy be
allowed to descend under the action of gravity, show that, when they are first at
rest, they are inclined at an angle cos~^ ^ to the horizon, that the points of contact
with tho sphere are the centres of oscillation of the rods relatively to the hinge,
that the pressure on the sphere at each point of contact equals one-fourth the
weight of either rod, and that there is no strain on the hinge.
5. Two circular discs are on a smooth horizontal plane; one, whose radius is n
times that of the other, is fixed : an elastic string %vraps round them so that those
portions of it not in contact with the discs are common interior tangents, the
natural length of the string being the sum of the circumferences. The moveable
disc is drawn from the other till the tension of the string is T, prove that if it be
now let go, the velocity acquired when it comes in contact with the fixed disc
will be
w
2 (n + l)ira.\
m
, where m is the mass of the moving disc, X the modu-
lus of elasticity, a the radius of the moving di^o.
6. Two straight equal and uniform rods are connected at their ends by two
strings of equal length a, so as to form a parallelogram. Cue rod is supported at
its centre by a fixed axis about which it can turn freely, this axis being perpendicu-
lar to the plane of motion which is vertical. Show that the middle point of the
lower rod will oscillate in the same way as a simple pendulum of length a, and that
the angular motion of the rods is independent of this oscillation.
* These examples are taken from the Examination Papers which have been set
in the University and in the Colleges.
IGO
MOTION IN TWO DIMENSIONS.
7. A flno strinpt in nttacbcd to two points A, If in tho same horizontal plftne,
and carries a weight W at its miihllo point. A rod whose length is AD and weight
W, has a ring at either end, through which tho string passes, and is let fall from
tho position AB. Show that the string must be at least I AB, in order that the
weight may ever reach tho rod.
Also if tho system bo in equilibrium, and tho weight bo slightly and vertically
displaced, the time of its small oscillations is 27r . / .
V %v/3
8. A fine thread is enclosed in a smooth circular tube which rotates freely
about a vortical diameter ; prove that, in tho position of relative equilibrium, the
inclination {0) to the vertical, of the diameter through tho centre of gravity of the
thread, will be given by tho equation cos = — --- — - , where w is the angular
aw* cos ^
Telocity of the tube, a its radius, and 20)3 the length of the thread.
Explain the
case in which the value of aw"' cos /3 lies between g and - g.
9. A smooth wire without inertia is bent into the form of a helix which ia
capable of revolving about a vertical axis coinciding with a generating line of the
cylinder on which it is traced. A small heavy ring slides down the helix, starting
from a point in which this vertical axis meets the helix: prove that the angular
velocity of the helix will 1)e a maximum when it has turned through an angle B
given by the equation cos" ^ + tan* a+ sin 2^=0, o being tho inclination of the
helix to the horizon.
10. A spherical hollow of radius a is made in a cube of glass of mass M, and a
particle of mass m is placed within. The cube is then set in motion on a smooth
horizontal plane so that the particle just gets round the sphere, remaining in con-
m
tact with it. If the velocity of projection bo F, prove that F*=5o(/ + 4a(;
M'
11. A perfectly rough ball is placed within a hollow cylindrical garden-roller at
its lowest point, and the roller is then drawn along a level walk with a uniform
velocity F. Show that the ball will roll quite roimd the interior of the roller, il
F* be > V (7 (6 - a), a being the radius of the ball, and h of the roller.
12. AB, BC are two equal uniform rods loosely jointed at B, and moving with
the same velocity in a direction perpendicular to their length ; if the end A be sud-
denly fixed, show that the initial angular velocity ol ABii three times that of BC.
Also show that in the subsequent motion of the rods, the greatest angle between
them equals cos'^ f , and that when they are next in a straight line, the angular
velocity of BO is nine times that of AB.
13. Three equal heavy uniform beams jointed together are laid in the same
right line on a smooth table, and a given horizontal impulse is applied at the
middle point of the centre beam in a direction perpendicular to its length ; show
that the instantaneous impulse on each of the other beams is one-sixth of the given
impulse.
14. Three beams of like substance, jeined together so as to form one beam,
are laid on a smooth horizontal table. The two extreme beams are equal in length,
and one of them receives a blow at its free extremity in a direction perpendicular to
its length. Determine the length of the middle beam in order that the greatest
possible angiilar velocity may be given to the third.
EXAMPLES.
161
intal plftne,
oiul weight
et fttU from
lor that the
id vertically
states freely
libriiim, the
avity of the
the angular
Explain the
lix which is
g line of the
iclix, starting
, the angular
h an angle &
aation of the
lass M, and a
1 on a smooth
[lining in cou-
,rden-roller at
,th a uniform
the roller, if
I moving with
end A be sud-
3S that of BG.
angle between
e, the angular
in the same
^pplied at the
length ; show
th of the given
rm one beam,
qualinl'jngth,
irpendicular to
,t the greatest
RetuU. If m bo the masn of either of the outer rods, /9m that of the inner rod,
P the momentum of the blow, w the angular velocity communicated to the third
8
rod, then mawf- + q + -jj = P. Hence when u ia a maximum p=\j3.
15. Two rough rods A, B are placed parallel to each other and in the same
horizontal plane. Another rou^h rod C is laid across thom at rit,'ht angles, its
centre of gravity being half way V)etwoon them. If C be raised through any angle a
and let full, detcrmlno the conditions that it may oscillate, and show that if its
length be equal to twice the distance between A and D, the angle through which
it will rise in the n"> oscillation is given by the equation sin tf = I - ; . sin a.
16. A rod moveable in a vertical plane about a hinge at its upper end has a
given uniform rod attached to its lower end by a hinge about which it can turn
freely in the same vertical plane as the u]>per rod ; at what point must the lower
rod be struck horizontally in that same vertical plane that the upper rod may
initially be imaffected by the blow ?
17. A ball spinning about a vertical axis moves on a smooth table and impinges
directly on a perfectly rough vertical cushion; show that the vis viva of the ball is
diminished in the ratio 10 + 14 tan' ^ : -? + 49itan'' 6, where e is the elasticity of the
ball and 6 the angle of reflexion.
18. A rhombus is formed of four rigid uniform rods, each of length 2a, freely
jointed at their extremities. If the rhombus be laid on a smooth horizontal table
and a blow be applied at right angles to any one of the rods, the rhombus will begin
to move as a rigid body if the blow be applied at a point distant a (1 - cos a) from
an acute angle, where a is the acute angl(<.
19. A rectangle is formed of four uniform rode of lengths 2a and 2& respectively,
which are connected by hinges at their ends. The rectangle is revolving about its
centre on a smooth horizontal plane with an angular velocity n, when a point in
one of the sides of len^h 2a suddenly becomes fixed. Show that the angular
velocity of the sides of length 26 immediately becomes ^ "tjv «• Find also the
change in the angular velocity of the other sides and the impulsive action at the
point which becomes fixed.
20. Three equal uniform inelastic rods loosely jointed together are laid in a
straight line on a smooth horizontal table, and the two outer ones are set in
motion about the ends of the middle one with equal angular velocities (1) in the
same direction and ('.!) in opposite directions. Prove that in the first case, when
the outer rods make . ue greatest angle with the direction of the middle one pro-
duced on each side the common angular velocity of the three is — , and in the
second case after the impact of the two outer rods the triangle formed by them will
move with uniform velocity — - , 2a tbeing the length of each rod.
o
21. An equilateral triangle formed of three equal heavy uniform rods of length
a hinged at their extremities is held in a vertical plane with one side horizontal uinI
the vertex downwards. If after falling through any height, the middle point of i/.a
R. D. 11
\\
162
MOTION IN TWO DIMENSIONS.
upper rod be suddenly stopped, the impulsive strains on the upper and lower hinges
will be in the ratio of sjl3 to 1. If the lower liinge would just break if the system
fell tnrough a height —p , prove that if the system fell through a height -.— the
lower rods would just swing through two right angles,
22. A perfectly rough and rigid hoop rollir.g down an inclined plane comes in
contact with an obstacle in the shape of a spike Show that if the radius of the
hoop=r, height of spike above the plane - and F= velocity just before impact, then
the condition that the hoop will surmount the spike is F*> V fl''' 1 1 - sin («+/:)(.
a being the inclination of the plane to the horizon.
Show that unless V^<'^t^ gr.sm(a + ^\ , the hoop will not remain in contact
with the spike at all.
If this inequality be satisfied the hoop will leave the spike when the diameter
through the p^lut of contact makes an angle with the horizon
=sm'
■MsLt+^^^K''-*-?)!
23. A flat circular disc of radius a is projected on a rough horizontal table,
which is such that the friction upon an element a is c F** ma where V is the velocity
of the element, m the mass of a unit of area : find the path of the centre of the disc.
If the initial velocity of the centre of gravity and the angular velocity of the
disc be Wo<^c' prove that tlu velocity m and angular velocity u at any sub. equent
tune satisfy the relation ( „ , — , . I = -^r- •
24. A heavy circular lamina of radius a and mass M rolls on the inside of a
rough circular arc of twice its radius fixed in a vertical plane. Find the motion.
If the lamina be placed at rest in contact with the lowest point, the impulse which
must be applied horizontally that it may rise as high as possible (not going all
round), without falling off, is Mj'iarj.
25. A string without weight is coUed round a rough horizontal cylinder, of
which the mass is M and radius a, and which is capaule of turning roimd its axis.
To the free extremity of the string is attached a chain of which the mass is m and
the length I ; if the chain be gathered close up and then let go, prove that if d be
the angle tlu'ough which the cylinder has turned after a time t before the chain is
fully stretched, Mae= j ( % - «^ )
26. Two equal rods AG, BC, are freely connected at C, and hooked to / and B,
two points in the same horizontal line, each rod being then inclined at an angle a to
the horizon. The hook B suddenly giving way, prove that the direction of the strain
'1 + Csiu«a 2-3(
at C is iustantaneouply shifted through an angle tan'
■i/l±«
Vl + 6
cos'' a
8 sin
icos'aX
a COB a J
ir binges
B system
^k the
V3
comes in
us of the
)act, then
In contact
B diameter
mtal table,
be velocity
of the disc.
city of the
sub, equent
inside of a
the motion.
)ulse which
)t going oil
cylinder, of
id its axis.
Iss is m and
Itbat if e be
the chain ia
EXAMPLES.
163
U
27. Two particles A , B are connected by a fine string ; A rests on a rough hori-
zontal table and B hangs vertically at a distance I below the edge of the table. If
A be on the point of motion and B be projected horiaontally with a velocity u, show
that A will begin to move with acceleration - -r-
/U+l I
and that the initial radius of
curvature of B's path will be (/x + 1) I, where n is the coefficient of friction.
28. Two particles {m, m') are connected by a string passing through a small
fixed ring and are held so that tlio string is horizontal ; their distances from the
ring being a and a', they are let go. If p, p' be the initial radii of curvature of
their paths, prove that = — , , and -- + - = -+-.
p P p p a a
29. A sphere whose centre of gravity is not in its centre is placed on a rough
table ; the coefficient of friction being p,, determine whether it will begin to slide or
to roll.
30. A circular ring is fixed in a vertical position upon a smooth horizontal
plane, and a small ring is placed on the circle, and attached to the highest point
by a string, which subtends an angle o at the centre ; prove that if the string be cut
and the circle left free, the pressurep on the ring before and after the string is cut
are in the ratio M+m sin^ a : il/coso, m and M being the masses of the ring and
circle.
31. One extremity C of a rod is made to revolve with uniform angular velocity
n in the chcumference of a circle of radius a, while the rod itself is made to revolve
in the opposite direction with the same angular velocity about that extremity. The
rod initially coincides with a diameter, and a smooth ring capable of sliding freely
along the rod is placed at the centre of the circle. If r be the distance of the ring
from C at the time t, prove »*=-v (e"'+e~"*) + t cos 2nt.
32. Two equal uniform rods of length 2« are joined together by a hinge at one
extremity, their other extremities being connected by an inextensiblo string of length
21. The system rests upon two smooth pegs in the same horizontal line, distant 2c
from each other. If the string be cut prove that the initial angular acceleration of
either rod will be g •
8an-' ■62a*c'>
d
+ ■
l^
- 8a-cl
33. A smooth horizontal disc revolves with angular velocity sjp. about a verti-
cal axis at which is placed a material particle attracted to a certain point of the disc
by a force whose acceleration is /u x distance ; prove that the path on the disc will
be a cycloid.
to / and B,
In angle a to
If the strain
I cos'' '
I a COB
a/
11—2
I
CHAPTER V.
MOTION OF A EIGID BODY IN THREE DIMENSIONS,
r.
Translation and Rotation.
182. If the particles of a body be rigidly connected, then
•whatever be the nature of the motion generated by the forces,
there must be some general relations between the motions of the
particles of the body. These must be such that if the motion of
three points not in the same straight line be known, that of every
other point may be deduced. It will then in the first place be
our object to consider the general character of the motion of
a rigid body apart from the forces that produce it, and to reduce
the determination of the motion of every particle to as few in-
dependent quantities as possible : and in the second place we
shall consider how when the forces are given these independent
quantities may be found.
183. One point of a moving rigid body being fixed, it is re-
quired to deduce the gen^-al relations between the motions of the
other points of the body.
Let be the fixed point and let it be taken as the centre of
a moveable sphere which we shall suppose fixed in the body.
Let the radius vector to any point Q of the body cut the sphere
in P, then the motion of every point Q of the body will be repre-
sented by that of P.
If the displacements of two points A, B, on the sphere in any
time be given as AA'y BE, then clearly the displacement of any
other point P on the sphere may be found by constructing on
A'B' as base a triangle A'F B similar and equal to APB. Then
PP' will represent the displacement of P. It may be assumed as
evident, or it may be proved as in Euclid, that on the same base
and on the same side of it there cannot be two triangles on the
same sphere, which have their sides terminated in one extremity
of the base equal to one another, and likewise those terminated in
the other extremity.
Let D and E be the middle points of the arcs AA\ BB', and
let DC, EG he arcs of great circles drawn perpendicular to AA',
f;v
I
IS re-
of the
mtre of
body,
sphere
repre-
in any
of any
med as
lie base
on the
tremity
atcd in
W, and
to AA',
TBANSLATION AND ROTATION.
165
BB' respectively. Then clearly CA = GA' and CB- Cff, and
therefore since the bases AB, A'B' are equal, the two triangles
ACB, A'CB are equal and similar. Hence the displacement of
C is zero. AIpo it is evident since the displacements of and G
are zero, that the displacement of every point in the straight line
OCia also zero.
Hence a body may be brought from any position, which we may
call AB, into another A'B' by a rotation about OC as an axis
through an angle POP' such that any one point P is brought into
coincidence luith its new position P'. Then every point of the body
will be brought from its first to its final position.
184. A body is referred to rectangular axes x, y, z, and the
origin remaining tlie same the axes are changed to x, y', s*, accord-
ing to the scheme in the margin. Show that this is equivalent
to turning the body round an axis whose equations are any two
of the following three:
{ai~\)x + a.^y + 0.^2=0,
hiX+ {\-l) y + h^z=0,
c^x+dy -\:{c^-\)z=0,
K, y', /
X
Oil
"2,
flj
y
K
K
&3
z
Cl,
Ca»
c.
through an angle $, where
8-4sin'- = ai + Ja + C3.
What is the condition that these three equations are consistent 7
Take two points one on each of the axes of z and z' at a distance h from the
origin. Their co-ordinates are (0< 0, h) {a^h, b^h, c^h) therefore their distance is
V2(l-(
But it is also 2h sin 7 siu - ;
.0
2 sin" -sin' 7=1-^3. We have by
a a
similar reasoning 2 siu2-8in3tt=l- a^ and 2 siu2,cBin'/3 = l-&j, whence the equa-
tion to find follows at once.
185. When a body is in motion we have to consider not
merely ita first and last positions, but also the intermediate posi->
16d
MOTION IN THREE DIMENSIONS.
tiona. Let us fhevi suppose AB, A'B' to be two positions nt any
indo^ litoly sraaM interval of time di. We see that wlien a bodv
I'uyvos about a fixed point 0, there is, at every instant of the
motion, a straight line OG, such that the displacement of every
point in it during an indefinitely short time dt is zero. This
straight line is called the instantaneous axis.
Let d9 be the angle through which the body must be turned
round the instantaneous axis to bring any point P from its posi-
tion at the time t to its position at the time t + dt, then the
ultimate ratio of dO to dt is called the angular velocity of the
body about the instantaneous axis. The angular velocity may
also be defined as the angle through which the body would turn
in a unit of time if it continued to turn uniformly ribout the
same axis throughout that unit with the angular velocity it had
at the proposed instant.
186. Let us now remove the restriction that the body is
moving with some one point fixed. We may establish the fol-
lowing proposition.
Every displacement of a rigid body may he represented hy a
combination of the two folloiuing motions, (1) a motion of trans-
lation whereby every particle is moved parallel to the direction of
motion of any assumed point P rigidly connected luith the body
and through the same space. (2) A motion of rotation of the wliole
body about some axis through this assumed point P.
It is evident that the change of position may be effected by
moving P from its old to its new position P' by a motion of trans-
lation and then retaining P' as a fixed point by movin^- any two
points of the body not in one straight liac with P into their
final positions. This last motion has bef^^ ■ iu od to be equivalent
to a rotation about some axis through P
Since these motions are quite independent, it is evident that
their order may be reversed, i.e. we may rotate the body fi'fst and
then translate it. We may even suppose them to take place
si multaneou sly.
It is clear that any point P of the body may be chosen as the
base point of the double operation. Hence the given displace-
ment may be constructed in an infinite variety of ways.
187,. To fnd the relations between the awes and angles of rota-
tion when different points P, Q are chosen as bases.
Le - the displacement of the body be represented by a rotation
6 about in axis Pli and a translation PP'. Let the same dis-
piaceraeni l>e also represeutud by a rotation 6' about an axis QS
tind ■ trvwlatio I Q(J'. It is clear that any point has two dis-
TRANSLATION AND ROTATION.
167
ition
(lis-
QS
dis-
K
placements, (1) a translation oqual and parallel to PF', and (2) a
rotation through an arc in a plane perpendicular to the axis of
rotation PR. This second displacement is zero only when the
point is on the axis PR. Hence the only points whose displace-
ments are the same as the base point lie on the axis of rotation
corresponding to that base point. Through the second base point
Q draw a parallel to PR. Then for all points in this parallel, the
displacements due to the translation Pl^', and the rotation 6
round PR, are the same as the corresponding displacements for
the point Q. Hence this parallel must bo the axis of rotation
correoponding to the base point Q. We infer that the axes of
rotaiinn corresponding to all base points are parallel.
188. The axes of rotation at P and Q having been proved
parallel, let a be the distance between them. The rotation 6
about PR will cause Q to describe an arc of a circle of radius a
and angle 6, the chord Qq of this arc is 2a sin ^ and is the dis-
placement due to rotation. The whole displacement of Q is the
resultant of Qq and the displacement of P. In the same way the
rotation & about QS will cause P to describe an arc, whose chord
Pp is equal to 2a sin — . The whole displacement of P is the
resultant of Pp and the displacement of Q. But if the displace-
ment of Q is equal to that of P together with Qq, and the dis-
placemexit of P is equal to that of Q together with Pp, we must
have Pp and Qq equal and opposite. This requires that the two
rotat'ons 9, & about PR and QS should be equal and in the same
direccion. We infer that the angles of rotation corresponding to
all base points are equal.
189. Since the translation QQ' is the resultant of PP and
Qq, we may by this theorem find both the translation and rotation
corresponding to any proposed base point Q when those for ' are
given.
Since Qq, the displacement due to rotation roimd PR, is per-
pendicular to PR, the projection of QQ' on the axis of rotation is
the same as that of PP'. Hence the projections on the axis of rota-
tion of the displacements of all points of the body are equ<d.
190. An important case is that in which the displacement is
a simple rotation 6 about an axis Pit without any translation. If
any p'jiiit Q distant a from PR be chosen as the base, the same
displacement is ropresencod by a translation of Q through a chord
Qq = 2a sin - in a diro^tion making an angle — ^— with the plane
QPi2 and a rotation which must be equal to 6 about an ax. which
^ ■■■>■ V >^V'i?^,J
hil
m\'-
1G8
MOTION IN THREE DIMENSIONS.
mast be parallel to PB. Hence a rotation about any axis may he
replaced by an equal rotation about any parallel axis together with
a motion of translation.
191. When the rotation is indefinitely small, the proposition
can be enunciated thus, a motion of rotation todt about an axis
PR is equivalent to an equal motion of rotation about any parallel
axis QS, distant a from PR, together with a motion of translation
awdt perpendicular to the plane containing the axes and in the
direction in which Q8 moves.
192. It is often important to choose the base point so that
the direction of translation may coincide with the axis of rotation.
Let us consider how this may be done.
Let the given displacement of the body be represented by a
rotation 6 about PR, and a translation PP'. Draw PN perpendi-
cular to PR. If possible let this same displacement be represented
by a rotation about an axis Q8, and a translation QQ' along QS.
By Arts. 187 and 188 QS must be parallel to PR and the rotation
about it must be 6. This translation will move P a length Q Q'
along PR, and the rotation about QS will move P along an arc
perpendicular to PR. Hence Q Q' must equal PIf and NP' must
be the chord of the arc. It follows that QS must lie on a plane
bisecting NP at right angles and at a distance a from PR where
a
NP' = 2a sin p; , or, which is more conveniont, at a distance y from
the plane NPP' where NP' = 2y tan ^ . The rotation 6 round QS
is to bring ^to P' and is in the same direction as the rotation Q
rouiid PR, Hence the distance -?/ must be measured from the
» •' , ^ ■ ;
lif!
1
!'■ \
nay he
',r with
(osition
m axis
parallel
islation
in the
so that
otation.
3d by a
3rpendi~
resented
ong QS.
rotation
gth QQ'
f an arc
P' must
I a plane
R where
e y from
»und QS
)tation
:om the
TRANSLATION AND EOTATION.
169
1'
middle point of NP' in the direction in which that middle point
is moved by its rotation round Pit.
Having found the only possible position of QS, it remains to
show that the displacement of Q is really along QS. The rotation
round PE will cause Q to describe an arc whose chord Qq is
g
parallel to P'N and equal to 2a sin ;^ . The chord Qq is therefore
equal to NP', and the translation NP' brings q back to its position
at Q. Hence Q is only moved by the translation PN, i.e. Q is
moved along QS.
193. It follows from this reasoning that any displacement of
a body can be represented by a rotation about some straight line
and a translation parallel to that straight line. This mode of con-
structing the displacement is called a screw. The straight line is
sometimes called the central axis and sometimes the axis of the
screw. The ratio of the translation to the angle of rotation is
called the pitch of the screw.
194. The same displacement of a body cannot be constructed
by two different screws. For if possible let there be two central
axes AB, CD. Then AB and CB by Art. 187 are parallel. The
displacement of any point Q on CB is found by turning the body
round AB and moving it parallel to AB, hence Q has a displace-
ment perpendicular to the plane ABQ and therefore cannot move
only along CB.
195. When the rotations are indefinitely small, the construc-
tion to find the central axis may be simply stated thus. Let the
displacement be represented by a rotation (odt about an axis PR
and a translation Vdt in the direction PP. Measure a distance
VsmP'PR
y= from P perpendicular to the plane P'PR on that
side of the plane towards which P' is moving. A parallel to PM
through the extremity of y is the central axis.
196. Ex. 1. Given the displacements AA', BB', CC of three points of a body
in direction and magnitude, but not necessarily in position, find the direction of
the axis of rotation corresponding to any base point P.
Through any assumed point draw Oa, 0/3, O7 parallel and equal to A A', BB',
CC If Op be the direction of the axis of rotation, the projections of Oa, 0/3, Oy
on Op are all equal. Hence Op is the perpendicular drawn from on the plane
a/37, ^^is O'^^o shows that the direction of the axis of rotation is the same for all
base points.
Ex. 2. If in the last example the motion be referred to the central axis, find
the translation along it.
It is clearly equal to Op.
170
MOTION IN THREE DIMENSIONS.
Ex. 8. Given the diflplncoments A A', BB' of two pointR A, B ot the body and
the direction of the contnil axis, find the position of the central axia. Draw planes
tlironph AA', BB' parallol to the central axis. BiHcct A A', UB'hy planes i)erpen-
diciila:' to these planes respectively and parallel to the direction of the central axis.
The'3e two last planes intersect in the central axis.
Composition of Rotations.
197. It is often necessary to compound rot.ations about axes
OA, OB which meet at a point 0. But as the only case which
occurs in Rigid Dynamics is that in which these rotations are
indefinitely small we shall first consider this case with some par-
ti alarity, and then indicate generidly the mode of proceeding
when the rotations are of finite magnitude.
198. To explain what is meant by a bodij having angular
velocities about more than one axis at the same time.
A body in motion is said to have an angular velocity to about
a straight line, when, the body being turned round this straight
line through an angle oodt, every point of the body is brought
from its position at the time t to its position at the time t + dt.
Suppose that during three successive intervals each of time dt,
the body is turned successively round three different straight lines
OA, OB, OG meeting at a point through angles (o^dt, to.jdt,
o)/H. Then we shall first prove tha,t the final position is the same
in whatever order those rotations are effected. Let P be any
point in the body, and let its distances from OA, OB, C, respect-
ively be Tj r^, r^. First let the body be turned round OA, then
P receives .% displacement a>j\dt. By this motion let r^ be in-
creased to 1\ + di\, then the displacement caused by the rotation
about OB will be in magnitude w^ (r^ + dr^ dt. But according to
the principles of the Differential Calculus we may in the limit
neglect the quantities of the second order, and the displacement
becomefii ta^rjit. So also the displacement due to the remaining
rotation will be wjrAt. And these three results will be the same
in whatever order the rotations take place. In a similar manner
we can prove that the directions of these displacements will be
independent of the order. The final displacement is the diagonal
of the parallelepiped described on these three lines as sides, and
is therefore independent of the order of the rotations. Since then
the three rotations are quite independent, they may be said to
take place simultaneously.
When a body is said to have angular velocities about three
different axes it is only meant that the motion may be determined
as follows. Divide the whole time into a number of small in-
tervals each equal to dt. During each of these, turn the body
Dody and
w planea
( perpen-
)ral axis.
mt axes
3 which
ons are
ne par-
iceediug
angular
ft) about
straight
brought
+ dt.
■ time dt,
ght lines
[dt, (o./it,
he same
be any
respect-
)A, then
, be in-
rotation
)rding to
he Umit
acement
jmaining
le same
manner
will be
diagonal
ides, and
nee then
said to
\\i three
ermined
mall in-
die body
COMPOSITION OF ROTATIONS.
171
round the three axes successively, through angles (o^dt, (o.jdt, m^dt.
Then when df diminishos without limit the motion during the
whole time will be accurately represented.
199. It is clear that a rotation about an axis OA may be
represented in magnitude by a length measured along the axis.
This length will also represent its direction if we follow the same
rule as in Statics, viz. the rotation shall appear to be in some
standard direction to a spectator placed along the axis so that
OA is measured from his feet at towards his head. This di-
rection of OA is called the positive direction of the axis.
200. If tv)o ancfidar velocities about two aires OA, OB he
represented in magnitude and direction by the tiuo lengths O A, OB ;
then the diagonal 00 of the parallelogram constructed on OA, OB
as sides will be the resultant axis of rotation, and its length will
represent the magnitude of the residtant angidar velocity. This
Prop, is usually called " The parallelogram of angular velocities."
Let P be any point in OG, and let PM, PN be drawn per-
pendicular to OA, OB. oince OA represents the angular ve-
locity about OA and PM is the perpendicular distance of P
from OA, the product OA . PM will represent the velocity of P
due to the angular velocity about OA. Similarly OB.PN will
represent the velocity of P due to the angular velocity about
OB. Since P is on the left hand side of OA and on the right-
hand side of OB, as we respectively look along these directions,
it is evident that these velocities are in opposite directions.
Hence the velocity of any point P is represented by
OA.PM-OB.PN
= OP [ OA . sin COA - OB . sin GOB]
= 0.
Therefore the point P is at rest and 0(7 is the resultant axis
of rotation.
Let «D be the angular velocity about OG, then the velocity
of any point A in OA is perpendicular to the plane AOB and is
represented by the product of o) into the perpendicular distance
of A from 00= ot . OA sin COA. But since the motion is also
VI
172
MOTION IN THllEE DIMENSIONS.
»1
:^1
f
»f
determined by the two given angular velocities about OA, OB, the
motion of the point A is also represented by the product of OB
into the perpendicular distance of A from 0B= OB. OA sin BOA ;
.-. o> = 0B
sin BO A
sin COA
OG.
Hence the angular velocity about C is represented in mag-
nitude by OG.
From this proposition we may deduce as a corollary "the
parallelogram of angular accelerations." For if OA, OB repre-
sent the additional angular velocities impressed on a body at
any instant, it follows that the diagonal OG will represent the
resultant additional angular velocity in direction and magnitude.
201. This proposition shows that angular velocities and an-
gular accelerations may be compounded and resolved by the same
rules and in the same way as if they were forces. Thus an an-
gular velocity to about- any given axis may be resolved into two,
(0 cos a and to sin a, about axes at right angles to each other and
making angles a and „ — a with the given axis.
If a body have angular velocities w^, w^, w^ about three axes
Ox, Oy, Oz at right angles, they are together equivalent to a
single angular velocity w, where w = Vwi* + «/ + w^, about an
axis making angles with the given axes whose cosines are re-
spectively
ft).
ft)„
w.
This may be proved, as in the corre-
_j
ft) ft) ft)
sponding proposition in Statics, by compounding the three angular
velocities, taking them two at a time.
It will however be needless to recapitulate the several propo-
sitions proved for forces in Statics with special reference to an-
gular velocities. We may use " t!.e triangle of angular velocities "
or the other rules for compounding several angular velocities
together, without any further demonstration.
202. A body has angular velocities a, w about two parallel
axes OA, O'B distant a from each other, to find the resulting
motion.
Since parallel straight lines may be regarded as the limit of
two straight lines which intersect at a very great distance, it
follows from the parallelogram of angular velocities that the two
given angular velocities are equivalent to an angular velocity
about some parallel axis 0"G lying in the plane containing OA,
O'B.
m
OB, the
L of OB
,nBOA;
in mag-
iry "the
B repre-
body at
esent the
rnitude.
3 and an-
the same
us an an-
into two,
3ther and
ihree axes
dent to a
about an
s are re-
le corre-
e angular
ral propo-
ce to an-
elocities "
velocities
'parallel
resulting
e limit of
Lstance, it
it the two
velocity
ning OA,
COMPOSITION OF ROTATIONS.
173
Let X be the distance of this axis from OA, and suppose it
to be on the same side of OA as OB. Let fl be the angular
velocity about it.
Consider any point P, distant y from OA and lying in the
plane of thr three axes. The velocity of P due to the rotation
about OA is wy, the velocity due to the rotation about OB is
o)'{y — a). But tlicso two together must bo equivalent to the
velocity due to the resultant angular velocity 11 about 0"G, and
this is fl (y — x),
.". 601/ + to' (y — a) =n (y — x).
This equation is true for all values of y, .*. H = w + co', x=-^ .
This is the same result we should have obtained if we had
been seeking the resultant of two forces w, co' acting along OA,
OB.
If 0) = — (I)', the resultant angular velocity vanishes, but x is in-
finite. The velocity of any point P is in this case wy-'ca!{y— a) =a&),
which is independent of the position of P.
The result is that two angular velocities, each equal to w but
tending to turn the body in opposite directions about two parallel
axes at a distance a from each other, are equivalent to a linear
velocity represented by aoi. This corresponds to the proposition
in Statics that " a couple " is properly measured by its moment.
We may deduce as a corollary, that a motion of rotation «a
about an axis OA is equivalent to an equal motion of rotation
about a parallel axis OB plus a motion of translation aw perpen-
dicular to the plane containing OA, OB, and in the direction in
which O'B moves.
203. To explain a certain analogy which exists between Statics
and Dynamics.
All propositions in Statics relating to the composition and
resolution of forces and couples are founded on these theorems :
1. The parallelogram of forces and the parallelogram of
couples.
2. A force F is equivalent to any equal and parallel force
together with a couple Fj), where p is the distance between the
forces.
Corresponding to these wo have in Dynamics the following
theorems on the instantaneous motion of a rigid body :
1. The parallelogram of angular velocities and the parallelo-
gram of linear velocities.
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174
MOTIO^T IN THREE DIMENSIONS.
2. An angular velocity « is equivalent to an equal angular
velocity about a parallel axis together with a linear velocity
equal to cop, where p is the distance between the parallel axes.
It follows that every proposition in Statics relating to forces
has a corresponding proposition in Dynamics relating to the
motion of a rigid body, and these two may be proved in the
same way.
To complete the analogy it may be stated (i) that an angular
velocity like a force in Statics requires, for its complete determina-
tion, five constants, and (ii) that a velocity like a couple in Statics
requires but three. Four constants are required to determine the
line of action of the force or of the axis of rotation, and one to
determine the magnitude of either. There will also be a conven-
tion in either case to determine the positive direction of the line.
Two constants and a convention are required to determine the
positive direction of the axis of the couple or of the velocity and
One the magnitude of either.
It is proved in Statics that a system of forces and couples is
generally equivalent to a single force and a single couple, and
that these may be reduced to a resultant JR, acting along a line
called the central axis, and a couple about that axis. Or they
may also be reduced to a resultant R of the same magnitude
as before, acting along any line parallel to the central axis at
any chosen distance c from it, together with a couple 0' about
an axis perpendicular to the line whose length is c, and in-
clined to the resultant It at an angle 0. Then we know that
G' = '^ G^ + M^c*, and is a minimum when c = 0, and also that
tan = -7y .
The same train of reasoning by which these results ware
established, will establish the following proposition. The ins'tan-
taneous motion of a body having been reduced to a motion of
translation and one of rotation, these are equivalent to a motion
of rotation to about a line called the central axis, and a trans-
lation V along that axis. Or they may also be reduced to a
rotation « of the same magnitude as before about any line par-
allel to the central axis, and at any chosen distance c from it,
together with a translation V along a line perpendicular to the
line c, and inclined to the axis of w at an angle 0. Then we
know that 1^' = V !'''*+ cW, and is a minimum when c = 0, and
C(0
V'
may be established.
X
e
f
r
a
n
&]
to
as
ar
also that tan = -^, In a similar manner many other propositions
i ft
204. Ex. 1. The locus of points in a body moving about a fixed point which
at any proposed instant have the same actual velocity is a circular cylinder.
m
» I :■
COMPOSITION OF ROTATIONS.
175
Ex. 2. The geometrical motion of a body is represented by angular velocities
inversely proportional to ^-y, y-a, o - /3 about three lines forming three edges
of a cube which do not meet nor are pai-allel. Prove that the body rotates about
the line
{P-y)x-aa = {y-a)7j-ap={a-p)z-ay,
2a being an edge of the cube, the centre being the origin, and the axes parallel to
the edges.
Ex, 3. A body has an angular velocity u about the axis
as-a_y-/3_ z-y
I m ~ n *
where ?' + m^ + n^ = l. The motion is equivalent to rotations lu, mu, nu about the
co-ordinate axes, and translations (my-np)u, (na-ly)<a, (^^ - ma) w in the direc-
tions of the axes.
This follows from the analogy of forces in Statics to angular velocities in
Dynamics. See Art. 203.
Ex. 4. A body has equal angular velocities about two axes which neither meet
nor are parallel. Prove that the central axis of the motion is equally inclined to
each of the axes.
205. When the rotations to be compounded are finite in magnitude, the rule to
find the resultant is somewhat more complicated. Let the given rotations be (1) a
rotation about an axis OA through an angle d; (2) a subsequent rotation abfuit an
axis OB through an angle 0', and let both these axes be fixed in space. Let lenirths
measured along OA, OB represent these rotations in the manner explained in
Art. 199.
Let the directions of the axes OA, OB out a sphere whose centre is at in .1
and B. On this sphere measure the angle BA C equal to ^ in a direction opposite
to the rotation round OA and also the angle ABC equal to - in the same direction
as the rotation round OB and let the arcs intersect in C. Lastly, measure the
angles BAG', ABC' respectively equal to BAC, ABC, but on the other side of AB,
The rotation 6 round OA wiU then carry any point P in OC into the straight
line OC and the subsequent rotation 0' about OB will carry the point P back into
00. Thus the points in OC are unmoved by the double rotation and OC is therefore
the axis of the single rotation by which the given displacement of the body may be
constructed. The straight line OC is called the resultant axis of rotation. If the
order of the rotations were reversed, so that the body is rotated first about OB and
then about OA, the resultant axis would be 00'.
; li
h
176
MOTION IN THREE DIMENSIONS.
n
'1 \ I
If the axes OA, OB were fixed in the body, the rotation about OA would bring
OB into a position OB'. Then the body may be brought from its first into its
last position by rotations 0, ff about the axes OA, OBf fixed in space. Hence the
same con'^truction will again give the position of the resultant axis and the rotation
about it.
To find the magnitude 0" of the rotation about the resultant axis 00 we notice
that if a point P be taken in OA, it is unmoved by the rotation about OA, and the
subsequent rotation 0" about OB will bring it into the position P', where PP' is
bisected at right angles by the plane OBC. But the rotation 0" about 00 must
give P the same displacement, hence in the standard case 0" is twice the external
angle between the planes OCA, 0C3. If the order of the rotations be reversed, the
rotation about the resultant axis OC would be twice the external angle at C", which
is the same as that at C. So that though the position of the resultant axis of rota-
tion depends on the order of rotation the resultant angle of rotation is independent
of that order.
206. A rotation represented by twice any internal angle of the spherical
triangle ^^Cis equal and O' nosite to that represented by twice the corresponding
external angle. For since the sum of the internal and external angles is v, these
two rotations only differ by 2ir ; and it is evident that a rotation through an angle
2jr cannot alter the position of any point of the body. This is merely another way
of saying that when a body turns about a fixed axis it may be brought from one
given position to another by turning the body either way round the axis.
207. The rule for compounding finite rotations may be stated thus:
If ABC he a spherical triangle, a rotation roitnd OA from C to B through twice
the internal angle at A, followed by a rotation round OB from A to G through twice
the internal angle at B is equal and opposite to a rotation round OC from B to X
through twice the internal angle at C.
It will be noticed that the order in which the axes are to be taken as we travel
round the triangle is opposite to that of the rotations.
As the demonstrations in Art. 205 are only modifications of those of Bodriguos,
we may call this theorem after his name.
208. Ex. 1. If two rotations 0, 0' about two axes OA, OB at right angles be
compoimded into a single rotation about an axis 00, then
6' fi 0^ th 0'
tan CO A = tan — coseo - , tan COB = tan - cosec ^ and cos ^ = cos - cos „- .
209. From Eodrigues' theorem we may deduce Sylvester's theorem by drawing
the polar triangle A'B'C, Since a side BC is the «upplement of the angle A, a
rotation represented in direction and magnitude by 2B'C' differs from that repre-
sented by 2A in the opposite direction by a rotation through an angle 2v. But a
rotation through 2t cannot alter the position of the body, hence the two rotations
2B'C' and 2A are equivalent in magnitude but opposite in direction. If therefore
A' EC be any spherical triangle, arotation represented by ttvice B'C followed by a
rotation tivice C'A' produces the same displacement of the body as a rotation twice
B'A'. By a rotation B'C is meant a rotation about an axis perpendicular to the
plane of B'G' which will bring the point B' to C".
210. The following proof of the preceding theorem was given by Prof. Donkin
in the Phil. Mag, for 1851. Let ABC be any triangle on a sphere fixed in space,
(Tould bring
rst into its
Hence the
the rotation
)C we notice
OA, and the
here PP' is
lat OC must
the external
reversed, the
at C, which
axis of rota-
independent
ho spherical
lorresponding
3S is ir, these
ingh an angle
r another way
ight from one
is.
through twice
through twice
from B to A
a as we travel
of Eodriguos,
ight angles he
e'
cos 2 .
m by drawing
le angle A, a
iin that repre-
le 2t. But a
two rotations
If therefore
followed by a
rotation twice
dicular to the
Prof. Donkin
fixed in space,
COMPOSITION OF ROTATIONS.
177
a/9Y a triangle on an equal and concentric sphere noveable abont its centre. The
sides and angles of ajSv are equal to those of A PC, but differently arranged, one
triangle being the inverse or reflection of the other. If the triangle afiy be placed
in the position I, so that the sides containing the angle a may be in the same great
circles with those containing A, it is obvious that it may slide along AB into the
position n, and then along £C into the position III; into which last position it
might also be brought by sliding along AC. To slide a/Sy along AB la equivalent,
to moving /3 and a each through an arc twice the arc AB about an axis perpen-
dicular to the plane ot AB. A similar remark applies when the triangle slides
along BC or AC. Hence, twice the rotation AB •followed by twice the rotation BC
produces the same displacement as twice the rotation A C.
211. If it be required to compound the rotations about two parallel axes, the
construction of Bodrigues requires only a slight modification. Instead of arcs
drawn on a sphere, let planes be drawn through the axes making with the plane
containing the axes the same angles as before; their intersection will be the
resultant axis. One case deserves notice, li d= - 0', the resultant axis is at
infinity. A rotation about an axis ; t infinity is evidently equivalent to a translation.
Hence a rotation $ about any axis 0.^ followed by an equal and opposite rotation
about a parallel axis O'B distant a from OA is equivalent to a translation 2a sin ^
a
perpendicular to a plane through OA making an angle ^ with the plane containing
the axes and in the direction of the chord of the arc described by any point in OA .
These results also follow easily from Art. 190.
212. Any given displacemertt of a body may be represented by two finite rotations,
one about any given straight line and the other about some other straight line which
does not neeessarily intersect the first. When a displacement is thus represented,
the axes are called conjugate axes and the rotations are called conjugate rotatiotts.
Let OA be the given straight line and let the given displacement be represented
by a rotation <p about a straight line OR and a translation OT. We wish to resolve
this rotation about OR into two rotations, one about OA to be followed by a
rotation about OB, where OB is some straight line perpendicular to OT. To do
this we follow the rule in Art. 205, we describe a sphere whose centre is and
radius unity and let it intersect OA, OR, OT in A, R and T. Make the angle ARB
R. D. 12
■ i>2ri'-ff-»^-»
178
MOTION IN THREE DIMENSIONS.
on the one side or the other of OT according to the direction of the rotation,
equal to the supplement of % and produce RB to B so that TB=^ and join AB.
By the triangle of rotations the rota/on is now represented by a rotation about
OA which we may call d, followed by a rotation about OB which we may call ff.
By Art. 211 the rotation 6' is equivalent to an equal rotation 0' about a parallel
axis CD, together with a translation, which may be made to destroy the translation
OT. This will be the case if the angle OT makes with the plane of OB, CD be
ir-J'
2
d'
and if the distance r between AB, CD be such that 2r sin -^OZ*.
The whole displacement has thus been reduced to a rotation aboiit OA followed
by a rotation ff about CD.
213. Analytically, we might reason thus — A screw motion is given when we
know (1) its axis, (2; the rotation about it, (3) the translation along it. The axis is
known when its inclination to two of the axes and the two co-ordinates of the point
in which it cuts the plane of xy are given. Thus six constants are required to
determine a screw.
Let a given screw be resolved into two screws. We have then twelve constants,
but since they are together equivalent to the given screw there are six relations
between the constants. We are therefore at liberty to choose any six relations we
please between these twelve constants. We might, for example, resolve a given
screw into two screws of any given pitches, the remaining four constants being
chosen to make the axis of one screw coincide with any given straight line. If the
given pitch of each screw be zero, the screws are reduced to simple rotations, and
thus any displacement can be reduced to two conjugate rotations. It has been
shown in the preceding article that the two rotations are real.
214. Ex. Show that any screw may be resolved into two real screws having
the axis of one in a given direction and the axis of the other intersecting the first
at a given angle.
215. Any two successive displacements of a body may he represented by two
successive screw motions. It is required to compound these.
Let the body be screwed first along the axis OA with linear displacement a and
V
COMPOSITION OF ROTATIONS.
m
join AB,
on about
i parallel
anslation
B, CD be
I rotation,
1 followed
1 when we
Che axis is
' the point
equired to
constants,
X relations
3lationB we
ve a given
ants being
ae. If the
ations, and
t has been
ews having
g the first
\ted by two
lent a and
M
angle of rotation 0, and secondly along the axis CD with displacement a' and angle
ff. Let OC be the shortest distance between OA and CD, and for the sake of the
perspective let it be called the axis of y. Let be the origin and let the axis of x
be parallel to CD, so that OA lies in the plane of xz. Let OC=r, and the angle
a'
AOx=a. Draw a plane xOT making with the plane of xz an angle ^ , and let it cut
Draw another plane AOR making with xz an angle ~ , and cutting the
yz in OT.
plane xOT in OR.
Produce ^0 to a poijit P, not marked in the figure, so that PO=a, and let us
choose P as a base pouit to which the whole displacement of the body may be
referred. The rotation 0' is equivalent to a rotation 0* about Ox together with a
0'
translation along 02'=2rsin~ by Art. 190
By Art. 205 the rotation about OA
followed by 0' about Ox is equivalent to a rotation about OR where is twice the
The whole displacement is now repre-
angle ART, so that sm ^=8m :r . . „
2 2 siuKx
sented by (1) a translation of the base point 'P to 0, (2) the rotation fi, (3) a further
0'
linear translation which is the resultant of the translations 2r sin - along OT and
o' along Ox. By Art. 186 these displacements may be made in any order, being all
connected with the same base point. They may therefore be compounded into a
single screw by the rule given in Art. 192. This is called the resultant screw, A
screw equal and opposite to the resultant screw will bring the body back to its
original position.
The angle of rotation of the resultant screw is and its axis is parallel tc OR
by Art. 187. It follows by Art. 206 that the sine of half the angle of rotation of
each screw is proportional to the sine of the angle between the axes of the other
two screws.
To find the linear displacement along the axis of the resultant screw, we must by
Art. 189 add together the projections on OR of the three displacements OT, a, a'. The
0*
projection of 0T=2r sin - cos TR = 2r cos Ty . cos TR which is twice the projection
of the shortest distance r on the axis of rotation. If T be the Unear displace-
ment, we have T=2r cos Ry + a cos RA + a' cos Rx.
216. If the component screws be simple rotations we have o=0, a'—O, and it
may be shown without diflSculty that T sin ^j = 2r sin ~ sin -j,- sin a. It has bean
shown in Art. 212 that any displacement may be represented by two conjugate
rotations in an infinite number of ways, but it now follows that in all these
r sin - sin -= sin a is the same. When the rotations are indefinitely small, and equal
to udt, w'dt respectively, this becomes \ ruu sin a; that is, the product of an angular
velocity into the moment of its conjugate angular velocity about its axis is the same
for all conjugates representing the same motion.
Ex. 1. If the component screws be simple finite rotations, show that the equa-
tions to the axis of the resultant screw are
0' 0' 0' 0' 0' 0'
-actan^' + j/Bin ^ + 2C085- = r8in-, ycos^ -28in-=rsin - cos^'cot-g ,
12 2
180
MOTION IN THREE DIMENSIONS.
where ^' is the angle xOR and is the resultant rotation. The first eqiiatiou
expresses the fact that the central axis lies in a plane which bisects at right angles
a straight line drawn from perpendicular to OR in the plane xOR to represent
the linear tranBlation in that direction. The second expiesses that the central nxis
lies in a plane parallel to TOR at a distance from it determined by Art. 192.
These equations may also be deduced from those of Bodrigues given in Art. 223.
To effect this we must write for (a, 6, c) the resolved parts of the translation along
OT. Since however the positive direction of the rotation in Bodrigues' formulas
has been taken opposite to that chosen in the preceding article, we must write for
{I, m, n) the direction cosines of OR with their signs changed.
The equations to the central axis of any two screws may be found by either of
these methods.
Ex. 2. Let the motion be constructed by two finite rotations 0, ff taken in
order round axes OA, CD at right angles to each other and let CO be the shortest
distance between the axes. Let the two straight lines OT, CP be drawn in the
plane DCO such that the angle POC= ^ and tan PCO= sin" s" cot ^ . Then if P bo
moved backwards by the rotation or forwards by the rotation d\ in either case its
new position is a point on the central axis.
Ex. 3. If OA , OS be the axes of two screws at right angles, with linear dis-
placements a and b, the point P is the intersection of two parallels to the straight
lines described in the last example ,* these parallels being drawn respectively at
distances ^t^ii^ and ^f l + cof'^'sin^g j , where </>, ^' ho angles the
-t ultant axis of rotations makes with OA and CD. Then u ^ screwed back-
Y r Is by the first screw or forwards by the sec.nd, in either case its new position is
n point on the central axis.
217. Ex. 1. If the instantaneous motion of a body be represented by two con-
jugate rotations udt and u'dt, the axis of the resultant screw intersects at right
angles the shortest distance between the conjugate axes. Let y, y' be the angles the
conjugate axes make with the axis of their resultant, a the angle they make with
each other ; c, c' the shortest distances between the conjugate axes and the axis of
the screw, V and C the linear and angular velocities of the screw, then prove that
sin a*
V
sm7'
cos 7'
sm7
c'w'
cos 7
sma
c tan 7'= c' tan 7 = — .
The first line follows from Art. 201. The second expresses the fact that the
direction of the linear motion of the point where the axis cuts the shortest distance
is along the axis of the screw.
Ex. 2. If one conjugate axis of an instantaneous motion is at right angles to
the central axis, the other meets it, and conversely.
Ex. 3. If one conjugate axis of an instantaneous motion is parallel to the
central axis, the other is at an infinite distance, and conversely.
\>
\: I I
[[nation
t angles
ipresent
;ral nxis
Irt. 223.
>u along
Eormulsa
Trite fot
either of
taken in
shortest
rn in the
1 if P be
ir case its
inear dis-
e straight
ictively at
ngles the
wed back-
position is
two con-
at right
ngles the
make with
;he axis of
ove that
t that the
it distance
; angles to
lei to the
FIXED AXES.
181
Ex. 4. The locus of tangents to the trajectories of different points of the s ime
straight line in the instantaneous motion of a body is a hyperbolic paraboloid.
Let AB he the given straight line, CD its conjugate. The points on AB axe
turning round CD and therefore the tangents all pass through two straight lines,
viz. AB and its consecutive position A'B', and are also all parallel to a plane which
is perpendicular to CD.
Ex. 5. If radii vectores be drawn from a fixed point to represent in direction
and magnitude the velocities of all points of a rigid body in motion, prove that the
extremities of these radii vectores at any one instant lie in a plane. [Coll. Exam.]
Motion referred to fixed axes.
218. The general equations of motion given in Art. 71 of
Chapter II. involve the diflferential coefficients t- , -r , -r- . -t-5 »
^ dt at at dtr
&c. It will now be necessary to express these in terms of the
instantaneous angular velocities of the body.
219. Let us suppose in the first instance that one point in
the body is fixed. Let us take this point as the origin of co-
ordinates, and let the axes Ox, Oy, Oz be any directions fixed in
space and at right angles to each other. The body at the time t
is turning about some axis of instantaneous rotation. Let its
angular velocity be fl, and let this be resolved into the angular
velocities w^, w,, Wj about the co-ordinate axes. We have to
dx dii dz
find the resolved velocities -^ , -^ , -^ of a particle whose co-
ordinates are x, y, z.
These angular velocities are supposed positive when they tend
the same way round the axes that positive couples tend in Statics.
Thus the positive directions of Wj, «,, w, are respectively from y
to z, from z to a;, and from a; to y.
I' I
I
1;
182
MOTION IN THREE DIMENSIONS.
Let US determine the velocity of P parallel to the axis of z.
Let PN be the ordinate z, and let PM be drawn perpendicular
to Ox. The velocity of P due to the rotation about Ox is clearly
(oPM. Resolving this along ^P we get w, Pi/ sin NPM = u)^y.
Similarly that due to the rotation about Oy is — &),^a; ; and that due
to the rotation about Oz is zero. Hence the whole velocity of P
parallel to Oz is
dz
dt
= (0^1/-(0^X,
and the velocities parallel to the other axes are
dx
dy
f^ = co,x-<o,z.
220. The quantities a^, co^, Wg are called the angular veloci-
ties of the body about the axes of x, y, z respectively, but they
must be carefully distinguished from the angular velocities of any
particular particle of the body about the same axes. Let P be
any particle of the body whose co-ordinates are x, y, z, and draw
PL = r perpendicular to the axis of z. Let 6 be the angle xON,
dd
dt
then the instantaneous angular velocity of P about Oz is
But "^^ -ji = ^ jt ~ y ~ji — ^^^ ~ ^zo>i — y^f^it by substituting
for -^ , -^ , their values just found ;
die
dV
xz
yz
«3-«l7--«2^«
Hence the angular velocity of a particle about Oz is the same
as that of the body when the particle lies in the plane of xy, or
<w.
when it lies in the plane given by ^ = — ic —
<»„
If the axes be themselves moving in any manner, these equa-
tions only give the linear velocities of the particle relatively to the
axes. Thus suppose the directions of the axes to be fixed in space,
but the origin to be in motion with a velocity F whose resolved
parts parallel to the axes are respectively w, v, w. Then the
velocities in space resolved parallel to the axes will be
It' = w + to^z
to.
w'=w + ft),;/ — w^x '
V —V + (o^x
^J
FIXED VXES.
183
is of z.
dicular
clearly
/ = o),y.
bat due
tyof P
r veloci-
but they
;s of any
.et P be
,nd draw
de xON,
. dd
istituting
the same
lof xy, or
bse equa-
lly to the
lin space,
1 resolved
Ihen the
I
221. The motion being given, as before, by the linear veloci-
ties (u, V, w) of some point O and the angidar velocities (w^, &>,,
(o^fjind the equations to the central axis.
Let the same motion be also represented by the linear ve-
locities u', v', w' parallel to the axes, of some other point 0' and
by angular velocities &>,', w^, rUg' about axes parallel to the co-
ordinate axes and meeting in 0'. Let (f, rj, f) be the co-ordinates
of 0'. We have now two representations of the same motion, both
these must give the same result for the linear velocities of any
point. Hence
u + w^z-(o^ = u' + «; (z-^)- to^' {y-v)^
V + co^x — (a^z = v' + (o^ (a;-|) — (u/ (2 — ^ r (1)>
1^+ w,y - (i)^x = w'+ to^' (y-v)- <»a' (^ - ^) .
must be true for all values of x, y, z.
This gives o)j'=6),, (o^ = to^, lo^' =a>^, so that whatever origin
is chosen, the angular velocity is always the same in direction and
magnitude. See Art. 188.
Also (f, r}, f) may be so chosen that the vek uty of 0' is along
the axis of rotation ; in this case we have {u', v, w) proportional
to (a),, Wj,, tUg). The equation to the locus of 0' is therefore
u + ft).^ g" - 6)3?; _ v + w^^—a>X _ w + to, 7; — <oJ^
ft).
<u»
ft>.
.(2).
By multiplying the numerator and denominator of each of
these fractions by g>j, &>,, 0)3 respectively, and adding them to-
gether, we see that each of them is
The motion of the body is thus represented by a motion of
translation along the straight line whose equations are (2) and
an angular velocity equal to fl about it.
This straight line has been called the central axis, and the
fraction just written down is equal to the ratio of the velocity
of translation along the central axis to the angular velocity about
it, i. e. the pitch of the screw.
If the motion be such that mWj + vwjj + ^6)3 = 0, and «,, a>^,
6)3 do not all vanish, each of the equalities in (2) is zero, and
hence by equation (1) it' =0, w' = 0, t«'= 0. The motion is there-
fore equivalent to a rotation about the central axis, without
translation. This is also evident from the analogy explained
in Art. 203.
222. When the rotations are finite the corresponding formulae are somewhat
more complicated. Let the given displacement of the body be a rotation through a
'■^"
184
MOTION IN THREE DIMENSIONS.
ii
I
; 1
finite angle about an axis passing through the origin whose direction cosines are
(I, m, n). It is required to find the changes produced in the coordinates [x, y, z) of
any point P.
Let PP* be tlie chord of the arc described by P and let Q be the middle point
of PP'. Let x + Sx, y + ij/, « + & be the co-ordinates of P' and f, ?j, f those of Q.
Since the abscissas of Q is the arithmetic mean of those of P and i" we have
dx
i'J
6z
f=a!+rr; similarly 7;=y+^, f=j+ „ . Let QM be a perpendicular from Q on
2*
e
the axis, then PP' = 2 QM tan ,.
Let (\, fi, v) be the direction cosines of PP', then since PP' is perpendicular to
the axis, we have A+m/t + nv = 0, and since it is also perpendicular to OQ we have
i\ + rifi + S;» = 0, hence
\ ft _ V
mt-nii~ nii-li~ lit-m^'
The sum of the squares of the denominators is
(f ' + »?' + f«) (J" + m» + n") - (/H »»»; + nf )«,
which is OQ' - 03f*= ^37*. Hence each of these latios is = ^^ .
Now ix is the projection of PP" on the axis of x,
8 Q
.'. Zx=2Q,M . tan 5 \ = 2 tan ^ (mf - nij) ;
similarly 5y=2tan ^ (n^-l^;), &=2 tan 5 {l-ri-m^), which are the required formulso.
If the origin have a linear displacement whose resolved parts parallel to the axes
are (a, 6, c), we must add those displacements to the values of 8ar, iy, 8« found by
solving these equations. Let the co-ordinates of the middle point of the whole dis-
placement of Pbe represented by f, 1;', f . Then we have, as before, k'=x + -^ &o.,
but since 805, Sy, Sz, are increased, by a,, b, c we must write f-H»'?'-o>f'~o
^0' f 1 ij. f« We thus obtain
8«.=a-!-2tan| jn^^f -|j -n ^ij'-0j ,
with similar expressions for Sy and Sz.
223. The equations to the central axis follow from these expressions without
difficulty. The whole displacement of any point in the central axis is along the
axis, so that (^', yj, f) the co-ordinates of the middle point of the displacement are
co-ordinates of a point in the axis, and 8x, Sy, Sz are proportional to {I, m, n) the
direction cosines of the axis. Hence
a + 2tan|j^(r-|)-n(v-|)j i + 2tan|] n(r-|) -^(^-0 j
I m
c^!iun|j.(^-|).-^(f-;)j
n
Each of these is evidently equal to la + nib + nc, which is the linear displacement
along the central axis. The results of this and the preceding Article are due to
Bodrigues.
FIXED AXES.
185
coslneA are
I {x, y, z) o!
iddle point
those of Q.
P' we have
From Q ou
ndionlar to
iQ we have
id formulae.
to the axes
iz found by
) whole dis-
Sx
, b ., e
"2'* ~2
ns without
along the
sement are
m, n) the
placement
ire due to
1
224. Ex. Let the restraiuta on a body be such that it admits of two motions
A and B each of which may be represented by a screw motion, and let m, m' be the
pitches of these screws. Then the body must admit of a screw motion compounded
of any indefinitely small rotations udt, u'dt about the axes of these screws accom-
panied of course by the translations mudt, m'u'dt. Prove that (1) the locus of the axes
of all those screws is the surface z {x' + y'*) = 2ary. (2) If the body be screwed along
any generator of this surface the pitch is c + a cos 20, whore c is a constant which is
the same for all generators and is the angle the generator makes with the uxis of
X. (li) The size and position of the surface being choben so that the two given
screws A and B lie on the surface with their appropriate pitch, show that only one
surface can be drawn to contain two given "rews. (4) If any three screws of the
surface be taken and a body be displaced by being screwed along each of these
through a small angle proportional to the si^e of the angle between the other two,
the body after the last displacement will occupy the same position that it did before
the first.
This surface has been called the cylindroid by Frof. Ball, to whom these four
theorems are due.
225. Ex. 1. If an instantaneous motion be given by the linear velocities
(u, V, w) along and the angular velocities {ui, w,, u^) about the co-ordinate axes,
show that the equations to the conjugate of
-/.
I
y-g
m
z-h
are
X
w,
I
X
/
m
y
9
z
W3
n
z
h
=/u + ni» + nw,.
= (/-a;)u + (5f-y)r + (/t-2)«.
The first equation follows from the fact that the direction of motion of any
point on the conjugate is perpendicular to the given axis, and the second from the
fact that the direction of motion is also perpendicular to the straight line joining
the point to (/, g, h).
Ex. 2. If an instantaneous motion be represented by a screw along the axis of
2, the linear and angular velocities being V and 0, prove that the equations to the
X - / y-g
conjugate of —j^ = - — - =
, - aiemx-hj + n-^=0 aaigx-fy--^(z-h)-0.
I m n II if
Ex. 8. The locus of the conjugates of all axes of- instantaneous rotation which
are parallel to a fixed straight line is a plane parallel to the central axis and to the
fixed straight line.
Ex. 4. The locus of the conjugates of all axes of instantaneous rotation
which pass through a given point is a plane. If two axes intersect, their conjugates
also intersect.
226. If the instantaneous motion of a body b« represented by two conjugate
rotatioas about two axes alright angles, a plane can be drawn through either axis
perpendicular to the other. The axis in the plane has been called the characteris-
tic of that plane, and the axis perpendicular to the plane is said to cut the plane in
its focus. These names were given by M. Ghasles in the Comptes Rendus for 1843.
Some of the following examples were also given by him, though without demonstra-
tions.
186
MOTION IN THREE DIMENSIONS.
Ex. 1. Show that every plane has a characteristic and a focaa.
Let the central axis cut the plane in 0. Besolve the linear and angular veloci-
ties in two directions Ox, Oz, the first in the plane and the second perpendicular to
it. The translations along Ox, Oz may be removed if we move the axes of rotation
Or, Oz parallel to themselves, by Art. 202. Thus the motion is represented by a
rotation about an axis in the plane and a rotation about an axis perpendicular to
it. It also follows that the chaiacteristic of a plane is parallel to the projection of
the central axis.
Ex. 2. If a plane be fixed in the body and move with the body, it mtersects
its consecutive position in its characteristic. The velocity of any point P in the
plane when resolved perpendicular to the plane is proportional to its distance from
the chMacteristic, and when resolved in the plane is proportional to its distance
from the focus and is perpendicular to that distance.
Ex. 3. If two conjugate axes cut a plane in F and G, then FG passes through
the focu;.
If two conjugate axes be projected on a plane, they meet in the characteristic of
that plane.
Ex. 4. If two axes CM, CN meet in a point C, their conjugates lie in a plane
whose focus is C and intersect in the focus c ' the plane CMN.
This follows from the foct that if a straight line cut an axis the direction of
motion of every point on it is perpendicular to the straight line only when it also
cuts the conjugate.
Ex. 5. Any two axes being given and their conjugates, the four straight lines
lie on the same hyperboloid.
Ex. 6. If the instantaneous motion of a body be given by the linear and angu-
lar velocities (m, v, lo) (wp Wj, a?a)i prove that the characteristic of the plane
is its intersection with
A (u + u^z- w^y) +B (v+ u^x- uiz) + C {w + u^y - u^) = 0,
and its focus may be found from
w + Ujy - WjO!
■ ^- .
For the characteristic is the locus of the points whose directions of motion are
perpendicular to the normal to the plane, and the focus is the point whose direction
of motion is perpendicular to the plane.
What do these equations become when the central axis is the axis of z ?
Ex. 7. The locus )f the characteristics of planes which pass through a given
strai-iiht line is a hyperboloid of one sheet ; the shortest distance between the given
straight line and the central axis being the direction of one principal diameter, and
the other two being the internal and external bisectors of the angle between the given
straight line and the central axis. Prove also that the locus of the foci of the
planes is the conjugate of the given straight line.
Ex. 8. Let any surface A be fixed in a body and move with it, the normal
planes to the trajectories of all its points envelope a second surface B. Prove that
if the surface B bo fixed in the body and move with it, the normal planes to the
M + WgZ - Wjy _v+ u^ - Wj^s
A ~ B
ular velooi-
jndicular to
of rotation
sented by a
mdicular to
rojection of
it intersects
it /* in the
iitauce from
its distance
euler's equations.
187
trajectories of its points will envelope the surface A : so that the surfaces A and B
have conjugate properties, each surface being the locus of the foci of the tangent
planes to the other.
Prove that if one surface is a quadric the other is also a quadric.
Ex. 9. A body is moved from any position in space to any other, and every
point of the body in the first position is joined to the same point in the second
position. If all the straight lines thus found be taken which pass through a given
point, they will form a cone of the second order. Also if the middle points of all
these lines be taken, they will together form a body capable of an infinitesimal
motion, each point of it along the line on which the same is situate. Gayley's
Report to the Brit. Assoc, 1862.
3es through
icteri&tic of
) in a plane
iirection of
'hen it also
•aight lines
and angu-
le
notion are
direction
;h a given
the given
leter, and
the given
)ci of the
e normal
'rove_ that
28 to the
Eulers Equations.
227. To determine the general equations of raotion of a body
about a fixed point.
Let the fixed point be taken as origin, and let x, y, z be the
co-ordinates at time t of any particle ni referred to any rectangular
axes fixed in space. Let Xm, Ym, Zm be the impressed forces
acting on this element parallel to the axes of co-ordinates, and
let L, M, N be the moments of all these forces about the axes.
Then by D'Alembert's Principle, if the effective forces m
dt\
X
m -A , m, -^ a ^6 applied to every particle m in a reversed direc-
tion, there will be equilibrium between these forces and the im-
pressed forces. Taking moments therefore about the axes, we have
H^^^-y^y^- ■ «.
and two similar equations.
To simplify these equations, let a>^, w^, w, be the angular velo-
cities about the axes.
dz
^ = <.^y-a>^x;
d'x
Then -£ = co^z-(o^,
dy
G),a; — a)j.«,
Je ~^ dt
d(o do), , . , .
y-^^^y \^^y ~ ^*^) ~ "» ("'^ ~ ^'^h
dhi dw, da>- . . . , .
183
MOTION IN THREE DIMENSIONS.
■. I
Substituting in equation (1) we get
Sm (a^ + /) — - tmyz
1.
K
— Xmxy . (ft)/ — cDj,') + Sm (cc' — j/^) m^o)^ — Xmyz . 6)^6),
+ Xmxz . &)j,ft), J
The other two equations may be treated in the same manner.
The coefficients in this equation are the moments and products
of inertia of the body with regard to axes fixed in space and are
therefore variable as the body moves about. Let us then take a
second set of rectangular axes OA, OB, OG fixed in the body, and
let ei)j, ft)^, 0)3 be the angular velocities about these axes. Since
the axes Ox, Oy, Oz are perfectly arbitrary, let them be so chosen
that the axes OA, OB, OG are passing through them at the
moment under consideration. Then 0)3, = a),, w^ — a)^, co,= (o^. If
the principal axes at the fixed point have been chosen as the set
of axes fixed in the body, and A, B, C be the moments of inertia
about them, the equation takes the form
C
di
ft).
dt
-(A-B)co,a,, = N,
in which all the coefficients are constants.
228. "We shall now show that -~ = -— . This may appear
at first sight to follow at once from the equation 0)3 = w^ But it is
not so; 0), denotes the angular velocity of the body about OG fixed
in the body, while ft>, denotes the angular velocity about a line Oz
fi^ed in space and determined by the condition that at the time t
0(7 coincides with it. At the time t-\-dt OC will have separated
from Oz and we cannot therefore assert a priori that the angular
velocity about OG will continue to be the same as that about Oz.
We have to prove that this is the case as far as the first order of
small quantities. Let OR, OR' be the resultant axes of rotation
at the times t and t-\-dt, i.e. let a rotation ^dt about OR bring
OG into coincidence with Oz at the time t, and let a further
rotation ^'dt about OR' bring OG into the position OC' in space
at the time t+dt. Then according, to the definition of a differ-
ential coefficient
da>
dt
5 = X'of
ft).
^'=Z'of
dt
n' cos R'C'-n cos RG
dt
CI' COS R'z — n COS Rz
~dt
Since a rotation about OR' brings OG from the position Oz to
OG', EG' and R'z differ by quantities of the second order, and
therefore these two diflferential coefficients are ultimately equal.
J
11
euler's equations.
189
^ = N.
! manner.
d products
ce and are
en take a
body, and
es. Since
! so chosen
3m at the
>, = «8- If
as the set
I of inertia
lay appear
But it is
1 00 fixed
a line Oz
the time t
separated
le angular
about Oz.
3t order of
if rotation
OR bring
a further
in space
a differ-
jon Oz to
Irder, and
1 equal.
229. The following demonstration of this equality has been
given by the late Professor Slesser of Queen's College, Belfast, and
is instructive as founded on a different principle. Let A, B, Che
the points in which the principal axes cut a sphere whose centre
is at the fixed point. Let OL be any other axis, and let fl be
the angular velocity about it. Let the angles LOA, LOB, LOG
be called respectively a, /8, 7. Then by Art, 201
fl = &)j cos a + ftjg co^ fi + f^s cos 7 ;
do. d
w.
i<w„
•■•-^T = ^^"^"+-df^'^^'^ +
d(i)„
dt
cos 7
- <w, sma^ - ft), sin /3-^ - WjSm 7
da
dt -2 — A- ^^ -,3^... , ^^.
Now let the line OL be fixed in space and coincide with OC
at the moment under consideration. Then a. = ^, 13= ^ , 7 = 0;
therefore
d%
m
dt
dt
— w.
dot
d^
dt "^ dt
Also -T- is the angular rate at which A separates from a
dt
d/3
fi^ed point at C, this is clearly (o^. Similarly -~~= — co^. Hence
dD, _ dcO^ mv ^w, _ dM^
^~W dt ~ dt ^
dw^
dt
= ^2
dt
dt
dt
do>^
dt
230. The three equations of motion of the body referred to
the principal axes at the fixed point are therefore
^da>.
dt
.d(o„
{B-C)co,co, = L,
B"^'-{C-A)co,co,=^M,
C^^f-iA-B)co,ay, = K
These are called Euler's equations.
231. We know by D'Alembert's principle that the moment
of the effective forces about any straight line is equal to that of
the impressed forces. The equations of Euler therefore indicate
that the moment of the effective force about the principal axes
at the fixed point are expressed by the left-hand sides of the above
equations. If there is no point of the body which is fixed in
space, the motion of the body about its centre of gravity is the
same as if that point were fixed. In this case, if A, B, G be the
principal moments at the centre of gravity, the left-hand sides of
Euler's equations give the moments of the effective forces about
190
MOTION IN THREE DIMENSIONS,
'I
the principal axes at the centre of gravity. If we want the
moment about any other straight line passing through the fixed
point, we may find it by simply resolving these moments by the
rules of Statics.
232. Ex. 1. If 2T=-Awj' + Buf^' + Cu)^' and be the moment of the impressed
forces about the instantaneous axis, the resultant angular velocity, prove that
dT
dt
= GQ.
Ex. 2. A body turning about a fixed point is acted on by forces which tend to
produce rotation about an axis at right angles to the instantaneous axis, show that
the angular velocity cannot be uniform unless two of the principal moments at the
fixed point are equal. The axis about which the forces tend to produce rotation is
that axis about which it would begin to turn if the body were placed at rest.
233. To determine the pressure on the fixed point.
Let X, y, z be the co-ordinates of the centre of gravity referred
to rectangular axes fixed in space meeting at the fixed point, and
let P, Q, R be the resolved parts of the pressures on the body in
these directions. Let /it be the mass of the body. Then we have
1"^ = ^-^^''^
d^x..
and two similar equations. Substituting for -5-j its value in terms
a>„ 6)^, G), we have ,
and two similar equations.
If we now take the axes fixed in space to coincide with the
principal axes at the fixed point at the moment under considera-
tion we may substitute for —j-" and -y- from Euler's equat.c as.
We then have
with similar expressions for Q and R.
234. Ex. If G be the centre of gravity of the body, show that the terms on
the left-hand sides of the equations which give the pressures on the fixed point are
the components of two forces, one O** . GH along GH which is a perpendicular on
the instantaneous axis 01, being the resultant angular velocity, and the other
0'*. GK perpendicular to the plane OGcK, where GK is a perpendicular on a straight
B-C C-A
line OJ whose direction cosines are proportional to -
w«w,
J "81
"S"!'
A-B
Wjw,, and 0'* is the sum ot the squares of these quantities.
i
euler's equ.mions.
191
want the
I the fixed
uts by the
lie impressed
', prove that
hich tend to
is, show that
iments at the
ce rotation is
t rest.
ty referred
point, and
le body in
1 we have
le in terms
•+2wX
le with the
considera-
I equat.c as.
JV \
the terms on
ted point are
jendicular on
|nd the other
^n a straight
C-A
235. To determine the geometrical equations connecting the
motion of the body in space with the angidar velocities of the body
about the three moving axes, OA, OB, 00.
Let the fixed point be taken as the centre of a sphere of
radius unity ; let X, Y, Z and A, B, C be the points in which the
sphere is cut by the fixed and moving axes respectively. Let ZC,
BA produced if necessary, meet in E. Let the angle XZG = y(r,
ZG = 0, EGA = <f). It is required to determine the geometrical
relations between 9, </>, y^, and Wj, tu^, Wj.
Draw CN perpendicular to OZ. Then since •x^ is the angle
the plane GOZ makes with a plane XOZ fixed in space, the velo-
city of G perpendicular to the plane ZOG is GN -T , which is the
same as sin ^ -^, the radius OG of the sphere being unity. Also
the velocity of C along ZG is
de
dt'
Thus the motion of G is re-
presented by -J- and sin 6 -^ respectively along and perpendi-
cular to ZG. But the motion of G is also expressed by the angular
velocities Wj and »„ respectively along BG and GA. These two
representations of the same motion must therefore be equivalent.
Hence resolving along and perpendicular to ZG we have
dO
sin ^-i'- = —
dt
J = ft>, sin + «i>a ^^^
Wj cos ^ + Wj sin
n
192
MOTION IN THREE DIMENSIONS.
i
i '
Similarly by resolving along CB and CA we have
(»j = -^ sin 9 — -r- sm cos 9
dt
■dO
dyjr
cOjj = -^ cos <j) + ~ sin 6 sin <^
These two sets of equations are precisely equivalent to each
other and one may be deduced from the other by an algebraic trans-
formation.
In the same way by drawing a perpendicular from E on OZ we
may show that the velocity of E perpendicular to ^^is -^ sin ZE,
and this is the same as v^ cos 6.
dt
Also the velocity of A relative
to E along EA is in the same way -— sin CA, and this is the
d^
dt
same as ^ . Hence the whole velocity of A in space along AB
But this motion is also ex-
is represented by -^ cos ^ + -jt ■
pressed by to^. As before these two representations of the same
motion must be equivalent. Hence we have
a,. = ^cos^+^^.
If in a similar manner we had expressed the -motion of any
other point of the body as B, both in terms of &)j, w^, 0)3 and
By <f>, "^y we should have obtained other equations. But as we cannot
have more than three independent relations, we should only
arrive at equations which are algebraic transformations of those
already obtained.
236. Ex. lip, q, r be the direction cosines of OZ with regard to the axes OA,
OB, 00, show that these equations may be put into the symmetrical form
dr -
dp
Any one of these may foe obtained by differentiating one of the expressions
p = -Bin.0coa<f>, g = 8intf8in^, r = coBO. The others may be inferred by the
rule of symmetry.
237. It is clear that instead of referring the motion of the body
to the principal axes at the fixed point, as Euler has done, we
may use any axes fixed in the body. But these are in general so
complicated as to be nearly useless. When, however, a body is
making small oscillations about a fixed point, so that some three
rectangular axes fixed in the body never deviate far from three
axes fixed in space, it is often convenient to refer tlie motion to
ii
!
EULERrt EQUATIONS.
193
Qt to each
)raic trans-
:on O^we
-^ sin ZE,
at
f A relative
this is the
e along AB
is also ex-
of the same
tion of any
as we cannot
should only
ons of those
to the axes OA,
form
,1 = 0.
the expreflsions
infened by the
,nof thehody
has done, we
in general so
er, a body is
it some three
tr from three
tie motion to
these even though they are not principal axes. In this case
«,, ft)j, Wg are all small quantities, and we may neglect their
products and squares. The general equation of Art. 227 reduces in
this case to
at at at
where the coefficients have the usual meanings given to them in
Chap. I. We have thus three linear equations which may be
written thus :
dt dt
dt^ dt
at
-E
(U,
dt
dt dt
238. It appears from Euler's Equations that the whole changes of Wj, u.^, wj
are not due merely to the direct action of the forces, but are in part due to the
centrifugal force of the particles tending to carry them away from the axis about
which they are revolving. For consider the equation
du, N A-B
= ?. + — /?- '^'Wj-
dt C
N
Of the increase du^ in the time dt, the part -r^ dt is duo to the direct action of
wjWjj dt is due to th. centrifugal
the lorces whose moment is X, and the part — ^^
force. This may bo proved as follows.
If a body he rotating about ■. n axis 01 with an angulir velocity w, then the
moment of the centrifugal forces of the -hole body about the axis Oz is {A -B) wiWj.
Let F be the position of any particle m and let x, y, z be its co-ordinates. Then
x = OR, y=RQ, z=QP. Let PS be a perpendicular on 01, let OS=u, and PS=r.
Then the centrifugal force of the particle m is wh-m tending from 01.
II. D.
13
} ;
ri
\
, i
.':
«•(
!' ■«
■ 1 1
194
MOTION IN THREE DIMENSIONS.
The force u'rm is evidently equivalent to the four forces bPxm, uhjm, u*zm, and
- w'um acting at P parallel to x, y, z, and u respectively.
The moment of ui^xm round Oz= - u'rym \
uhjm = u^xym y,
w'zm =0 )
these three therefore produce no effect.
The force - u'um parallel to 01 is equivalent to the three, - ww, um, - wwj um,
-UW3UVI, acting at P parallel to the axes, and their moment round Oz is evidently
wum(«iy-«ax). Now the direction cosines of 01 being — , —7, -f, we get by
u
u u
<•>,
Wfl
projecting the broken line *, y, z on 01, u=-^x+ — y+ —z; therefore sub
u
u
Btituting for u, the moment of centrifugal forces about Oz is
= (ujij - Wjx) {ujx + w^y + u^z) m,
= (wi'iry + WjWay' + u^"^^ - w^w^a;^ - w^xy - WjjWaiKa) m.
Writing S before every term, and supposing the axes of a;, y, z, to be principal
axes, then the moment of the centrifugal forces about the principal axis Oz
= WiW32»i(y* - a:*) = w^Wj (A — B).
Let the moments of the centrifugal forces about the principal axes of the body
be represented by L', M\ N', so that
L' = (B-C} W4W3, M'={C-A) W3W1, N'=(A- B) Wj Wg,
and let G be their resultant couple. The couple G is usually called the centrifugal
couple.
Since L'w-^-\-M'u^-\-N'u.^=Q, it follows that the axis of the centrifugal couple is
at right angles to the instantaneous axis.
Describe the momeutal ellipsoid at the fixed point and let the instantaneous
axis cut its surface in I. Let OH be a perpendicular from on the tangent plane
at I. The direction cosines of OH are proportional to Au^, Bua, Cwg. Since
Aw^L' + BuiM' + CW3N' =0, it follows that the axis of the centrifugal couple is at
right angles to the perpendicular OH.
The plane of the centrifugal couple is therefore the plane lOH.
If /ik^ be the moment of inertia of the body about the instantaneous axis of
rotation we have *"= -y^, and T=ijJc^u)^ is the Vis Viva of the body. We may
then easily show that the magnitude G of the centrifugal couple is G=T tan <l>,
where <f> is the angle lOH.
This couple will generate an angular velocity of known magnitude about the
diametral line of its plane. By compoun-liug this with the existing angular velocity,
the change in the position of the instantaneous axis might be found.
Expressions for Angular Momentum.
239. We may now investigate convenient formuloB for the
angular momentum of a body about any axis. The importance
of these has been ah-eady pointed out in Art. 77. In fact, the
general equations of motion of a rigid body as given in Art. 71,
I
EXPRESSIONS FOR ANGULAR MOMENTUM.
195
ym, u*zm, aud
urn, -wu^um,
Oz is evidently
-3, we get by
u
therefore sub-
to be principal
ixis Oz
es of the body
the centrifugal
fugal couple is
instanianeous
tangent plane
Cwg. Since
al couple is at
aneous axis of
)dy. We may
is G'^Ttan^,
iude about the
gular velocity,
lie for the
importance
n fact, the
in Art. 71,
cannot be completely expressed until these formulae have been
found.
When the body is moving in space of two dimensions about
either a fixed point, or its centre of gravity regarded as a fixed
point, the angular momentum about that point has been proved in
Art. 88 to be Mk^o) where AW is the moment of inertia, and w
the angular velocity about that point. Our object is to find cor-
responding formulae when the body is moving in space of three
dimensions. Following the same order as in Euler's Equations,
we shall first find the angular momentum about any fixed straight
line in space, taken as the axis of z and passing through the
fixed point; secondly, the momentum about any fixed straight line
in the body and also passing through the fixed point, and lastly, we
shall show how the angular momenta about other axes may be
found.
240. A body is turning about a fi<ced point in any manner, to
determine the moments of the momentum about the axes, i.e. to find
the areas conserved round those awes. See Chap, ii; Art. 78.
Let {x, y, z) be the co-ordinates of any particle m of the body
referred to axes fixed in space meeting at the fixed point. Let
w^, Wj,, ft>, be the angular velocities of the body about the fixed
axes. Then the moment of the momentum about the axis of z is
L = 2m \x
dy <faj\
tt'^dij'
Substituting for -^ , -— their val
ues
1
dx
di/
we have Ji^ = %m («' + y^) o\ — {l,mxz) co^ — (^myz) w^.
241. The coefficients of w^, w^, w^ are the moments and pro-
ducts of inertia of the body about the axes, and if the axes be
fixed in space, these will generally be variable. In some cases it
will be found more convenient to take as axes of reference three
straight lines fixed in the body.
Let Wj, G).^, (It be the angular velocities of the body about rect-
angular axes Ox , Oy, Oz fixed in the body and meeting at the
fixed point 0. Since in the sxpression given above for hg the
fixed axes may be any whatever, let them be chosen so that the
moving axes coincide with them at the time t. Then, (0^= a^,
13—2
t ■
/I
■; i]
1
190
MOTION IN THREE DIMENSIONS.
W»=a>8»
(o, — a>,, and the moment of the momentum about the
moving axis of z will be expressed by the form
where C7=2w(ar" +y«). E = Xmx2\ D^Xmy'z.
These will be constant throughout the motion, and their values
may be found by the rules given in Chapter I.
If the axes fixed in the body be principal axes, t\. i die pro-
ducts of inertia will vanish. The expressions for the moments of
the momentum will then take the simple forms
hi — -4a),
A,' = Bu)^
K = ^^z
where A, B, C are the principal moments of the body.
Let the direction-cosines of the axes fixed in space but moving
with reference to axes fixed in the body be given by the following
X
y
z
a:, y,
o,, a,
K
c.
a'
a.
&o, K
'8'
1'
'U*
diagram ; where, for example, \ is the cosine of
the angle between the axes of z and y'. It has
just been proved that the resultant of the mo-
menta of all the particles of the body is equiva-
lent to the three "couples" h^, h^, h^ about the
axes Ox', Oy\ Oz'. Hence the moment of the
momentum about the axis of z which is fixed in space may be
v/ritten in the form
which will be frequently found useful.
242. It may be required to find the moment of the momen-
tum about axes neither fixed in space nor in the body, but moving
in any arbitrary manner. This will be expressed by the same
form as if the axes were fixed. If at^, (Oy, tw, be the angular
velocities about these axes, the moment required will be
= Xm (a;* i-y^) w, — (Zmxz) a>^ — {Xmyz) a^.
If the axis of z coincide with the instantaneous axis of rotation,
«,, = (), (Uy=0, and ft)j is the resultant angular velocity. The ex-
pressions for the moments of the momentum or areas conserved
about the axes of x, y, z become respectively
- i^mxz) w„ - (Xmyz) &>,, tm {x^ f ?/") w,.
The axis of the couple which is the resultant of the moments
of the momentum about the axis is sometimes called the resultant
axis of angular momentum and sometimes the resultant axis of
areas. It is to be remarked that this axis does not in general
ON MOVING AXES AND RELATIVE MOTION.
197
1 about the
their values
. i cUe pro-
moments of
but moving
be following
X, y, z
ace may be
he m omen-
but moving
Y the same
le angular
of rotation.
The ex-
conserved
e moments
e resultant
mt axis of
in general
coincide with the instantaneous axis of rotation. The two are
coincident only when the axis of rotation is a principal axis. If a
body be turning about a straight lino, which we may call the axis
of z, as instantaneous axis, it is a common mistake to suppose that
the angular momentum about a perpendicular axis is zero. We
see from the last remark that this is not generally true.
If it be required to find the moment of the momentum about
the axis of ^ of a rigid body moving in any manner in space, we
may use the principle proved in Chapter II. Art. 76.
In the case of a system of rigid bodies, the moment of their
momenta may be found by adding up the separate moments of the
several bodies.
Ex. 1. A triangnlar nroa ACB whose mass is M is turning roimd the side CA
with an angular velocity u. Show that the angular momentum about the side CB
is -^ Mob sin- Cw, where a and b are the sides containing the angle C.
Ex. 2. Two rods OA, AB, are hinged together at A and suspended from a
fixed point 0. The system turns with angular velocity u about a vertical straight
line through so that the two rods are in a vertical plane. If 0, <p be the inclina-
tions of the rods to the vertical, a, b their lengths, M, M' their masses, show that
the angular momentum about the vortical axis is
w [( J i»f + M' ) a" sin" d + M 'ab sin tf sin + J i/ 6" sin* </>.]
Ex. 3. A right cone, whose vertex is fixed, has an angular velocity u com-
municated to it about its axis OC, while at the same time its axis is set moving in
space.
The semi-angle of the cone is - and its altitude is h.
If be the inclii
tion of the axis to a fixed straight line Oz and f the angle the plane zOC makes
with a fixed plane through Oz, prove that the angular momentum about Os is
f Mh^u (sin' -j^ + i cos 0), where M is the mass of the cone.
Ex. 4. A rod AB is suspended by a string from a fixed point and is moving
in any manner. If {I, m, n) {p, q, r) be the direction cosines of the string and rod
referred to any rectangular axes Ox, Oy, Oz, show that the angular momentum
about the axis of z is
dp
.,.«/, <^'» dl\ .,a^ I dq dp\ , .,ab f dm dp , ,dq dl\
'^^Vw-'''dt)'-^jVi-^i)-'''2[pdt-'''i+'Tr^di}'
where M is the mass of the rod, and a, b the lengths of the rod and string.
On Moving Axes and Relative Motion.
243. In many cases it will be found convenient to refer the
motion of the body under consideration to axes moving in space in
some manner about a fixed origin. If we refer the motion of these
axes to other axes fixed in space we shall have the inconvenience
of two sets of axes. For this reason their motion at any instant is
sometimes defined by angular velocities {0^, 6^, 0^) about them-
I
I
! ;
:<
198
MOTTON IN THREE DIMENSIONS.
selves. In this case wo are to regard the axes as if they were
a material system of three straight lines at right angles whose
motion at any instant is given by three coexistent angular velo-
cities about axes instantaneous! v coincident with them.
When the axes are moving we may suppose the motion of the
body to be determined by the three angular velocities a>,, ta , tw,
about the axes, in the same manner as if the ax'^s were fixed for
an instant in space. The position of the body at tho time t + dt
may be constructed from that at the time t by turning the body
through the angles (o.dt, (o^dt, (o,^dt successively round the instan-
taneous poHition of the axes. liut it must be remembered that
w^dt does not now give the angle the body has been turned
through relatively to the plane xz, but relatively to some plane
fixed in space passing through the instantaneous position of the
axis of z. The angle turned through relatively to the plane o*" xz
is {o>,- 0,) dt
244. To find the resolved part of the velocity of any particle
parallel to the moving axes.
The resolved parts of the velocity of any point whose co-
ordinates are {x, y, z) are not given by ^ » T/f » ^ • These are
the resolved velocities of the particle relatively to the axes. To
find the motion in space we must add to these the resolved veloci-
ties due to the motion of the axes themselves. If we supposed the
particle to be rigidly connected with the axes, it is clear that its
velocities would be expressed ^y the forms given in Art. 219 with
^,, Q^, 6^ substituted for w,, w^, «g. So that the actual resolved
velocities of the particle are
dX y. n
*'=^-«^i + «^^3.
dt
dZ y, ^
245. To find the accelerations of any particle p)arallel to the
axes we may proceed thus.
The velocities of the particle at the time t resolved parallel to
the axes Ox, Oy, Oz are respectively («, v, w). At the time t-\- dt,
the axes have been turned into the position Ox, Oy, Oz by
rotations equal to d^dt, d^dt, d^dt round the axes Ox, Oy, Oz
respectively, and the velocities of the particle parallel to the axes
in their new position are
, dn ,. , dv ,^ dw ,
w + -J- «<, V 4- -Tzdt, w + y dt.
at dt dt
ON MOVINQ AXES AND RELATIVE MOTION.
199
<^,„ (o.
xz
the
Describe a sphere of unit radius whose centre is at the fixed
origin and let all these axes cut the sphere in the points x, y, z,
x\ y , z respectively. Thus we have two spherical triangles xyz
and x'y'z', all whose sides are right angles. The resolved part of
the velocity of the particle at the time t + dt along the axis of z is
H'r
w + ^dt)
COS zz .
at dt) COS zx' + (v + -£dt) COB zy' + (v., ^^
By the rotation round Oy, x' has receded from z by the arc O^dt,
and by the rotation round Ox, y' has approached z by the arc 6^dt.
Therefore
zx =zx + $,^ dt,
zy' = zy — 0^ dt.
Also the cosine of the arc zz differs from unity by the squares
of small quantities. Substituting these we find that the compo-
nent velocity of the particle at the time t + dt parallel to the axis
of z is ultimately
w + -r-dt — u9^ dt + vd^ dt.
But the acceleration is by definition, the ratio of the velocity
gained in any time dt to that time. Hence if Z be the acceleration
resolved parallel to the axis of z, we have
Similarly if X and Y be the accelerations parallel to the axe ^
of X and y, we have
r= *-,„(>.+<.
246. Ex. 1. Let the motion be referred to oblique moving axes so that the
sides of the spherical triangle xyz are a, b, c and the angles A, £, C. Let the equal
quantities sin a sin 5 sin C, sin 6 sine sin J, sin c sin a sin ^ be called /i. Prove
that if the velocity be represented by the three components u, v, lo parallel to these
axes, then the resultant acceleration parallel to the axis of z is
_ dw du , dv ^ ^
Z = -TT + -i7C0s6 + 3- cosa-«tf„u+vtf,u,
at at at
with similar expressions for X and T.
This may be done by the use of the spherical triangles xyz, afy'sf, by first proving
that za! =b + O^dt 3in c Bin A, zy'=a-6idt sine Bin B, and then substituting as before.
Ex. 2, Prove in the same way that it x, y, z be the co-ordinates referred to
oblique axes, and u', v', w' the resultant velocities parallel to the axes,
, dz dx , dy
w =^+ -TrCOsJ+^cosa-x^gM + y^iA*,
with similar expressions for «' and f '.
('
I, '.
( <
' I
r
\A
• :-!■
1 ii^;
1:1
I •
200
MOTION IN THREE DIMENSIONS.
Ex. 3. Prove also that the equations connecting u, v, w with the co-orJinatea
are
sin'c
-cotfi
-cot^
h
Ox
e.
z
X
y
with two similar expressions for u and v.
Since w' is the resolved velocity parallel to 2 of (m, v, w,) we have
« cos 6 + V cos a + to= w',
with similar expressions for u' and 1/, By solving these we get the required values
of W, I', w.
Ex. 4. If the whole acceleration be represented by the three components
X, Y, Z parallel to the axes, prove that the expressions for these in terms of uvw,
may be obtained from those given in the last example by changing x, y, z into u, v, w
and M, V, to into X, Y, Z.
247. To express *lie geometrical conditions that a straigitt line
whose equations with reference to the moving axes are given is fixed
in direction in space.
Let the equation to the given straight line be
V 9.
r
and let the equations be so prepared that {p, q, r) are the
direction cosines of the line. Let a straight line be drawn through
the origin parallel to this given straight line and let a point Pbe
taken on this at any given distance L from the origin 0. Then
the co-ordinates of P are pL, qL, rL respectively. Since the
straight line OP is fixed in direction in space, the resolved parts
of the velocity of P parallel to the axes are zero. Hence we have
dip
dt
- Lq9^+ Lrd,^^ 0,
and two similar equations. The required geometrical conditions
are therefore
f^-qe,+re, = o,
§->-^. + ?'^3 = o,
%-pe,+qe, = o.
When it is necessary to refer the motion of those moving axes
to otl.cr axes fixed in space, we may cither use tlie equations of
this article or those of Art. 23o. Taking the notation of the
ON MOVING AXES AND RELATIVE MOTION.
201
30-orJinates
lirecl values
omponents
ns of uvw,
into M, V, w
\ight line
is fixed
diti
ous
g axes
oils of
f the
are the
through
Dtp be
Then
ice the
',/
d parts
. '■
e have
■■'
'■:]
article referred to, it is obvious (the axes being treated as a body
consisting simply of three straight lines) that we shall have the
results
-^ sin ^ = — ^j cos <f> + 0^ sin <f)
de
.- = 6^ sin ^ + 6.^ cos ^
These equations will determine 6, 0, sjr in terms of the angular
velocities 6^,6^, 6^.
248. To express the geometrical conditions that a point ivhose
co-ordinates tuith reference to the moving axes are (x, y, z) is fixed
in space.
This may be done by equating to zero the resolved parts of the
velocity of the point as given in Art. 244. If the origin of the
moving axes be fixed, the conditions are
and two similar equations. If the origin be in motion, lot u^, %, w^
be the resolved parts of its velocity parallel to the axes, then the
required conditions are clearly
and two similar equations.
249. Ex. Let the direction cosines of a straight line OM fixed relatively to the
moving axes be (X, /*, /') and let it be required to refer the motion of OM to some
straight line OL fixed in space whoso direction cosines at the time t are (p, q, r).
Jjet the angle LOM be and let \p bo the angle the plane LOM makes with any
fixed plane in space passing thi-oiigh OL. Then show that
COB 9 =p\ + qiJ, + rv, \
sin" Y* = ^1 (/) - \ cos 0) + 03{q-fi cos 0) + 0.i{r-v cos 0)1 '
If 0,, 0„ be the resolved parts of the angular velocities about OL, OM respec-
tively, the last equation may bo written in the form
Biu«^^=fl,-^„C08tf.
I If the straight line OM be not fixed relatively to the axes, then (X, fi, v) will b(
variable and we must add to the right-hand side of the second equation the deter-
minant
r dn d\\ ( dv dit\ ( d\ . dv\
[''di-''di)''[''dt-''di)p + Vrt-^d-t)'^'
i n
; s
u
\ ■•■'
ft
■I l[
t
I -, I
202
of t«o- 1-- . .
»«» will !,„ ,";;'' 6'™" hy Art. 2M, ai.,1 fw _.. ■^' "' ™l«»«e» poral.
I8l lo .1, "^ l"^' "' 'I !» tile «<, „„i • , Wlooifj
tiie moment about OL
« we effect thene substituH '^^ ~^^+^'^^ + ^'-.
230 7'„ ■ . '"''"y.ft-.n.e
• ^^ '"P^^""^ method of clu,«„- ^
S/ri^?™poS ;S Pr* a„d „e have e.
^Kt.*o 1 0,
To determine the rp]o+- , .7
*hus. Let (?/ . " ^'*^^^^^ § and ^ ,.
niovinc.ayp.rn ^^ ''^^^3^ straio-ht Jjnp ^ f . ^^ ^^ "^^3^ Proceed
fi = - CO. ' ™ "^ ■'" ^f '•"IS "^ °'
^
(^^
tf(
^-«« + ^^cos;3 + ^
f/7
ON MOVING AXES AND RELATIVE MOTION.
203
Since OL is any fixied line in space, lot it be so chosen that the
movinjf axis of z coincides with it at the time t
Then a = '^,
Since o is the angle OL
B— ^ , and 7 = 0, also -7- will be - ''
'^2 ' at dt
d%
makes with the moving axis of x, -r. is the rate at which the axis
of X is separating from a fixed straight line coincident with the
. . dB
axis of z and this is clearly 6,^. Similarly -^
da>. do). n , a
Similarly
— ^j, hc^ce
da^
dt
d(o^
~dt
d(o,, da)„ n n
If we substitute these expressions in the given general equation
we shall have the corresponding equation referred to moving axes.
If the moving axes be fixed in the body, and move with it, we
have ^j = o>j, 6^=-(o^, d^=o)^. In this case the relations will
become -"^^ -—' d^y_d^, ^^ _ ^ as in Art 290
become ^^- ^^, dt~ dt' dt - dt ' ^^ '"^ ^^^- '^-'^•
The preceding proof of the relation between —j^ and -~
IS a
simple corollary from the parallelogram of angular velocities. The
result will therefore be true for any other magnitude which obeys
the " parallelogram law." In fact the demonstration is exactly the
same. Now linear velocities and linear accelerations do obey this
law. Hence the expressions obtained in Arts. 244, 245, for the
velocities (m, v, w) and the accelerations {X, Y, Z) may be deduced
from the one proved above.
If the general equation i/r = should contain the velocity or
acceleration of any particle of the body, then to obtain the corre-
sponding equation referred to moving axes, we must substitute for
these velocities or accelerations the expressions found in Arts. 244
and 245.
251. If the general equation sliould contain - — j' or any other second differen-
tial coefficients, the expressions to be substituted for them become more compl •
catod.
'I
I
204
MOTION IN THREE DIMENSIONS.
'■. u;
i ii
Since -^ , -r^ , -y-' , being angular accelerations, follow the parallelogram law,
ftt »C («C
we have
dt
= (-^ - Wa^3+ ta-iOa) COS0+ f ^ - wA + '^1^3) ''"^ ''^ ( ^' " "1^8 + <^A) COS 7.
We may repeat the same reasoning and we shall finally obtain
So we may proceed to treat third and higher differential coefficients.
2.52. A body is tuiiiing about a fixed point in any manner,
to determine the moments of the effective forces about the axes.
Let {x, y, z) be the co-ordinates of any particle m of the body
referred to axes fixed in space and meeting at the fixed point,
and let h^, h^, h^ be the moments of the momentum about the
axes.
The moment of the effective forces about the axis of z is
0.
and this may be written in the form
dh^
It
Thus the moments of
the effective forces about axes Ox, Oy, Oz fixed in space are
respectively ,S ->,-, -7.-, where A,, \, h^ have the values
found in Art. 240.
Let A,', ^/, /ig' be the moments of the momentum, found by
Art. 242, about axes Ox', Oy, Oz moving in space about the fixed
origin. Let d^, 6^, 0^ be the angular velocities of these i xes about
their instantaneous directions. Then since moments or *.uaples
follow the parallelogram law, we see by the proposition of Art. 250
that the moments of the effective forces about the moving axes
are respectively
ff-V^3 + W.
^-h:e,^hx,
'J^-h;d, + h^e,.
If the moving axes be fixed in the body, wo have 0^ = ro^ ,
0g=ft)j, ^g — (Wg, and the equations admit of some simplification.
If the axes be the principal axes we have h^ — Aw^, h^'=Ba)^,
|1 (:
ON MOVING AXES AND RELATIVE MOTION.
205
Ag' = Ca)g, and the moments of the effective forces take the simple
forms
dt
(A -B) (0^(0^,
where A, B, C are the principal moments. See Art. 230.
If it be required to find the moment about the axis of ^ of the
effective forces on a rigid body moving in any manner in space,
we may use the principle proved in Chap. ll. Art. 72.
In the case of a system of rigid bodies, the moment of their
effective forces may be found by adding up the separate moments
of the several bodies.
253. To obtain the general equations of motion of a system of
rigid bodies.
These equations have been already obtained in Chap. Ii. Art. 83,
when the system is referred to axes fixed in space. If the axes be
moveable we must replace the accelerations -n > 'jit ~jf. t>y the
corresponding forms in Art. 245 and the couples -y,*, -y^, -^
by the expressions in Art. 252.
Thus, suppose we refer the motion to three axes moving in
space about a fixed origin 0. Let X, Y, Z be the impressed
forces on any rigid body of the system, including the unknown
reactions of the other bodies of the system. Let L, M, N be the
moments of these forces about axes drawn through the centre of
gravity of the body parallel to the co-ordinate axes. Let m be the
mass of the body. Then if we adopt the notation of Arts. 245
and 252, the equations of motion for the rigid body under con-
sideration will be
at ^ VI
dv n n Y
dt * ' m
dw a y a ^
dt * ' m
-:i
r 4
^i^.tM'uJlm
I \l
>.ll
206
and
MOTION IN THKEE DIMENSIONS.
dh;
dt
-Ke, + h:0, = L,
dt
- h:d, + k;e, = M,
dh:
dt
- hX + K^^ = ^^.
where h^, hj, h^ have the values given in Art. 240 *.
Similar equations will apply for each body of the system.
Besides these dynamical equations there will be the geome-
trical equations expressing the connections of the system. As
every such forced connection is accompanied by some reaction,
the number of geometrical equations will be the same as the
number of unknown reactions in the system.
Thus we have sufficient equations to determine the motion.
254. If two of the principal moments at the fixed origin
are equal, it is often convenient to choose as axis of z the axis
OG of unequal moment, and as axes of x, y two other axes OA,
OB moving in any manner round OG. Let ;^; be the angle the
plane oi AOG makes with some plane fixed in the body and pass-
ing through OG. Then we have 0^= Wj, 6^= w^, and d^ = 0)3+ -^ .
Also by Art. 241, we have h^'=A(o^, A„' = Aco^y h^ = Cto^
equations of motion of Art. 253 now become
dt
The
dt
In this case the most convenient geometrical equations to
express the relations of these moving axes to straight lines fixed
in space will be those given in Art. 235.
flfv
Since -^ is arbitrary, it m?.y be chosen to simplify either the
dynamical or the geometrical equations.
• The equations of Art. 253 were first given in this form by Prof. Slesser to
whom the equations of Art. 254 had been shown by the author. It appears however
that similar results had been previously published in Liouville's Journal in 1858.
em.
5 geome-
,em. As
reaction,
e as the
otion.
origin
' the axis
axes OA,
ingle the
and pass-
dy
'3'
The
Itions to
les fixed
ther the
ISlesser to
h however
In 1858.
ON MOVING AXES AND RELATIVE MOTION.
207
First, we may put -^ — ~^v The dynamical equations ihen
become
^'dt
--=iV.
dx
Secondly, we may so choose -^ that = 0. In this case the
plane COA always passes through a straight line OZ fixed in
space. The geometrical equations then become,
dd
d^ . ^
_dx d±
dr dt""^^^-
(O.
S"
d(o^_M
dt~ A'
da)^_N
255. If three principal moments at the fixed origin be
equal, there are three sets of axes such that when the motion is
referred to them, the equations take a simple form.
First. We may choose axes fixed in space. Since every axis
is a principal axis in the body, the general equations of motion
become
da>^ _ L
~dt~J.* dt ~ A' dt ~ A
The geometrical equations of Art. 235 are not required.
Secondly. We may choose one axis as that of OC fixed in
space and let the other two move round it in any manner, then as
in Art. 254, the equations of motion become
da). dy M
dt
d
CO.
dt
N
A
Thirdly. We can take as r xes any three straight lines at right
angles moving in space in any proposed manner. The equations
of motion are then by Art. 253
dt
day,
dt
-0)3(9,4-0)^^3=^,
. - 7V"
n
■ 'til
20S
MOTION IN THREE DIMENSIONS.
i' :
The geometrical equations will then be the same as those
givon in Art. 235 or Art. 247.
256. Ex. An ellipsoid, whose centre is fixed, contracts by cooling and being
set in motion in any manner is under tbo action of no forces. Find the motion.
The principal diameters are principal axes at throughout the motion. Iiet us
take them as axes of reference. The expression? for the angular momenta about
the axes are by Art. 241 h^'=Auy, hs=Bu^, h^'=Cu^. The equations of Art. 263
then become
d
dl
d
dt
d
dt
{Bu)^-{C -A) W3Wi =
Multiplying these equations by A Wj, Bu^, C'wg, adding and integrating we see
that A^u^-\-B^(j)^ + C'^<j).^ is constant throughout the motion. To obtain another
integral, let A = Af,f[t), B = Sof{t), C-Caf{t) where /(«) expresses the law of cool-
ing which has been supposed such that the body changes its form very slowly. Let
"if{f)~^n w^/CO -^ai "3/(0=^3, and put - -. =-rr-:, then the equations, become
ill J (t)
(B^-C,)Q.fl., = 0,
and two similar equations. These may be treated as in the chapter on the motion
of a body under no forces. Liouville's Journal.
257. The theory of relative motion is best understood by
viewing it in as many aspects as possible. We shall, therefore,
now consider a method of determining the motion which is more
elementary, and does not, in the result, make an exclusive use of
Cartesian co-ordinates.
Let it be required to refer the motion of a particle P to any
given system of moving axes. The motion of these axes during
any interval of time dt may be constructed by a sere \v -motion
along and round some straight line 01. Let Udt be the transla-
tion along and D.dt the rotation round 01, Let P^ be the position
of P at the time t, and let P^ be attached to the given axes and
move with them during the interval dt. Let / represent the
acceleration of P^ in direction and magnitude. The particle P
will, of course, separate from P^ ; but, as is explained in Dynamics
of a Particle, the actual acceleration of P in space is the resultant
of its acceleration relative to P^ treated as a fixed point, and the
acceleration f of P„.
To find the acceleration relative to P^, we must treat P^ as a
fixed point. Draw P^z parallel to 01 and let P^y be the projec-
tion of the direction of the relative motion of P on a plane perpen-
dicular to P^z, and let P^cc be perpendicular to P^y and P^z.
Tliese axes are taken for the purposes of description, and but little
ON MOVING AXES AND llELATIVE MOTION.
209
nation!, become
)ter on the motion
and Imt little
use will be made of co-ordinates. Let these axes move during the
time dt, so as to preserve unchanged the angles they make with
the given axes of reference. Let I\P^ be the displacement of F
relative to P^, and let P^Pi make an angle with P^z', so that
P P sin ^ is the projection of the relative displacement on the
plane of x'y'. Since these axes, in the interval of time dt, have
turned round P^z through an angle fldt, the x co-ordinate of P,
after that interval, is greater than what it would have been if
referred to axes fixed in space by Po^i ^^^ O^dt, while the y and z
co-ordinates are unaltered. We have here, according to the
rules of the Differentia^ Calculus, retained only the lowest powers
of the small quantities which occur. Hence, if the acceleration
of P relative to these axes be compounded with an acceleration
equal and opposite to that which would produce a displacement
PjPj sin dndt, we shall have the acceleration of P relative to axes
whose directions are fixed in space, but having the moving point
Pq as origin. Let F'be the velocity of the particle relative to the
moving axes, then PJP^= Vdt in the limit, and therefore tho
change hx in the x co-ordinate of P is hx' = VH sin 6 (dt)^. If
/' be the acceleration corresponding to this displacement, we
have 8x' = ^/' (dty. Comparing these two expressions we sec
that /' = 2 Vh sin 0. This acceleration must be supposed to act
along the positive direction of the axis of x'.
The general conclusion is that the acceleration of P in space is
the resultant of the accelerations /, — /', and the acceleration
relative to the given moving axes.
The equations of motion of a particle being comprised in tho
formula, "acceleration in any fixed direction ec^uals the impressed
force divided by the mass," it is nxore convenient to transpose the
terms / and — /' to the other side of the equation with opposite
signs, we then have the following theorem :
In finding the motion of a particle of mass m tuith reference
to any moving axes, ive may treat the axes as if they tvere fixed in
space, provided we regard the particle as acted on, in addition to the
impressed forces, by two forces:
(1) a force equal and opposite to that which would constrain
the particle to remain fixed to the moving axes, and which is mea-
sured by mf where f is the reversed acceleration of the j^oint of
moving space occupied by the particle,
(2) a force perpendicidar to both the direction of relative
motion of the particle and to the central axis or axis of rotation of
the moving a^es, and which is rneasured by 2mVn sin 0, where V
is the relative velocity of the particle, H the resultant angular velocity
of the moving axes, and the angle between the direction of the
velocity and the axis of rotation.
IV; 1
"I 1
i ! i I
m
I
R. D.
14
210
MOTION IN THREE DIMENSIONS.
To find the direction of this last force, we notice that in the
investigation, the rotation H lias been supposed to be, as usual,
from the positive direction of x to the positive direction of y\ and
that the positive direction of y is a tangent to the projection of
the relative velocity of P. Since the force acts along the positive
direction of x, we have this rule : Stand with the back along the
axis of rotation, so that the rotation appears to be in the direction
of the hands of the watch ; then vietuing the particle recedinj from
the axis of rotation, the force acts on the left Jiand. We may call
these forces respectively the fo7xe of moving space, and the corn-
pound centrifugal force of the particle.
268. This method of determining the relative motion of a particle was first
given by Clairaut in 1742, and afterwards the same rule was demonstrated in a
different manner by Coriolis. The arguments of the former were criticized and
improved by M. Bertrand in a paper published in the nineteenth volimie of the
Journal Polytechnique. We have here followed, with but slight variations, M.
Bertrand's mode of proof, as being the most different of any from the analytical
methods given in this chapter. But it will be important to perceive the connection
between the two methods of expressing the relative motion, and this will be
explained in the next article.
259. Let us refer the motion of P to any moving axes having
a fixed origin, and let X, Y, Z be the impressed forces on the
particle resolved parallel to the axes. If we eliminate u, v, w
from the equations of Art. 244 and Art. 245 we get
X_ d^x _ dy
m df dl
^^0, + 2^^^e, + Ax-\-By+Cz,
with similar expressions for Y and Z. Here A, B, C are functions
of 6^, 6 J, ^3 and their differential coefficients with regard to t,
which it is unnecessary to write down. If x, y, z were constants,
all the terms of X would disappear except the three last. These
then with the corresponding terms in Y and Z e.. press the acce-
leration of a point P^ rigidly attached to the axes, but occupying
the instantaneous position of P. The second and third terms of
X taken together, with the corresponding terms of Y and Z,
express the resolved parts of an acceleration perpendicular both
to the resultant axis of the rotations d^, O^y 0^, and to the direc-
tion of the velocity which is the resultant of tt , -^ , -57 . By
adding up the squares we easily find the magnitude of the re-
sultant acceleration to be 212 V sin 0, where 12, V and have the
meaning given in Art. 2-57*.
* Another demonstration by the use of polar co-ordinates is given in Vol. xir. of
the Quarterly Journal of Matliematics, by the Kev. H. W. Watson.
ON MOVING AXES AND RELATIVE MOTION.
211
In in Vol. xn. of
To determine the manner in which these forces should bo
applied, we must transpose the terms which represent them to the
other sides of the equations. The first equation will t
then become
«|f = A'+2™(|«.-*«,)-,«(J. + % + (7.),
and the other two will take similar forms. These are the equa-
tions of motion of a particle referred to fixed axes, moving under
the same impressed forces as before, but with two additional forces.
These are, first, a force equal and opposite to that represented by
fnf, where / is the acceleration of the point of moving space occu-
pied by the particle ; and secondly, a force whose magnitude has
been shown to be 2»iFfl sin 6. To determine the direction of this
force, let the axis of z be taken along the instantaneous axis of
rotation of the moving space, and let the plane of yz be parallel to
the direction of motion of the particle, then Q^ = 0, ^^ = and
= 0. We then easily see that this force disappears from the
dt
A
(fz
equations giving m -— and m t-jj ; while in that giving m
dy
we have the single term 2w -^. By
The magnitude of this force is
obviously 2m Vil sin 6, and it acts along the positive direction of
the axis of sc. This is the left-hand side when the receding parti-
cle is viewed from 'the axis of rotation and the rule given at the
end of Art. 257 is therefore established.
When these equations have been integrated, the arbitrary con-
dx
stants are to be determined from the initial values of .r, y, z, -,- ,
^ dt '
-4-, -t: . These differential coefficients are clearly the components
[of the initial velocity of the particle, taken relatively to the mov-
I ing axes.
260. Ex. If the particle be constrained to move along a curve which is itself
j moving in any manne", the compound centrifugal force, being pt rpendiciilar to the
jdirection of the relative velocity of the particle, may be included in the reaction of
Ithe ciurve. The only force which it is necessary to impress on the particle is the
jfcrco of the moving space. If the curve be turning about a fixed axis with an
jangular velocity fi in the manner described in Art. 181, the components of the
|accclerating force of moving space are clearly fi'j- tending directly from the axis of
rotation, and , r perpendicular to the plane containing the particle and the axis,
lere r, as in the article referred to, is the distance of the particle from the axis.
261. In finding the compound centrifugal force it will be
found useful to remember, that we may resolve the angular velo-
14—2
ill- -I
I - ' -
I )
_ \
I
I
IB'
212
MOTION IN THREE DIMENSIONS.
city n or the linear velocity V in any manner tlu-t wo please,
and find the forces due to each of the components separatt^ly.
Though we have th\i8 more than two forces which must be applied
to the particle, yet, by making a proper resolution, some of these
may produce either no effect, and may therefore be omitted, or
may produce an effect which it may be easy to take account of.
262. When we wish to determine the motion of a rigid body
by this method, we mtist consider each particle to be acted on by
the two forces corresponding to the position and velocity of that
particle. This will generally require an integration to be per-
formed ; which, though not difficult, is not always convenient.
The forces of moving space for any body are the same as the
effective forces of an imaginary body occupying the instantaneous
position of the real body, and moving with the space occupied by
it. The resultant of these forces may, therefore, be found by the
method indicated in Art. 83.
The components of the compound centrifugal forces on any
particle are, by Art. 259, algebraic functions of -^ , -4:,-t.' We
may, therefore, use Art, 14 to help us in finding the resultants
of the compound centrifugal forces of the whole body. If M
be the mass of the body, V the velocity of its centre of gravity^
n the angular velocity of the moving space, the angle between
the direction of F and the axis of fl, then the compound centri-
fugal forces of all the particles of the body are equivalent to a
force 2i/Fn sin 6 acting at the centre of gravity perpendicular
both to its direction of motion and the axis of 12, together with
the compound centrifugal forces of the body after the centre of
gravity has been reduced to rest.
To find these latter forces, let us refer the body to the princi-
pal axes at the centre of gravity as axes of co-ordinates. Let
ft),, ft),, ft)3 be the resolved angular velocities of the body, £i^,£i^, H,
the resolved parts of 12 about these axes ; A, B, C the principal
moments of inertia at the centre of gravity. Then, by Art. 259,
the compound centrifugal forces on any particle of the body whose
co-ordinates are {x, y, z) and mass m, are
X = m-
with similar expressions for Y and Z. The centre of gravity being
at the origin, the resultant forces of these are easily seen by inte-
gration to be all zero, while the resultant couples about the axes
are
with similar expressions for M and N.
ON MOVING AXES AND RELATIVE MOTION.
213
203. Ex. 1. A disc of mass 3/ is constrained to movo in a piano tinder any
forces while the plane turuH about a straight lino parallel to the piano and distant
a from it with angular velocity 0. Show that in finding tlie motion of the disc, wo
may regard the plane as fixed, provided wo impress on the disc in addition to tho
given forces, (1) a force Milh'- Ma . acting through tho centre of gravity tending
directly from tho projection of tho axis of rotation on the plane, where r is tho
distance of the centre of gravity from tho projoction, (2) a conide FU^ where F is
the product of inertia ahout two roctanguliir axes in tho plane intorsccting at tho
centre of gravity, and respectively i)arallol to the axis and perpendicular to it.
The constants of integration are to be determined from the initial conditions taken
relatively tu tho moving plane.
Ex. 2. A disc of mass M is constrained to movo in a piano under nny forces
while the plane turns witli angular velocity 12 about a straight lino perpendicular to
its plane and cutting tho plane in the point 0. Show that we may regard tho plane
as llxed provided wo impress on tho disc (1) a force AtU^r acting at the centre of
gravity and tending directlj fiom tho axis, where r is the distance of tho centre of
gravity from the axis, (2) a force Mr -j- acting at the centre of gravity perpendicular
to r in the direction opposite to the rotation, (3) a couple Mlfi-r , where Mh"^ is tho
moment of inertia of tlie disc about an axis through its centi-o of gravity perpen-
dicular to its plane, (4) a force 2M VQ acting at tho centre of gravity perpendicu-
lar to its direction of motion, where V is the velocity of the centre of gravity.
Ex. 3. A sphere of mass M moves in space, show that tho compound centri-
fugal forces of all its elements are equal to («1) a resultant force 2M FJi sin acting
at the centre of gravity, where V is the velocity of tho centre of gravity and fi the
angular velocity of the moving space and $ the angle the direction of V makes with
tho axis of 0, (2) a couple Mh^Qu sin (j>, where w is the angular velocity of the
sphere, the angle its instantaneous axis makes with the axis of 0, and the plane
of the couple is parallel to the axes of n and u.
On Motion relative to the Earth.
2G4. The motion of a body on the surface of the earth is not
exactly the same as if the earth were at rest. As an illustration
I of the use of the equations of this chapter, we shall proceed to
I determine the equations of motion of a particle referred to axes of
I co-ordinates fixed in the earth and moving with it.
Let be any point on the surface of the earth whose latitude
lis \. Thus \ is the angle the normal to the sr^rface of still water
lat makes with the plane of the equator. Let the axis of z be
irertical at and measured positively in the direction opposite to
[gravity. Let the axes of x and y be respectively a tangent to the
leridian and a perpendicular to it, their positive directions being
Irespcctively south and west. In the figure the axis of y is dotted
m
v,**
jiT.rcssaKt-. -s^rxr^-^z
5ra=::.-SiL,K-B.t-i.i_,.
!;„L..»j.-i.ji-i.i..'_'.'-,'...i'-..i..„jvj.:;.'j..-
214
MOTION IN THREE DIMENSIONS.
! '!
): Iff
to indicate that it is perpendicular to the plane of the paper. Let
ft) be the angular velocity of the earth, b the distance of the point
from the axis of rotation.
We may reduce the point to rest by applying to every
point under consideration an acceleration equal and opposite to
that of 0, and therefore equal to (o^b and tending from the axis of
rotation. We must also apply a velocity equal and opposite to
the initial velocity of 0. This velocity is tab. The whole figure
will then be turning about an axis 01, parallel to the axis of
rotation of the earth with an angular velocity m.
When the particle has been projected from the earth it is
acted on by the attraction of the earth and the applied accelera-
tion oy'b. The attraction of the earth is not what we call gravity.
Gravity is the resultant of the attraction of the earth and the
centrifugal force, and the earth is of such a form chat this resultant
acts perpendicular to the surface of still water. If it were not so,
particles resting on the earth would tend to slide along the sur-
face. It appears, therefore, that the force on the particle, after O
has been reduced to rest, is equal to gravity. Let this be repre-
sented by g. Besides this there may be other forces on the par-
ticle, let their resolved parts parallel to the axes be X, Y, Z.
Since the earth is turning round 01 with angular velocity w,
the resolved part about Oz is a sin\, since the angle lOz is the
complement of w; since the rotation is from west to east, the
resolved angular velocity is from ?/ to x, which is th« negative
direction, hence 0^ = — (i) sin X. The resolved angular velocity
round Ox is &> cos \ and is from y to z, which is the positive
direction, hence d^ = (o cos \. Also since 01 is perpendicular to
Qj^^ 0^ = 0. Hence, by Art. 244, the actual velocities of any
particle whose co-ordinates are {x, y, z), are
i I
the paper. Let
ice of the point
ilying to every
Ltid opposite to
Tom the axis of
md opposite to
le whole figure
1 to the axis of
le earth it is
)plied accelera-
e call gravity.
earth and the
this resultant
it were not so,
along the sur-
larticle, after O
this be repre-
ces on the par-
X, Y, Z.
w
t
liar velocity w,
gle lOz is the
st to east, the
s the negative
gular velocity
s the positive
rpendicular to
ocities of any
ON MOTION RELATIVE TO THE EARTH.
M = -^- +0) sin\y
213
dt
v = -^ — a) coaXz— a sinXa?
at
dz , -
«; = -,- + &) cosX?/
dt ''
To find the equations of motion it is only necessary to substitute
these in the equations of Art. 245.
The resulting equations may be simplified if we neglect such
small quantities as the difference between the force of gravity at dif-
ferent heights. If a be the equatorial radius of the earth and g' the
force of gravity at a height z, we have g' =g\\ j nearly. Now
ft)*a is the centrifugal force at the equator, which is known to be
1 z
-— g. Hence if we neglect the small terra ^r - we must also
neglect ti^z. The equations will therefore become
<«>
2a) cos X ^r — 2g> sin \ ^r = Y \ ,
at dt
de
^ + 2a,cos\^ = -^ + ^
.^ = -.
where the terms (X, Y, Z) include all the accelerating forces,
except gravity, which act on the particle. These equations agree
with those given by Poisson, Journal Polytechnique, 1838.
265. If we do not neglect the term containing to, the equa-
tions of motion are
-jTa^ + 2g> sin \ -^ — ft)' sin'Xa? — m^ sin \co%\z — X,
de'
2(0 cos X -57 — 2a) sin X -rj — a)'y = Y,
dt
dt
TTg + 2w cos X s? — w' cos'Xa — o)' sin X cosXaj = — ^r + Z.
266. As an example, let us consider the case of a particle dropped from a
height h. The initial conditions are therefore «, «, -^, ~, -r^ all zero, and
dt at at
z=h. As a first approximation, neglect all the terms containing the small factor w.
Thenwehave«=0, y=0, « = h-5»/<».
\\\
v\
).-;M
m
i
n
216
MOTION IN THREE DIMENSIONS.
For a second approximation, we may substitute these values of (x, y, z) in the
small terms. We have after integration
fi 1
a;-0, 2/= - wcosX^ , z = h- gf^.
Thus there will be a small deviation towards the east, proportional to the cube
of the time of descent. There will bo no southerly deviation, and the vertical
motion vviU be the same as if the earth were at rest.
An elementary demonstration of this resiJt will make the whole argument
clearer. Let the particle be dropped from a height li vertically over 0. Then
being reduced to rest, the particle is really projected eastwards with a velocity
w/i cos X. Hence, if the direction of gravity did not alter owing to the rotation of
the earth about 01, the particle would describe a parabola and the easterly deviation
would be (w/t cos X) t where t is the time of falling. Since h=-\()t^, this deviation is
u cos \fj - . The rotation w about 01 is etiuivalent to w sin X about Oz and <a cos \
about Ox. The former does not alter the position of 00 the normal to the surface
of the earth, which is the direction of gravity. The latter turns OG in any
time t through an angle w cos \t. Thus gravity gradually changes its direction
as the particle falls. The particle is therefore acted on by a westerly component
= f) sin (w cos X<), which, since w< is small, is nearly equal to grw cos \i. Let ij be the
distance of the particle from the position of the plane xz in space at the moment
when the particle began to fall, and let y' be measured positively to the west. The
equation of motion of the particle in space is therefore
dhj'
dt^''
Integrating this and remembering that as explained above ,j = -uh cos \ when
=ffwt cos X.
< = 0, we get
y'= - uht cos X + ^r/wfi cos X.
When the particle reaches the ground we have y'=y very nearly and h=lgi',
thus the deviation westwards is - ug - cos X, which is the same as before. If it be
o
not evident that y'=y, it may be shown thus. In the time t Oy, Oz have turned
through a very small angle d = u cos \t, hence, as in transformation of axes,
y'=y cos 0-zsinO,
which gives y' = y when we reject the squares of 0.
2G7. In many cases it will be found convenient to refer the
motion to axes more generally placed. Let be the origin, and
let the axes be fixed relatively to the earth, but in any directions
at right angles to each other. Let 0^, 6^, 0^ be the resolved
parts of ft) about these axes, then 0^, 0^, 0^ are known constants.
After substituting from Art. 244 in the equations of motion given
in Art. 245 we get
P>-'%^>=^'
d'y
de
dz
dx
-25''.+^s'''=^>
of (at, y, 2) in the
)out Oz and u cos \
= -uh cos X wlien
ON MOTION RELATIVE TO THE EARTH.
217
d'z dx dy ^ _
For example, if we wished to determine the motion of a projectile, it will be
convenient to take the axis of z vertical and the plane of xz to be the plane of
projection. Let the axis of x make an angle ^ with the meridian, the angle being
measured from the south towards the west. Then
^i=wcosXcos/3, Sj = - w cos X sin j3, tf3=-wRinX.
These equations may be solved in any particular case by the
method of continued approximation. If we neglect the small
terms we get a first approximation to the values of {x, y, z). To
find a second approximation we may substitute these values in the
terms containing w and integrate the resulting equations. As
these equations are only true on the supposition that &>* may be
neglected, we cannot proceed to a third approximation.
268. Ex. 1. A particle is projected with a velocity F in a direction making an
nngle a with the horizontal plane, and such that the vertical plane through the
direction of projection makes an angle /3 with the plane of the meridian, the angle j3
being measured from the south towards the west. If x be measured horizontally in
the plane of projection, y be measured horizontally in a direction making an angle
j3 + - with the meridian, and z vertically upwards from the point of projection,
prove that
x=7cosa<+( r sin at* -^firt'j wcosXsin/S,
y= [ Vsinot'-;r(7<^ j wcosXcos/3+ Vcosot'wsinX,
z— Fsinoi-ggf**- Fcosat'wcosXsin/S,
where X is the latitude of the place, and w the angular velocity of the earth about
its axis of figure.
Show also that the increase of range on the horizontal plane through the point
of projection is
y^ /I \
4w -J sm /3 cos X sin a ( ^ sin' a - cos' a 1 ,
and the deviation to the right of the plane of projection is
r sin Q.
4w -J sin' a (cos Xcos /3 — j.— + sin X cos a).
Ex. 2. A bullet is projected from a gun nearly horizontally with great velocity
BO that the trajectory is nearly flat, prove that the deviation is nearly equal to
iJtosinX, where R is the range, and the other letters have the same meaning as in
the last question. The deviation is always to the right of the plane of firing in the
Northern hemisphere, and to the left in the Southern hemisphere. It is asserted
(Qom-^iti Rendus, 1866) that the deviation due to the earth's rotation as calculated
by this formula is as much as half the actual deviation in Whitworth's gun.
'i
|: \
r:ii
. i.i
'I
! Ml
218
MOTION IN THBEE DIMENSIONS.
Ex. 3. A spherical bullet is projected with so great a velocity that the resistance
of the air must be taken into account. The resistance of the air being assumed to
be i^— , and the trajectory to be flat, prove that, neglecting the effects oi the
k
rotation of the earih,
2wRinX * X ,.
z=xtma-§-yl-2l-l)-
?-"-?^-"^-^H«(el-|-l).
These are given by Poisson, Journal Poly technique, 1838.
269. Let us apply these equations to determine the effect of
the rotation of the earth on the motion of a pendulum. In this
as in some other cases, it will be fouud advantageous to refer the
motion to axes not fixed in the earth but moving in some known
manner. Let the axis of z be vertical as before and let the axes
of X and y move slowly round the vertical with angular velocity
to sin \ in the direction from the south towards the west. In this
case we have
6^ — a) cos \ cos j3, 6^= — a) cos \ sin fi,
and ^3 = — G> sin \ + 0) sin X, = 0,
where /9 is the angle the axis of x makes with the tangent to the
meridian, so that -^ = a> sin \. If, as before, we neglect quanti-
ties which contain the square of a> as a factor, the terms which
riff (10
contain -jJ and -r^ must be omitted. Hence the required equa-
tions may be obtained from those of Art. 267, by putting 0^ = 0.
If m be the mass of the particle, I the length of the string,
and T the tension ; these equations are
d'x „ -y ' adz T X
-J- — 2a> cos X sm iS ^ = -y
dv dt ml
^ — 2a)COs\cosi8 J- = ^
dr "^ dt m I
d^^ , n % • yo^^ . o N ady T Z
-Ts + "«o COS \ sm a -j- -1- 2o) cos \ cos iS -^ = — <7 ^
dv dt dt ^ m t
the origin being taken at the point of suspension.
If the oscillation be sufficiently small z will differ from I by
small quantities of the order a* where n is the semi-angle of oscil-
lation. The last equation then shows that T differs from mgi by
quantities of the order aa at least. If then we neglect terms of the
ON MOTION RELATIVE TO THE EARTH.
219
order wa' and a', we may put mg for i in the two first equations
and neglect the uerms containing to -^ . The equations of motion
thus become the same as for a pendulum attached to a fixed
point. The solutions of the equations are clearly
x = A cos
(\/f'+^)' y-BA.y\t^D).
The small oscillations of a pendulum on the earth referred to
axes turning round the vertical with angular velocity w sin \ are
therefore the same as those of ai imaginary pendulum suspended
from an absolutely fixed point.
Let us then suppose the pendulum to be drawn aside so as to
make with the vertical a small angle a and then let go. Relacively
therefore to the axes moving rou. \ the vertical with angular
velocity o) sin \ we must suppose the particle to be projected with
a velocity Z sin a o) sin \ perpendicular to the initial plane of dis-
placement. We have then when i = 0, x = hy y = 0, -,- = 0,
^ = laco sin \. It is then easy to see that in the above values
clt
of a; and y, G and D are both zero and that the particle de-
scribes an ellipse, the ratio of the axes being to sin ^ a/"* "^^^
effect of the rotation of the earth is to make this ellipse turn
round the vertical with uniform angular velocity w sin \ in a
direction from, south to west. If the angle a be not so small
that its square may be neglected, it is known by Dynamics of a
particle that, independently of all considerations of the rotation
of the earth, there will be a progression of the apsides of the
ellipse. It is therefore necessary for the success of the experi-
ment that the length I of the pendulum should be very great.
This motion of the apsides depending on the magnitude of a is in
the opposite direction to that caused by the rotation of the earth
and cannot therefore be mistaken for it.
It also appears that the time of oscillation is unaffected by the
rotation of the earth, provided the arc of oscillation be so small
that the effects of forces whose magnitude contains the factor coa*
may be neglected.
270. In Chapter iv. we have considered the motion of a system ?* bodies
constrained to remain in a fixed plane. Since no plane can be found which does
not move with the earth, it is important to determine what effect the rotation of the
earth will have on the motion of these bodies. Let us treat this as an example of
the method of Coriolis given in Art. 257.
Let the plane make an angle a with the axis of the earth. Let a point in
this plane be on the siuface of the earth and let it be reduced to rest. Then, as
M
1 ,}';
220
MOTION IN THREE DIMENSIONS.
proved iu Art. 2C4, the moving bodies w^ile in the neighboiirhood of are acted on
by their weights in a direction normal lo the surface of the eai'th. The earth ia
now turning round an axis through parallel to the axis of figure with a constant
angular velocity w. Let this angular velocity be resolved into two, viz., u sin a
about an axis perpendicular to the plane and w cos a about an axis in the plane.
Now the square of w is to be rejected, hence by the principle of the superposition of
small motions, we may determine the whole effect of these two rotations by adding
together the effects produced by each separately.
It is a known theorem that if a particle be constrained to move in a plane which
turns round any axis in that plane with a constant angular velocity u> cos o, the
motion may be found by regaiding the plane as fixed and ''"pressing an accelera-
tion ui^r cos^ a on the particle, where r is the distance of the particle from the axis.
This may be deduced, as iu Art. 260, from the theorem of Coriolis. This impressed
acceleration is to be neglected beeaiise it depends on the square of w. The angular
velocity u cos a has therefore no sensible effect.
If the bodies be free to move in the plane, the effect of the rotation u sin a is to
turn the axes of reference round the normal to the plane drawn through the point
0. If then we calculate the motion without regard to the rotation of the earth,
taldng the initial conditions relative to fixed space, the effect of the rotation of the
earth may be allowed for by referring this motion to axes turning round the normal
with angular velocity w sin a. For example, if the body be a heavy particle sus-
pended by a long string from a point fixed relatively to the earth, it is really
constrained to move in a horizontal plane, and the reasoning given above shows
that the plane of oscillation will appear to a spectator on the earth to revolve with
angular velocity a sin a round the vertical
If the bodies be constrained to revolve with the plane, it " vill be required to find
the motion relatively to that plane. We must therefore apply to each particle the
force of moving space and the compound centrifugal force. If r be the distance of
any particle of mass mi from 0, the former is mrw" sin^ o. This is to be neglected
because it depends Oxi the square of u. The latter is therefore the only force to be
considered. By Art. 262, the compound centrifugal forces on all the particles of a
body are equivalent to a force at the centre of gi'avity and three couples. In our
case these couples are easily seen to be zero. For if the plane be taken as the plane
of x>j, we have 0^=0, ^,=0, «i=0, w, = 0. Hence L, M, N are all zero. If, there-
fore, m be the mass of a body, V the relative velocity of its centre of gravity, the
effect of the rotation of the earth may be found according to the rule given in Art.
2o7, by impressing on the body a force equal* to 2nirwsina, acting at the centre of
gravity, iu the plane of motion and perpendicular to the direction of motion of the
centre of gravity.
The ratio of this force to gravity for a particle moving S2 feet per second, is at
most , which is less than a five thousandth. This is so small that, except
under special circumstances, its effect will be imperceptible.
271. Ex. 1, In Foucault's experiment, a long pendulum is suspended from a
point over the centre of a circular table, and the arc of oscillation is seen to pass
from one diameter to another. Show that the arc of the circidar rim of the table
described by the plane of oscillation iu one day is equal to the difference in length
betwcr two parallels of latitude one through the centre and the other through the
ON MOTION RELATIVE TO THE EARTH.
221
northern or southern extremity of the rim. This theorem in due to the late Prof.
Young.
Ex. 2. A heavy particle is suspended from a Jixed point of support by a string
of length a. It performs elliptic oscillations whose major and minor semi-axes are
b and c. If 6 and c be small compared with a, prove that the apses will advance,
3 be
in each complete revolution of the particle, through an angle — 2jr nearly. If b
O U"
and c be not small compared with a but be very nearly equal, the apse will advance
through an angle
^ -1^2^,
V^l-^sin«« /
b .
vhere sina= in each complete revolution of the particle.
Ex. 3. A pendulum, at rest relatively to the earth, is started iu any direction
with a small angular velocity, show that the oscillations will take place in a vertical
plane turning uniformly round the vertical so that the pendulum becomes vervical
once iu each half oscillation.
Ex, 4. Let be the angle a pendulum of length I makes with the vertical, and
^ the angle the vertical plane containing the pendnlum makes with a vertical plane
which turns round the vertical with uniform angular velocity w sin \ in a direction
from south to west. Prove that when terms depending on u^ are neglected the
equations of motion become
(§)•"'"'«
\dtj I
COS0 + A,
— I sm" ^ ^ 1 =
dt\ dtj
do
2 sin- 6 cos (0 4-/8) w cos \ -j- ,
where A is an arbitrary constant, and the other letters have the meanings given to
them in Art. 267. See M. Quet in Liouville'a Journal, 1853.
These equations will be found convenient in treating the motion of a pendulum.
They may be easily obtained by transforming those given in Art. 239 to polar co-
ordinates.
Ex. 5. A semi-circular arch ACB is fixed with its plane vertical on a horizontal
wheel at A and B, and may thus be moved with any degree of rapidity from one
azimuth to another. A rider slides along the inner edge of the arch which is
graduated and may be fixed at any degree marked thereon. A spiral spring by
means of which a slow vibration is obtained with comparatively a short length ia
attached at one end to a pin in the axis of the semicircle so that the point of
attachment may- be in the axis of rotation and at the other end it is fixed to a
similar pin in a paraUel position fixed to the rider. The vertical semicircle is not
placed in a diameter of the horizontal wheel but parallel to it at such a distance as
not to interrupt the eye of the observer from the vertical plane passing through the
diameter, and in which plane the wire in all its positions remains.
If the rider be placed at an angular distance $ from the highest point of the
arch and the wire set in vibration in any plane, show that the plane of vibration of
the wire will make a complete revolution relatively to the arch while the arch turns
round sec 6 complete revolutions. This is best observed by fixing the eye on a line
i
1- .)!
:f
: 6
' I'^i
1 ■ it';|
1 '.
A
m
222
MOTION IN THREE DIMENSIONS.
in the eame plane with the wire while walking round with the wheel during its
rotation. This apparatus was devised by Sir C. Wlieatstone to illustrate Foucault's
mechanical proof of the rotation of the earth. Proceedings of the Royal Society,
May 22, 1851.
272. Hitherto we have considered chiefly the motion of a
single particle. The eftect of the rotation~of the earth on the
motion of a rigid body will be more easily understood when the
methods to be described in the following chapters have been read.
If, for example, a body be set in rotation about its centre of
gravity, it will not be difficult to determine its motion as viewed
by a spectator on the earth, when we know its motion in space.
It seems, therefore, sufficient here to consider the peculiarities
which these problems present, and to seek illustrations which do
not require any extended use of the equations of motion.
273. The effect of the rotation of the earth is in general so
small compared with that of gravity, that it is necessary to fix the
centre of gravity in order that the effects of the former may be
perceptible. Even when this is done, the friction on the points of
support and the other resistances, cannot be wholly done away
with. If, however, the apparatus be made with care that these
resistances should be small, the effects of the rotatiuu of the earth
may be made to accumulate, and after some time to become
sufficiently great to be clearly perceptible.
If a body be placed at rest relatively to the earth and free to
turn about its centre of gravity as a fixed point, it is actually in
rotation about an axis parallel to the axis of the earth. Unless
this axis be a principal axis, the body would not continue to rotate
about it, and thus a change would take place in its state of
motion. By referring to Euler's equations, we see that the change
in the position of the axis of rotation is due to the terms
(^— ^)&),ft)g, (5 — (7) tOgWg, {C — A)(o^a>^. The body having
been placed apparently at rest, m^, eo^, Wj are all small quan-
tities of the same order as the angular velocity of the earth ; these
terms are, therefore, all of the order of the squares of small quan-
tities. Whether they will be great enough to produce any visible
effect or not will depend on their ratio to the frictional forces
which could be called into play. But since those frictional forces
are just sufficient to prevent any relative motion, these terms will
in general be just cancelled by the frictional couples introduced
into the right-hand sides of Euler's equations. The body will,
therefore, continue at rest relatively to the earth.
In order that some visible effect may be produced, it is usual
to impress on the body „ very great angular velocity about some
axis. If this be the axis of w^, the terms in Euler's equations,
which are due to the centrifugal forces, and which contain co^ as a
factor,
The gr
will be,
body 1
If
.sufficie
24x()0
In thes(
of the
velociti€
lected.
As a
selected
an elem<
274.
while th(
relatively
aocis of J
Let I
gravity b
the earth
an angle
on the pli
called the
about its
with the
given by
Let ft),
[moving a
[centre of
figure is i
[parallel t(
lapplicatio:
[axes are i
ISubstituti
ON MOTION RELATIVE TO THE EARTH.
223
factor, become greater than when Wj had no such initial value.
The greater this initial angular velocity, the greater these terms
will be, and the more visible we may expect their effects on the
body to be.
If the angular velocity thus communicated to the body be
sufficient to turn it only once in a second, it will be still
24 X GO X 60 times as great as the angular velocity of the earth.
In these problems, therefore, we may regard the angular velocity
of the earth as so small, compared with the existing angular
velocities of tlie body, that the square of the ratio may be neg-
lected.
As an example of the application of these principles, we have
selected one case of Foucault's pendulum, which seems to admit of
an elementary solution.
274. The centre of gravity of a solid of revolution is fixed,
while the axis of figure is constrained to remain in a plane fixed
relatively to the eaHh. The solid being set in rotation about its
axis of figure, it is required to find the motion.
Let us refer the motion to moving axes. Let the centre of
gravity be the origin, the plane of yz the plane fixed relatively to
the earth. Let the axis of figure be the axis of z, and let it make
an angle ^ with Ihe projection of the axis of rotation of the earth
on the plane oi yz. Let this projection, for the sake of brevity, be
called the axis of %. Let p be the angular velocity of the earth
about its axis, a the angle the normal to the plane of yz makes
with the axis of the earth. The motion of the moving axes is
given by
j 6^=p co8a + -^, ^2 ^i' s^"^ * ^^'^ X' C^=p sin a cos X-
Let Wj, o)j, Wgbe the angular velocities of the body about the
[moving axes; A, A, G the principal moments of inertia at the
centre of gravity. Let Ji be the reaction by which the axii of
figure is constrained to remain in the fixed plane, then R acts
parallel to the axis of x. Let h be the distance of its point of
I application from the origin. The angular momenta about the
[axes are respectively
h^ = A(0^,
h^ = Aa>^y
h^=Ca>^.
[Substituting in Art. 230, the equations of motion are
A^^^^^-C<.,e, + A^J, = Rh
fl 9
dt
^ft>,^,+ ^a),^, =
,iii !
■'^m
224
MOTION IN THREE DIMENSIONS.
>
i"
Since the axis of z is fixed in the body, we see by Art. 243,
that o), = ^,, (a^ = 6^. The last equation of motion, therefore,
shows that w, is constant. It should however be remembered that
eog is not the apparent angular velocity of the body as viewed by
a spectator on the earth. If VL^ be the angular velocity relatively
to the moving axes, we have by Art. 243, llj = 0)^—6^, so that
fig + p sin % cos X — constant.
Thus the body, if so small a difference could be perceived, would
appear to rotate quicker the nearer its axis approached the pro-
jection of the axis of the earth's rotation on the fixed plane.
The first equation of motion after substitution for w^, w^, 6^, 0^,
their values in terms of x> becomes
A ~j^ — Ap' sin'^ a sin % cos ;^ + Cnj) sin a sin ;^ = 0,
where n has been written for tWg.
The second term may be rejected as compared with the third,
since it depends on the square of the small quantity p. We have,
therefore,
d'x G . .
By Art. 92, this is the equation of motion of a pendulum
under the action of a force constant in magnitude, and whose
direction is along the axis of v, i.e. the projection of the axis of
rotation of the earth on the fixed plane. The body being set in
rotation about its axis of figure, we see that that axis will imme-
diately begin to approach one extremity or the other of the axis of
X with a continually increasing angular velocity. When the axis
of figure reaches the axis of ^t its angular velocity will begin to
decrease, and it will come to rest when it makes an angle on the
other side of the axis of v equal to its initial value. The oscilla-
tion will then be repeated continually.
The axis of figure will oscillate about that extremity of the
axis of x> which, when ^ is measured from it, makes the coefiBcient
on the right-hand side of the last equation negative. This extre-
mity is such, that when the axis of figure is passing thro"g^ i^
the rotation n of the body is in the same direction as the resolved
rotation p of the earth.
275. If we compare bodies of different form, we see that the 1
C
time of oscilldtion depends oidy on the ratio -^ . It is otherwise
independent of the structure or form of the body. The greater M
this ratio the quicker will the oscillation be. For a solid of m
revolution, it appears from the definitions in Art. 4, that this
ON MOTION RELATIVE TO THE EARTH.
225
ratio is greatest when Swu' = 0. In this case tho ratio is equal
to 2, and the body is a circular disc or ring.
27G. If we compare the different planes in which the axis may
be constrained to remain, we see that the motion is the same for all
planes making the same angle with the axis of the earth. It is
therefore independent of the inclination of the plane to the horizon
at the place of observation. The time of oscillation will be least,
and the motion of the axis most perceptible when a= ^,i.e. when
the plane is parallel to the axis of rotation of the earth. If the
plane be perpendicular to the axis of the earth, the axis of figure
will not oscillate, but if the initial value of -7^ is zero, it will
at
remain at rest in whatever position it may be placed.
277. Ex. 1. Show that a person furnished with the particular form of Fou-
caiUt'H pendulum just described, could, without any Astronomical observations,
determine the latitude of tho place, the direction of the rotation of the earth, and
the length of the sidereal day. This remark is due to M. Quet, who has given a
different solution of this problem in Liouville'a Journal, vol. xviii.
Ex. 2. If the body be a rod, and its centre of gravity supported without friction,
prove that it could rest in relative equilibrium either parallel or perpendicular to
the projection of the earth's axis on the plane of constraint. If it be placed in any
other position, its motion will be very slow, depending on j)', but it will oscillate
about a mean position perpendicular to the projection of the earth's axis.
Ex. 3. If the axis of figiwe be acted on by a frictional force producing a
retarding couple, whoso moment about the axis of x bears a constant ratio ft, to the
moment of the reactionol couple about tho axis of y, and if the fixed plane bo
1 parallel to the axis of the earth, find the small oscillations about the position of
equilibrium. Show that the position at any time t is given by
X=Zc-^'cos[(^-X«)*<+m].
jwhere 2A\=fk{Cn~2Ap) and L and M are two constants dependhig on the initial
conditions.
Ex. 4. The centre of gravity of a solid of revolution is fixed, while the axis of
Sgure is constrained to remain in the surface of a smooth right cone fixed relatively
\q the earth. Show that the axis of figure will oscillate about the projection of the
is of rotation of the earth on the surface of the cone, and that the time of a com-
A Bine
^.r<sin/3 '
I the semi-angle of the cone, /3 the inclination of its axis to the axis of the earth,
ad the other letters have the same meaning as befOTe. This result is due to
Quet.
blete small oscillation about the mean position will be 2r . /^
where e
Ex. 5. Two equal heavy rods CA, CB are connected by a hinge at C, with a
pring so that they tend to make a known angle with each other. The free ends
and B are then tied together and the whole is suspended by a string OC attached
R. D. 15
h'l
if
: ■ 1 » 1
\i
i
22G
MOTION OF TWO DIMENSIONS.
to the binge. The Bystem is left to itsoU until It is at rest relatively to the earth.
If the string which fastens A and J3 be now out, the arms separate from each other.
Show that the system will immediately have an apparent angular velocity round
the vertical equal to p sin \, where /, /' are the moments of inertia of the
system about the vertical OC respectively before and after the string joining A and
£ was cut, p is the angular velocity of the earth about its axis and X is the latitude
of the place. In which direction will the system turn? This apparatus was
devised by M. Poinsot who considered that the experiment would be so effective
that the latitude of the place could be deduced from the observed angular velocity.
See Comptea Itendus, 1851, Tome xxxii. page 206.
Ex. 6. If a river is flowing due north, prove that the pressure on the eastern
bank at a depth z is increased by the change of latitude of the running water in
the ratio gz + bvu sin I : gz, where h is the breadth of the stream, v its velocity, I the
latitude and u the angular velocity of the earth about its axis. [Math. Tripos, 1875.]
CHAPTER VI.
ON MOMENTUM.
278. The terra Momentum has been given as the heading of
this Chapter, though it only expresses a portion of its contents.
The object of the Chapter may be enunciated in the following
problem. The circumstances of the motion of a system at nny time
tf, are given. At the time i, the system is moving under other
circumstances. It is required to determine the relations -which
may exist between these two motions. The manner in which
these changes are eft'ected by the forces is not the subject of
enquiry. We only wish to determine what changes have been
effected in the time t^ — t^. If the time t^ — t^ be very small, and
the forces very great, this becomes the general problem of im-
pulses. This also will be considered in the Chapter.
Let us refer the system to any fixed axes Ox, Oy, Oz. Then
the six general equations of motion may, by Art. 71, be written in
the form
Integrating these from < = ^„ to < = ^j, we have
Let a force P act on a moving particle m during any time
ti — t^, and let this time be divided into intervals each equal
to dt. At the middle of each of these intervals let a line be
drawn from the position of m at that instant, to represent, at the
same instant, the value of mPdt both in direction and magnitude.
Then the resultant of these forces, found by the rules of Statics,
may be called the whole force expended in the time t^ — t^. Thus
I mZdt is the whole force resolved parallel to the axis of Z.
These equations then show that
15—2
V\
Ifl
•M
f. HI,
Nil
i
828
MOMENTUM.
(1) The change produced by any forces in the resolved part
of the momentum of any system is equal in any time to the whole
resolved force in that direction.
(2) The change produced by any forces in the moment of the
momentum of the system about any straight line is, in any time,
equal to the whole moment of these forces about that straight line.
When the interval t^^ — t^ is very small, the " whole force "
expended is the usual measure of an impulsive force, and the
preceding equations are identical with those given in Art. 86.
It is not necassary to deduce these two results from the equa-
tions of motion. The following general theorem, which is really
equivalent to the two theorems enunciated above, may be easily
obtained by an application of D'Alembert's principle.
279. If the momentum of any particle of a system in motion
he compounded and resolved, as if it luere a force acting at the
instantaneous position of the particle, according to the rules of
Statics, then the momeru'-', of all the particles at any time t^ are
together equivalent to the momenta at any previous time t^ together
with the whole forces which have acted during the interval.
In the case in which no forces act on the system, except the
mutual actions of the particles, we see that the momenta of all
the particles of a system at any two times are equivalent ; a result
which has been already enunciated in Art. 72. The two princi-
ples of the Conservation of Linear Momentum and Conservation of
Areas may be enunciated as follows.
If the forces which act on a system be such that they have no
component along a certain fixed straight line, then the motion
is such that the linear momentum resolved along this line is
constant.
If the forces be such that they have no moment about a cer-
tain fixed straight line, then the moment of the momentum or
area conserved about this straight line is constant.
It is evident that these principles are only particular cases of
the results proved in Art. 79.
280. Ex. Suppose that a simple particle m describes an
orbit about a centre of force 0. Let v, v' be its velocities at any
two points P, P' of its course. Then mv' supposed to act along
the tangent at P' if reversed would be in equilibrium with mv
acting along the tangent at P together with the whole central
force from P to P'. If p, p be the lengths of the perpendiculars
from on the tangents at P, P', we have, by taking moments
about 0, vp = v'p', and hence vp is constant throughout the
MOMENTUM.
229
id part
i whole
, of the
,y time,
ht line.
force "
md the
86.
le equa-
s really
le easily
I motion
g at the
rules of
ie \ are
J together
ccept the
ita of all
a result
princi-
vation of
have no
motion
line is
it a cer-
Qtum or
cases of
ribes an
at any
ct along
vith mv
central
Idioulars
loments
lout the
motion. Also if the tangents meet in T, the whole central force
expended must act along the line TO, and may be found in terms
of V, V by the rules for compounding velocities.
Ex. Two particles of masses m, m' move about the same centre of force. If
h, h' be the double areas described by each per unit of time, prove that nh + m'h'
is unaltered by an impact between the particles.
281. Ex. Suppose three particles to start from rest attracting
each other, but under the action of no external forces. Then the
momenta of the three particles at any instant are together equiva-
lent to the three initial momenta and are therefore in equilibrium.
Hence at any instant the tangents to their paths must meet in
some point 0, and if parallels to their directions of motion be
drawn so as to form a triangle, the momenta of the several parti-
cles are proportional to the sides of that triangle.
If there are n particles it may be shown in the same way that
the n forces represented by mv, m'v', &c. are in equilibrium, and if
parallels be drawn to the directions of motion and proportional to
the momenta of the particles beginning, at any point, they will
form a closed polygon.
If F, F', F" be the resultant attraction on the three particles,
the lines of action of F, F', F" also meet in a point. For let
X, Y, Z be the actions between the particles m!m\ m"m, mm',
taken in order. Then F is the resultant of — F and Z; F' oi — Z
and X; F" of -X and Y. Hence the three forces F, F, F'
are in equilibrium*, and therefore their lines of action must meet
in a point 0'. Also the magnitude of each is proportional to the
sine of the angle between the directions of the other two. This
point is not generally fixed, and does not coincide with 0.
If the law of attraction be proportional to the distance, the
two points 0, 0' coincide with the centre of gravity G, and are
fixed in space throughout the motion. For it is a known propo-
sition in Statics that with this law of attraction, the whole attrac-
tion of a system of particles on one of the particles is the same as
if the whole system were collected at its centre of gravity. Hence
0' coincides with Q. Also, since each particle starts from rest,
the initial velocity of the centre of gravity is zero, and therefore,
by Art. 79, C? is a fixed point. Again, since each particle starts
from rest and is urged towards a fixed point O, it will move in the
straight line joining its initial position with O. Hence coin-
cides with O. When the law of attraction is proportional to the
distance, it is proved in Dynamics of a Particle, that the time of
reaching the centre of force from a position of rest is independent
* This proof is merely an amplification of the following. The three forces
F, F\ F", being the internal re-actions of a system of three bodies, are in equili-
brium by D'Alembert's Principle,
; I!
t
p |1'
K^te;
; ^^
i
II
»
230
MOMENTUM.
of the distance of that position of rest. Hence all the particles of
the system will reach G at the same time, and meet there. If !Sm
be the sum of the masses, measured by their attractions in the
1 Stt
usual manner, this time is known to be -r ,-— : .
282. Ex. Three particles whose masses are m, m', m", mutu-
ally attracting each other, are so projected that the triangle formed
by joining their positions at any instant remains always similar to
its original form. It is required to determine the conditions of
projection.
The centre of gravity will be either at rest or will move uni-
formly in a straight line. We may therefore consider the centre
of gravity at rest and may afterwards generalise the conditions of
projection by impressing on each particle an additional velocity
parallel to the direction in which we wish the centre oi gravity to
move. Let be the centre of gravity, P, P, F" the positions of
the particles at any time t. Then by the conditions of the ques-
tion the lengths OP, OP', OP" are always to bo proportional, and
their angular velocities about are to be equal. Since the moment
of the momenta of the system about is always the same, we
have
mr'^n + m'r'^n + rr^'r'^n = constant,
where r, r', r" are the distances OP, OP', OP", and n is their
common angular velocity. Since the ratios r : r' : r" are con-
stants, it follows from this equation that mr\i is constant, i.e. OP
traces out equal areas in equal times. Hence by Newton, Section ii,
the resultant force on P tends towards 0.
Let p, p, p" be the sides P'P", P"P, PP' of the triangle
mass
(disty*
formed by the particles, and let the law of attraction be
Then since the resultant attraction oim', m" on m passes through 0,
m
m
^ sm FPO=^ sin P"PO,
P P
but since is the centre of gravity,
m'p" sin P'PO = m"p' sin P"P0.
Hence either the three particles are in one straight line or
If A; = — 1 the law of attraction is "as the distance."
If k be not = - 1, we have p = p", and the triangle must be 'Equi-
lateral.
Conversely, suppose the particles to be projected in directions
making equal angles with their distances from the centre of
gravity with velocities proportional to those distances, and sup
pose also the resultant attractions towards the centre of gravity tc
p =p
'it
MOMENTUM.
f>31
be proportional to those distances, then in all the three cases the
same conditions will hold at the end of a time dt, and so on con-
tinually. The three particles will therefore describe similar orbits
about the centre of gravity in a similar manner.
First, let us suppose that the three particles are to be in one
straight line. To fix our ideas, let m' lie between m and w", and
between m and r/i. Then since the attraction on any particle
must be proportional to the distance of that particle from 0, the
three attractions
m
m
m
m
m
m
{PPy^ {PFY {F'P'f {PF)
k>
{ppy {ppy
PF'
PF
must be proportional to OP, OF, OF'. Since 'ZmOP^ 0, these
two equations amount to but one on the whole. Let z =
OP _ m'+'m"{l + z) OF _ -m + m"z
sotbat-pp- ^^^'^^'' > PP'-m + m' + m"'
Then we have
which agrees with the result given by Laplace, by whom this
problem was first considered.
In the case in which the attraction follows the law of nature
k = 2 and the equation becomes
ms» {(1 + zY - 1} - m' (1 + zY (1 - «') - ni!' {(1 + zf - z'] = 0.
This is an equation of the fifth degree, and it has therefore
always one real root. The left side of the equation has opposite
signs when z = and ^ = oo , and hence this real root is positive.
It is therefore always possible to project the three masses so that
they shall remain in a straight line. Laplace remarks that if m
be the sun, m' the earth, and m" the moon, we have very nearly
z = a/ — K = TTxTL • If then originally the earth and moon had
been placed in the same straight line with the sun at distances
from the sun proportional to 1 and 1 + rrrr^ , and if their velocities
had been initially parallel and proportional to those distances, the
moon would have always been in opposition to the sun. The
moon would have been too distant to have been in a state of
continual eclipse, and thus would have been full every night. It
has however been shown by Liouville, in the Additions d la
Gonnaissance des Temps, 184o, that such a motion would be un-
stable.
i; Pi
t ',
I t
t i.ll
I
i;''
232
MOMENTUM.
!
if
The paths of the particles will be similar ellipses having the
centre of gravity for a common focus.
Secondly. Let us suppose that the ' iw of attraction is " as the
distance." In this case the attraction on each particle is the
same as if all the three particles were collected at the centre of
gravity. Each particle will describe an ellipse having this point
for centre in the same time. The necessary conditions of projec-
tion are that the velocities of projection should be proportional to
the initial distances from the centre of gravity, and the directions
of projection should make equal angles with those distances.
Thirdly. Let us suppose the particles to be at the angular
points of an equilateral triangle. The resultant force on the par-
ticle m is
^, cos FPO + ^cos F'PO.
P P
The condition that the forces on the particles should be pro-
portional to their distances from shows that the ratio of this
force to the distance OF is the same for all the particles. Since
m'p" cos FPO + m"p' cos P"FO ={m + m' + m") OP,
it is clear f^iat the condition is initially satisfied when p = p = p".
Hence, by the same reasoning as before, if the particles be pro-
jected with equal velocities in directions making equal angles with
OP, OP, OP' respectively, they will always remain at the angular
points of an equilateral triangle.
Ex. 1. Show that if the three particles attracted each other according to the
law of nature, the paths of the particles, when at the comers of an equilateral
triangle, are equal ellipses having for a common focus. Find the periodic time.
Ex. 2. If four particles he placed at the eorners of a quadrilateral whose sides
taken in order ore a,,h,c,d and diagonals p, p', then the particles could not move
under their mutual attractions so as to remain always at the corners of a similar
quadrilateral unless
(/)y » - 6»d») (c™ + o») + («"(!» - /jV) (6" + d") + (6"d'» - a'^c") (p^ + />'") = 0,
where the law 'J attraction is the inverse (w- l)ti» po'^ jt of the distance.
Show also that the mass at the intersection of b, e divided by the mass at
intersection of c, dia equal to the product of the area formed by a, p', d divided by
the area formed by a, b, p and the difference -7ji--j^ divided by the difference
P Cv
p» ~ i" ■
These results may be conveniently arrived at by reducing one angular point as
A of the quadrilateral to rest. The resolved part of all the forces which act on each
particle perpendicular to the straight line joining it to A will then bo zero. The
case of three particles may be treated in the same manner. The process is a little
shorter than that given in the text, but does not illustrate so well the subject of the
chapter.
MOMENTUM.
233
283. When the system under consideration consists of rigid
bodies we must use the results of Art. 75 to find the resolved part
of the momentum in any direction. The moment of the momentum
about any straight line may also be found by Art. 76 in Chap, ii,
combined with Art. 123 in Chap, iv, if the motion be in two
dimensions, or Art. 240 in Chap. V, if the motion be in three
dimensions.
284. Ex. A disc of any form is moving in its own plane in
any manner. Suddenly any point O in the disc is fixed, find the
angular velocity of the disc about O.
Let us suppose that just before became fixed the centre of
gravity O was moving with velocity V, and that p is the length of
the perpendicular from on the direction of motion. Also let to
be the angular velocity of the body about its centre of gravity.
Just after has become fixed, let the body bo turning about
with angular velocity w'. Let M¥ be the mou:ont of inertia of
the disc about the centre of gravity, and 'et 00 = r.
The change in the motion of the disc is produced by impulsive
forces acting at during a short time t^ — t^. These forces have
no moment about 0. Hence the moment of the momentum about
is the same just after and just before the impact. Just before
became fixed, the moment of the momentum about G was
Mk^(o (Art. 123), and the moment of the momentum of the whole
mass collected at was MVp. Hence the whole moment of the
momentum about is the sum of these two (Art. 76). Just after
has become fixed the body is turning about 0, hence by Art. 123
the moment of the momentum about is M{k'^ + r") w'. Equating
these we have
M (F + O 0)' = Mk'co + MVp ;,
,_ k^co+Vp
••«- ;fc« + ^8 .
Ex. A circular area is turning about a point A on its circumference. Suddenly
A is loosed and another point B also on the circ amference is fixed. Show that if AB
is a quadrant, the angular velocity is reduced to one-third its value, and if il ^ is a
third of the circumference, the area will be reduced to rest.
285. Ex. A disc of any form is turning about an axis Ox
sitiwited in its own plane with an angidar velocity co. Suddenly
this axis is let free and another axis Ox, also situated in the plane
of the disc, becomes fi^ed, it is required to find the new angular
velocity to' about Ox'.
The change in the motion of the disc is caused by the action
of the impulsive forces due to the sudden fixing of the axis Ox'.
These act at points situated in Ox' and have no moment about
,i I
ili
1
'iii
Ii
m
f
t • h
1 t
{■'?
n
1 1
:•■ IS
avMiMMi
234
MOMENTUM.
1
Ox'. Hence the moment of the momentum about Ojj' is the same
just before and just after Ox' is fixed.
tauces
Let d<T be any element of the area of the disc ; y, y' its dis-
ces from Ox, Ox'. Then yw, y'to are the velocities of d<x just
before and just after the impact. The moments of the momentum
about Oaf just before and just after are therefore yy'wdcr and
y'^wda: Summing these for the whole area of the disc, we have
(o'Xy'^da- = a^yy'da- (1).
First, let Ox, Ox be parallel, so that the point O is at in-
finity. Let h be the distance between the axes, then y' =y — h.
Hence we have
a>"ty'^d(T = a> {Xy'da — JiXyda] .
Let A, A' be the moments of inertia of the disc about Ox,
Ox' respectively, y the distance of the centre of gravity from Ox,
M the mass of the disc. Then we have
A'(o' = (o{A-Mhy).
Secondly, let Ox, Ox' not be parallel. Let be the origin
and the angle xOx = a, then y =ycosa. — x sin a. Let F be the
product of inertia of the disc about Ox, Oy where Oy is perpen-
dicular to Ox. Then by substitution in (1) we have
A'a>' = G) (^ cos a — jPsin a).
Ex. 1. An elliptic area of eccentricity e is turning about one latiis rectum.
Suddenly this latus rectum is loosed and the other fixed. Show that the angular
velocity is
of its former value.
Ex. 2. A right-angled triangular area ACB is turning about the side AC.
Sudde ily ^C is loosed and BC fixed. If C be the right angle, the angular velocity
is q-.p of its former ynlue.
286. A rigid body is moving freely in space in a known
manner. Suddenly either a straight line or a point in the body
becomes fixed. To determine the initial subsequent motion.
MOMENTUM.
235
This proposition will include the last two articles as par-
ticular cases. It is obvious that all the impulsive actions on the
body pass through the fixed straight line or the fixed point.
Hence the moments of the momentum of the body about the
fixed axis in the first case or about any axis *arough the fixed
point in the second case are unaltered by the impulsive forces.
First. Let a straight line suddenly become fixed. Let it be
taken as the axis of z.
Let MK* be the moment of inertia of the body about the axis
of z, and H the angular velocity after the straight line has become
fixed. Suppose that the body when moving freely was turning
with angular velocities w^., at^, w, about three straight lines Ox,
Gy' Oz' throuQfh the centre of gravity parallel to the axes of co-
ordinates. And let the co-ordinates of the centre of gravity be
i, y,z.
Then
0'o>. - (Sm^V) 0). - {tmz'y') a>^^M(x^^-y^^ = MK\ fl,
where C is the moment of inertia of the body about Oz', and
"^mz'x, %mz'y' are calculated with reference to the axes Ox ,
Oy', Oz'.
Secondly. Let a point in the moving body be suddenly
fixed in space. Take any three rectangular axes Ox, Oy, Oz,
and three parallel axes Ox', Oy, Oz through the centre of
gravity 0. Let (o^, w^, ©, be the known angular velocities of
the body about the axes Gx, Oy', Oz before the point became
fixed, Ilj,, Oj,, O, the unknown angular velocities about Ox, Oy,
Oz after nas become fixed.
Then, following the same notation as before, we hav« by
Art. 240,
A'ta^ - (Sm x'y) >, - (2m xz) w, ■^rtm\v~-z -J j
= A^^ - {tm xy) Oy - (2m xz) H,.
B'a^ - (2m y'z') (o, - (2«i y'x) o>^ + 'Zm\z~~x ^ j
= B[ly - (Zmyz) SI, - (tm yx) fl^.
O'w, - (2m z'x) m^ - (2m z'y') <a^ + '^'^ (^ -£ " ^ -£)
= Cn, — {Xm zx) Q,^ — (Zm zy) Sly.
These equations determine fl^, Xl„, fl,, It is obvious that
they may be greatly simplified by so choosing the axes that one
!
'I
;■
■II
M'
; I
(!
1 10
t
f
1 '
I
. n
236
MOMENTUM.
. I
of the two sets Ox, Oy, Oz or Gjs, Gy\ Oz may be a set of
principal axes.
287. If the body be turning about an axis 01 through the
centre of gravity O just before the point is fixed, the terms
containing the co-ordinates of the centre of gravity disappear
from the equations. They now admit of an easy geometrical
interpi-etation. The equation to the momental ellipsoid at the
centre of gravity is
A'X^ + 5' r' + C'Z" - 2:Zmyz' YZ- 2Sm a'a;' ZX
-^tmx'y'XY^'Me*.
It is therefore clear that the left-hand sides of these equations are
proportional to the direction-cosines of the diametral plane of a
straight line whose direction-cosines are proportional to {a>^, to , a,).
In the same way if we construct the momental ellipsoid at 0, the
right-hand sides are proportional to the direction-cosines of the
diametral plane of the axis (O^, fl^,, H,). Thus the instantaneous
axes of rotation, before and after is fixed, are so related that
their diametral planes with regard to the momental ellipsoids at
G and respectively are parallel.
"We may also deduce this result, without difficulty, from
Art. 117. The motion of the body about the axis GI may be
produced by an impulsive couple in the diametral plane of GI
with regard to the momental ellipsoid at 0. Let us then suppose
the body at rest and fixed, and let it be acted on by this couple.
It follows from the same article, that the body will begin to turn
about an axis OT which is such that its diametral plane with
regard to the momental ellipsoid at is parallel to the plane of
the couple.
The direction of the blow at may also be easily found. The
centre of gravity being at rest suddenly begins to move perpen-
dicular to the plane containing it and the axis 01'. This is
obviously the direction of the blow.
288. Ex. 1. A sphere in co-latitude 6 is hung up hj a point in its surface in equi-
librium under the action of gravity. Suddenly the rotation of the earth is stopped, it
is required to determine the motion of the sphere. [Math. Tripos, 1867, j
Let be the centre of the sphere, its point of suspension, and a its radius.
Let C be the centre of the earth. Let us suppose the figiire so drawn that the
sphere is moving away from the observer.
Let w= angular velocity of the earth, then if CQ^/m, the sphere is turning
about an axis Gp parallel to CP, the axis of the earth with angular velocity u, while
the centre of gravity is moving with velocity ^o- sin ^ . u.
Let OC, Op, and the perpendicular to the plane of 00, Op be taken as the axes
of X, y, z respectively, and let Oj., Oj,, 0, be the angular velocities about them just
after the rotation of the earth is stopped.
set of
MOMENTUM.
237
By Art. 286, the angular momenta about Ox, just before and juat after tbo
rotation was stopped, are equal to each other ;
where Mk* is the moment of inertia of tlie sphere about a diameter.
Again, the angular momenta about Oy are equal to each other ;
.-. - Mk^w Bisi0 + Mna^ w sm0=M {k'<' + a^)Qy.
Lastly, the angular momenta about Oz are equal.; .•• 0=Mk'''il,.
Solving these equations, we get
.0],=wsintf -,» — ~ = uBm.O — ^r— ^.
* F + a« 7
But Ox= u cos 9. Adding together the squares of Qj.,ily, 0, we have
O" = <o« |cos« ^ + r^-^^y sin" j ,
where Q is the angular velocity of the sphere about its instantaneous axis.
Ex. 2. A particle of mass M, without velocity, is suddenly attached to the sur-
face of the earth at the extremity of a radius vector making an angle with the
axis of the earth. If E be the mass of the earth before the addition of M, A and G
its principal moments of inertia at the centre, u the angular velocity about its axis,
prove that
a "^{E+M) AC+ EMCr' cos"* '
C0t<» = C0t^ + — s— . TFT-- — 2 a)
^ E Mr* Bin 0cob6'
where Q is the initial angular velocity about an axis parallel to the axis of the earth
and (p is the angle the initial axis of rotation makes with the axis of the earth.
Ex. 3. A body having a point fixed is turning with angular velocity u about
an axis 01 whose direction cosines referred to the principal axes at are {I, m, n).
Suddenly, an axis OF whose direction-eosines are {l', m', vi!) is fixed. Show that
the angular velocity about 01' is given by
{AV^ + Bm!* + (7n'a) m' = {AU + Bmrn! + Cm^) w,
where A, B, C are the principal moments at 0,
' i,
: 'l
I 'i
.1.
\\
I ',
1
! (J.
I
III!'
W
r
238
MOMENTUM.
Ex. 4. A regular homogeneous pi ism whoBe normal section is a regular polygon
of n sides rolls on a perfectly rough piano. Prove that, when the axis of rotation
changes from one edge to another, the angular velocity is reduced in the ratio
(
2 + 7 COS —
n
•iir
8 + COS —
289. In these examples the changes produced in the motion
were sudden, but the method of proceeding is the same if the
changes are gradual
Ex. 1. A bead of mass m, sUdes on a circular wire of mass M and radius a,
and the wire can iwna. freely aoout j, vertical diameter. Prove that, if u, fl be the
angular velocities of the wire when the bead is respectively at the extremities of a
horizontal and vertical diameter, - = 1 + 2 -rj .
Ex. 2. If the earth gradually contracted by radiation of heat, so as to be always
similar to itself as regards its physical constitution and form, prove that when every
radius vector has contracted an n^ part of its length, where n is small, the angular
velocity has increased a 2n"' part of its former value.
Ex. 3. If two railway trains each of mass M were to travel in opposite direc-
tions from the pole along a meridian and to arrive at the equator at the same time,
2M€?
prove that the angular velocity of the earth would be decreased by -=, , where ois
the equatorial radius of the earth and El? its moment of inertia about its axis of
figure.
What would be the effect if one train only were to travel from the pole to the
equator ?
Ex. 4. A fly alights perpendicularly on a sheet of paper lying on a smooth
horizontal plane and proceeds to describe the curve r=f{6) traced on the sheet of
paper, the equation to the curve being referred to the centre of gravity of the paper
as origin. Supposing the fly to be able to prevent himself from slipping on the
paper, show that his angular velocity in space about the common centre of gravity
of the paper and fly is equal to -~ — r-~ ; -=- , where M and m are the masses
of the paper and the fly and h is the radius of gyration of the paper about its centre
of gravity. Hence find the path of the fly in space.
Ex. 6. Suppose the ice to melt from the polar regions twenty degrees round
each pole to the extent of something more than a foot thick, enough to give 1^ feet
over those areas or -066 of a foot of water spread over the whole globe, which would
in reality raise the sea-level by only some such undiscoverable difference as f th of
an inch or an inch, then this would slacken the earth's rate as a time-keeper by one-
tenth of a second per year. This and the next example are taken from the Phil.
Mag. They are both due to Sir W. Thomson.
If £ be the mass of the earth, a its radius, Je its radius of gyration about the
polar axis, w its angular velocity before the melting, then we have by the principle
THE INVARIABLE PLANE.
230
[e to the
masses
centre
of angular momentum — = -5-=j-oC0Stf (l + oo8«), where M is the mass of the ice
W OJiK''
melted and is twenty degrees. Substituting for the letters their known numerical
values, the value of 8u is easily found.
Ex. 6. A layer of dust is formed on the earth h feet thick, where h is small, by
the fall of meteors reaching the earth from all directions. Show that the change in
the length of the day is nearly — f)^^ ^ ^^7 whore a is the radius of the earth
in feet, /> and D the densities of the dust and earth respectively. If the density of
the dust be twice that of water and A= ^V express this in numbers.
The InvariaUe Plane.
290. It is shown in Art. 72 of Chap, ii, that all the momenta
of the several particles of a system in motion, are together equi-
valent to a single resultant linear momentum at any assumed
origin 0, represented in direction and magnitude by a line O'V,
together with an ungular momentum about some line passing
through 0, represented in direction and magnitude by a line OH.
Let h^, h^, hg be the moments of the momenta of the particles
about any rectangular axes Ouj, Oy, Oz meeting in 0, so that
with similar expressions for A-,, h^, and let
h'
K
-T? and
n
the an-
Then the direction-cosines of OH are
gular momentum itself is represented by h.
If no external forces act on the system then by Art. 72 or Art.
279 h^, h^, Ag are constant throughout the motion, hence OH is
fixed in direction and magnitude. It is therefore called the in-
variable line at 0, and a plane perpendicular to OH is called the
invariable plane at 0.
If any straight line OL be drawn through making an angle 6
with the invariable line OH at 0, the angular momentum about
OL is ^cos^. For the axis of the resultant momentum-couple
is OH, and the resolved part about OL is therefore OH cos 0.
Hence the invariable line at may also be defined as that axis
through about which the moment of the momentum is greatest.
At different points of the system the position of the invariable
line is different. But the rules by which they are connected are
the same as those which connect the axes of the resultant couple
of a system of forces when the origin of reference is varied. These
pi I
■ i
ii^:
f;
i
\\\
240
MOMENTUM.
have been already stated in Art. 203 of Chap. V, and it is un-
necessary here to do more than generally to refer to them.
291. The position of the invariable plane at the centre of
gravity of the solar system may be found in the following manner.
Let the system be referred to any rectangular axes meeting in the
centre of gravity. Let co be the angular velocity of any body
about its axis of rotation. Let Mk^ be its moment of inertia
about that axis and (a, /9, 7) the direction-angles of that axis.
The axis of revolution and two perpendicular axes form a system
of principal axes at the centre of gravity. The angular momentum
about the axis of revolution is Mk^oa, and hence the angular mo-
mentum about an axis parallel to the axis of z is Mk''oi cos 7. The
moment of the momentum of the whole mass collected at the
centre of gravity about the axis of « is ilf [ a? -^ — y -^ j ,
have
hence we
A3 = Sl/F(BC0S7 + Sif^a;^-
dx
y-dt
lx\
dtj'
The values of h^y h^, may be found in a similar manner. The posi-
tion of the invariable plane is then known.
292. The Invariable Plane may be used in Astronomy as a
standard of reference. We may observe tbe positions of the
heavenly bodies with the greatest care, determining the co-ordi-
nates of each with regard to any axes we pilease. It is, however,
clear, that unless these axes are fixed in space, or if in motion
unless their motion is known, we have no means of transmitting
our knowledge to posterity. The planes of the ecliptic and the
equator have been generally made the chief planes of reference.
Both these are in motion and their motions are known to a near
degree of approximation, and will hereafter probably be known more
accurately. It might, therefore, be possible to calculate at some
future time, what their positions in space were when any set of
valuable observations were made. But in a very long time some
error may accumulate from year to year and finally become con-
siderable. The present positions of these planes in space may also be
transmitted to posterity by making observations on the fixed stars.
These bodies, however, are not absolutely fixed, and as time goes
on, the positions of the planes of reference would be determined
from these observations with less and less accuracy. A third
method, which has been suggested by Laplace, is to make use of
the Invariable Plane. If we suppose the bodies forming our
system, viz. the sun, planets, satellites, comets, &c., to be subject
only to their mutual attractions, it follows from the preceding
articles that the direction in space of the Invariable Plane at the
centre of gravity is absolutely fixed. It also follows from Art. 79
THE INVARIABLE PLANE.
241
that tlio centre of gravity is either at rest or moves uniformly ia
a straight line. We have here neglected the attractions of the
stars. These, however, are too small to be taken account of in
the present state of our astronomical knowledge. We may, there-
fore, determine to some extent the positions of our co-ordinate
planes in space, by referring them to the Invariable Plane as being
a plane which is more nearly fixed than any other known plane in
the solar system. The position of this plane may be calculated at
the present time from the present otate of the solar system, and at
any future time a similar calculation may be made founded on the
then state of the system. Thus a knowledge of its position cannot
be lost. A knowledge of the co-ordinates of the Invariable Plane
is not, however, sufficient to determine conversely the position of
our planes of reference. We must also know the co-ordinates of
some straight line in the Invariable Plane whose direction is also
fixed in space. Tliie, as Poisson has suggested, is supplied by the
projection on the Invariable Plane of the direction of motio.i
of the centre of gravity of the system. If the centre of gravity
of the solar system were at rest or moved perpendicularly to
the Invariable Plane, this would fail. In any case our knowledge
of the motion of the centre of gravity is not at present sufficient
to enable us to make much use of this fixed direction in space.
293. If the planets and bodies forming the solar system can
be regarded as spheres whose strata of equal density are concen-
tric spheres, their mutual attractions act along the straight lines
joining their centres. In this case the motions of their centres
will be the same as if each mass were collected into its centre of
gravity, while the motion of each about its centre of gravity
would continue unchanged for ever. Thus we may obtain another
fixed plane by omitting these laHer motions altogether. This is
what Laplace has done, and in his formula the terms depending
on the rotations of the bodies in the precedii)g values of A,, h.^, h^
are omitted. This plane might be called the Astronomical Invari-
able Plane to distinguish it from the true Dynamical Invariable
Plane. The former is perpendicular to the axis of the momentum
couple due to the motions of translation of the several bodies,
the latter is perpendicular to the axis of the momentum couple
due to the motions of translation and rotation.
The Astronomical Invariable Plane is not strictly fixed in
space, because the mutual attractions of the bodies do not strictly
act along the straight lines joining their centres of gravity, so that
the terms emitted in the expressions for A,, h^, h^ are not abso-
lutely constant. The effect of precession is to make the axis of
rotation of each body describe a cone in space, so that, even though
the angular velocity is unaltered, the position in space of the Astro-
nomical Invariable Plane must be slightly altered. A collision
between two bodies of the system, if such a thing were possible,
R. D. 16
■i
■ n
. i
ti
:
PI'
242
MOMENTUM.
or an explosion of a planet similar to that by which Olbers sup-
posed the planets Pallas, Ceres, Juno and Vesta, &c., to have been
produced, might make a considerable change in the sum of the
terms omitted. In this case there would be a change in the
fosition of the Astronomical Invariable Plane, but the Dynamical
nvariable Plane would be altogether unaffected. It might be
supposed that it would be preferable to use in Astronomy the
true Invariable Plane. But this is not necessarily the case, for
the angular velocities and moments of inertia of the bodies form-
ing our system are not all known, so that the position of the
Dynamical Invariable Plane cannot be calculated to any near
degree of appro x'mation, while we do know that the terms into
which these vmknown quantities enter are all very small or nearly
constant. All the terms rejected being small compared with
those retained, the Astronomical Invariable Plane must make
only a small angle with the Dynamical Invariable Plane. Al-
though the plane is very nearly fixed in space, yet its intersection
with the Dynamical Invariable Plane, owing to the smallness of
the inclination, may undergo considerable changes in course of
time.
In the M^canique Celeste, Laplace calculated the position of
the Astronomical Invariable Plane at the two epochs, 1750 and
1950, assuming the correctness for this period of his formulae for
the variations of the eccentricities, inclinations and nodes of the
planetary orbits. At the first epoch the inclination of this plane
to the ecliptic was 1"." 089, and longitude of the ascending node
114''.3979; at the second epoch the inclination will be the same as
before, and the longitude of the node 114!''.3934!.
294. Ex. 1. Show that the invariable plane at any point of space in the
straight line described by the centre of gravity of the solar system is parallel to
that at the centre of gravity.
Ex. 2. If the invariable planes at all points in a certain straight line are
parahei, then that straight line is parallel to the straight line described by the
centre of gravity.
Impulsive Forces in Three Dimensions.
295. To deterw.'He the general equations of motion of a body
about a fixed point undir the action of giuen impulses.
Let the fixed point be taken as tlie origin, and let the axes,
of co-ordinates be rectangular. Let (il„, ft^, flj, (tw^, Wy, m,) bo
the angular velocities of the body just before and just after the
impulse, and let the differences w^ — D,^, tw,^ — 11^, &>, — H, bo
called ft)/, ft)/, ft)/. Then ft)/, coj , coj are the angular velocities
generated by the impulse. By D'Alombcrt's Principle, see Art. 87,
IMPULSIVE FORCES.
24*3
the difference between the moments of the momenta of the par-
ticles of the system just before and just after the action of the
impulses is equal to the moment of the impulses. Hence by
Art. 240,
Aw J — (Xmxj/) (oj — (Xmxz) oaj = L \
Bwy - (Zrmjz) coj - (Xmi/x) coj =mI (1),
Cft)/ — (Zmzx) coj — (%mzy) coy = N J
where L, M, N are the moments of the impulsive forces about the
axes.
These three equations will suffice to determine the values of
(oj, (Oy, ft)/. These being added to the angular velocities before
the impulse, the initial motion of the body after the impulse is
found.
296. Ex. 1. Show that these equations are independent of each other.
This follows fi'om Art. 20 where it is shown that the climiuant of the equations
cannot vanish.
Ex. 2. Deduce those equations from the general equations of motion referred to
moving axes given in Art. 253.
Ex. 3. Show that if the body be acted on by a finite number of given impulses
following each other at infinitely short intervals, the final motion is independent of
their order.
297. It is to be observed that these equations leave the axes
of reference undetermined. They should be so chosen that the
values of A, Xmxy, &c. may be most easily found. If the posi-
tions of the principal axes at the fixed point are known they will
in general be found the most suitable.
lu that case the equations reduce to the simple form
AcoJ = L
B(o.:=M
(2).
The values of coj, cdJ, &)/ being known, we can find the pres-
sures on the fixed point. For by D'Alembert's Principle the
change in the linear momentum of the body in any direction is
equal to the resolved part of the impulsive forces. Hence if
F, G, H, be the pressures of the fixed point on the body
d;
,(3).
XX + F=M.'~ by Art. 8Q
= il/(«;i-a,;^)byArt. 219
tY+G = M{<o:x-<oJz)
lZ+H = M(wJ^-to;x)
298. Ex. A uniform disc bounded by an arc OP of a parabola, the axis ON,
and the ordinate PN, /ia» its vertex fixed. A bloto B is given to it perpendicular
16—2
ii'i
I jt >i
i r\
!l i 1
\M
ln]<
244
MOMENTUM.
to its plane at the other extremity P of the curved boundary. Supposing the disc to
be at rest before the application of the blow, find the initial motioii.
Let the equation to the parabola be y^=iax and let the axis of zbe perpendicular
to its plane. Then S>fta;2=0, i:myz = 0. Let /i be the mass of a unit of area and
let ON=c. Also
2mxy—ixjjxydxdy--/il x—dx=2/xl ax^dx=-nac^.
16 " *
^=i/* / y^dx=:^iMa\"' , B =
^^H
£=fi rxhjdx=- AuiV and C=A+£ by Art. 7.
The moments of the blow B about the axes are L = bJ^, M=-Bc, iV=0. The
equations of Art. 295 will become after substitution of these values
16 It 2 , oTji^T
^ixa c uy--iJMC^u„ = -Bc I
«,=0 J
i ■■
I I
I '■]
From these «,, uy may be found. By eliminating B we have — =7rz — — . Hence
Ujl iO C
7
if NQ, be taken equal io^rpNP, the disc will begin to rotate about OQ. The re-
Jo
75 B
Bultant angular velocity will be -^ — -^ OQ.
299. When a body free to turn about a fixed point is acted on by any number
of impulses, each impulse is equivalent to an equal and parallel impulse acting at
the fixed point together with an impulsive couple. The impulse at the fixed point
can have no effect on the motion of the body, and may therefore be left out of con-
sideration if only the motion is wanted. Compounding all the couples, we see that;
the general problem may be stated thus: — A body moving about a fixed point is
acted on by a given impulsive couple, find the change produced in the motion. The
analytical solution is comprised in the equations which have been written down iu
Art. 295. The ioUowing examples express the result in a geometrical form.
Ex. 1. Show from these equations that the resultant axis of the angular
velocity generated by the couple is the diametral line of the plane of the couple
with regard to the momental ellipsoid. See also Art. 117.
IMPULSIVE FORCES.
245
Ex. 2. Let be the magnitude of the couple, p the perpendicular from the
fixed point on the tangent plane to the momental ellipsoid parallel to the plane
of the couple Q. Let fi be the angular velocity generated, r the radius vector of
the ellipsoid which is the axis of fi. Let Mi* be the parameter of the ellipsoid.
Prove that 7: = — .
Q pr
Ex. 3. If Qx> fij/i ^a l>e angular velocities about three conjugate diameters of the
momental ellipsoid at the fixed point, such that their resultant is the angular
velocity generated by an impulsive couple G, A', B', C the moments of inertia
about these conjugate diameters, prove that
4'fije=Gcosa, B'Qg=0 coap, C'Q^ = G cob y,
where o, ft 7 are the angles the axis of G makes with the conjugate diameters.
Ex. 4. If a body free to turn about a fixed point be acted on by an impulsive
jouplu 0, whose axis is the radius vector r of the ellipsoid of gyration at 0, and if p
be the perpendicular from on the tangent plane at the extremity of r, then the
axis of the angular velocity generated by the blow will be the perpendicular p and
the magnitude fi is given by G = MprQ.
Ex. 5. Show that if a body at rest be acted on by any impulses, we may take
moments about the initial axis of rotation, according to the rule given in Art. 89, as
if it were a fixed axis.
i ,ii
. %
\'4
111
s ■
4^ ':<
300. Ex. 1. When a body turns about a fixed point the product of the moment
of inertia about the instantaneous axis into the square of the angular velocity is
called the Vis Viva. Let the vis viva generated from rest by any impulse be 2T
and let the vis viva generated by the same impulse when the body is constrained to
tiirn about a fixed axis passing through the fixed point be 2T'. Then prove that
T'=Tcos^d, where d is the angle between the eccentric lines of the two axes of
rotation with regard to the momental ellipsoid at the fixed point,
Ex. 2. Hence deduce Lagrange's theorem, that the vis viva generated from
rest by an impulse is greater when the body is free to turn about the fixed point,
than when constrained to turn about any axis through the fixed point.
Ex. 3. If a body be moving in any manner about a fixed point and an axis
through the fixed point be suddenly fixed, show that if the vis viva 2r be changed
into 22", we have T=Tcoa^$, where 6 has the same meaning as before.
301. To determine the motion of a free body acted on by any
given impidse.
Since the body is free, the motion round the centre of gravity
i:i the same as if that point were fixed. Hence the axes being any
three straight lines at right angles meeting at the centre of
gravity, the angular velocities of the body may still be found by
equations (1) and (2) of Ai't. 295.
To find the motion of the centre of gravity, let {U, V, W),
{> , V, vj) be the resolved velocities of the centre of gravity just
:t Jl
i:-:
1
u ;■■!
ii !
111
ii.
'I
246
MOMENTUM.
before and just after the impulse. Let X, F, Z be the com-
ponents of the blow, and let M be the whole mass. Then by
resolving parallel to the axes we have
M{u-TJ)^X, M{v- V)=Y, M{w-W) = Z.
If we follow the same notation as in Art. 295, the differences
u— U, V —V, w — Wma,y be called u, v, w'.
302. Ex. 1. A body at rest is acted on by an impulse whose components parallel
to the principal axes at the centre of gravity are (X, Y, Z) and the co-ordinates of
whose point of application referred to these axes are (p, q, r). Prove that if the
resulting motion be one of rotation only about bome axis,
A (B - C)'pYZ + B{G- A) qZX+ C(A-B) rXY=0.
Is this condition sufficient as well as necessary ? See Art. 221.
Ex. 2. A homogeneous cricket-ball is set rotating abotit a horizontal axis in
the vertical plane of projection with an angular velocity iJ. V,1ien it strikes the
ground, supposed perfectly rough and inelastic, the centre is moving with velocity
F in a direction making an angle a with the horizon, prove that the direction of the
motion of the ball after impact will make with the plane of projection an angle
-, where a is the radius of the ball.
tan~i
6 Fcosa
303. The equations of Art. 301 completely determine the
motion of a free body acted on by a given impulse, and from these
by Art. 219 we may determine the initial motion of any point of
the bod3^ Let (p, q, r) be the co-ordinates of the point of appli-
cation of the blow, then the moments of the blow round the axes
are respectively qZ — rY, rX—pZ, pY—qX. These must be
written on the right-hand sides of the equations of Art. 295. Let
ip'> q > '>"') he the co-ordinates of the point whose initial velocities
parallel to the axes are required. Let {u^, v^, wj, {u^, v^, w^) be
its velocities just before and just after *he impulse. Let the rest of
the notation be the same as that used in Art. 295. Then
Mjj - w, = w' -f ft)/r' - o,'q,
with similar equations for v^—v^, lu^ — w,. Substituting in these
equations the value of u', v, w', aj, <w ', coj given by Art. 301 we
see that u.^ — u^, v^ — v^, w,^ — iv^ are all linear functions oi X, Y,
Z of the first dcgreo of the form
«2 - ^1
FX-^GY+IIZ,
where F, 6r, //are functions of the structure of the body and the
co-ordinates of the two points.
304, When the point whose initial motion is required is the
point of application of the blow, and tlie axes of reference the
principal axes at the centre of gravity, these expressions take the
simple forms
iifti#«£
IMPACT OF ROUGH ELASTIC BODIES.
247
w.
The right-hand sides of these equations are the differential
coefficients of a quadratic function of X, F, Z, which we may call
E. It follows that for all blows at the same point P of the same
body the resultant change in the velocity of the point F of appli-
cation is perpendicular to the diametral plane of the direction of
the blow with regard to a certain ellipsoid whose centre is at P,
and whose equation is ^= constant.
The expression for E may be written in either of the equiva-
lent forms
1
^^^x^l±z^^
M
M
ABO
-, [(Ap' + Bq'+ Cr') {AX' + BY'+ CZ")
-{ApX+BqY+CrZf]
+ 1 {qZ- rYy + \ (rX -pZ) + ^{p Y-qXf.
In this latter form we see that it is
which is the vis viva of the motion generated by the impulse.
Impact of Rough Elastic Bodies.
305. The problem of determining the motion of any two bodies
after a collision involves some rather long analysis and yet there
are some points in which it differs essentially from the same
problem considered in two dimensions. We shall, therefore, first
consider a special problem which admits of being treated briefly,
and will then apply the same princijjios to the general problem
iu tliree dimensions.
306. Tmo rough ellipsoids moving in any manner impinge on
each other so that the extremity of a 2)^'iucii}al diameter of one
strikes the extremity of a principal diameter of the other, and at
that instant the three principal diameters of one are parallel to
those oftlie other. Find the motion just after impact.
Lot us refer the motion to co-ordinate axes parallel to the prin-
cipal diameters of either ellipsoid at the beginning oF the impact.
Tiien since the duration of the impact is indefinitely small and
the velocities arc finite, the bodies will not have time to change
i '!■
i^ !K
iu:
\m
i.
I !
248 MOMENTUM.
their position, and therefore the principal diameters will be par-
allel to the co-ordinate axes throughout the impact.
Let U, V, W be the resolved velocities of the centre of gravity
of one body just before impact; u, v, w the resolved velocities at
any time t after the beginning of the impact, but before its termi-
nation. Let ftj,, 12,, n, be the angular velocities of the body just
before impact about its principal diameters at the centre of gravity ;
ft),, ft) J,, ft), the angular velocities at the time t. Let a, h, e be the
semiaxes of the ellipsoid, and A, B, C the moments of inertia at
the centre of gi*avity about these axes respectively. Let M be the
mass of the body. Let accented letters denote the same quan-
tities for the other body. Let the bodies impinge at the extremi-
ties of the axes of c, c'.
Ijet P, Q, R be the resolved parts parallel to the axes of the
momentum generated in the body M by the blow during the time
t. Then —P, — Q^ — R are the resolved parts of the momentum
generated in the other body in the same time.
The equations of motion of the body M are
^(G),-n,) = Qc .
5K-nj = -Pci (1),
C (ft), - ft,) = )
M{v-V) = q\ (2).
M{w-W)=^R)
There will be six corresponding equations for the other body
which may be derived from these by accenting all the letters on
the left-hand sides and writing — P, — Q, — R and — c for P, Q, R
and c on the right-hand sides of these equations. Let us call these
new equations respectively (3) and (4).
Let S be the velocity with which one ellipsoid slides along the
other, and the angle the direction of sliding makes with the
axis of X, then
>S' cos ^ = U 4 C(oJ — M + CO), (5),
Si^md = v' ~c'(oJ —V + Cft)j, (6).
Let C be the relative velocity of compression, then
C=w' — w (7).
Substituting in these equations from the dynamical equations
we have
Scoa e= 8, cos e,~pP. (8),
/S'sin^=/8;sin^„-l7^ (9),
C=C,~rR (10).
If'.S
where
IMPACT OF ROUGH ELASTIC BODIES.
8, cos 0,^u'+ c'n; -u+cci
s,sme,= v'-c'aj-v+cnl
C = W'-W
249
.(11),
^=i+
,'«-«
1^
M
(12).
These are the constants of the impact. S^, C^ are the initial
velocities of sliding, and 0^ the angle the direction of initial sliding
makes with the axis of x. Let us take as the stando.rd case that
in which the body M is sliding along and compressing the body M,
so that /S'j and G^ are both positive. The other three constants
p, q, r are independent of the initial motion and are essentially
positive quantities.
307. Exactly as in two dimensions we shall adopt a graphical
method of tracing the changes which occur in the frictions. Let
us measure along the axes of x, y, z three lengths OP, OQ, OB to
represent the three reactions P, Q, R. Then if these be regarded
as the co-ordinates of a point T, the motion of T will represent
the changes in the forces. It will be convenient to trace the loci
given by /Si = 0, C=0. The locus given by /S> = is a straight
line parallel to the axis of R, which we may call the line of no
sliding. The locus given by C= Ois a plane parallel to the plane
P, Q, which we may call the plane of greatest compression. At
the beginning of the impact one ellipsoid is sliding along the other,
so that according to Art. 144 the friction called into play is limit-
ing. Since P, Q, R are the whole resolved momenta generated in
the time t\ dP, dQ, dR will be the r-esolved momenta generated
in the time dt, the two former being due to the frictional, and the
latter to the normal blow. Then the direction of the resultant of
dP, dQ must be opposite to the direction in which one point of
contact slides over the other, and the magnitude of the resultant
must be equal to fidR, ■where ^i is the coefficient of friction. We
have therefore
.(13),
e^Q " -So sin ^„ - jQ.
{dPf+{dQy = fi^{dRf (14).
The solution of these equations will indicate the manner in
which the representative point T approaches the line of no sliding.
[I
t I
Vm
1
:
1 '*
\
j I
1
1 •
■ .
'm
; i
i ,1
il
250 MOMENTUM.
The equation (13) can be solved by separating the variables.
We get
11
{8, cos 0, -pPy> = OL [S, sin e, - qQ) 5.
where a is an arbitrary constant. At the beginning of the motion
P and Q are zero, hence we have
/ 8, cos 8, -pP \i _ f 8, sin 0,-gQ \\ . .
V 8,cose, J [ 8,Bin6, J ^ ' ^'
which may also be written
(Br'^Am)^ (-)•
This equation gives the relation between the direction and the
velocity of sliding.
308. If the direction of sliding does not change during the
impact 6 must be constant and equal to 6^. We see from (16)
that if p = q, then d = 0Q; and conversely if 6=6^, /S would be
constant unless jp — q. Also if sin 6^ or cos 6^ be zero, 8 woidd
be zero or infinite unless Q=6^. The necessary and sufficient
condition that the direction of frict^'or. should not change during
the impact is therefore ^ = 2^ or sir 2^o = 0. The former of these
two conditions by (12) leads to
«' (i- 2) +<'"(t -!'}=» (i«)-
If this condition holds, we have by (13) P= Q cot 6^ and
therefore by (14)
^""f"^^"} (19).
It follows from these equations that when the friction is limit-
ing, the representative point T moves along a straight line making
an angle tan"' /a with the axis of P, in such a direction as to meet
the straight line of no sliding.
309. If the condition p^q docs not hold, we may, by dif-
ferentiating (8) and (9) and eliminating i*, Q, and 8, reduce the
determinaLcn of It in terms of Q to an integral.
By substituting for 8 from (17) in (8) and (9), we then have
P, Q, li expressed gs functions of 9. Thus we have the equations
to the curve along which the representative point T travels.
The curve along which 2' travels may more conveniently be
I l!
IMPACT OF ROUGH ELASTIC BODIES.
251
defined by the property that its tangent by (14) makes a constant
angle tan"^/u,with the axis of R and its projection on the plane
of FQ is given by (15). And it follows that this curve must
meet the straight line of no sliding, for the equation (15) is satis-
fied by 2>P =■ ^0 cos ^0 , qQ= 8^ sin 0^.
310. The whole progress of the impact may now be traced
exactly as in the corresponding problem in two dimensions. The
representative point T travels along a certain known curve, until
it reaches the line of no sliding. It then proceeds along the line
of no sliding, in such a direction that the abscissa li increases.
The complete value R^ of R for the wliole impact is found by
multiplying the abscissa R^ of the point at which T crosses the
plane of greatest compression by 1 + e so that R.^ = R^{l+e), if e
be the measure of the elasticity of the two bodies. The complete
values of the frictions called into play are the ordinates of the
position of T corresponding to the abscissa R^R^. Substi-
tuting these in the dynamical equations (1), (2), (3), (4), the
motion of the two bodies just after impact may be found.
311. Let us consider an example. Since the line of no
sliding is perpendicular to the plane of PQ, P and Q are constant
when T travels along this line. So that when once the sliding
friction has ceased, no more friction is called into play. If there-
fore sliding ceases at any instant before the termination of the
impact as when the bodies are either very rough or perfectly rough,
the whole frictional impulses are given by
7 ■
Q = -«l^-^«.
If o- be the arc of the curve whose equation is (15) from the
origin to the point where it meets the line of no sliding, then the
representative point T cuts the line of no sliding at a point whose
<T a C
abscissa is R= -. If the bodies be so rough that -<—"-, the
point T will not cross the plane of greatest compression until after
it has reached the line of no sliding. The whole normal impulse
in this case is therefore given by R = — (l + e). Substituting
these values of P, Q, R in the dynamical equations, the motion
just after impact may be found.
312. Ex. 1. If be the angle the direction of sliding of one ellipsoid over the
other malces with the axis of x, prove that continually increases or continually
decreases throughoiit the impact. And if the initial value of lie between and
2'
then approaches ^ or zero according as p is
> or < q. Show also that the
representative point reaches the line of no sliding when has either of those values.
I'l
"i X
Pi
I '•!
V '
;' i
,i
252
MOMENTUM.
Ex. 2. If the bodies be such that the direction of sliding continues unchanged
diu'ing the impact and the shdiug ceases before the termination of the impact,
the roughness must be such that u> 7^ — ,^ — .
CXl + e)
Ex. 3. If two rough spheres impinge on each other, prove that the direction of
sliding is the same throughout the impact. This proposition was first given by
Coriolis. Jeu de billard, 1835.
Ex. 4. If two inelastic solids of resolution impinge on each other, the vertex
of each being the point of contact, prove that the direction of sliding is the same
throughout the impact. This and the next proposition have been given by
M. Phillips in the fourteenth volume of Liouville's Journal.
Ex. 5. If two bodies having their principal axes at their centres of gravity
parallel impinge so that these centres of gravity are in the common normal at the
point of contact and if the initial direction of sliding be parallel to a principal axis
at either centre of gravity, then the direction of sliding will be the same throughout
the impact.
Ex. 6. If two ellipsoids of equal masses impinge on each other at the extremi-
ties of their axes of c, c', and if aa'=bb' and ca'~bc', prove that the direction of
friction is constant throughout the impact.
313. Two rough hodie$ moving in any manner impinge on each other. Find tJie
motion just after impact.
Let us refer the motion to co-ordinate axes, the axes of x, y being in the tangent
plane at the point of impact and the axis of z along the normal. Let U, V, W be
the resolved velocities of the centre of gravity of one body just before impact,
u, V, w the resolved velocities at any time i after the beginning, but before the
termination of the impact. Let fi^, Qy, Q, be the angular velocities of the same
body just before impact abo^it axes parallel tc the co-ordinate axes, meeting at the
centre of gravity; w„ Uy, w, the angular velocities at the time t. Let A, B, C, D,
E, F be the moments and products of inertia about axes parallel to the co-ordi-
uate axes meeting at the centre of gravity. Let JTbe the mass of the body. Let
accented letters denote the same quantities for the other body.
Let P, Q, R be the resolved parts parallel to the axes of the momentum
generated in the body M from the beginning of the impact, up to the time t. Then
-P, -Q, -R are the resolved parts of the momentum generated in the other body
in the same time.
Let (a;, y, z) {x', y', z') be the co-ordinates of the centres of gravity of the two
bodies referred to the point of contact as origin. The equations of motion are
therefore
A (w^ -U^-F (wy -Qy)-E (w, - Q,)= -yR + zQ.
-F(u,^~n^)+B(u,y-ny)-D(w,-n,)=-zP+xR[ (i).
-JJK-O^)- I>{u,y-Qy) + C(w,-Q,)=-xQ + 7jP)
M(v- r) = Q\ (2).
M(w-W) = R)
We have six similar equations for the other body, which differ from those in
having all the letters, except P, Q, R, accented, and in having the signs of P, Q, R
changed. These we shall call equations (3) and (1).
IMPACT OF ROUGH ELASTIC BODIES.
253
Let S bo the Telocity with which one body Rlides along the other and let $ be the
angle the direction of sliding makes with the axis of x. Also let C be the relative
velocity of compression, then
(S cos = U' - ujz'+ W.y - M + UyZ-UJ/K
Ssin0 = v' - Wj'a;' + u^'z' -v +u^-Ujz\
C=V)'- Wx'y' + Vy'sC^ - \W + Uj^ - WyX]
If we imbstitute from (1) (2) (3) (4) in (5) we find
SQeoa0o-Scose = aP+fQ + eR-.
SaBm0o-SBme=:fP + bQ + dRl...
Co-C=cP + dQ + eR)
(5).
(6),
where Sg, $ot C'o ^^^ ^^^ initial values of S, 0, G and are found from (5) by writing
for the letters their initial values. The expressions for a, h, c, d, e, f are rather
complicated, but it is unnecessary to calculate them.
314. We may now trace the whole progress of the impact by the use of a
graphical method. Let us measure from the point of contact 0, along the axes of
co-ordinates, three lengths OP, OQ, OR to represent the three reactions P, Q, R.
Then if, as before, these be regarded as the co-ordinates of a point T, the motion of
T will represent the changes in thb forces. The equations to the line of no sliding
are found by putting ^j=0 in the first two of equations (G). We see that it is a
straight line.
The equation to the plane of greatest compression is found by putting (7=0 in
the third of equations (6).
At the beginning of the impact one body is sliding along the other, so that the
friction called into play is limiting. The path of the representative point as it
travels from is given, as before, by
dP
dQ
cos sin
=HdR.
•(7).
When the representative point T reaches the line of no shding, the sliding of
one body along the other ceases for the instant. After this, only so much friction is
called into play as will suffice to prevent sliding, provided this amount is less than
the limiting friction. If therefore the angle the line of no sliding makes with the
axis of R be less than tan~Vi the point T wUl travel along it. But if the angle be
greater than tan"^/*, more friction is necessary to prevent sliding than can be called
into play. Accordingly the friction will continue to be limiting, but its direction
will be changed if S changes sign. The point T will then travel along >i, curve given
by equations (7) with d increased by ir.
The complete value B^ of i? for the whole impact is found by multiplying the ab-
scissa R of the point at which T crosses the plane of greatest compression by 1 -f c,
where e is the measure of elasticity, so that Ri=Ri (1-f f). The complete values
of P and Q are represented by the ordinates corresponding to the abscissa R^. Sub-
stituting in the dynamical equations, the motion just after impact may be found.
315. The path of the representative point before it reaches the line of no
Blidiug must be found by integrating (7). By differentiating (C) we have
d (S cos e) _ adP+fdQ + edB _ an cos 6 +fix sin g -f e ^
d{SBW.e) ~ fdP + bdQ + ddit ~ fn cos e + b/iBind + d'
H
:?!
254
MOMENTUM.
which reduces to
, .„ ^^ + '~ coB20+fBin29 +- COB 0+ -sine
S dd _ a-b ^^ 2^ ^.ycoa 2tf + - cos ^ - * bin
From this equation wo may find S as a function of in tho form S=Af(d), tho
constant A being determined from tho condition tliat 8=8^ when 0=Oa- Diileron-
tiatiug tho first of equations (0) and substituting from (7) we got
-Ad {ooa0f (0)] = {na 003 + fif am + c)dIl,
whence wo find R=AF{0) + B, tho constant B being dotormiued from the condition
that 22 vanishes when 0=0^, By substituting these values of >Si and R in tlie first
two equations of (6) we find P and Q in terms of 0. The three equations giving
P, Q, R as fimctions of are the equations to the path of the representative point.
It should be noticed that the tangent to the path at any point makes with tho axis
of i2 an angle equal to tan~' fx.
816. If the direction of friction does not change during tho impact, is con-
stant and equal to 0,,, so that d cannot be chosen as the independent variable. In
this case P=nRooa0Q, Q=iiRBia0Q and the representative point moves along a
straight line making with the axis of R an angle tan~^ n. Substituting these values
of P and Q in the first two of equations (6) we have
— ^ sin 2^0 +/COS 2^0 + - cos ^0 - - sin ^o = 0,
2 " M A*
as a necessary condition that the direction of friction should not change. Conversely
if this condition is satisfied the equations (6) and (7) may all be satisfied by making
constant. In this case it is also easy to see that the path of the representative
point intersects the line of no sliding. If Sq be zero, and if more friction is neces-
sary to prevent sliding then oan be called into play, the initial value of is im-
known. But if 0^ be taken equal to that root of the above equation which makes S
positive, and if d be supposed constant, the equations (6) and (7) are all satisfied.
817. Ex. 1. Let 0=
A
-F
-E
yR-
-zQ
-
-P
B
-D
zP-
■xR
-
-E
-D
C
xQ-
■yP
yR
-zQ
zP-xR
xQ-yP
and let A be the determinant obtained by leaving out the last row and last column.
Let G', A' be the corresponding expressions for the other body. Then a, b, c, d, e, f
are the coelBcients of P», Q", Ji«, 2QR, 2RP, 2PQ in the quadrio
(F + i)(^'+^''+^')+^^^-*--^'-
■ 2E,
A" ■ A'
where 2E is a constant, which may be shown to be the sum of the vires viva) of
the motions generated in the two bodies, as explained in Art. 304.
This quadric may be shown to be an ellipsoid by comparing its equation with
that given in Art. 28, Ex. 3.
Show also that a, b, e are necessarily positive and ab>f', bc> d\ ca > e^.
Show that by turning the axes of reference round the axis of R through the
proper angle we can make / zero.
EXAMPLES.
255
Ex. 2. Provo that the line of uo sliding is parallel to the oonjngato diameter of
the plane containing the frictions I', Q. And the plane of greatest compression is
the diametral plane of the reaction li.
Ex. 3. The line of no sliding is the intorsoction of the polar planes of two
points situated on the axes of P and Q and distant respectively fiom the origin
2E 2£
and „ • a • The plane of greatest compression is the polar plane of a
2E
pomt on the axis of li distant -p^ frrm the origin.
Ex. 4. The plane of PQ cuts the eUipsoid of Ex. 1 in an ellipHo, whose axes
divide tlie plane iiito four quadrants; the lino of uo sliding cuts the plane of PQ io
that quadiant in which the initial sliding Sq occurs.
Ex. 6. A parallel to the line of no sliding through the origin cuts the plane of
groatost compression, in a point whoso abscissa It has the same sign as C^, Honce
show, from geometrical considerations, that the representative point T must cross
the plane of greatest compression.
EXAMPLES*.
1. A cone revolves round its axis with a known angular velocity. The altitude
begins to diminish and the angle to increase, the volume being constant. Show
that the angular velocity is proportional to the altitude.
2. A circular disc is revolving in its own plane about its centre ; if a point in
the circiunference become fixed, find the new angular velocity.
3. A imiform rod of length 2a lying on a smooth horizontal plane passes
through a ring which permits the rod to rotate freely in the horizontal piano. Tho
middle point of tho rod being indefinitely near the ring any angular velocity ia
impressed on it, show that when it leaves the ring the radius vector of the middle
point will have swept out an area equal to
6
4. An elliptic lamina is rotating about its centre on a smooth horizontal table.
If Wj, Wj, W3 be its angular velocities respectively when tho extremities of its major
axis, its focus, and the extremity of the minor axis become fixed, prove
7 6
w,
Wo
5
+ —
w.
6. A rigid body moveable about a fixed point at which the principal momenta
are i, 5, C is struck by a blow of given magnitude at a given point. If the angular
velocity thus impressed on the bod' be the greatest possible, prove that (a, b, c)
bemg the co-ordinates of the given p int referred to the principal axes at 0, and
(I, m, n) the duection cosines of the blow, then
al + bm cii = 0,
I \B-' c) "^ m \G^ Ay ■*" n \A-' B-) '
* These examples are taken from the Examination Papers which have been set
iu the University and in tho Colleges.
I !
t .
W
||!
l'.\
'* '-I
!■...■
i
256
MOMENTUM.
6. Any triangular iamina ABC has the angular point C fixed and is capable of
free motion about it. A blow is struck at B perpendicular to the plane of the
triangle. '^' ow that the initial axis of rotation is that trisector of the side AB
which is furthest from £,
7. A. cone of mass m and vertical angle 2a can move freely about its axis, and
has a fine smooth groove cut along its surface so as to make a constant angle /3 with
the generating lines of the cone. A heavy particle of mass P moves along the
groove under the action of gravity, the system being initially at rest with the
particle at a distance c from the vertex. Show that if be the angle through
which the cone has turned when the particle is at any .distance r from the vertex,
then
mk'' + Pir^siia?a _ jie sin o . cot^
mk''' + Pc'^ sin* a
= €
it being the radius of gyration of the cono about its axis.
8. A body is turning about an axis through its centre of gravity, a point in the
body becomes suddenly fixed. If the new instantaneous axis be a principal axis
with respect to the point, show that the locus of the point is a rectangular
hyperbola.
9. A cube is rotating with angular velocity u about a diagonal, when one of its
edges which does not meet the diagonal suddenly becomes fixod. Show that the
angular velocity about this edge as axis =7- ._,
10. Two masses m, m! are connected by a fine smooth string which passes
round a right circular cylinder of radius a. The two particles are in motion in one
plane under no impressed forces, show that if A be the sum of the absolute areas
swept out in a time « by the two unw apped portions of the string,
^A
[2^1 _ 1 /I 1 \
T being the tension of the string at any time.
11. A piece of wire in the form of a circle lies at rest with its plane in contact
with a smooth horizontal table, when an Insect on it suddenly starts walking along
the arc with uniform relative velocity. Show that the wire revolves round its
centre with uniform angular velocity wliile that centre describes a ciicle in space
itb imiform angular velocity.
12. A uniform circular wire of radius a, moveable about a fixed point in its
circumference, lies on a smooth horizontal plane. An insect of mass equal to that
of the wire crawls along it, starting from the extremity of the diameter opposite to
the fixed point, its velocity relative to the wire being uniform and equal to V.
Prove that after a time t the wire will have turned through an angle
5 P tan"i I —
2a ^-6
\J'i
tan
'2a)'
13. A small insect moves along a uniform bar of mass equal to itself, and
length 2a, the extremities of which are constrained to remain on the circumference
ol a fixed circle, whose radius is — -
Supposing the insect to start from the middle
ble of
)f the
leAB
s, and
j3 with
Qg the
th the
hrough
vertexi
,t in the
pal axis
tangular
,ne of its
that the
jh passes
on in one
ute areas
jn contact
dng along
[round its
in space
bint in its
lal to that
[)posite to
3ual to V.
litself, and
lumference
Ithe midiile
EXAMPLES.
257
point of the bar, and its velocity relatively to the bar to be uniform and equal to V;
1 Vt
prove that the bar in time t will turn through an angle —.- tan~^ — .
14. A rough circular disc can revolve freely in a horizontal plane about a vertical
axis through its centre. An equiangular spiral is traced on the disc having the
centre for pole. An insect whose mass is an n^ that of the disc crawls along the
curve starting from the point at which it cuts the edge. Show that when the insect
reaches the centre the disc will have revolved through an angle — log ( 1 + - | ,
where a is the angle between the tangent and radius vector at any point of the
epiral.
15. A uniform circular disc moveable about its centre in its own plane (which
is horizontal) has a fine groove in it cut along a radius, and is set rotating with
an angular velocity w. A small rocket whose weight is an Hth of the weight of the
disc is placed at the inner extremity of the groove and discharged ; and when it has
left the groove, the same is done with another equal rocket, and so on. Find the
angular velocity after n of these operations, and if n be indefinitely increased, show
that the limiting value of the same is we~'.
16. A rigid body is rotating about an axis through its centre of gravity, when a
certain point of the body becomes suddenly fixed, the axis being simultaneously set
free; find the equations of the new instantaneous axis; and prove that, if it be
parallel to the originally fixed axis, the point must he in the line represented by
•62)-=0; the prin-
the equations aVx + Vmy + c'm = 0, (6''
.c^)^+(c^-a^)l + (a^.
cipal axes through the centre of gravity being taken as axes of co-ordinates, a, b, c
the radii of gyration about these lines, and /, m, n the direction-cosines of the
originally fixed axis referred to them.
17. A solid body rotating with uniform velocity w about a fixed axis contains
a closed tubular channel of small uniform section filled with an incomprecdible fluid
in relative equilibrium ; if the rotation of the solid body were suddenly destroyed
the fluid would move in the tube with a velocity - - , where A is the area of the
t
projection of the axis of the tube on a plane perpendicular to the axis of rotation
and I is the length of the tube.
18. A gate without a latch in the form of a rectangular lamina is fitted with a
universal joint at the upper corner and at the lower corner there is a short bar
normal to the plane of the gate and projecting equally on both sides of it. As the
gate swings to either side from its stable position of rest, one or other end of the
bar becomes a fixed point. If h be the height of the gate, h tan a its length and 2/3
the angle which the bar subtends at the upper corner, show that the angular
veloci .y of the gate as it passes through the position of rest is impulsively dimin-
ished in the ratio
sin!" a - tan" fi
, and the time between successive impacts when tho
Bin«o-htan»^
oscUIations become small decreases in the same ratio, the weights of the bar and
joint being neglected.
R. 1).
17
-1 1
i iM
'
■rv.
* '-' \
t ■ - :
( :
! i|
liii
" s
II
■i
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CHAPTER VII.
VIS VIVA.
The Force-function and Work.
318. If r>. particle of mass m be projected along the axis of x
with an initia^ velocity V and be acted on by a force i^ in the
same direction, the motion is given by the equation m -,.^= F.
Integrating this with regard to t, if v be the velocity after a
time t, we have,
m
{y- V)=fFdt.
•'0
If we multiply both sides of the differential equation of the
dx
second order by -7 and integrate, we get*
dt
^m {v' - V) = f Fdw.
* It is seldom that Matliematicip'" , can be foimd engaged in a controversy
such as that which raged for forty years in the last century. The object of the
dispute was to determine liow the force of a body in motion was to be measured.
Up to the year 1686, the measure taken was the product of the mass of the body
into its velocity. Leibnitz, however, tlionght he perceived an error in the con;mon
opinion, and undertook to show that the proper measure should be, the product of
the mass into the square of the velocity. Shortly all Europe was divided between
the rival theories. Germany took part with Leibnitz and Bernoulli ; while Eng-
land, true to the old measure, combated their arguments with great success.
France was divided, an illustrious lady, the Marquise du Chatulet, being first a
warm supporter and then an opponent of Leibnitzian opinions. Holland and Italy
wore in general favourable to the German philosopher. But what was most strange
in this great dispute was, that the same problem, solved by geometers of opposite
opinions, had the same solution. However the force was measured, whether by
the first or the second power of the velocity, the result was the same. The argu-
ments and replies advanced on both sides are briefly given in Montucla's Ilistory,
and are most interesting. For this however we have no space. The controversy
was at last closed by D'Alembort, who showed in his treatise on Dynamics that the
whole dispute was a mere question of words. When we speak, he says, of the force
of a moving body, we either attach no clear meaning to the word or we understand
only the property that certain resistances can be overcome by the moving body. It
FORCE-FUNCTION AND WORK.
259
The first of these integrals shows that the change of the mo-
mentum is equal to the time-integral of the force. By applying
similar reasoning to the motion of a dynamical system we have
been led in the last chapter to the general principle enunciated in
Art. 279, and afterwards to its application to determine the changes
produced by very great forces acting for a very short time. The
s'^oond integral shows that half the change of the vis viva is equal
to the space-integral of the force. It is our object in this chapter
to extend this result also, and to apply it to the general motion of
a system of bodies.
.319. For the purposes of description it will be convenient to
give names to the two sides of this equation. Twice the left-hand
side is usually called the vis viva of the particle, a term introduced
by Leibnitz about the year 1695. Half the vis viva is also called
tlie kinetic energy of the particle. Many names have been given,
to the right-hand side at various times. It is now commonly
called the woi'k of the force F. When the force does not act
in the direction of the motion of its point of application the term
"work" will require a more extended definition. This we shall
discuss in the next article.
320. Let a force F act at a point A of a body in the direction
AB, and let us suppose the point A to move into any other po-
sition A very near A. If be the angle the direction AB oi
the force makes with the direction AA' of the displacement of
the point of application, then the product i^ . ^^' . cos </> is called
the work done by the force. If for ^ we write the angle the
direction AB of the force makes Avith the direction A' A opposite
to the displacement, the product is called the work done against
the force. If we drop a perpendicular A'M on AB, the work done
hy the force is also ecjual to the product F.AM, where AM is to be
estimated as positive wh^n in the direction of the force. If F' be
the resolved part of F in the direction of the displacement, the
work is also equal to F. A A'. If several forces act, we can in the
same way find the work done by each. The sum of all these is
the work done by the whole system of forces.
is not then by any simple coiisiilorations of merely the mass and the velocity of the
body that we must estimate this force, but by the natm'e of the obstacles overcome.
The greater the resistance overcome, the greater we may say is the force ; provided
we do not understand by this word a pretended existence inherent in the body, but
simply use it as an abridged mode of expressing a fact. I^Alembert then points
out that there are different kinds of obstacles and examines how their different
Iduds of resistances may be used as measures. It will perhaps be sufficient to
observe, that the resistance may in some cases be more conveniently measured
liy a space-integral and in others by a time-integral. See Jlontucla's Hhlorij,
Vol. III. and Whewell's Hhtory, Vol. ii.
17—2
i
Ffjl
; 1
■f: ■ .
'.
l-i
.
'■ 'I
i
i! W
I
^iM
! i
260
VIS VIVA.
I !..
I' I
■ ' ;
Thus defined, tho work done by a force, corresponding to any
indefinifply small displacement, is the same as the virtualmoff* int
r.f trc force. In ,Statics, we are only concerned -with the small
ljy[»otbetical displacements, we give the system in applying the
priiiciple of Virtual Velocities, and this definition is therefore
suihcient But in Dynamics the bodies are in motion, and we
must extend our definition of work to include the case of a dis-
placement of any magnitude. When the points of application of
the forces receive finite displacements we must divide the path
of each into elements. The work done in each element may be
found by the definition given above. The sum of all these is the
whole work.
It sbould be noticed that tbe work done by given forces as the
body moves from one given position to another, is independent
of the time of transit. As stated in Art. 318, the work is a space-
integra'l and not a time-integral.
321. If two systems of forces be equivalent, the work done hy
one in any small displacement is equal to that done hy the other.
This follows at once from the principle of Virtual Velocities in
Statics. For if every force in one system be reversed in di-
rection without altering its point of application or its magnitude,
the two systems will be in equilibrium, and the sum of their
virtual moments will therefore be zero. Restoring the system of
forces to its original state, we see that tlie virtual moments of the
two systems are equal. If the displacements are finite the same
remark applies to each successive element of the displacement,
and therefore to the whole displacement.
322. We may now find an analytical expi'cpsion for the work
done by a system of forces. Let {x, y, z"\ the rectangular
co-ordinates of a particle of the system ai «i !< ohe mass of this
particle be m. Let (A' Y, Z) be the accLluiating forces acting
on the particle resolved parallel to the axes of co-ordinates. Then
mX, mF, mZ are the dynamical measures of the acting forces.
Let us suppose the particle to move into the position x -f- dx,
y -t- dy, z-\- dz; then according to the definition the work done by
the forces will be
2 {mXdx + m YJy + m.Zdz) (1 ),
the summation extending to all the forces of the system. If the
bodies receive any finite displacements, the whole work will be
'.m
j{Xdx+ Ydy + Zdz) (2),
tlie limits' v'.' the integral bf vng determined by the extreme
positiunr of liie sys'cTfi.
FCxvClil-FUNCTION AND WORK.
261
<
>.
323. Vr hen the forces are such as fff^nerally occur in riature,
it will be proved that the summation (1) of tb'^ last Article is a
t,omplete differential, i.e. it can bo iiit«^grated independently of any
relation between the co-ordinates x, ;y, z. The summation (2) can
therefore be expressed as a function of the f^o-ordinates of the
system. When this is the case the indejhiite integral of the
summation (2) is called the force-function. This name was given
to the function by Sir W. R. Hamilton and Jacobi independently
of each other.
If the force-function be called U, the work done by the forces
when the bodicf^ move from one given position to another is the
definite integral b\— U^, where tf^ and Z/^ ave the values of U,
corresponding to the two given positions of the bodies. It follows
that the work is independent of the mode in which the system
moves from the first given position to the second. In other words,
the work depends on the co-ordinates of the two given extreme
positions, and not on the co-ordinates of any intermediate posi-
tion. When the forces are such as to possess this property, i.e.
when they possess a force-function, they have been called a con-
servative system of forces. This name was given to the system
by Sir W. Thomson.
324, There will he a force function, first, iiilien the external forces
tend to fixed centres at finite distances and are functions of the
distances from those centres ; and secondly, when the force due to
the mutual attractions or repulsions of the particles of the system
are functions of the distances between the attracting or repelling
particles.
Let ?n^ (r) be the action of any fixed centre of force on a
particle m distant r, estimated positive in the direction in which r
is measured, i.e. from the centre of force. Then the summation
(1) in Art. 322 is clearly Xm<f) (r) c?r. This is a complete differ-
ential. Thus the force-function exists and is equal to Xm |(^(r)cZr.
Let mm'^ (r) be the action between two particles m, m' whose
distance apart is r, and as before let this force be considered
positive when repulsive. Then the summation (1) becomes
"Himm' (f)(r) dr. Tlie force-function therefore exists, and is equal
to Xmm' / <f) (r) dr.
If the law of attraction be the inverse square of the distance,
</) (r) = — -^ and the integral is - . Thus the force-function differs
from the Potential by a constant qua':itity.
32.5. It is clear that there is r.othing in the definition of the
force-function to compel us to use Cartesian Co-ordinates. If
11 !■ t^ifi
' ■ 'i.
;l
».ii
262
VIS VIVA
P, Q, &c. be forces acting on a particle, dp, dq, &c. their virtual
velocitieg, m the mass of the particle, then the force-function is
?/=: tm \[Pdp + Qdq -V &c.),
the summation extending to all the forces of the system.
Ex. 1. If {p, <f>, z) be the cyliudrical or semi-polar co-ordinates of the particle
m ; P, Q, Z the resolved parts of the forces respectively along and perpendicular to
p and along z, prove that dU'=^m(Pdp + Qpd(p + Z(]z).
Ex. 2. If (r, 0, (/>) be the polar co-ordinates of the particle m ; P, Q, R the
resolved parts of the forces respectively along the radius vector, perpendicular to it
in the plane of and perpendiciUar to that plane, prove that
d f7= 2m {Pdr + Qrd0 + Itr sin ed<p).
Ex. 3. If {x, y.,z) bo the oblique Cartesian co-ordinates of ni; A', Y, Z the
components along the axes, prove that
dU=^'2.m[X{flx + vdy+ii.dz)+ Y {vdx + di/ + \dz) + Z {/j.dx + \dij + dz)},
where (X, ,u, v) are the cosines of the angles between the axes yz, zx, xy respectively.
This example is due to Foiusot.
Ex. 4. If the system be referred to rectangular axes moving about a fixed
origiu, show that the force-function may be found by writing for dx, dy, dz, in
Art. 322 the values of udt, vdt, wdt given in Art. 244.
326. If a system receive any small displacement ds parallel to
a given straight Iviie and an angular displacement dd round that
line, then the partial differential coefficients -r- and -j^ represent
respectively the resolved part of all the forces along the line and the
moment of the forces about it.
Since dU is the sura of the virtual moments of all the forces
due to ar^ displacement, it is independent of any particular co-
ordinate axes. Let the straight line along which ds is measured
be taken as the axis of z. laking the same notation as before,
d U== Sw {Xdx + Ydy + Zdz).
But dx =0, dy = 0, and dz = ds, hence we have
dU
dU ~ds .XmZ: .". -,- =XmZ.
ds
Here dU means the -honge produced in U by the single dis-
placement of the svRtoi;:, tai en as one body, parallel to the given
straight line, through a space Is.
Again, the moment of fifl the forces about the axis of z is
Sm {xY — yX), but dx — — ydO, dy ~ xdd, and dz = 0. Hence the
above moment
rdy-\- Xdx + Zdz _dU
de Id'
= 2
m
wm
FORCE-FUNCTION AND WORK.
263
Here dU is the change produced in U by the single rotation
of the system, taken as one body, round the given axis through
an angle dd.
327. As considerable use ■will be made of tlio force-function, the student will
find it advantageous to acquire a facility in writing down its form. The following
examples have therefore been given.
Ex. An elastic htring whose unstretched length is I is stretched, find the work
done by the tension when the string is stretched from a length r to a length r'.
Let p be any length intermediate between r and / and let E be the coefBcient of
elasticity. The tension is T=E- and acts opposite to the direction in which p
is measured. The work done while p becomes p + dp is therefore equal to - Tdp,
The force-function is therefore - JTdp. If this be integrated and taken between
the limits r to r', we find the required work equal to - k", { (r' - /)' - ()• -lY).
It follows from this that the work required to stretch an elastic string from one
length to another is the product of the arithmetic mean of the initial and final
tensions iuJ;o the extension of the string.
328. Ex. 1. A system of bodies falls under the action of gravity. If ilf be
the whole mass, h the space descended by the centre of gravity of the whole system,
the work done by gravity is Myh.
Let the axis of z be vertical and let the positive direction bo downwards. Then
in the summation (1) of Art, 322, Z=0, Y=0 and Z=g. Hence dU—I,mgdz. If z
be the depth of the centre of gravity below the plane of xy, and C be any constant,
we find U=Mgz + C. Taking this between limits we easily obtain the result given.
The theoretical imit of work is the work done by a dynamical unit of force
acting through a tmit of space. We may use the result of this example to supply a
practical imit. The work required to raise the centre of gravity of a given mass a
given height at a given place may be taken as the unit of work. English engineers
use a pound for the mass and a foot for the height, and the unit is then called afoot-
pound. The term Horse-potcer is used to express the work done per unit of time.
The unit of horse-power is usually taken to be 33000 foot-pounds per minute. The
duty of a steam-engine is the actual work done by the consumption of a unit quan-
tity, usually a bushel, of coal.
Ex. 2. A force communicates to a particle whose mass is equal to that of a
cubic foot of water a velocity of one foot per minute. Find the work done in foot-
pounds.
Ex. 3. Prove that the amount of work required to raise to the surface of the earth
the homosienoous contents of a very small conical cavity whose vertex is at the
centre of the earth, is equal to that which would be expended in raising the whole
mass of the contents, through a space equal to one-fifth of the earth's radius from
the surface, supposing the force of gravity to remain constant. [Coll. Exam.]
329. Ex. 1. If m, m! be the masses of two particles attracting each other with
a force
mm'
where r is the distance between them, show that the work done when
TYVHIh
thoy have moved from an infinite distance apart to a distance r is — ,
This follows from Art. 321.
^|.^^
*!
ISTTiTiii rjjrT'Xvi
II
I! I
i
2C4.
VIS VIVA.
Ex. 2. If the particles oompobing any mass were separated from each other,
work might be obtained from their mutual attractions by allowing the particles to
approach eacli other. The work thus obtained is greatest when the particles are
collected together from infinite distances. If dv be an element of volume of a solid
mass attracting according to the law of nature, p the density of the element, V the
potential of th^ solid mass at the element dv, prove that the work performed iu
collecting the particles composing the mass from infinite distances is - A^dr,
Let Mj, JHj, m,, &o. bo the masses of any particles, r,2, r^^, &o. the distances
between the masses m^, nig, mj, m-^, &o. in any arrangement. Then as before
the work done in collecting them from infinite distances is U— - *—" + —^—*+ &c.,
'IS
'S3
which may be written U-=^
mm
Now if Fi be tbo potential at the particle m^ of
7?? 4M
all the particles except m^ in the given arrangement, V^— ~ + —+... li Vg, Fg, &o.
'la
'13
have similar meanings wo may write the work in the form
ir= ^:i ( Fjmi + FaOTa +...) = i 2) Vm.
In finding the potential of aniy solid mass at any point P we may omit the
matter within any indefinitely small element enclosing P if its density be finite.
For, since "potential is mass divided by distance," and the mass varies as the cube
of the linear dimensions, it follows that the potential of similar figures at points
similarly situated must vary as the square of the linear dimensions and must vanish
when the mass becomes elementary and the distance indefinitely small. In
applying, therefore, the form 17= - S Fm to a solid body we may write pdv for m and
take F to be the potential of the whole mais at the element dv.
The problem of determining how much work can be obtained from the bodies
forming the solar system by allowing them to consolidate into a solid mass has
been considered by several philosophers. Sir W. Thomson has calculated that the
potential energy or the ^ 'ork which can be obtained from the existimg solar system
is 380,000 X 1033 foot-pounds. Edin. Trans. 1854.
Ex. 3. The particles composing a homogeneous sphere of mass M and radius
r were originally at infinite distances from each other. Prove that the work done
3M2
by their mutual attractions is
momentum
5 r
Ex. 4. The pai'ticles of a homogeneous ellipsoid whose mass is 3/ and semiaxes
a, b, c are collected from infinite distances, show that the work done ia
>r
d\
sJ(a'^ + \)(b^ + \)(c' + \}
330. Ex. 1. An envelope of any shape and whose volume is v, contains gas
at a uniform pressure p. Assuming that the pressure of the gas per unit of area
is some function of the volume occupied by it, prove that the work done by the
fb
pressures when the volume increases from v = o to t' = 5 is I pdv.
reasoning as
FORCE-FUNCTION AND WORK.
265
Divide the surface into elementary areas each equal to d<r, then pd<T ia the
pressure on d<r. When the volume has increased to v + dv, let any element da take
the position da' and let dn be the length of the perpendicular drawn from the
central point of da' on the plane of da, then pdadn is the work done by the pressure
on da and p jdadn ia the work done over the whole area. But dadn is the volume
of the oblique cylinder whose babe is da and opposite face da'; so that jdadn is the
whole increment of volume. The whole work don© when the volume increases by
do is therefore pdv.
Ex. 2. A spherical envelope of radius a contains gas at f-essure P, assuming
that the pressure of the gas per unit of area is inversely propoi tional to the volume
occupied by it, prove that the work required to compress the envelope into a sphere
n
of radius 6 is iirceT log t • I
Ex. 3. An envelope of any shape contains gas and the shape ia altered '.vitLout
altering the volume. Show that the work done over the whole surface is zero.
331. Ex. 1. An impulsive force acts on a body in a fixed direction in space. Show
that if F be the whole momentum communicated by the force ; u^, u^ the velocities
of the point of application resolved in the direction of the force, just before
and just after the impulse, then the work done by the impulse is -'' ^ F. This
proposition is given in Thomson and Tait's Natural Philosophy.
Let us regard the impulse as the limi* of a finite force acting in the fixed direc-
tion for a very short time T. Let the direction of the axis of x be taken parallel to
the fixed direction and let X be the whole momentum communicated during a time
t measured from the commencement of the impulse. Here t is any time less than
T and X varies from zero to i^ as « varies from to T. Also, since A' is the whole
momentum up to the time t, -^ is the moving force on the body at the time t. Let
w be the resolved velocity of the point of application at the time (, then Uq and Mj
are the values of u when t=0 and t = T. Since udt is the space described in the
time dt by the point of application of the force -r- , the work done in the time T ia
I — - udt. This is the same as I udX. Now, when the time t is indefinitely small,
jo dt Jo
the velocity u is known by Art. 8o3 to be a linear function of X, so that we may write
M =Uo + ZiC where L is a constant depending on the nature of the body. Substi-
tuting this value of w, we have the work equal to I {Uq + LX) dX=VQF+L
1'"'"
ps
But
ttj = u„ + LF. Eliminating L we find that the work = 5 (Uj + u^) F.
Ex. 2, Find the work done by an impulse whose direction is not necessarily
the same during the indefinitely short duration of the force.
Let X, Y, Z be the components of the whole momentum given to the body in
any time t measured from the commencement of the impulse. Let «, v, w be the
resolvea velocities of the point of application at the time t. Then, by the same
reasoning as before, the work done - \ ( -j- " + -jT " + -r, '" ) dt. But by Art. 30 1
• W 1
1
■111
260
VIS VIVA.
i
h
It:
when T is indefinitely small u=Mo +■!-„, «=t'o+7pi «'=Wo+ j, , wbero E is a
known quadratic function of {X, Y, Z) depending on the nature of the body. Sub-
stituting we bave
work = Huyi + Vori + Wo^i+ \\Jx '^ dY dZ )
= n^X^^■v^T^ + n'^Z^■^E^,
wbere X^, Fj, Z^, E^ are the values of .Y, Y, Z, E when t = T.
We may eliminate tbe form of tbe body and express the work in terms of tbo
resolved velocities of the point of application just after tbe termination of the im-
pulse. Since E^ is a homogeneous quadratic function of X^, Y^, Z^ we have
'iE^=fi^^ X, + ^^1 Y,+ ^^Z, = (n,- V,) X, + {v,- v,) Y, + (w, - w,) Z.
Substituting we find
wor
, _"o + Wl V . ''o+^i K , Wo + Wl 7
332. A spherical membrane is stretched into a sphere whose radius is r. Let
Tds be the tension across any elementary arc ds when tbe membrane is stretched,
where T is a known function of r depending on the nature of tbe material. Then
the work done by the tensions, when the membrane is stretched into a sphere of
fb
radius 6 is Sir | Trdr.
■f
Let the centre of the sphere be taken as origin and let us refer any point on the
sphere to polar co-ordinates (r, B, (p). The adjacent sides of an elementary area
are rdd, r sin dd<p. The tensions across rd$ and the opposite side are each equal to
Trdd. When the radius r increases bj dr, the distance between these sides is
increased by dr sin dd<p, this being tbe differential of an adjacent side. Hence the
work done by these tensions is Trdd . dr sin dd<p. Let us now consider the remain-
ing two sides of the element. The tensions across r sin Odtp and the opposite side
are each equal to Trsva6d<f>. When the radius r increases by dr, the distance
between these sides is increased by drdO. Hence the work done by these tensions
is Tr sin 6d(t> . drdd. The work done by tbe tensions on the four sides of the
element is therefore 2Trdr sin 6ddd<f>. Integrating this from 0-=O to 2w, 6=0 to w,
we find that tbe work done over tbe whole sphere when the radius increases by dr
is 8irTrdr.
If the membrane be such that we may apply Hooke's law to the tension T, wo
have T=E , where a is the natural radius of the membrane and E is the co-
a
efficient of elasticity. Substitutiug this vaVie of T we find that the work done by
4:E
tbe tensions when the radius increases from a to 6 is - - (6 -a)* (26-ha).
O Of
If we assume that for a sopp-bubble T is constant, we find that the work done
when the radius increases from ;* io 6 is iwT (6' -a').
If we suppose the spherical membrane to be slowly stretched by filling it with
gas at a pressure^, we have by a theorem in Hydrostatics j)r=2r. In this case the
r 4
work required has been shown to be pdv, and since v = q7rr' this leads to the same
resulL as before.
M
FORCE-FUNCTION AND WORK.
207
833. Ex. 1. A roil originally straight Ih bent in one piano, if L be the stresfi
couple at any point, p the radius of curvature, it is known both by experiment and
thcoiT that Z = - where ^ ia a constant depending on the nature of the material
P
and the form of a ticction of the rod. ABsuming this prove that the work done iu
t '* f 'i
bending the rod is o I tt £?*■
Let PQ be any element of the rod and lot its length be dn. As PQ is being bent,
let t// be the indefinitely small angle between the tangents at its extremities, then
the stress couple ia E J-. Ah f increases from to — the work done is -^ / ^dr//,
which is the same as
dn'
The work done on the whole rod is therefore
Ij?''-
Ex. 2. A uniform heavy rod of length I and weight w is supported at its two
extremities so as to be horizontal. Show the work done by gravity in bending
it IS
240E '
Conservation of Vis Viva and Energy.
n34<. Def. The Vis Viva of a particle is the product of its
mass into the square of its velocity.
If a system he in motion under the action of finite forces, and if
the geometrical relations of the parts of the system he expressed hy
equations tvhich do not contain the time explicitly, the change in the
vis viva of the system in passing from any one position to any other
is equal to twice the corresponding work done hy the forces.
In detei-mining the force-function all forces may be omitted
which would not appear in the equation of Virtual Velocities.
Let X, y, z be the co-ordinates of any particle m, and let
X, Y, Z be the resolved parts in the directions of the axes of the
impressed accelerating forces acting on the particle.
The effective forces acting on the particle m at any time t are
m
dt;'
a »
m
d^y
di
a »
m
d^z
df
If the effective forces on all the particles be reversed, they will be
in equilibrium with the whole group of impressed forces by Art. 67.
Hence, by the principle of virtual velocities,
Xm
(X-
df
Sx+iY
-f)v
+ Z-
dh
di
Zz\ = 0,
where hx, By, Sz are any small arbitrary displacements of the par-
ticle m consistent with the geometrical relations at the time t.
'H^
Mi
I 1
i! I
•7
m
1 4
I
IMAGE EVALUATION
TEST TARGET (MT-3)
1.0
1.1
11.25
[jKii 122
^ "^ 1^'^
■it
u
HiotograiJiic
Sciences
Corporaaoii
23 VVIST MAIN STMIT
WItSTIR.N.Y. I4SM
(71«)I72-4S03
\
S'
V
<^
^.
^.\
4^
\
hi
■I '
u
',
\i
268
VIS VIVA.
Now if the geometrical relations be expressed by equations
which do not contain tlie time explicitly, the geometrical relations
which hold at the time t will hold throughout the time Bt ; and,
therefore, we can take the arbitrary displacements Bx, By, Bz to be
respectively equal to the actual displacements -^ Bt, -^ Bt, -j- Bt
of the particle in the time Bt
Making this substitution, the equation becomes
fd'x dx . d^y dy d?z dz
[dt;* dt "^ df dt ^ dt^ dt
Integrating, we get
Sm(=^-^- +
= 2?n
4l+^
^ A. 7^\
dt^ dt) :
^» |(s)* + (IT + ©} = <^+ 2S»/(X^+ Yd,^Zd.).
where C is a constant to be determined by the initial conditions
of motion.
Let V and v be the velocities of the particle m at the times t
and t'. Also let U^, U^ be the values of the force-function for the
system in the two positions which it has at the times t and <'.
Then
335. The following illustration, taken from Poisson, may show
more clearly wliy it is necessary that the geometrical relations
should not contain the time explicitly. Let, for example,
^ [xy y, z, t) = 0.
.(1)
be any geometrical relation connecting the co-ordinates of the
particle m. This may be regarded as the equation to a moving
surface on which the particle is constrained to rest. The quanti-
ties Bx, By, Bz are the projections on the axes of any arbitrary
displacement of the particle m consistent with the geometrical
relations which hold at the time t They must therefore satisfy
the equation
ts-+^s,+^-*a.= o.
dx
dy
dz
The quantities -j- Bt, -^ Bt, -vr Bt are the projections on the
axes of the displacement of the particle due to its motion in the
time Bt. They must therefore satisfy the equation
«</> dx
dx dt
d<f) dy
dy it
^:::;:8t + "-^'^Bt +
dz dt dt
i :
VIS VIVA AND ENERGY.
269
Hence unless
-^ is zero throughout the whole motion we can-
dx ^, dy r,. dz
hot take Zx, Bt/, Sz to be respectively equal to -7- Bt, -j-. Bt, -r. Bt.
fJth
The equal ion jj = expresses the condition that the geometrical
dt
equation (1) should not contain the time explicitly.
836. If a system be tmcler the action of no external forces, we have X=0, Y=0,
Z=0, and hence the vis viva of the system is constant.
If, however, the mutual reactions between the particles of the system are such
as would appear in the equation of virtual moments, then the vis viva of the system
will not be constant. Thus, even if the solar system were not acted on by any
external forces, yet its vis viva would not be constant. For the mutual attractions
between the several planets are reactions between particles whose distance does not
remain the same, and hence the sum of the virtual moments will not be zero.
Again, if the earth be regarded as a body rotating about an axis and slowly con-
tracting from loss of heat in course of time, the vis viva will not be constant, for
the same reason as before. The increase of angular velocity produced by this
contraction can be easily found by the conservation of areas.
837. Let gravity 'be the only force acting on the system. Let the axis of z be
vertical, then we have X=0, Y=0, Z= -g. Hence the equation of vis viva become3
Imv'^ - I,mv^ = - 2Mg (Z - 2).
Thus the vis viva of the system depends only on the altitude of the centre of
gravity. If any horizontal plane be drawn, the vis viva of the system is the same
whenever the centre of gravity passes through the plane.
338. The equation of Virtual Velocities in Statics is known
to contain in one formula all the conditions of equilibrium. In
the samQ way the general equation
Xm (^^Bx + ^Btf + ^l^ Bz) = Xm{XBx+ YB^f + ZBz),
may be made to give all the equations of motion by properly
choosing the arbitrary displacements Bx, By, Bz. In Article 334
we made one choice of these displacements and thus obtained an
equation in an integrable form.
If we give the whole system a displacement parallel to the
axis of a we have Bx = 0, fi w = 0, and Bz is arbitrary. The equa-
tion then becomes Sm ~ = tmZ, which represents any one of the
three first general equations of motion in Art. 71.
If we give the whole system a displacement round the axis of
of z through an angle B$, we liave Bx — — yB0, By = xBd, Bz => 0.
't
I.:
ii
■ T'lf ,
m
I ■: ''il
I ';•■
270
VIS VIVA.
'.
111 fr
1 i 'I'
(cPiJ (TxS
X -^.^ — y -jTi ] — ^in {x Y— i/X),
which represents any one of the three last general equations of
motion in Art. 71.
339. The principle of Vis Viva was first used by Huyghens
in his determination of the centre of oscillation of a body, but in
a form different from that now used. See the note to pige 69.
The principle was extended by John Bernoulli and applied by
his son, Daniel Bernoulli, to the solution of a great variety of
problems, such as the motion of fluids in vases, and the motion of
rigid bodies under certain given conditions. See Montucla, Histoire
de Mathematiquef Tome ill.
The great advantage of this principle is that it gives at once a
relation between the velocities of the bodies considered and the
variables or co-ordinates which determine their positions in space,
so that when, from the nature of the problem, the position of all
the bodies may be made to depend on one variable, the equation
of vis viva is sufficient to determine the motion. In general the
principle of vis viva will give a first integral of the equations of
motion of the second order. If, at the same time, some of the
other principles enunciated in Art. 278 may be applied to the
bodies under consideration, so that the whole number of equa-
tions thus obtained is equal to the number of independent co-
ordinates of the system, it becomes unnecessary to write down
any equations of motion of the second order.
340. Ex. If a system in motion pass through a position of equilibrium, t. e. a
position in which it would remain in equilibrium under the action of the forces if
placed at rest, prove that the vis viva of the system is either a maximum or a
minimum. Courtivron's Theorem, Mem, de VAcad. 17i8 and 1749.
341. Suppose a weight mg to be placed at any height h
above the surface of the earth. As it falls through a height z,
the force of gravity does work which is measured by mgz. The
weight has acquired a velocity i>, half of its vis viva is ^mw" which is
known to be equal to mgz. If the weight fall through the re-
mainder of the height h, gravity may be made to do more work
measured by mg{h—z). When the weight has reached the
ground, it has fallen as far as the circumstances of the case
permit, and no more work can be done by gravity until the weight
has been lifted up again. Throughout the motion we see that
when the weight has descended any space z, half its vis viva,
together with the work that can be done during the rest of the
descent is constant and equal to the work done by gravity during
the whole descent h.
If we complicate the motion by making the weight work
some machine during its descent, the same theorem is still true.
! n
3 re-
work
the
case
jight
that
viva,
the
iring
kvork
Itrue.
VIS VIVA AND ENiSRGY.
271
By the principle of vis viva, proved in Art. i334, half the vis viva
of the particle, when it has descended any spiice z, is equal to the
work mgz which has been done by gravity during this descent,
diminished by the work done on the machine. Hence, as before,
half the vis viva together with the difference between the work
done by gravity and that done on the machine during the re-
mainder of the descent is constant and equal to the excess of the
work done by gravity over that done on the machine during the
whole descent.
Let us now extend this principle to the general case of a
system of bodies acted on by any conservative system of forces.
342. Let us select some position of a moving system of bodies
as a position of reference. This may be an actual final position
passed through by the system in its motion, or any position
which it may be convenient to choose, into which the system
could be moved. Suppose the system to start from some position
which we ma}' call A, and at the time t, to occupy some position P.
Then at the time t, half the vis viva generated is equal to the
work done from A to P. Hence half the vis viva at P together
with the work which can be done from P to the position of refer-
ence is constant for all positions of P.
To express this, the word energy has been used. Half the vis
viva is called the kinetic energy of the system. The work which
the forces can do as the system is moved from its existing position
to the position of reference is called the potential energy of the
system. The sum of the kinetic and potential energies is called
the energy of the system. The principle of the conservation of
energy may be thus enunciated : —
When a system moves under any conservative forces, the sum of
the kinetic and potential energies is constant throughout the motion.
343. The distinction between work done and potential energy
may be analytically stated thus. The force-function has been
defined in Art. 323 to be the indefinite integral of the virtual
moment of the forces. As the system moves the work done is
the definite integral taken with its lower limit fixed and its upper
limit determined by the instantaneous position of the system.
The potential energy is the definite integral taken with its upper
limit fixed and its lower limit determined by the instantaneous
* Coriolis, Helmholtz and others have suggestecl that it would be more con-
venient if the Via Viva were defined to be half the sum of the products of the
masses into the squares of the velocities. See Phil. Trans. 1854, p. 89. But this
change in the meaning of a term so widely established in Europe would bo very
likely to cause some oonfusion. It seems better for the present to use another
name, such as kinetic energy.
Pill
:1^
, 1,
Ml
ill I ■
I
H
I
•
'■;i
272
VIS VIVA.
position of the system. The terms potential energy and actual
energy are due to Prof. Ra kine.
844. Ex. 1. A particle describes an ellipse freely about a centre of force in its
centre. Find the whole energy of its motion.
Let m be the masp of the particle, r its distance at any time from the centre,
nr the accelerating force on the particle. If coincidence of the particle with the
centre of force be taken as the position of reference, the potential energy by Art. 343
18
-l>
nifir) dr = 5 m/ir^.
If r' be the semi-conjugate of r, the velocity of the
particle is ojfii^ and the kinetic energy is therefore -mur'^. As the particle de-
scribes its ellipse round the centre of force, the sum of the potential and kinetic
energies is eqnal to -mn (a'+ 6'') where a and 6 are the semi-axes of the ellipse.
Ex. 2. A particle describes an ellipse freely about a centre of force in the
centre. Show that the mean kinetic energy during a complete revolution is equal
to the mean potential energy; the means being taken with regard to time.
Ex. 3. If in the last example the means be taken with regard to the angle
described round the centre, the difference of the means is ^mn{a- 6)'.
Ex. 4. A mass M of fluid is running round a circular channel of radius a with
velocity «, another equal mass of fluid is running round a channel of radius 6 with
velocity v, the radius of one channel is made to increase and the other to decrease
until each has the original value of the other, show that the work required to pro-
duce the change is ^ f Ij - ^^ ] (6" - a") M. [Math. Tripos, 1866.]
345. In applying the principle of vis viva to any actual cases, it will be im-
portant to know beforehand what forces and internal reactions may be disregarded
in forming the equation. The general rule is that all forces may be neglected
which do not appear in the equation of Virtual Velocities. These forces may be
enumerated as follows :
A. Those reactions whose virtual velocities are zero.
1. Those whose line of action passes through an instantaneous axis ; as rolling
friction, but not sliding friction nor the resistance of any medium.
2. Those whose line of action is perpendicular to the direction of motion of
the point of application; as the reactions of smooth fixed surfaces, but not those of
moving surfaces.
B. Those reactions whose virtual velocities are not zero and which therefore
would enter into the equation, but which disappear when joined to other re-
actions.
1. The reactions between particles whose distance apart remains the same ; as
the tensions of inextensible strings, but not those of elastic strings.
2. The reaction between two rigid bodies, parts of the same system, which roll
on each other. It is necessary however to include both these bodies in the same
equation of vis viva.
li
I
VIS VIVA AND ENERGY.
273
rolling
jtion of
Ithose of
lerefcre
Ibher re-
fime; as
liioh roll
le tame
C. All tensiona which act along inextensibld striags, even though the strings
are bent by passing through smooth fixed rings.
For let p. string whose tension is T connect the particles m, m', and pass through
a ring distant respectively r, r' from the particles. The virtual velocity is clearly
Tdr+rSr', because the tension acts along the string. But since the string is
iuextensible 5r + 5/=0; tlierefore the virtual velocity is zero.
346. To determine the vis viva of a rigid body in motion.
If a body move in any manner its vis viva at any instant is
equal to the vis viva of the whole mass collected at its centre of
gravity, together with the vis viva round the centre of gravity con-
sidered as a fixed point : or
The vis viva of a body = vis viva due to translation
+ vis viva due to rotation.
Let X, y, z be the co-ordinates of a particle whose mass is m
and velocity v, and let x, y, 2 be the co-ordinates of the centre of
gravity of the body. Let x = x + ^, y=y + r}, z = 'z+^. Then
by a property of the centre of gravity 2m| = 0, "Zmrj = 0, %m^= 0.
Hence S»n ^ = 0, Sw J? ="0, 2m J* = 0. Now the vis viva of a
body is
Smij' = 2m
{§)'-(l)'-(S)}-
Substituting for x, y, z, this becomes
All the terms in the last line vanish as they should, by
Art. 14. The first term in the first line is the vis viva of the
whole mass 2wt, collected at the centre of gravity. The second
term is the vis viva due to rotation round the centre of gravity.
This expression for the vis viva may be put into a more con-
venient shape.
347. First. Let the motion be in two dimer^sions. Let v he
the velocity of the centre of gravity, r, 6 its polar co-ordinates
referred to any origin in the plane of motion. Let ?•, be the
distance of any particle whose mass is m from the centre of gravity,
and let Vj be its velocity rela/tively to the centre of gravity. Let
to be the angular velocity of the whole body about the centre of
gravity, and Mk^ its moment of inertia about the same point.
.
J I ii
H. D.
18
-TS^TTTrMTVSm
274
VIS VIVA.
The vis viva of the whole mass collected at is Mv*, which
may by the Diflferential Calculus be put into either of the forms
«'=^l(S)"-(l)}=^{(i)'-ni)}-
The vis viva about G is Swv,'. But since the body is turning
about O, we have v^ = r^a. Hence Xmv' = eo" . '^mr' = w" . Mk\
The whole vis viva of the body is therefore
'S^mv' = Mv' + MkW.
If the body be turning about an instantaneous axis, whose
distance from the centre of gravity is r, we have v = ra>. Hence
Smr» = ilf a>» (r« + A;') = Mk'W,
where Mk'' is the moment of inertia about the instantaneous axis.
348. Secondly. Let the body he in motion in space of three
dimensions.
Let V be the velocity of G ; r, 6, <f> its polar co-ordinates re-
ferred to any origin. Let a>x, (o^, co^ be the angular velocities
of the body about any three axes at right angles meeting in G,
and let A, B, C be the moments of inertia o^ 'he body about
the axes. Let ^, ;;, ^ be the co-ordinates of a ^le m referred
to these axes.
The vis viva of the whole mass.^ collected at G is Mv\ which
may be put equal to
according as we wish to use cartesian or polar co-ordinates.
The vis viva due to the motion about G is
x«v=.»{(D'.(§)V(f)].
Substituting these values, we get, since A = Sm {rf + ^),
5 = 2m(r + r), G=Xmi^-' + v%
Xmv,' = AcoJ" + Bco^' + Ceo,'
— 2 (%m^T)) (OgWy — 2 i^mr]^) w^w, - 2 (Sw^|) w.o)^.
If the axes of co-ordinates be the principal axes at G, this re-
duces to
Swy^' = Jw/ + Ba>J' + Co)/.
t)l
1
fixed
way
when
point
about
849,
its posij
0, </>, }f/
in the n
21
where
ac
Show
takes the
This resu
Ex.2,
fluence of
If the law
perature
that the
where A, 2
Ex. 3.
equation
Let the
that if X, y,
axes fixed or
with regard
Thus the
coefficients a
semi vis viv
round the cej
Ex.4.
same express]
the origin is
VIS VIVA AND ENERGY.
275
If the body be turning about a point 0, whose position is
fixed for the moment, the vis viva may be proved in the same
way to be
where A\ B\ C are the principal moments of inertia at the
point 0, and w^, w^, w, are the angular velocities of the body
about the principal axes at 0.
849. Ex. 1. A rigid body of mass M is moving in space in any manner and
its position is determined by the co-ordinates of its centre of gravity and the angles
d, if>, ^ which the principal axes at the centre of gravity make with some fixed axes
in the manner explained iu Art. 235. Show that its vis viva is given by
2r = Jf (x'« + y'« + 2'2) + C{<i>' + y}/ cos tf)» + {A sin" + 5 cos« 0) 0'*
+ BVD?0{A cos''0+5 sin2 0) ^'« + 2 (B-A) sm S sin ^ cos ^^'f',
where accentj denote differential coefiScients with regard to the time.
Show also that when two of the principal moments A and B are equal, this
takes the simpler form
2r = 3f («'!>+ y* + z'S) + C (0' + y}/ cos BY + A {0" + sin« tf ^'«).
This result will be often found useful.
Ex. 2. A body moving freely about a fixed point is expanding under the in-
fluence of heat so that in structure and form the body is always similar to itself.
If the law of expansion be that the distance between any two particles at the tem-
perature d is equal to their distance at temperature zero multiplied hyf{0), show
that the vis viva of the body =AwJ' + Buy'> + Cu,' + ^(A + B + C)(^^^~^\ ,
where A, B, '^ are the principal moments at the fixed point.
Ex. 3. A body is moving about a fixed point and its vis viva is given by the
equation
2T=Au^'+Buy' + Cu,^ - 2Du)yU,-2EuiUg,-2FugUg.
Show that the angular momenta about the axes are 5 — ,
dT dT
dUy
dT
dwg'
Let the body be moving freely and let 27*0 be the vis viva of translatioc. Prove
that if X, y, z be the co-ordinates of the centre of gravity referred to any rectangular
axes fixed or moving about a fixed point, and if accents denote differential coefficients
with regard to the time, then the linear momenta parallel to the axes will be
dT,
dx"
d7\
dy-
dTo
dz'
Thus the vis viva, like the force-function, is a scalar function whose differential
coefficients are the components of vectors. See Art. 240 and 326. In the case of the
semi vis viva, these are the resultant linear momentum and angular momentum
round the centre of gravity.
Ex. 4. A body is moving about a fixed point and its vis viva is given by the
same expression as in the last example. Show that if the axes are fixed in space and
the origin is at the fixed point, the equations of motion may be written iu the form
■dt dwx '
!,: (I
18—2
27G
VIS VIVA.
^Mi
i\l
" :
i
I i f;
',■: .1
'I
with two similar equations for the axes of y and t. In these eqnations A, B, &e.
will gonerally bo variable.
If the axes move in the manner explained in Art. 243, tht equations of
motion are
d dT dT . dT
'•Og diOy *
"Stdiir d(i>« * du, "
01, = L,
with two similar equations. See Art. 253.
If the centre of gravity of a body moving freely bo referred to axes moving about
a fixed origin and if 27^ be the vis viva of translation, show that the equations of
motion of Art. 245 may be written
ddTo dT„ dr„
aidx' ' dy'^^'^'dl"''-'^'
with two similar equations.
850. Ex. 1. A circular wire can turn freely about a vertical diameter as a fixed
axis, and a bead can slide freely along it under the action of gravity. The whole
system being set in rotation about the vertical axis, find the subsequent motion.
Let M and m be the masses of the wire and bead, u their common angular
velocity about the vertical. Let a be the radius of the wire, Mk^ its moment of
inertia about the diameter. Let the centre of the wire be the origin, and let the
axis of y be measured vertically downwards. Let be the angle the radius drawn
from the centre of the wire to the bead makes with the axis of y.
It is evident, since gravity acts vertically and since all the reactions at the fixed
axis must pass through the axis, that the moment of all the forces about the vertical
diameter is zero. Hence, taking moments about the vertical, we have
Jf Pw + ma' sin* 0u=h.
And by the principal of vis viva,
Mh^uP + »/i I a" ( -^ j + a' sin" ^w' | = C+ 2mga cos 0.
These two equations will suffice for the determination of j- and w. Solving
them, we get
%m~. — TT ■ -t n + w*"^ ( J. 1 =C-\- 2mga cos 0.
Mk^ + ma^ sm* \ dt) ''
This equation cannot be integrated, and hence cannot be found in terms of f .
To determine the constants h and C we must recur to the initial conditions of
motion. Supposing that initially 0=it, and ^ = and w=a, then A = A'ifc'o and
Ex. 2. A lamina of any form rolls on a perfectly rough straight line under the
action of no forces ; prove that the velocity v of the centre of gravity is given by
r-=c' aTijt ' 'wliere r is the distance of G from the point of contact, and i is the
radius of gyration of the body about an axis through G perpendicular to its plane,
and c is some constant.
Ex. 3. Two equal beams connected by a hinge at their centres of gravity so as
to form an X are placed symmetrically on two smooth pegs in the same horizontal
VIS VIVA AND ENEROY.
277
line, the distance between which in b. Show that, if the beams ho perpendicular to
each other at the commencement of the motion, tlie velocity of their centre of
gravity, when in the line joining the pegs, is equal to a/ t^ ' , where k is the
radius of gyration of either beam about a line porpeudioular to it through its centre
of gra\'ity.
Ex. 4. A uniform rod is moving on n horizontal table about one extremity, and
driving before it a particle of mass equal to its own, which starts from rest in-
definitely near to the fixed extremity ; show that when the particle has described r
distance r along the rod, its direction of motion makes with the rod on angle
* [Christ's Coll.]
tun~i
Vr'+'A'
Ex. 5. A thin uniform smooth tube is balancing horizontally about its middle
point, which is fixed; a uniform rod such rs just to fit the base of the tube is placed
end to end in a line with the tube, and then shot into it with siich a horizontal
velocity that its middle point shall only just reach that of the tube ; supposing the
velocity of projection to bo known, find the angular velocity of the tube and rod at
the moment of the coincidence of their middle points. [Math. Tripos.]
Bciult, If wi be the mass of the rod, m! that of the tube, and 2a, 2a' theu* re-
spective lengths, V the velocity of the rod's projection, « the • oquired angular
velocity, then w«=- „ , ,, .
Ex. 6. The centre C of a ciicular wheel is fixed and the rim is constrained to
roll in a uniform manner on a perfectly rough horizontal plane so that the plane of
the wheel makes a constant angle a with the vertical. Bound the circumference
there is a uniform smooth canal of veiy small section, and a hea\y particle which
just fits the canal can slide freely along it under the action of gravity. If m be the
particle, B the point where the wheel touches the plane and 0=lBCm, and if n be
the angular rate at which 27 describes the cu'cular trace on the horizontal plane,
prove that ( 77 ) = ~ cos a cos ^ - n* cos- o cos' 6 + const, where a is the radius of
the wheel. Aimales de Gergonne, Tome xix.
Ex. 7. If an elastic string, whose natural length is that of a uniform rod, be
attached to a rod at both ends and suspended by the middle point, prove by means
of vis viva that the rod will sink until the strings are inclined to the horizon at an
B 8
angle $, which satisfies the equation cot' ;, - cot - - 2n=0, where the tension of the
string, v/hen stretched to double its length, is n times the weight. [Math. Tripos.]
Ex. 8. A regular homogeneous prism, whose normal section is a regular polygon
of n sides, the radius of the circumscribing circle being a, rolls down a perfectly
rough inclined plane whose inclination to the horizon is o. If w„ be the angular
velocity just before the n* edge becomes the instantaneous axis then
g sm g
8 + cos
2ir
n
asm- 5 + 4cos —
n n
. 8-hC03^''\
, gsing 11 1
a sni -5+1 cos — /
« u I
I
■ I \
I it
Si' I
1:1
m
^■.*r\
*■ il
y^
*lil
278
VIS VIVA.
I
I '!
881. The eqnation of Vis Viva may be applied to the case of relative motion in
the following manner*. Suppose the system at any imtant to become fixed to the
set of moving axes relative to johich the motion is required, and calculate what would
then be the effective forces on the system. If we apply these as additional impressed
forces to the system bxit reversed in direction, we may use the equation of Vis Viva to
determine the relative motion €u if the axes were fixed in space.
We may reduce the origin of the moving axes to rest by applying to every
particle an acceleration equal and opposite to that of 0, in the manner explained
in Art. 174. As these will be included as part of the additional forces mentioned
in the enunoiotion it will be sufficient to prove the theorem for axes moving about
a fixed point.
If we follow the notation of Art. 259, the accelerations of any point P resolved
paroUol to rectangular moving axes having a fixed origin are
with two similar expressions for y and z. The three last terms, with the corre-
sponding terms in the othsr expressions, are the resolved accelerations of a point Pg
rigidly attached to the axes but occupying the instantaneous position of P. Let us
call these A'o, Yq, Zq.
Recurring to the proof of the principle of vis viva given in Art. 334 we see that we
d"x
have to substitute these expressions for -r-^ , &c. in the general equation of virtual
velocities. After substitution for dx, Sy, Sz, it is clear that the terms containing
— , J-, -J ail disappear. The equation after integration then becomes, as before,
^'^\(j)'+(^^y+{^y\=^^^f^(^-^o)dx+{Y-T,)dy + (Z-Z,)dz} + C.
The theorem of Coriolis really follows at once from that of Clairaut given in
Art. 257. The above mode of proof has the advantage of recurring to first
principles.
352. Ex. 1. A sphere rolls on a perfectly rough plane which turm with a uniform
angular velocity n about a horizontal axis in its own plane. Supposing the motion
of the sphere to take place in a vertical plane perpendicular to the axis of rotation,
find the motion of the sphere relative to the plane.
Let Ox be the trace described by the sphere as it rolls on the plane, and let Oy
be drawn through the axis of rotation perpendicular to Ox in the plane of motion of
the sphere. Let nt be the angle Ox makes with a horizontal plane through the axis
of rotation. Let ^ be the angle that radius of the sphere which was initially
perpendicular to the plane makes with the axis of y. Let {x, y) be the co-ordinates
of P the centre of the sphere, and Mk^ the moment of inertia of the sphere about a
diameter.
If the sphere were fixed relatively to the plane its effective forces would be Mn^x
and Mn-y parallel to the axes, and Mk'^ ^7 =^ round the centre of gravity. Also the
* This theorem is due to Coriolis, see the Journal Polyteeh. 18ul.
VIS VIVA AND ENERGY.
279
impressed foroe, (gravity, is equivalent to ffeinnt and -pcoant parallel to the
moving axes. Houce tbo cciaation of Via Viva for relative motion beoomoa
Id {/dxy /duy ,,/d<i>y) , dx , dv , dx rftf
Here -. =a — and -t(=0. Wo have therefore
at Ut dt
04:)
d^x
dt^
r^ni
n^x+g Bin nt.
This equation miglit also have been derive " from the formulae for moving axes
2
given in Art 179. If i'=g a», this equation leads to
where A, £ are two constants which depend on the initial conditions of the
question.
353. To determine the change in the vis viva of a moving
system produced by any collisions between the bodies or by any
explosions. (Carnot's Theorem.)
Let v^, v^y v„ vj, vj, V,' be the resolved parts of the velocities
of any particle m of the system before and after the impulse.
Then the momenta m {vj — vj, m {vj — v^), m (v/ — u,), being
reversed and taken throughout the whole system, are by
D' A.lembert's Principle in equilibrium with the impulses. But
these last are themselves in equilibrium. Hence the former
set are also in equilibrium. Therefore by Virtual Velocities,
tm {(vj - V,) tx + (V - vj hj + « - V.) Zz] ^ 0,
where Zx, Zy, Zz arc any small arbitrary displacements of the
particles impinging on each other, which are consistent with the
geometrical conditions of the system during the time of action of
the impulse.
During the impact, it is one geometrical condition that the
particles impinging on each other have no velocity of separa-
tion normal to the common surface of the bodies of which they
form a part.
First Let the bodies be devoid of elasticity. Then the
above geometrical conditon will hold just after the moment of
greatest compression as well as during the impact. Hence we
can put Zx = vJBt, Zy = vJZt, Zz = vJZt. The equation now be-
comes
Sm {{vj - V,) vj + {vj - v^) vj + « - V,) <} = ;
.-. %m «' + <» + v,") = Im (v^vj + ty-; + v.v.').
i '
i'ij
' I
ii I
;
iJj
' I
I !
n
^1 11 1!
r i
280
VIS VIVA.
This may be put into the form
tm («;» + vj' + v,") - tm (y/ + v; + v')
= - 2m {(vj - v^)' + « - v^y + {v; - tO«|.
Therefore in the impact of inelastic bodies vis viva is always
lost.
Secondly. Let an explosion take place in any body of the
system. Then the geometrical equation above spoken of will
hold just before the impulse begins as well as during the ex-
plosion, but it will not hold after the particles of the body have
separated. Hence we must now put hx = v^St, By = v^,St, Bz = v,Bt.
As before, we have
and
r^
= + 2m {{vj - v:f + [v^ - v,Y + {v: - vn
Therefore in cases of explosion vis viva is always gained.
Thirdly. Let the particles of the system be perfectly elastic.
Then the whole action consists of two parts, a force of compres-
sion as if the particles were inelastic, and a force of restitution of
the nature of an explosion. The circumstances of these two forces
are exactly equal and opposite to each other. By examining
these two expressions it is easy to see that the vis viva lost in
the compression is exactly balanced by the vis viva gained in the
restitution.
354. It should be noticed that Oarnot's demonstration does
not exclusively apply to collisions, but to all impulses which are
such as do not appear in the equation of Virtual Velocities.
Let a system be moving in any way, and let us suddenly intro-
duce some new restraints, by which some of the particles are
compelled to tak": new courses. The impulses which produce this
change of motion are of the nature of reactions, and are such
that in the subsequent path their virtual moments are zero. It
follows from Carnot's first theorem, that vis viva will be lost, and
the amount of vis viva lost is equal to the vis viva of the relative
motion.
Let there be two systems at rest, in a,ll respects the same
except that one is subject to some restraints from which the
other is free. Let both these be set in motion by equal im-
pulses, and let X) K, Z be the components of these. Then, if
1, .'
VIS VIVA AND ENERGY.
281
accented letters I'efer to the more free system and twice accented
letters to the other, we have
2m (vJSx + &c.) = t {XBiv -{- &c.))
%m {vJ'Bx + &c.) = E {XBx -f &c.)j '
where Bx, By, Bz are any arbitrary displacements consistent with
the geometrical conditions. Since both systems may be displaced
in the manner in which the less free system actually begins to
move, we mc!,y put Bx = vJ'Bf, &c. We therefore have
Xm {vjvj' + &c.) = Sm (vj" + &c.).
It again follows from Carnot's first demonstration that the
vis viva of the constrained system is less than that of the free.
Generally, the greater the constraints impressed on a system at rest,
the less will he the vis viva generated by any given impulses. Tliis
theorem is in part due to Lagrange, it has been generalized by
Bertrand in his no-tes to the MeGanique Analytique.
355. Let two systems be in all respects the same and moving in the same
manner. Let us suppose that suddenly some of the constraints are removed from
one system and at the same instant let both be acted on by equal impiUses. Then
following the same notation as before, we have
2 m {{Vx - Vt) &>; + &C.1 = S {XSx + &c.),
2wi {(t'j;" - Vx) Sx + &c.} = S (A'5j! + Ac).
If we make Sx=Vg"St, &c. we obtain
2wi (f 2 V' + &c.) = 2?>i {v/^ + &c.),
and we may deduce from this equation theorems similar to those of the last article.
Let us now give these two systems any other displacement which is permitted
by the geometrica,l relations common to both. Let this displacement be represented
by Sx=Vx"'U, &c. Then as before we have
2ot (f>/' + &c.) = Sm {vj'v^" + Ac).
From this and the last eq.uation we easily find
2hi {« - v^y + &c.} = 2/(t {{vj - vj'f + &c.} + 2m {(r/ - v/O** + &c.}.
Let Oj, a^, &c. be the positions of the particles m^, m.^, &c. just bef^^re the action of
the impulses ; a/, a./, &c. , o/', aj", &c. their positions just after, in the more free
and constrained systems respectively, a^'", a^"', &o. their positions after any hypo-
thetical displacement. Then
Zvi (aWy = 2?» {a'a"y + 1m, (a"a"')«.
Hence we infer that the motion of the more constrained system is such that
2nt [a.'a"Y is less than if the particles took any other coiu'ses, consistent with all
the geometrical relations.
If we suppose the systems to be acted on by a series of indefinitely small im-
pulses, these impulses may be regarded as finite forces. We may therefore infer
the following theorem, which is called Gauss' principle of least cntmtra int.
The motion of a system of material points connected by any geometrical nln-
tious is always as nearly as possible in accordance with free motion; i.e. if the
1
1
: ': M
'H
ill
1
m
X I
i! i -aU
ili
i t
• n
^. m
282
VIS VIVA.
constraint during any time dt is measnred by the stira of the products of the mass
of each particle into the square of its distance at the end of that time from the
position it would have taken if it had been free, then the actual motion during the
time dt is such that the constraint is less than if the particles had taken any other
positions.
M. Gauss remarks that the free motions of the particles when they are incom-
patible with the geometrical conditions of the system are modified in exactly the
same way as geometers modify results, which have been obtained by observation,
by applying the method of least squares so as to render them compatible with the
geometrical conditions of the question.
356. To determine the mean vis viva of a system of inaterial points in stationary
motion. Clausius' Theorem*.
By stationary motion is meant any motion in which the points do not continually
remove further and further from their original position, and the velocities do not
alter continuously in the same direction, but the points move within a limited
space and the velocities only fluctuate within certain limits. Of this nature are all
periodic motions, such as those of the planets about the sun and the vibrations of
elastic bodies, and further, such irregular motions as are attributed to the atoms
and molecules of a body in order to explain its heat.
Let X, y, z be the co-ordinates of any particle in the system and let its mass
be m. Let X, Y, Z be the components of the forces on this particle. Then
We have by simple differentiation,
and therefore
dt^ ~
m
2
„df dx\ „ /rfx\« „ d'x
-''dt[^di)=^[dt)+^''dr^'
fdxy 1 m#(x')
Let this equation be integrated with regard* to the time from to t and let the
integral be divided by t, we thereby obtain
m
2t
[i/dxy,^ If^,, mrd(x^) fd{x')\~\
in which the application of the suffix zero to any quantity implies that the initial
value of that quantity is to be taken.
The left-hand side of this equation and the first term on the right-hand side are
1
and - - xX during the time t. For a periodic
evidently the mean values of -^ ( 77 ) 9 •
motion the duration of a period may be taken for the time t ; but for irregular
motions (and if we please for periodic ones also) we have only to consider that the
time t, in proportion to the times during which the point moves in the same direc-
tion in respect of any one of the directions of co-ordinates is very great, so that in
the course of the time t many changes of motion liavo taken place, and the above
expressions of the mean values have become sufiiciently constant. The last term
of the equation, which has its factor included in square brackets, becomes, when
the time is periodic, equal to zero at the end of each period. When the motion is
* This and the next article are an abridgement of Clausius' paper in the Phil,
Mag., August, 1870.
1 . '-
VIS VIVA AND ENERGY.
283
le Phil
not periodic, bnt irregularly varying, the factor in brackets does not so regularly
become zero, yet its value cannot continually increase with the time, but can only
fluctuate within certain limits ; and the divisor t, by which the term is affected,
must accordingly cause the term to become vanishingly small with very great values
of t. The same reasoning will apply to the motions parallel to the other co-ordi-
nates. Hence adding together our results for each particle, we have, if v be the
velocity of the particle m,
1 1
mean - Smt>*= - mean ^ S (Xx + Yy + Zz).
The mean value of the expression - ^ S (Zx + Yj/ + Z£) has been called by Clausiua
the virial of the system. His theorem may therefore be stated thus, t)ie mean
temi vis viva of the system is equal to its virial.
357. In order to apply this theorem to heat, let us consider a body as a system
of material particles in motion. The forces which act on the system will in general
consist of two parts. In the first place, the elements of the body exert on each
other attractive or repulsive forces, and secondly, forces may act on the system from
without. The virial will therefore consist of two parts, which are called the
internal and external virial.
If <t> (r) be the law of repulsion between two particles whose masses are m and m'.
we have Xx + X'x'= -^(r)
X X J f \ ^ ^ f , t \ \P^ ~ ^)
x-^{r) -—-x'=(t>(r)
And since for the
r ■ - ■ r • • - J.
two other co-ordinates corresponding eqtuations may bo formed, we have for the
mtemal vii-ial - ,-j S {Xx+ Yy + Zz) = - 2r0 (r).
As to the external forces, the case most frequently to be considered is where the
body is acted on by a uniform pressure normal to the surface. If p bo this pres-
sure, d<T an element of the suiface, I the cosine of the angle the normal makes with
the axis of «, - ^ SA'ac^^ Ap ld<T=^JJxdydz. If F be the volume of the body this
1 3
is jtJjF, and therefore the whole external vii'ial is \:pV.
Ex. Show that the virial of a system of forces is independent of the origin
and the directions of the axes supposed rectangular.
The first result is clear, since in stationary motion SA' = 0, &c. The second
follows from the equality Xx + Yxj + Zz = Rp, where II is the resultant of A', 1', Z, and
p is the projection of the radius vector on the dii'ection of E.
Netuton'a Principle of Similitude.
358. What are the conditions necessary that two systems of
particles which are initially geometrically similar should also be
mechanically similar, i.e. the relative positions of the particles in
one system at time i should also be similar to the relative posi-
tions in the other syc+em at time t', where t' bears to < a constant
ratio ?
284
VIS VIVA.
si i
i
In other words, a model is made of a machine, and is found to
work satisfactorily, what are the conditions that a machine made
according to the model should work as satisfactorily ?
Since all the equations of motion may be deduced from the
principle of Virtual Velocities, that principle seems to afford the
simplest method of investigating any general theorem in Dyna-
mics. It has also the advantage of not requiring us to consider
the unknown reactions, if there be any in the system. This mode
of proof is given by M. Bertrand in Calder xxxii. of the Journal
de I'ecole Poly technique.
359. Let (a*, y, z) be the co-ordinates of any particle of mass
in in one system referred to any rectangular axes fixed in space,
and let (A', Y, Z) be the resolved part of the impressed moving
forces on that particle. Let accented letters refer to correspond-
ing quantities in the other system.
Then the principle of Virtual Velocities supplies the two
following equations:
•jfX— w-T-2 j hx + &C. f =0,
')aa;' + &c.[ = 0.
t\X
It is evident that one of these equations will be changed into
the other if we put X' = FX, Y' = FY, &c., x' = Ix, y = ly, &c.,
m —fim, &c., i = Tt, &c., where F, I, fi, r are all constants, pro-
vided fil = Ft^. In two geometrically similar systems, we have
but one ratio of similarity, viz. that of the linear dimensions, but
in two mechanically similar systems we have three other ratios,
viz. that of the masses of the particles, that of the forces which
act on them, and that of the times at which che systems are to ^^^
compared. It is clear that if the relation just established hold
between these four ratios of similitude, the motion of the two
systems will be similar.
Suppose then the two systems to be initially geometrically
similar, and that the masses of corresponding particles are pro-
portional each to each, and that they begin to move in parallel
directions with like motions and in proportional times, then they
will continue to move with like motions and in proportional times
jirovided the external moving forces in either system are propor-
, , mass X linear dimensions ... ,, ^ ^ ^ •,-
tional to r-. — rr, . biuce the resolved velocities
(time)"'
dec
of any particle are -j- , &c., it is clear that in two similar systems
the velocities of corresponding points at corresponding times arc
H
PRINCIPLE OF SIMILITUDE.
28"
jTstems
cs arc
proportional to
linear dimensions
time
If we eliminate the time
between these two relations, we may state, briefly, that the con-
dition of similitude between two systems is that the moving
p . , ^. 1 X mass X (velocity)'
forces must be proportional to ,-. ,4 .- .
hnear dimensions
360. .M. Bertrand remarks, that in comparing the working of
a model with that of a large machine, we must take care that all
the forces bear their proper ratios. Supposing the model to be
made of the same material as the machine, the weights of the
several parts will vary as their masses, and therefore as the
ciibes of the linear dimensions. Hence we infer that the velocity
of working the model must be made to be proportional to the
square root of its linear dimensions. The times of describing
corresponding arcs will also be in the same ratio.
If tlicre be any forces besides gravity which act on the model,
these must bear the same ratio to the corresponding forces in the
machine, if the model is to be similar to the machine. Hence the
impressed forces must be made to vary as the cubes of the linear
dimensions. For example, in the case of a model of a steam-
engine, the pressure of the steam on the piston varies as the
product of the area of the piston into the elastic force. Hence,
the elastic force of the steam used must be proportional to the
linear dimensions of the model.
Supposing the impressed forces in the two systems to have,
each to each, the proper ratio, the mutual reactions between the
parts of the systems will, of themselves, assume the same ratio. For
if, by giving proper displacements according to the principle of
Virtual Velocities, we form equations of motion to find these reac-
tions, it is easy to see that they will be, each to each, in the same
ratio as the forjes. Since sliding friction varies as the normal
pressure, and is independent of the areas in contact, these frictions
will bear their proper ratio in the model and machine. This, how-
ever, is not the case with rolling friction. Recurring to Art. 150,
we see that the rolling friction varies inversely as the diameter of
the wheel, and will, therefore, bear a greater ratio to the other forces
in the model than ^n the machine. If the resistance of the air be
proportional to the product of the area exposed into th.. square of
the velocity, this resistance will bear the proper ratio in the
model and the machine.
861. As an example, let ns apply the principle to the case of a rigid body
oscillating about a fixed axis under the action of gravity. That the motions of two
pendulums may be similar they must describe equal angles, correspoudiug times
are therefore proportional to their times of oscillation. Since the forces vary as the
mass into gravity, we see that when u pendulum oscillates through a given angle.
.■< ^.
; S
t: < j
I]
ii
■
Ii I
1^ <. 11
f ' 1 1< ■
' y
'.'i *\
286
VIS VIVA.
■ W
the sqnare of the time of oscillation must Tary as the ratio of the linear dimensions
to gravity.
As a second example consider the case of a particle describing an orbit round
the centre of attraction whose force is equal to the product of the inverse square
of the distance into some constant //. The principle at once shows that the square
of the periodic time must vary as the cube of the distance directly and as n in-
versely. This is Kepler's third law.
362. In the twenty-ninth volume of the Annates dc Chimie (Paris, 1825) Savart
describes numerous experiments which he made on the notes sounded by similar
vessels centaining air. He says that if we construct cubical boxes and set the air
in motion as is ordinarily done in organ pipes we find that the number of vibrations
in a given time is proportional to the reciprocals of the linear dimensions of the
masses of air. This law was verified between extreme limits, and its truth tested
with a great many notes. He says he frequently used the law during his researches,
and never once found it led him wrong. This result having been obtained for
cubes, it was natural to examine if the same law held for prismatic tubes on
square bases. After a great many experiments he foimd the same law to be true.
He then tested the law with conical pipes in which the opening was always
of the same solid angle, ihen with cylindrical pipes, then with pipes whose
bases were equilateral triangles. These he made to sound in different ways, put-
ting the mouth- piece for instance at different points of the length of the tube. In
all cases the same law was found to hold, for tubes whose diameters were very
small compared with their lengths as well as for those whose diameters were very
great. This law he again found applicable t^^ masses of air set in motion by communi-
cation from other vibrating bodies. Hence he infers this general law which he
enunciates as an experimental fact.
When masses of air are contained in two simUar vessels, the number of vibra-
tions in a given time [i. e. the pitch of the note sounded] is proportional inversely
to the linear dimensions of the vessel.
This theorem of Savart'e follows at once from the principle of Similarity. Divide
the similar vessels into corresponding elements, then the motion of these elements
will be similar each to each if the forces vary as —. — ^, ' . But by Mar-
itime)^
riotte's law the force between two elements varies as the product of the area of
contact into the density. Hence the times of oscillation of corresponding particles
of air must vary as the linear dimensions of the vessel.
863. The first person who gave a theoretical explanation of Savart's law was
Cauchy, who showed, in a Memoire presented to the Academy of Sciences in 1829,
that it followed from the linearity of the equations of motion. He refers to the
general equations of motion of an elastic body whose particles are but slightly dis-
placed even though the elasticity is different in different directions. These equa-
tions which serve to determine the displacements (f, 77, f) of a particle in terms of
the time t and the co-ordinates {x, y, z) are of two kinds. One applies to all points
of the interior of the elastic body and the other to all points on its surface. These
are to be found in all treatises on elasticity. An inspection of these equations
shows that they wUl continue to exist if we replace f, ij, f, x, y, z, t by k|, k% xf, kx,
Ky, KZ, Kt, where k is any constant provided we cl^^^. tj^e accelerating forces in the
ratio K to 1. Hence if these accelerating forces are zero, it will be sufficient to
measure
V
PRINCIPLE OP SIMILITUDE.
287
•1|
increase the dimensiouB of the elastic body and the initial vahiea of the displice-
ments in the ratio 1 to k, in order that the general valnes of ^, 17, ^ and the dura-
tions of the vibrations should vary in the same ratio. Hence we deduce Cauchy's
extension of Savart's law, viz. if we measure the pitch of the note given by a body,
by a plate or an elastic rod, by the number of vibrations produced in a unit of time ;
the pitch will vary inversely as the linear dimensions of the body, plate or rod, sup-
posing all its dimensions altered in a given ratio.
364. These results may be also deduced from the theory of
dimensions. Following the notation of Art. 318, a force F is
measured by m -v-a . We may then state the general principle,
that all dynamical equations must be such that the dimensions of
terms added together are the same in space, time and mass, the
,. . - » ^ - . 1 .1 mass . space
dimensions of force being taken to be — j-. — \5 — .
* (time)''
Let us apply this to the case of a single pendulum of length I,
oscillating through a given angle a, under the action of gravity.
Let m be the mass of the particle, F the moving force of gravity,
then the time t of oscillation can be a function only of F, I, m
and a. Let this function be expanded in a series of powers of
F, I and m. Thus
T^l^AFn'mT,
where A being a function of a only is a number. Since t is of no
dimension in space, we have p+ q = 0. Also t is of one dimen-
sion in time ; .'. —2p = l. FinalK t is of no dimensions in mass;
:. p + r = 0. Hence p = — ^, q = r~^, and since p, q, r have
each only one value, there is but one term in the series. We
infer that in any simple pendulum r = A \/ ^ where A is an
undetermined number.
1 ■ I
IIM
365. Ex. 1. A particle moves from rest towards a centre of force whose attrac-
tion varies as the distance in a medium resisting as the velocity, show by the
theory of dimensions that the time of reaching the centre of i'orce is independent of
the initial position of the particle.
Ex. 2. A particle moves from rest in vacuo towards a centre of force whose
attraction varies inversely as the n^^ power of the distance, show that the time of
reaching the centre of force varies as the
particle.
n + 1
tb power of the initial distance of the
. i 1-
288
VIS VIVA.
Lagrange's Equations.
306. Our object in this section is to form the general equation
of motion of a dynamical system freed from all the unknown
reactions and expressed, as far as is possible, in terms of any kind
of co-ordinates which may be convenient in the problem under
consideration.
In order to eliminate the reactions we shall use the principle
of Virtual Velocities. This principle has already been applied to
obtain the equation of Vis Viva by giving the system that par-
ticular displacement which it would have taken if it had been left
to itself. But since every dynamical problem can, by D'Alembert's
principle, be reduced to one in statics, it is clear that by giving
the system proper displacements, we must be able to deduce, as
in Art. 338, not Vis Viva only, but all the equations of motion.
307. Let {x, y, z) be the co-ordinates of any particle m of the
system referred to any fixed rectangular axes. These are not
independent of each other, being connected by the geometrical
relations of the system. But they may be expressed in terms of
a certain number of independent variables whose values will de-
termine the position of the system at any time. Extending the
definition given in Art. 73, we shall call these the co-ordinates
of the system. Let these be called 6, ^, -^j &c. Then x, y, z, &c.
are functions of 0, ^, &c. Let
x=fit,0,<f>,&e.) ...(1),
with similar equations for y and z. It should be noticed that these
equations are not to contain -^ , ~ , &c. The independent
variables in terms of which the motion is to be found may be any
we please, with this restriction, that the co-ordinates of every
particle of the body could, if required, be expressed in terms of
them by means of equations which do not contain any differ-
ential coefficients with regard to the time.
The number of independent co-ordinates to which the position
of a system is reduced by its geometrical relations, is sometimes
spoken of as the number of the degrees of freedom of that body.
Sometimes it is referred to as being the number of independent
motions which the system admits of.
In the following investigations total differential coefficients
with regard to t will be denoted by accents. Thus -r. and -vj
will be written x' and x'.
But
laoranoe's equations.
289
If 2T be the vis viva of t.ie system, we have
22'=Sm(aj"+y+0 (2);
we also have, since the geometrical equations do not contain
& t <f>, &c.,
rim: ///*• _. ffm
(3),
dx dx yy da; ., g
dt dd
with similar equations for y' and z'. In these the differential co-
efficients jt » jB » &c. are all partial. Substituting theie in the
expression for 2 T, we find
2T=F{f,e,<l>,&c.d',<f>',&c\
When the system of bodies is given, the form of F will be
known. It will appear presently that it is only through the form
of F that the effective forces depend on the nature of the bodies
considered ; so that two dynamical systems which have the same
i'^are dynamically equivalent.
It should be noticed that no powers of ff, <f>', &c. above the
second enter into this function, and when the geometrical equa-
tions do not contain the time explicitly, it is a homogeneous
function of ff, ^', &c. of the second order.
368, To find the virtual moments of the momenta of a system,
and also of the effective forces corresponding to a displacement prO'
duced by varying t>ne co-ordinate only.
Let this co-ordinate be 0, and let us follow the notation al-
ready explained. Let all differential coefficients be partial, unless
it be otherwise stated, excepting those denoted by accents. Since
x, y, z' are the components of the velocity, the virtual moment of
the momenta will be Xm (x'Sx + y'By-^ z'Bz), where Sx, By, Bz are
the small changes produced in the co-ordinates of the particle m
by 'I variation Bd of 6. This is the same as
s™(4^^'l+4.)«''-
If 22' be the vis viva given by (2) of the last article
dT ^ f 'dx' . \
But differentiating (3) partially with regard to 0', we see
UiOS dot*
that -jTv = -j;; . Hence the virtual moment of the momenta is
dd do
equal to -Tff^^'
R. D.
19
I- i\'
'.' i
I 1
■ ■
\'\
1
■ \\\
:m
I'/
290 VIS VIVA.
The virtual moment of the effective forces will be
This may bo written in the form
4&c.)-2m(ar'^^J + &c.),
d ^ f ,dx
where the t- represents a total differential coefficient with regard
to t We have already proved that the first of these terms is
-j: jQi- I^ remains to express the second term also as a differ-
ential coefficient of T. Differentiating the expression for 22*
partially with regard to d
dT
= 2w [■^' -ja +&C.J,
But differentiating the expression for x' with regard to 6
dx d^x . d'
X
(fa;
dd ~ dddt ^de'^"^ ded<t> '^' "*■ *°'
and this is the same ^s ^ -^^ . Hence the second term may be
written -52 , and the virtual moment* of the effective forces is
WJ
therefore (^§'-§)s^.
i'l
• The following explanation will make the orgnment clearer. The virtual
moment of the effective forces is clearly the ratio to dt of the difference between
the virtual moments of the momenta of the particles of the system at the times
t + dt and t, the displacements being the same at each time. The virtual moment
of the momenta at the time ( is first shown to be -73 h9. Hence I t:;; + ^ ^-r; dt ) 55
d9 \M dt do' J
is the virtual moment of the momenta at the time t + dt corresponding to a dis-
placement SO consistent with the positions of the particles at that time. To
make the displacements the same, we must subtract from this the virtual moment
of the momenta for a displacement which is the difference between the two displace-
dx
ments at the times t and t + dt. Since Sx=^r:S$, this difference for an abscissa is
dO
-|- f ^ J dt 55. We therefore subtract on the whole 2m J a/ -r [ -j|] dt + &c. 1 55, Jind
dT
this is shown to be ,- d( SO.
do
LAORANOE*S EQUATIONS.
291
virtual
Between
|e times
lomeiit
a dis-
18. To
lomeut
displace-
scissa is
SS.tod
869. To deduce the general equations of motion referred to any
co-ordinatea.
Let U he the force-function, then CT" is a function of 6, <f>, &c.
and t. The virtual moment of the impressed forces corresponding to
a displacement produced by varying only is -rrf^O' But by
D'Alembert's principle this must be the same as the vutual
moment of the effective forces. Hence
ddT
dtdef
dT
dd'
dU
dd'
o. M , , d dT dT dU
Similarly we have -^-^ = -^-,
&c. = &c.
It may be remarked that if V be the potential energy we
must write — V for U. We then have
±dT_dT dV_
dt dff dd '^ de~ '
with similar equations for <^, -^, &c.
In using these equations, it should be remembered that all the
differential coefficients are partial except that with regard to t.
These are called Lagrange's general equations of motion. Lagrange only con-
siders the case in which the geometrical equations do not contain the time ex-
plicitly, but it has been shown by Yieille, in Liouville'i Journal, 1849, that the
equations are still true when this restriction is removed. In the proof given above
\v6 have included Vieille's extension, and adopted in part Sir W. Hamilton's mode
of proof, PhU. Trant., 1834. It di£Fers from Lagrange's in these respects ; firstly, he
makes the arbitrary displacement such that only one co-ordinate varies at a time,
and secondly, he operates directly on T instead of 2mx'*.
370. To deduce the general equations of motion for Im-
pulsive forces.
Let 8Z7j be the virtual moment of the impulsive forces pro-
duced by any displacement of the system. Then from the geo-
metry of the system, we can express BU^ in the form
BU, = F8d+QS<f>+.„
The virtual moment of the momenta imparted to the par-
ticles of the system is
Xm{{x:-x:)Bx+{y:^y:)8y + {z:-z:)Bz},
where (a-,', y/, z^), (aj/, y/, z^) are the values of {x, y\ z) just
before and just after the action of the impulsive forces.
19—2
ii
:^'!|
rf:
\ li!
^•1
m
292
VIS VIVA.
t
Let 0^, ^;, &c. 0^, <f>^, &c. bo the values of ^, <f>', &c. nist
before and just after the impulse, and let T^, T^ be the values
of T when these are substituted for ff, «f>', &c. Then as in
(fJT tIT \
j^\ — 1/1° ) 8^.
The Lagrangian equations of impulses may therefore be written
de; dd;~-^*
with similar equations for ^, and ^, &e.
371. If we compare this equation with the general principle
of Art. 295, viz. that the momenta of the particles just after an
impulse compounded with the reversed momenta just before are
equivalent to the impulse, we sec that it will be convenient to
call jTv the component of the momenta with regai'd to 6, a name
only slightly altered from that suggested in Thomson and Tait's
Natural Philosophy. More briefly we may say that the ^-com-
dT
ponent of the momentum is -^, . In the same way we may
d dT dT
define the 6 component of the effective forces to be ^ itv n? .
^ dt da do
872. These eqnations for impulsive forces are not given by Lagrange. They
seem to have been first deduced by Proi, x'tiven from the Lagrangian equation
ddT dTdU
dt dff~ dd~ de'
We may regard an impulse as the limit of a very large force acting for a very
short time. Let Iq, t^ be the times at which the force begins and ceases to act. Let
us integrate this equation between the limits t = tQ to 1=1^. The integral of the first
r" dT~^ti d'P
term is I :j^, I which is the difference between the initial and final values of -jr, .
L"" J^o d9
The integral of the second term is zero.
dT
For Tfl is a function of d, <p, &c. ff, </>', Ac.
which though variable remains finite during the time t^ - tf.. If A be its greatest
value during this time, then the integral is less than A (<^ - to) which ultimately
vanishes. Hence the Lagrangian equation becomes I jw I ' = -i^ • S^o a paper
in the Mathematical Messenger for May, 1867.
373. Other expressions for the virtual moments of the momenta and of the
effective forces may be found when T is expressed in terms of the angular velocities
of the bodies of the system instead of the co-ordinates. Thus taking any one body,
if {x,y, z) be the co-ordinates of its centre of gravity, w,, w^, u, the angular velocities
about rectangular axes meeting at the centre of gravity, M its mass, A, B, C, Ac.
its moments and products of inertia, v the velocity of its centre of gravity, then by
Art. 348,
2 r= Mv^ + A'ug^ + £uy^ + Cw,* - 2 Z>«^w, - 2^w,w, - 2Fu^Uy.
I I M
laoranoe's equations. 293
Tho Tirtaal moment of the momouta will then be by Ex. 8. of Art. 819
dT, (IT, dT, dT ,^ dT ,^ dT , ,
dx dy •' dz du)ji dug dw,
and by Ex. 4 the virtual moment of the ufleotive forces will be if the directions of
tho axes arejixed in space
d dT, ^ d dT ,„ ,
dtdUji
Jtdaf'
where ix, 8y, tz are the linear displacements of tho centre of gravity and S0, 80, 8^
tho angular diuplacoments of tho body a^out the axes of Uj^, Wy, u,. If tho axes be
moving wo have merely to substitute for the coefficients of Sx, &c. the corresponding
expressions given in Axo example just rotcred to.
874. Before proceeding to discuss some properties of Lagrange's equations, let
us illustrate their use by the following problems.
A body, two of whose principal moments at the centre of gravity are equal, turns
about a fixed point situated in the axis of unequal moment under the action of
gravity. To determine the conditions that there may be a simple equivalent
pendulum,
Jkf. If a body be suspended from a fixed point under the action of gravity,
and if the angular motion of tho straight line joining to the centre of gravity be
the some aa that of a string of length / to the extremity of which a heavy particle is
attached, then I is called the length of the simple equivalent pendulum. This is (in
extension of the definition in Art. 92.
liot OC be the axis of unequal moment, A, A, the principal moments at the
fixed point, and let the rest of the notation be the same as in Art. 819, Ex. 1. Then
2T=A(e'^-^ sin* ^f «) + C (0' + ^' cos d}*,
V= Mgh cos 6 + constant,
where h is the distance of the centre of gravity from the fixed point, and gravity is
supposed to act in the positive direction of the axis of z. Lagrange's equations will
be found to become
^ {Ae')-A sin $ cos e^'* + C^' (0' + f cos 0) sin fl = - Mgh sin 9,
|^{(7(0' + f cose)} = 0,
d
^j {C (0' + f cos e) cos tf + ^ sin" tf f } = 0.
Integrating the second of Lagrange's eqi;ation3 we have
0'+ ^'cos B=n,
where n is some constant expressing the angular velocity about the axis of unequal
moment. Integrating the third we have
dip
Crt cos tf + 4 sin» e -j^ = o,
where a is another constant expressing the moment of the momentum about the
vertical through 0. ' ' '
■\l
i „
I'i
w\
t . 1 !' t
m
m
II
29-^/
VIS VIVA.
; I
r.'
I'l
B
There is an error, Bometimes made in nsing Lagrange's equations, which we
should here guard against. If u^ be the angular velocity about OC, we know by
Euler's equations, Art. 230, that u^ is constant. If n be this constant, the Vis Viva
of the body might have been correctly written in the form
2T= A (^» + sin" eip'^ + Cn\
But if this value of T be substituted in Lagrange's equations, we should obtain
results altogether erroneous. The reason is, that, in Lagrange's equations, all the
differential coeffcients except those with regard to t are partial. Though Wg is
constant, and therefore its total differential coefQcient with regard to t is zero, yet
its partial differential coefficients with regard to 6, <f>, &c. are not zero. In writing
down the value of T, preparatory to nsing it in Lagrange's equation, no properties
of the motion are to be assumed which involve differential coefficients of the co-
ordinates as indicated in Art. 367. But we must introduce into the expression any
geometrical relations which exist between the co-ordinates and which therefore ro-
duoe the number of independent variables.
Instead of the first equation^ we may use the equation of vis viva, which gives
To determine the arbitrary constants a and /3 we must have recourse to the
, ~ be the initial values of d, f, y ;
initial values of 6 and i^. Let Og, ^o> 777 » 'jT ^ *^^ initial values of ^, ^i ^ .
-^ , then the above equations become
dt
On
A
Bin' ^ -7; + "r cos 9 = sin'-* 60 ^" + ^ cos $0
dt
Cn
1
em'
•(f)^(f)*=*-.(t)^ (f )'^^^'<— «l
.(1).
I
id
These equations, when solved, give and ^ in terms of t, and thus determine
the motion of the line OG. The corresponding equations for the motion of the
simple equivalent pendulum OL are found by making (7=0, A^MP, aiiih=l,
where I is the length of the pendulum. This gives
sin«<?^=Bin«e„^^«
dt " dt
(2).
In order that the motions of the two lines OG and OL may be the same, the two
equations (1) and (2) must be the same. This will be the case if either Cn=0, or
O=0Q. Li the first case, we must have n=0, or C=0, so that the body must either
have no rotation about OG, or else the body must be a rod. In the second case, we
must have throughout the motion 6 and -J^ constant, so that the body must
be moving in steady motion making a constant angle with the vertical. In
either case, the two sets of equations are identical if I =
formula which was obtained in Art. 92.
Mh
This is the same
LAGRANGE'S EQUATIONS.
295
875. Ex. 1. Show how to deduce Etiler'e equationt, Art. 280, from Lagrangc'i
equations.
Taking as axes of reference tlie principal axes at the fixed point,
We cannot take {(j\, <<;,, Wg) as the independent variables because the co-ordinates of
every particle of the body cannot be expressed in terms of them without introducing
differential coefficients into the geometrical equations. Let U8 therefore express
Ui, ua, u, in terms of 0, 0, ^. By Art. 235, we have
c<>j = 0'sin^-^Bin0cos0 \
Wj=tf'cos^ + ^sinff sin^ \.
u^=<p'+\j/'(ioa0 )
As it will be only necessary to establish one of Euler's equations, the others follow-
ing by symmetry, we need only use that one of Lagrange's equations which gives
the simplest result. Since 0' does not enter into the expressions for wj, ua, it will
be most convenient to use the equation
d_dT dTdU
dt dip' ~ d</>~'d(p'
„ dT _ dw. ^ , dT . dw, _ da>a . „
Now^,=C«3^^»=C7«3, and -=i«,_V5«.-=i«,«,-5a;,«„ a8 may
be seen by differentiating the expressions for u^, wj. Also by Art. 326, if if be
the moment of the forces about the axis of (7, ■t-=N.
d<t>
Substituting we have
j^(Cw3)-(^-5)WiWg = i\r,
which is Euler's equation.
Ex. 2. A body turns about a fixed point and its vis viva is given by
2 r= J wi« + Bw^ + Cms* - 2DuiO)^ - 2^WjWi - 2 f Wjo;,.
Show that if the axes are fixed in the body, Euler's equations of motion may be
generalized into
d dT dT dT
dt rfwi dwj ' du.
U^r=L,
with two similar equations. This result is given by Lagrange.
376. Ex. Dcdwe the equation of Vis Viva from Lagrangc''s equations.
If the geometrical equations do not contain the time explicitly, 7 is a homo-
dT dT
goneous function of 6', 0', &c. of the second degree. Hence 2T=-t^$' + -t-, <p'+ ...
Differentiating this totally, we have ^-rt~^ di ItW') "*" dff ^ " + **''»
where the &o. implies similar exprespions for 4>, yp, &c. If we now substitute on
the right-hand side from Lagrange's equations, we have
„d7' dT^. dT .„ dU ., ,
^di = do''-de''^de'^''''
^n
1
,!
t
: 1
1 <i
'■ i
if
if :
i
i . I
« m
': : ';•':
i ::|
$
T
}
V
>i
ilf
296
VIS VIVA.
dr dT dT
But since T ifl a fonction of 0, 6', 4>, <f>', &e., t, = j^ ^ + ,t5' ^' + *"•»
'3^'
subtracting this from the last expression we have
dT dU^dU^,,
di=de^^d^'^^-
Integrating, we have the equation of Vis Viva
T- U=h,
where A is an arbitrary constant, sometimes called the constant of Vis Viva.
377. Ex. As an illustration of the application of Lagrange's equations to
impulsive forces, let us consider the example already discussed in Art. 154.
Let X be the altitude of the centre of gravity of the rhombus at any time, then «
and a may be taken as the independent variables.
We have
Let P be the impulsive action between the rhombus and the plane, then the
virtual moment of the impulsive forces is
8 ?7= /'3 (as - 2a cos a) = P5j! + 2a sin a PJa.
The Lagrongian equations are therefore
4(Xi'-0=P
4 (fc« + a") (o/ - O = 2a P sin o
!•
Now the initial and final values of x' are x„'= - V, x^'— - 2a sin au ; those of a'
are Oo'=0, ai'=w. Hence eliminating P we have
la =
3 V sin a
2 al + Ssin^a'
the same result as before.
378. Sir W. R. Hamilton has put the general equations of
Lagrange into another form, which is found to be more con-
venient for the investigation of the general properties of a dyna-
mical system. This transformation may be made to depend on
the following lemma.
Let Tj he a function of 6, <f>, &c., 0', <f), &c., stick that there
are no powers of the accented letters above the second. Let
dT dT
-TKT = M, ^ t! = w, &c., then 6\ ^', &c. may he found in terms of
6, <f>, &c. and u, v, &c. from these equations of the first order.
Let
T, = -r, + M^'+i'f +&C., '
and let T^ he expressed in terms of 0, <f), &c., u, v, &c. 07ihj,
& , (j), &c. being eliminated. Then -~ = — -j^ , -^, " = 0', with
do do du
similar equations for <f), yfr, &c.
tions -
laqrange's equations. 297
To prove this let us take the total differential of T,, we have
dT
^T^^-^-^de^-(--jQ}+t!\d& + e'du + kc.
By the conditions of the lemma, the quantity in brackets vanishes,
and therefore -]^J = --Jl\ ^^' = 6'.
ad do du
It should be noticed that if T, be a homogeneous quadratic
function of (6', ^', &c.) then u6' +v(l>' + &c. = 22\, and therefore
T^= T^, but diffeicntly expressed, T^ being a function of 6', <f>',
&c. and 0, (f>, &c., T^ a function of u, v, &c. and 0, <f), &c. In
this case T^ is a homogeneous quadratic function of u, v, &c.
As this process of eliminating ff, <f>', &c. and introducing
M, V, &c. will have to be frequently performed, it will be con-
venient to have a name for the result. We shall call T^ the
reciprocal function of T^, because 2\ may be derived from T^ by a
nearly similar process.
If T, be the vis viva of a dynamical system, this process is
equivalent to changing from the component velocities to the com-
ponent momenta and conversely.
879. Ex. If (0', (f>', \f/), (m, V, w) be regarded as the Cartesian co-ordinates of
two points and T^ be a homogeneous quadratic function of ($', 4>', ^), then 2^=^ is
the equation to a quadric. Prove that its polar reciprocal, with regard to a sphoro
whose radius is ijh, may be found by eliminating {d', ip', ^') by means of the equa-
tions-n^; =
dT. dT,
— ?=« i
.„,—«, -j^=v, j-7/='*'' Hence show geometrically that, if r,r=A be the
reciprocal quadric, — -=^', — '=</>', -r-' =•(/''.
dvL dv dw ^
S80. To express the Lagrangian equations in the Ilamiltonian
form.
If a system be acted on by any impulses, the Lagrangian
equations of motion may be written in the typical form f ,^, )=Py
where the bracket implies that 0' — 0^, </>,' — ^^, «&;c. are to bo
written for ff, <^', &c. after differentiation, using the rame notation
sis, before. Let H be the reciprocal function of T. Then these
equations take the typical form 0^ — 0J= ( y- ) > where the bracket
on the right-hand side implies that (P, Q, &c.) are to be written
for (w, V, &c.) after differentiation.
^
5 rt;
i'
ii
\
In
(' I i
)■■ \
i
(l i
V ! t.
; !il
i^*l
y'i
ill 'I
293
VIS VIVA.
381. If a system be acted on by any finite forces, tlie La-
grangian equations of rilotion may be written in the typical form
d^ dL dL _^
~dt dd'~W~ '
where L = T+ U, so that L is the difference between the kinetic
and potential energies. Since U does not contain {d\ <f>', &c.) the
equations of transformation may be written in the form
_dL_dT
^~ d&~ d&*
_dL_dT
^~d<l>'~d<l>'*
Also Lagrange's equations may be written in the form
«' =
dL
dd'
/ dL .
Let II be the reciprocal function of L, then these equations
6'
change into
dH
du'
dv
, dH , dH
which are called the Biamiltonian equations.
When the geometrical equations do not contain the time ex-
plicitly, r is a homogeneous quadratic function of {ff, </>', &c.), and
therefore
uB' + vf + &c. = 2r.
Hence n=- L^uO' -^-vj) +&.c. = T-U.
Thus H is the sum of the kinetic and potential energies, ar^d
is therefore the whole energy of the system.
882. Ex. To deduce the equation of Vis Viva from the Hamiltonian equa-
tions.
Since ZT is a function of (0, 0, &c.), (m, v, &c,) we have, if accents denote total
differential coefficients with regard to the time,
„, dH dll„, All , ^ dn
dt do du dt
so that the total diiierential coefficient of B with regard to t is always equal to the
partial differential coefficient. If the geometrical equations do not contain the
time explicitly, this latter vanishes and therefore we have n=h, where A is a con-
stant.
383. Ex. 1. To deduce Euler's equations of motion from the Hamiltonian
equations.
LAGRANGE'S EQUATIONS.
299
Taking the samo notation as in the corresponding proposition for Lagrange's
equations, Art. 376, we have
u=^^,=AuiBm,p + Buncos <f,, v = ^, = Cu^,
dT
»=y7>=(-ilwiC03 + £wjsin^) sin^ + Cw^cos 0.
To express T in terms of (m, v, w) we must find (wj, Wg, Wj). Wo have
1 • . I /. V COS <t>
Au, =« sin + (V cos O-w) —. — ^ ,
Rin0
Also
£ci>.=:U cos A - (v cos ^ - W) -;— ^ .
" ^ Sin tf
An
As we only require one of Euler's equations, let us use -,- = -v',
The former of these gives ^Wi-r-i + 5wa -7^ - s-r = - G -^l %
dll
dv
= 0'.
^Wj
^Wi rfU
, dWa
which is the same as Au, ^' - Bu/-^ _ '-^ = - c ~ ,
* ^ * a d(f> dt
and this leads at once to the third Euler's equation in Art. 230. Tiie latter of the
two Hamiltonian equations leads to one of the geometrical equations of Art. 235.
Thus the six Hamiltonian equations are equivalent to all the three dynamical and
the three geometrical Eulerian equations.
334. Ex. 1. The position in space of a body, of mass M, is given by (a;, y, z) the
rectangular co-ordinates of its centre of gravity, and (6, <p, ^) the angular co-ordi-
nates of its principal axes at the centre of gravity, as used in Art. 235. If two of its
principal moments are equal and if (|, rj, f, m, v, w) be the {x, y, z, 0, <p, f) com-
ponents of the momentum, prove that the Hamiltonian function H ib given by
M AC Aa.n'^d
Ex. 2. If the vis viva be given by the general expression
22'i=^,itf!' + 2^i2«y+
snow that the reciprocal function of Ti may be written in the form
T»=-
2A
w t) .
V A
la
^22 •
where A is the discriminant of T^. Thus the coefTicionts of 11^, v', 2nv, &c. in T.^
are the minors, after division by A, of the corresponding terms in 1\. See also
Art. 28, Ex. 3.
885. To explain how Lagrange's equations arc to be used when some of the forces
are non-comervative.
Lagrange's equations in the form given in Art. 309 can only be used when tlio
forces which act on the system have a force-function. If however P5d be tho
virtual moment of tho impressed forces obtained by varying only, Q3^ the vir-
'i
■\-> ,y
1 1
! , i
• 1
. hi 1
i i]
N#.l
I
I! i
I
) i|
'--■■11
iii.
itt
> ■ a
i
:J
I'i'
t I'li
300
VIS VIVA.
tual moment obtained by varying ^ only and so on, it ia clear that Lagrange's
equations may be written in the typical form t- ^-, ~ 'dO~^'
386. It will often be convenient to separate the forces which act on the system
into two sets. Firstly those which are conservative. The partb of P, Q, &o. due to
these forces may be found by differentiating the force-function with regard to 0, (p,
&c. Secondly those which are non'Conservative, such as friction, some kinds of
resistances, &c. The parts of P, Q, &o. due to these must be found by the usual
methods given in Statics for writing down virtual moments.
Though these non-conservative forces do not admit of a force-function, yet
sometimes their virtual moments may be represented by a differential coefficient of
another kind. ThuF suppose some of the forces acting on any particle of a body to
be such that their resolved parts parallel to three rectangular axes fixed in space are
proportional to the velocities of the particle in those directions. The virtual
moment of these forces is
S {/j^x'Sx + fi^'Sy + fi^z'Sz),
where ni, /j^, fi^ are three constants which are negative if the forces are resistances.
For example, if the particles be moving in a medium whose resistance is equal to
the velocity multiplied by a constant k, then fi^, fi.^, /Xg are each equal to - k. Put
^^'° %=^ (^'»'^+&«-) =2:(At,x'g+&c.) .
by Art. 3C8. Hence
dP
dff
5^-i.^304.&c.=sj^x'(g5^-Hg80+...) + &c.j
=S(/ttja;'8a; + &c.).
H!
In this case, therefore, if U be the force-function of the conservative forces, F the
function just defined, 055, <l>50, <fec. the virtual moments of the remaining forces,
Lagrange's equations may be written
d^dT_dT_dU dP^
dtde' dd~ de'^ de''^ '
with similar equations for <p, f , &o. The use of this fanction was suggested by
Iiord Eayleigh in the Proceedings of the London Mathematical Society, June, 1873.
The function F was called by him the Dissipation Function.
387. Ex. 1. If any two particles of a dynamical system act and react on each
other with a force whose resolved parts in three fixed directions at right angles are
proportional to the relative velocities of the particles in those directions, show that
these may be included in the dissipation function P. If V„ Vy, V^ be the com-
ponents of the velocities, Mi^a? Ma^yi /^3^» the components of the force of repulsion,
the part of P due to these is ^ S {/ji^Vx^ + /tj V^' + n^ F,'). This example is taken from
the paper just referred to.
Ex. 2. A solid body moves in a medium which acts on every element of tho
surface with a resisting force partly frictional and partly normal to the surface.
laorange's equations.
301
Each of these wlien referred to a anlt of area is equal to the velocity resolved in ita
own direction multiplied by the same cons, it k. Show that these resistances may
be included in a dissipation function P,
F=-^{ff («9 + 1;» + «>») + Aug* + Buy' + Cw,» - 2DuyU, - 2^w,w, - 2Fu;cUy),
where c is the area; A, B, &c. the moments and products of inertia of the surface
of the body and (u, v, w) the resolved velocities of the centre of gravity of a.
388. To explain how Lagrange's equations can he used in
some cases when the geometrical equations contain differential
coefficients with regard to the time.
It has been pointed out in Art. 3G7, that the independent
variables 6, <f), &c. used in Lagrange's equations must be so
chosen that all the co-ordinates of the bodies in the system can
be expressed in terms of them without introducing 6', ^', &c.
But when we have to discuss a motion like that of a body rolling
on a perfectly rough surface, the condition that the relative
velocity of the points in contact is zero may sometimes be ex
pressed by an equation which, like that given in Art. 127, may
necessarily involve differential coefficients of the co-ordinates.
In some cases the equation expressing this condition is integrabie.
For example ; when a sphere rolls on a rough plane, as in
Art. 133, the condition is x—ad' = 0, which by integration
becomes x—ad=h where h is some constant. In such cases we
may use the condition as one of the geometrical relations of the
motion, thus reducing by one the number of independent vari-
ables.
But when the conditions cannot easily be cleared of differ-
ential coefficients, it will be often convenient to introduce the
reactions and frictions into the equations among the non-con-
servative forces in the manner explained in Art. 386. Each
reaction will have an accompanying equation of condition, and
thus we shall always have sufficient equations to eliminate the
reactions and determine the co-ordinates of the svstem.
The elimination of the reactions may generally be most easily
effected by recurring to the general equation of Virtual Velocities,
and giving only such displacements to the system as may make
the virtual moments of these forces disappear. Suppose, to fix
our ideas, a body is rolling on a perfectly rough surface. Let
6, 6, &c. be the six co-ordinates of the body, then by Art. 127,
there will be three equations of the form
L, = A,e' + D,<i>'+...^0
(1),
the other two being derived from this by writing 2 and 3 for the
suffix. These three equations express the fact that the resolved
"
'll
\
I
■}'i
V
Mm.
I li
'' (
, )
!: >i
m
■1 ,
? i
*'li
!■ .'T'l
^
ili
II !■
302
VIS VIVA.
velocities in three directions of the point of contact are zero,
equation of virtual velocities may be written
The
\dt dff " dd)
dU
hd + &c. = "" S^ + &c (2),
where V is the force-function of the impressed forces. Since the
virtual moments of the reactions at the point of contact have been
omitted, this equation is not true for all variations of 6, (j), &c.,
but only for such as make the body roll on the rough surface.
But the geometrical equations i,, L^, L^ express the fact that
the body rolls in some manner, hence B6, B(f), &c. are connected
by three equations of the form
A^Bd + B^S(f>+...=0 (3).
If we use the method of indeterminate multipliers*, the equa-
tions of virtual velocities will.be transformed in the usual manner
into
d dT dT dU.^dL,. dL„ . dL^ .^.
dt d(y dd
dd'
!••••••••
with similar equations for the other co-ordinates cf), yjr, &c. These
joined to the three equations L , L^, L^ are sufficient to determine
the co-ordinates of the body and \, fi, v.
This process will be very much simplified, if we prepare the
geometrical equations 2/,, L^, L^ by elimination, so that one dif-
ferential coefficient, as 6', is absent from all but the first equation,
another, as ^', absent from all but the second, and so on. When
this has been done, the equation for 6 becomes
d^dT^
dt dd'
dT^_dU
dd~ dd'
dL,
(5).
Thus \ is found at once. The values of //. and v may be found
from the corresponding equations for ^, '^. We may then sub-
stitute their values in the remaining equations.
389. The method of indeterminate multipliers is really an
introduction of the unknown reactions into Lagrange's equations.
* If we multiply the geometrical equations (3) by X, ii, v respectively and sub-
tract them from (2) we get
^ldtdd'-de-T0-^d¥-''dff-''de'y^=^-
Now there will be as many indeterminate multiples X, /*, v as there are geome-
trical equations (3) connecting the quantities S6, 50, &c., i.e. there are as many
multipliers as there are dependent variations. By properly choosing X, n, v the
coeflScients of these variations may be made to vanish, and then the coefficients of
the independent variations must vanish of themselves. Hence the coefficient of
each variation in this summation will be separately zero.
LAGRANGE'S EQUATIONS.
303
an
IS.
Ibhe
of
of
Thus let B,
i»
-Sa» -^3
be the resolved parts of the reaction at the
point of contact in the directions of the three straight lines used
in forming the equations L , L^, L^. Then L^, L^, L^ are propor-
tional to the resolved relative velocities of the points of contact.
Let these velocities be /c,2/,, k^L^, k^L^. Then if 6 only be varied
the virtual velocity of R^ is k^A^O which may be written
dL
dff
K,
B0. Similarly the virtual velocities of i2, and B^ are
dL,
%' ^^ ^^^ "^ dff
B6. Hence, by Art. 385, Lagrange's equa-
tions are
d^dT dT_dU J.
dt dff dd~ dd^ "' '
-^ + kB
dff ^ « "
dL^ P dL^
dff ^ '"» « dff
Comparing this with the equations obtained by the method of
indeterminate multipliers we see that X, fjL, v are proportional to
the resolved parts of the reactions. The advantage of using the
method of indeterminate multipliers is that the reactions are
introduced with the least amount of algebraic calculation, and in
just that manner which is most convenient for the solution of the
problem.
The method of indeterminate multipliers may sometimes be
used with advantage when the geometrical equations do not
contain ff, <f>', &c., but are too complicated to be conveniently
solved. Thus if
f{t,e,<f>,...) =
be a geometrical equation, connecting 6, <f>, &c., we have, as in
Art. 335,
|8« + Js^-|....=0.
This may be treated in the same manner as the equations
Z^, Xg, Lg in the preceding theory. We thus obtain the equation
d^dT_dT
dt dff dd ''
^+X^ +
with similar equations for <^, '^, &c.
390. Ex. Form by Lagrange's method the equations of motion of c lomoje-
neous sphere rolling on an inclined plane under the action of gravity.
Let the axis of x be taken down the plane along the line of greatest slope and
let the axis of y be horizontal and that of 2 normal to the plane. Let (x, y, a) be
the co-ordinates of the centre of gravity of the sphere, d, (f>, ^ the angular co-ordi-
nates of three diameters at right angles fixed in the sphere in the manner explained
in Art. 235. Then, if the mass bo taken as imity, the Via Viva is by Art. 319
2r =«" + 7/'« + *« {(0' + ^' cos ^)2 -h «'«-(- sin" 5f 2}.
! ! V ;
n i (I
r. 1 I
J
'\ m
li.^i!
Ill !l
304
VIS VIVA.
The resolved velooities parallel to the axes of x and y of the point of the sphere
in contact with the plane are to be zero. These conditions will be found to lead
to the equations L^= xf - a6^ ooa^- a^' sin sin ^ = 0,
Zj=y + a0'Bin^-a^'Bin0co8 0=O.
Also if ^ be the resolved part of gravity along the plane and C any constant
U=gx+C.
The general equation of motion is
dtd<i' dq ~'dq ^ dq''^'^ dq' '
whore q stands for any one of the five co-ordinates x, y, 0, yf/, <l>. Taking these in
turn we have j!'=g + \, y"=n,
** {$"+<t)f sin 6) = - Xo cos ^ +/ita sin
(0' cos + \^ == - Xa sin sin ^ - /Mt sin cos ^
*"
dt
l»"(0'+^cos5)=O
The last equation shows that 0'+^'cosd is constant. From this we infer that
the angular velocity of the sphere about a normal to the plane is constant through-
out the motion. Eliminating ju from the two preceding equations and substituting
for \p" from the last, we find
-^=0" cos <p + }l/' Bin Oain<f>- $'<p' sin ^ f 0'f' sin ^ cos ^ + O'}/ cos 9 sin <p.
x" tut w"
But this is — . In the same way we find -•« = — • Substituting these values
of X and /x in the first two of Lagrange's equations, we have
These are the equation of motion of a projectile. Hence the centre of gravity
describes a parabola as if it were under a constant acceleration equal to
tending along the line of greatest slope.
If we had used some of the other expressions for the virtual moments given in
Art. 373, the solution of this problem would have been much simplified. Thus let
Ujc, Uy, u, be the angular velocities of the sphere about axes meeting i the centre
of gravity parallel to the co-ordinate axes. Then
2T=x'^ + y'^ + k* {uj> + V + w<«),
and the equations of condition are
x-a(i>y=0, y' + aux=0.
Displace the sphere by rolling it along a small arc parall&l to the axis of x
through an angle dd. Then we have
dtdx' dtduy dx '
.'. ax" + h*-^=ga.
Similarly rolling the sphere parallel to the axis of y and twisting it round tha
axis of w„ we have
-ay" + l*'^'=0, andA;»^'-«
dt
dt
These, by elimination of Ug, wg, u,, lead to the same result as before.
..I*,-.' ■ ■ . '.-.fc^
^.-rrr- . .-»-•*■ ■
a--' + Jfc«
Ind tiie
LEAST ACTION AND VARYINa ACTION. 305
Principles of Least Action and Varying Action.
391. Let (q^, q^, q^, &c.) be the co-ordinates of a system of
bodies, and let q stand for any one of these. Let 2 T be the vis
viva of the whole system and U the force-function, and let
L = T+ U. As before let accents denote differential coefficients
with regard to the time.
Let us imagine the system to be moving in some manner,
which we will call the actual motion. Then q^, q^, &c. are all
functions of t, and it is generally our object to find the form of these
functions. Let us suppose the system to move in some slightly
different manner, i.e. let q^, q,^, &c. be functions of t slightly
different from their actual forms. Lot us call the motion thus
represented a neighbouring motion. We may pass, in our minds,
from the actual motion to any neighbouring motion by the process
called variation in the calculus of that name. By the fundamental
theorem in that calculus
^.[s|(8,-,'s*)];;.
where the letter S implies summation for all the co-ordinates
q^, q^, &c. and, as implied by the square brackets, the terms
outside the integral sign are to be taken between limits.
The co-ordinates being independent of each other, each sepa-
rate term under the integral sign vanishes by Lagrange's equa-
tions, and we have therefore
-[-
mt + t
where H is the reciprocal function of L, by Art. 378.
The integral I L dt has been called by Sir W. R. Hamilton
the principal function, and is usually represented by the letter S.
If the geometrical equations do not contain the time explicitly,
we have H=T — U. In this case the equation of vis viva will
hold, and if h be the constant of vis viva we have
hi'^L dt = -h {8t, - SO + h ^rkT-
R. D.
20
I v\ A
m
306
VIS VIVA.
392. Otlier functions may be used instead of 8. Let us put
The function Fis called the characteristic fiincti(m.
If the geometrical equations do not contain the time explicitly,
we have // = h, where h is a constant which may be used to repre-
sent the whole energy of the system. In this case
V=8+h{t,-t,)
=.f\T+U)dt+r(T-U)
= 2 r TJt.
dt
The function V therefore expresses the whole accumulation of the
vis viva, i.e. the action of the system in passing from its position
at the time t^ to its position at the time t^.
393. In the proof of these theorems we have supposed that all the forces are
conservative. If in addition to the impressed forces there are any reactions, such
as rolling friction, which cannot be taken account of by reducing the number of
independent co-ordinates, wo must use Lagrange's equation in the form
d dL dL
dtdq'~dq~ '
where, as explained in Art. 385, PSq is the virtual moment of these reactions corre-
sponding to a displacement dq. In this case the quantity under the integral sign
will not vanish unless the variations are such that
SP(Sq-q'5t)=0.
Now q being the value of any co-ordinate in the actual motion at the time t,
q + Sq is its value in a neighbouring motion at the time t + St. But q'St is the
change of q in the time St, hence q + Sq- q'St is the value of the co-ordinate in the
neighbouring motion at the time t. The neighbouring motions must therefore be
such that the virtual moments of the reactions corresponding to a displacement of
the system from any position in the actual motion into its position in a neighbour-
ing motion at the same time is zero. With this restriction on the variations, the
two equations which express the variations of S and V will still be true.
394. The two fundamental equations, giving the values of
B8 and 6V, will be found to lead to many important theorems
which we shall now proceed to considei*.
Let us call the positions of the system at the times t^ and t^ the
initial and terminal positions, and let us suppose these fixed, so
LEAST ACTION AND VARYING ACTION.
807
that the actual motion and rll its neighbouring motions are to
have the same initial and terminal positions. In this case Bq
vanishes at each limit, and the two fundamental equations take
the form*
■'!':
8 Ldt^-h{Zt^-hQ,
28
{t-t,)hh.
V I
• We may easily establish these theorems without the use of Lagrange's
theorems. Let (x, y, z) be the rectaufrular co-ordinates of any particle and let m be
the mass of this particle. Lot -Y, }', Z bo the components of the impressed accele-
rating forces on it. Then
and by the fundamental theorem in the Calculus of Variations
«j;>=i"o^x>(£-*")"'-^"""*[4'(''-'''"]::-
If vre substitute for L and remember that T is a homogeneous function of
«', y', z', this becomes
i f''Ldt = [(U-T)8t + Tmx'5r:{' + f*'Zm{X-x") {Sx-x'St)dt.
''to to •'U
If we consider the positions of the system at the times (q and t^ to bo given, 8x
is zero in the part taken between limits.
If the time of transit be given it is unnecessary to vary the time. Putting St=0,
the part under the integral sign vanishes by the principle of virtual velocities. The
part outside the integral sign is also zero and therefore 8 / ^Ldt=0.
J to
If the time be varied, Sx - x'St is the projection on the axis of x of the displace-
ment of the particle m from its position in the actual motion at time t to its position
in a neighbouring motion at the same time. Hence the part under the integral
sign vanishes as before by the principle of virtual velocities. Lot us suppose that
the geometrical conditions do not contain the time explicitly, then T - U— h and
L=2T-h. The equation then becomes
28 f^'Tdt - [S(ht)]*' = [ - hStf .
•'to to to
ptl
If h be giveii ;ri have 8 / Tdt^O.
•'to
From the general value of the variation in Cartesian co-ordinates we can also
deduce the values of 5<S and iV given in the text. For the term 2wia;' is clearly the
(CT
virtual moment of the momenta, and this by Art. 368 is — , Sq, The method
followed in the text seems however to be preferable.
Lagrange has given a general view of his transformation from Cartesian co.
ordinates which seems worthy of notice. Let L be any function of x, x', <fec.,
J/, y', kc and of t, and let the variables x, y, Sec. be transformed into others
20—2
^[i
!.'.;
:il
308
VIS VIVA.
M '•
!' I
where it has been supposed that the geometrical equations do not
contain the time explicitly.
If the time of transit of the system from its initial to its
terminal position be also given, ■we have Bt^ = S<j, and therefore
rti , .
Hence I Ldt is either a maximum or a minimum. It cannot
be the former, since by causing the bodies to take circuitous
paths we may make it as large as we please. It is therefore a
minimum.
If the constant h be given, or which is the same thing (since
the terminal position is given) if the energy of the system be
given, we have Bh = 0, and therefore B \ Tdt = 0. We may
now infer the two following theorems.
Let any two positions of a dynamical system be given, the
actual motion is such that I Tdt is less than if the system were
constrained, without violating any geometrical conditions, to
move in some other manner from the one position to the other
with the same energy; these other motions being such that,
throughout, T is the same function of the co-ordinates and their
differential coefficients.
This is called the principle of Least Action.
5i, gj, &c. by writintj for x, y, &o. any functions of q■^, q^, &o. and of *. The funo-
tion L is thus expressed in two ways, and by comparing the two vahxes of 5 / 'Ldt
given by the Calculus of Variations, we see that the integral of
H(S-*»>-(f-*">«i
may be completely found. Hence this expression must be a perfect differential
with regard to t, quite independently of the operation 5. But this cannot be unless
it vanishes, because it contains only the variations Sx, dq, &c. and not the
differential coefficients of these variations. We have therefore the general equa-
tion of transformation
d dL
dtd^'^^')^^'
\dx dtdx' ') \dq
where the ^' implies summation for all the variables x, y, &c. or q^, q^, &o.
If a;, y, <fec. be Cartesian co-ordinates the left-hand side of this equality vanishes
by virtual velocities. Hence 2 ( -7- - &c. J Sg niust also vanish. The j's being all
independent, we are led to Lagrange's equations.
ti
LEAST ACTION AND VARYING ACTION.
309
In the same way if the system moves in the varied motion,
not with the same energy, but in the same time, from the one
given position to the other, then l Ldt
IS a mmimum.
395. Maupertuis conceived that he could establish ct priori by theological argu-
ments, that all mechanical changes must take place in the world so as to occasion
the least possible quantity of action. In asserting this, it was proposed to measure
the action by the product of velocity and space ; and this measure being adopted,
mathematicians, though they did not generally assent to Maupertuis' reasonings,
found that his principle expressed a remarkable and useful truth, which might be
established on known mechanical grounds. Whewell's History of the Inductive
Sciences, Vol. ii. p. 119.
396. Conversely, from either of these theorems we may deduce
the motion of any system, by making I Ldt or i Tdt a minimum
J to J to
according to the rules of the Calculus of Variations*. That this
* Lagrange's equations are the ordinary equations supplied by the Calculus of
Variations when we make f Ldt a minimum under known conditions. Su: W.
Hamilton put these equations under a form (see Art. 381) which is very useful in
Dynamics. It is an interesting question to determine what is the corresponding
transformation when X is a function of differential coefl&cients higher than the first.
This was considered by Ostrogradsky in a M&moire sur les equations differentielles
relative au probl&me des Isop6rimetres, published in the Memoirs of the Academy of
Sciences at St Petersburgh in 1850. The Memoir is rather difficult on account of
the immense length of the algebraical transformations. The following short ac-
count may therefore prove useful.
Let L be a function of * and of m variables, of which q is any one, and let it be
a function of the first n differential coefficients of q with regard to t.
d*'(i
Let Qfc stand for the partial differential coefficient of L with regard to — , and
let Q*=Q*-QVi + Q"*4a- »
where, as usual, accents denote differential coefficients with regard to t, and let k
accents be denoted by (/t). The relations between these variables are, therefore.
and so on np to
and the last is
Qo
_dL
~ dq
■Q\
Qi
dL
-dq'
Q'a
, ,
dL
«n-l =
"dr/<»-
l)~^n
^~m
dL
Q»=
di'»>
■a).
By the principles of the Calculus of Variations, the minimum is given by the
typical equation Qq=0, [When
! ■
m
\ !'i ■ 1
1
i!^: 1,
1 . ■ ■
' 1
. 1
■i-
,
!
. ,
j
'V
i\k
310
VIS VIVA.
process will really lead to the equations of motion may be seen by
simply reversing our steps. Thus granting that 8 j Ldt=0 under
the known conditions, we have
TMien L contains no differential coefficient above the first, Sir W. Hamilton
eliminated the m first differential coefficients typified by q' by introducing m new
variables typified by Q^=-—, Let us in the same way eliminate the highest differ-
ential coefficients typified by 2^"' and introduce instead the m new variables typified
byQ„. Let
H=L-'S (Q,q'+ Q„2"+ ... + Q„q'%
where the 2 refers to summation for all the q'a. Let g*"' be found from the equa-
tion Q„= •— —- and let its value be substituted in this expression for H so that H is
now a function of t, q, <?'... 5^""^', Qi, Q3...Q,,. Since L was originally a function of
t, q, ^'...g'") it is now a function of t, q, g'...g"'~i> and Q„.
We have by differentiation
dff
provided i + 1 is not n. In that case
(IH _ dL^ dq(^>
dQ^~dq^"UQ„
_g(fc+l)=: _
d
dt'
>(«:)
■(2),
but the first and third of these terms destroy each other, so that the theorem (2) is
also true when /t + 1 = jj. Also
dH ^ dL_ dL_ d^^ n -O ^
d«f>~ dq'-''^'^ df''^dq'i^^~^'' ^'dg<*>*
Here the second and fourth terms destroy each other. The first and third, by
d -
(1), become Q\+^ or -7 Qj^Jrv Thus all the equations may be written in the typical
dt
Eamiltonian form
dH
^-=-dt^
(It)
*-di^''+^
dH
dgi*'
which are true for all values of k from k=Q to k=n-l. Thus there are 2n equa-
tions corresponding to each g.
"We may show in the same way as in Art. 382, that the total differential coeffi-
cient of H with regard to t is equal to its partial differential coefficient. Bo that
when L, and therefore H, are not explicit functions of t, we have as one integral
if =/t, where A is a constant. Writing this at length it becomes
L = 2(Q,q'+Q^q"+...) + h,
which is the integral continually used in the Calculus of Variations. We see the';
this integral corresponds to the equation of 'Vis Viva in Dynamics,
LEAST ACTION AND VARYING ACTION. 311
for all variations. The Bq^s being all arbitrary and independent,
each coefficient under the integral sign must vanish separately,
and this leads to the typical Lagrange's equation.
Ex, 1. There is another method of deducing Lagrange's equations from the
principle of Least Action which is worthy of notice. We are to make f ' Tdt a
minimum, subject to the condition T- U—h, By Lagrange's rule in the Calculus
of Variations we are to make
sf{T+\{T-U--h)]dt=0,
without regard to the given condition, and afterwards make \ such a function of I
that the given condition is satisfied. This will be found an excellent exercise in
the Calculus of Variations.
The solution maybe indicated as follows. Putting W—T+\{T-U) vre have
with the same notation as before
and this must be equal to h8 jxdt. The integrals are to be taken between the
limits, which are omitted for the sake of brevity.
First, let us consider the part outside the integral sign. Tlie initial and final
positions being given 52=0, and wo have
WSt - S ^p q'St ■= hif\dt = h\St.
This equation is satisfied by 8t=0, but since the time of transit is not to be the
same in the actual and varied motions, this factor must be rejected. Also 2* is a
dT
homogeneous function of the ^'s, hence li ~—,q'=2T. Substituting for W its
value and using this equation we find (1 + X) 2'+Xl7+AX=0. But X is such that
T-U=h, hence{l + 2\)2'=0 and .-. Xrn-^.
Next, let us consider the part under the integral sign. By the rule in the
Calculus of Variations this gives at once the typical equation
dW _ ddW^
dq dt dq'
Substituting for W we have the typical Lagrange's equation.
Ex. 2. If we add to the conditions given in the 'principle of Least Action, the
condition that the time of transit is to be always the same, show that the minimum
does not in general lead to Lagrange's equations. Following the notation of the
1 A
last Article, show that the minimum for a given time is determined by X= - ^ -f »,•
I I <l
!'■
I
i
mi
312
VIS VIVA.
!
li
. A'
M\
^ ■
where ^ is an arbitrary constant to be chosen so that the constant of vis viva has
its given value, while the absolute minimum is determined by X= - ^ .
397. When the geometrical equations do not contain the time explicitly the
symbol H or h may be used to express the energy of the system. If we represent
(he energy by E, Sir W. B. Hamilton's fundamental equation may be written
This equation has been applied to the motion of a system of bodies oscillating
in such a manner that the motion repeats itself in all respects at some constant
interval. Let this interval be i. Suppose that f ^me disturbance is given to the
system by the addition of a quantity of energy 5^. Let the system be such that
the motion still recurs after a co.iBtant interval, and let this interval be now
i+5i. The symbols of variation in Hamilton's equation maybe used to imply a
change from one kind of motion to the other. If the time t be taken equal to the
period i of complete recurrence, the initial and terminal stat )s of motion are the
same and therefore the last term vanishes when taken between the limits. The
equation reduces to 25 f Tdt=idE. Let T^ be the mean vis viva of the system
during a period of complete recurrence of the motion, then / Tdt=iT^. We there-
•'0
5{iTJ
fore have 7^ =2
iT„
This equation may be put into another form. Let P,^ be the mean potential
energy of the system during a period of complete recurrence ; then we have
6P.
'm"
which serve to determine the change in the mean potential and kinetic energies
v/hen any additional energy 5E is added to the system.
These or equivalent equations have been applied by Bolzman, Clausius and
Sr:"iy to the Dynamical Theory of Heat. The papers of the two latter are in
various numbers of the Philosophical Magazine extending from 1870 to the present
time. The second of the equations above written may be called Clnusius' eqiiation.
398. Ex. 1. If the period of complete recuxrence of a dynamical system be not
altered by the addition of energy, prove that this additional energy is equally dis-
tributed into potential and kinetic energy.
Ex. 2. A quantity of energy dE is communicated to a system whose mean
semi-vis-viva during a period of complete recurrence is T^. This is repeated
continually, so that at last the mean vis viva and the period of complete recurrence
"dE
are the same as at first. Prove that /"^ =0.
This example is due to M. Szily, and is important in the Dynamical Theory of
Heat.
GENERAL EQUATIONS OF MOTKN. 313
On the Solution of the General Equations of Motion.
399. Sir W. R. Hamilton has applied his fundamental theo-
rem expressing the variation of the Principal and Characteristic
functions to obtain a nev/ method of solving dynamical problems.
Let (oti, a/, ttj, a' &c.) be the values of {q^, q^, q^, q^, &c.)
when t = t^, and let 1\ be the same function of (a^, a,', &c.) that
T is of {q^, q!, &c.). We have then when t is written for the
upper limit
h8=^^^hq-^^fZa- ^m + H,K
BV=t^,Bq-X^~na + tSE-t,8E,.
It is clear that both 8 and V may be regarded as functions of
the time and the initial conditions of the system of bodies, i. e. we
may regard either of these quan:ities as a function of t, a^, a^, &c.,
a/, ttj', &c. Also the co-ordinates q^, q^, &c. are functions of t and
the same initial conditions. Though these functions are in general
unknown, yet we can conceive the initial velocities a,', aj, &c.
eliminated, so that S and Fare now functions of t, and a^, a^, &c.,
q^, q^, &c. the co-ordinates of the system at the times t^ and t.
Let 8 be thus expressed, then, by the equation for BS, we have
the typical equations
dq dq ' da da '
Since T is not a function of q", the first of these equations
contains no differential coefficient of a co-ordinate higher than the
first. This equation, therefore, represents typically all the first
integrals of the equations of motion.
Since T^ contains only the initial co-ordinates and the initial
velocities, the second equation has no differential coefficient of
any co-ordinate in it. This ecjuation, therefore, represents typically
all the second integrals of the motion.
Besides these we have the two equations
d8
dS rr
-dt--^^
dt
= H.
0>
where, if the geometrical equations do not contain the time ex-
plicitly, we may put h for H, h being a constant. In this case
i li.
! i 'I
1 II-
! i ' I
hf
1
:S 1
■ ,■ . !*'
■
■
::i 1
hn
J
■';:l
: ' ii !
\\\
II-
314
VIS VIVA.
the integrals may be used to connect the constant of vis viva with
the constants (a, a, &c.).
Comparing Art. 394 with these results we see that ;S^ is such
a function, that all the equations of motion and their integrals are
included in the statement that 8S is a known function of the
variation of the limits. If we keep the limits fixed, we get
Lagrange's equations; if we vary the limits we get the integrals.
400. In just the same way, if we regard q^', q', &c. as
functions of t, the initial co-ordinates r-ud their initial velocities,
we may eliminate t also by means of the equation
which reduces to H=T— ?7 when the geometrical equations do
not contain the time explicitly.
Let us suppof'e V to be expressed in this manner as a function
of the initial co-ordinates, the co-ordinates at the time t, and of
H. Then, by the equation for 8 F,
dV dT dV dT^ dV
da
da" dli
= t.
dq dq
Supposing V to be known, the first of these equations gives in
a typical form all the first integrals of the equations of motion.
The second supplies as many equations as there are co-ordinates
{q^, q^, &c.). When the geometrical equations do not contain the
time explicitly these do not contain t, but they all contain h.
One of them, therefore, reduces to the relation between this
constant and the constants (a, a', &c.). The equation —rj- = t will
give another second integral of the equations of motion containing
the tim"^.
401. Ex. ^iQ=f (Sqp' + E) dt, vrliere p=j-,, jirove th&t SQ = imt + 'SqSj)]* .
Thence show that if Q be expressed as a function of the initial and terminal
components of momentuji, viz. (6^, b^, &c.) an;' (py, p^, &c.) and of the time, then
-- = «, ^=-a. ^=H. ThisresuItisduetoSirW.R. Hamilton.
dp do at
402. Ex. 1. A homogeneous sphere of unit mass rolls dovm a perfectly rough
fixed inclined plane. If the position of the sphere is defined by the distance q of
the point of contact from a fixed point on the inclined plane, show that
„ 7 (q-a)^ 1 , V , 5 , ,
where g is the resolved part of gravity down the plane and tQ=0.
GENERAL EQUATIONS OF MOTION.
315
Thence obtain by substitution the Hamiltonian first and second integrals of the
equation of motion.
B 7
We easily find, as in Art. 133, that q=a + a't + jjgt''. Also ^=iq9''> ^=92-
pi
To find S, we substitute in &= / (T+ U) dt. After integration we must eliminate
*'o
a' by means of the equation for q,
Ex. 2, Taking the same circumstances of motion as in the last example, show
2 /ii " P
that r= — W -=-^iOi + '0' - (f/« + '0^}' Thence also deduce the Hamiltonian first
and second integrals.
Ex. 3. Show how to deduce the equation (f vis viva, from the Hamiltonian
integrals.
We have V a function of </„ </«, &c. and H, Hence ,=S-t^(/' + -—-,-?- »
' dt dq dH dt
which becomes by Hamilton's integi-als 2T=:^-y-q'+t-f , Wlien T is a homo-
dq dt
JJT
geneous quadratic function of (q^, q^', &c.) this gives -^- =0, or Ji'=constant. The
equation of vis viva may also be deduced from Hamilton's principal function.
Ex. 4. When the geometrical equations do not contain the time explicitly,
show that no two of the Hamiltonian integrals can be the same and no one can be
deduced from two others.
dT dT
If it were possible that two should be the same, the ratio of ^— , to , -, must be
dqi dq^
some constant m. Integrating this partial differential equation we find T to be a
homogeneous quadratic function of q^ - mq^, q./, &c. It would, therefore, be possi-
ble to set the system in motion, with values of q^' and q^' which are not zero, and
yet so that the system is without vis viva.
403. By the preceding reasoning all the integrals of a dy-
namicrJ system of equations can be expressed in terms of the
differential coefficients of a single function. But the method
supplies no means of discovering this function d priori. We shall
now show that this function must always satisfy a certain differ-
ential equation, so that the solution of all dynamical problems
may be reduced to the integration of thi^ one equation.
Let us, for the sake of brevity, suppose that the geometrical
equations do not contain the time explicitly. We have then
H=T —U. If we follow the process indicated in Art. 378, we
dT dT
put ^—, =Pi, -t— / =i>2' ^^- ^^^^ eliminate q^\ q^, &c. Let the
reciprocal function of H thus found be
H:=F{q„p^,q^,p,,&c.).
I i
I
i < I
r
i
■It
'.' i I
j ■■Ik
' Pi
t ^1
1
H
M
' ■ ■ -: .i ,
li;i
816 VIS VIVA.
But Pi = ;t— > Pa=-r— ' *^^' ^^^ ^~~"^' Hence 8 must
satisfy the equation
dS ^ r^f dS dS , \ ^
In just the same way, p^= -j- , p^ = - , &c. and the equa-
tion of vis viva gives H = h. Hence Fmust satisfy the equation
T.( dV dV , \ ,
If we consider the initial value of T, we shall have another
equation of a similar form with a^, a^, &c. written for q^, q^, &c.,
and ^, for t. It is necessary that the functions should satisfy both
these equations.
Ex. Taking the same circumstance of motion as in Ex. 1 of Art. 402, show
that the difierential equation to find ^ ^'^ yI\ t) ~91 — ^^' Integrate this equa<
tiou and thence find the motion.
404. When there are several independent variables, the equation to find V is
of the form
.2^"(^J-'^"^¥."''^''=^-''' <^^'
where (5^, B^^, &c.) are functions of q^, q^, &c. only. The left-hand side of this
equation, by Ex. 2 of Art. 384, may be written in the form of a determinant. We
dV dV
have only to replace w, v, &c. by their values — , -j— , &c.
We thus have, in general, a partial differential equation to find F, and
Sir W. Hamilton gave no rule to determine which integral is to be taken. This
rule has been suppUed by Jacobi in the following proposition.
Suppose a solution to have been found containing n-1 constants* besides h, and
the constant lohich may he introduced by simple addition to the function V. These
need not be tlie initial values o/q,, q2-<ln> but may be any constants whatever. Let
them be denoted by Oj, aj...an-i» *° *'"**
V = f (qi, qa-qn. Oi, aj...On_i) + on (2).
Then the integrals of the dynamical equations will be
rar^- '''■u-^r^^-^ (3)'
ffi=*+' <4).
• An integral of a partial differential equation has been called by Lagrange
"complete," when it contains as many arbitrary constants as there are independent
variables. It is implied that the constants enter in such a manner into the inte-
gral that they cannot by any algebraic process be reduced to a smaller number.
For instance, if two of the cougtauts enter in the form aj + a,, they amount on the
whole to only one.
where
the equi
Let
where tl
iber.
the
GENERAL E'^UATIONS OF MOTION.
317
where Pi, /9j.../9n_i ""'^ « ^^^ " w^'" arbitrary constants. And the first integrah of
the equations may be written in the form
dqi dqi' ' dq^ ~ dqj' '
Let the expression for the semi-vis-viva be
5).
^=s^ii'?i"' + ^ii.'7iV+&c.
.(6),
where the coefficients A^^, A^^, &o. are functions of q^, jj, &c. only.
Let Qi, Qg...Q^he such functions of 17,, ^g.-.^n and the constants, that they
may satisfy identically the n equations
df
,(7).
^=AiiQi+A,^Q^+...
&C. = &C.
Then from the mode in which the differential equation to find V has been
formed, in Art. 403, we know Qj, Q.^ will also satisfy identically the equation
^ + '^=2^iiQi' + ^i2<?i«a+ (8).
Firstly, we shall prove that Qi = qi, Qi=qi, &c., it will then follow that the
equations (5) are satisfied. Differentiating equations (3) and (4), we have
dy dq^ d'f dq,
+ J — 5 3r +
dui dq^ at dui dq^ dt
dhdq^ dt dhdq.j
=
=0 J-
t-... = l
(9).
These are the equations to find -y , -y^, (fee.
But differentiating (7) with regard to o^, we have
^L = A ^-^ + A ^^ +
doj dqi ^^ dai ^* da^
dai dq
dQi
&c. = &c
(10),
because A^yAi^, &c. are not functions of the constants. Multiplying these equations
by Qv Qa ■; and adding, we get
Si, «^ + dZ^ ««+•••= di ii ^M^-^^^Q^^ ■
Since the equation (8) is an identical equation the quantity in brackets on the
right-hand side does not contain Oi, being equal to V+h. Hence the expression on
the left-hand side vanishes. Thus we have an equation connecting Q^, Qg... ex-
actly similar to the first of equations (9). Similarly by differentiating equations (7)
with respect to a,,.. A successively, we shall have equations similar to the second,
&c. and last of equations (9). We have therefore exactly the same equations to
find Qi, Qj... and g,', g,'.... Hence Q, = 3i', $3=39'. &o«
't
I t
!]1M!
I ,1
!1 'i
1' Ul
i III
318
VIS VIVA.
Secondly, we shall prove that (3) and (4) satisfy the equations of motion. Let
lis consider the equation of Lagrange *,
dt ilqi dqi ~ dqi '
When qy, q^.-.q^ have been expressed in terms of q^, ffj- ?n ^^^ *^® coiistant
by means of equations (5), we have identically
Therefore, differentiating partially,
dU IdA
11 « '2
dA
ii„ i„ I
dqx
I rt '*J L l* /* 'n ' I-
+ i'^ij
dqa
<
dqi "* rfgi
g,' + &o.
But differentiating (5) written at length, with regard to q^, we have
^+A ^
+ ...=
d'f .dAn
.dA
IS
'''Wi'^""'dfi'^'~dq'i'~'^' dqi~''' dq^ •••
^" If, ^^'^ dq, ■*" - - dq^dq^'^^ dq, *« dq^, -
Hence, substituting,
&C.=:&0.
dH
^ d^f ,
dq,~ dq,^^^ "^ dq.dq^
IdA,
2 rfg,
3i"
dA,
q^'+.„ _ ' ^:;^i «.'2 _ "^13 „ >„ >
dq.
q{q,-...
* We may also show that the Jacobian integrals satisfy the Hamiltonian form
of the equations of motion. The peculiar relation of the differential equation to
the Hamiltonian function H adapts it to this process. If we substit.ute the value
of F given by (2) in the differential equation (1), the result is an idantical equation.
dV
Differentiating this identity with regard to each of the n constants and replacing ,—
dq
I, * I' , ^x, t dH dH dH dH A X ^ J
by p, we get n equations of the form -j—^-—- + -,— -^ — ~ + ...=0 to find
dp, dq, da op, dqg da
J- , y- , &0. These are the same as the equations (9) in the text, hence
——=q'. Again, differentiating H partially with regard to g,, we have
dp
dH dHd^ dH d^f ^^
dq, dp,dq;^ dp^dq^dq^
But all the terms of this equation except the first are together equal to the total
dp.
differential coefficient
dt
Hence t- = ~ -jz' The investigations of Hamilton and
dq, dt
Jacobi apply to a system of free particles mutually attracting each other referred to
Cartesian co-ordinates. In the text the reasoning has been applied to a system of
bodies referred to any co-ordinates.
GENERAL EQUATIONS OF MOTION.
319
Next let U8 consider the expression for T\ we see that the partial differential
coefficient
f,. -9 .1:7 9i + ,,„ Qi<li+--'
d'/i 2 dq^
(iQi
is the same as the latter part of the expression for
dU
dqi'
Also ,- -7 = V^ I therefore taking the total differential coefficient, we have
±dT_d^, dj ,
dt dqj^ ~ dq^' *1 ■*■ dq^dq.^ 1i + -
which is the same as the first part of the expression for
tial equation of motion is satisfied.
We have also, since T is homogeneous,
dU
Hence the different
df
-_, dT , dT , df , df ,
where the differential coefficient is total. This shows that the function / represents
the whole accumulated "action " in the time t. See Art. 392.
405. Jacobi has extended his theorem to the case in which the geometrical
equations do contain the time explicitly. But for this we ht^ve no space. It is no
part of the plan of this book to enter on Theoretical Dynamicii. We cannot there-
fore do more than allude to Professor Donkin's theorem that a knowledge of half
the integrals of the Hamiltonian system will in certain cases lead to a determi-
nation of the rest.
In Boole's Differential Equations it is shown that when the Hamiltonian equa-
tions are four in number, and one integral besides Vis Viva is known, both the
remaining integrals can be found by integrating an exact differential equation.
Miscellaneous Exercises, No. 15.
■iii
W
Variation of the Elements.
406. Let the integrals of a dynamical problem be
<!i=fi{Pv9vPi>Qvth
Ci=fi{Pi, 2vPi, flu."- m (1).
&o. =&c. )
where p, q, ... are some variables which determine the position an \ motion of the
system, and which are such that the equations of motion may bu written in the
forms
, dH , dll ,„.
^ = -d^' «=rf^ ^2>'
in the manner explained in Art. 381. Let the equations of motion of a second
dynamical problem be
dH dK , dH dK ...
^^-Tq-di^ ^ = d^ + d^ (^^'
I m
320
VIS VIVA.
where K is nome function of p,q,...t. If we consider fj, r„ ... the constants of the
solution of tlie first problem to be functions of p, q, and (, we may suppose the
solution of the second problem to be represented by integrals of the same form
(1) as those of the first problem. It is therefore our object to discover what func-
tions fp c.^, ... are oi p, q, and t. The function K is called "the disturbing func-
tion," and is usually small as compared with //.
Since the equations (1) are the integrals of the diflorontial equations (2), we
shall obtain identical expressions by substituting from (1) in (2). Hence dif-
ferentiating (1), and substituting for y and q' their values given by (2), we got
Q^_dc,dJI_^dr,dn^^^dc,
dp dq dq dp
dt
(4).
0=«&c.
uj, ... are considered as variables, the equations (1) are the integrals
of the differential equations (3). Hence repeating the same process, we have
But when e,, c.
rfCj _ dc^ dn dc^ dH
dt dp dq dq dp
dc, dK de, dK
— ' f. — 1 J.
dp dq dq dq
dc^
"di
de^_
dt
= &0.
where the differential coefficients on the left-hand side are total, and those on the
right-hand side partial.
Hence, using the identities (4), we get
•(5),
dcy _ rfcj dK dc^ dK
dt ~ dp dq dq dp
dc
with similar expressions for -j^ , &o.
If K be given as a function of p, q^ &c. and f, we have -f^ , &c. expressed ad
functions of p, q, &c. and t. Joining these equations to those marked (1) we find
Cj, Cj ... as functions of t,
continue thus,
dK
If K be given as a function of c^, c^, ... and t we may
dK dc, dK dc,
dp rfcj dp rfcj dp
dKdKdc^ dK de^
dq ~ dc^ dq dc, dq
Substituting in the expression for -^, we get
dc»
de^ d£g~j dK
dqj dc.
dt Ldq dp ~ dp dq J dcg \_dq dp dp dq.
where the 2 means summation for all values of ^, q, viz. p^, q^, p,, q^, &c.
Since by hypothesis Cj, Cj,... are supposed expressed as functions of pj, q^, &c.
and «, these coefficients may be found by simple differentiation. It will, of course,
be more convenient to express them in terms of Cj, Cj, &c. and t by substituting
for jjj, ji, &c. their values given by the integrals (1).
407. On effecting this substitution it will be found that t disappears from the
expressions. This may be proved as follows. Let A be any coefficient, so that
/1 = S
of2)i, .
"l. Cj,
...(5).
^1, &c.
course,
titutiDg
rom the
BO that
GENERAL EQUATIONS OF MOTION. 321
^ = S I . '- .---/' ^- I * wo have to prove that A heing regarded as a (tmctioQ
d A
of j)j, 2i, &o. and t, the total differential coefficient -* — ia zero. Now
d.A dA dA , dA ,,
dt dp^ dq^
dt
The letters p^, 9j, &o. enter into the expression for A only throngh e^ and e,.
Let us consider only the part of -\ due to the variation of c^, then the part duo
to the variation of c, may be found by interchanging «j and c„ and changing tho
sign of the whole. The complete value of
d.A
d.A
dt
ia the sum of those two parta.
The part of - ' - due to tho variation of c^ ia
\_dp {dq dt dpdq dq dq^ dp "") dq (dp dt dp^ dq dpdq dp *"iJ*
dc
If we substitute for -.^ its value given by the identity (4), we get
I, dp \dp dq* dq dpdq)~ dq \ dp dp dq dq dp* ) J '
If we now interchange c^ and c, we get the same result. Hence when the two
d. A
parts of -~~ are added together, the sigps being opposite, the bu:j1 is zero.
408. Let the expression S ["^ ^ - ^i J?"] , where tho S
means summa-
tion for all the values of p, q, be represented shortly by (Cj, Cj). Then in any
dynamical problem if iT be the disturbing function, the variations of bhe parameters
"i* Cj,
dcj_ ,
dK
dK
are given by -r} = [c^, Cj) ^— + (01,03)^— + ..., where all the coefficients are
dt
dc
dc.
functions of the parameters only and not of (.
This equation may be greatly simplified by a proper choice of the constants
e^, c,, ... In the M^canique Analytique of Lagrange, it is shown that if the con-
stants chosen be tho initial values of pj, j)^, ... and q^, Jt,..., viz. a, /3, 7,... and
\, H, y, ... respectively, then the equations become
da_ _ dK d^_ _ dK
dt~ d\' dt~ dfi'
d\ dK d/i dK
&0.
dt
da' dt
d^'
&c.
It is assumed in the demonstration that iT is a function of 7^, 17,,... only. This
simplification has been extended by Sir W. Hamilton and Jacobi to other cases, but
for this we must refer the reader to books which treat on theoretical dynamics.
409. It foUows from the investigation in Art. 407, that if two integrals of a
dynamical problem be found, viz, Cj=o, Cj=/3, where c^ and c, stand for some
functions of p^, Qi, jpj, q^, ... and t, and a and /3 are oonstaats, then (Cj, Cj) is also
constant. So that (c^, Cj) = 7, where 7 is a constant, is either a third integral of
the equations of motion or an identity. If it is an integral it may be either a
new integral or one derivable from the two c^ and Cj already found.
R. D. • 21
ilH
m
'4
m
322
VIS VIVA.
I
EXAMPLES*.
1. A screw of Archimedes is capable of turning freely about its axis, which is
fixed in a vertical position : a heavy particle is placed at the top of the tube and
runs down through it ; determine the whole angular velocity commimicated to the
screw.
Sesult. Let n be the ratio of the mass of the screw to that of the particle,
a=the angle the tangent to the scrpv' makes with the horizon, h the height
descended by the particle. Then the angular velocity generated is
v:
2(lh cos'' a
a^(/H-l)(n4- sin's aj'
2. A fine circular tube, carrying within it a heavy particle, is set revolving
about a vertical diameter. Show that the difference of the squares of the absolute
velocities of the particle at any two given points of the tube equidistant from the
axis is the same for all initial velocities of the particle and tube.
3. A circular wire ring, carryint^ a small bead, lies on a smooth horizontal
table ; an elastic thread the natural length of which is less than the diameter of
the ring, has one end attached to the bead and the other to a point in the wire ;
the bead is placed initially so that the thread coincides very nearly with a diameter
of the ring ; find the vis viva of the system when the string has contracted to its
original length.
4. A straight tube of given length is capable of turning freely about one ex-
tremity in a horizontal plane, two equal particles are placed at different points
within it at rest, an angular velocity is given to the system, determine the velocity
of each particle on leaving the tube.
5. A smooth circular tube of mass M has placed within it two equal particles
of mass m, which ai i connected by an elastic string whose natural length is f of
the circumference. The string is stretched until the particles are in contact and
the tube is placed flat on a smooth horizontal table and left to itself. Show that
when the string arrives at its natural length, the actual energy of the two particle.^
is to the work done in stretching the string as
2(M^ + Mm + m''):{M+2m){2M+m).
6. An endless flexible and inextensible chain in which the mass for unit length
is fi through one continuous half and jw' through the other half is stretched over
two equal perfectly rough uniform circular discs (radius a, mass M) which can turn
freely about their centres at a distance b in the same vertical line. Prove «hat the
time of a small oscillation of the chain under the action of gravity is
, />/ + (
^W — i
ira+b) (fi + fJ.')
2 {n-fi)g '
7. Two particles of masses »i, in' are connected by an elastic string of length «.
The former is placed in a smooth straight groove and the latter is projected in a
* These examples are taken from the Examination Papers which have been set
in the University and in the Collegesi.
('
^of
length a.
loted in n
been set
EXAMPLES.
323
direction perpendicular to the groove with a velocity V. Prove that the particle m
will oscillate tlxrough a space — , , and if m he large compared with m' the time
of oscillation is nearly
2wa
m + m
8. A rough plane rotates with uniform angular velocity n about a horizontal
axis which is parallel to it but not in it. A heavy sphere of radius a being placed
on the plane when in a horizontal position, rolls down it under the action of
gravity. If the centre of the sphere be originally in the plane containing the
moving axis and perpendicular to the moving plane, and if x be its distance from
tliis plane at a subsequent time t before the sphere leaves the plane, then
2^35
(||'_84«-60c)(eV?"'-e-
v
^nt
, 5 ft . ;
c being the distance from the axis to the plane measured upwards.
9. The extremities of a imiform heavy beam of length 2a slide on a smooth
wire in the form of the curve whose equation is r=a (1-cos^) the prime radius
being vertical and the vertex of the curve downwards. Prove that if the beam
be placed in a vertical position and displaced with a velocity just sufficient to
1 ( iJ^ I iJ^i
bring it into a horizontal position tan^=- ]e''2a - p-^ •!«.'
through which the rod has turned after a time t.
2a - e~^ ao ' I , where B is the angle
10. A rigid body whose radius of gyration about G the centre of gravity is Ic, is
attached to a fixed point C by a string fastened to a point A on its surface. CA{=b)
and AG{=a) are initially in one line, and to G is given a velocity V at right angles
to that line. No impressed forces are supposed to act, and the string is attached
so as always to remain in one right line. If be the angle between A G and A C
at time t, show that
it
m
y^k'^-iabain^^
b^
2 sin"
2sl^
be very small, the period is
2nbk
and if the amplitude of 0, i.e.
V^a{a + b)'
11. A fine weightless string having a particle at one extremity is partially
coiled round a hoop which is placed on a smooth horizontal plane, and is capable
of motion about a fixed vertical axis through its centre. If the hoop be initially at
rest and the particle be projected in a direction perpendicular to the length of the
string, and if s be the portion of the string unwound at any time t, then
'm + ix
rH' + 2Vat,
where b is the uiitial value of s, m and n the masses of the hoop and particle, a the
radius of the hoop and V the velocity of projection.
12. A square formed of four similar uniform rods jointed freely at their ex-
tremities is laid upon a smooth horizontal table, one of its angular points being
fixed : if angular velocities w, u' in the plane of the table be communicated to the
two sides containing this angle, show that the greatest value of the angle (2a)
5(w--wr
■« u^ + u' *
21—2
between them is given by the equation cos 2a
ill
1
i!!
;lf
i 1 '
\
\ ;
324
VIS VIVA.
18. Two particles of masses m, m/ lying on a smooth horizontal table are con*
nected by an inelastic string extended to its full length and passing through a small
ring on the table. The particles are at distances a, a' from the ring and are pro-
jected with velocities v, v' at right angles to the string. Prove that if »iv'a'=mW'
their second apsidal distances from the ring will be a', a respectively.
14. If a imiform thin rod PQ move in consequence of a primitive impulse
between two smooth curves in the same plane, prove that the square of the angular
Telocity varies inversely as the difference between the sum of tho squares of the
normals OP, OQ to the curves at the extremities of the rods, and ^^ of the square
of the whole length of the rod.
15. A small bead can slide freely along an equiangular spiral of equal mass
and angle a which can turn freely about its pole as a fixed point. A centre of
repulsive force F is situated in the pole and acts on the particle. If the system
start from rest when the particle is at a distance a, show that the angular velocity
of the spiral when the particle is at a distance i from the pole is
where mJc^ is the moment of inertia of the spiral about its pole.
m*2{l + 2cot''a)
16. The extremities of a uniform beam of length 2a, sUde on two slender rods
without inertia, the plane of the rods being vertical, their point of intersection
fixed and the rods inclined at angles j and - j t;o the horizon. The system is set
rotating about the vertical line through the point of intersection of the rods with
an angular velocity u, prove that if be the inclination of the beam to the vertical
at the time t and a the initial value of d
m-
(3cos'a + Bin''a) »
3cos''tf + Bin''^
> 6(7
w'= (3oos'a + sin'o)w' + — (sino-sintf).
17. A perfectly rough sphere of radius a is placed close to the intersection of
the highest generating lines of two fixed equal horizontal cylinders of radius c the
axes being inclined at an angle 2a to each other, and is allowed to roll down be-
tween them. Prove that the vertical velocity of its centre in any position will be
sin a cos </>
jlOgr (a +c)(l- sin «/>))*
I 7-5coB'»0cos''ar"
, where <p is the inclination to the horizon of tho
radius to the point of contact.
d^x dT
18. Let a complete integral of the equation -^ = -j- in which T ia & function
at (t'30
otxhex=X, X being a known function of a and b two arbitrary constants and t.
d'^x
Then the solution of ^ — . • , ,
+ , , R being a function of x may also be repre-
sented by a; =X provided a and b are variable quantities determined by tho equations
J - = A; :jr I J, = - ft -r- 1 where k is a function of a and 6 which does not contain the
dt db dt da
time explicitly.
4
moti
or so
the (
wher
the s;
the S(
4]
and ii
puttii
form
ys
whew
the ir
of th
motio
exprci
CHAPTER VIII.
ON SMALL OSCILLATIONS.
Oscillations with one degree of freedom.
410. When a system of bodies admits of only one independent
motion and is making small oscillations about some mean position,
or some mean state of motion, it is in general our object to reduce
the equation of motion to the form
d^ce , dx ,
where co is some small quantity which determines the position of
the system at the time t. This reduction is effected by neglecting
the squares of the small quantity x.
411. It will generally happen that a, h, c are all constants,
and in this case we can completely determine the oscillation. By
c -
putting a! = T + ^e~i , we reduce the equation to the well known
form
iH-f)f=o-
a'
When h — -j-is positive, we therefore have
at
x = r + -^e" a sin
{^MnB}.
where A and B are two undetermined constants which depend on
the initial conditions of the motion. The physical interpretation
of this equation is not difficult. It represents an oscillatory
277-
motion. If we write for t, t + . , we have exactly the same
expression for x with A^ written for J, where A^r=Ae v^-V, wo
: 1
r
\ -
I i
\'
■ '■ I
,^26 SMALL OSCILLATIONS.
therefore infer that the time of a complete oscillation is
27r
The central position about which the system oscillates is deter-
rained by a; = v . To find the times at which the system comes
momentarily to rest we put -^7 = 0. This gives
dt
*^K\/^-T^+^^)
^4>h
a
a
The extent of the oscillations on each side of the central position
may be found by substituting the values of t given by this equa-
tion in the expression for x — j. Since these must occur at a
constant interval equal to
IT
y-?
we see that the extent of the
oscillation continually decreases, and that the successive arcs on
each side of th:^ position of equilibrium form a geometrical pro-
ait
gression whose common ratio is e ^i'>-a'.
a
If 5 — -J is negative, the sine must be replaced by its expo-
nential value, and the integral becomes
X
.. ^,/-|.Vf.).^^/-«-V.%)._
where A' and B' are two undetermined constants. The motion is
now no longer oscillatory. If a and h are both positive, or if the
initial conditions are such that the coefficient of the exponential
which has a positive index is 2ero, x will ultimately become equal
to r and the system will ultimately continually approach the
position determined by this value of x.
a'
If J — — = 0, the integral takes a different form and we have
aj = |+(^"< + 5")e-?,
where A" and B" are two undetermined constants. If a is
positive, the system will ultimately continually approach the
position determined by a; = r .
ONE DEGREE OF FREEDOM.
327
When the value of x as given by these equations becomes
large, the terms depending on a;* which have been neglected in
forming the equation may also become great. It is possible that
these terms may alter the whole character of the motion. In
such cases the equilibrium, or the undisturbed motion of the
system as the case may be, is called unstable, and these equations
can represent only the nature of the motion with which the
system begins to move from its undisturbed state.
d^x dx
Ex. Show that the complete solution of -j-j + a-Tr + bx =/(«) is
x=e
hat I .Bin 67 , ,,, a . ,,, ) 1 /*« -%{t-t') . ,,,, .,..,.,. ,,
^ ' ~ ,' -^,— + .^0 (cos b't + 2^, sm i") + j, j « sm 6' (« - 1') / (f) dt',
a? d"
where i'* = 6 - — and oja, x'a ai*o the values of x and - ,- "hs^" c -: 0.
[Math. Tripos, 1876.]
412. It will be often found advantageous to trace the motion
of the system by a figure. Let equal increments of the abscissa
of a point P represent on any scale equal increments of the time,
and let the ordinate represent the deviation of the co-ordinate x
from its mean value. Then the curve traced out by the repre-
sentative point F will exhibit to the eye the whole motion of the
In the case in which a and h — — are both positive the
system.
curve takes the form
4
i; t'
\'\
» :: i ;
at
The dotted lines correspond to the ordinate ±Ae ^ . The repre-
sentative point P oscillates between these, and its path alternately
touches each of them. In just tiie same way we may trace the
representative curve for other values of a and b.
I'M
328
BMALL OSCILLATIONS.
The most important case in dynamics is when a = 0. The
motion is then given by
x-l=±Asm(^/bt + B).
The representative curve is then the curve of sines. In this
case the oscillation is usually callea harmonic.
'i 1
418. Ex. 1. A system oscillates about a r^ean position, and its deviation is
measorcd by x. If x^ and Xq be the initial valacs of c and ^- , show tbe system
will never deviate from its m«an position by so much as f ? 'r\ 9 P if 4&
is greater than a\
Ex. 2. A system oscillates about a position of equilibrium. It is required to
find by observations on its motion the numerical values of a, h, c.
Ajiy three determinations of the co-ordinate x at three different times will gene-
rally supply sufficient equations to find a, b, c, but some measurements can be
made more easily than others. For example the values of x when the system
comes momentarily to rest can be conveniently observed, because the system is
then moving slowly and a measurement at a time slightly wrong will cause an
error only of the second order, while the values of t at such times cannot be con-
veniently observed, because, owing to i;he slowness of the motion, it is diiHcult to
dx
determine the precise moment at which — vanishes.
If three successive values of x thus found be x-^, «j, x^, the ratio ©I th« tw« suc-
cessive arcs x^-Xi and x^ - x, is a known function uf a and h and one equation can
thus be formed to find the constants. If the position of equilibrium is unknown,
c c
we may form a second equation from the fact that the three arcs a^-r, a^-ri
x^-rr also form a geometrical progression.
In this way we find t whieh ffi the
value of X corresponding to the position of equilibrium.
The position of equilibrium being known, the interval bet ffeen two successive
passages of the system through it is also a known function of a and &> and thui) a
third equation may be formed.
Ex. 3. A body performs rectilinear vibrations in a medium whose resistance is
proportional to the velocity, under the action of an attractive force tending towards
a fixed centre and proportional to the distance therefrom. If the observed period
of vibration is T and the co-ordinates of the extremities of three consecutive semi-
vibrations arep, q, r; prove that the co-ordinate of the position of equilibrium and
the time of vibration if there were no resistance are respectively
f^S,r''V*^^{'<^^V ■
[Math. Tripos, 1870.]
ONE DEGREE OF FREEDOM.
829
414. When the coefiSoients are functions of the time, the equation can be
integrated only by some artifice suited to the particular case under consideration.
Let the equation be
d*x dx
then a few nseful methods of solution will be indicated in the following examples.
Ex. 1.
n^ I dp
If g-j-g-^isa positive constant, viz. n', prove that the successive
oscillations of the system will bo performed in the same time, though the extent of
the oscillations may follow any law.
This may be proved by clearing the equation of the second term in the usual
way, i.e. put x=^e-^-^'^'
o d 1
Ex. 2, If r=0 and -.z. - y. — r=a, where a is a constant> prove that
V2 ''^ Vg
x^e'^^y^^A sin j ^1 - jf*s/qdt + b\ .
Thence show that if / i^Jqdt does not become infinite, the time of oscillation is
independent of the arc of oscillation but the successive oscillations are not per-
formed in the same time.
This may be proved by writing «=^(\), and then so choosing the form of \p
that the coefficient of x in the differential equation becomes unity or some constant.
Ex. 3. A system oscillates about a position of equilibrium and its motion is
determined by the equation ■-^ + qx-0, whore g is a known function of t, which
during the time under consideration always lies between /S' and /y, the latter being
the greater. If the system be started with an initial co-ordinate tK, and an initial
velocity Xq in a direction away from the position of equilibrium, show that the
system will begin to return before x becomes so great as */ x^' + "^j-. If ± »i,
TWl
^.
be two ^ccesfiive maximum values of st^ prove that m' cannot be bo great as ^- m,
and that the time trom one maximum to the next lies between ^ and -^ •
P P
415. When the arc of oscillation is not small, the equation cannot always be
reduced to the linear form, and no general rule can be given for its solutiooy In
many caaes it is important to ascertain if the mo ion of the system is tautochro-
nous. Various methods of determining this will be shown iu the following
examples.
Ex. 1. Show that if the equation of motion be
dt«
= ( a homogeneous function of -j- and x of the first degree) ,
then, in whatever position the system is placed at rest, the time of arriving at the
position detei mined by iii;=0 is the same.
• i
I;
1: >
\'l
i.
■
1
;
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1 1
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1 '
I i^^;
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I ill
^ - A.']
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11
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330
SMALL OSCILLATIONS.
Let the homogeneous function be written ^f {--/}] • ^^^ ^ ^'^^ f ^^ t^^ co-
ordinates of two systems starting from rest in two different positions, and let a! = a,
f = (ca initially. It is easy to see that the differential equation of one system is
changed into that of the other by writing | = Kir. If therefore the motion of ono
system is given by x — <p{t, A^ B), that of the other is given by ^—K4>(t, A', B').
To determine the arbitrary constants, A, B and A', B', we have exactly the same
conditions, viz. when t = 0, tj> = a and ~- ■■
=0. Since only one motion can follow
from the same initial conditions wo have A'=A, and B'—B. Hence throughout
the motion }^ — kx and therefore as and { vanish together. It follows that the
motions of the two systems are perfectly similar.
This result may also be obtained by integrating the differential equation. If we
put ";^=p, we find x=A(t>{t + B). When ( = 0, -;,=0, and therefore 0'(«) = O.
30 (tt etc
Thus B is known and x vanishes when (p(t + B) = whatever be the value of A .
Ex. 2. If the equation of motion of the system bo
= - ( -r^ I >V r + ( ^ homogeneous function of — and/(.c) of the first degree j,
where /(«) is any function of x, show that in whatever position the system is placed
the time of arriving at the position determined by a; = is the same.
This is Lagrange's general expression for a force which makes a tautochronoua
motion. The formula was given by him in the Berlin Memoirs for 17C5 and 1770.
Another very complicated demonstration was given in the same volume by D'Alem-
bert, which required variations as well as differentiations. Lagrange seems to
have believed that his expression for a tautochronous force was both necessary and
sufficient. But it has been pointed out by M. Fontaine and M. Bertrand that
though sufficient it is not necessary. At the same time the latter reduced the
demonstration to a few simple principles. A more general expression than
Lagrange's has been lately given by Brioschi.
In practice it will be more convenient to apply Bertrand's method than
d'X I (2'jc\
Lagrange's rule. Suppose the equation of motion to be - = i^* f ar, y J . Put
x—ipiy) and if possible so choose the form of <(>, that "^ becomes a homogeneous
function of y and ~ of the first degree. If this can be done, the motion is, by Ex. 1,
tautochronous.
Ex. 3. If the motion of any system is tautochronous according to Lagrange's
formula in vacuo, it will also be tautochronous in a resisting medium, if the effect
of the resistance is to add on to the differential equation of motion a term propor-
tional to the velocity. This theorem is due to Lagrange.
Ex. 4. A particle, acted on by a repulsive force varying as the distance and
tending from a fixed point, is constrained to move along a rough curve in a medium
rorsisting as the velocity, find the curve that the motion may be tautochronous by
Lagrange's rule.
Let V bo the velocity, s the arc to be described, r the radius vector of the
particle, it the perpendicular on the tangent, p the radius of curvature. Let ar bo
FIRST METHOD OF FORMING THE EQUATIONS OF MOTION. 331
the repulsive force, b the coeflSoient of friction. Then omitting the resistance by
Ex. 3, the equations of motion are
j^-ap + It J
Eliminating the pressure It, we have
-^^ = h~ + ahp-ajr^-p\
By Lagrange's rule, the motion is tautochronous if, when f{s) = ab2>- a Jr'^-p'*, we
b f (s)
find - = - -J— I . This will be found to give p = {l + b')p, which is on epicycloid.
( I
' .{ 1
First Method of forming the Equations of Motion.
416. When the system under consideration is a single body,
there is a simple method of forming the equation of motion which
is sometimes of great use.
First, let the motion be in two dimensions.
It has been shown in Art. 175, that if we neglect the squares of
small quantities we may take moments about the instantaneous
centre as a fixed centre. Usually the unknown reactions will be
such that their lines of action will pass through this point, their
moments will then be zero, and thus we shall have an equation
containing only known quantities.
Since the body is supposed to be turning about the instan-
taneous centre as a point fixed for the moment, the direction of
motion of any point of the body is perpendicular to the straight
line joining it to the centre. Conversely when the directions of
motion of two points of the body are known, the position of the
instantaneous centre can be found. For if we draw perpendiculars
at these points to their directions of motion, these perpendiculars
must meet in the instantaneous centre of rotation.
The equation will, in general, reduce to the form
Mh'^ (/*^ _ /moment of impressed forces about\
dt^* \ the instantaneous centre / '
where is the angle some straight line fixed in the body
makes with a fixed line in space. In this formula Mk^ is the
moment of inertia of the body about the instantaneous centre,
and since the left-hand side of the equation contains the small
factor -j.^ we may here suppose the instantaneous centre to have
i I
) !
^i
332
SMALL OSCILLATIONS.
its mean or undi&turbed position. On the right-hand side there is
710 small factor, and we must therefore be careful either to take
the moment of the forces about the instantaneous centre in its
disturbed position, or to include the moment of any unknown
reaction which passes through the instantaneous centre.
Ex. If a body with only one independent motion can be in eqnilibriam in
the same position under two different syBtems of forces, and if Lj, L, are the
lengths of the simple equivalent pendulums for these systems acting separately,
then the length L of the equivalent pendulum when they act together is given by
111
417. Ex. A homogeneous hemisphere performs small oscillations on a perfectly
rough horizontal plane : find tJie motion.
Let C be the centre, O the centre of gravity of the hemisphere, N the point of
contact with the rough plane. Let the radius = a, CG=c, 0=^ NCO.
Here the point N is the centre of instantaneous rotation, because the plane
being perfectly rough, sulBcient friction will be called into play to keep N at rest.
Hence taking moments about N
{k^ + GN*)'l^^ = - go. Bine.
Binoe we can put GN=a-e in the small terms, this reduces to
{i? + (a-ty]^,+gt.0=O.
illation is = 2ir a. / ^'- ,
^ eg
Therefore the time of a small oscillation
2 g
It is clear that fc' + c' = (rad.)' of gyration about C= -= a' and c = ^a.
o o
If the plane had been smooth, M would have been the instantaneous axis, GM
being the perpendicular on CN. For the motion of iV is in a horizontal direction,
because the sphere remains in contact with the plane, and the motion of is
vortical by Art. 79. Hence tho two perpendiculars GM, NM meet in the instanta-
neous axis. By reaseniug similar to the above the time will be found to bo
^ eg
418. A cylindrical surface of any form rests in stable equi-
librium on another perfectly rough cylindrical surface, the axes
1,
riRST METHOD OP FORMING THE EQUATIONS OF MOTION. 383
of the cylindera being parallel. A small disturbavce being given
to the upper surface, find the time of a small oscillation.
Let BAP, B'A'P be the sections of the cylinders perpendicular
to their axes. Let OA, CA! be normals at those points -4, A'
which before disturbance were in contact, and let a be the angle
A makes with the vertical. Let OPG be the common normal
at the time t. Let Q be the centre of gravity of the moving body,
then before disturbance A'O was vertical Let AQ=.r.
Now we have only to determine the time of oscillation when
the motion decreases without limit. Hence the arcs AP, A'P will
be ultimately zero, and therefore C and may be taken as the
centres of curvature of AP, A'P. Let p = OA, p = CA', and let
the angles A OP, A' CP be denoted by 0, <f>' respectively.
Let d be the angle turned round by the body in moving from
the position of equilibrium into the position B'A'P. Then since
before disturbance, A'G and AG were in the same straight line,
we have = ^ CDE= <f> + <f>', where GA' meets OAE in 3. Also
since one body rolls on the other, the arc J.P=arc^'P, .•. p^=p'^',
-,e.
P+P
Again, in order to take moments about P, we require the
horizontal distance of Q from P; this may be found by projecting
the broken line PA' +A'G on the horizontal. The projection of
PA' = PA' cos {oL + 6) = p(j} cos a when we neglect the squares of
small quantities. The projection of A'G is rd. Thus the hori-
zontal distance required is [ ——> cos a - r j ^.
1
H
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II
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334
SMALL OSCILLATIONS.
If k be the radius of gyration about the centre of gravity, the
equation of motion is
(i'+0^')^f = -^s(V^,cosa-r).
If L be the length of the simple equivalent pendulum, we
have
L p + p
-, cos a—r.
Along the common normal at the point of contact A of
the two cylindrical surfaces measure a length A8 = s where
- = - + -, and describe a circle on AS as diameter. Let -4^,
^ P P.
produced if necessary, cut this circle in N. Then GN= s cos a — r,
the positive direction being from N towards A. The length L of
the simple equivalent pendulum is given by the formula
k' + GA'
= GK
It is clear from this formula, if G* lie without the circle and
above the tangent at -4, X is negative and the equilibrium is
* Let Jl be the radius of curvature of the path traced out by G as the one
cylinder rolls on the other, then we know that R= - . f >, . so that all points with-
NG
out the circle described on AS as diameter are describing curves whose concavity is
turned towards A, while those within the circle are describing curves whose con-
vexity is turned towards A. It is then clear that the equilibrium is stable,
unstable, or neutral, according as the centre of gravity lies within, without, "or on
the circumference of the eircle.
FIRST METHOD OF FORMING THE EQUATIONS OF MOTION. 335
unstable, if within L is positive and equilibrium is stable,
circle is called the circle of stabiliti/.
This
419. It may be noticed that the prccodinp; problem is perfectly
general and may be used in all cases in wliich the locus of the
instantaneous axis is known. Thus p is the radius of curvature of
the locus in the body, p that of the locus in space, and a the incli-
nation of its path to the horizon.
If dx be the horizontal displacement of the instantaneous centre
produced by a rotation cW of the body, then the equation to find
the length of the simple equivalent pendulum of a body oscillating
under gravity may be written
L dd
This follows at once from the reasoning in Art. 418. It may also
be easily seen that the diameter of the circle of stability is equal
to the ratio of the velocity in space of the instantaneous axis to
the angular velocity of the body.
Ex. 1. A homogenoous sphere makes small oscillations inside a fixed sph tg bo
that its centre moves in a vertical piano. If the roughness be sufficient to prevent
all sliding, prove that the length of the equivalent pendulum is seven- fifths of the
difference of the radii. If the spheres were smooth the length of the equivalent
pendulum would be equal to the difference of the radii.
Ex, 2. A homogeneous hemisphere being placed on a rough fixed plane, which
is inclined to the horizon at an angle sin~i — ;- , makes small oscillations in a
vertical plane
equivalent pendulum is (
2^/2
Shew that, if a is the radiui of the hemisphere, the length of the
46 J^'
5 ~ 4
i).
420. If the body be acted on by any force which passes
through the centre of gravity, the results must be slightly modi-
fied. Just as before the force in equilibrium must act along the
straight line joining the centre of gravity O to the instantaneous
centre A. When the body is displaced the force will cut its
former line of action in some point F, which we shall assume to be
known. Let -4i*^=/, taking / positive when O and F are on
opposite sides of the locus of the instantaneous centre. Then it
may be shown by similar reasoning, that the length L of the
simple equivalent pendulum under this force, supposed constant
and equal to gravity, is given by
1^ + r^ pp fr
^ P + P /+»•
where a is the angle the direction of the force makes with the
normal to the path of the instantaneous centre.
■•
1 i
1 ;
if,.
1.
336
SMAIL OSCILLATIONS.
If we measure along the line AG & length AO' so that
111
-T-Ty, = j-75 + -jc'> tlisn the expression for L takes the form
i" + r»
= G'iV.
^1
■I f
The equilibrium is therefore stable or unstable according as G'
lies within or without the circle of stability.
421. Two points k, "& of a body are eonstrained to describe given curves, and
the body ia in equilibrium under the action of gravity. A small disturbance being
given, find the time of an c <cillation.
Let C, D be the centres of curvature of the given curves at the two points A, B.
Let AC, BD meet in 0. Let be the centre of gravity of the body, GE a perpen-
dicular on AB. Then in the position of equilibrium OG is vertical. Let i, j be
the angles CA, BD make with the vertical, and let a be the angle AOB. Let
A', ^'...denote the positions into which vl, 5... have been moved when the body haa
been turned through an angle 0. Let ACA' = <f), BJ)B'=<j>'. Since the body may
be brought from the position AB into the position A'B' by turning it about
through an angle 6, we have -^ '- = ~-?r~- = 0. Also GG' is ultimately perpen-
(JA V-tS
dicular to 00, and we have GG' = 0G,6i. Also let x, y be the projections of 00' on
the horizontal and vertical through 0, Then by projections
X cos j + y sin j = distance of 0' from OD = OD . <p',
h
.1 'J
sccost-y sin i— distance of 0' from OC=OC. <f>;
0D.suii.<l)'+0C.nmj.(p
, '. x~
BUI a
FIRST METHOD OF FORMINQ THE EQUATIONS OF MOTION. 337
Now taking raomeuts about 0' as the centre of instantaneous rotation, we have
(l-« + OG«)|^=-j;.(GG'+a!)
.( .„ OD. OB Bin i OC.OAfimj\
= -90 [0G+ -^^- . ^- + - ^^ - ^ ,
where k is the radius of gyration about the centre of gravity.
Hence if L be the length of the simple equivalent pendulum, we have
k^+OG' OD.OB Bini , OC.OA sinj
L BD sin a AC sin a
Cor. If the given curves, on which the points A, B are constrained to move, bo
straight lines, the centres of curvature C and D are at infinity. In this case, wo
may put r,-^= - 1| 77,= - 1. and the expression becomes
IS Lf ACf
L Bina sin a
If OA and 0^ be at right angles, this takes the simple form
^'^^-^' = 00.20F,
where F is the projection on OG of the middle point of AB.
422. A body oscillates about a position of equilibrium under the action of
gravity, the radius of curvature of the path of the centre of gravity being known,
find the time of oscillation.
Let A be the position of the centre of gravity of the body when it is in its
position of equilibrium, G the position of the centre of gravity at the time (. Then
since in equilibrium the altitude of the centre of gravity is a maximum or mini-
mum, the tangent at A to the curve AG is horizontal. Let the normal GC to the
' i''!l
Si
cm've at G meet the normal at .1 in V. Then when the oscillnlion becomes iudpll-
uitely small 6' is the centre of curvature of the curve at A. Lot AG — t, the angle
ACG=\f/, and lot R be the ratUus of curvature of the curve at A.
Let 6 be the angle turned round by the body in moving from the position of
equilibrium into the position in which the centre of gravity is at (? ; then —- is the
ctt
angular velocity of the body. Since G is moving along the tongent at G, the
R. D. 22
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I It
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838
SMALL OSCILLATIONS.
In'' '
M .'
! i
! U
'I '
if
centre of instantaneous rotation lies in the normal GC, at such a point 0, that
Let ilf ^-^ be the moment of inertia of the body about its centre of gravity, then
taking moments about 0, we have
Now ultimately when the angle is indefinitely small ^= t|'= d ; •'• the
equation of motion becomes
Hence if L be the length of the simple equivalent pendulum we have
li = (l + ^.)^.
423. When the system of bodies in motion admits of only one independent
motion, the time of a small OBcillation may frequently be deduced from the equa-
tion of Vis Viva. This equation will be one of the second order of small quantities,
and in forming the equation it will be necessary to take into account small quanti-
ties of that order. This will sometimes involve rather troublesome considerations.
On the other hand the equation will be free from all the unknown reactions, and
we may thus frequently save much elimination.
The method of proceeding will be made clear by the following example, by
which a comparison may be made with the method of the last article.
Tlie motion of a body in space of two dimensiom is given by the co-ordinates x, y
of its centre of gravity, and the angle which any fixed line in the body makes with
a li7ie fixed in space. The body being in equilibrium under the action of gravity it
is required to find the time of a small oscillation.
Since the body is capable of only one independent motion, we may express (x, y)
as functions of 0, thus
«=F(e), y=f{0).
Let Mk' be the moment of inertia of the body about its centre of gravity, then the
equation of Vis Viva becomes
where C is an arbitrary constant.
Let a be the value of when the body is in the position of equilibrium, and
suppose that at the time t, 0=a + (t>. Then, by M'Laurin's theorem,
y=yo+yo4>+yo"-2 + •••'
where 3/0', y,," are the values of — , -r~ when = a. But in the position of equili-
brium y is a maximum or minimum; .'.yg'^O. Hence the equation of Vis Viva
becomes
■4
>\ ere
;ii
>t 0, that
vity, then
; .-. the
idependent
the equa-
qaautities,
lall quanti-
iiderations.
BtionB, and
sample, by
iinates x, y
makes with
f gravity it
brium, and
I of equili-
[)f Vis Viva
FIRST METHOD OF FORMING THE EQUATIONS OF MOTION. 339
dx
where x^' is the value of -^ when 0=a; differentiating we get
d9
(*o''+*»)^t=-fl^VV-
ir L be the length of the simple equivalent pendulum, we have
where for 9 we are to write its value a after the differentiations have been effected.
It is not difficult to see that the geometrical meaning of this result is the same m
that given in the last article.
This analytical result was given by Mr Holditch, in the eighth volume of tho
Cambridge Tramactiom, It is a convenient formula to use when the motion of the
oscillating body is known with reference to its centre of gravity,
424. When a body moves in space with one independent
motion there is not in general an instantaneous axis. It has,
however, been proved in Art. 186 that the motion may always be
reduced to a rotation about some central axis and a translation
along that axis.
Let I be the moment of inertia of the body about the instan-
taneous central axis, fl the anfrular velocity about it, Fthe velocity
of translation along it, M the mass of the body, then by the prin-
ciple of vis viva ^ ICl" + ^ MV'^ = U+ C, where U is the force-
function, and C some constant. Differentiating we get
dt '^2 dt'^ il dt ~lldt'
Lot L be the moment of the impressed forces about the in-
stantaneous central axis, then L= ^r-r- by Art. 326.
Let p be the pitch of the screw-motion of the body, then
V^pQ. The equation of motion therefore becomes
If the body be performing small oscillations about a position of
equilibrium, we may reject the second and third terms, and the
equation becomes
If there be an instantaneous axis p = 0, and we see that we
may take moments about the instantaneous axis exactly as if it
vere fixed in space and in the body.
22—2
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340
SMAWi OSCILLATIONS.
Mi ;!
425. Ex. A heavy hody oscillates in three dimensions, with one degree of
freedom, on a fixed rouyh surface of any form in such a manner that there is no
rotation about the common normal. Find tlie motion.
(1) Let be the point of contact, Og the common normal, Oy a tangent to the
arc of rolling determined by the geometrical conditions of the question, 01 the
instantaneous axis. Then 01, Oy are conjugate diameters in the relative indi-
catrix.
The relative indicatrix is a conic having its centre at and lying in the com-
mon tangent plane at 0, such that the difference of the curvatures of the normal
sections through any radius vector OR varies as - .-^ .
(2) Let p, p' be the radii of curvature of the normal sections through Oy,
taken positively when the curvatures are in opposite directions, and let - = - + - .
Then s mny be called the radius of relative curvature.
Measure a length « . sin^ yOI along the common normal Oz, and describe a
cylinder on it as diameter, the axis being parallel to 01. If the centre of gravity
of the body be inside, the equilibrium is stable; if outside and above the plane
of xy, imstablo. This cylindei may therefore be called the cylinder of stability.
(3) Let G be the centre of gravity, and let 00 produced out the cylinder of
stability in V; then if K be the radius of gyration about 01, the length L of the
simple equivalent pendulum is given by -^ = GV. sin* GOI. This equation may
also be written in tho form -jr =s coa Goz. Bin' yOI- 00 .sin^ GOI.
M
This result may be obtained by taking moments about the instantaneoiis axia,
TiCt 0' be the point of contact, G' the position of the centre of gravity at the
time t and let 07' be the instantaneous axis. In the small terms we may con-
Hider these as coincident with 0, G and 01 respectively. If be the angle turned
round the instantau')ous axis, it may be shown that the arc 00' rolled over is
Os^inyOI. Let this be called <r. To find the moment of tbo weight we resolve
gravity parallel and perpendicular to 0'/'. The former may be neglected, the
latter is ff sin GOI'. Let this force act parallel to seme line KO. The moment
ri'i [Hired is the product of resolved gravity into the difference of the projections
of 00' and OG' on a plane through OT perpendicular to KO. The projection of
tlio former is o-sinyO/cos A'O^. The projection of the latter is tf . 0(? . sin (?07.
The result then follows by the same reasoning as in Art. 418.
(4) The motion of the upper body is the same as if the fixed surface were plane
and the curvature of the upper body at the point of contact altered so that the
rolav'.ve indicatrix remain the same as before. This supplies an easy method of
liiiding the oscillations Ln any particular case.
426. Ex. A heavy cone of any form oscillates on a fixed rough conical surface,
tlui vertices being coincident. Let bo the common vertex, 01 the lino of contact
ill the position of equilibrium, the centre of gravitj'. Let A' be the radius of
pyvation about 01, z= inclination of 01 to the vortical measured in the direction
mposite to gravity. Let OG~h, and tlie angle GOJ — r. Let n bo the inclination
'A ihe vertical plane GOI to the normal plane to the two cones along 01. Let p, p
•ugh Oy,
1 1
= - + -,.
P P
SECOND METHOD OF FORMING THE EQUATIONS OF MOTION. 341
be the semi-angles of the two right oironlar osculating con69 of contact along 01
taken positively when the curvatures are in opposite directions. Then the length
L of the simple equivalent pendulum is given by
K' . , , sin o sin p'
r-=- = sm (z - r) cos n .— 7 7r-binrsinz.
If the upper body be a right cone of semi-angle p, and if it be on the top of any
conical surface, the preceding expression takes the form
IP _ Bm {z+p')Bia.'p
hL~ sin (p + p')
Second Method of forming the Equations of Motion.
427. Let the general equations of motion of all the bodies be
formed. If the position about which the system oscillates be
known, some of the quantities involved will be small. The squares
and higher powers of these may be neglected, and all the equations
will become linear. If the unknown reactions be then eliminated
the resulting equations may be easily solved.
If the position about which the system oscillates be unknown,
it is not necessary to solve the statical problem first. We may by
one process determine the positions of rest, ascertain whether they
are stable or not, and find the time of oscillation. The method of
proceeding will be best explained by an example.
428. Ex. The ends of a uniform heavy rod AB of length 21
are constrained to move, the one along a horizontal line Ox, and the
other along a vertical line Oy. If the tvhole system turn round Oy
with a uniform angular velocity «, it is required to find the ])osi-
tions of equilibrium and the time of a small oscillation.
Let X, y be the co-ordinates of G the middle point of tlio
rod, 6 the angle OAB which the rod makes with Ox. Let li, W
be the reactions at A and B resolved in the plane xOy. Let the
mass of a unit of length be taken as i^e unit of mass.
li
'. V
342
SMALL OSCILLATIONS.
Ml
•I
\6
!
, III
1(1
The accelerations of any element dr of the rod whose co-
ordinatep are (^, rj) are -5J — w'f parallel to Ox,-z-j (^w) parpen-
dicular to the plane xOt/ and — parallel to Oi/.
As it will not be necessary to take moments about Ox, Oy, or
to resolve perpendicular to the plane xOy, the second acceleration
d?P
will not be required. The resultants of the effective forces ~^ dr
and -^ dr, taken throughout the body, are 2Z -^ and 2i -^ acting
.cT^
at G, and a couple 2ll^-^ tending to turn the body round O. The
resultants of the effective forces m^Pdr taken throughout the body
is a single force acting at 6^ = I w' (a? + r cos 6) dr = wV; . 21, and a
couple* round 6^= I w" (oj + r cos ^) r sin ^ <?r = ©' . 2L „ sin (9 cos^,
the distance r being measured from 6^ towards A.
Then we have, by resolving along Ox, Oy, and by taking
moments about Q, the dynamical equations
2/.^=-^ + a)*a;.2i
or
2l.^ = -B+g.
df
21
21
J2Q n
.A;'.-p =jRa; — ^'y — a>'.2L^sin^co8^
.(i).
d^ "" "^ 3
We have also the geometrical equations
x = lcos0j y = lsin0
Eliminating R, K^ from the equations (1), we get
rfV d^x . ,^d^0
df
y'W^^^^l^~9^'~ ^"^y ~ **' 5 sin ^ COS 6
di
dt
.(2).
.(3).
• If a body in one plane be turning about an axis in its own plane with an
angular velocity w, a general expression can be found for the resultants of the
centrifugal forces on all the elements of the body. Take the centre of gravity G as
origin and the axis of y parallel to the fixed axis. Let c be the distance of G from
the axis of rotation. Then all the centrifugal forces are equivalent to a single
resultant force at G
=/(ir' (c + x) dm - ul^ . Mc, since 5 = 0,
and to a single resultant couple
=/«' (e + x) ijdw., = uPJxydm, since y = 0.
?
SECOND METHOD OF FORMINO THE EQUATIONS OP MOTION. 343
To find the position of red. We observe that if the rod were
placed at rest in that position it would always remain there, and
that therefore •;r^ = 0, -j^ = 0, "rj=0. This
dt
df
gx — (axy
de
gives
0)* s sin ^ cos ^ = 0.
.(4).
Joining this with equations (2), we get ^ = ^ , or sin 6 =-t^j,
and thus the positions of equilibrium are found. Let any one of
these positions be represented by ^ = a, a: = a, y = 6.
To find the motion of oscillation. Let x = a + x\ y — h-\-i/,
6 = 0. + ff, where x, y', & are all small quantities, then we must
substitute these values in equation (3). On the left-hand side
since -^ , — ^, -^ , are all small, we have simply to write a, h, a,
for X, y, 6. Ou the right-hand side the substitution should be
made by Taylor's Theorem, thus
da
db
dx
We know that the lirst term f{a, h, a) will be zero, because
this was the very equation (4) from which a, h, a were found.
We therefore get
^'^~^Tfi^'^~M='^-' ^^) ^ " ^^li - w Q cos 2a . d .
de
de
de
3
But by putting ^ = a + ^ in equations (2), we get by Taylor's
Theorem as' = -- i sin a . ^, y' = ^ cos a . ff.
Hence the equation to determine the motion is
(P + T^)~ + {ghina + | oiT cos 2a) ^ = 0.
4
Now, if gl sin a + K w'^ cos 2r = w be positive when either of the
o
two values of a is substituted, that position of equilibrium is stable,
and the time of a small oscillation is 27r a/ — — .
If n be negative the equilibrium is unstable, and there can be
no oscillation.
li(o*>~ there are two positions of equilibrium of the rod. It
will be found by substitution that the position in which the rod is
inclined to the vertical is stable, and the other position unstable.
If
"
! .
II
i ■ ;
I ■ :m
m
- »t, u m--m»-^
ir
344
SMALL OSCILLATIONS.
1/1
If ©' < -^ the only position in which the rod can rest is vertical,
and this position is stable.
If n = 0, the body is in a position of neutral eqailibrium. To
determine the small oscillations we must retain terms of an order
higher than the first. By a known transformation we have
de y d('~dt V dtj'
jia
Hence the left-hand si'^a of em itioi (3) becomes (i' + A;') t,j .
The right-hand side becom, *^^ • tylor's Theorem
50" (^'
cos a —
sm
1.2
+ &C.
TT
When M = 0, we have a = ^ and <»' = -^ . Making the neces-
sary substitutions the equation of motion becomes
Since the lowest power of & on the right-hand side is odd
and its coefficient negative, the equili' ,/ium is stable for a displace-
ment on either side of the position of equilibrium. Let a be the
initial value of ff , then the time T of reaching the position of
etiuilibrium is
j0L*-d'
put ^ = a^, then
V gl 'J,jl-(b*'a'
Jl -<}>*
Hence the time of reaching the position of equilibrium varies
inversely as the arc. When the initial displacement is indefi-
nitely small, the time becomos infinite.
Thia definite integral may be otherwise expressed in terms of the Gamma
.04
function. It raay be easily shown that
Jo 'Ji-'P*
4n/27i
429. This problem might have been easily solved by the
first method. For if the two perpendiculars to Ox, Oy at
A and B meet in N, N is the instantaneous axis. Taking mo-
ments about Nf we have the equation
e neccs-
OSCILLATIONS WITH TWO OR MORE DEGREES OF FREEDOM. 345
(f + k') -~ = gl COS ^ _ J^ «« (i + ry
4P
sin 6 cos 6
dr
= gl cos ^ — -^ to' sin ^ cos ^
=f(e). .
Then the position of equilibrium can be found from the equa-
tion /(a) = Oand the time of oscillation from the equation
^,^,^^o;^m,_
df doL
430. Ex. 1. If the mass of the rod AB is M show that the magnitude of the
couple which constrains the system to turn round Oy with uniform angular velocity is
Would the magnitude of this couple be altered if Ox or Oy had any mass ?
Ex. 2. The upper extremity of a uniform beam of length 21 is constrained to
elide on a smooth horizontal rod without inertia, and the lower along a smooth
vertical rod through the upper extremity of which the horizontal rod passes : the
system rotates freely about tha vertical rod, prove that if a be the inclination of the
beam to the vertical when in a position of relative equilibrium, the angular velocity
of the system will be ( /r— "^ — ) . m^^ if t^e beam be slightly displaced from this
position show that it will make a small oscillation in the time
47r
-y (sec o+ 3 cos a) I
4
[Coll. Exam.].
In the example in the text the system is constrained to torn round the vertical
with uniform angular velocity, but in this example the system rotates freely. The
angular velocity about the vertical is therefore not constant, and its email variations
must be found by the principle of angular momentum.
^-ti-f
I' I : I
i' ') jl
, ,1 I
Oscillations with two or more Degrees of Freedom.
431. When the position of a system of bodies depends on
several independent co-ordinates, the equations to determine the
motion become rather complicated. In order to separate the
difficulties of analysis from those of dynamics, we shall consider
the case in which the system depends on two independent co-
ordinates, though the remarks about to be made will be for the
most part quite general, and will apply, no matter how many
co-ordinates the system may have. In the sequel we shall con-
sider Lagrange's general method of forming the equations when
the system has ?i co-ordinates.
Iriiii-I
i.i
, f
1 I
11
I, I
346
SaiALL OSCILLATIONS.
432. The equations of motion of a dynamical system per-
forming small oscillations with two independent motions are of
the form
„cPx
df
dt
de
dt
A'-.^ + B'^+C'x + F'^UG'^^+H'y = 0.
dt
de
dt
To solve these, we eliminate either a? or y ; \i D stand for
we have
dt'
AD" +BD + C, FI)'+ GD + H
A'I/ + B'D+C\ rD^+O'D+H'
x = 0,
with a similar equation for y. If AB stand for the determinant
A B . .
a' jy this biquadratic becomes, when x is omitted.
AFD*+(A G+BF)D'+(AII+Ba+CF)D'+{BH+CG) D+GH=0.
If the roots of this biquadratic be m^, m^, m^, m^, we have by
the theory of Linear Differential Equations
X = Mj^^^* + M^el^ii + M^€l^>^ + M^e"^**,
where J/,, M^, Jfg, M^ are arbitrary constants. Similarly we have
The ilf's are not independent of the Jf s, for by substituting in
either differential equation and taking any M and M as typical
of all,
{Am^ + Bm+C)M^--{F'm? + Gm-\-E)M'.
There are therefore just four arbitrary constants, and these are to
be determined by the initial values oi x, y, -jj , -^ .
433. If the position of the system depends on three indepen-
dent co-ordinates x, y, z, we shall have three equations of motion
similar to the two at the beginning of this article. These may be
solved in the same way. In this case we obtain a subsidiary equa-
tion of the sixth degree to determine the exponentials which
occur in the variables. The relations between the coefficients of
corresponding exponentials can be, found by substitution in any
two of the equations of motion.
In certain cases it may be more convenient to choose x or y
to be itself a differential coefficient of a co-ordinate. In this case
the biquadratic or sextic equation will reduce to a cubic or
quintic.
A
dt'
OSCILLATIONS WITH TWO OR MORE DEGREES OP FREEDOM. 347
' 434. It appears from this summary that the character of the
motion depends on the forms of the roots* of this biquadratic.
* If the general character of the motion is required it will be necessary to
analyse the biquadratic. Rules by which this is made to depend on a cubic
equation are given in most of the books on the theory of equations, but aa the final
results are not stated, it will be useful to give here a short analysis for reference.
Let the biquadratic be
ox* + 46iic» + 6<a!« + 4rfa! + e = 0,
so that the invariantB are /=a«-4Jd+3e' and J=aee+%cd-ad}-eh''-cr*. This
last may also be written in the form of a determinant. It will generally be found
convenient to dear the equation of the second term. Let the equation so trans-
formed be
where H=b*-ac and 0=2lfi-Sabc+a''d, By using the invariants or by actual
transformation, it is easy to see that Ia^-P+3H' and a^J=UP-G^~a^III.
Let A be the discriminant, i. e. \=1*-27J', then it is proved in all books on
the theory of equations that if A is negative and not zero the biquadratic has two
real and two imaginary roots. If A is positive and not zero, the roots are either ail
real or all imaginary.
Usually we can distinguish whether the roots are all real or all imaginary by
ascertaining if the biquadratic has or has not a real root, thus if a and e have opposite
signs one root is, and therefore all the roots are, real. In any case we may use the
following criterion. Let Ka*= 9ff« -F= 12if « - Ia«. Then if o, /S, 7, 8 be the roota
of the transformed equation it is easy to prove
If all the roots are real // must be finite and positive. Since the arithmetic
mean of four positive quantities is greater than their geometrical mean, it is clear
th'it K is also positive, and can vanish only when all the squares of the roots are
equal. If all the roots are imaginary, let them be 2) ijj'V -^> ~P^l' »/-^' We
then have
o« ~ 2 I
^=(r^) -v (!>'»+ 2'")
J
If JT is positive or zero, it is easy to see that K must be negative. If therefore
// and K are both positive, the four roots are real, if either is negative or zero, tho
four roots are imaginary.
If the discriminant A is zero, but / and / not zero, it is known that the
biquadratic has two roots equal. If two of the roots are real and equal and the other
f ^ \v
I
I
It
V ■ ,'
i^
1: >.
v..,-. Ill
343
SMALL OSCILLATIONS.
If any one of the roots is real and positive, x and y will ultimately
become large, unless the initial conditions are such that the term
depending on this root disappears from the values of x and ;/. If
the roots are all real and negative, the motion will gradually
disappear and the system will come to rest at the end of an
infinite time.
If two of the roots are imaginary, we have a pair of imaginary
exponentials with imaginary coefficients, which can be rationa-
lized into a sine and a cosine. This rationalization will be however
unnecessary if, as usually happens, only the character of the oscil-
lations is required. Suppose the roots to be o ± jj VC- 1). we have
X = c"' (iVj cos 2yt + N"^ sin pt) + &c.,
where iVj, N^ are arbitrary constants. There will be a similar
expression for y with N' written for N". Thus the period of the
. 27r
oscillation is — . The oscillation will ultimately become very
large or vanish away, according as a is positive or negative. If
a = 0, the oscillations will continue throughout of the same mag-
nitude.
> I: 1-1
i
If it be required to find not merely the character of the motion,
but also the motion resulting from given initial conditions, it will
be necessary to determine the relations between the arl>itrary
constants which enter into the expression for x and y. This may
be effected very easily in the following manner. Let D^ +fO +
be the factor which eqiiated to zero gives the imaginary roots,
then /and g are known in terms of a and p. Iict us now substi-
tute —fD — g for D* in the two first equations of Art. 432. They
reduce to equations of the form
dt
dt
(B;| + a.> + ((j,'^+ff;)2, = o
t »
Hi
- 1
two imaginary, we see by putting g' zero that if IT is positive or zero, A' must bo
negative. Hence if // and K are both positive all the roots are real, if // or K is
negative or zero, two roots are real and two imaginary. If G is zero, there are then
two pairs of equal roots. In this case K is zero, and these roots are all real if //
is positive, all imaginary if H is negative.
Lastly if A is zero and also both I and J zero. The biquadratic has three roots
equal, and therefore all the roots are real. If H=0 also, the four roots are all equal
and real.
OSCILLATIONS WITH TWO OR MORE DEGREES OF FREEDOM. 349
where B^, C,, &c. are some constant coefficients. Eliminating -~
from these equations, we have an equation of the form
where K and L are constants, so that when the two terms of x,
which depend on this factor are known, tlie corresponding terms
of 1/ can be found immediately. If there be another pair of
imaginary roots, wo obtain by a similar process a similar e(juation
with different constants for A' and L, to find the corresponding
terms in i/.
If two of the roots are equal, say m^ = m^, then, by the theory
of Linear Equations, we know that
where N^ and N,, &c. are arbitrary constants. If three roots
are equal, there will be a term with f and so on. The expres-
sion for ^ will of course contain similar terms. Let it be
The terms containing i as a factor will at first increase with t,
and if ??^, is positive or zero will become very great, but if wi, is
negative, they will ultimately vanish. The motion will, in the
latter case, be stable if the initial increase of the terms is not such
that the values of x and y become large, i e, if the system is not
at first so much disturbed that the motion cannot be considered
as a small oscillation.
In some cases the relations between the constants in the ex-
pressions for X and y are such that the coefficients of both the
terms containing the factor t vanish*. When this occurs the four
* To prove this let us find the relations between the constants. Substituting
the values of x and y in the two first equations of Art 432, we find
{A nti" + Bm^ + N^ = - (Fm^^ + Gm^ + H) N.^,
(4 mi" + Z?j»i + C)Ni + (2iwii + B) N^ = - {Fiiii^ + Cfm^+E) iV/ - {2Fmi + G) N^,
with two Rimilar equations which may be obtained from these by accenting the
letters A, B, C, F, 0, H. If the.
ylHii" + 2?% + C - 1 JPmjS + Om^ + // = )
A'mi' + B'm^ + C" = \ ' F'm^^ + U'n\ + i?'= i '
while the two expressions
(2^1mi + B) (2 F'm^ + 6') and (24 'nt^ + B) {^Fm^ + G)
are unequal, we have N„, N^' both zero, and A',, N^' both arbitrary. If the two
expressions just written down were equal also, it may be shown that the biquadratic
tij liiid D would have three equal root^.
t J !
II
i
1''
P'
4 .
Ill;
i;
I i
ii
V
350
SMALL OSCILLATIONS.
arbitrary constants will be JV,, N' il/g and 31 ^. In such cases the
motion is stable for all initial conditions.
435. lue most important case is that in which there are no
real exponentials in the values of x and y. If AG+ BF and
BH-\- CG both vanish, there v/ill be no odd powers in the sub-
sidiary biquadratic. The biquadratic may now be regarded as
quadratic in L^. If its roots are real and negative, let them be
—p^ and — j". The expression for x will then take the form
a; = JV, sin (jpt + v^ + N^ sin {qt + v^,
where iV^, N^, v,, v^ are arbitrary constants. The corresponding
terms in y may be found by the rule just given. Eliminating
— between the two given equations of motion, let the result be
A'^t^S^+G'x+F^ + H'y^^O.
de
dt
d?
df
Then writing —p^ for -7-g , we have
df
C'-Ay
X —
B dx
y~ ' H'-Fp*'" JI'-Fy dt
C'-Ay .r - f ., ^ •»> XT /
= - w^y ' ^'" ^^ "*" "^^ - W^fy ^' "^^^ ^^* "^ ''»^
C'-AV xr • / *^ N ^^ AT / . N
436. In many cases it will be found impracticable to solve
the biquadratic on which the character of the motion depends.
If however we only wish to ascertain whether the position of
equilibrium, or the steady motion about which the system is iu
osr lation, is stable or unstable, we may proceed without solving
tht, Diquadratic.
With the reservations in the case of equal roots mentioned in
Art. 434, the necessary and sufficient conditions for stability are,
that the real roots and the real parts of the imaginary roots should
be all negative. It is proposed here to investigate a method of
easy application to decide whether the roots are of this character.
Let the biquadratic be written in the form
Let us form that symmetrical function of the roots which is the
OSCILLATIONS WITH TWO OR MORE DEGREES OF FREEDOM. 351
product of the sums of the roots taken two and two. If this be
called V, we find*
a
X=hcd-ad^-eV
= i
2a h c
b d
c d 2e
It will be convenient to consider first the case in which X is
finite.
Suppose we know the roots to be imaginary, say o ± pV— 1,
and ^±q V^.
Then f = 4a/3 {(a + /3)» + (p + qY] {(a + ^)» + (;> - 2/}.
Thus, oijS always takes the sign of — , and a + /9 always takes the
Of
sign of — . Thus the signs of both a and fi can be determined ;
and if a, h, X have the same sign, the real parts of the roots are
all negative.
Suppose, next, that two of the roots are real and two imagi-
nary. Writing c[ V— 1 for q, so that the roots are o.±p \f—\ and
P + ^'i we find
X
~ = 4a^ {[(a + /3)« +/ - g'7 + ^pY)-
a
* This value of X may be found in several ways more or less elementary. If
we substitute J)ssE±Z in the given biquadratic and equate to zero the even and
odd powers of Z, we have
aZ*+{loaE*+SbE + c)Z*+AE*+bF' + cE^ + dE+e = 0)
{iaE + b)Z3 + (iaE^+QbE^-^2cE + d)Z = o\'
Rejecting the ix>ot Z = and eliminating Z we have
64o»JS«+ +bcd-acP-eb^=0,
where only the first and last terms of the equation are retained, the others not
being required for our present purpose. Since x = ^± Z it is clear that each value
of E is the arithmetic mean of two values of x. We have an equation of the sixth
degree to find E because there are six ways of combining the four roots of the
biquadratic two and two. The product of the roots of the equation in E may be
deduced in the usual manner from the first and last terms, and thence the value
of X is seen to be that given in the text.
If we eliminated E we should obtain an equation in Z whose roots are the
nrithmetic means of the differences of the roots of the given equation talicu two
ftud two.
1 I
-1
■f
n
'M I
U =M
U
fi, >i
\H i
352
SMALL OSCILLATIONS.
);i i
Just as before, 0/3 takes the sign of — , and a + ^ takes the sign
it
b e
of . Also, 0^ — q" takes the sign of the last term - of the bi-
quadratic. This determines whether /8 is numerically greater or
less than q\ If, then, a, h, e, and X have the same sign, the real
roots and the real parts of the imaginary roots are all negative.
Lastly, suppose the roots to be all real. Then, if all the
coefficients are positive, we know, by Descartes' rule, that the
roots must be all negative, and the coefficients cannot be all posi-
tive unless all the roots are negative. In this case, since X is the
product of the suras of the roots taken two and two, it is clear that
will be positiv >.
Whatever the nature of the roots may be, yet if the real roots
and the real parts of the imaginary roots are negative, the biquad-
ratic must be the product of quadratic factors all whose terms are
positive. It is therefore necessary for stability that every coeffi-
cient of the biquadratic should have the same sign. It is also
clear that no coefficient of the equation can be zero unless either
some real root is zero or two of the imaginary roots are equal and
opposite.
Summing up the several results which have just been proved,
we conclude that if X is finite, the necessary and sicfficient con-
ditions that the real roots and the real paHs of the imaginary roots
shoidd he negatine or zero are that every coefficient of the biquad-
ratic and. also X shoidd have the same sign.
Tlie case in which X = does not present any difficulty. It
follows from the definition of X, that if X vanishes two of the
roots must be equal with opposite signs and conversely if two
roots are equal with opposite signs X must vanish. Writing
— D for Z) in the biquadratic and subtracting the result thus
obtained from the original equation we find bl)^ + dD = 0. The
equal and opposite roots are therefore given by Z) = + a/ — j- . If
h and d have opposite signs these roots are real, one being positive
and one negative. If b and d have the same sign, they are a pair
of imaginary roots with the real parts zero.
js
The sum of the other two roots is equal to — and their product
be
ad'
We therefore conclude that if X = 0, the real roots and the
real parts of the negative roots luill be negative or zero if every
coefficient of the biquadratic is finite and has the same sign.
e sign
the bi-
ater or
le real
ive.
ill tlio
at the
11 posi-
'. is the
ar that
il roots
biquad-
rms are
^ cooffi-
; is also
s either
ual and
proved,
nt con-
•?/ roots
^iquad-
Ity. It
of the
if two
Writing
lit
0.
thus
Th(!
If
d
b'
positive
re a pair
product
and the
if every
OSCILLATIONS » WITH TWO OR MORE DEGREES OF FREEDOM. 353
If either a or e vanishes, the biquadratic reduces to a cubic.
Putting e zero, we have
-v, = hc — ad.
ad
If the coefficients have all the same sign it is easy to see that
it is necessary for stability that be — ad should be positive or zero.
If a and e be not zero and one of the two b, d vanish, the other
must vanish also, for otherwise X could not have the same sign as
a. In this case X vanishes, and the biquadratic reduces to the
quadratic
aD* + cD' + e = 0.
As this equation admits of an easy solution, no difficulty can
arise in practice from this case. It is necessary for stability that
the roots of the quadratic should be real and negative. The con-
ditions for this are, ^firstly the coefficients a, c, e must all have the
same sign, secondly that c' > 4ae.
437. If the equation on which the motion of the system depends is of the fifth
degree, we may proceed in the same way. Let the equation be
<l>{D) = aD^ + bD* + cD3 + dD^+sD+f^0,
and let us sitppose the coefiBcient a to he positive.
Form, as before, the product of the sums of the roots taken two and two. If this
, X „ , X= be -ad be-af
be -; , we find , , , \,
a*' he-af de-cf '
Lot us consider first the case in which X is finite.
Suppose that there are four imaginary roots a±p^-l, /3=fc7^-l, and one
real root y. Then y has the sign opposite to /, and o/3 takes the sign of X, while
2(a + ^)+7=--. If then / be positive, y is negative. If b be positive and
(p ( — ) negative, the root y is numerically less than - , so that a + /3 is negative.
If therefore a, b, f, X, and -i>i — j be all positive, a, |3, y will bo all negative.
Suppose that there are two imaginary roots aJt^piJ-1, and three real roots
/3, 7, 5. Then, if all the coefiicients are positive, /3, y, 5 are nrf^ative, and X takes
tlio sign opposite to a; so that, if X be also positive, a, /3, y, S vill bo all negative.
Suppose all the roots to be real; then, if all the coefiicients be positive, the
roots will be all negative, and not otherwise ; and it is also clear that X, being the
product of ten negative quantities, will be positive.
In both those cases, if the real roots and the real parts of the imaginary roots
be negative, it is clear that <;!> ( — ) must have the sign opposite to a.
Conversely, if all the real roots and the real parts of the imaginary roots bo
negative, the factors of tho equation, and therefore the equation itself, must havo
all tho coefiicieutu of the same sign.
R. D. 88
m
■;ti
p^
1 :
'■
' ii
1
■
t|
]
Hi
M
i
u
!■
<
r
*>-,4
iij* SMALL OSCILLATIONS.
We therefore conclude that n is necessary and sufficient for stability of equili-
brium that every fneffioient of tTie equation, ■ <p( ) , and also X, should be posi-
tive.
T}.u< cane in which X is zero may be treated in the same manner as in the
biquadrittic.
As it is very seldom that equations beyond the fourth or fifth degrees present
themselves in dynamics, it is unnecessary to consider any other cases in detail.
A more general method of proceeding will be inoicated in a note.
438. It will be often found advantageous to trace the more
complicated cases of motion by the help of a figure. There are
various methods of eflfecting this, some being more suited to illus-
trate one kind of motion, others to illustrate another. We might,
for instance, follow the method indicated in Art. 412. Let the
abscissa of a point P represent on any scale the time elapsed since
some epoch, and let the ordinate represent the value of x. In the
same way the curve tracad out by another point Q will represent
the changes of y. Suppose, for example, we wished to trace the
motion represented by
X = N sin pt-\- N sin 2pt,
the coefficients being equal in magnitude. There will be no
difficulty in tracing the two curves x^ = N' sin pt and x^ = Nsin 2pt
Let these be the two dotted lines. We obtain the required curve
by adding the ordinates corresponding to each abscissa. Let this
be the continuous line.
In the figure the axis of the abscissae is not drawn. It clearly
joins the iA,'o extrem- points on the right and left hand sides.
We poe from n. simple inspection of the figure, that the motion
consi.-its of a vio/cit oscillation to each side of the mean position
i
wht'i
!S.
oscillation; Wilrf TWO OR MORE DEGREES OF FREEDOM. 355
fallowed by a very slight one, and so on aKernately. Thir; figure
r sembles that used in Astronomy to trace the ci.anges of magni-
t Lide of the equation of time throughout the year.
439. Ex. 1. Show that the motion represented by x^Nnrnpt + Nsmiipt
consists of two large oscillations to one side of the mean position followed by two
equally large ones to the oiher side and so on continually.
Ex. 2. Trace the motion represented by x=NBin2pt + NBin iipt, and point out
th3 difference between the two parts of the large oscillation.
440. Let us trace the motion represented by x=N-^ sin {pt+v-^ + N.^ sin (qt + v.^),
whore N^ and N^ are both positive, firstly when p and q are nearly equal, and
secondly when p is sxupII compared with q.
In the first cas e, consider any time at which pt + v^ and qt-\-v.^ differ i'rom each
other by an even multiple of w. At this instant the two trigonometrical terms
have the samn sign, and, since p and q are nearly equal, they will increase and
decrease together for several oscillations, how many will depend on the nearness of
p and q to each other. The value of x will therefore vary between the limits
±(A''j-f iVg). Next consider any time at which pt + v^ and qt + v^ differ by an odd
multiple of ir. The two trigonometrical terms have opposite signs and will continue
to have opposite signs for several oscillations. The value of x will therefore vary be-
tween the limits ± (iVj - N^). We see that the motion of that part of the dynamical
system which depends on the co-ordinate x undergoes a periodic change of character.
At one time, this part of the system is oscillating with an arc N^ + N^, after an
interval equal to , the arc of oscillation is N^-N^. If N^ and N^ are nearly
equal, this last arc may be so small, that the motion is invisible to the eye.
Thus there will be alternate periods of comparative activity and rest. This alter-
nation is sometimes called beats. Usually the two co-ordinates x and y will be so
lelated that the period of comparative rest in one will coincide with the period of
comparative activity in the other. When this is the case there will be an alternate
transference of energy from one part of the system to another and back again.
441. E.K. Show that, if p and q be unequal, x may bo written in the form
x = N sin ,y{pt + v^ + qt + v^ + S),
where IP=Nj' + N^^ + 2xVi N^ cos (pt + v^-qt- v.^,
^ 5 iV,-Ar 1
**" 2 = nI+% ^''^ 2 (^' + "1 - 5« - "«)•
Tlience show that when p and q are nearly equal, the oscillation will appear to the
eye to be harmonic, but the arc of oscillation vill slowly vary between the limits
^i±J^a.
442. Next, let p be small compared with q. In this case qt + v^ increases by
2ir while pt + Cj alters only by - 2ir, so that tho second trigonometrical term goes
through all its changes while the first is only very slightly altered. The system
will therefore appear to oscillate about u mean position determined by the instan-
23—2
'i'l
I
Pi
f I
!!
v. !"
Ji
! Mil
•Mlhf>li[BM
I'i
|l ' i
III)
I!
^^1
'•111
I t
I;
I i;
i
356
SMALL OSCILLATIONS.
taneonn value of the first trigonometrical term, Tims the oscillationa tcill appear
to be. hirmomc to the eye, while the apparent mean position will travel Jirst to one
side and then to the otJier of tlie real mean.
443. Ex. Investigate the following geometrical construction to represent the
motiou
i= X = N^ Bin pt + N^ din qt.
Let q be gieater than p in the standard case and let x have a sign such that N^ is
positive. Describe a circle with centre and radius equal to ^^^iV,. Let another
circle with centre C and radius equal to - N^ toucli the first circle externally at a
point A. Measure CP equal to AT^ in the direction 00, so that if N^ is negative
CP must be measured in the opposite direction. If the second circle be now made
to roll on the first, the point P traces out an epitrochoid. If C and P' be cor-
responding positions of the centre of the moving circle and the generating point,
then the distance of P' from the fixed straight line OA is the value of x, while
the angle CO A is equal to pt.
Apply this to trace the motion when p and q are nearly equal.
The third or Lagrange's metliod of forming the equations of motion.
444. Let a system of bodies be in equilibrium under any con-
servative forces. When disturbed into any otlier position let Z7be
the force function, 2jrthe vis viva. Let the position of the system be
defined by n co-ordinatos 6, (j), &c., which are such that they vanish
in the position of equilibrium. Then if the system oscillate about
the position of equilibrium, 0, <f), &e. will be small throughout thij
rvhole motion. As before, let accents denote differential coeffi-
cients with regard to t.
Let us suppose that the geometrical equations do not contain
the time explicitly, then by \rt. 307 T may be expressed as a
homogeneous function of 6', ^', &c., of the form
2T^A,,e" + 2A^^0'<l>' + A,,<l>"-\-&c (1).
Here the coefficients A^^, &c. are all functions of 6, <}>, &c., and
've may suppose them to Lo expanded in a series of some powers
of these co-ordinates, "f the oscillations of the system are so small
that we may reject a?! pow is of the small quantities 0, <f>, &c.
except the lowest which orcMr, v e may reject all except the con-
stant terms of these stvies. Wo shall therefore regard the coeffi-
cients A^^, &c. as constants.
In the same way we ukiv expand U in &, series of powers of
0, <f), &c. In tliis srries the terms coiitaining the first powers will
vanish, because bv the principle of virtual velocities
. dU=--^m{ Xdx + Ydy + Zdz)
LAGRANGE'S METHOD.
357
vanishes in the position of equilibrium. Hence we may put
2U=2U, + a,,e' + 2a^J<f)-¥&c (2),
where U^ is a constant, which is easily seen to be the value of U
in the position of equilibrium. It is necessary for the success of
Lagrange's method that both these expansions should be possible.
We have now to substitute these values of T and U in the n
Lagrange's equations
d_dr_dT_dU ,„.
dtdd' dd~ dd ^ ''
with similar equations for 0, i^. Since the expression for T does
not contain 6, (f), &c., we have
dT ^dT ^ ,
dd = ^'d4> = ^'^'- .
The n equations (3) therefore become
&c. = &c.
(4).
These are Lagrange's equations to determine the small oscillations
of any system about a position of equilibrium, under any conserva-
tive forces, provided the geometrical equations do not contain the
time explicitly, and are not functions of the differential coeffi-
cients of the co-ordinates.
These equations may be obtained in a variety of ways. In
many cases it is more convenient to use the process of taking
moments and resolving. The advantage of using Lagrange's method
is that the whole motion is made to depend on one function only,
viz. T^-U.
445. We shall now proceed to the solution of the equations.
We notice that these equations do not contain any differential
coefficients of the first order. This will be the case when a dyna-
mical system oscillates about a position of equilibrium under con-
servative forces. This peculiarity greatly simplifies the solution.
Instead of using exponentials, as in Art. 432, which (when we want
anything moi'e than the periods) have afterwards to be ration-
alized, we may now conveniently introduce the trigonometrical
expressions at once. Let us then put
6 = L^ sin {ii^ -h a,) -t- L^ sin (j)./ + 7^) + &c.
= il/j sin (;?,< -I- a,) -I- M^ sin \i\t + a,) -f &c. ^ (5),
&c. = &c.
r ,1
'-«MnM|iW*
K
If
fi i
n
^V
' Si
I' !
;4l
358 SMALL OSCILLATIONS,
which may be written in the typical form
6^ = Z sin (2)t + a), = M sin {jit -f a), &c.
If we substitute in equations (4) we have
(J„ p' + aj L + (^l,,p'^ + a J Af+ &c. = 01
(0).
&c.
&c.
=
EHminating L, M, &c., we have a detcrminantal e(juation
A^^p''■Va^„A^,Jy' + a^^,kc. = (7),
&c. &c. &c.
which, it will be observed, is symmetrical about the leading
diagonal. This equation Js of the ?i*'' degree to find JJ^ It will be
presently shown that its roots are real. Taking any root positive
or negative, the equations (6) determine the ratios of J/, N, &c. to L ;
and we notice that these ratios will also be all real. If all the
roots are positive, the equations (5) will give the whole motion,
with 2m arbitrary constants, viz. Zj, L^...L,^\ a,, a5,....a„. These
have to be determined by the initial values of 6, (f), &c,, 6', ^', &c.
If any root be negative, the corresponding sine will resume its
exponential form, the coefficient being rationalized by giving the
coefficient L an imaginary form.
That the determinant should contain no odd powers of p is
just what we should have expected a piiori. In our preliminary
assumption (5) each sine is really the sum of two exponentials
with indices of oppc-ite signs. The equation therefore of Art. 432
to determine p shoulu here give pairs of equal roots of opposite signs.
The equation (7) may be written down without difficulty as
soon as the values of T and U have been expanded in powers of
6' , &c., 6, (fee, respectively. In finding the times of oscillation of a
system about a position of equilibrium, it is not necessary to go
through all the intermediate steps; we may, if we please, write
down at once the detcrminantal equation. The rule will be as
follows. Omitting the accents in the expression for' T, and the
canstant term in U, equate to zero the discriminant of p'^T + U.
2'he roots of the equation thus formed are the values of p. If we
require the motion as well as the periods, we shall require e(|ua-
tions (6). But these may be also very simply found in the follow-
ing manner. Omitting accents as before and taking any of the
values of ^ j^ist found, form the equations*
* These cqimtioii.s arc given by Lagrange.
■i
■t
\
r«
laorange's method.
359
(8).
The 0, ^, <i-c. in these equations may he replaced hy the coejfficients
required in the equations (5).
If we solve these equations we see that the ratios of L, M, &c.
are equal to the ratios of the minors of the constituents of any one
line in the determinant (7).
Ex. 1. A rod AB. whose length is 2a and mass m is suspended from a fixed
point by a string OA the length of which is I. The rod oBcillates under gravity
in a vertical plane, find the periods of the small oscillations.
Let 0, (f> bo the angles the string and the rod make with the vertical. Proceed-
ing as in Art. 136, we find that when powers of and higher than the second are
neglected,
r=^ m {l^0'^ + 2al0'</>'+ (r f flS) <p'%
Forming the discriminant ot p-T+ U and dividing out the common factor in, we
have
pH^-gl alp* j=0.
alp^ p^(l? + a^'\-ag \
This quadratic gives two values oip'^. If these hep^ and p^, we have
tf = ij sin {p^t + oi) + 1,2 Sin {psf + a^).
alpi
alp,^
■2« + «2)j
Ex. 2. Show that when the determinant (7) :i zero, the ratios of the minors of
the constituents of any one line are equal to the ratios of the corresponding minors
of the constituents of any other line.
Ex. 3. If (T^, Uj), {T^, U^), &c. be the values of T and U- U^ when (L^,
Ml, &c.), (Zig, M3, &o.) are substituted for (0', </)', &c.) or {0, <j>, &c.), prove that
rjyj2+ Ui=o, T,p.,^+ u^=o, &c.
This follows from equations (8) by Euler's theorem on homogeneous functions.
446. In order to determine the values of p^, it will often be necessary to expand
the determinant. This may be done by the use of Taylor's theorem. Let A be
the discriminant of T and let IT represent the operation
n=a
11
d
ilA
+ «,
11
d
dA,
+ a,
m
d
dA
+ &C.,
23
then the determinant when expanded becomes
A^)'-^" + n (A) ^2.»-2 + n* (A)2)2'«-i + . . . = 0.
If A' be the discriminant of U and 11' the operation 11 when the great and small
letters are interchanged, we may write the equation in the form
A' + n' (A')23H n'2 (A')i)* + . . . = 0.
When ther<3 are only three independent co-ordinates, we may adopt the notation
used in tte chapter on Invariants in Dr Salmon's Conies.
IS- '
riil
■.III
111
i' i I
■'V
:^
i!
i •
360
SMALL OSCILLATIONS.
Ex. 1. If a system be in a position of equilibrium, sbow thnt the equi-
librium will be stable if - n(A), IFCA), -Il'HA), &o. be all powitive.
Firstly, we may show that A is necessarily positive, and secondly that these
are then the conditions that the roots of the equation (7) are all real.
Ex. 2. If S^ bo the sum of the products of each itth minor of the discriminant
A' into the conjugate minor of A, prove that .S'^ is the coefficient of /»*.
Ex, 3. The same dynamical system can oscillate about the same position of
eciuilibrium under two different sets of forces. If p^, pa... and o-j , (T3 . . . be the
jieriods of the oscillations when the two sets act separately, i?i, Ri... the periods
when they act together, prove that S „ + S — , = S -^^ .
p' a^ ii"
This follows from the fact that ri(A) contains A^-^ Ac. only in their first powers.
Ex. 4. Two difTeront Kystems of bodies wlien acted on by the same set of forces
oscillate in periods p,, pa... and ctj, arj ... If JJj, Eg... be the periods when they are
both set in oscillation by the same set of forces, prove that Zp' + So-'^SiJ'.
Ex. 5. Prove that the equation giving the periods of the oscillations may bo
expressed as a determinant of 2/i rows and columns by using Sir W. Hamilton's
equations given in Art. 381.
447. If we refer the motion of the system to any other co-ordinates {, ij, f, i&o.
which vanish in the position of equilibrium, it is clear that when d, (f>, ^, &c. aro
expressed in terms of ^, <&c. and the squares of small quantities neglected, wo shall
have equations of the form
.(0).
= /"if + ^•^'7 + »'«■'<'• [
&c. =&c. J
Now 0, <f>, &c. being expanded in a series of sines as in equations (5) it is clear
that f, t), &c. will bo expanded in a series of the same sines but with different co-
efficients. Hence the values of p^ found from the determinantal equation will bo
the same whatever co-ordinates the system is referred to. The ratio of tho
coefficients of the several powers uf p are therefore invariable.
If fjL be the determinant of transformation, wo know that A becomes /x^A. Henco
all the other coefficients will be altered in the same ratio. The quantities A, 11(A),
n°(A), &o. are therefore called the invariants of the dynamical system.
448. To show that the values ofp^ are all real*.
Since T is essentially a positive quantity for all values of 6',
<^', &c. the coefficients of 6"\ 0'^ &c., viz. A^^, A^, &c., must be all
positive. Let us collect ^•--^: ether the terms containing Q'^, 6', and
complete the square by adding and subtracting the proper qua-
dratic function of 0', i/r', &c. We have
This theorem seems to have been first discovered by Sii' \V. Thomson.
1 . I
wher^
LA.aRANGB:'S METHOD.
S61
u
'It
and since A^^ is positive, this transformation is real. In the same
way B„ muwst be positive, and we may repeat the process. We
thus have
where
23
and it is clear that this process may be repeated continually.
We may take f, rj, &c. as co-ordinates of the system because
they arc; independent of each other and vanish in the position of
e(juilibrium. We thus have
2r=r + V"+... 1
2(f^-f^„)=/„r+2/,f^+...|'
where /„, /,,, &c. are all real constants. The detorminantal
equation now takes the form
&c. &c. &c.
= 0.
When there are only three co-ordinates, this is the discrimi-
nating cubic used in Solid Geometry, and we know that its roots
are all real. When there are more than three co-ordinates, it is
proved in Dr Salmon's Higher Algebra, Lesson VI., that the roots
are all real.
449. To explain what is meant by the principal co-ordinates
of a dynamical system.
When we have two homogeneous quadratic functions of any
number of variables, one of which is essentially positive for all
values of the variables, it is known that by a real linear trans-
formation of the variables we may clear both expressions of the
terms containing the products of the variables, and also make the
coefficients of the squares in the positive function each equal to
unity. If the co-ordinates 6, (f), &c. be changed into ^, rj, &c. by
the equations (9) of Art. 447, we observe that 6', <f>', &c. will Ije
changed into f', r)', &c. by the same transformation. Also the
vis viva is essentially positive. Hence we infer that by a proper
choice of new co-ordinates, we may express the vis viva and force
function in the form
ff
!'!f
li
.' I
I ■■(
■ (ii J
ru
; i'
u :Hm
^.
IMAGE EVALUATION
TEST TARGET (MT-3)
1.0
I.I
us
■u
u
11-25 W 1.4
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FhotDgraphic
Sciences
Corporalion
23 WIST MAIN STRHT
WIISTn,N.Y. 14StO
(71«)«72-4503
e^
-I
^k
I s
K
i
3C2
SMALL OSCILLA'nONS.
These new co-ordinates ^, ij, &c. are called the principal co-
ordinates of the dynamical system. A great variety of other
names have been given to these co-ordinates ; such as Harmonic,
simple and normal co-ordinates.
450. When a dynamical system is referred to principal co-
ordinates, Lagrange's equations take the form
rf^f
ef77
so that the whole motion is given by
^=Esm{pJ; + aJ, rj = F sin (pjt + a^), &c.,
where E, F, &c., Oj, a^, &c. are arbitrary constants to be deter-
mined by the initial conditions and p^ = — 6„, p^ = — b.^, &c.
When the initial conditions are such that all the principal
co-ordinates are zero except one, the system is said to be per-
forming a principal or harmonic oscillation.
451. The physical peculiarities of a principal oscillation are :
1. The motion recurs at a constant interval, i.e. after this
interval the system occupies the same position as before, and is
moving in exactly the same way.
2. The system passes through the position of equilibrium,
twice in each complete oscillation. For taking f as the variable
co-ordinate, we see that ^ vanishes twice while p^t increases by
27r.
3. The velocity of every particle of the system becones zero
at the same instant, and this occurs twice in every complete
dB .
oscillation. For -^ vanishes twice while /),< increases by 27r. These
may be called the extreme positions of the oscillation.
4. The system being referred to any co-ordinates, 0, <^, yjr,
&c., which are all variable, the ratios of these co-ordinates to each
other are constant thi'oughout the motion*. For referring to
the equations (9) of Art. 447, we see that when r), f are all zero,
and only ^ is variable,
_<li
/*!
... = ?.
* Tills property is mentioned by Lagrange, who on several occasions uses
principal co-ordinatos though not the name.
ion are :
nous uses
LAGRANGE'S METHOD.
363
452. When some of the roots of the equation to find ']^ are
equal, we know by the theory of linear differential equations
that either terms of the form {At + B) sin yt enter into the
values of 6, <f>, &c., or else there must be an indeterminateness
in the coefficients L, M, &c. given by equations (8). Referring
the system to principal co-ordinates we see that the first alter-
native is in general excluded. If two values of jt>' were equal,
say J„ = 6- J, the trigonometrical expressions for ^ and tf have
equal periods, but terms which contain < as a factor do not make
their appearance. The physical peculiarity of this case is that
the system has more than one set of principal, or harmonic
oscillations. For it is clear that, without introducing any terms
containing the products of the co-ordinates into the expressions
for T or If, we may change ^, t) into any other co-ordinates ^,,1;,,
which make ^ + »?* = f ,* + Vx^ *he other co-ordinates 5", &c. re-
maining unchanged. For example we may put f = ^. cos ol — tj^ sin a
and 17 = ^1 sin a -I- 1;^ cos a, where a has any value we please.
These new quantities ^j, rj^, ^, &c. will evidently be prmcipal
co-ordinates, according to the definition of Art. 449.
One important exception must however be noticed, viz. when
one or more of the values of p are zero. If, for example, &„ =
we have ^ = At + B, where A and B are two undetermined con-
stants. The physical peculiarity of this case is that the position
of equilibrium from which the system is disturbed is not solitary.
To show this, we remark that the equations giving the position
/JTT fiTT
of equilibrium are -^ = 0, — = 0, &c., where U has the value
given at the end of Art. 449. These in general require that f,
VI, &c. should all vanish, but if 6„ = they are satisfied whatever
^ may be, provided 97, f, &c. are zero. These values of f must
however be very small, because the cubes of ^, rj, &c. have been
rejected. It follows therefore that there are other positions of
equilibrium in the immediate neighbourhood of the given position.
Unless the initial conditions of disturbance are such as to make
the terms of the form At-¥ B zero, it may be necessary to examine
the terms of the higher order to obtain an approximation to the
motion.
453. The motion being referred to any co-ordinates 6, <f>..,
it may be required to find the principal oscillation. This may be
done by finding \, /i, &c. in equations (9) Art. 447, by the analy-
tical process of clearing the two quadratic expressions of the terms
containing the products, in the manner explained in Art. 449.
We may also proceed thus. Let the system be performing the
principal oscillation whose period is — . Then in the equations
(5), Zjj, ilig, &c., Z3, il/3, &c. are all zero, hence 0, </>, sp-, &c. arc in
I: ^'i
f t
It^
■ '■ 1
* i'i'
Ik
h
M
i
, I
364
SMALL OSCILLATIONS.
the ratio X,, M^, &c. But these ratios are given by (6) or (8), in
the form
.(8).
j0{p^'T+U)=O, ^^{p,'T+U) = 0,&c
where the accents in T have been omitted. These equations give
the relations between 0, <f), &c., when the system is performing a
principal oscillation.
454. When the dynamical system has but three co-ordinates, we may obtain a
geometrical interpretation of this process. If we regard 0, <f>, \f/ as the Cartesian
co-ordinates of a point P, it is clear that the position of P at any instant will give
the position of the system. Omitting the accents in T and the constant term in Z7,
the equations T=a, 17= -j8, where a and /3 are any constants, represent two
quadric surfaces which have their centres at the origin. These have a common set
of conjugate diameters wluch may be found by the following process. Let 0, </>, \jf
be the co-ordinates of any point on one of the three conjugates. Then, since the
diametral planes of t^<'s point in the two qnadrics are parallel, we have
dT _
dU
dd'
dT
MjT =
^dJ7 dT^dU
'd<i>~d<p' '*drf/~d\l/'
Comparing these with equations (8), we see that when the system is performing
a principal oscillation, the representative point P oscillates on one of the common
conjugate diameters of the two quadrics.
By Euler's theorem on homogeneous functions we clearly have ij,T=U. Applying
the same reasoning to equations (8) we see that fi= -p^. Let the diameter con-
sidered out the quadrics T and U in the points D and D' and let be the origin.
Putting the point P at D we have r= o and since Uia homogeneous U= - I ^ r)» ) /3»
Hence p^= -M = (,^/) • The period of the oscillation corresponding to the
diameter ODiy is therefore 2n-
OP' /
odV\
Since 0, <f>, ^ contain only a single trigonometrical term (Art. 450) when the
system is performing a principal oscillation, we see that the representative point P
moves with en acceleration tending to the origin and varying as the distanc3 there-
from.
455. As an example of this geometrical analogy let us consider the following
problem. A rigid body, free to move about a fixed point 0, is under the action of
any forces and makes small oscillations about a position of equilibrium; find the
principal oscillations.
Let OA, OB, OC be the positions of the principal axes in the position of
equilibrium, OA', OB', OC their positions at the time t. The position of the body
maybe defined by the angles between (1) the planes AOC, AOC, (2) the pla'^ds
BOG, BOC, (3) the planes GOA, COA'. Let these be called 0, <p, ^ respectively.
Then 0, i>, ^ aro angular displacements of the body about OA, OB, OG. Taking
theise as the axes of co-ordinates in the geometrical analogy ; a small displacement
of P from the origin to a point 0, f, f represents a rotation of the body about the
• I
le oommon
LAGRANGE'S METHOD.
365
straight line desoribed by P and whose magnitude is measured by the distance
traversed by P.
U A, B, C he the principal moments of inertia at O, the vis viva of the body is
clearly
2T=Aff* + B<p'^ + Cf'^.
Omitting the accents, the qnadrio T=a is evidently the momental tiilipsoid at the
fixed point.
Let the work of the forces as the co-ordinates change from zero no d, </i, \J/ be, as
in Art. 444,
t7= aii«« + ai,»0 + &c.
Then, following the analogy, as P moves along a radius vector OD' of the quadric
U= - p, the work is - ( ^-^ j /3. Hence this quadric possesses the property that
the work done by the forces when the body is twisted through a given angle round
any radius vector varies inversely as the square of that radius vector. If the equi-
Ubriom is stable, the work due to a rotation about every diameter must be negative,
the quadric must therefore be an ellipsoid.
It now follows from the general theorem that the body will perform a principal
oscillation if it is set in rotation about any one of the three conjugate diameters of
the momental ellipsoid and the ellipsoid U, and will therefore continue to oscillate
as if that diameter were fixed in space.
The quadric U has been called the ellipsoid of the potential. This name was
given to it by Prof. Ball, who arrived at the theorem just proved by a dift'erent
course of reasoning. See his Theory of Screws, Art. 126. The following application
is also due to him.
456. Wlicn the only force acting on the body is gravity, the ellipsoid of the
potential is u surface of revolution about a vertical axis. For the inverse square of
any radius vector measures the work done in turning the body through a given
small angle about that radius "ector. But the work is also proportional to the
vertical distance through which the centre of gravity has been elevated from its
position in equilibrium vertically under the point of support. Hence all radii
vectores which make the same angle with the vertical are equal. Further the
vertical radius vector is infinite, for the work done in rotating the body about
a vertical axis is zero. The ellipsoid of the potential i? therefore a right circular
cyUnder with its axis vertical.
The common conjugate diameters of these two quadrics are obviously the
vertical and the two common conjugate diameters of the two ellipses in which the
diametral plane of the vertical with regard to the momental ellipsoid intersects the
momental ellipsoid and the cylinder.
The principal oscillation about the vertical conjugate is performed in an infinite
time and would therefore cause the body to depart far from the position of equi-
librium. But this is contrary to supposition. The initial axis of rotation must
therefore be in the plane of the other two conjugates, i.e. must be in the diametrnl
plane of the vertical with regard to the momental elUpsoid, and it will remain in
this plane throughout the whole of the subsequent motion.
Since these conjugate diameters project into the conjugate diameters of the
^1;
i i
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1}
I
h'f
Id
1
ll'
f!
i>
!1H
I-
' ' .'t '
f;
3G6
SMALL OSCILLATIONS.
horizontal section of tho oylindor, it is clear that two vertical planes each contain-
ing one of the principal or harmonic axes are at right angles to each other.
457. Ex. Show that thd mean kinetic energy of a dynamical system oscillating
about a position of equilibrium is equal to the mean potential energy, tho mean
being taken for any long period, and the position of eqoilibrium being the position
of reference.
Befer the motion to principal co-ordinates and let
22'=f'» + »,"' + <&c„ 2{U- f7o)= -Pi^'^-PtW-Ao-
Then we find ^=E8in{p{t + aj), ti = F sin {p.2t + a). Substituting these in T and
Z/q - 17 we have the instantaneous kinetic and potential energies. Tho means of
these are obviously the same, and equal to j (E^Pi^ + F^p^^ + Sec.).
If the system remain in the position of eqmlibrium the Hamiltonian character-
istic funoLon 5= UqU If the system be disturbed and after any time t again pass
through the position of equilibrium, the value of S for these two neighboimng
modes of passing from one position to another in the same time must be equal.
Hence / (T+V)dt= U^t, i.e. the mean values of the kinetic and potential energies
•'0
arc equal.
458. Ex. Find the energy of a dynamical system oscillating about a position
of equilibrium referred to any co-ordinates.
By referring the system to its principal co-ordinates, we can easily show that
the energy is the sum of the energies of its principal oscillations. Let the system
be referred to any co-ordinates 9, ip, &o. and let it perform the principal oscillation
whose type is, by equation (5),
j^ = ^ = &0. = sin {p^t + oi).
Substituting in the expression for T, we have T=TiPi^coB!^{pit + aj). Bepeating
this for all the principal oscillations, we have
kinetic energy = T^pj^ cos^ (Fi* + Oi) + T^p^^ cos' {p^t + Og) + &c.
where Ti, Tj, Ac. are the values of 2* when L^, M^, Ac, Xj, M^, &c. are substituted
for 0', <t>', &o. Similarly we find when the position of equilibrium is taken as the
position of reference
potential energy = - t/j sin' (^, ( + a^) - f/g sin' (j) j< + Og) -H &o.
Adding these two, we have by Art. 445, Ex. 3,
whole energy = T^p^' + T^p^^+...
459. Ex. 1. A new constraint is introduced into a dynamical system, so that the
general co-ordinates 0, ^, &o. are constrained to vary in the ratio I, m, &o. If we
put d — lain{p't + a), <p=mBin{p't + a), &c., and if 2*, V be the values of T and
U-Uq when I, m, &o. are substituted for 0', ^', <&c., or 0, ^, &c., prove that
rp'*+u'z=o.
A theorem similar to this is given by Lord Bayleigh in the Proceedings of the
Mathematical Society, No. 63, 1873,
LAGRANGE'S METHOD.
367
Ex. 2. Show how to find the possible displacements of a system which have a
given time of oscillation.
Ex. 3. Show that all possible times of oscillation of a system caused by
introducing any new constraints lie between the greatest and least of the times of
its principal oscillations.
460. When a system starts from rest under the influence
of any forces we may use Lagrange's equations to find the initial
motion. Let the system be referred to any co-ordinate 3 6, <t>, &c.
which however do not necessarily vanish in the positioa of rest.
As in Art. 444, let
where J.,^, &c. are functions of $, <^, &c. Since the system starts
from rest, 6', <f>', &c. will all be very small quantities in the be-
ginning of the motion. If we reject all powers of ff, 0', &c.
except the lowest which occur, we may regard -4„, &c. as con-
stants whose values are found by substituting for 0, 0, &c. their
initial values. Further, since the initial position of the system
is not a position of equilibrium, the first differential coefficients
of U with regard to d, <f>, &c. will not be zero. Let the initial
values of these differential coefficients be respectively a^, a^, &c.
The equations of motion are now
AJ"+AJ>" + ...=a^
&c. = &c.
From these equations we may determine the initial values
of $", <f>\ &c. If X, y, z be the co-ordinates of any particle m
of the system referred to any rectangular axes fixed in space,
we have, by the geometry of the system, these co-ordinates ex-
pressed as known functions of 0, <f), yfr, Sec, Art. 367. Thus if
x=f{0, <f>, &c.), we have initially
X
db'
with similar expressions for y and z. The quantities oj", y'\ z"
are evidently proportional to the direction cosines of the initial
direction of motion of m. In this way the initial direction of
motion of every part of the system may be found.
Ex. A systeni has three co-ordinates 0, <p, ^ and starts from rest in a position
in which these co-ordinates are all zero. Show that the representative point P
(Art. 464) begins to move along the diametral line of the plane OiO+a^f/i + a^^^Q
with regard to the ellipsoid g^A^^ff'+A y^Otp + (fee. = a.
^f.'i
■-%{.>
f:^?
%9i
i i;i
t!
(I
«
368
SMALL OSCILLATIONS.
'*'fil. When the geometrical equations contain differential
coefficients with regard to the time, or when we do not wish to
express T and U in terms of independent co-ordinates, the La-
graugian equations must be modified in the manner explained in
Art. 38S. The equations (3) of Arc. 444 must be replaced by
the equations (4) of Art. 388. Since we reject all powers of the
small quantities 6, <^, &c. except the lowest which occur, we may
still use the expression for T given in (1) Art. 444, and treat the
coefficients as constants. But, in making the position of the system
depend on the quantities 0, <^, &c. (Art. 3G7), we may not have
used all the available geometrical conditions, and therefore the
first powers of 0, ^, &c. in the expansion of U may not be absent.
Let
U=Uo + a^0 + aj^ + &c. + J a„^+ a^^0<l> + &c.
Also let the geometrical equations which are to be introduced
by the method of indeterminate multipliers be
H& + K<i>'+... = (A (10),
&c. =
where E, H, &c. are in general functions of 0, <f>, &c., each of
which may be expanded in the form
i:=E^ + i;^0 + E^<f>^
The equations of motion of Art. 388 iO
A^^0" + &c. = aj + a J- + &c. + \E+ fiH+ &c.
A,^0"-^&c. = a, + aJ + &c. + XF + fjtK-^-&eX (11).
&c. = &c.
Since the system has been disturbed from a position of equi-
librium, these equations are satisfied by ^ = 0, ^ = 0, &c. We
thus obtain the equilibrium values of X, fi, &c. Let these be
\, H^, &c., then
0=a, + \E,+fjL^IT^ + &c.l
= &c. J ^12>
Let \ = \„ + \j, fi=fi^+fi^, &c. so that X,, /*,, &c. are small
quantities of the same order as 0, (f>, &c. The equations of oscil-
Ifition then become
A^/'+&c. = aJ+&c.+X, {E^0 + E^<l> + &c.) + \E, + &c.
&c. = &c.
}...(13).
Joining these to equations (10) we have a sufficient number
of linear equations to find 0, <f>, &c., X^, /Aj, &c. in terms of t. The
solutions of these equations may evidently be conducted as in
Art. 445.
ENERGY TEST OF STABILITY.
3G9
[ferential
t wish to
the La-
lained in
iaced by
rs of the
we may
treat the
le system
not have
efore the
)e absent.
itroduced
....(10),
, each of
.. (11).
a of eqiii-
&c. We
these be
,...(12).
are small
of oscil-
...(13).
it number
of t The
ted as in
i
The equations will be greatly simplified if the equilibrium values
of \, /*, &c. are all zero. This will generally be the case if 6, <^, &c.
can be so chosen that the first powers in the expansion of U are
absent. In this case E^, E^, &c. disappear from the equations,
so that it is unnecessary to calculate the geometrical equations
(10) beyond terms of the first order. The coefficients will then
be constant, and the equations can be integrated. As explained
in Art. 388, we may now reduce the number of variables B, <}>, &c.
to the proper number of independent co-ordinates. We may
therefore proceed as in Art. 444, without introducing \, /*, &c.
into the equations.
If, however, we prefer to retain the quantities X , /a,, &c., we
Boe by equations (10) and (13) that we may obtain the periods
exactly as in Art. 445, by equating the discriminant oi p'T+ U'
to zero, where
. u'=^u+ \ {E^e + F,<i>+...)+^^ {H^e + iir„<^+ ...) +«&c.
The determinant thus obtained has as many rows as there are
quantities 0, (f>, &c., \, fi^, &c.
The Energy test of Stability.
462. The principle of the Conservation of Energy may be
conveniently used in some cases to determine whether a system
of bodies at rest is in stable or unstable equilibrium.
Let the system be in equilibrium in any position and let V^ be
the potential energy of the forces in this position. Let the system
b*^ displaced into any initial position very near the position of
equilibrium and be started with any very small initial kinetic
energy T^, and let V^ be the potential energy of the forces in this
position. At any subsequent time let T and V be the kinetic and
potential energies. Then by the principle of energy
T+ v= 7;+ V, ; (1).
Let V be an absolute minimum in the position of equilibrium,
so that Fis greater than V^ for all neighbouring positions. The
initial disturbed position being included amongst these, it follows
that Fj — F, is a small positive quantity. Now the kinetic energy
T is necessarily a positive quantity, and since F is > F^, the
equation (1) shows that T is < T^ + F, — F„. Thus throughout the
subsequent motion the vis viva is restricted between zero and a
small positive quantity, and therefore the motion of the system
can never be great.
Also, since T is necessarily positive, the system can never
deviate so far from the position of equilibrium that F should
become greater than T^ + Fj. These two results may be stated
thus.
n. D.
24
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11
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370
SMALL OSCILLATIONS.
If a system he in eqnilihnum in a position in which the potential
enerrfy of the forces is a minimum or the work a maximum for all
displacements, then the system if slightly displaced will never acquire
any large amount of vis viva, and will never deviate far from the
position of equilibrium. The equilibrium is then said to be stable.
4C3. If the potential energy be an absolute maximum in the
position of equilibrium, V is less than V^ for all neighbouring
positions. By the same reasoning we see that T is always greater
than 7^^+ V^— V^, and the system cannot approach so near the
position of equilibrium that V should become greater than 7\ + V^.
So far therefore as the equation of vis viva is concerned there is
nothing to prevent the system from departing widely from the
position of equilibrium. To determine this point we must examine
the other equations of motion*.
If any principal oscillation could exist, let the system be placed
at rest in an extreme position of that oscillation, then the sys-
tem will describe that principal oscillation and will therefore pass
through the position of equilibrium. But if T^ be zero, V can
never exceed V^, and can therefore never become equal to V^.
Hence the system cannot pass through the position of equilibrium.
It is unnecessary to pursue this line of reasoning further, for
the argument will be made clearer in the next proposition.
4G1<. We may also deduce the test of stability from the equa-
tions which determine the small oscillations of a system about a
position of equilibrium. Let the system be referred to its prin-
cipal co-ordinates, and let these be 6, (j), &c. Then we have
2T=d^ + <i>"+
2iU-U,)=^h,e' + b,^'+
where b^, h^, &c. are all constants, and U^ is the value of U in the
position of equilibrium. Taking as a type any one of Lagrange's
equations •
.ddT_dT_dU
dtdd' dd~dO'
we have
e"-b,0 = O,
• This demonstration is twice given by Lagrange in his Mecanique AnahjUquf.
In the form in which it appears in the first part of that work, 7 is expanded in
powers of the co-ordinates, which arc supposed very small ; bnt in Section vi. of
the second part, this expansion is no longer used, and the proof appears almost
exactly as it is given in this treatise up to the asterisk. The demonstration in the
next proposition is simplified from that of Lagrange by the use of principal
co-ordinates.
EXERQY TEST OF STABILITY,
371
with similar equations for <f>, \ , (%c. If J, is positive, this equation
will give d in terms of real exponentials, and the equilibrium will
bo unstable for all disturbances which affect $, except such as
make the coefficient of the term containing the positive exponent
zero. If i, is negative, d will be expressed by a trigonometrical
term, and the equilibrium will be stable for all disturbances which
affect only. In this demonstration the values of b^, \,&c. are
supposed not to be zero.
If in the position of equilibrium U is a. maximum for all
possible displacements of the system, we must have 6,, J,, &c. all
negative. Whatever disturbance is given to the system, it will
oscillate about the position of equilibrium, and that position is
then stable. If Z7 is a maximum for some displacements and a
minimum for others, some of the coefficients b^, \, &c. will be
negative and some positive. In this case if the system be dis-
turbed in some directions, it will oscillate about the position of
equilibrium; if disturbed in other directions, it may deviate more
and more from the position of equilibrium. The equilibrium is
therefore stable for all disturbances in certain directions, and un-
stable for disturbances in other directions. If f^ is a minimum
in the position of equilibrium for all displacpimento, the coefficients
ij, 6 , &c. are all positive, the equilibrium >vill then be unstable
for displacements in all directions. Briefly, we may sum up the
results thus,
The system will oscillate about the position of equilibnum for
all disturbances if the potential energy is cu minimum for all dis-
placements. It will oscillate for some disturbances and not for others
if the potential energy is neither a maximum nor a minimum. It
will not oscillate for any disturbance if the potential energy ia a
maximum for all displacements.
It appears from this theorem that the stability or instability of
a position of equilibrium does not depend on the inertia of the
system but only on the force function. The rule is, give the
system a sufficient number of small arbitrary displacements, so
that all possible displacements may be compounded of these. By
examining the work done by the forces in these displacements we
can determine whether the potential energy is a maximum or
minimum or neither.
Ex. 1. A perfectly free particle is in equilibrium under the attraction of any
number of fixed bodies. Show that if the law of attraction be the inverse square,
the equilibrium is unstable. [Earnshaw^s Theorem.]
Let be the position of equilibrium. Ox, Oy, Oz any three rectangular axes,
then if V be the potential of the bodies, 6j = — , 62 = -v-^ , h- s^' ^^^ ^^'^^'^
the sum of these is zero, &j, b^, 63 cannot all have the same sign.
Ex. 2. Hence show that if any number of particles, mutually repelling each
24—2
1
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i
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1
1
1
(
1
■
1
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'4
372
SMALL OSCILLATIONS.
)!
r. I h
other, be contained in a vcBSfl, and be in eqnilibrinm, the equilibrium will be
unstable udIchr thoy all lie on the containing surface. [Sir W. Thomson, Camb.
Math. Journal, 1845.]
405. We may in certain cases apply the energy criterion to determine when
a given motion is ntable. Let a dynamical system be in motion in any
manner under a conservative system of forces, and let E be its energy. Then
J? is a known function of the co-ordinates 6, 4>, Sea. and their first differential co-
efficients ef, 0', iS;c. ; this is constant and equal to h for the given motion. Sup.
pose that either some or all of the other first integrals of the equations of motion
are also known, let these be
/"i {e, ff, Sea.) = Cj , F, (», ff*, &c.) = C,, *c. =&c.
For the purposes of this proposition, lot us regard 6 and 0', and 4>', &o. as inde<
pendent variables, except so far as they are connected by the equations just written
down. Then if E be an absolute maximum, or an absolute minimum, for all
variations of 0, ff, &c. (those corresponding to the given motion making E con-
stant), the motion is stable for all dibturbances which do not alter the constants
Ci,C„ Ac.
This result follows from the same reasoning as in Art. 462, which we may
briefly recapitulate thus. Let as many of the letters as is possible be found
from the first integrals in terms of the rest, and substituted in the expression for
E. Let ^, \(/', &o. be these remaining letters, then we have
•E = / (^, f , &c., Ci , C, , &c.) = h.
Let the system be started in some manner slightly different from that given, then
the constant k is altered into h + S/t. First let f be a minimum along the given
motion, then any change whatever of the letters \j/, f, ko. increases E, and it
follows that the disturbed motion cannot deviate so far from the given motion
that the change in E becomes greater than Sh. Similarly, if £ be an absolute
maximum, the same result will follow.
The same argument will apply to any first integral of the equations of motion,
besides the energy integral. If any one of the functions F,, F^, Ac, which con-
tains all the letters, be an absolute maximum or miuimiim, tlion the motion is
stable for all displacements which do not alter the constants of the other integrals
used.
When the system is disturbed from a position of equilibrium which is defined,
as in Alt. 444, by the vanishing of the co-ordinates 0, <f>, &o., we have
E=^Aiie'^ + Ai^B'4>' + &c.~ U,
where A^^, Ai^, &o. are all constants, and U is independent of ff, <f>', &o. Here
the terms which constitute the kinetic energy, being necessarily positive and
vanishing with ff, <j>' &c., are evidently a minimum for all variations of ff, 0', &c.
We see, without the use of any other integrals, that if - Z7 be a minimiim for
all variations of 6, <j>, &c., £ will be an absolute minimum, and that therefore
the eqiiilibrium is stable.
466. It often happens that the expression for the energy is not a function of
some of the co-ordinates, though it is a function of the differential coefficients of
all the co-ordinates with regard to the time. When this is the case, the system
admits of what we shall call a steady motion. Let x, y, &o. be the co-ordinates
which ore absent from the expression for the energy E, and let ^, ■>!, &o. be the
ilNEROY TEST OF STABILITY.
373
renifiininr; co-ordinates, then E in & function of f, 17. Ac, f, V. Ac., x', y', Ac. If
we form the 0(iuation8 of motion by Lagrange's rule (Art. 309), these equations
will contain (, ri, i',if\ f", V'. x',y', x'\if, Ac, Ac. Since those equations do
not contain t o\i»licitly, they may be satisfied by putting x'=^a, rj = h, Ac, f = a,
i;-/3, Ac, where a, b, Ac, a, /3, Ac. are constants to be determined by substituting
in the equations. If 6 stand for any one of the co-ordinates, it is evident that
,^ and ..,, will both bo constants after the substitution is made. The constanta
av ad
must therefore satisfy the tj-pical equation -— j^ — ' = (Art. 3C9). Siuce «, y, Ac.
da
are absent from the expressions for T and U, this is an identity if we write any of
these co-ordinates for 0. Hence we ^ave as many equations, viz.
d(T+U)_ d{T+U)
0).
fts there are co-ordinates (, 17, Ac. present in the expressio is for T and U. The
quantities a, b, Ac. are therefore undetermined except by the initial conditions,
while a, ff, Ac. may be found in terms of a, b, Ac by these equations. These
equations may be conveniently remembered by the following rule. In the Lagran-
gian function, which is the difference between the kinetic and potential energies,
trrite for the differential coefficients, their assumed constant values in the steady
motion, viz. x'— a, &o., ^=0, &o. Differentiating the result partially with regard
to each of the remaining co-ordinates, we obtain the equations of steady motion.
467. To determine if this motion is stable, we must by Art. 465 use the integrals
Let
-r-7=«, x-/ = ''i &c., where «, v, Ac. ore constants,
dx dy
T = ^ {XX) x'^ + (a;^) x'f' -f Ac.
.(2),
where the coefficients of the accented letters, viz. the quantities in brackets, are
all known functions of {, 17, Ac, but not of x, y, Ac. The integrals may then be
written in the form
(sKc) a;' -I- (a;y) y -f . . . = u - (a;f ) f ' - (a!ij) V - &c. I
(«y)a!'+(yy)y'+- = ''-(ys^)£'-t'/'»)V-&4 (3).
Ac. =Ac.
7' -Ac J
For the sake of brevity, let us call the right hand sides of these equations u-X,
v-Y, Ac Since T is a quadratic function of the accented letters, we may write
it in the form
T=lmr+{iv)^r,' + &o. + lx'{u+X) + ly'{v+Y) + &o.
If we substitute in the terms after the first Ac. the values of x', y' given by (3)
we obtain the determinant
2A
0, u-\-X, v+Y, Ac
u-X, (xx) {xy), Ac.
v-Y, {xy), {yy), &o.
Ac
where A is the discriminant of T, when {', ij', Ac. have been put zero. If we change
the signs of A', Y, Ac, this determinant is unaltered, hence when expanded such
terms as uX, vX, Ac. cannot occur. If therefore, we put
•i!
i
t
1 !
\ 1
if
!!
f-
i
,
'i I
m
*., :«i
m
m
■
fl;:
1'
■|
■I 1|i. ■
.*^r ' .
I i
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374
SMALL OSCILLATIONS.
F=
1^
2A
M W
u (xx) (xy)
.(4),
aud expand the first determinant, we have
(5),
vhere the terms after F express some homogeneous quadratic function of (', if, &o.
When f , 7)', &e. are prt zero, the process of finding F is exactly that described
in Art. 378, as the Hamiltonian method of forming the reciprocal function.
Following the same proof* as in that Article, we may show that if ^ be any letter
JTT JET
contained in T, we have ^ = - j^ • Hence the equations of steady motion (1) may
also be written in the fjrm
d{F-V)
._rHF-U)
''- du '
d{F-U)
dr,
=0
y =
d(F-V)
do
(0).
where F - U is the energy expressed rs a function of u, v, &c. instead of x', y', Ac,
the other accented letters, viz. ^', rj, &c. heing put equal to zero either before or
after differentiation.
Further T is essentially positive for all values of a/, y', &c. and therefore for
such as make m, v, &o. all zero. Hence the quadratic expression Bu^'* + &c. is a
minimum when ^', n)', &c. are zero. If then the function F -TJ is a minimum for all
variations of f, ij, &c., the steady motion given by (6) is stable for all disturbances
which do not alter the momenta u, v, &c.
468. If the energy be a function of one only of the co-ordinates, though it is a
function of the differential coefficients of all of them, we may show conversely that
the steady motion will not be stable unless F -^ U wo minimum.
Let { be this single co-ordinate, then following the same notation as before, we
have by Vis Viva
Is^.^'^ + F-U^h.
Differentiating with regard to t, and treating J5n as constant because we shall
neglect the square of f*, we obtain
* Taking the notation of Art. 378, the proof is as follows. The total differential
of T^ when all the letters vary is
^-^^de-'^l^-
do t/|
dT,= -'^de- -— » di+(-~^ + tA dff+ 6'du + &c. ;
as before, the quantity in brackets vanishes, and hence when T, is expressed as a
(IT
function of 9, </>, &o., w, v, &e. and {, wo have --.'=
"4
dT^
di '
w,
(5),
)f {', V. &0.
it desoribed
1 function,
e any letter
ion (1) may
(fi),
fx',y',&o.,
'.r before or
herefore for
'» + &c. is a
mum for all
iiattirbances
ough it is a
versely that
i before, we
se we sball
differential
rcsBcd as a
. OSCILLATIONS ABOUT STEADY MOTION.
To find the oscillation, let f =a + p, then by (6) we have
3V5
d^p r dHF-U) -t
^^^dr^ + l—de~V'
where a is to ,be written for { after differentiation in the quantity in square
brackets. The motion is clearly stable or unstable according as the coefiSciont of p
is positive or negative, i.e. according as i^- U^ is a minimum or maximum.
469. Ex. 1. Let us consider the simple case of a particle describing a circular
orbit about a centre of attraction whose acceleration at a distance r is /ur". If 6 be
the angle the radius vector r makes with the axis of x, we have her« a steady motion
in which /=0 and ^ is constant. Also
1 ur"
n + i*
We notice that is absent from this expression, hence by the rule we eliminate
0' also by the integral rW=h, where h is the constant called u in Art. 467. We
have then
1 ,„ 1 /i" /tr"+i
£=tir" + -■-,, + '
2r^ ' n + 1
Putting the remaining accented letters equal to zero according to the rule, we
have in steady motion
dr J..1 <- '
and since
this steady motion is stable or unstable according as n + 3 is positive or negative
for all disturbances which do not alter the angular momentum of the particle.
Ex. 2. Taking the example considered in Art. 374, show that a state of steady
motion is given by $ constant and that it is stable if C^ii^ + iMgJiA cos d is positive.
Hence ii d < ^ the motion is stable for all values of n.
Ex. 3. A solid of revolution moves in steady motion on a smooth horizontal
plane, so that the inclination of its axis to the vertical is constant. Prove that
the angular velocity fi of the axis about the vertical is given by
Cn Mg dz
'* Adoso'^'^ ABVuOao^edd
=0,
where z is the altitude of the centre of gravity above the horizoital plane, n the
angular velocity of the body about the axis, C, A and A the principal moments
of inertia at the centre of gravity and M the mass. Find the least value ol n which
makes /x real and determine if the steady motion is stable.
Oscillations about Steady Motion.'
470. The oscillations of a system about a state of steady
motion may be found by methods analogous to those used in the
oscillations about a position of equilibrium. Let the general equa-
tions of motion of the bodies be formed by any of the methods
already described. If any reactions enter into these equations it
4
■
m
''if
Wi
Ml
I !
\
1 '
: 1
376
SMALL OSCILLATIONS.
will be generally found advantageous to eliminate them as ex-
plained in Art. 428. Let the co-ordinates used in these equations
to fix the positions of the bodies be called 0, <^, &c. Suppose the
motion, about which the oscillation is required, to be determined
by =f\t), <f> = F(t), &c Then exactly as in Art. 428, we substi-
tute =f[t) +x, ^ = F{t) + y, &c., in the equations of motion.
The squares of x, y, &c. tjeing neglected, we have certain linear
equations to find x, y, &c. These equations can, however, seldom
be solved unless we can make t disappear explicitly from them.
When this can be done the linear equations can be solved by the
usual known methods, and the required oscillations are then found.
In what follows we shall first illustrate the method just de-
scribed by forming the equations in a few interesting cases from
the beginning. We shall then generalize the process and obtain a
determinantal equation analogous to that given by Lagrange for
oscillations about a position of equilibrium. This equation will be
adapted to all cases which lead to differential equations with
constant coefficients.
471. Ex. 1. To find the motion of the balls in WatVs Oovernor of the steam
engine.
The mode in which this works to moderate the fluctuations of the engine is well
kno\vn. A somewhat similar apparatus has been used to regulate the motion of
clocks, and in other cases where uniformity of motion is required. If there be any
increase in the driving power of the engine, or any diminution of the load, so that
the engine begins to move too fast, the balls, by their increased centrifugal force,
open outwards, and by means of a lever either cut off the driving power or increase
the load by a quantity proportional to the angle opened out. If on the other hand
the engine goes too slow, the balls fall inward, and more driving power is called
into action. In the case of the steam engine the lever is attached to the throttle-
valve, and thus regulates the supply of steam. It is clear that a complete adapta-
tion of the driving power to the load cannot take place instantaneously, but the
machine will make a series of small oscillations about a mean state of steady
motion. The problem to be considered may therefore be stated thus : —
Two equal rods OA, OA', each of length I, are connected with a vertical spindle
by means of a hinge at which permits free motion in the vertical plane AOA'. At
A and A' are attached two balls, each of mass m. To represent the inertia of the
other parts of the engine we shall suppose a horizontal fly-wheel attached to the
spindle, whose moment of inertia about the spindle is /. When the machine is in
uniform motion, the rods are incIiueJ at some angle a to the vertical, and turn
round it with uniform angular velocity n. If, owing to any disturbance of the
motion, the rods have opened out to an angle Q with the vertical, a force is called
into play whose moment about the spindle is - /3 (0 - a). It is required to find the
oscillations about the state of steady niotiou.
Let be the angle the plane AOA' makes with some vertical piano fixed in
space. The equation of angular momentum about tlio spindle is
l^^,2,n„^0)'^.
■P(0-a).
(1).
THE GOVERNOR.
377
where mk^ is the moment of inertia of a rod and ball about a perpendicular to the
rod through 0, the balls being regarded as indefinitely small heavy particles. The
semi Vis Viva of the system is
and the moment of the impressed forces on either rod and ball about a horizontal
through perpendicular to the plane A OA' is g -j^ = -mghBin 0, where h is the dis-
tance of the centre of gravity of a rod and ball from O. Hence by Lagrange's
d dT dT dU
dt '
where a has been written for
sin cos 6
i2
m-i'^' (%
This equation might also have been obtained by
taking the acceleration of either ball, treated as a particle, in a direction perpen-
dicular to the rod in the plane in which d is measured.
To find the steady motion we put 0=a, 37=n> ^^^ second equation then gives
n» cos tt = - . To find the oscillations, we put d = a + x,-^=n-\-y. The two equa-
tions then become
(1+ 2mlc^ sin« a) :^ + 2mJfln sin 2a -j- = - fix
* ' /** dt
dt
• n sin 2oy = ( n* cos 2a - - cos o j «
To solve these equations, we must write them in the form
(sin2a2> + 2-'|^) nx + (^-^^ sin»a)z>,=0) ^
(Z)' + n* sin* a)x-n sin 2ow = 0.'
(Z)' + n* sin* a)x-n sin 2oy = (
where the symbol D stands for the operation ^ . Eliminating y by cross multipli-
cation we have
\_\2mh
^^.^ + sin''a)D'> + n*sin''a^l + 3cos«a + 2^)D
+ 2l«F^"""^"]*=^-
The real root of this cubic equation is necessarily negative because the last term
is positive. The other two roots are imaginary because the term D* has dis-
appeared between two terms of like signs. Also the sum of the three roots being
zero, the real parts of the two imaginary roots must be positive. Let these roots
therefore be - 2p and j)±:qj -1. Then
X = He-^»* + AV sin (g« + L),
where H, K, L are three undetermined constants depending on the nature of thg
initial disturbance. Thus it appears that the oscillation is unstable. The balls
will alternately approach and recede from the vertical spindle with increasing
violence.
i'
:fi
X
js
V
l\
i: /
if
PlA\
"Ay
I:
I '%
m
^
;'i
I •!
1
378
SMALL OSCILLATIONS.
472. A common defect of governors is that they act too quickly, and thua
produce considerable oscillation of speed in the engine. If the engine is working
too violently, the governor cuts off the steam, but owing to the inertia of the parts
of the machinery, the engine does not immediately take iip the proper speed.
The consequence is that the balls continue to separate after they have reduced
the supply of steam to the proper amount, and thus too much steam is cut off.
SimQar remarks apply when the balls are approaching each other, and a con-
siderable oscillation is thereby produced. This fault may be very much modified
by applying some resistance to the motion of the governor.
In the same way when the motion of clock-work is regulated by centrifugal
balls, it is found as a matter of observation that there is a strong tendency to
irregularity. If the balls once receive in the slightest degree an elliptic motion,
the resistance p {0-a) by which the motion of the balls is regulated may tend to
render the elliiiso more and more elliptical. To correct this some other resistance
must be called into play. This resistance should be of such a character that it
does not affect the circular motion and is only produced by the ellipticity of the
movement.
One method of effecting this has been suggested by Sir G. Airy. The elliptic
motion of the balls may be made to cause a slider on the vertical spindle to rise
and fall. If this be connected with a horizontal circular plate in a vertical
cylinder of slightly greater radius, and filled with water, the sUder may be made
to move the plate up and down by its osl Illations. Thus the slider may be
subjected to a very great resistance, tending to diminish its oscillations, while its
place of rest, as depending on statical, or slowly altering forces, is totally un-
affected. Memoirs of the Astronomical Society of London, Vol. xx., 1851.
Tho general effect of the water will bo to produce a resistance varying as the
velocity, and may therefore be represented by a term -y-fr on the right hand of
equation (2).
form
rit
The solution beuig continued as before, the cubic will now take the
/3
If the roots of this cubic are real, they are all negative, and the value of x takes the
form
x = Ae-''^ + Be-''i+Ce-''*,
where -X, -/i, -v are the roots, and A, B, C are three undetermined constants.
If one root only is real, that root is negative, ana if the other two be jp ± g v' - 1 tho
value of X takes the form
X = lie. - ♦■« + Ke^t sin (2< + L) ,
where 11, K, L as before are undetermined constants.
In order that the motion may be stable it is necessary that p should be negative.
The analytical condition* of this is
• If the roots of the cubic aji? + bx'^ + cx + d=Ohex=a:i^PyJ{-l) and y, we have
-- = 2a + 7,-=27a + a2 + |3=, _ - == (aS + /S^) 7, whence wo easily deduce ^-^~
= - 2a{(a-f7)* + /32}; hcuco be - ad and a have always opposite signs. See Art. 436.
'
* •
THE GOVERNOR.
3^
(9
7(l + 3coB«a + 2-i;j^,)>2,^^,
2coto,
If
7 be
sufficiently great
this condition may be
satisfied.
The
uniformity
of
motion of the rods round the vertical will then be disturbed by an oscillution whose
magnitude is continually decreasing and whose period is — . By properly choosing
the magnitude of I when constructing the instrument, the period may sometimes
be so arranged as to produce tlie least possible ill effect. If the period bo made
very long the instrument will worlf smoothly. If it can be made very short tliere
will be less deviation from circular motion.
In tliis investigation no notice has been taken of the frictions at the hinge and
at the mechanical appliances of the Governor, which may not be inconsiderable.
These in many cases tend to reduce the oscillation and keep it within bounds.
473. In the case of Watt's Governor if any permanent change be made in the
relation between the driving power and the load, the state of uniform motion which
the engine will finally assume is different from that which it had before the change.
Thus, when the engine is driving a given number of looms, let the rods OA, OA' of
the Governor be inclined to each other at an angle 2a and be revolving about the
vertical with an angular velocity n. If some largo number of the looms is sud-
denly disconnected from the engine, the balls will separate from each other, and the
rods will become inclined at some other angle 2a'. In this case, if n' be the angulir
velocity about the vertical, n'*cos a' = Mucosa. The rate of the engine is therefore
altered, it works quicker with a leua load than with a greater. This is a great
defect of Watt's Governor. For tliis reason it has been suggested that the term
Governor is inappropriate, the instrument being in fact only Vi, moderator of the
fluctuations of the engine.
This defect may be considerably decreased by the use of Huyghens' parabolic
pendulum. In this instrument the centres of gravity ^ , ^4' of the balls are made to
move along the arc of a parabola whose axis is the axis of revolution. Let AN ho
an ordinate of the parabola, A the normal, then NG is constant and equal to L,
where 2Z is the latus rectum. Regarding the balls as particles, and neglecting the
inertia of the rods which connect them with the throttle valve, we see by tlio
triangle of forces that the balls will rest in any positions on the parabola, if
n^L=g, where n is the angular velocity of the balls about the vertical through 0.
It is also clear that when the angular velocity is not that given by this formula, the
balls ^unless placed at the vertex) must slide along the arc. Let us now consider
how this modification of the governor affects the working of the engine. When the
load is diminished the engine begins to quicken ; the balls separate and the steam is
cut off. It is clear that equilibrium will not be established until the quantity of
steam admitted is just such as to cause the engine to move at exactly the same rate
as before.
Ex. Show that when the inertia of the rod and ba^^s are taken account of,
the centre of gravity of either ball and rod must be constrained to describe a
parabola whose latus rectum is independent of the radius of the ball, if the
Governor is to cause the engine always to move at a given rate.
474. The reader who may be interested in the subject of Governors may refer
to an article by Sir G. Airy, Vol. XI. of the Memoirs of the Astronomical Society,
18-10, where four different constructions are considered Ho may also consult an
' \'
I 111
; f
P.n
Il
ll
\ *
380
SMALL OSCILLATIONS.
article by Mr Siemens in the Phil. Tram, for 1866, and a brief sketch of several
kinds of governors by Prof. Maxioell in the Phil. Mag. for 1868. An account of
some experiments by Mr Ellery, on Huyghens* paraboUo pendulnm, may be found
in the A8tro7wmical Notices for December, 1875.
475. Ex. 2. It has been shown in Art. 282 that if three particles be placed at the
comers of an equiangular triangle and properly projected, they will move under
their mutual attractions so as always to remain at the angular points of an equi-
lateral triangle. These we may call Laplace's three particles. It is our present
object to determine if this motion is stable or unstable*.
Let the mass M of the particle to be reduced to rest be taken as unity, and let
m, m' be the masses of the other two. Let r, r', R be the distances between the
particles Mm, Mrri, mm'; and let 0', ^, \j/ be the angles opposite to these distances.
If 0, d' be the angles r, r' make with a straight line fixed in space, and if the law of
attraction be the inverse xth power of the distance, the equations of motion are '
rdt\ dt)
\b m' cos d) „
R"
m' sin \f/ m' sin (f>
~~r^ ^^
=0
'I
with two similar equations for the motion of nt'.
Let us now put r=a+x, r'=a+x+ X, and let the angle between these radii
vectores be ^ + T, also let 0==nt+y, where x, y, X and Y, are all small quantities
whose squares are to be neglected. It should be noticed that a variation of x, y
alone, X and Y being zero, will represent a variation of steady motion in which the
particles always keep at the corners of an equilateral triangle, while a variation of
X, Y will represent a change from the equilateral form. The former of these we
know by Art. 282 is a possible motion, hence the equations can be satisfied by some
values of x, y joined to X=0, Y—0. By this choice of variables we may hope to
discover some roots of the fundamental determinant previous to expansion, and
thus save a great amount of numerical labour,
tions will now become
If D stand for ^ , the four equa-
l6Z)'- (K+l)(l + TO + m')U-2a6»i)i/-|»i,'(K + l)2C- jmV + l)ai'=0,
V3
3
1
2hnBx-k- a62)>--^-m'((c + l)X+jm'(K + l)ar=0,
6Z)»-(K + l)(l+OTH-j>i')|«-2«6>i2)i^+}62)--(/c+l)(l+| + m')jX-J2a6wZ)+^m(/c + l)ajr=0,
2hnDx+ abD''y+ \2bnD-^~(K+l)mlx+ |a62)2- Jm(K + l)aj F.-^O.
* In a brief note in JuUien's Problems, Vol. ii. p. 29, it is mentioned that this
question has been discussed by M. Gascheau in a These de M^canique, the particles
being supposed to attract each other according to the law of nature. The result
arrived at is that the motion is stable when the square of the sum of the masses is
greater than 27 times the sum of the products of the masses taken two and two.
No reference is given to where M. Gascheau's work can be found, and the author is
therefore unable to give a description of the process employed.
these radii
Laplace's three pauticles.
381
476. To solve tbese we put a; = .4 e^', y = Be**, X=Gc'^', r=i7e^'. Substituting
and eliminating the ratios ot A, B, and H we obtain a dcterminantal equation
whose constituents are the coefficients of x, y, X and 1" with X written for D. This
equation will give six values of \. We see at once that one factor is \, This m:ght
have been expected, because we know that a variation of y with x, X and Y all zero,
is a possible motion. Again, some variation of x and y with X and Y both zero is
also a possible motion, hence some factor of the determinant can be found by ex-
amining the first two columns. By subtracting from the first 2n times the second
column we find that this factor is 6\* - (k - 3)(1 + m + m')=0.
To find the other factors we divide the determinant by the factors alrea-ly
found. Then subtracting the first row from the third and the second from the
fourth wo have three zeros in the first column and two in the second. The
expansion is then easy. We see that there is another factor X, also
6«XH 6X''(3 - /c)(l + m + m') + 1(1 + K)2(m + m' + mm') = 0.
The two zero roots give x=Ai + A^t with similar expressions y, X and Y. But
K + 1 A
by substitution in the equations of motion we see that x=A^, y—Bj^ — x- * nt,
X=0 and F=0. These roots therefore indicate merely a permanent change in the
size of the triangle. On examining the other values of X*, we find (1) The motion
cannot be stable unless k is less than 3. (2) The motion is stable whatever the
masses may be, if the law of force be expressed by any positive power of the dis-
tance or any negative power less than unity. (3) The motion is stable to a first
approximation if
(Af+wi+m'js
:-GHy.
Mm + Mm' + mm'
where M, m, m,' are the masses. To express the co-c*dinates in terms of the time,
we must return to the diffe rential equations of the s> cond order. The results are
rather long, and it may be Ki>f .' lient to state that when, as in the solar system, two
of the masses are much smallev than the third, the inequalities in their angular
distances, as seen from the large body, have much greater coefficients than their
linear distances from the same body.
477. To form the general equations of oscillation of a dynami-
cal system about a state of steady motion.
Let the system be referred to any co-ordinates 6, <^, ■^, &c.
Let the state of motion about which the system is oscillating be
determined by 6 =f (t), <f> = F (t), & c, then as explained in Art. 470
we shall put d=f{t)+x, ^ = F't)+y, &c. Let the Lagrangian
function L (see Art. 381) be exjjanded in powers of x^ y^ &c., as
follows :
i = i„ + A^x + A^y + &c. 4- B^x + B^' + &c.
^ \ {A,,x- + ^A,,xy + &c.) + \ {BJ' + ^IBJy' + &c.)
+ C.^xx + C^^xy + C^^yx + &c.
■ i
I >
I ! 1
; i I
1
si M
j
lit' .
J
f^
i.
III
I- 'm
382
SMALL OSCILLATIONS.
We shall now define a steady motion to be one in which all the
coefficients in this expansion are independent of the time. The
physical characteristic of such a motion is that when referred to
l^roper co-ordinates the same oscillations follow from the same dis-
turbance of the same co-ordinate at whatever instant it may be
applied to the motion. If the coefficients are not constant for the
co-ordinates chosen it may be possible to make them constant by
a change of co-ordinates. There are obviously many systems of
co-ordinates which may be chosen, and a set may generally be
found by a simple examination of the steady motion. If there are
any quantities which are constant during the steady motion, such
as those called ^, 17, &c. in Art. 466, these may serve for some of
the co-ordinates, others may be found by considering what quanti-
ties appear only as differential coefficients or velocities, for example
those called x, y, &c. in the same Article, If none of these are
obvious, we may sometimes obtain them by combining the existing
co-ordinates. Practically these will be the most convenient
methods of discovering the proper co-ordinates.
478. To obtain the equations of motion we must now substi-
tute the value of L in the Lagrangian equations
ddL_dL
dt dx dx
= 0, &c. = 0,
and reject the squares of small quantities. The steady motior.
being given by x, y, &c. all zero, each of these must bo satisfied
when we omit the terms containing a;, y, &c. We thus obtain the
equations of steady motion, viz.
A^ = 0, ^2 = 0, &c. = 0,
which by Taylor's theorem are the same as the equations (1) of
steady motion give i in Art. 466.
Omitting these terms and retaining the first powers of all the
small quantities we obtain the equations of small oscillations, of
which the following is a specimen :
+ |b„|' + (C., - C„) ^ - ^..} 2 + &c. = 0.
To solve these we write x = L^*, y = Me^^, &c. Substituting and
eliminating the ratios of L, M, &c. we obtain the following deter-
minantal equation
ABOUT STEADY MOTION.
383
^u'^' - ^u
&c.
■??.^' - K
&c.
As^^ - ^33
&c.
&c.
&c.
&c.
&c.
= 0.
If in this equation we write — X ^or \ the rows of the new
doterminant are the same as the columns of the old, so that the
determinant is unaltered. When expanded the equation contains
only even powers of \.
479. Regarding this as an equation to find X", we notice that
if the roots are all real and negative, each of the co-ordinates cc, y,
&c. can be expressed in a series of trigonometrical terms having
different periods; the motion will therefore be stable. If any one
of the roots is imaginary or if any one is real and positive, there
will be both positive and negative real exponentials entering into
the expressions for x, y, &c. and therefore the motion will be un-
stable. The condition of dynamical stability is therefore that the
roots of this equation must all be of the form \ = + fjbs/ — 1, where
/A is some real quantity.
480. It follows also that when a system, under the action of
forces which have a potential, oscillates about a stable state of
steady motion, the oscillations of the co-ordinates are represented
by trigonometrical terms of the form A sin (\t + a.) which are not
accompanied by any real exponential factors such as those which
occurred in the problem of the Governor.
We see further that there will in general be as many finite
values of X" and therefore as many trigonometrical terms of differ-
ent periods as there are co-ordinates. It often happens, as ex-
plained in Art. 477, that some of the co-ordinates are absent from
the expression for L, appearing only as differential coefficients.
Suppose for example 6 to be absent; then A^^, A^^, &c. are all
zero, and we may divide X both out of the first line and the first
column of the fundamental determinant. We therefore have two
zero values of X, while at the same time the number of finite
values of X** is diminished by unity. Hence the number of trigo-
nometrical terms of different periods cannot exceed the number of
1^:
.
'\ !
a I
i '' ' I'
Mf
V
■I.. I,
i -hfl
384
SMALL OSCILLATIONS.
co-ordinates which explicitly enter irito the Lagrangian function.
For example in Art. 374, the function T— f^has only the co-ordi-
nate 6 explicitly expressed, the others 0' and >^' appearing only as
differential coefficients. It follows that if a top is disturbed from
a state of steady motion, there will be but one period in the
oscillation.
481. The relations between the coefficients / ice. in the
exponential values of x, y, &3. may bo obtained wi>.iiout difficulty
if we remember that the several lines of the fundamental determi-
nant are really the equations of motion. Taking any one line ;
multiply the first constituent by L, the second by M, &c. and
equate the sum to zero. We thus obtain as many equations as
there are co-ordinates. On the whole we shall have, exactly as in
Art. 445, twice as many arbitrary constants as there are co-ordi-
nates, all the other constants being determined by the equations
just found. The arbitrary constants are determined by the initial
values of the co-ordinates and their differential coefficients.
But, unlike Art. 445, the quantity \ occurs in the firsf power
in each of these equation.s, so that the ratios of L, M, Sic. thus
found may be imaginary. The expressions for the co-ordinates
when rationalized may therefore take the form
■x=A^ sin {\t + a,) + A^ sin (\< -f- ot^) + . ..
y=B^ sin {\t -F )9J -h i?., sin (\< + ^J + . . .
z = &c.
where a^ is not necessarily equal to ^^, nor ofj, to y3^, &c., though
they are connected together.
482. When the initial conditions are such that every co-
ordinate is expressed by a trigonometrical term of one and the
same period, the system is said to be performing a principal or
harmonic oscillation. Thus each trigonometrical term corresponds
to a principal oscillation, and any oscillation of the system is
therefore said to be compounded of its principal oscillations. The
physical characteristic of a principal oscillation is that the motion
of every part of the system is repeated at a constant interval.
48.3. The stability of the motion depends on the nature of the
roots of the fundamental determinant. If we expand the determi-
nant we may use the methods given in the theory of equations to
discover if the roots are all of the proper form. This however is
often tedious and we may sometimes settle the point by a simple
examination of the determinant as it stands.
;i!
ABOUT STEADY MOTION.
385
In practice it frequently happens that the determinant is
reducecl to two rows. If the invariants be written
A = A^,A„
■^ii>
s=^.A,-K'>
= AA + AA-2AA
the conditions of stability are
(1) A is positive.
(2) (C^ai ~ CJ' - is positive and greater than 2 VZ/y.
These conditions may also be expressed thus. Let a and /S be
the roots of the quadratic formed by omitting the terms containing
(7,g and C„. Then by Art. 448, a and ^ are real. If a and /3 are
both negative the motion is stable. If both are positive, the
C " C
motion is stable or unstable according as -",— " is numerically
greater or less than sja + i^^, the roots being taken positively. If
a and /3 have opposite signs, the motion is unstable.
Whatever maybe the number of co-ordinates, it may be shown
that the motion cannot be stable unless the discriminant of
A^^x^ + ^A^^xy + &c. is positive or negative according as the
number of rows is even or odd.
The following theorem is also useful. Beginning with the
fundamental determinant we may form a series of determinants,
each being obtained from the preceding by erasing the first lino
and the first column. As we may supplement the fundam ital
determinant with a row and a column of zeros added on at the
bottom and right-hand side with unity at the right-hand bottom
corner, we may suppose the series of determinants to terminate
with unity. Let us substitute in the series any negative value of
X"" and count the number of Variations of sign in the series. Then
as \' changes from — oo to 0, there cannot be fewer negative roots
between any two given values of \' than there are losses in the
number of variations of sign corresponding to the two values of \'.
If there be more negative roots than losses the excess nmst be an
even number.
484. Ex. A homogeneous sphere of unit mass and radius a is suspended from
a fixed point by a string of length h, and is set in rotation about the vertical diame-
ter. When the sphere is slightly disturbed, let hx, hy and b be the co-ordinates of
the point on the surface to which the string is attached; hx+af, by -i-arj, and b + a
the co-ordinates of the centre, the fixed point being the origin and the axis of z
being vertical and downwards. Also let x='P + ^ where ^ and ^ have the same
meaning as in Art. 235, so that before disturbance x'=n. Prove that the La-
grangian function is
R. D. 25
11
i
1 ,
1
,
1
1
j 1
'i i
' 1
1
!
i
ii
i it[|
H
) ;i
I M
386
SMALL OSCILLATIONS.
If the motion of tho centre of gravity be roprosontod by a BoriM of tonua of tho
form 31 COB (jit T N), prove that tlio voIuub of m are given by
(.•-») (^'---rO'i--
Bliow that wliatovor sign n may have thia equation has two positivo and two
negative roots, which ore separated by tho routs of either of the factors ou the loft-
hand side.
f! 1
Application of the Calculus of Finite Differences.
485. We shall give some examples to illustrate the use of the
Calculus of Finite Differences in cases in which there are an in-
definite number of bodies similarly placed.
48G. Ex. A string of length (n + 1) 1, and insensible mass,
stretched between two fixed points with a force T, is lauded at
intei'vals 1 with n equal masses m not under the influjnce of gravity
T ...
aiid is slightly disturbed ; if f~ = c', prove that the periodic times
of the simple transversal vibrations which, in general coexist are
given by the formula — cosec-x-. — — -rv on putting in succession
issl, 2, 3...n.
Let At B be the fixed points; y,, ^^,'..t/^ the ordinates at
time t of the n particles. The motion of the particles parallel to
AB is of the second order, and hence the tensions of all the strings
must be equal, and in the small terms we may put this tension
equal to T. Consider the motion of the particle whose ordinate
is y^ The equation of motion is
^ J/tc _ .Vn-1 *" Vk rp Vk "" Vk-i rp .
^dt'" I ^ r ^'
.•.g* = c'(2/,,.-22r, + y,J (1).
Now the motion of each particle is vibratory, we may therefore
expand y^ in a series of the form
y, = -^1 sin (pt + a) (2),
where 2 implies summation for all values of ^x
lUB of tho
'0 and two
1 the loft-
Be of the
■e an iu-
hle mass,
oiided at
f gravity
)dic times
\€odst are
siLccession
inates at
arallel to
he strings
is tension
ordinate
.... (1).
therefore
,...(2),
CALCULUS OF FINITE DIFFEBENCES.
387
As there may be a term of the argument pt in every y, let
L^, L^, ... ho their respective coefficients. Then substituting,
wo have
A+i ~ 2Zft + A-i == - ^ A
■(»).
To solve this linear equation of (liffcrencoa wo follow the usual
rule. Putting L^ — Aa^, where A and a are two constants, we get
after substitution and reduction a — 2 + - = — ( ^- ) , or
^a-^ =^ V^J and y/a + -)~ = ± 2 a/i - f^'V;
\Ja c ^ \/a y \2cJ
Let these roots be called a^ and a^, then
is a solution, and since it contains two arbitrary constants, it is the
general solution ;
.-. y, = 2[^a,* + Z?a,*]sin(p« + a) (4).
The equations (1) and (3) will represent the motion of every
particle from ^ = 1 to ^• = n, provided wo suppose y^ and t/,,^, both
zero, though there are no particles corresponding to values of k
equal to and « + 1. Since y = when A; = for all values of t,
every term of the series must vanish; .•. -4+i? = 0. Alsoy=0
when A; = « + 1 for all values of « ; .'. Ja,"*' + i^V' = 0. These
equations give a^*^ = a^*^. But if |- > 1, the ratio of a^ to a,^ is
2c
real and different from unity. Hence wo must have |- < 1. Let
2c
then ^ = sin ^ ; and therefore a = cos 2^ + sin 2^ V— 1.
2c
Hence, by what we proved before,
(cos 20 + sin 29 V- I)'"* = (cos 26 - sin 26 V^)"'' ;
W ITT
.•.sin2(n+l)^ = 0, or |^ = sin ^^-^^ ,
and the period of any term =
If m and I be indefinitely small and n indefinitely large, tho
loaded string may bo regarded as a uniform string of Ici.gtli
{n-\-l) 1= L and mass nm = M stretched between two fixed points
25—2
111 m
•I! 1
?li
,1 i.
iit
I I II
m
m
ivi !
ll \
388 SMALL OSCILLATIONS.
with a tension T. In this case the expression just found reduces
toi)=7rty^.
487. If we substitute these values of in the expressions for a^ and a^, we
easily find
y4=SC<sin
kiir
n + 1
sin ] 2ct sin
»ir
2 (n+1)
+ ai
where d has been written for 2 A -J^, 0( for a, and the symbol 2 implies summa-
tion for all integer values of i from i = l to t=n. This expression has n terms,
and thus we have 2n arbitrary constants, viz. Cj, Cj ... C„ and Oj, Oj ... a„. These
are to be determined by the known initial values of yi,y^, &c. and —^ , -^, &o.
To find these it will be more convenient to write the expression in the form
iir
Vie = zEi sm — r sin i 2ct sin ^r- -. .
'^ n + 1 ( 2{n + l)
I + SF; sin — "!, cos ] 2ct sin
' n + 1
2 (n+1)
Putting t = 0, we have the two typical ectuationa
Mir
[yt]o = Si^<sin
n + 1'
tir
2c \_dt Jo ' n + 1 2 (n+1)
It is a theorem in Trigonometry that if i, V be any integers between and
n+1, the sum of the neries 2 sin —7^ sin— rr; taken from Jt=l to h=n is zero
w+1 n+1
n + 1
when i is different from i' and the sum is equal to — ~- when i=f'. This may be
proved by expressing the general term of the series as the difference of two cosines,
thus separating the given series into two series, each consisting of cosines of angles
in arithmetical progression. Summing these from i=0 to fc=n when i and i' are
both even or both odd, and from ^■=l to k=n when i is even and i' odd, we easily
fi'id the whole sum to be zero when i and i' are unequal. This change in the limits
of the summation only adds a term which is zero to one end of the original series
and therefore does not affect the sum. When i and i' are equal the value of the
series may be found in a similar manner.
This theorem will at once enable us to find the general values of Et and Fi.
Let us multiply both sides of the first typical equation by the coeflicient of Fi and
sum all the series of which it is the type. We have
<« (r , . Uir ) n + 1 ^
S|[y*]oSinjj^^j=-^-ii',.
where 2 implies summation for all values of Ic from fc = l to h=n. . Treating the
second equation in the same way, we have
2c sin •
iir
2l[tl
. km ) n+1 _
sm — , > = -— - Ei.
n + 1 2
2 (n+1)
488. Lagrange in his Mdcanique Anahjtique has applied his general equations
of motion to the solutioii of the preceding problem. He has also determined the
THE CAVENDISH EXPERIMENT.
389
oscillations of an inextensible string charged with any number of weights, and
suspended by both ends or by one only. Though several solutions of these pro-
blems had been given before his time, he considers that they were all more or less
incomplete.
489. Ex. 1. A light elastic string of length nl and coeflScient of elasticity E ia
loaded with n particles each of mass vi, ranged at intervals I along it beginning at
one extremity. If it be suspended by the other extremity, prove that the periods of
its vertical oscillations will be given by the formula ir a/
i=0, 1, 2 ... n- 1 successively. Hence show that the periods of vertical oscillation
of a heavy elastic string will be given by the formula „. — r- k/-~
2i + X ▼ iJ
length of the string, M its mass, and i is zero or any positive integer.
Tripos, 1871.]
Im 2t + 1 IT ,
■ , where L is the
[Math.
Ex. 2. An infinite number of equal particles, each of mass m, are placed in a
row at distances each equal to I and mutually repel each other so that the force
between any two is nfifip), where D is tlie distance between those two, A disturb-
ance is given to the system such that each particle makes oscillations in the direc-
tion of the row whose extent is very small compared with I. Show that the
disturbance of the li^ particle, counting from any one particle, is given by the series
Stt cos Y" (fti ftJ), where S implies summation for all values of \, and
'wa.1(i\^
K = ls!Tn jlV'W (~)%2V'(2/0 ('-^ ) +&C.
J +&C.
\
and 9= ;r .
h
Thence show that all very long waves travel with the same velocity.
If /(2)=/x2~", show that V is infinite unless n is greater than 3. [Phil. Mag.]
The Cavendish Experiment.
490. As an example of the mode in which the theory of small
oscillations may be used as a means of discovery we have selected
the Cavendish Experiment. The object of this experiment is to
compare the mass of the earth with that of some given body. The
plan of effecting this by means of a torsion-rod was first suggested
by the Rev. John Michell. As he died before he had time to
enter on the experiments, his plan was taken up by Mr Cavendish,
who published the result of his labours, in the Phil, Trans, for
1798. His experiments being few in number, it was thought
proper to have a new determination. Accordingly in 1837, a
grant of £500 was obtained from the Government to defray the
expenses of the experiments. The theory and the analytical
formuliB were supplied by Sir G. Airy, while the arrangement
of the plan of operation and the task of making the experiments
were undertaken by Mr Baily. Mr Baily made upwards of two
thousand experiments with balls of different weights and sizes,
and suspended in a variety of ways, a full account of which is
I
! :
I !
m
iil-i; .■pill
390
SMALL OSCILLATIONS.
given in the Memoirs of the Astronomical Society, Vol. xiv.
The experiments were, in general, conducted in the following
manner.
491. Two small equal balls were attached to the extremities
of a fine rod called the torsion-rod, and the rod itself was sus-
pended by a string fixed to its middle point C. Two large
spherical masses A, B were fastened on the ends of a plank
Avhich could turn freely about its middle point 0. The point
was vertically under C and so placed that the four centres of
gravity of the four balls were in one horizontal plane.
i ^
I !
A 1
1 r ^,
i
First, suppose the plank to be placed at right angles to the
torsion-rod, then the rod will take up some position of equilibrium"
called the neutral position, in which the string has no torsion.
Let this be represented in the figure by Col. Now let the masses
A and B be moved round into some position J5,-4,, making a
not very large angle with the neutral position of the torsion-rod.
The attractions of the masses A and B on the balls will draw the
torsion-rod out of its neutral position into a ncAV position of equi-
librium, in which the attraction is balanced by the torsion of the
string. Let this be represented in the figure by CE^. The angle
of deviation EJ^x and the time of oscillation of the rod about this
position of equilibrium might be observed.
Secondly, replace Hhe plank AB at right angles to the neu-
tral position of the rod, and move it in the opposite direction until
the masses A and B come into some position AJi^ near the rod
but on the side opposite to B^A^. Then the torsion-rod will
perform oscillations about another position of equilibrium CE,^
under the influence of the attraction of the masses and the torsion
of the string. As before, the time of oscillation and the deviation
EjOa might be observed.
In order to eliminate the errors of observation, this process
was repeated over and over again, and the moT,u results taken.
•I ill
THE CAVENDISH EXPERIMENT.
S9l
The positions B^A^ and A^B^, into which the masses were alter-
nately put, were as nearly as possible the same throughout all the
experiments. The neutral position Ca of the rod very nearly
bisected the angle between -BjJ.j and A^B^, but as this neutral
position, possibly owing to changes in the torsion of the string,
was found to undergo slight changes of position, it is not to be
considered in any one experiment coincident with the bisector
of the angle -4j0^2-
Let Cx be any line fixed in space from which the angles may
be measured. Let 6 be the angle xCci, which the neutral position
of the rod makes with Gx ; A and B the angles which the al-
ternate positions, B A and A^B^, of the straight line joining the
A + B
centres of the masses, make with Cx ; and let a = — ^ — • -A^so
let oo be the angle which the torsion-rod makes with Cx at the
time t.
Supposing the masses to be in the position A^B^, the moment
about GO oi their attractions on the two balls and on the rod will
be a function only of the angle between the rod and the line A^B^',
let this moment be represented by ^ (A— x). The whole appa-
ratus was enclosed in a wooden casing to protect it from any
currents of air. The attraction of this casing cannot be neglected.
As it may be different in different positions of the rod, let the
moment of its attraction about GO be "^{x). Also the torsion of
the string will be very nearly proportional to the angle through
which it lias been twisted. Let its moment about CC^ be E{oa—h).
If then / be the moment of inertia of the balls and rod about
the axis CO, the equation of motion will be
df
<f>{A'-x) + ylr(x)'-Ji:{x-h).
Now a-'X is a small quantity, let it be represented by f.
Substituting for ca and expanding by Taylor's theorem in powers
of ^, we get
-/^|=</.(^-a) + ^(a)-^(a-&) + [</>'(^-a)-.|r'(a) + ^}|.
Let
< f>'(A^a) -^lr'{a) + E
It — ' -r " 1
fi' i
11 hi
II -11
\ [
and
.- . a. < l>(A-(i) + ir(a)-E{a-h)
Then x = e + Lsin {nt + L'),
where L and L' arc two arbitrary constants. Wo sec therefore
that in the position of equilibrium the angle the torsiou-n d
%
!i
m
1 '..-V
1
'^1 I
392
SMALL OSCILLATIONS.
makes with the axis of x is e, and the time of oscillation about
the position of equilibrium is
n
Let us now suppose the masses to be moved into their alternate
position A^B ; the moment of their attraction on the balls and
rod will now he —^{x — B). The equation of motion is therefore
if=-^(..
B) + ylr(w)^i:(x-b).
Let a = £B — ^, then substituting for B its value 2a — A, we
find by the same reasoning as before
x = e' + Nsia(nt + N'),
where n has the aame value as before and
/-„ , -<f>(A-a)+ ylr{a)^Eia-b)
In
In these expressions, the attraction yjr (a) of the casing, the
coefficient of torsion E and the angle b are all unknown. But
they all disappear together, if we take the difference between
e and e. We then find
<f> (A — a) _e — e
m-
•(A),
where T is the time of a complete oscillation of the torsion-rod
about either of the disturbed positions of equilibrium. Thus the
attraction ^{A — a) can be found if the angle e — e' between the
two positions of equilibrium and also the time of oscillation about
either can be observed.
492. The function ^(A—a) is the moment of the attraction
of the masses and the plank on the balls and rod, when the rod
has been placed in a position Cf, bisecting the angle A,CB^ be-
tween the alternate positions of the masses. Let M be the mass
of either of the masses A and B, m that of one of the small balls,
m that of the rod. Let the attraction of M on m be represented
by [I ra- , where D is the distance between their centres. If
{p, q) be the cf'-ordinates of the centre of A^ referred to Cfam the
axis of X, the moment about C of the attraction of both the masses
on both the balls is
= 2iJiMm\- ^-^r
oq
\[{p-cr + q'\^ {{p + cf-i-qfr
where c is the distance of the centre of either ball a, b from the
centre C of motion. Let this be represented by fiMmP. The
moments of the attraction of the masses on the rod may by into-
n about
Itemate
alls and
lerefore
-A, we
iing, the
m. But
between
....(A),
Irsion-rod
bus the
;veen the
on about
ttraction
the rod
,^J5, be-
the mass
all balls,
)resented
)tres. If
tyas the
e masses
from the
IP. The
by intc-
THE CAVENDISH EXPERIMENT.
393
gration be found =tiMm'Q, where ^ is a known function of the
linear dimensions of the apparatus. The attraction of the plank
might also be taken account of.
Thus we find
<^{A-a)= fiM{mP + mQ).
If r be the radius of either ball, we have
/=2»i
H']
+ w
,{o-rY
which may be represented by /= mP-\- m'Q', where P' and Q' are
known functions of the linear dimensions of the rod and balls.
Hence we find by substituting in equation (A)
^ mP+m'Q _ e-^ (W
^^'mF^mq~ 2 \t)'
Let E be the mass of the earth, JB its radius and g the force
of gravity, then g = ii-ai' Substituting for /^, we find
E
e-e' /27rY _1^
2 '[Tj-gP''
m
m
P' + Q:
m
P + Q
m
The ratio — , was taken equal to the ratio of the weights of
the ball and rod weighed in vacuo, but it would clearly have been
more accurate to have taken it equal to the ratio weighed in air.
For since the masses attract the air as well as the balls, the pres-
sure of the air on the side of a ball nearest the attracting mass is
greater than that on the furthest side. The difference of these
pressures is equal to the attraction of the mass on the air displaced
by the ball.
493. By this theory the discovery of the mass of the earth
has been reduced to the determination of two elements, (1) the
time of oscillation of the 'orsion-rod, and (2) the angle e — e'
between its two positions of equilibrium when under the influence
of the masses in their alternate positions. To observe these, a
small mirror was attached to the rod at C with its plane nearly
perpendicular to the rod. A scale was engraved on a vertical
plate at a distance of 108 inches from the mirror, and the image
of the scale formed by reflection on the mirror was viewed in
a telescope placed just over the scale. The telescope was fur-
* In Baily's experiment, a more accurate value of g was used. If e be the ellip-
ticity of the earth, m the ratio of centrifugal force at the equator to equatoreal
gravity, and \ the latitude of the place, we have ^ = M^il-2e + (^nt -el cos' \
i, :
'i\ ii
H -:!'
394
SMALL OSCILLATIONS.
nished with three vortical wires in its focus. As the torsion-rod
turned on its axis, the image of the scale was seen in the telescope
to move horizontally across the wires and at any instant the
number of the scale coincident with the middle wire constituted
the reading. The scale was divided by vertical lines one-thirteenth
of an inch apart and numbered from 20 to 180 to avoid negative
readings. The angle turned through by the rod when the image
of the scale moved through a space corresponding to the interval
of two divisions was therefore ^5 • tt^ • o '='73" 4 6. But the
lo lUo A
division lines were cut diagonally and subdivided decimally by
horizontal lines ; so that not only could the tenth of a division
be clearly distinguished, but, after some little practice, the frac-
tional parts of these tenths. The arc of oscillation of the torsion-
rod was so small that the sqiiare of its circular measure could be
neglected ; but as it extended over several divisions it is clear
that it could be obsei-ved with accuracy. A minute description
of the mode in which the observations were made would rot find
a fit place in a treatise on Dynamics, we must therefore refer the
reader to Baily's Memoir.
In this investigation no notice has been taken of the effect
of the resistance of the air on the arc of vibration. This was,
to some extent at least, eliminated by a peculiar mode of taking
the means of the observations. In this way also some allowance
was made for the motion of the neutral position of the torsion-rod.
494. The density of water in which the weight of a cubic
inch is 252725 grains (7000 grains being equal to one pound
avoirdupois) was taken as the unit of density. The final result
of all the experiments was that the mean density of the earth
is 5-6747.
495. Two other methods of finding the mean density have
been employed. In 1772 Dr Maskelyne, then Astronomer Royal,
suggested that the mass of the earth might be compared with
that of a mountain by observing the deviation produced in a
plumb-line by the attraction of tlie latter. The mountain chosen
was Schehallien, and the density of the earth was found to be
a little less than five times that of water. See I'hil. Trans.
1778 and 1811. From some observations near Arthur's Saat, the
mean density of the earth is given by Lieut.-Col. Juuies, of the
Ordnance Survey, as 5'316. See Phil. Trans, 185G.
The other method, used by Sir G. Airy, is to compare the
force of gravity at the bottom of a mine with that at the surface,
by observing the times of vibration of a pendulum. In this way
the mean density of the earth was found to be GoGG. Sec Phil.
Trans. 1856.
i
•sion-rod
ielescope
;aiit the
istituted
lirteenth
negative
le image
interval
But the
raally by
division
the frac-
B torsion-
could be
; is clear
ascription
1 rot find
refer the
the effect
This was,
of taking
allowance
rsion-rod.
a cubic
ne pound
nal result
the earth
sity have
ler Royal,
ired with
iced in a
tin chosen
ind to be
il Trans.
gsat, the
es, of the
nparc the
le surface,
this way
See FliiL
OSCILLATIONS OF THE SECOND ORDER. 395
OscillaiiQ"'^ of the Second Order.
496. The equations of small oscillations are formed on the following principle.
Some small quantities are selected as the co-ordinates of the system, and all powers
of these above the first are neglected. The assumption is tacitly made that the
order of magnitude of the terms is not materially altered by the process of solving
the equations ; so that a small term, which should by the rule be neglected in
forming the differential equations, cannot become of importance in the final
integrals. This assumption, however, is not strictly correct. In the Lunar and
Planetary theories, where something more is wanted than the mere periods of
oscillations, there are many instances of small terms in the differential equations,
which become of great magnitude in the result. Wo require some rule to dis-
tinguish the small terms v/hich become of importance from those which remain
insignificant. For the sake of simplicity we shall consider the case in which the
system depends on two independent co-ordinates, though the remarks are for the
most part quite general.
497. Referring to Art. 432, let PsinXt be some small periodic term which
occurs on the right-hand side of the first of the two differential equations of
motion. To simplify the solution, let us write for the trigonometri al term its
exponential value, and fix our attention on the part — p=. P^^"^ or, as we shall
2 1^ — 1
write it, Qe**'. Let/{Z)) stand for the determinant which is the operator on as in
the third equation of Art. 432. Also let F(D} be the minor of the leading con-
stituent ; the value of x is then known to be
The term Qe*^' in the differential equation is the analytical representation of
some small periodical force which acts on the system. The first term of the
expression for x is the direct effect of the force, and is sometimes called the
forced vibration in the co-ordinate x. The quantities m,, mj, &c. being generally
imaginary, the remaining terms are also trigonometrical and are sometimes called
the free or natural vibrations in the co-ordinate. In the analytical theory of linear
differential equations, the forced vibration is called the particular in'egral and the
free vibration the compleiiumtary function.
498. If we examine the coefficient of the forced vibration in x we shall see that
it is large only if /(/x) is very small or zero. Since the roots of the equation
/ (/t) = are m^, m^, &c. the rule may be simply stated thus : any sviall periodical
term lohose coefficient in the dijf'erential equation is less than the standard of quantities
to be neglected may rise into imporvance if its period is nearly equal to one of the
free vibrations of the system.
Suppose the dynamical system to have two of its free periods equal and let it
be acted on by a small force whose period is nearly equal to this free period. The
divisor/ (/t) of the forced vibration will be a small quantity of the second order and
the magnitude of the terra may be much greater than if the free periods were
unequal. When such a case occurs in the Lunar theory, the term is said to rise
tivo orders.
I
I'
I "I i
I ^jllll
t. i !.j
u
■iff
Mi
396
SMALL OSCILLATIONS.
Is •
499. This principle admits of an elementary explanation in some cases. Let a
system oscillating with one degree of freedom be acted on by a small periodical
force at some point A, The force will act sometimes to accelerate the motion of A
and sometimes to retard it, and thus the maguitiide of the vibration will not become
very great. But if the period of the force be equal to that of the point A, the force
may continually act to increase the motion of A in whatever direction A is moving.
Thus the extent of the vibration will be continually increasing. For example,
every one knows how a heavy swing can be set in violent oscillation by a series
of small pushes and pulls applied at the proper times.
If the period of the force be only nearly equal to that of the point A, a time
will come when the force acts continually to decrease the motion of A. Thus the
oscillation will not increase indefinitely/ but will alternately slowly increase and as
slowly decrease.
600. A remarkable nse of this principle was made by Gapt. Eater in his
experiments to determine the length of the seconds' pendulum. It was important
to determine if the support of his pendulum was perfectly firm. He had recourse
to a delicate and simple instrument invented by Mr Hardy a clockmaker, the
sensibility of which is such that had the slightest motion taken place in the support
it must have been instantly detected. The instrument consists of a steel wire,
the lower part of which is inserted in the piece of brass which forms its support,
and is flattened so as to form a delicate spring. On the wire a small weight slides
by means of which it may be made to vibrate in the same time as the pendulum
to which it is to be applied as a test. When thus adjusted it is placed on the
material to which the pendulum is attached, and should this not be perfectly firm,
the motion will be communicated to the wire, which in a little time will accompany
the pendulum on its vibrations. This ingenious contrivance appeared fully adequate
to the purpose for which it was employed, and afforded a satisfactory proof of the
stability of the point of suspension. See Phil. Trans. 1818.
601. It generally happens that the small terms rejected in the equations of
motion are functions of the co-ordinates and their differential coefficients. To
take account of these terms we proceed by successive approximation. Suppose the
co-ordinates x, y to determine the oscillation about some state of steady motion, and
to be zero for that motion. As a first approxunation we obtain (Art. 432)
a; = JJ/^e*"'* + JJ/,c'"'* +
with a corresponding expression for y, where m^, mt, &c. give the free periods, and
jl/j, 3/a, &c. are all small quantities of the first order. If we now substitute these
values of x and y in any small term of a high order which occurs in the differential
equation, it becomes a series of exponentials of the form
where p, q, &c. are positive integers whose sum is equal to the order of the term.
By the principle explained in Art. 498, the corresponding forced vibration cannot
be important unless pwij + ^nij + . . . is very nearly equal to one of the quantities
«ii, Wig, &c. In the same way, in any approximation, if the periods of the terms
are not such that an equality of this nature can be very nearly true, the next
approximation to the motion will not produce any important terms. Even if such
a relation does approximately hold, yet, if the order of the term to be examined is
great, the term will probably remain insignificant.
OSCILLATIONS OF TEE SECOND ORDER.
S97
502. As an oxample let Tis consider the case of a planet describing a circle
about the sun, considered as fixed in the centre. If slightly disturbed the changes
in the radius vector and longitude will be very small and will correspond to what
we have called x and y. From the theory of elliptic motion, we know that these
will be approximately
«=a-a<cog (n< + o), y = J< + c + 2csin(H< + a),
2ir
where a, &, c and e are all small quantities, and — is the period of the planet. Com-
paring these with the expressions for x and y given in Art. 432, we see that the
free periods for x are given by m=0, m=±n\/-i, and for y, by m=0, m=0,
m= ±n \/ - 1, one period being absent from x. We infer that any small periodical
force may produce a considerable disturbance both in the radius vector and in tho
longitude of the planet, if its period is nearly equal to that of the planet or is very
long. Since there are two equal free periods in the longitude corresponding to
m=0, those small forces whose periods are very long may be expected to rise two
orders in the longitude. If any such forces act on the planet it will be necessary to
examine into their effects. Small forces, whose periods are different from these,
and whose magnitude is beneath the standard of quantities to be retained, may be
disregarded.
608. If the period of the small disturbing force Qe*^' be equal to one of the free
periods, the solution changes its character. The forced vibration now takes the
form ~f\ Qji^^. This may indicate that the motion of the system will, after a
time, become very different from that which we took as a first approximation. We
may have therefore to amend our first approximation by including in it the effect
of this force. We may then enquire how far this modified first approximation
indicates that the undisturbed motion is stable or unstable. When this force is
included in the equations, the equations will probably be no longer linear, and it
may be impossible to solve them or to find a solution sufficiently accurate to serve
as a first approximation throughout the whole motion.
504. In many cases however the effects of some of these forces may be included
in the first approximation by slightly pltering the free periods. Referring to Art.
432, let us suppose that on substi+ucmg our first approximation in the small terms,
we have on the right-hand side of the two first equations
JBie™'< + i?ae«»«'+...J ^
These are supposed to have ariden from some relations of the form
i)mi + 3»Jg+...=mi (2).
Let us take as our amended first approximation
a; = iVie'»i' + iVje«»*+...| „.
y = N,'e»^* + N^e'^^*+...]
where N-^, &c. N{, &c. are, as before, small quantities oi the first order, and
ni=wii + 5HJ^, n^-m.i + Bm.2, &c. where 5%, Sm^, &c. are quantities of the order
Qi, &c. i^i , &c. If we substitute the amended values of x, y in the small terms,
they will become
i?/e"'< + 2?./e««'+...| ■ '
! i
%^
■^ i
WW
m
1 'T'.
:
p
r
1^^ !
S ;
398
SMALL OSCILLATIONS.
instead of (1), provldod the relations represented by (2) apply also to tlie indices
Hj, n,, &o. Hero Q/, &c. R^', «fco. differ from Q^, &c. 7?,, &c. by quantities of the
order Q^h Substituting the values (8) in the differential equations of Art. 482,
and rejecting the squares of Qp &o. H■^,ko., we obtain
(.1 n» +Bn + C)N+ {Fn' + Gn + ir)N'=Q\
)N'=Il]'
.(6).
{A'n* +B'n + C')N+ {F'n' + G'n + If) .
where the suffixes have been dropped for the sake of generality. These two equa-
tions determine n and N\ leaving N to bo determined by the initial conditions.
The toBt of the success of the amended first approximation is that tho values of 7t
thus found satisfy the relation (2).
605. The condition may also bo stated thus. Consider the dctemunant given
in Art. 432, which when expanded is equal to / (D). After substitution of the first
approximation in the small terms of the higher orders in tho equations, perform on
these equations the operations indicated by the minors of the constituents in the
first column, and add the results together. We have an equation of tho form
/(D)a;=Ale'»'''^■Aae'^*+...
where the coefficients Aj, Ag, &c. are all functions of A/,, M^, &c., m,, m^, A'o.
Following the same reasoning as in the last Article, and amending our first approxi-
mation, we find
__4. _ «., ^
Jmi =
MJ'{m^y
8j?Io =
M^f (TOj) '
&c.
If these Batisfy the relations typified by
•pSm^ + qhm^+ ... =8mi,
the e£ioct of the disturl Ing cause is to modify the free periods of the system without
afitibting the btability jf the undisturbed motion.
506. Having in this way amended the first approximation, wo may proceed to
the second by substitution in the small term, and so on. If the several stages can
be so arranged thut no iorm makes its appearance which can become greater than
our previourj approximation, we may consider that we have obtained a correct repre-
sentation o.i the motion,
607. Ex. 1. A pendulum swings in a very rare medium, resisting partly as the
velocity and partly as the square oftlie velocity, to find the motion.
Let 6 be the angle the straiglit line joining the point ot support to the centre
of gravity Q of the pendulum makes with the vertical. Then tho equation of
motion is
(1).
d'^9 . ^ ^ d0
-(^y
where I is the length of the simple equivalent pendulum, 2k and fi tho coefficients
of the resistance divided by tho moment of inertia of the pendulum about tbo
axis of suspension. Let g=ln\ Since 6 is small we may write the equation in the
iorm
-(:
doy , e»
Since k and are very small, we might at first suppose that it would be
sufficient as a first approximatiou to reject all the terras on the light-haud side.
D indicoB
eH of tlie
Lrt. 482,
....(5),
wo equa-
nditions.
luOB of n
mt given
I the first
jrform ou
tits in the
orm
t approxi-
m without
proceed to
stages can
atcr than
rect repre-
rtJy as the
the centre
juation of
...(1).
ocfficicntfl
about the
tion iu the
wonld he
laiiil side.
OSCILLATIONS OF THE SECOND ORDER. 309
This gives O^aainnt, the origin of moasnromont of ( being ho chosen that ( and
vanish together. If wo substitute this in the small tcrn^s we get
which gives
j,„ + n'tf = - ixn .aeoant ¥s n'o' sin nt + &c. ,
at* o
= a sin nt -Ka.t sin nt + , ,, na't cos nt + &o.
lo
These additional terms contain t as a factor, and show that our first approximation
was not sufficiently near the truth to represent the motion except for a short
time. To obtam a suilloiently near first approximation wo must include in it the
(10
small term 2k -7- , we have therefore
sin mt, whore for the sake of brevity we have put n' - k* =m'.
^ «!.-«'
This gives = ae
In oiu* second approximation wo shall reject all terms of the order a? or a'«
unless they are such that after integration they rise in importance in the manner
explained in Art. 498. We thus get
^ + 2K^+n^e^- ':^e-2'''(l + COB 2»U) + f a^ ~ (3 sin mt - sm 3mt)
at^ at 2 4
- fia'Ke
V ~ 2 "*■ 2
cos 2m,t + m sin 2mt
)•
where all the terms on the right-hand side after the first are of tho third order, and
are to be rejected unless they rise in importance. To solve this, let us first cousidur
the general case
S "*■ ^^ ^■'" "''^= «"'"'' • (^ si" ™« + ^ COS rmt).
Put 9=e "**' (L sin rmt + M cos rmt). Substituting we get
£ {(p - 1) V + nt« (1 - r») } + 2 (2) - 1) otbM = 4
M{(p ~ l)V+m«<l - T^)) - 2 (iJ - 1) KrmL ■■
M=
Now K is very small. If then r be not equal to unity, we have L=
B , , -B „ A
nearly ; but if r = 1 , we have L = ,
, M=,
nearly.
"m« (1 - r2) """"•' ' -, ..- -c.^ 2 (2, - 1) /cm ' " 2 {p - 1) Km
The case of ^ = 1 does not occtur in our problem. It appears that those terms only
in the differential equation which have r = 1 give rise to terms in the value of x
which have the small quantity k iu the denominator. Hence in the differential
equation the only term of the third order which should bo retained is tho first.
We thus find, putting successively r = 0, r= 2, r = l,
^= ae-*^ sin vit - "-f e " 2"' + ^-^ e " 2"' cos 2mt + ^' e ' ^"^ cos mt,
2 32/c)?t
This equation determines the motion only during any one swing of the pendu-
lum ; when the pendulum turns to go back n changes sign. Let us suppose tho
pendulum to bo moving from left to right, and lot us find tho lengths of tho arcs of
descent and ascent. Tci do this, we must put j -■ 0. Let the equation be written
is
>'. I
I
I i
I I " ■ 'ft
400
8MALL OSCILLATIONS.
i,'i
il:;
f:'i
\i
in the form 9^f{t), tlion if wo negloot all the Hmall torme, ^ vanisboB when
mt= ± „ . Put then mt= -^ + x where jc id a small quantity, we have
Now
f.fji f Off Qtn \
/'(e) = ae-«'(mcosmt-/{sinn»t)-^^-c~2«'( - 2(c + -^ cos 2m« + - Bin2mn
+ „„ -«"'*'{ -m sin mt-8/c cos mf).
A fluiBciently near approximation to the valne of f"(t) may be fonnd by differ-
K 4 iiCtfC 7t CL
entiatincr the first term of the value of/' (t). We thus find x= ^- _„ - ;
the second of those terms being smaller than the other two might be neglected.
We also find as the arc of descent
Kir Kir Kir Smr
Similarly to find the arc of ascen^^^ we put mt=^ + y. This gives w = -—— ,
and the arc of ascent is
= ae
Kir
" Sni 2
(fir
* « >« I ~ am , w'a" ~ am
3*"
Ism \ '
In those expressions for the arcs of descent and ascent the terms containing x
and y are very small, and assuming k not to be extremely small, these terms will be
neglected *.
Kow a is different for every swing of the pendulum, we must therefore eliminate
Kir
a. Let M„ and w„4.j be two successive arcs of descent and ascent, and let \=e ^m ,
so that X is a little less than unity. Then we have
1 2 „ 1 ^ 2 ,,,
eliminating a we have very nearly
where c = 5- r--^ = -. — nearly.
2/x 1 + X" 4/xm
• If these terms are not neglected tho equation connecting the successive arcs of
descent and ascent becomes
M« Mn4.i 3 ^ ' 32Kni \
'n "n+l
2Kir
Now 1-X*= — nearly, so that this additional term is very small compared with
m
that retained.
!S8ivc axes of
SMALL OSCILLATIONS. 401
The auoeeosiTe »roB are, thereforoi anob tb»t - 4- ia the general term of n
geometrical eeriea whoae ratio ia e^ . Tbe ratio of any arc u^ to tbe following arc
«.
'»+i
is
which ooutinually decreases with the arc. In any scrioR of oscillations tbe ratio is
at first greater and af terwarda lesa than its mean value. This result seems to agree
with experiment.
Tojlnd the time of oneillation. Let r„ f, bo the timfs nt which the pendulum is
at the extreme left and right of its arc of oscillation. Then
ir K
HI 'a
TT
2
K
m
n-a*
32w« '
The time of oscillation from one extreme position tu the other is f, - t^ which is
equal to — . This result is independent of the arc, so that tbe time of oscillation
m
remains constant throughout the motion. The time is however not exactly thn
same as in vacuo, but is a littlo longer ; the difference depending on the square of
the small quantity k.
Ex. 2. If in Art. 418 a first approximation to the motion in $=A sin {nt + B),
show that a second will bo
<>= .4 sin («e + ^) + J (ft + c) i4« + K36 + c) ^' cob 2 {at + B)
where
b =
rs sm a
1
COR a da sin 2a
sin
7
na)
k^ + r^ ' ^~2 scoBa-r
and ff is the length of the arc of either cylinder.
A general method of solving problems of this kind, both for two and three
dimensions, is given in tbe Proceeding* oj the London Mathematical Society, Vol. v.
page 101, 1874.
Ex. 8. A rigid body is Rnspended by two eqnal and parallel threads attached
to it at two points symmetrically situated with respect to a principal axis through
the centre of gravity which is vertical, and being turned round that axis through a
small angle is left to perform small finite oscillations. Investigate the reduction to
infinitely small oscillations. [Smith's Prize.]
EXAMPLES*.
1. A uniform rod of length 2e rests in stable equilibrium with its lower end
at tbe vertex of a cycloid whose plane is vertical and vertex downwards, and passes
through a small smooth fixed ring situated in the a;as at a distance h from the
vertex. Show that if the equilibrium be slightly d'sturbed, the rod will perform
* These examples are taken from the Examination Papers which have been set
in the University nud in the Colleges.
R. D. 26
*!:
r
4
\:m
1
■MRem
sl'i
hi
fiii
!■!
402 SMALL OSCILLATIONS.
email oscillations with its lower end on the arc of the cycloid in the time
47r
cf}
ac)
, where 2a is the lengtl: of the axis of the cj'cloid.
2. A small smooth ring elides on a circular wire of radius a which js con-
strained to revolve about a vertical axis in its own plane, at a distance c from the
centre of the wire, with a uniform angular volocity a/ _l_Vr_ • show that the ring
^ Cs/2 + a
will he in a position of stable relative equilibrium when the radius of the circular
wire passing through it is inclined at an angle 45* to the horizon ; and that if the
ring be slightly displaced, it wiU perform a small oscillation in the' time
< 9 c\/8 + a'
V8 + <
3, A uniform bar of length 2a suspended by two equal parallel strings each of
length b from two points in the same horizontal line is turned throiigh a small
angle about the vertical line through the middle point, show that the time of a
small oscillation is 27r ,
4. Two equal heavy rods connected by a hinge which allows them to move
in a vertical plane rotate about a vertical axis through the .hinge, and a string
whose length is twice that of either rod is fastened to their extremities and
bears a weight at its middle point. If M, M' be the masses of a rod and tlio
particle, and 2a the length of a rod, prove that the angular volocity about tho
vertical axis when the rods and string form a square is \/ — --—
M + 2M'
and
2a J2 ' M
if the weight bo slightly depressed in a vertical direction the time of a small
'" V 15^ 'M + 2M''
oscillation is 2?!
5. A ring of weight W which slides on a rod inclined to tho vertical at an angle
a is attached by means of an elastic string to a point in the plane of the rod so
situated that its least distance from the rod is equal to the natural length of tho
string. Prove that if 6 be the inclination of the string to the rod when in
W
equilibrium, cot - cobO = coso, where to is the modulus of elasticity of tho
10
string. And if the ring be slightly displaced the time of a small oscillation will be
2ir A / — :— — , where I is the natural length of the string.
V wff 1 - mu?d
6. A circular tube of radius a contains an elastic strmg fastened at its highest
point equal in length to v: of its circumference, and having attached to its other
o
extremity a heavy particle which hanging vertically would double its length. Tho
system revolves about the vortical diameter with an angular velocity a /•-• Find
the position of relative equilibrium and prove that if the particle be slightly dis-
turbed tho time of a small oscillation is - ^ * / - •
^/7r + 4 V J/
he time
ill is con-
from the
,t tlie ring
le circular
hat if the
igfl each of
^h a small
5 time of a
m to move
id a string
amities and
•od and tho
f about tho
^W
M
of a small
and
at an angle
tho rod so
Ingth of tho
rod when iu
picity of tho
ition will be
It its higliest
to its other
length. Tho
Find
a
sligbtly dia-
EXAMPLES.
403
7. A heavy uniform rod AB has its lower extremity A fixed to a vertical
axis and an elastic string connects B to another point C in the axis such that
AO = — - = a ; the whole is made to revolve round AC with such angular velocity
that the string is double its natural length, and horizontal when the system is in
relative equilibrium and then left to itself. If the rod be slightly disturbed in a
4^
21</
, the weight
vertical plane, prove that the time of a small oscillation ia 2ir a/,
of the rod being sufficient to stretch the string to twice its length.
8. Three equal elastic strings A B, BC, CA surround a circular arc, the end A
being fixed. At B and C two equal particles of mass wi are fastened. If I be tho
natural length of each string supposed always stretched and X the modulus of
elasticity, show that if the equilibrium be disturbed the particles will be at equal
distances from ^ after intervals T/v/ — .
9. A particle of mass M is placed near the centre of a smooth circular
horizontal table of radius a, strings are attached to the particle and pass over n
smooth pullies which are placed at equal intervals round the circumference of the
circle ; to the other end of f>ach of these strings a particle of mass M is attached ;
show that the time of a small oscillation of tho system is 2ir \ V- .
\ n 0/
10. In a circular tube of uniform bore containing air, slide two discs exactly
fitting the tube. The two discs are placed initially so that tho lino joining their
centres passes through the centre of the tube, and the air in the tube is initially of
its natural density. One disc is projected so that the initial velocity of its contro
is a small quantity 7i'. If the inertia of the air be neglected, prove that tho point
on the axis of the tube equidistant from tho centre of tho discs moves uniformly
-T-p- , where M is the
mass of each disc, a the radius of the axis of tube, P tho pressure of air on the disc
in its natural state.
and that the time of an oscillation of each disc
is2.y^
11. A uniform beam of mass M and length 2a can turn round a fixed horizontal
axis at one end ; to the other end of the beam a string of length I is attached and
at the other end of the string a particlo of mass m. If, during a small oscillation of
tho system, the inclination of the string to the vertical is always twice that of tho
beam, then M(3l-a) — Cmi(l + a).
12. A conical surface of semivertical angle a is fixed with its axis inclined at
an angle to the vertical, and a smooth cono of semiverticnl- angle /3 is placed
within it so that the vertices coincide. Show that the time of a small oscillation
=:27r / f "^"("~P ) where a is the distance of the centre of gravity of tlio cono
from the vertex.
13. A number of bodies, tho particles of which attract each other with forces
varying as the distance, are capable of motion on certain curves and surfaces.
Trove that if A, B,C bo the moments of inertia of tho system about xhreo axes
mutually at right angles through its cent'-o of gravity, tho positions of stable
equiUbrium will bo found by making A+B + C si minimum.
20—2
[' i
lli*''
) m
■ m
fmmmmm
CHAPTER IX.
MOTION OF A BODY UNDER THE ACTION OF NO FORCES.
■J ;
Solution of Elder's Equations.
608. To determine the motion of a body about a fixed point,
in the case in which there are no impressed forces.
The equations of motion are by Art. 230,
^^|i-{5-0)a>.a,3 =
multiplying these respectively by «»,, «„ w^; adding and inte-
grating, we get
ila>,« + 5<+Ca)3«=!r. (1),
where T is an arbitrary constant.
Again, multiplying the equations respectively by -4a),, Ba>^,
Cwg, we get, similarly,
A\' + ffco,' + Cto* = (P (2),
where is an arbitrary constant.
To find a third integral, let
o>i' + Wj* + ©a" = a>'
9 9
■(3);
G).
da)
~dt
day,
d(o.
' + ««-:7r + '»»8-:7*' = <"
dt
dt
aoy
di'
(0. &)„ O).
then multiplying the original equations respectively Tt)y -f , -^i yf ,
and adding, we get
dot fB- C . C^A A-B\
da I
B-C ^ C^A A
C
-ja),a),a)3
W
{B-C)(C-A){A'-B)
ABC
WjWjWg.
SOLUTION OF EULEllS EQUATIONS.
But solving the equations (1), (2), (3), we get
405
< =
< =
CA
.(-\ + a,»)
8
{B-A){B-C)
AB / ^ . .N
iC-B){C-A)'^^^''^'^^J
T (B + C) — G*
where \ = — ^^ — ^^^ , with similar expressions for X, and \.
(5),
BG
Substituting in equation (4), we have
da
to
^ = V(\-«^)(\-o)^)(\-a,«)
dt
(6).
The integration of equation (6) * can be reduced without diffi-
culty to depend on an elliptic integral. The integration can be
effected in finite terms in two cases ; when A=B, and when
G* = TB, where B is neither the greatest nor the least of the three
quantities A, B, C. Both these cases will be discussed further on.
Ex. If right lines are xneasored along the three principal axes of the hody from
the fixed point, and inversely proportional to the radii of gyration round those axes,
the sum of the squares cf the velocities of their extremities is constant throughout
the motion.
509. It will generally be supposed that A, B, C are in order of magnitude, so
that A is greater than B, and B than C. The axis of B will be called the axis of
mean moment. If we eliminate w^ from the equations (1) and (2), we have
AT-G^=B{A-B)u^^+C[A-C)u^,
which is essentially positive. lu the same way we can show that CT- CP is nega«
tive. Thus the quantity -^ may have any value lying between the greatest and
least moments of inertia.
The three quantities Xi , Xg, \ in Art. 508 are all po^dtive quantities; for since
B-^C-Avi positive, and -ys<A,ii follows that \ is positive. The numerators of
\ and \ are each greater than that of \, and are therefore positive, the denominators
TG — C^
are also positive ; hence \ and Xj are both positive. Also X^ - X « = . „. ^- (A-B),
with similar expressions for \-\ and Xg-X^. It easily follows that X, ia
G*
the greatest of the three, and X^ or X3 is the least according as -^ is > or < B.
It follows from equations (6) that throughout the motion w' must lie between X,
and the greater of the quantities \ and X3.
* Euler's solution of these equations is given in the ninth volume of the Quarterly
Journal, p. 361, by Prof. Cayley. Kirohhoff's and Jacobi's integrations by elliptic
functions are given in an improved form by Prof. Oreenhill in the fourteenth
volume, pages 182 and 265. 1876.
il V
Irta
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11
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i' '
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406
MOTION UNDER NO FORCES.
510. The solution in terms of elliptic integrals has been effected in the follow-
ing manner by Kircbboff. If we put
'' sin'*
A{i>)-^Jl-ii'smP<p, F{4>)
=Jo Vl-A^i
then k is called the modulus of F, and must be less than unity if F is to be real for
all values of 0. The upper limit is called the amplitude of the elliptic integral
J*' and is usually written am F. In the same way sin <f>, cos <p, and A {<j)) are written
sin am F, cos am F, and A am F.
We have by differentiation
dCOB(p
dF
-sin0^=-sin0A(^)
.(J).
d sin . d<l> , . , >
dA (0) h? sin cos rf0 , ., .
These equations may be made identical with Euler's equations if wo put
i''=X(«-T)and
o)i—aAam\(t-T)
W3 = 6sinam\ (<-t) I (2),
Wj^ccos amX (t-r) J
A-B c\ A-C b\ B-C ,.a\
.(3).
C
ah'
B
cd"
A
-;&»
he-
We have introduced here six new constants, viz. a, 6, c,\ h and t. With these
we may satisfy the three last equations and also any initial values of Wj, w.,, w^.
The I olution if real will also be complete.
Whcn<=r wehavefrom (2) Wj=a, Wj, = 0, and W3=c. Hence by Art. 508
Aa^ + Cc'^=T, A-^a'^ + C-'c^ = G"'\
G^-CT
c'=
AT-G*
A [A-C)' ~C(A-q' _ '
Dividing the second of equations (3) by the first, we have
■ ' b^_A-CO ,3_ AT-G'
c" A-BB' •■• '' ~£{A-B)'
Multiplying the first and secondof equations (3), we obtain
The ratios of the right-hand sides of (3) are as c^ : b^ : khi^, and those have just
been found. Hence ii' the signs of a, b, c, \ be chosen to satisfy any one of the
tliree equalities, the signs of all will bo satisfied.
Dividing the last of equations (3) by either of the other two, we find
' ■ A-BG-'-CT' •'■ ^-(rr77)((,«-cT)'
poinsot's and mac cullagh's constructions.
407
If 0^ > BT and A, B, C are in descendiUg order of magnitude, the values of
a', 6", c* and X" rre all positive. Also I? is positive and less than unity. The
solution is therefore real and complete.
If G' < BT we must suppose J , B, (7 to be in ascending order of magnitude to
obtain a real solution. If we may anticipate a phrase used by Poiusot, and which
will be explained a little further on, we may say that the expression for Wj in this
solution is to be taken for the angular velocity about that principal axis which is
enclosed by the polhode.
UCP=BT we have F= 1 and
Jo
cos<l>
1 . 1 + sin (f>
2 ^^'l-siu^'
sin amF=
pf-i
e*'+e-
Snbstiiutin{j in equations (2) the elli^itic functions become exponential.
If B — we have F=0 and in this o tse F=(t>, so that amF= F. If we again
substitute in equations (2) the elliptic functions become trigonometrical.
The geometrical meaning of this solution will be given a little further on.
Poinsot's and MacCallaglis comtructiom for the motion.
511. The fundamental equations of motion of a body about a
fixed point are
V + ^V/+CV=<^*.
V(o^ + Bw,
(1),
+ c<=r. (2).
These have been already obtained by integrating Euler's
equations, but they also follow very easily from the principles of
Angular Momentum, and Vis Viva.
Let the body be set in motion by an impulsive couple whose
moment is O. Then we know by Art. 279, that throughout the
whole of the subsequent motion, the moment of the momentum
about every straight lino which is fixed in space, and passes
through the fixed point 0, is constant, and is equal to tho mo-
ment of the couple G about that line. Now by Art. 241, the
moments of the momentum about the principal axes at any
instant are A(t\, Ba,^, Ga^. Let a, /8, 7 be the direction angles
of the normal to the plane of the couple G referred to these
principal axes as co-ordinate axes. Then we have
-4ft)j= 6^ cos a
' i?w^ = (? cos /9 • (3),
C(W3= G cosy
adding the squares of these we get equation (1).
s
; I
i '
1 ,;
1 .■;
I:
. 1
' j!
! ^;
■ I
■ i
!ki
S , J'! W
] H
MM
(•r
408
MOTION UNDER NO FORCES.
Throughout the subseque^it motion the whole momentum of
tlie body is equivalent to the couple O. It is therefore clear
that if at any instant the body were acted on by an impulsive
couple equal and opposite to the couple O, the body would be
reduced to rest.
512. It follows from Art. 290, that the plane of this couple
is the Invariable plane and the normal to it the Invariable line.
This line is absolutely fixed in space, and the equations (3) give
the direction cosines of this line* referred to axes moving in the
body.
It appears from these equations, that if the body be set in
rotation about an axis whose direction cosines are (^ m, n) when
referred to the principal axes at the fixed point, then the direction
cosines of the invariable line are proportional to Al, Bm, Cn. If
the axes of reference are not the principal axes of the body at the
fixed point, the direction cosines of the invariable line will, by
Art. 240, be proportional to Al — Fm — En, Bm— Dn — Fl, and
On — El — Dm, where the letters have the meaning given to them
in Art. 15.
513. Since the body moves under the action of no impressed
forces, we know that the Vis Viva will be constant throughout the
motion. Hence by Art. 348, we have
where T\ is a constant to be determined from the initial values
of Wj, 0),, Wj.
The equations (1), (2), (3) will suffice to determine the path
in space described by every particle of the body, but not the posi-
tion at any given time.
* That the straight line whose equations referred to the moving principal axes
na 9J 2
are -t— = -~- = jz- is absolutely fixed in space may be also proved thus, if we
Au-^^ JSu^ C(>>3
assume the truth of equation (1) in the text. Let x, y, z be the co-ordinates of
any point P in the straight line at a given distance r from the origin, then each of
the equalities in the equation tc the straight line is equal to ^ and is therefore con-
stant. The actual velocity of P in space resolved parallel to the instantaneous
position of the axis of x is
_dx
~ (ft'
But this is zero, by Euler's equation. Similarly the velocities parallel to the other
axes are zero.
f It should be observed that in this Chapter T represents the whole vis viva of
I'ji body. In treating of Lagrange's equations in Chapter vii. it was convenient to
let T represent halj the vis viva of the system.
■yuz + zaa=-AA -^'-{B- C) WjWj,|
to the other
POINSOTS CONSTRUCTION.
409
514. To explain Poinsot's representation of the motion hi/
means of the momental ellipsoid.
Let the momental ellipsoid at the fixed point be constructed,
and let its equation be
Let r be the radius vector of this ellipsoid coinciding with the
instantaneous axis, and p the perpendicular from the centre on
the tangent plane at the extremity of r. Also let a be the an-
gular velocity about the instantaneous axis.
The equations to the instantaneous axis are — = ^l = — and
®i <»a <»8
if (a?, y, z) be the co-ordinates of the extremity of the length r,
each of these fractions is equal to — .
Substituting in the equation to the ellipsoid, we have
I T r
Again the expression for the perpendicular on the tangent
plane at (a;, y, z) is known to be —^ = ^^j^ , substi-
tuting as before we get
JfV
1 _^V + ff<-}-CV t-J^ ¥t.
P'
JfV
TiJi
a>
a MY ' T '
p
^MT ,
Q
. €.
The equation to the tangent plane at the point (a;, y, z) is
Ax^ + Byr)+ Cz^=M€\
substituting for (a?, y, z) we see that the equations to the perpen-
dicular from the origin are
A(o^ Boii^ Co),'
but these are the equations to the invariable line. Hence this
perpendicular is fixed in space.
From these equations we infer
(1) The angular velocity about the radius vector round which
the body •"" turning varies as that radius vector.
H
til
i.'i^
I If
Ji
I
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fi^i)
kil:
m ii
410
MOTION UNDER NO FORCES.
*' .1
U4h ;
(2) TJie resolved part of the angular velocity about the per-
pendicular on the tanr/ent plane at the extremity of the instan-
taneous axis is constant. Tliis theorem is due to Lagrange.
For the cosine of the angle between the perpendicular and
n
tlie radius vector = - . Hence the resolved angular velocity is
n T . .
= &)- = >., which is constant.
r G
(3) The perpendicular on the tangent jtlane at the extremity
of the instantaneous axis is fixed in direction, viz. normal to the
invariable plane, and constant in length.
The motion of the momental ellipsoid is therefore such that,
its centre being fixed, it always touches a fixed plane, and the
point of contact, being in the instantaneous axis, has no velocity.
Hence the 'motion may he represented by supposing the momental
ellipsoid to roll on the fixed plane with its centre fixed.
515. Ex. 1. If the body while iu motion be acted on by any impulsive couple
whose plane is perpendicular to the invariable line, show that the momental ellipsoid
will continue to roll on the same plane as before, but the rate of motion will be
altered.
Ex. 2. If a plane be drawn through the fixed point parallel to the invariable
plane, prove that the area of the section of the momental ellipsoid cut off by this
plane is constant throughout the motion.
Ex. 3. The sum of the sqiiares of the distances of the extremities of the princi-
pal diameters of the momental ellipsoid from the invariable line is constant through-
out the motion. This result is due to Poinsot.
Ex. 4. A body moves about a fixed point under the action of no forces. Show
that if the surface Ax^ + Bif + Cz' ?= M{x^ +y^ + z^Y ^^ traced in the body, the principal
axes at being the axes of co-ordinates, this surface throughout the motion will
roll on a fixed sphere.
51G. To assist our conception of the motion of the body, let
us suppose it so placed, that the plane of the couple G, which
would set it in motion, is horizontal. Let a tangent plane to the
momental ellipsoid be drawn parallel to the plane of the couple G,
and let this plane be fixdd in space. Let the ellipsoid roll on this
fixed plane, its ceotre remaining fixed, with an angular velocity
which varies as the radius vector to the point of contact, and let
it carry the given body with it. We shall then have constructed
the motion which the body would have assumed if it had been
left to itself after the initial action of the impulsive couple G*.*
* Prof. Sylvester has pointed out a dynamical relation between the free rotating
body and the ellipsoidal top,. as he calls Poinsot'a central ellipsoid. If a material
the per-
e instan-
;e.
jular and
Blocity is
extremity
>ial to the
mch tliat,
, and the
) velocity.
momental
ilsive couple
Qtal ellipsoid
tion will be
he invariable
^ ofE by this
»f the princi-
ant through-
rces. Show
the principal
motion will
body, let
G, which
me to the
couple G,
11 on this
velocity
t, and let
nstructed
lad been
eG*:
ree rotating
a material
poinsot's construction.
411
The point of contact of the ellipsoid with the plane on which
it rolls traces out two curves, one on the surface of the ellipsoid,
and one on the plane. The first of these is fixed in the body and
is called the polhode, the second is fixed in space and is called the
herpolhode. The ecpiations to any polhode referred to. the prin-
cipal axes of the body may be found fronti the consideration that
the length of the perpendicular on the tangent plane to the ellip-
soid at any point of the polhode is constant. Hence its equations
are
Eliminating ij, we have
A {A - B) x' + C{C-n) z'=[^^^-B\ I
Me*
Hence if B be the axis of greatest or least moment of inertia,
the signs of the coefficients of x^ and z' will be the same, and the
projection of the polhode will be an ellipse. But if B be the
axis of mean moment of inertia, the projection is an hyperbola.
A polhode is therefore a closed curve drawn round the axis of
greatest or least moment, and the concavity is turned towards the axis
of greatest or least moment according as -„,- is greater or less than
the mean moment of inertia. The boundary line which separates
the two sets of polhodes is that polhode whose projection on the
plane pei*pendicular to the axis of mean moment is an hyperbola
whose concavity is turned neither to the axis of greatest, nor to
the axis of least moment. In this casv. G'=BT, and the projec-
tion consists of two straight lines whose equation is
A{A-B)x'-G{B- G)z' = 0.
This polhode consists of two ellipses passing through the axis
of mean moment,' and corresponds to the case in which the per-
pendicular on the tangent plane is equal to the mean axis of the
ellipsoid. This polhode is called the separating polhode.
Since the projection of the polhode on one of the principal
planes is always an ellipse, the polhode must be a re-entering
curve.
ellipsoidal top be constructed of uniform density, similar to Poinsot's central cllip-
sold, and if with its centre fixed it be set rolling on a perfectly rough horizontal
plane, it will represent the motion of the free rotating body not in space only, but
also in time : the body and the top may be conceived as continually moving round
the same axis, and at the same rate, at each moment of time. The reader is referred
to the memoir in the rhilosophical Transactions for 18G6.
il!
i!M
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^
i'
i
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yii
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w
412
MOTION UNDER NO FOKCES.
517. To find the motion of the extremity of the instantaneous
axis along the polhode which it describes we have merely to sub-
stitute from the equations
w, _ a)^ _ Wg _ w _ / y 1^
in any of the equations of Art. 508. For example we thus obtain
dx_ ITB-Cyz
X
,• —
M A
BG
~,&c., &c.,
{A-a){A-B)
(-V + r"), &c., &c.
Ex. 1. A point P moves along a polhode traced on an ellipsoid, show that the
length of the normal between P and any one of the principal planes at the centre
is constant. Show also that the normal traces out on a principal plane a conic
bimilar to the fooal conic in that plane. Also the measure of curTature of an
ellipsoid along any polhode is constant.
Ex. 2. Show that the line OJ used in Art. 234 to find the pressnre on the
fixed point is at right angles to the invariable line, and parallel to the tangent
plane to the momental ellipsoid at the point where the invariable line cuts it.
8howalsothatO'^=-c^ + a,'^^^'-^'^^-^''^-<^'^«7»)^^^^^»^%herep,.^„l>3
are the sum of the products A, B, C taken respectively one, two and three together.
518. Since the herpolhode is traced out by the points ef
contact of an ellipsoid rolling about its centre on a fixed plane,
it is clear that the herpolhode must always lie between two circles
which it alternately touches. The common centre of these circles
will be the foot of the perpendicular from the fixed centre on
the fixed plane. To find the radii let OL be this perpendicular,
and / be the point of contact. Let LI= p. Then we have
MAC cullagh's construction.
413
The radii will therefore be found by substituting for w' its
greatest and least values. But by Art. 509, these limits are \
and the greater of the two quantities X,, Xj.
The herpolhode is not in general a re-entering curve ; but if
the angular distance of the two points in which it successively
touches the same circle be commensurable with 27r, it will be re-
entering, i.e. the same path will be traced out repeatedly on the
fixed plane by the point of contact.
619. To explain Mac Cullagh's representation of the motion
hy means of the ellipsoid of gyration.
This ellipsoid is the reciprocal of the momental ellipsoid, and
the motion of the one ellipsoid may be deduced from that of the
other by reciprocating the properties proved in the preceding
Articles. We find,
(1) The equation to the ellipsoid referred to its principal
axes is
A^ B^ C~ M'
(2) This ellipsoid moves so that its superficies always passes
through a point fixed in space. This point lies in the invariable
line at a distance -r- from the fixed point. By Art. 509 we
know that this distance is less than the greatest, and greater than
the least semi-diameter of the ellipsoid.
(3) The perpendicular on the tangent plane at the fixed point
is the instantaneous axis of rotation, and the angular velocity of
the body varies inversely as the length of this perpendicular.
1 /T
lip be the length of this perpendicular, then ^ — 'K/ll'
(4) The angular velocity about the invariable line is constant
and = ^ .
The corresponding curve to a polhode is the path described on
the moving surface of the ellipsoid by the poin*j fixed in space.
This curve is clearly a sphero-conic. The equations to the sphero-
conic described under any given initial conditions are easily seen
to be
^ x^ t/" z^ 1
MT
^+f + ^=U^n
A'^ B'^C M'
These sphero-conics may be shown to be closed curves round
the axes of greatest and least moment. But in one case, viz.
5 ;
; if
(f: ;
■'
4U
MOTION XTNDER NO FOllCEH.
» i: I .
i 'i
I
m
i
' '
when 7„ = Ji, whore B is neither the greatest nor least mo-
ment of inertia, the sphero-conic becomes the two central circular
sections of the ellipsoid of gyration.
The motion of the body may thus be constructed by means of
cither of those ellipsoids. The momcntal ellipsoid resembles the
general shape of the body more nearly than the ellipsoid of gy-
ration. It is protuberant where the body is protuberant, and
compressed where the body is compressed. The exact reverse of
this is the case in the ellipsoid of gyration.
C20. MacCullagb has uncd tlio ellipsoid of g^'ration to obtain n gcomotrioal
intcrprotatiou of the solution of Euler'H equations in terms of elliptic integrals.
The ellipsoid of gyration moves so as always to touch a point L fixed in space.
Let us now project the point L on a plane passing through the axis of mean
moment and making an angle a with the axis of greatest moment. This projection
may be effected by drawing a straight line parallel to either the axis of greatest
moment or least moment. We thus obtain two projections which we will call
P and Q. Those points will bo in a plane PQL which is always perpendicular to
the axis of moan moment. As the body moves about the point L describes on
the surface of the ellipsoid of gyration a sphero-conic KK\ and the points P, Q
describe two curves pp', qq' on the plane of projection OBD. If the sphero-conic
as in the figure enclose the extremity A of the axis of greatest moment, the curve
inside the ellipsoid is formed by the projection parallel to the axis of greatest
moment, but if the sphero-conic enclose the axis of least moment, the inner curve
is formed by the projection parallel to that axis. The point P which describes the
inner curve will obviftiisly travel round its projection, while the point Q which
describes the outer curve will oscillate between two limits obtained by drawing
tangents to the inner projection at the points where it cuts the axis of mean
moment. . . ...
inner curve
is of mean
maccullagh's construction. 415
Since the direction cosines of OL are proportional to Au^, liu^, Cu^ it is easy to
see that, it x, y, zaxo the co-ordinatca of L,
Au^ liu.i Cwj G ^MT
Let OP=p, 0(1= p', and let the antjlcs those radii vcctores make with the pluno
contiiining the axes of greatest and leaat moment ho <p and <p' measured in the
lUrcction BD so that DOP= ~<t>, BOQ- -0': we then have
- p sin <p=y = nw.,{MT)-!>)
pco3<pBina = z-^Cwn(MT)-i \ ^''''
-p'Bm<j>' =y=Bu.,{M'I')-n ^ ''
It is proved in treatises on solid geometry that, if tlio plane on which the
projection is made is one of the circular sections of the ellipsoid, the projections
will he circles. This result may ho verified by finding p or p' from these equations,
licmcmboring that p and p' are constants, let us substitute in Eulcr's equation
from (2) and the first of equations (3). We have
P -ji= jp iJmT pp' sin a cos a cos 0'.
Since p' cos 0' is the ordinate of Q, we see that the velocity of V varies as the
ordinate of Q, and in the same way the velocity of Q varies as the ordinate ofV,
To find the constants p, p' we notice tl I p is the value of y obtained from
the equations to the sphero-conic when s=0. Wo thus have
s = '^AT-Ct'>')B ,j ^ {,G" - CT)B
^ MT{A-Ji)' ^ MT{Ii-C)*
the latter being obtained from the former by interchanging the letters A and C.
Hence
( velocity \ ^A -B i^, — j^ /ordinate \
521. Since p' sin 0' = p sin (p, wo have by substitution
where X' has the same value as in Art. 510. Let us suppose ^ expressed in terms
of t by the elliptic integral •
X((_r)= \ ,
so that 0=amX(<-T). Substituting this value of <p in equations (2) or (3), we
obtain the values of Wj, w^, Wj expressed in term a of the time,
Ex. Investigate the corresponding theorem for the momental ellipsoid.
41G
MOTION UNDER NO FOKCES.
522. If a body be set in rotation about any principal axis at
a fixed point, it will continue to rotate about that axis as a per-
manent axis. But the three principal axes at the fixed point do
not possess equal degrees of stability. If any small disturbing
cause act on the body, the axis of rotation will be moved into a
neighbouring polhode. If this polhode be a small nearly circular
curve enclosing the original axis of rotation, the instantaneous
axis will never deviate far in the body from the principal axis
which was its original position. The herpolhode also will be a
curve of small dimensions, so that the principal axis will never
deviate far from a straight line fixed in space. In this case the
rotation is said to be stable. But if the neighbouring polhode be
not nearly circular, the instantaneous axis will deviate far from
its original position in the body. In this case a very smell dis-
turbance may produce a very great change in the subsequent
motion, and the rotation is said to be unstable.
If the initial axis; of rotation be the axis OB of mean mo-
ment, the neighbouritij; polhodes all have their convexities turned
towards B. Unless, tiiierefore, the cause of disturbance be such
that the axis of rotation is displaced along the separating polhode,
the rotation must be unstable. If the displacement be along the
separating polhode, the axis may have a tendency to return tO its
original position. This case will be considered a little further on,
and for this particular displacement the rotation may be said to
be stable.
If the initial axis of rotation be the axis of greatest or least
moment, the neighbouring polhodes are ellipses of greater or less
eccentricity. If they be nearly circular, the rotation will certainly
be f^tab'e ; if very elliptical, the axis will recede far from its initial
position, and th'e rotation may be called unstable. If OC be the
axis of initial rotation, the ratio of the squares of the axes of the
A(A — C)
neighbouring polhode is ultimately -77771 — Tyi • It is therefore
necessary for the stability of the rotation that this ratio should not
differ much from unity.
It is well known that the steadiness or stability of a moving
body is much increased by a rapid rotation about a principal axis.
The reason of this is evident from what precedes If the body
be set rotating about an axis very near the principal axis of
greatest or least moment, both the polhode and herpolhode will
generally be very small curves, and the direction of that principal
axis of the body will be very nearly fixed in space. If now a
small impulse/ act on the body, the effect will be to alter slightly
the position of the instantaneous axis. It will be moved from one
polhode to another very near the former, and thus the angular
position of the axis in space will not be much affected. Let fi
be the angular velocity of the body, w that generated by the im-
THE INVARIABLE AND INSTANTANEOUS CONES.
4,17
pulse, then, by the parallelogram of angular velocities, the change
in the position of the instantaneous axis cannot be greater than
0)
sin"' jy . If therefore H be great, w must also be great, to produce
any considerable change in the axis of rotation. But if the body
has no initial rotation fl, the impulse may generate an angular
velocity a about an axis not nearly coincident with a principal
axis. Both the polhode. and the herpolhode may then be large
curves, and the instantaneous axis of rotation will move about
both in the body and in space. The motion will then appear
very unsteady. In this manner, for example, we may explain
why in the game of cup and ball, spinning the ball about a ver-
tical axis makes it more easy to catch on the spike. Any motion
caused by a wrong pull of the string or by gravity will not produce
so great a change of motion as it would have done if the ball had
been initially at rest. The fixed direction of the earth's axis in
space is also due to its rotation about its axis of figure. In rifles,
a rapid rotation is communicated to the bullet about an axis in
the direction in which the bullet is moving. It follows, from
what precedes, that the axis of rotation will be nearly unchanged
throughout the motion. One consequence is that the resistance
of the air acts in a known manner on the bullet, the amount of
which may therefore be calculated and allowed for.
1 i
id 'i
1 1
On the Cones described by the Invariable and Instantaneom Axes.
523. It is clear from what precedes that there are two im-
portant straight lines whose motions we shoidd consider. These
are the invariable line and the instantaneous axis. The first of
these is fixed in space, but as the body moves the invariable line
describes a cone in the body, which by Art. 519 intersects the
ellipsoid of gyration in a sphero-conic. This cone is usually called
the Invariable Cone. The instantaneous axis describes both a
cone in the body and a cone in space. By Art. 514, the cone de-
scribed in the body intersects the momental ellipsoid in a polhode,
and the cone described in space intersects the fixed plane on
which the momental ellipsoid rolls in a herpolhode. These two
cones may he called respectively the instantaneous cone and the
cone of the herpolhode.
524. There are various ways in which we may study the
properties of these cones. We may have recourse to solid geo-
metry. We may examine their curves of intersection with the
momental ellipsoid or the ellipsoid of gyration, as Poinsot and
MacCullagh have done. We may also examine by the help of
spherical trigonometry their curves of intersection with a sphere
i{. 1). 27
i\ ■■ ■
f\ I
1
■: I.
I- I
! n
418
MOTION UNDER NO FORCES.
I
.1
I* I
I: . :
! > ^ i
7
11 '.
II
i!
whose centre is at the fixed point, and which is either fixed in the
body or fixed in space at our pleasure. This will be found con-
venient when we wish to use a diagram.
525. Let the principal axes at the fixed point be taken as the
axes of co-ordinates. The axes of reference are therefore fixed in
the body but moving in space. By Art. 512, the direction-cosines
of the invariable line are
L«.
G
B(o. a
G
(o.
a
to.
ft).
cosmes of the instantaneous axis are — , -^
(0 (O
equations (1) and (2) of Art. 511, we easily find
and the direction-
From the
Wo
0)
A(o^' + Bay,' + Geo,' = (^ V + ^"< + ^O ^-2 .
•
If we take the co-ordinates x, y, z to be proportional to the
direction-cotines of either of these straight lines and eliminate w,,
Wg, &)g by the help of this equation, we obtain the equation to the
corresponding cone described by that straight line. In this way
we find that the cones described in the body by the invariable
line and the instantaneous axis are respectively
AT-G' , BT-G'
« +
r
GT-G'
,2 —
= 0,
A *" ' B ^ ' G
A {A T- G') x'' + B{BT-G')y'+C {CT- G') z' = 0.
These cones become two planes when the initial conditions are
such that G' = BT.
Ex. 1. Show that the circular sections of the invariable cone are parallel to
those of the ellipsoid of gyration and perpendicular to the asymptotes of the focal
conic of the momental ellipsoid.
526. There is a third straight line whose motion it is sometimes convenient to
consider, though it is not nearly so important as either the invariable line or the
instantaneous axis. If x, y, z be the co-ordinates of the extremity of a radius vector
of an ellipsoid referred to its principal diameters as axes and if a, 5, c be the semi-
X tJ z
axes, the straight line whose direction-cosines are - , r > - is called the eccentric line
a c
of that radius vector. Taking this deAnition, it is easy to see that the direction-
cosines of the eccentric line of the instantaneous axis with regard to the momental
ellipsoid are "j. / = , <>>ax/f> '^»\/f' ^^^^^ ^^^ ^^^° *^^^ directioi-cosir
of the eccentric line of the invariable line with regard to the ellipsoid of gy/ation.
This straight lino may therefore be called simply the eccentric line and the c( no
described by it in the body may be called the eccentric cone.
Ex. 1. The equation to the ecoentiic cone referred to the principal axes at the
fixed point is
(A T - cr^) x^+{nr-G^) i/ + (ct - c') c« = o.
■cosines
1 in the
nd con-
in as the
fixed in
i-cosines
irection-
rom
the
lal to the
linate Wj,
on to the
this way
invariable
= 0.
litions are
parallel to
of the focal
onvenient to
le line or the
radius vector
be the semi-
cccentric line
le direction-
le momental
ictior -cosines
of gy^'ation.
and the C( no
1 axes at the
THE INVARIABLE AND INSTANTANEOUS CONES.
419
This cone has the same circular sections as the momental ellipsoid and cuts that
ellipsoid in a sphero-conic.
Ex. 2. The polar piano of the instantaneous axis with regard to the eccentric
cone touches the invariable cone along the corresponding position of the invariable
line. Thus the invariable and instantaneous cones are reciprocals of each other
with regard to the eceontric cone,
6.27. Let a sphere of radius unity be described with its centre
at the fixed point about which the body is free to turn. Let
this sphere be fixed in the body, and therefore move with it in
space. Let the invariable line, the instantaneous axis, and the
eccentric line cut this sphere in the points L, I, and J5/ respectively.
Also let the principal axes cut the sphere in A, B, C. It is clear
that the intersections of the invariable, instantaneous, and eccen-
tric cones with this sphere will be three sphero-conics which are
represented in the figure by the lines KK\ JJ', DD\ respectively.
The eye is supposed to be situated on the axis OA, viewing the
sphere from a considerable distance. All great circles on the
sphere are represented by straight lines. Since the cones are co-
axial with the momental ellipsoid, these sphero-conics are sym-
metrical about the principal planes of the body. The intersections
of these principal planes with the sphere will be three arcs of
great circles, and the portions of these arcs cut off by any sphero-
conic are called axes of that sphero-conic. If we put a = in the
equations to any one of the three cones, the value of - is the
tangent of that semi-axis of the sphero-conic which lies in the
plane of xy. Similarly, putting y = 0, we find the axis in the
plane of xz. If (a, h), (a, J'), (a, /3) be the semi-axes of the
invariable, instantaneous, and eccentric sphero-conics respectively,
we thus find
27—2
I I..
It •
iHlli
>Sif
lil';'!
f v<tm\
■*■
t:
;^f'
i -
II!
M; P
In
illv
I .
Ill
ii1
iitti
i
I '
i
i I
420
MOTION UNDER NO FORCES.
tan a tan a
B
tan 6
A
tan 6'
tana
^AT-G' 1
IG'-BT^AB"
^tan/3^ VZr -Q^' 1
The first of these two sets gives the axes in the plane AOB,
the second those in the plane AOG. The former will be imagi-
nary if G'<BT. In this case the sphere- conies do not cut the
plane AOB. The sphero-conics will therefore have their con-
cavities turned towards the extremities of the axes OA or 00, i.e.
towards the extremities of the axes of greatest or least moment
according as 0^ is > or < BT.
sin^ b
Ex. 1. If we put l-e3=-r-s— we may define e to be the eccentricity of the
Bin" a •'
sphero-conio whose semi>axes are a and b. If e and e' be the eccentricities of the
AB-C
BA-C
and
invariable and eccentric sphero-conics respectively, prove that e^ =
B — C
^—'J~n ^^ *^*t ^oth theue eccentricities are independent of the initial conditions.
Ex. 2. If the radius of the sphere had been taken equal to ( wy,) instead of
nnity, show that it would have intersected the ellipsoid of gyration along the invari-
—7f^- \ , it would have intersected the
momental ellipsoid along the eccentric ellipse.
Ex. 3. A body is set rotating with an initial angular velocity n about an axis
which very nearly coincides with a principal axis 00 at a fixed point O. The
motion of the instantaneous axis in the body may be found by the following
formulae. Let a sphere be described whose centre is 0, and let / be the extremity
of the radius vector which is the instantaneous axis at the time t. If {x, y) be the
co-ordinates of the projection of I on the plane AOB referred to the principal axes
OA, Ob, then
* = V-B (B - C) i sin {pnt + M),
y=jA(A~ C) L cos {pnt + M),
IB — Cf\ (A — C)
where p'*=- -^ , and L, if are two arbitrary constants depending on the
initial values of x, y.
Ex. 4. If in the last question L be the point in which the sphere cuts the
invariable line, if (p, B) be the spherical polar co-ordinates of C with regard to
L as origin, and a the radius of the sphere, then
P^ r.n^^^ L^ \2AB^C (A + n) + (A -D)Cco62 (pnt + M)U
= ^^t +
CO J p
aPdt
neAOB,
le imagi-
t cut the
aeir con-
: 00, i.e.
, moment
icity of the
eitiea of the
AB-C
and
BA-C
I conditions.
instead of
g the invari-
jrsected the
bout an axis
nt 0. The
le following
le extremity
(x, y) be the
rincipal axes
ding on the
lere cuts the
th regard to
THE INVARIABLE AND INSTANTANEOUS CONES.
421
528. To find the motion of the invariable line and the instan-
taneous axis in the body.
Since the invariable line OL is f.xed in space and the body
is turning about 01 as instantaneous axis, it is evident that the
direction of motion of OL in the body is perpendicular to the
plane 10 L. Hence on a sphere whose centre is at the arc IL
is normal to the sphero-conic described by the invariable line. This
simple relation will serve to connect the motions of the invariable
line and the instantaneous axis along their respective sphero-
conics.
529. Lot V be the velocity of the invariable line along its
sphero-conic, then since tbe body is turning about 01 with an-
gular velocity &», and OL is unity, we have t; = w sin LOT. But
T . . T
by Art. 514 ^ = to cos i OL Elirctittatmg o) we have v = ^ tan LOT.
530. Produce the arc IL lo cut the axis AK in N, so that
LN \s &. normal to the sphero-conic described by the invariable
line. Taking the principal axes at the fixed point as axes of
reference, the direction -cosines of OL and 01 are respectively
proportional to ^w,, Bm^, Cq>^, and Wj, a^, Wg. The equation to
the plane LOT is
{B - C) a^w^x + {C-A) (0^(0 J/ + {A-B) co^a^z = 0.
This plane intersects the plane of xy in the straight line ON^
hence putting 2 = 0, we find the direction-cosines of ON to be
proportional to {A — G)o)^, {B— C) a>^, and 0. Hence
,o,LON^i^^.zSM±m^S>L.
Gsl{A-Cfo>,' + {B-Crftoy
The numerator of this expression is easily seen to be 0^ — CT.
Expanding the quantity under the root we have
A\' + B'co^'-2G{Aa>^'+B(o^')+C'{<o^'+a>,%
which is clearly the same as
G« _ C V - 2 C (T - (7a,/) + C (a,» - to,').
Substituting we find
coaLON=
G'-CT
G^/G''-2GT+G
Q>
Um L ON =
C\/OW~ T'
G'-CT' '
t! Ill
* ni
il ii
l|!f
1
If I.
i J
t ■ '
it' f
i- . i
ii
-.1 jrf
Ji l!
422
MOTION UNDER NO FORCES.
But jy = 0) COS L 01,
'. tan LOI^'^l^'—^. Hence the
T
.. iBxiLOI G^-CT , . ,, . . . .1 I .
ratio ~ — y^^ = — j:^ — , and is therefore constant throughout
the motion.
Treating the other principal planes in the same way, we see
that this proposition supplies us with a geometrical meaning for
G^ G^ G*
the three expressions -r-ji— 1, 'vfff~'^> ^^^ 'PT~ ^'
Combining this result with that given in the last Article, we
see that the
velocity of L I ^ G'-CT
along its conic] ~ CQ '
where n is the angle LON. If we adopt the conventions of
spherical trigonometry, n is also the length of the arc normal to
the sphero-conic intercepted between the curve and the principal
plane AB oi the body.
Ex. 1. If the focal lines of tlie invariable cone cwt the sphere in S and S', these
points are called the foci of the sphero-conic. Prove that the velocity of L
resolved perpendicular to the arc 8L is constant throughout the motion and equal
l\(G^- BT)(AT-(P) \k
If LM be an arc of a great circle perpendicular to the
axis containing the foci, and p be the arc SL. prove also that
*°0r AB
dp_ G \ (A-C){B-0 )i
dt " c\ AB \
sm LM.
'if
if
; J i
Ex. 2. Prove that the velocity of L resolved perpendicular t& the central radius
AT-GP
vector AL is — -,-pi — cot AL.
Ex. 3. If r, /, r" bo the lengths of the arcs joining the extremity A of a princi-
pal axis to the extremities £, I, E of the invariable line, instantaneous axis, and
eccentric line respectively ; 0, &, 0" the angles these arcs make with any priucipftl
plane A OB, prove that
CO?: " _ cos r' _ cos r"
tan (y tan ff'
B
sjBC'
where f^^^aroi/. Tliis theorem will enable us to discover in what manner the
motions of the three points L, I, E are related to each other.
Ex. 4. Show that the velocity of the instantaneous axis along it» sphero-conic
is TT, — -T-jT' ^^^ '*' '^'^^ ^' ^^^^'^ '*' ^^ ^^^^ length of the normal to the instantaneous
sphero-couio intercepted between the curve and the arc AB, and f-arc LI.
liculor to the
sentral radins
maiiuer tlio
THE CONE OF THE HERPOLHODE.
423
Comparing thia result with the corresponding formula for the motion of L given
in Art. 630, we see that for every theorem relating to the motion of L in its sphere-
conic there is a corresponding theorem for the motion of /. For example, if S' be a
focus of the instantaneous sphero-conic, we see that the velocity of / resolved per-
pendicular to the focal radius vector S'l bears ;' constant ratio to cos LI, This
constant ratio is^j<i?:z^|!i:^j*.
Show that the velocity of the eccentric line along its sphero-conic is
. tan n", where n" is the length of the arc normal to the sphero-conic inter-
Ex. 5.
G»-cr
cepted between the curve and the principal arc A B.
Ex. 6. Prove that (velocity of E)'^ - (velocity of £)"= constant. Show also that
this con8tant=^ ^^-^^^ '-.
Ex. 7. The motion of L along its sphero-conic is the same as that of a particle
acted on by two forces whose directions are the tangents at L to the arcs LS, I^S'
joining L to the foci of the sphero-conic and whose magnitudes are respectively
proportional to sin LS cos LS' and sin LS' cos LS,
531. The instantaneous axis describes a cone in space, whicli
has been called the cone of the herpolhode. The equation of
this cone cannot generally be found, but when it can be determined
we have another geometrical representation of the motion. For
suppose the two cones described by the instantaneous axis in
space and in the body to be constructed. Since each of these
cones will contain two consecutive positions of their common
generator, they will touch each other along the instantaneous
axis. Then the points of contact having no velocity the motion
will be represented by making the cone fixed in the body roll on
the cone fixed in space.
532. To find the motion of the instantaneous axis in space.
Since the invariable line OL is fixed in space, it will be con-
venient to refer the motion to OL as one axis of co-ordinates.
Let the ang'e the instantan'^ous axis 0/ makes with OL be called
f, and let ^ be the angle ttie plane lOL makes v/ith any plane
passing through OL and fixed in space.
During the motion the cone described by 01 in the body rolls
on the cone described by GI in space. It is therefore clear that
the angular velocity of the Instantaneous axis in space is the
same as its angular velocity in the body. Describe a sphere
whose centre is at and radius unity, and let this sphere be
fixed in the body. Let L, i be the intersections of the invariable
line and instantaneous axis with the sphere at the time t, L', I'
their intersections at the time t + dt. Then TL, 11/ are con-
secutive normals to the sphoro-conic /v7v' traced out by the in-
variable line and therefore intersect each other in some point V
III!
f, ii
» :'!■■
'H
III
■'!
ii
'J I
H
I s
!i
I ill
1 1
ili
424
MOTIOX UNDEU NO FORCES.
which may be regarded as a centre of curvature of tlic sphero-
conic. Let p = PL. Then clearly
velocity of / resolved") _ /velocity N sin (p + ^)
perpendicularly to ILj \ oi L ) ' sin p
T
= ^ tan f (cos ^+ oot p sin ^) ;
. <^0^y^ , tann
' ' dt G\ tan p) '
J. S
But it may be proved that in any sphero-conic tanp = - — ^ ,
tan (
where n is the length of the normal intercepted between the
curve and that axis which contains the foci, and 11 is the length
of the ordinate through either focus, and is usually called the
latus rectum. Substituting for tan p, and remembering that
tan? (P-CT , . . .«. , , , tan'^J
= — j^Trfi — , by Art. 530, and tan I =
tanw CT ''
T T^( P-CTY /tan^Y
CT J ' Vtan a)
dt 0^G\
tana
cot* f .
, we get
If we substitute for tan a and tan h their values, we get .
d<ly_T (AT- (?) {BT- G') (CT- (P) ^
dt G^ ABCGr "^^^ ^•
This result was first discovered by Poinsot.
533. Since the resolved angular velocity about the invariable
T
line is constant, we easily find to = ^ sec f. Substituting this
value of 0) in equation (6) of Art. 508, we find a relation between
5" and -7, , which however is too complicated to be of much use.
spliero-
tan'n
jen the
length
led the
%t
jet
rariable
mg this
)ctween
use.
THE ROLLING AND SLIDING CONE.
425
1 ( -ju
The values of -,- and -k in terms of t have now both been
at at
found ; from i^hese the motion of the instantaneous axis in space
can be deduced.
'-"''•=2S)(^-^^+^-4)
Ex. 1. Sliow that the angular velocity v' of the instantaneous axis in space or
in the body is given by
u '
where u is thr resultant angular velocity of the body and \, Xg, Xg have the mean-
ings given to them in Art. 608, This result is due to Foinsot.
Ex. 2. The length of the spiral between two of its successive apsides, described
in absolute space, on the surface of a fixed concentric sphere, by the instantaneous
axis of rotation, is equal to a quadrant of the spherical ellipse described by the same
axis on an equal sphere moving with the body. This is Booth's Theorem.
Ex. 3. If the eccentric line intersect in the point E the unit sphere which is
fixed in the body and has its centre at the fixed point, prove that
/ velocity V Td4>. -
534. Let be the fixed point, 01 the instantaneous axis.
Let the angular velocity oi about 01 be resolved into two, viz.
T
a uniform angular velocity -p about the invariable line OL, and
an angular velocity to sin lOL about a line OH lying in a plane
fixed in space perpendicular to the invariable line, and passing
through the fixed point 0. Let this fixed plane be called the
invariable plane at 0. As the body moves, OH will describe a
cone in the body which will always touch this fixed plane. The
velocity of any point of the body lying for a moment in OH is
unaffected by the rotation about OH, and the point has therefore
only the motion due to the uniform angular velocity about OL.
We have thus a new representation of the motion of the body.
Let the cone described by OH in the body be constructed, and
let it roll on the invariable plane at with the proper angular
velocity, while at the same time this plane turns round the in-
T
variable line with a uniform angular velocity yy . The cone de-
scribed by OH in the body has been called by Poinsot the Boiling
and Sliding Cone.
535. To find a construction for the sliding cone. Its generator
OH is at right angles to OL, and lies in the plane lOL. Now
OL is fixed in space ; let OL' be the line in the body which, after
an interval of time dt, will come into the position OL. Since the
body is turning about 01, the plane LOL' is perpendicular to the
plane LOT, and hence OH is perpendicular to both OL and OL'.
That is, OH is perpendicular to tlie tangent plane to the cone
426
MOTION UNDER NO FORCES,
,:
4
■!t:
described by OL in the body. Tlio cone described by OH in the
body is therefore tlie reciprocal cone of that described by OL.
The equation to the cone described by OL has been shown to be
AT- O^ , . BT- G" , CT- CP , ^
— :r— ^ + — 5— 2/' + — 77— «'= 0.
Hence the equation to the cone described by Oil is
A . B . G
AT-G
,a!' +
BT-G'
y' +
CT-(P
«' = 0.
The focal lines of the cone described by OH are perpendicular
to the circular sections of the reciprocal cone, that is the cone
described by OL. And these circular sections are the same as
the circular sections of the ellipsoid of gyration. Hence the focal
lines lie in the plane containing the axes of greatest and least
moment, and are independent of the initial conditions.
This cone becomes a straight line in the case in which the
cone described by OL becomes a plane, vi/. when the initial con-
ditions are such that G^ = BT.
53G. To find the motion o/OH in space and in the body.
Since OL, OH and 01 are always in the same plane the
motion of OH in space round the fixed straight line OL is the
dt
m
same as that of 01, and is given by the expression for
Art. 532.
To find the motion of OH in the body it will be convenient
to refer to the figure of Art. 532. Produce the arcs PL, PL
to H and H' so that LH and L'H' are each quadrants. Then
// and H' are the points in which the axis OH intersects the
unit sphere at the times t and t + dt.
We have therefore
/velocity\ _ /velocityN
V of // ; ~ V of Z }
sm
( P + 2; T
^ = -p tan if cot p.
smp
Substituting for tan p as before we may express the result in
terms of §" or n at our pleasure.
Since the cone described by OH in the body rolls on a plane
which also turns round a normal to itself at 0, it is clear that the
angular velocity of OH in the body is loss than the angular
velocity of OH in space by the angular velocity of the plane, i. e.
T
G'
/velocity\ _ rZ0
V of ^ J~dt
r/ in the
by OL.
n to bo
mdicular
the cone
same as
the focal
md least
hich the
itial con-
dy.
lane the
•Z is the
dt
ir
m
nvenient
PL, PL'
Then
ects the
result in
a plane
that the
angular
ne, i. e.
MOTION OF THE PRINCIPAL AXES.
Motion of the Piinclpal Axes.
427
537. To find the angular motions in space of the pnncipal
axes.
Since the invariable line OL is fixed in space it will be con-
venient to refer the motion to this straight line as axis of z.
Let OA^ OB, OG be the principal axes at the fixed point 0, and
let, as before, a, /3, 7 be their inclinations to the axis OL or OZ.
Let X, fi, V be the angles the planes LOA, LOB, LOC make
with some fixed plane LOX passing through OL. Our object is
to find J- and -r- with similar expressions for the other axes.
This problem is really the same as that discussed in Art. 235, but
it will be found advantageous to make a slight variation on the
demonstration.
Describe a sphere whose centre is at the fixed point, and
whose radius is unity. Let the invariable line, the instantaneous
axis and the principal axes cut this sphere in the points L, I,
A, B, C respectively. The velocity of A resolved perpendicular
to LA will then be sin a -^ . But since the body is turning round
01 as instantaneous axis, the point A is moving perpendicularly
to the arc lA^ and its velocity is w sin lA. Resolving this per-
pendicular to the arc LA, we have
sin a -rr = ft) sin AI cos LAI
at
= 0)
cos LI— cos LA cos I A
sin Lxi '
by a fundamental formula in spherical trigonometry. But w cos LI
is the resolved part of the angular velocity about OL, which is
T
equal to -^ > ^^^ ^ cos lA is the resolved part of the angular
!
til
m
f i?<
I. -^ !>■
428
MOTION UNDEB NO FORCES.
Mi
velocity about OA, which is a>^, Wo have therefore
. ^ tl\ T
8in a Ti = -7=; — w, cos a,
at O ^
a result wliich follows immediately from Art. 249.
G cos a = Au)^, we have
dX^ T G'cos'g
This result may also be written in the form
AT-a''
Since
sin' a
.(1).
d\_T
dt a"^
AG
cot' a
.(2).
da.
538. To find -^ we 'may proceed in the following manner.
We have cos a =
lO),
G'
cos /3 = -^,
cos 7 = -Tj\
Substituting in Euler's equation
^ Tt
dfx
we have
sm a
dt
= ( -^ - -^JG cos ^ cosy (3).
But by Art. 508 cos a, cos/9, cos 7 are connected by the equations
G"
cos'a
cos^ cos''7
AT ■*■ ~B~ "^ n7~
cos'a + cos'/S + cos''7 = 1
.(4).
If we solve these equations so as to express cos /8, cos 7
in terms of cos a, we easily find
Bin
, fday G^ (CP-CT A-C „ \/
G^-BT A-B
G«
A
COS'
.)..
(5).
539. Since the left-hand side of equation (6) is necessarily real, we see that the
values of cos* a are restricted to lie between certain limits. If the axis whose
motion we are considering is the axis of greatest or least moment let B be the axis
0^-CT A
of mean moment,
G^~BT A
In this case cos^ a must lie between the limits
and
G« A-G
if both be positive. By Art. 509 the former of these two is positive
G* A-B
and less than unity ; this is easily shown by dividing the numerator and the de-
nominator by A C(P. If the latter is positive the spiral described by the principal
axes on the surface of a sphere whose centre is at the fixed point lies between two
concentric circles which it alternately touches. If the latter limit is negative cos a
lias no inferior limit. In this case the spiral always lies between two small circles
on the sphere, one of which is exactly opposite the other.
MOTION OP THE PRINCIPAL AXES.
429
COS 7
If tltA axis couHidoroil is tho axiH of moan moment, coh> a muHt lio outiide tlio
Bamu two liuiitH an befui-o. Both these are positive, but one In greater and the
otlier leHH tbau unity. Tho spiral thoroforo lies between two amall circIoB oiJpoHito
each other.
In order that ,. may vanish we must havo G'co8'a = iir, but this by substitu-
tion makes t- imaginary. Thus t- always keeps one sign. It is easy to see that
G"
if tho initial conditions arc stoh that -=^ is less than the mcmont of inertia about
tho aids which describes tlio hi iral wo are considering, tho angular velocity will bo
greatest when the axis is nearest tho invariable hue and least when tho axis is
furthest. The reverse is the case if -yp is greater than tho moment of inertia.
640. Ex. 1. Let OM be any straight line fixed in tho body and passing
through and let it cut the eUipsoid of gyration at in Ihe point M. Let OM' bo
the perpendicular from on the tangent plane at Hf. If OM~r, OM'-p, and if
i, i' be the angles OM, OM' make with the invariable lino OL, prove that
Bin* 1 4^ ■■
at
TO..,
; cos I cos 1 ,
Q pr
where j is tho angle the plane LOM makes with some plane fixe i in space passing
through OL. This follows from Art. 249 or from Art. 537.
Ex. 2. If KLK' be the spiral traced out by the invariable line in the manner
described in Art. 527| show that
% r^ ^A /vectorial area\
where X is the angle described by the plane containing the invariable line and the
principal axis OA,
Ex. 3. If xj/ be the angle described in space by the plane containing the invari-
able line and any straight line OM, fixed in the body, passing through and
cutting the sphere in M, prove that
. T -A /vectorial area \
where MN is any spherical arc fixed in the body and cutting in N the sphero-conic
described by the invariable line.
Ex. 4. If we draw three straight lines OA, OB, OC along the principal axes at
the fixed point of equal lengths, tho sum of the areas conserved by these lines on
the invariable plane is proportional to tho time. [Poinsot.]
Ex. 5. If the lengths OA, OB, OG bo proportional to tho radii of gyration
about the axes respectively, the sum of the areas conserved by these lines on the
invariable plane will also be proportional to the time. [Poinsot.]
ill
I ;i
i'll
ti'sli
" .-i
430
MOTION UNDER NO FORCES.
il I
r
I.'
Motion of the hody when two principal axes are equal.
541. Let the body be rotating with ,an angular velocity ta
about an instantaneous axis 01. Let OL be the perpendicular
on the invariable line. The momental ellipsoid is in this case a
spheroid, the axis of which is the axis of unequal moment in the
body. Let the equal moments of inertia be A and B. From
the symmetry of the figure it is evident that as the spheroid rolls
on the invariable planes, the angles L OG, L 01 are constant, and
the three axes 01, OL, OC are always in one plane. Let the angles
LOC = %IOC=i.
Following the same notation as in Art. 508, we have
ft). = ft) cos I,
ft), + &>a = ft) sm I,
T=(AsmU+CcosU)o>\
AVe therefore have
Cftjg _ C cos t
cos 7 =
jA^shiH+V'coFi'
This result may also be obtained as follows. In any conic if
i and 7 be the angles a central radius vector and the perpendicular
on the tangent at iio extremity make with "^he minor axis, and if
a, b be the semi-axes, then tan 7 = — a ta,n i. Applying this to the
momental spheroid, we have
tan 7=7^ tan i.
The angle i being known from the initial conditions, the angle 7
can be found from either of these expressions. The peculiarities
of the motion will then be as follows.
The invariable line describes a right cone in the body whose
axis is the axis of unequal moment, and whoso semi-angle is 7.
The instantaneous axis describes a right cone in the body
whose axis is the :\xis of unequal moment, and whose semi-angle
is I.
The instantaneous axis describes a right cone in space, whose
axis is the invariable line, and whose ^emi-angle is i ~ 7.
The axis of uneqiial moment describes a right cone in space
whose axis is the invariable line, and whose semi-angle is 7.
The angular velocity of the body about the instantaneous
axis varies as the radius vector of the spheroid, and is therefore
constant.
MOTION WHEN A = B.
431
542. The rate of motion of the invariable line and the
instantaneous axis in the body may be found most readily by
referring to the original equations of motion in Art. 508. We have
in this case
>-=ol
A-r^ — {A — C) (0^(0 COS 1 =
■ A-^-\-{A-C)(o^cocosi=0
Solving these by differentiating the first and eliminating w^,'
we find
a)^ = i cos I — -^ — (ot COS z
ft)j=-Fsinr
^ (otco&z + n,
A
A-C
where i^ and /are arbitrary constants. Let the projection of either
the instantaneous axis or the invariable line on the plane per-
pendicular to the axis of unequal moment make an angle ;^ with
any fixed straight line which may be taken as axis OA. Then
tan ;^ = — ^ Hence we find
__ 2
dt
A- G
A
(o cos I.
543. To find the common rate of motion in space of the
instantaneous axis and the axis of unequal moment.
Let G be the extremity of the axis of figure of the momental
ellipsoid, and let H be the rate at which the plane LOG is turning
round OL. Let CM, CN be perpendiculars on GL and CI.
Then since the body is turning round GI, the velocity of G is
GN.(o. But this is also CM M. Since GM=OGsmy,
CN= (9(7 sin i, we have at once
fl sin 7 = G) sin i,
whence fl can be found.
544, Ex, 1. If a right circular cone whose altitude a is double the radius of
its base turn about its centre of gravity as a fixed point, and be originally set in
motion about an axis inclined at an angle a to the axis of figure, the vertex of the
cone will describe a circle whose radius is -r asin o,
4
[Coll. Exam.]
Ex. 2, A circular plate revolves about its centre of gravity as a fixed point. If
an angvdar velocity w were originally impressed on it about an axis making an angle
a with its plane, a normal to the plane of the disc will make a revolution in space in
27r
time — ; — [Coll. Exam.]
w>/l + 3sin*o
:!■
Il!|
lir
!
n
432
MOTION UNDER NO FORCES.
Ex. 3. A body wliich can turn freely about a fixed point at wbich two of the
principal moments are equal and less than the third, is set in rotation about any
axis. Owing to the resistance of the air and other causes, it is continually acted
on by a retarding couple whose axis is the instantaneous axis of rotation and whose
magnitude is proportional to the angular velocity. Show that the axis of rotation
will continually tend to become coincident with the axis of unequal moment. In
the case of the earth therefore, a near coincidence of the axis of rotation and axis
of figure is not a proof that such coincidence has always held. Astronomical
Notices, March 8, 1867.
Motion when G' = BT.
' if
I H !r
1 : I
545. The peculiarities of this case have been already alhuleil
to in Art. 508. When the initial conditions are «uch that this
relation holds between the Vis Viva and the Momentum of the
body the whole discussion of the motion becomes more simple*.
The fundamental equations of motion are
Solving these, we have
B-C G^-B'ay.^
= bt]
(o' =
A-C
AB
, A-B G'
B'- ''
ft)„
BC
But
d(o„ C-A
dt
B
<»i<»3;
.(1).
(2).
k\
do),
dt
--w^
B) { B- G) G' - R
AG ' B*
<»„
When the initial values of w^ and w, have like signs, {G- A) &>,&•,
d
CO.
is negative and therefore -rj' must be negative, hence in this
expression the upper or lower sign is to be used according as the
initial values of cd,, Wg have like or unlike signs.
B'
" G'- B%
} dt
= V
{A-B){B~G )
AG
» •'
III
* This case appears to have been considered by nearly every writer on tins
subject. As examples of different methods of treatment the reader may consult
Lvrjmdrc, Traite den Fonctious EllqHiques, 1825, Vol. I. page 382, and Poimot
Theorie Nonvelle dc la Rotation des coqjs, 1852, patrc 104.
tvo of the
about any
tally acted
md whoso
f rotation
ment. In
a and axis
ronomical
alliKle<l
,hat this
n of the
iple*.
.(1).
.. (2).
-A)
(0.a\
in this
ig as the
er on this
iiy consult
1(1 Poinsvt
MOTION WHEN G' - BT.
433
If we put + n for the right-hand side and integrate we have
where E is some undetermined constant.
G
E.e
T-j^nt
+ 1
G
As t increases indefinitely, w^ approaches T-^ aa its limit and
therefore by (2) Wj and Wg approach zero.
The conclusion is that the instantaneous axis ultimately ap-
proaches to coincidence with the mean axis of principal moment,
but never actually coincides with it. It approaches the positive
or negative end of the mean axis according as the initial value
of {0—A) Wj, <Ug is positive or negative.
546. To find what the cones traced out in the body by the
invariable line and instantaneous aods become when (a = BT.
Eliminating w, from the fundamental equations of the last
Article we have
A{A-3)(o^' = C{B-C)a>,'.
Taking the principal axes at the fixed point as axes of refer-
ence, the equations of the invariable line are -j — = -^r— = 77— .
^ A(o^ xjo), C«»3
Eliminating a>^ and a^ the locus of the invariable line is one of
the two planes
/A-B , /B-G
OS 71 Si
The equations of the instantaneous axes are — = -^ = — .
^ ft), G)j 6)3
Eliminating ft), and g), the locus of the instantaneous axis is one
of the two planes
^A{A-B)x=± ^G{B-C) z.
In these equations since — follows the sign of -^ the upper
or lower sign is to be taken according as the initial values of
ft),, 6)3 have like or unlike signs. These planes pass through the
mean axis, and are independent of the initial conditions except
sofarthat (?' = i?r.
R. D. 28
ill
r. ,
'! 1
n
f
T' 1
t
1^
3i> t
434
MOTION UNDER NO FORCES.
The rolling and sliding cone is the reciprocal of that described
by the invariable plane, and is therefore the straight line perpen-
dicular to that plane which is traced out by the invariable line.
Ex. 1. Sbow that the planes described by the invariable line coincide with the
central circular sections of the ellipsoid of gyration and are perpendicular to the
asymptotes of that focal conic of the momeutal ellipsoid which lies in the plane of
the greatest and least moments.
Ex. 2. The planes described by the instantaneous axis are perpendicvdar to the
umbilical diameters of the ellipsoid of gyration and are the diametral planes of
the asymptotes of the focal conic in the momental ellipsoid.
547. The relations to each other of the several planes fixed
in the body may be exhibited by the following figure. Let
A, By C be the points in which the principal axes of the body
cut a sphere whose centre is 0, and radius unity. Let BLK',
BIJ' be the planes traced out by the invariable line and the
instantaneous axis respectively. Then by the last Article
tanCA =V6"2^i5'*^^^'^=V2-^5-
Hence v<e find
tan K'J' = tan LBI
v
{B -C){A-B)
AG
This is the quantity which has been called n in Art. 545.
Exactly as in Art. 528 the direction of motion of L is perpen-
dicular to IL and hence the angle ILB is a right angle. Thus
the spherical triangle ILB has one angle right, and another
constant and independent of all initial conditions.
Exactly as in Art, 528, the velocity of L along LB is equal to
described
3 perpen-
e line.
de with the
Eular to the
)he plane of
ciilar to the
1 planes of
mes fixed
lire. Let
the body
et BLK',
5 and the
H5.
s
pcrpen-
le. Thus
anotlier
c(pinl t()
MOTION WHEN G^ = BT.
435'
T
(o&mlL which, by Art, 514, is equal to ^tan/Z. But from he
spherical triangle ILB
n sin BL - ■ tan IL.
If then we put as before /5 = BL, we have
^^=±^nsin^.
If the initial values of w , Wg have the same sign, the body
is turning round / from K to B. Hence, since L is fixed in
space, BL is increasing and therefore the upper sign must be
used in this figure. See also Art. 545.
We may also find an expression for /9 in terms of the time.
Since cos /S = ■— we have, by Art. 545,
l+C0Sy8_ rr^'^Jl^
1-^s^"^^
.•.cot|=VIe'i"'.
Ex. Show that the eccentric line describe3 a great circle passing through Canil
cutting AC in some point D' where tan^ CD' = tan CJ' tan CK'. If E bo the inter-
section of the eccentric line with the sphere, show that the area BE and BL are
always equal.
548. To find the motion of the body in space.
We have already seen that the motion is such that a plane
fixed in the body, viz. the plane BK\ contains a straight lino
fixed in space, viz. the invariable line OL. Since the body is
brought from any position into the next by an angular velocity
T
w cos lOL = 7^ about OL, and an angular velocity o> sin lOL
about a perpendicular to OL, viz. OH, it follows that the plane
fixed in the body turns round the line fixed in space with a
T C
uniform angular velocity ^ or ^ . At the same time the plane
moves so that the line fixed in space appears to describe the
plane with a variable velocity w sin lOL, If /8 be the angle BL,
T
this has been proved in the last Article to be ^ n sin fi.
549. The cone described by OH in the body is the reciprocal
cone of that described by OL, and from it we may deduce re-
ciprocal theorems. The motion is therefore such that a straight
line fixed in the body, viz. OH, describes a plane fixed in space,
viz. the plane perpendicular to OL. The straight line moves
28—2
' !
I'M
M'
436
MOTION UNDER NO FORCES.
!
\:
is ■
i
' if!
T G
along this plane with a uniform angular velocity equal to ^ or ^ ,
■'vhile the angular velocity of the body about this straight line
is +-^Jisin^.
550. The motion of the principal axes may be deduced from
the general results given in Art. 537. But we may also proceed
thus. Since the body is turning about 01, the point B on the
sphere is moving perpendicularly to the arc IB. Hence the
tangent to the path of B makes with LB an angle which is the
complement of the constant angle IBL. The path traced out
by the axis of mean moment on a sphere whose centre is at is
a rhumb line which cuts all the great circles through L at an
angle whose cotangent is ± n.
65i. To find the motion of the instantaneous axis in spac3.
This problem is the same as that considered in Art. 532. We
may however deduce the result at once from Art. 548. The angle
ILB is always a right angle, it therefore follows that the angular
velocity of / round L is the same as that of the arc BL round L.
T
But the angular velocity of the latter is constant and equal to ^.
If then be the angle the plane hOI containing the instanta-
neous axis arid the invariable line makes with some fixed plane
passing through the invariable line, we have 7^ = 75 •
652. To find the equation of the cone described by the
instantaneous axis in space, we require a relation between if and <^,
where f is the arc IL on the sphere. From the right-angled
triangle ILB we have n sin y9 = tan 5", and by Art. 547,
cot| = V£'e ^ •
Eliminating ^, we shall have an expression for §" in terms of U
We find
o
-?^ = cotf+tanf = V:^e
tan ^22
By the last Article (f> = ^t + F, where F is some constant.
Let us substitute for t in terms of <f>j_and let us choose the plane
from v.hich <f> is measured so that s/Ee^^^— 1.
The equation to the cone traced out in space by the instan-
taneous axis is
2/icotf=e«* + e-»*.
T G
ight line
iced irom
proceed
B on the
ence the
ich is the
aced out
is at is
L at an
i32. We
rhe angle
e angular
round L.
T
ual to ^.
instanta-
ced plane
by the
1 ^ and <}>,
ht-angled
jrms of t.
constant,
ihe plane
le instan-
CORRELATED AND CONTRARELATED BODIES. 437
When ^ = 0, we have tan(;'=n. Therefore the plane fixed in
space from which (f> is measured is the plane containing the axes
of greatest and least moment at the instant when that plane
contains the invai'iable line.
On tracing this cone, we see that it cuts a sphere whose centre
is at the fixed point in a spiral curve. The branches determined
by positive and negative values of <f> are perfectly equal. As <f>
increases positively the radial arc ^ continually decreases, the
spiral therefore makes an infinite number of tuins round the
point L, the last turn being infinitely small.
2mb
Ex. In the herpolhode
'=/-«»
+c-'»*, if the looua of the extremity of the
polar subtangent of this curve be foand and another carve be similarly generated
from this locus, the curve thus obtained will be similar to the herpolhode. [Math.
Tripos, 1863.]
On Correlated and Contrarelated Bodies,
553. To compare the motions of different bodies acted on hy
initial couples whose planes are parallel.
Let a, /S\ 7 be the angles the principal axes OA, OB, 0(7 of
a body at the fixed point make with the invariable line OL.
Then by Art. 511, Euler's equations may be put into the form
dcosa , ^/l 1\ „ f. .-V
— ^^-+G'f-g--^jco3/3cos7 = (1),
with two similar equations. Let \, fi, v be the angles the planes
LOA, LOB, LOG make with any plane fixed in space, and passing
through OL. Then
. » <?\ T (rcos'a ,»v
^^^«d-^ = G^ AT (2)'
with similar equations for fi and v.
If accented letteiv. denote similar quantities for some other
body, the corresponding equations will be
^^'+G"(-^,--^)cos^cos7'-0 (3).
If then the bodies are such that
.
i'^
;lj,3
;i!
!^i
! :, . '\i
438
MOTION UNDER NO FORCES.
I
the equations (1) to find a, /9, 7 are the same as the ecjuations (3)
to find o', ^, 7'. Therefore if these two bodies be initially placed
with their principal axes parallel and be set in motion by impulsive
couples whose magnitudes are G and 0\ and whose planes are
parallel, then after the lapse of any time t the principal axes of
the two bodies will still be equally* inclined to the common axis
of the couples.
The equations (5) may be put into the form
O _G' _G_G' _G G' f
Since by Art. ' 'ntlK
B B' C C"
\a \ iva is given by
J.
€'.'
,a,
«c cosjS cos'7
.r ■*■ B ^ ~cr
(7),
T T'
wc see that each of the express:.oij ^ in (C) is equal to -p; — j^.
It immediately follows by subtracting equations (2) and (4)
and dividing by sin' a that
dX
dt
dt
G
G"
with similar equations for 11 and v. Thus the two bodies being
started as before with their principal axes parallel each to each,
the parallelism of the princi^jal axes may be restored by turning
* In order tbat the angles which the principal axes make with the axis of tho
conple may be the same in each body, it is necessary that the cones described by
the axis OL in the body should be the same. Hence by Art. 525, the two ellipsoids
of gyration must have the same circular sections, or which is the same thing, the
two momental ellipsoids must have the same asymptotes to their hyperbolic focal
conies. Also in Oxder that the cones may be the same we must have
L T^ 1. ^ 1 ^
A ~ G" _ B ~ G' _ £~G2
1^ T ~ J_ 2 " ~ 1 T' ■
If we put each of these equal to some quantity r , we easily find
i ^ I _ 1 i i.
A~ B ~B C C~ A
A'~ B' B'~ C C'~ A'
If in VM two bodies the angles between the principal axes and the axis of the couple
ai-e to be equal each to each at tho same time, tho equations (1) and (3) of Art. f'5a
show that we must have in addition ~ j = r. This leads to the generalization of Prof.
Sylvester's theory given in the text.
ations (3)
Ily placed
impulsive
>1anes are
al axes of
imon axis
...#.(C).
(7),
' O"
) and (4)
dies being
1 to each,
y turning
> axis of tho
described by
NO ellipsoids
10 thing, the
erbolic focal
)f the couplo
of Art. r>53
lion of Prof.
CORRELATED AND CONTRARELATED BODIES.
43.9
the body whoso principal axes are A', B, C about tho cora-
mon axis of the impulsive couples through an angle [jy — Trijt
in the direction in which positive impulsive couples act*.
554. When the couples G and 0' are equal the condition (6)
becomes
A A' B lj:~ C Cf ~ G-' '
the bodies are then said to be correlated. If m omental ellipsoids
of the two bodies be taken so that the moment of inertia in each
bears the same ratio to the square of the reciprocal of the radius
vector these ellipsoids are clearly confocal.
When the couples G and G' are equal and opposite, the
equation (6) becomes
1 1 _ 1^ 1 _ 1 1 _ r+ r
A^ A:~ B^ B~'G^C~ G* *
and the bodies are said to be contrarelated.
555. To compare the angular velocities of the two hodit xt
any instant.
Let ft) be the angular velocity of one body at any insta ^ M:en
following the usual notation we have
If the same letters accented denote similar quantities for the
other body
'« r"2 /cos
(o =G (-^
a cos* /3
r +
cos' 7'\
2?" ' 6"*
Bat remembering the condition (G) these give
..-.■.=(f4)[..«(^f,).co.,(«4).cosv|H-g;)].
* Since the cones described by the invariable line in the two bodies are identical,
their reciprocal cones, 1. e. Poinsot's rolling and sliding cones, are also identical in
the two bodies. Thus in the two bodies, the rolling motions of these cones are
equal, but the sliding motions may be different. The si. ding motions represent
T T'
angular velocities about the invariable line respectively equal to ^ and ^, . Hence
we have
dt ~ (It ~ dt " (it dt~' dt G~ G"
This remark on the former note is due to Prof. Cayley. '_
' t
m
M
?5i
440
MOTION UNDER NO FORCES.
!
11
By referring to (7) the quantity in square brackets is easily
T T
seen to be ^ + T77 1
Ex. If two bodies be so related that their ellipsoids of gyration are confooal, and
bo initially so placed that the angles (a, /3, 7) (o', ^, 7') their principal axes mako
with the invariable lino of each are connected by the equations
cos a
cos a' cos /3
Cos/S* cos y cos 7'
J A' ' Jb Jb' ' Jc ^/C" '
and if these bodies bo set in motion by two impulsive couples 0, 0' respectively
proportional to iJaBG and Ja'B'C', then the above relations will always hold be-
tween the angles (a, /3, 7) (a', /3', 7'). If p and p' be the reciprocals of -3; and -r- ,
then Op-Q'p' will bo constant throughout the motion, where \ X', &o., are the
angles the planes LOA, L'O'A' make at the time t with their positions at the
time (=0.
556. When a body turns about a fixed point its motion in
space is represented by making its momental ellipsoid roll on a
fixed plane. This gives no representation of the time occupied
by the body in passing from any position to any other. The
preceding Articles will enable us to supply this defect.
To give distinctness to our ideas let us suppose the momental
ellipsoid to be rolling on a horizontal plane underneath the fixed
point 0, and that the instantaneous axis 01 is describing a polhode
about the axis of A. Let us now remove that half of the ellipsoid
which is bounded by the plane of BG, and which does not touch
the fixed plane. Let us replace this half by the half of another
smaller ellipsoid which is confocal with the first. Let a p^ane
be drawn parallel to the invariable plane to touch this ellipsoid
in /' and suppose this plane also to be fixed in space. These two
semi-ellipsoids may be considered as the momental ellipsoids of
two correlated bodies; If they were not attached to each other
* This result may also bo obtained in tho following manner. By Art. 534 the
T
angular velocity w of one body is equivalent to an angular velocity ^ about the
invariable line and an angular velocity 12 about a straight liuo Oil which is a gene-
rator of the rolling and sliding cone. Hence w^ = ^o + 0". A similar equation with
accented letters will hold for the other body. Since in the two bodies the angles
between the principal axes and tho invariable line are equal each to each through-
out the motion, the rolling motions of the two cones must be equal, hence Q=R'.
It follows immediately that w'-«'»= -p, - ;^t„.
Or' Cr ^
is easily
mfooal, and
axes mako
respectively
,y3 hold be-
i\ , d\'
&c., are the
tious at the
notion in
roll on a
occupied
ler. The
nomental
the fixed
a polhode
ellipsoid
not touch
f another
a plane
ellipsoid
!'hese two
ipsoids of
ich other
Irt. 534 the
r
about the
1 is a gene-
uation with
3 the angles
h through-
ence Si=0'.
CORRELATED AND CONTRARELATED BODIES.
441
and were free to move without interference, each would roll tho
one on the fixed piano which touches at /, and the other on that
which touches at /'. By what has been shown the upper ellipsoid
(being the smallest) may be brought into parallelism with tho
lower by a rotation ^M j ~ "^') about the invariable line. If then
the upper plane on which the upper ellipsoid rolls be made to
turn round the invariable line as a fixed axis with an angular
velocity ^( t~'t)' *^® ^^^ ellipsoids will always be in a state
of parallelism, and may be supposed to be rigidly attached to each
other.
Suppose then the upper tangent plane to be perfectly rough
and capable of turning in a horizontal plane about a vertical axis
which passes through the fixed point. As the nucleus is mado
to roll with the under part of its surface on the fixed plane below,
the friction between the upper surface and the plane will cause
the latter* to rotate about its axis. Then the time elapsed will
be in a constant ratio to this motion of rotation, which may be
measured off on an absolutely fixed dial face immediately over the
rotating plane.
The preceding theory, so far as it relates to correlated and
contrarelated bodies, is taken from a memoir by Prof. Sylvester
in the Philosophical Transactions for 1866. He proceeds to in-
vestigate in what cases the upper ellipsoid may be reduced to a
disc. It appears that there are always two such discs and no
more, except in the case of two of the principal moments being
equal, when the "olution becomes unique. Of these two discs
one is correlatetx and the other contrarelated to the given body,
and they will be respectively perpendicular to the axes of greatest
and least moments of inertia.
Poinsot has shown that the motion of the body may be con-
structed by a cone fixed in the body rolling on a plane which
turns uniformly round the invariable line. If, as in the preceding
theory, we suppose the plane rough, and to be turned by the
cone as it rolls on the plane, the angle turned through by the
plane will measure the time elapseu.
* As the ellipsoid rolls on the lower plane, a certain geometrical condition must
be satisfied that the nucleus may not quit the upper plane or tend to force it
upwards. This condition is that the plane containing 01, 01', must contain
the invariable line, for then and then only the rotation about 01 can be resolved
into a component about Or and a component about the invariable line. That this
condition must be satisfied is clear from the reasoning in tho text. But it is
also clear from the known properties of coufocal ellipsoids.
' 1
I I:
I
442
MOTION UNDER NO POUCES.
•
EXAMPLES*.
1. A right cone the base of whioli is an ellipso is supported at O the centre of
gravity, and has a motion oommnnicatcd to it about an axis through per])cndicu-
lar to the line joining G, and the extremity li of the axis minor of the base, and in
the piano through B and the axis of the cone. Determine the position of the in-
variable plane.
liegult. The normal to the invariable plane lien in the plane passing through
the axis of the cone and the axis of instantaneous rotation, and mokes uu angle
2. A spheroid has a particle of mass m fastened at each extremity of the axis of
revolution, and the centre of gravity is fixed. If the body be set rotating about any
axis, show that the spheroid will roll on a fixed plane during the motion provided
— = r^fl--jj, where 31 is the mass of the spheroid, a and c are the axes of the
generating ellipse, e being the axis of figure.
8. A lamina of any form rotating with an angular velocity a about an axis
through its centre of gravity perpendiciUar to its plane has an angular velocity
a \/ B^^p impressed upon it about its principal axis of least moment, A, B, C
being arranged in descending order of magnitiide : show that at any time t the
angular velocities about the principal axes are respectively
„o<
and that it will ultimately revolve about the axis of mean moment.
4. A rigid body not acted on by any force is in motion about its centre of
gravity: prove that if the instantaneous axis be at any moment situated in the
plane of contact of either of the right circular cylinders described about the central
ellipsoid, it will be so throughout the motion.
If a, b, c be the semi-axes of the central ellipsoid, arranged in descending order
of magnitude, Cj, e^, e^ the eccentricities of its principal sections, Oj, 0^, R, the
initial component angular velocities of the body about its principal axes, prove that
the condition that the instantaneous axis should be situated in the plane above
- ., - . 0, ahit,
described is -^ = -= — = .
5. A rigid lamina not acted on by any forces has one point fixed about which
it can turn freely. It is started about a line in the plane of the lamina the moment
of inertia about which is Q. Show that the ratio of the greatest to the least angular
velocity is hJa + B : Jb + y, where A,Boxq the principal moments of inertia about
axes in the plane of the lamina.
* These examples are taken from the Examination Papers which have been set
in the University and in the Colleges.
e contro of
peqiemlicn-
ittBu, and iu
of the iu'
[ng through
IS un angle
f tho axis of
g about any
on provided
axes of tho
)nt an axis
lor velocity
!ut, A,B, C
r time ( the
EXAMPLES.
443
;s centre of
ated iu the
the central
nding order
fig, fig the
, prove that
)lane above
6. If tho earth were a rigid body acted on by no force rotating about a diameter
which is not a principal axis, show that tho latitudes of places would vary and that
(ho values would recur whenever J A - li J A - V Ju^dt is a multiple to 2wJli(J.
If a man were to lie down when his latitude is a minimum and to rise when it be-
comes a maximum, show that he would iucrease the vis viva, and so cause the polo of
the earth to travel from the axis of greatest moment of inertia towards that of least
moment of inertia.
7. If do bo the angle between two consecutive positions of the instantaneous
sxf!t, prove that
8. If n be the angular velocity of the plane through the invariable lino and
the instantaneous axis about the invariable line and X tho compouout angular
velocity of the body about the invariable line, prove that
as)'^<»-'("-!)("-i)(-')=»-
0. If a body move in any manner, and all the forces pass through tho contro of
gravity, prove that
T-^2|(loga,,)^^aogc.4jlog«,)=0.
where w,, «g, Wj are the angular velocities about the principal axes at tho centre of
gravity, and w is the resultant angular velocity.
! M
; i
ibout which
the moment
3ast angular
aertia about
ivo been set
1' m
Il <
il
CHAPTER X.
MOTION OF A BODY UNDER ANY FORCES.
557. In this Chapter it is proposed to discuss some cases
of the motion of a rigid body in three dimensions as exo.mples
of the processes explained in Chapter V. The reader will find
it an instructive exercise to attempt their solution by other
method," , for example, the equations of Lagrange might be
applied with advantage in some cases.
i
f
Motion of a Top.
658. A body two of whose principal moments at the centre
of gravity are equal moves about some fixed point in the axis
if unequal moment under the action of gravity. Determine the
motion. See Art. S?-*.
To give distinctness to our ideas we may consider the body
to be a top spinning on a perfectly rough horizontal plane.
Let the axis OZ be vertical. Let the axis of unequal moment
at the centre of gravity be the axis OG and let this be called
the axis of the body. Let h be the distance of the centre of
gravity of the body from the fixed point and let the mass
of the body be taken as uuity. Let OA be that principal axis
at which lies in the plane ZOO, OB the principal axis perpen-
dicular to this plane.
If we take moments about the axis OC we have by Euler's
equations (Art. 230),
C^-{A-B)co,<o, = K
But in our case A = B, and since the centre of gravity lies
in the axis OG, we have N= 0. Hence co^ is constant and equal
to its initial value. Let this be called n.
Let us measure along the axis OC in the direction 00 &
MOTION OF A TOP.
445
length OP = -r . Then, by Art. 92, P is the centre* of oscillation
of the body. This length we shall call I, Let be the inclina-
tion of the axis 00 to the vertical, yfr the angle the plane ZOG
makes with some plane fixed in space passing through OZ. Then
by the same reasoning as in Art. 235 we find that the velocities
of P resolved
perpendicular to plane ZOC=— lay^ = lain6-^
parallel to plane ZOG = la)^ = l
dO
dt
dt
I
.(1).
V TV
""'-••.._
Tl-"'
M
^'^^
P
/
*«, ^\/
^r .
'\ y/N,
B
/ *'
>^*» \
/ ,'
r-s/ \ \
/ ^'
/\/\ • ^
' /
jction OQ ^
It is clear that the moment of the momentum about OZ
will be constant throughout the motion. Since the direction-
cosines of OZ referred to OA, OB, OG are — sin^, and cos^,
this principle gives
-Aw^amd + Gncos0 = E (2),
where E is some constant depending on the initial conditions,
and whose value may be found from this equation by substituting
the initial value of a, and 0.
The equation of Vis Viva gives
A {(o * + (o^') + Cn^=F-2gh cos (3),
where F is some constant, whose value may be found by substi-
tuting in this equation the initial values of w^, w,, and ^ t,
* To avoid confusion in the figure, the body which is represented by a top
is drawn smaller than it should be.
t If we eliminate Wj, Wj from equations (1), (2), (3) we have two equations from
which and ^ ^^7 be found by quadratures. These were first obtained by
Lagrange in his Mccanique Analijtique, and were afterwards given by Poisson in
his Trait4 de Blecanlquc, The former passes them over with but slight notice,
and proceeds to discuss the email oscillations of a body of. any form suHpeudod
under the action of gravity from a fixed point. The latter limits the equations to
;;lf!
t<
!h
. I''
446
MOTION UNDER ANY FORCES.
650,
Let ns measure along the vertical OZ, in the direction opposite to parity
as the positive direction, two lengths 0^/^= T^, 0F=-^ ".-"-'. These lengths
Cn 2gh
we shall write briefly OU=a, and OV=b. Draw through U and V two horizontal
planes, and let the vertical through P intersect these planes in M and N. Then
the equations (2) and (3) give by (1),
■(4).
horizontal velocity) Cn , „„.,
ofP \=-f^tmPUM
(velocity of i')2= 2*; PJV (5).
Thus the resultant velocity of P is that due to the depth of P below the horizontal
plane through V, and the velocity of P resolved perpendicular to the plane ZOP
is proportional to the tangent of the angle PU makes with a horizontal plane.
It ap;"iears from this last result that when P is below the horizontal plane
through U, the plane POT turns round the vertical in the same direction as the
body turns round its axis, i.e. according to the rule in Art. 199, OF and OP are
the positive directions of the axes of rotation. When P passes above the horizontal
piano tiirough U, the plane POV turns round the vertical in the opposite direction.
If P be below both the horizontal planes through O and U these results are still
true, but if a top is viewed from above, the axis will appear to turn round the
vortical in the direction opposite to the rotation of the top. lu all the cases
in which P is below the plane UAf the lowest point of the rim of the top moves
round the vertical in the same direction as the axis of the top.
If we substitute for u^, Wj, E and F in (2) and (3) their values, we easily obtain
P
hi sin" e '/ + Cn cos e = Cn^
at I
(»)•
These equations give in a convenient analytical form the whole motion. We
sec from the last equation, >vliat is indeed obvious otherwise, that b - 1 cos 6 is
always positive. The horizontal plane through V is therefore above the initial
position of P and remains above P throughout the whole motion.
Ex. 1. If w be the resultant angular velocity of the body and v the velocity of P
show that a»*=n' + (y) .
Ex. 2. Show that the cosine of the inclination of the instantaneous axis to the
^. , . £+ (A -C)ncQS0
vertical is ^-^ ,
Au
560. As the axis of the body goes round the vertical its
inclination to the vertical is continually changing. These changes
the case in which the body has an initial angular velocity only about its axis, and
applies them to determine directly the small oscillations of a top (1) when its axis
is nearly vertical, and (2) when its axis makes a nearly constant angle with the
vertical. His results arc necessarily more liinitsd than those given in this
treatise.
to gravity
ese lengths
horizontal
IN. Then
(4).
(5).
3 horizontal
plane ZOP
plane.
ontal plane
ition as the
and OP are
e horizontal
te direction,
ilts are still
a round the
11 the cases
e top movea
isily obtain
(6).
notion. We
- 1 COR is
G the initial
velocity of P
IS axis to the
ertical its
se changes
ita axis, and
when its axis
gle with the
ivcn in this
MOTION OF A TOP.
447
dt
may be found by eliminating -J^ between the equation (6). We
thus obtain
(i^^ 9 /; T m C^i' fa -I cos e\
.(7).
I am 6
It appears from this equation that 6 can never vanish unless
a = l, for in any other case the right-hand side of this equation _
would become infinite. This may be proved otherwise. Since
J is equal to the ratio of the angular momentum about the vertical
to that about the axis of the body, it is clear the axis could not
become vertical unless the ratio is unity.
Suppose the body to be set in motion in any way with its
axis at an inclination i to the vertical. The axis will begin to
approach or to fall away from the vertical according as the initial
value of -77 or a^ is negative or positive. The axis will then
oscillate between two limiting angles given by the equation
= 2ghr (h ~ I cos 0) (1 - cos"*^) - CV (a - I cos fff (8).
This is a cubic equation to determine cos 6. It will be neces-
sary to examine its roots. When cos ^ = — 1 the right-hand side
is negative; when cos ^ = cose", since the initial value of [-Ji) is
essentially positive, the right-hand side is either zero or positive ;
hence the equation has one real root between cos ^ = — 1 and
cos ^=cos i. Again, the right-hand side is negative when cos^= + l
and positive when cos d= oc . Hence there is another real root
between cos 6 = cos i, and cos ^ = 1 , and a third root greater than
unity. This last root is inadmissible.
5C1. These limits may be conveniently expressed geometrically. The equation
(7) may evidently be written in the form
v2 . _.. C-'h"-' /P3I\->
('3"
-^■"--^iZf
Describe a parabola with its vertex at I', its axis vertically downwards and its
Intus rectum equal to —r-., . Ijet the vertical PMN cut this parabola in H, wc then
have
^ff
('")'■
20.MN
1 1
PM "*" PR
.(10).
The point P oscillates between the two positions in which the harmonic mean
of PM and PJi is equal to - 2 . MN, In the figure T is drawn above U, and in
tliis case one of the limits of P is above CM, and the other below the pnrabola. If
wc take U as origin and UO the axis of x, we have PM — r, I'M-y. Let 2^)1 be the
f
•1
1
1; ■
* ■
1
iMf
1
r '!^
448
MOTION UNDER ANY PORCF^.
latas rectum o^ the parabola, and lJV=e, then the axis oi the bcV;
bptw3»?n the two ;;;«nition8 in which P liea on the cubic curve
o-:.CiJi-'M'''f
y«(a!+c)=2pfo« (11)
When c is positive, i. e. when V is above 17, the form of the carve is Lcdica+'j.'
in the figure by the dotted line. The tangents at U cut each other at a finite
angle and the tangent of the angle either makes with the vertical is f — j . When
e is negative the curve has two branches, one on each side of the vertical, with a
conjugate point at the origin. It is clear from what precedes that the upper
branch will lie above, and the lower branch below, the initial position of P,
and that P must always lie between the two branches.
662. In the case of a top, the initial motion is generally given
by a rotation n about the axis. We have initially oa =0, u>^ — 0,
and therefore by (2) and (3) E= Cn cos i, and F— Cii = 2gh cos i.
■■ 2pl, as before, the roots
This gives a = 5 = Z cos i. Putting
^gft'
of equation (8) are cos 6 = cos i, and cos ^ = ja — Vi — 2^ cos i + ^/^
The value co3d=p + '^1 — 2pcosi+p^ is always greater than
unity, for it is clearly decreased by putting unity for coai, and
its value is then not less than unity. The axis of the body will
therefore oscillate between the values of just found.
Since a=b, the horizontal planes through 17 and V coincide, and c— 0. T'jo
cubic curve which determines the limits of OBcillation becomes the parabola ril
and the straight lino UM. The axis of the body will thon oscillato b(;tween the two
positions in which P lies on the horizontal through C and on the parabolrt.
Generally the angular velocity n about the axis of figure is
very great. In this case p is very great, and if we reject the
squares of - we see that cos 6 will vary between the limits cos i
f
and cos i — ,, cjIii' i.
2p
If the initial value of i is zero, we see that the two limits of
cos { are the same. The axis of the body will therefore remain
vertical.
663. Ex. 1. When the limiting angles between which varies are equal to
each other, bo that 6 is constant throughout the motion and equal to a, show that
tan' (p - tan rf> tan a H — ^- tan' o =0,
where <p is the angle PUM.
Ex. 2. A top is set in motion on a emooth horizontal plane with an initial
y esultaut angular velocity about its axis of figure. Show that the path traced out
!»y tlie ajv) on the horizontal plane lies between two circles, one of which it touches
and the other it cuts at rigiit angles. [M. Finck, Nouvcllen A nnalct de Ma theinallques.
Toi:;. )v, 1850. J
iJOHlTltOF.
..(11)
a finito
'. "When
1, with a
he upper
m of P,
y given
gh COS I.
he roots
er than
OS I, and
)ody will
6=0. 'The
jabola riZ
;tn the two
)lft.
figure is
RJect the
aiits CO!" *
limits of
•e remain
ire equal to
show that
th an initial
b traced out
cli it touches
tMmaiiques.
MOTION OF A TOP.
440
fiQ'. A body, two of tvhone principal momenta at the centre of gravity C arr
equal, turns about a fixed point in the axi of unequal moment viidtr the action of
gravity. The axis OG being inclined to the ver"cal at an angle a, and revolving
about it with a uniform angular velocity, find the condition that the motion may be
steady, and the time of a small oscillation.
The equetions (2) and (3) of Art. 558 contain the solution of this problem. But
if we use the equation of Vis Viva in the form (3) we shall have to take into account
the squares of small quantities. It will be found more convenient to replace it by
one of the equations of the second order from which it has been derived. The
simplest method of obtaining this equation is to use Lagrange's P.ule as in Art. 874.
We thus obtain
d^
dt^'
A cos tf sin ^
m
dxp
+ Cn sm 9 -- =gh sin d
dt
(12).
This equation might also have been obtained by differentiating both (2) and (3)
and eliminating
df^'
i^^.
When the motion is steady both 9 and ^y are constants.
Let 9=a, -~=n, then
the equation (2) only determines the constant E and (12) becomes
sin a ( - ^ cos an^ + Cnix -gh)=0 (13) ,
This indicates two possible states of steady motion, one in which o=0 or tt, and
the other in which
Cn i JC'ii^ - ighA cos o
.(14),
2A cos a
a relation which does not necessarily hold when a = or jt.
In the former of these two motions the axis of the body will oscillate about
the vertical and ~ will not be small or nearly constant. It will therefore be
dt
more convenient to discuss the oscillations about this state of steady motion with
other co-ordinates than 6 and tp.
^oliA cos (L
In the latter of these motions, we must have n" not less than - — ~, — . When
a and n are given we can make the body move with either of these two values of /*
by giving the proper initial angular velocities to the body. By eqiiations (1) we see
that the conditions of steady motion are w,= -^sina, Wj=0. When a top is set
in motion by unwinding a string from the axis, the value of n is very great wl 'le
the initial values of Wj and w^ are zero. The steady motion about which thr )p
makes small oscillations will therefore have /* small. Hence the radical in (14) will
gh
have the negative sign. We have therefore very nearly fi=-— .
565. To find the smM oscillation. Let 9=a + 9', and -.
djt
dt
= M + TT . whf 'e ff and
' , are small quantities whose squares are to be neglected. Let a and /* be such
dt
that they contain the whole of tlio constant parts of 9 and -^ , so that 9' and -j-
coutaiu only trigonometrical terms. Then when wo siibstituto these vahies in
('Huations (2) and (12), the constiiut parts must vanish of themselves. The equa-
R. D. 29
■n
^^■^■.
J.gJvvV ^j. .
» <
\ti
r
f '':i
Hi WI l ■■ ■Wiii -•■■< w . -
450
MOTION TNDER ANY FORCES.
tions thuB obtained determine E and n, and show that their values are the same as
those determined when the motion is steady. The variable parts of the two equa-
tions become, after writing for Cn its value obtained from (13),
df
Afi sin a
d'd'
dt
(gh- A fiP cos a.) 0'=O
d<f>'
Afi'— + Bin a (gh- A fj? cos o) '^ + n^A sin» a0'=O\
To solve these, put ^= Psin (pt+f), and ^'= G cos (pt+f).
Substituting, we have
- An am a. pG ={gh-Afi* cos a) F \
{A ftp* - iJ?A sin" o) F= - {gh - Ay? cos o) sin a . <rp |
Multiplying theso equations together, we have
ilV* - '^9^A cos om" + fif'A"
It is evident that p^ is always positive, and there-
and the required time is — .
V
fore both the values of ^l given by (14) correspond to stable motions.
It is to be observed that this investigation does nut appty if a be very small, for
in that case some of the terms rejected are of the same order of magnitude as those
retained. A different mode of investigation is therefore required, this case will be
considered in Art. 569.
» ;
I - i\
V I
Iti
566. We may also determine the steady motion very simply by another process,
which will be found usefid when we come to consider Precession and Nutation. Let
OC be the axis of the body, 01 the instantaneous axis of rotation, OZ the vertical.
Then when the motion is steady, these three must be in one vertical plane which
revolves a'lout OZ with a uniform angular veloci^.y /«. Let w be the angular velocity
about 01, then wcosI(7=n. Let OB be the horizontal axis about which gravity
tends to turn the body, then OB is perpendicular to the plane ZOC.
Since gravity generates an angular velocity - — ^ — dt in the time dt about OB,
therefore by the parallelogram of angular velocities, the instantaneous axis 01 has
moved in the time l: through an angle -— :- — dt in Sk plane perpendicular to the
Au
plane ZOI. Hence the angular velocity of I round Z due to the action of the forces
dfy _ ghBina 1
'inrrz'
IS
dt
Au
Also, sine ^ the angular velocity of the body about OB is zero, the moments of
the centrifugal forces about the axes OA, OC are zero. The moment about OB
is {A - C'j ii -jsin!
A~ C
■ (I'i and this generates an angular velocity — — nu sin IC dt about
OB, IT;nce tho angiuir velocity of I round Z due to the centrifugal forces of the
'■::a,JZ'
body is — ~ = - -
^ dt A
* This expre-sion was given by the Rev. N. M. Ferrers of Gonville and Cains
Coll '<e, as the rei lit of a problem proposed by him for solution in the Mathemati-
cal Tripos, 185y.
MOTION OF A TOP.
Tlio whole angular velocity is the sum of these twr>, i.p.
fgh sin a
4-.1
/gh sin _ t T^ "
u— { ''—, — cot IC+ -
\ An
A -C \ sin 70
A J sin IZ '
ve, and there-
But when the motion is steady OZ, 01 and OC are all in one plane. Now the
angular velocity of C round I is w, and therefore its angular velocity round Z is
But wcos IC=n, hence, tan7C= — ". Substituting this value of
fi=U
smJC
^mZO'
n
gh
tan IC in the value of n, we get ~-Cn-An cos o, the same expression as before.
567. Ex. A top two of whose principal moments at are ennal is set in rota-
tion about its axis of figure viz. OC with an angular velocity «, the point being
fixed. If OC be horizontal, and if the proper initial angular velocity be communi-
cated to the top about the vertical through O, prove that the top will not fall down,
but that the axis of figure will revolve round the vertical, in steady motion, with an
angular velocity fi=~ , where h is the distance of the centre of gravity of the top
from 0, and C is the moment of inertia about the axis of figure. Show also that if
the top be initially placed with OC nearly horizontal and if a very great angular
velocity be communicated to it about OC without any initial angvdar velocity about
OA or OB, then OC will revolve round the vertical remaining very nearly in a hori-
zontal plane with an angular velocity /* given by the same formula as before, and
the time of the vertical oscillations of OC about its mean position will be — -^ .
Cn
568. A body tvhose principal momenta of inertia are not neces-
sarily equal has a point fixed in space and, moves about O under
the action of gravity. It is required to form the general equations
of motion.
Let OA, OB, OG he the principal axes at the fixed point 0,
and let these be taken as axes of reference. Let h, k, I be the
co-ordinates of the centre of gravity G, and let the mass of tiie
body be taken as unity. Let F be drawn vertically upivard^
n'j
^-.-^^
'-' f
lii:'l;J
:yi
452
MOTION UNDER ANY FORCES.
and let p, q, r be the direction-cosines of OF referred to OA,
OB, OC. Then we have by Euler's equations
.(1).
,(2).
A^^^{D-C)<o,c., = -g{hr-lq)
C^s-{A-B)co,a>,=:~g{hq-kp)
Also p, q, r may be regarded as the co-ordinates of a point
in OV, distant unity from 0. This point is fixed in space, and
therefore its velocities as given by Art. 248 are zero. We have
dp
da
dr
^^ = a>,p-^^q
It is obvious that two integrals of these equations are supplied
by the principles of Angular Momentum and Vis Viva. These
give
A(o^p + Bco^q + Cw^r = E,
2g{ph+qk + rT),
where E and F are two arbitrary constants. The first of these
might also have been obtained by multiplying the equations (1)
by p, q, ^ respectively, and (2) by Ato^, Ba>^, Gto^, and adding all six
results. The second might have been obtained by multiplying
the equations (1) by «j, Wj,, c»g respectively, adding and simpli-
fying the right-hand side by (2j.
669. A body whose principal moments of inertia at the centre of gravity G are
not necessarily equal, has a point in one of the principal axes at Q fixed in space
and moves about under the action of gravity. Supposing the body to he performing
small oscillations about the position in which OG is vertical, find the motion.
Referring to the general equations of Art. 568, we see that in this case/i=0,
jfc=0. Since OC remains always nearly vertical, w^ and u^ are small quantities', wo
may therefore reject the product w^w,, in the last of equations (1). This gives Wj
constant. Let this constant value be called lu For the same reason r = 1 nearly
and p, q are both small quantities. Substituting we get the following linear
equations,
Ato^' + B(o^^+C(o^^ = F-
A~'-(B-C)nio, = lgq
J^-t!- {C-A)nu^= -Igp
at
.(3),
dp_
di~'
dq
di
= -pn + ui
.(4).
MOTION OF A TOP.
453
To solve these, assume
wi = ii'8in(\<+/)|
Wj,=Gcos(\t+/) \'
Substituting, we get
A\F-{B-a)nG=glQ)
n\0~(A-C)nF=glp\
.(5).
p = PBin(\t+f))
q = Q COS (\t+f))'
\P = Qn-G
\Q=Pn-F
•{«)•
Eliminating the ratios F : G : P : Q vfo have
X'n'>{A + B-C)^={gl+A\^+ {B-C)n^^ {gl + B\^ + (A~C)n'].
If the values of X thus found should be real, the body will make small oscillations
about the position in which OG is vertical. If C be the greatest moment, and n'
Bufiiciently great to make bothjjfZ- (C - A) n'^ and gl - [C - B) n^ negative, then all
the values of X are real and the body will continue to spin with Off vertical. If G
be beneath 0, I is negative and it will be sufficient that OC should be the axis of
greatest moment.
In order that the values of X' may be real, we must have
{gl(A + B)+n^AC+BC-2AB-C^)}^>M(B-C)n''+gl]{{A-C)n^+gl\ AB,
and in order that the two values o* a^ may have the same sign we must have the
last term of the quadratic positive ;
.'. {(B-C)n'^+gl}{(A-C)n!'+gl]= a positive quantity,
and in order that the values of X" may be both positive, we must have the coefficient
of X^ in the quadratic negative ;
.:gl(A-\-B)<nHB-C){A-C).
In the particular case in which A = B, each side of the quadratic becomes a
perfect square and we have
A\^:t:{2A-C)n\+ (A-C)n' + gl=Oi
2A
.-. X==F
C Jc^n^-Ugl
2A " 2A ■
With the reservations mentioned in Art. 434, the necessary and sufficient
2\/.ZvJ
condition of .stability is in this case n> - ■ ■ . By referring to equations (5) and
(f)) it will be seen that when ^1 = .fi we have F = G and P=Q. If X^, \ be the two
values of X found above, we have
jp = Pi sin (\t +/i) + Pa sin (X,* +/») )
2 = Pi cos (\t +/i) + Pj cos (Xa< +/») J "
Let 9 be the angle OC makes with the vertical, then r^=coa* = 1-0^, ..nd
lienco
0» = ^s + 23^Pj2+P.^2+ 2PiP, cos J(\i - X,) (+/i -/,}.
■I ! I
il
•'1
i J.
4.H
MOTION UNDER ANY FORCES.
I' I
^;i
•\-t
Also if, an in Art. 285, we let <f> be the angle the plane containing OA, OC makcH
with the plane containing OC and the vertical OV, we have j>= -biutfcos^,
and q = sinO sin ^, and henoe
'jCOS^(Vj-/,)
,8in(V+/,)'
Also since $ ia very small we have, following the notation of the Bume Article,
where o is some constant, depending on the position of the arbitrary plane from
which f is measured* .
• In order to understand the relation which exists between these results
and those of Art. 56-5, it will be necessary to determine the oscillations by some
process which holds both when a is large and very small. This may be done as
follows. We have by Vis Viva the equation (see Art. 558)
fdey f E - Cn cone y _ F'- 2qh cos $
\dt) ■'"V ^Biu<# ) ~ A ^^''
where F' has been put for F- C«*. If we put 3= cos d, this takes the form
^'(^O +(^'-tW = ^(i^'-V«'2){l-s'') (2).
Let us assume as the solution of this equation
2 = cos o + P cos (\t -I-/) (3),
where P is so small that on substituting in the above equation we may neglect P'.
Substituting and equating to zero the coefficients of th§ several powers of cos (Xt +/)
we get
A-P-'S? + (E - Cn cos a)^=A(F'- 2fih cos a) (1 - cos' o) ^
-(E -Cn cos a) Cn= -ghA-AF' cos a + '6(/hAcos,'^ a \. (4).
- .4-\a + Cm" =-AF + &ghA cos a )
Now let us change the constant E inio another fi by putting — - — —„- '''=u+yP',
A sin- a
where y is to be so chosen as to remove the term A'^P^X^ in our fiist equation.
Since
d^ _E- Cn coiiJ9^
'di A~^in^e ^''''
we see that, when is not small, /« differs from the constant part of /^ only by
quantities depending on the squares of the small oscillation, and which are
neglected in the text. Substituting for E and eliminating F' between the first
and second equations we get 6rt/*=^ cos an^ + gh.
Eliminating F' between the first and tliird of equations (4) and substituting for n
we get
. . H*A^- 2ffhA cos om" + g^h'
X» == j^^ - .
This process gives the period of the small oscillation in cos 0. When is finite
this is the same as the oscillation in 0, since cos tf =cos o - sin a0'. When is very
0^
small, cos tf=l-- and the time of oscillation in cos ^ is the same as that in 0\
With this imderstanding it will be seen that there is a perfect agreement between
the results of Arts. 565 and 569, when o is put equal to zero.
MOTION OF A TOP.
455
570. A bodij whoa principal mamentt at the centre of gravity are not neceisarily
equal is free to turn about a fixed point O, and it in equilibrium under the action of
gravity. A small disturbance being given, find the oscillations.
Beforriug to the general equations in Art. 668 we see that in this case w,, w,, Wg,
are sinall, honoe in equations (1) wo may omit the terms containing the products
WjWg, 01,0*3, "i'^y -^^^^ since in equilibrium OG is vertical, p, q, r are always
nearly in the ratio h:k:l; hence if 00 = a, we may write-, -, - for », q, r on the
a a a
right-hand sides of equations (2). The six equations are now all linear. To soIto
these we put
Ui =H Bin (\t + /i) and|) = - + Pcos(X« + /i) (8),
o>3, (i>3, q and r being represented by similar expressions with K and L written for
Jf; Q, k and It, I written for P and h. Substituting these in the equations we get
six linear equations. Eliminating P, Q, li we have
.(4).
(-AX' + k*-\-A£r-hkK-lhL=0
■-hkH+(-B\i + P + hA F~lkL=0
- IhH - IkL + (~ C\» + fc» + fc«) £ =0
Eliminating the ratios of 11, K, L we have an equation to find X*. One root is
X'=0, tho others are given by the quadratic
v.(5!±i?.!!i-,*Hi.^2,.,^4^w^„ ,,,
To ascertain if the roots are real we must apply the usual criterion for a quad-
ratic. This requires that
{A [B- C) A« + B [C-A) k^-C(A-B) ?«}«+44B(J3- C) [A-0 h^k^ (6)
should be positive. Since A, B, C can be chosen to be in descending order, we see
that the condition is satisfied. See also Art. 448.
If G is above 0, a is positive and the values of X' are both negative. The equi-
librium is therefore unstable. If G is below 0, a is negative and the values of X'
are both positive. If the roots are equal, the two positive terms in (6) must be
separately zero, this gives k=0 and A(B-C)h'^=C (A- B)l^, i.e. the centre of
gravity lies in the asymptote to the focal hyperbola of the momental ellipsoid. In
this case we find X'= --J. The case in which k=0, 1=0, B--C has been con-
sidered in Art. 664.
If the values of X' are written 0, "K^, \^ we have
«i = Ho + -fffl't + -^1 sin (Xit + /*i) + H^ sin (\t + /i,),
with similar expressions for Wj, o>a. Equations (2) then give p, q, r. But substitut-
ing in (1) we find that all the non-periodic terms which contain ( are zero.
Bemembering that 2>' -h 3' -f r' = 1 we have finally
Wj = n ^ + Hj sin {\t -H /*i) + Jfj sin (Xj< + jUj) ,
ll'I
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aa WIST MAM STRHT
WIUTIR,N.V. t4StO
(7U)*n-4503
^^ ^\. WrS
<6f
450
MOTION UNDER ANY FORCES.
w, and W3 being ropresented by similar expressions with k, K and {, L written for
h, H. The values of K^, L^ and K^, L^ are determined by equations (4) in terms of
H^ and Ht respectively. We also have
P=a'^-^ 'cos(V + Mi)+-^^^ — ?cos(Xjt + /«j,),
with similar expressions for q and r. There remain five constants viz. fi, Hi, //g,
Ml, /*» to be determined by the initial values of w^ w^, W3, r and q.
When the roots are equal the equations depending on p, r, Wj separate from those
depending on q, u^, Wj, forming two sets; we find
Wi=0- + //sin (Xt + /Wi)
>,
a\
^-A'Aco8(Xt + /.,)
A solution of this problem conducted in a totally different manner has been
given by Lagrange in his SIScanique Anahjtique. His results do not altogether
agree with those given here.
If we substitute the values of «i, u.^, ua, p, q, r in the equatiou of angular
momentum of Art. 568 and neglect the squares of small quantities, we evidently
obtain
(Ah^ + Bk^+Cl^) Q=Ea>, AHh + £Ak + CLl=0.
The first of these equations shows that Q vanishes when the initial conditions
are such that the angular momentum about the vertical is zero. In this case the
problem reduces to that considered in Art. 455.
571. A body whose principal moments of inertia are not necessarily equal has a
point fixed in space and moves about O tinder the action of gravity. It is required
to find what cases of steady motion are possible in which one principal axis OC at
describes a right cone round the vertical while the angular velocity of the body about
OC is constant; and to find the small oscillations.
Referring to the general equations of Art. 568, we see that r and w, are given to
be constants. In this case the first two equations of (1) and (2) form a set of linear
equations to find the four quantities p, q, u^, Wj. The solution of these equations
is therefore of the form
Wi=J'o + Fisin{X«+/))
Wa = G'o + G,cos(Xt+/)i '
p = Po+PiSin(Xt+/)|
But these must also satisfy the last of equations (1).
there will be a term on the left side of the form
-^(A-Ji)F^GiBm2(\t+f).
Substituting wo see that
But there will be no such term on the right side. Hence we must have either
A = B, Fi=0 or Gi=0. The motion in the case in which 4 = Z? has already been
considered m Art. 564. Again, substituting in the last of equations (2) and equat-
ing to zero the coefficient of sin 2 (X( +/) wc find
, written for
in terms of
;e from those
ner has been
ot altogether
m of angtdar
we evidently
ial conditions
this case the
'y equal hat a
It is required
axis OC at O
\he body about
f3 are given to
set of linear
lese equations
ig wo see that
It have either
already been
2) and equat-
MOTION OF A SPHERE.
457
Substituting in the first two of equations (1) and equating to zero the coefficients
of cos {\t +/) and sin (\t +/), we find
A\Fi-{B-C)nGi=glQi
- B\Gi -(C-A) nFi = - glP^ ;
from these equations we have F^, G^, P^, Q^ all equal to zero and therefore «i, Wj,,
p, q are all constant as well as the given constants w, and r.
In this case the equations (2) give
p q r '
so that the axis of revolution must be vertical. Let w be the angular velocity about
the vertical. Then u^^pu, w.2=qu, (03= rw. Substituting m equations (1) we get
.(3).
h_A(^_k Bu* I Cu^
P 9 ~Q~ 9 ~r g
Unless, therefore, two of the principal moments are equal, it is necessary for
steady motion that the axis of rotation should be vertical and the centre of gravity
(hkl) must lie in the vertical straight line whose equations are (8).
This straight line may be constructed geometrically in the following manner.
Measure along the vertical a length F= ^ and' draw a plane through V perpeu-
w
dicular to F to touch an ellipsoid confocal with the ellipsoid of gyration. The
centre of gravity must lie on the normal at the point of contact.
To find the small oscillations about the steady motion, i.e. to determine whether
this motion be stable or not, we must put
^ - cos a + Po sin \t + Pi cos \t,
with similar expressions for q, r, Wj , «a, W3. Substituting we shall get twelve linear
equations to determine eleven ratios. Eliminating these we have an equation to
find X. It is sufficient for stabihty that all the roots of tliis equation should be real.
Motion of a Sphere.
572. To detervtine the motion of a spliere on any perfectly rough surface under
the action of any forces whose resultant passes through the centre of the sphere.
Let Q bo the centre of gravity of the body and let the moving axes GC,GA,GH
be respectively a normal to the surface and some two lines at right angles to be
afterwards chosen at our convenience. Let the motions of these axes be de-
termined by the angular velocities 0„ ^„ ^3 about their instantaneous positions
in the manner explained in Art. 243. Let «, v, w be the velocities of G resolved
parallel to the axes so that «>=0, and Wj, w.j, Wj the angular velocities of the body
about these axes. Let F, F' be the resolved parts of the friction of the perfectly
rough surface on the sphere parallel to the axes, GA, GB, and let B be the normal
reaction. Let X, Y, Z be the resolved parts of the impressed forces on the centre
of gravity. Let * be the radius of gyration of the sphere about a diameter, a its
radius, and let its mass be unity. The equations of motion of the sphere arc by
Alts. 264 and 245,
',v '■
wl
458
MOTION UNDER ANY FORCES.
.(1),
du
It
-Bm
=X+i?
= Y+F'
-0^u + 6jV =Z+]t J
and since the point of contact of the sphere and surface is at rest, we have
«-awo=0)
■(2).
«-aWj,=0)
(3).
EUminating F, F, u^, «, from these equations, we get
-etv=
.X+
jfc"
it"
O^auj
du
di
(4).
573. The meaning of these equations may be found as follows. They are the
two equations of motion of the centre of gravity of the sphere, which we should
have obtained if the given surface had been smooth and the centre of gravity had
been acted on by accelerating forces -^ — r. O^au^ and -r— r. 0>flu^ along the axes
GA, GB, and by the same impressed forces as before reduced in the ratio
d^ + k^'
The motion therefore of the centre of gravity in these two cases with the same
initial conditions will be the same. More convenient expressions for these two
additional forces may be found thus. The centre of gravity moves along a surface
formed by producing all the normals to the given surface a constant length equal
to the radius of the sphere. Let us take the axes GA, OB to be tangents to
the lines of curvature of this surface and let p^, p^ be the radii of curvature of the
normal sections through these tangents respectively. Then
Pa
e,=
u
Pi
.(5).
If G be the position of the centre of gravity at the time *, the quantity 0.^dt is
the angle between the projections of two successive positions of GA on the tangent
plane at G. Let Xp Xa he the angles the radii of the curvature of the lines of
curvature at G make with the normal. The centre of the sphere ma; be brought
from to any neighbouring position G' by moving it first from G to H along one
line of curvature and then from H to G' along the other. As the sphere moves
from G to //, the angle turned round by GA is the product of the arc GH into
the resolved curvature of GH in the tangent plane. By Meuuier's theorem, the
curvature is , multiplying this by sin x\ to resolve it into the tangent plane
Pi ''OS Xi
we find that the part of 6a due to the motion along GH is tan Xi> Treating the
Pi
MOTION OF A SPHERE.
459
(1).
kve
.(2).
(3).
■(4).
hey are the
ii we should
gravity had
g the axes
th the same
ir these two
ig a surface
eugth equal
tangents to
ture of the
(5).
ntity 0.^dt is
the tangent
the lines of
be brought
'I along one
here moves
re OH into
leorem, the
ugeut plane
'reating the
arc HG' in the same way, we have
u V
0, = ~ tan Yi+ - tan Xi
(6).
We have also an expression for Wj given by equations (1). Substituting for
Ui, U] from the geometrical equations (3) we get
dt
=uv ( )
\Pa Pi/
.(7).
The solution of the equations may be conducted as foUo'^s. Let (a;, y, z) be tlie
co-ordinates of the centre of the sphere. Then u, v may be found from the
equation to the surface in terms of
dx dy dz
dt' dt' dt
-J by resolving parallel to the axes
of reference. If we eliminate «, v, 0^, 0^, 0^ by means of (4), (6), and (6), we shall
get three equations containing x, y, z, a>g, and their differential coefficients with
respect to t. These together with the equation to ihe surface will be sufficient to
determine the motion at any time. One integral can always be found by the
principle of Vis Viva. Since the sphere is turning about the point of contact as an
instantaneously fixed point we have
where <p is the force function of the impressed forces. This is the same as
"•"■+. ^^"^''^.-iTF*
.(8),
and the right-hand side of this equation is twice the force function of the altered
impressed forces.
574. It will sometimes be more convenient to take the axis GA to be a tangent
to the path. Then v=0 and therefore Wi = 0. If U be the resultant velocity of
the centre of the sphere we have u = U. Also if R be the radius of torsion of a
geodesic touching the path at G and p the radius of curvature of the normal
section at G through a tangent to the path, we have 0i= -p and 0^= -.
R
In these
expressions, as elsewhere, R is estimated positive when the torsion round GA is
from the positive direc ion of GB to the positive direction of GC. If x ^6 the
angle the radius of curvature of the path makes, with the normal, we have as before
tfj=-tanx« The equations (4) become
dU
dt ''
A' +
fc" V
a' + h^R
-p««3
(IV).
■3 J
.(VII).
The expression for wg given by equations (1) now takes the form
"-Tt- ~1i
It may be shown by geometrical considerations that this form is identical with
that given in (7).
575. To find the pressttre on the surface we use the last of equations (2). This
may be written in either of the forms
= - + - = -Z-if.
Pi Pi
,(9).
'M
!. I';
J 'ill,
:■ M
ill
4C0
MOTION UNDER ANY FORCES.
The sphere will leave the surface when R changes sign. This will generally
occur when the velocity of the centre of the sphere is that due to one half of the
projection of the radi. " of curvature of the normal section on the direction of the
resultant force.
576. Ex. 1. Show that the angular velocity of the sphere ahout a normal to
the surface, viz. u,, is constant when the direction of motion of the centre of
gravity is a tangent to a line of curvature, and only then.
Ex. 2. A sphere is projected without initial angiUar velocity ahout the radius
normal to the surface, so that its centre begins to move along a line of curvature.
Show that it will continue to describe that line of curvature if the force transverse
to the line of curvature and tangential to the surface is equal to seven-fifths of the
centrifugal force of the whole mass collected into the centre, resolved in the tangent
plane to the surface.
Ex. 8. If the sphere be homogeneous and be not acted on by any forces, show
that
J/'f tan'x+ s) = constant, 0^3=5 (Jtanxi
^log(tan»x+^)=-|tanx.
Show also that the path will not be a geodesic unless the path is a plane curve.
577. If the given surface on which the sphere rolls be a plane, we have p^ and p,
both infinite, hence tfj, 0^ are both zero. If therefore a homogeneous sphere roll
on a perfectly rough plane under the action of any forces whatever of which the
resultant passes through the centre of the sphere, the motion of the centre of
gravity is the same as if the plane were smooth, and all the forces were reduced in
5
ratio - . And it is also clear that the plane is the only surface which possesses this
property for all initial conditions.
Ex. A homogeneous sphere is placed upon an inclined plane sufficiently rough
to prevent sUdiug and a velocity in any direction is communicated to it. Show
that the path of its centre will be a parabola, and if V be the initial horizontal
velocity of the centre of gravity, a the inclination of the plane to the horizon, the
14 V*
latus rectum will be
5 g sin o*
578. If the given surface on which the sphere rolls be another sphere of radius
6 - ct, we have Pi — p^- h. Hence W3 is constant ; let this constant value be called w,
and let U be the velocity of the centre of gravity. Since every normal section ia
a principal section, let us take GA a tangent to the path. Hence the motion of
the centre of gravity is the same as if the whole mass collected at that point were
/fc' anU
acted on by an accelerating force -5 — v-j —. — in a direction perpendicular to the
a'
path, and all the impressed forces were reduced in the ratio ^ — , j . According to
the usual convention as to tiie relative positions of the axes QA, GB, GC it is
clear that if the positive direction of GA be in the direction of motion, the angular
velocity n should bo estimated positive when the part of the sphere in front is
moving to the right of GA find the additional force when positive will also act
MOTION OF A SPHERE.
4G1
lane curve.
ssesses this
toward the right-hand side of the tangent. Since this additional force acts per-
pendicular to the path, it will not appear in the equation of Vis Viva. Hence the
velocity of the centre of gravity in any position Ib the same as if it had arrived
there simply under the action of the reduced forces. Let be the centre of the
fixed sphere, B the angle OQ makes with the vertical OZ, and \j/ the angle the plane
ZOG makes with any fixed plane passing through OZ. Then by Vis Viva we have
where F is some constant to be determined from the initial conilitions. This also
follows from equation (8).
Also taking moments about OZ, we have
\an-
de
Bm0dt\ dtj a^ + k'^'"'dt*
an equation which will be found to be a transformation of the second of equations
(4). Integrating this equation we have
BinS 0^ = E - -r—r, -J- cos d,
dt af'+k^ b '
where E is some constant. These two equations will suffice to determine — and ~
dt dt
under any given initial conditions.
If the sphere have no initial angular velocity about the normal to the surface it
is clear that n==0 and the additional impressed force is zero. In this special case
the motion of the sphere may be very simply found by treating it as a particle acted
OQ by the reduced impressed forces.
Ex. A homogeneous sphere rolls under the action of gravity in any manner on
a perfectly rough fixed sphere whose centre is 0. Prove that throughout the motion
(1) the velocity of the centre G of the moving sphere is that due to = ths of its depth
below a fixed horizontal plane ; (2) the moving sphere will leave the fixed sphere
when the altitude of its centre above ^ is ^7 ^hs of the altitude of the fixed plane
above the same point ; (3) the horizontal velocity of is proportional to the tangent
of the angle GU makes with the horizon, where ^ is a fixed point on a vertical
through 0.
579. If the surface on wJiich the sphere rolls be a cylinder the lines of curvature
are the generators and the transverse sections. Lot the axis OA be directed paral-
lel to the generators, then pj is infinite and p^-a ia the radius of curvature of the
transverse section. We have 01= — , ft^-O, and since Xa=0' ^3=0- The equations
Pi
(4) and (7) therefore become
du a" „ A"
V
di ~ a^+k^
djau-i) _ Ml)
^t Pa
From these equations the motion may be found.
'id:
, )
a
if,'
462
MOTION UNDER ANY FORCER.
V
!i
Tile Beeond of these gives tbe motion transverse to the generators of the cylinder,
and if Y be the same for all positions of the sphere on the same generator, this
equation may be solved independently of the other two. The transverse motion of
the centre of the sphere is therefore the same under the same initial circumstances
as that of a smooth sphere constrained to blide in a plane perpendicular to the
generators on the transverse section of the cylinder and acted on by the same im-
pressed forces but reduced in the ratio -jTTi '
Having found v we may proceed thus ; let tf> be the angle the normal plane to
the cylinder through a generator and through the centre of the sphere makes with
some fixed plane passing through a generator, then v=/)j ^ .
the first and third equations then become
If -T- be not zero,
at
du P
aw,=
«" Pa
rt= + k^v
d^
X
If X be the same for all positions of the sphere on the same generator these
equations can be solved without diflBculty. For v and p, being known in terms of </>,
we have in this case two linear equations to find w and auy If X be zero, and
i« = =^, wefind
o
au,
^ = ABUx(/s/^i> + B\, U = AA^^(SOB(A^-i> + £\
where A and B are two arbitrary constants to t
mii.ed by the initial values of
u and u,
a-
If X be not the same for all pcsitions of the sphere on the same generator, let (
be the space traversed by the sphere measured along a generator. Then
~dt~ di/>pf'
Substituting this value of u, we have two equations to find ( and au^ in terms
of <p. One integral of these is equation (8) of Art. 573 which was obtained by the
principle of Vis Viva.
Ex. A sphere rolls under the action of gravity on a perfectly rough cylindrical
surface with its axis inclined at an angle a to the horizon. The section of the
cylinder is such that when the sphere rolls on it, the centre describes a cycloid with
its cusps on the same horizontal line. If the sphere start from rest with its centre
at a cusp, find the motion.
Let the position of the sphere be defined by | the space described along a gene-
rator and 8 the arc of the cycloid measured from the vertex. If 46 be the radius of
curvature of the cycloid at its vertex, we have
8=ibcos
v/
5// cos a
"286~
I.
Since v-
difficulty
= — and p,' + s' = 166' we find that — is constant.
dt Pa
This gives without
e cylinder,
irator, this
motion of
umstances
lar to the
e same im-
1 plane to
makes with
9 not zero,
erator these
I terms of 0,
6 zero,
and
tial values of
nerator, let f
m
flWj in terms
tained by the
;h cylindrical
ction of the
cycloid with
ith its centre
along a gene-
the radius of
jives witliont
MOTION OF A SPHERE.
sin a /356o 1 , 1 Ihg cos a i
"»= - -^ V cos« 1 1 - ««« 7 V - -26- ' I •
463
u = Bma
The relation, - = constant, holds whenever (1) the forces acting at the centre of
the sphere, and the form of the section of the cylinder, are so related that the tan-
gential component bears a constant ratio to />, ^ , and (2) the sphere starts from
rest at a point where />, is zero. In such a case, the normal plane to the section
through the centre of the sphere has a constant angular velocity in space and the
resolved motion of the sphere perpendicular to the generators is independent of
that along the generators.
Ex. A sphere rolls on a perfectly rough right circular cylinder who; e radius is
e under the action of no forces, show that the path traced out by the point of con-
tact becomes the curve a;=ii sin . / = - when the cylinder is developed on a plane.
This result shows that the sphere cannot be made to travel continnaUy in one
direction along the length of the cylinder except when the point of contact de-
scribes a generator.
580. If the surface on which the tphere rolls be a cone, the lines of curvature
are the generators and their orthogonal trajectories. Let the axis GA be directed
parallel to the generator, then p^^ is infinite and p, - a is the radius of curvature of
a normal section perpendicular to the generators. Also ^i= — , ^,=0. Let the
Pa
position of the sphere be defined by the distance r of its centre from the vertex of
the cone on which the centre always lies and by an angle (p such that d<t> is the
angle between two consecutive positions of the distance r, d<p being taken as positive
when the centre moves in the positive direction ot GB. If the cone were developed
on a plane it is clear that r and would be the ordinary polar co-ordinates of a
point G. We have
0.
d<t>
"di '
dr
v=r
dt'
The equations (4) and (7) become therefore
HP " *■ V d«y ~ a» + fc" a^ + k> p^ '^'^^ dt
rdt\ dt) a^+fc^
d (fl W3) _ r d(f> dr
dt pj dt dt
If the impressed forces have no component perpendicular to the normal plane
through a generator, r=0, and we have r" ^ = h, where h is some constant depend-
ing on the initial val\ es of r and v.
If also the component X of the forces along a generator be a function of r
only, another integral can be found by the principle of Vis Viva, viz.
1' f.
i\
{■ V
\'l
»;,:•
.1! J y
iJiil
» i'
464 MOTION UNDETl ANY FORCES.
where 7t' is another constant depending on the initial values of u, v and r.
If, further, the cone be a right cone, /),=r tan a wheid a is the semi-angle, and we
have
h cot a , „
where h" is a third constant depending on the initial values of w, and r. The equa-
tions of the motion of the centre of the sphere resemble those of a particle in central
forces. Hence r and ^ will be found as functions of the time if we regard them as
the co-ordinates of a free particle moving in a plane under the action of a central
force represented by
where ta^ has the value just found.
Ex. A sphere rolls on a perfectly rough cone such that the equation to the cone
on which the centre G always lies is —=F(d>). If the centre is acted on bv a force
Pa
tending to the vertex, find the law of force that any given path may be described.
If the equation to the path be -=/(^), prove that the force X is
where w, is given by
— — = J! 3— .
a0 a a<p
581. Let the given rough surface be any surface of revolution placed with
its axis of figure vertical and vertex upxcards, and let gravity he the only
impressed force. In this case the meridians and parallels are the lines of curvature.
Let the axis of figure be the axis of Z. Let 6 be the angle the axis 00 makes
with the axis of Z, f the angle the plane containing Z and GO makes with any
fixed vertical plane.
dij/ dff d\l/
Then e,= -sme /^, 6,=-, <».=cos#-^^.
Hence the equations (4) become
du „d^ o' . . h* , ad}!/ ,.^
dt-'^'^i''=^^Tk^^''''^-a'+T^""''"'^'It W,
dv ^dyp i« d9 ....
and equation (8) becomes
where E is some constant, and p is the radius of curvature of the meridian. Also
we have by (7)
dctfj, uv /I sinflX
-di^-liKp'-r) <'^)'
MOTION OF A SPHERE.
405
where r is the distanco of the '•ontre of the sphere from the axis of t. The
geometrical equations (5) become
do drf/
To solve these, we may put (ii) into tlxc form
.(v).
dv . d\p
which by (v) becomes
dv p cos it'
de + ^r-'^a^+k*''"
■i>
diit'erentiating this, we have by (iv),
d^v p cos rfc
d^' + '^T- d^ + ^''=<^
(vi),
where
de\ r )^l^' + a^\ r'J'
Now p and r may bo found from the equation to the meridian curve as functions
of 0. Hence P is a known function of 0. Solving this linear equation we have v
found as a function of 0. Then by (iv) we have
dw^
10
_ V f p sin \
and thence having found Wj we have u by equation (iii). Knowing u and v ; and
Vfr may be found by equations (v).
582. A heavy xphere rotating about a vertical axia is placed in equilihrium on
the highest point of a surface of any form and being slightly disturbed mahes small
oscillations, find the motion.
Let be the highest point of tlie surface on which the centre of gravity G
always lies. Let the tangents to the lines of curvature at be taken as the axes of
X and y, snd let (x, y, z) be the co-ordinates of Q, We shall assume that is not
a singular point on the surface. In order to simplify the general equations of
motion (4) we shall take as the axes GA and GB the tangents to the lines of
curvature at G. But since G always remains very near 0, the tangents to the
lines of curvature at G will bo nearly parallel to those at 0. So that to the first
order of small quantities we have
e,= ~
Idy
. 1 dx
''* = " rff
dx
dt'
dif
at'
padt 'Pi
and ^3 will be a small quantity of at least the first order. Also since the sphoro
is supposed not to deviate far from the highest point of the surface, we have Wj
constant, let this constant be called n.
I^e equation to the surface on which G moves, in the neighbourhood of
1 fx^ y'^\
the highest point, is «=-g( — + — )• The equation to the normal at x, y,
'^SPi Ps/
i— = i-^ - — - . Hence the resolved parts parallel to the axes of the normal
_ i« _ 3/ - 1
Pi Pi
pressui-e R on the sphere are Jt - , It and R. The equations of motion (4)
Px Pi
Z 13
R. 1).
80
^n
Hiii
m
j
\ 1
4GG
tbcrofuro become
MOTION UNDER ANY FORCES.
rf*x
dfl
dfl
_ 36 _ I* d^ an '
a^ + k* p, o« + it«rft7»,
Pi
i» rfoc an
p, a" + ^'' (it Px
(It).
But 2 is a small quantity of the second order, hence the last equation gives
R=g, To solve these equations, vre put
a;=fcos(\e+/), y = OBin(Xf+/).
These give
a\n
The equation to find X is therefore
g'X'H*
PiPj
This is a quadratic equation to determine X'. In order that the motion may
be oscillatory it is necessary and suillcient that the roots should be both positive.
If pj, p, be both negative, so that the sphere is placed like a ball inside a cup, the
roots of the quadratic are positive for all values of n. If pj, />, have opposite signn
the roots cannot bo both positive. If pj, p^ be both positive the two conditions of
stability will be found to reduce to
, a' + F , /-, /-,,
"*> -fA-OWPi+is/ps)*'
If pi be infinite, it is necessary that p^ should be negative, and in that case
the two values of X' are — rm ^^^ ^®'^' w^ich are both independent of «.
Ct T" ft Pa
If Pi=Pa, we have F=G. In this case ii 0he the inclination of the normal to the
x'^ + tfl
vertical, we have 6"= —~ and, as in Art. 569, we find
P
e'=F,' + F,^ + 2F,F, cos {{\ - Xj) t+/, -/,],
where X^, X, are the roots of the quadratic
i* an. a* g
X«±
''-"X +
aHi" p o^ + Pp
= 0.
This problem may also be solved by Lagrange's method in the ma? ler explained
in Art. 388. Let the axes of reference Ox, Oy, Oz be the same as before. Let GG
be that diameter which is vertical when the sphere is in equilibrium on the summit.
Let GA, GB be two other diameters forming with GC a system of rectangular axes
fixed on the sphere. Let the position of these with reference to the axes fixed in
space be defined by the angular co-ordinates 0, <f>, f in the manner explained in
Art. 235. The Via Viva of the sphere may then be found as in Art. 349, Ex. 1. If
MOTION OF A SPIIEl'.E.
467
(iv).
tion gives
notion may
)th positive,
e a cup, tbe
)poBite BigiiH
pnditions of
in that case
ndent of n.
ormal to the
ler explained
Dre. Let GC
the summit,
angular axes
axes fixed in
explained in
9, Ex. 1. If
yre put sin cos ^ = ^ sin sin ^ = ir, <t> + \p = x, and reject all small quantities above
the second order, we find that the Lagraugiau function ia
L=
l{x'' + y'') + lk^\x'*-x'{iv'-^r,) + i'* + r,'*]-^lg(^%^^y
It is easy to see by reference to the fi<^ure in Art. 23S that { and i) ore the cosines
of the angles the diameter OC meikea with the axes Ox, Oy.
If Ug, uy, u, are the angular velocities of tlio sphere about parallels to the axes
fixed in space, the geometrical equations are
x' ■ 1 ( Wj, - w, - ) =
These are found by making tlio resolved velocities of the point of contact in the
directions of the axes of x and y equal to zero ; see Art. 219. The angular velocities
Ujc, Uy, u, may be expressed in terms of d, 0, ^ by formulns analogous to those in
Art. 235. Bee also the note. Thus
Wj.= -tf'sin }{/ + <f> mi cos ^j
Uy= tf'cos ^ + 0'sin^ sin^> .
w,= ^'costfj-f '
Substituting and expressing the result in terms of the now co-ordinates {, r), x. the
geometrical equations become
^.=-^xVK'-x';,.
A- | + x'|-y-x';; = ol
The equations of motion are given by
dtlLdL djjj dL^
dtdi ~dq'' d<^ '^ ^ dq"
where q stands for any one of tlie five co-ordinates x, y, {, ?j, x- Tlie steady motion
is given by x, y, f, i; all zero and x'""* Taking q =« and q-y and giving the
several co-ordinates their values in the steady motion, we find that \ and /u are botli
zero in the steady motion.
To find the oscillations, we write for q in turn x, y, Xi ^ and ij, and retain the
first powers of the small quantities, Romembpriiig that \ and n are small quanti-
ties (Art. 4C1), we find
X
X
Pi a
A-V'=oJ
These and the two geometrical equations L^ and L^ are all linear, and may be
solved in the manner explained in Art. 432. If we put x'=m and eliminate first \
and M and then | and t) we get two equations to find x and y, which are the same as
those marked (iv) in the first solution.
80—2
i\\
I
i!!
1;
i'm
'I
I-
468
MOTION UNDER ANY FORCES.
I'll
! f'^
: I
■:
Ex. A perfectly rough sphere is placed on a perfectly rongh fixed sphere
near the highest point. The upper sphere has an angular velocity n about the
diameter through the point of contact; prove that its equilibrium will be stable
if n2> _?i^iLJ , where 6 is the radius of the fixed sphere, and a the radius of the
moving sphere.
683. A perfectly rough surface of revolution is placed with its axis vertical.
Determine the circumstances of motion tliat a heavy sphere may roll on it to that its
centre descri-hes a horizontal circle. And this state of steady motion being disturbed,
find the small oscillations.
In this case we must recur to the equations of Art. 581, and let us adopt the
notation of that article, except that to shorten the expressions we shall put for >?
its value z, a\
o
dy(f
To find the steady motion. We must put a, v, Wj, 0, ^ all constant. Let
dt
#,
a, /t and n bo the constant values of 0, -J and W3. Then we have tt=0, v=hfx,
where h is the constant value of r. The equation (i) becomes
The other dynamical equations are satisfied without giving any relation between
the constants. If the motion be steady, we have therefore
5 g ^7b .
n=- — + ji-/iCota;
2 aft. 2 o
thus for the same value of rt we have two values of n, which correspond to different
initial values of v.
We have the geometrical relation au^ = - v, so that u^ and n have opposite
signs. Hence the axis of rotation which necessarily passes through the point of
contact of the sphere and the rough surface makes an angle with the vertical less
than that made by the normal at the point of contact.
By inspecting the expression for n, it will be seen that it is a minimum when
5 g Ihn
a - s ~ cot a,
2 ail 2 a
and '■ lerefore
hg
Ad
n'=35 -^' cot a, ii?=^ i tano
cr lb
To find the small oscillation.
Tut d-a^O', J = fi-^- ^ , where a and ju are supposed to contain all the con-
stant parts of and -j- , so that 6' ani -J- only contain trigonometrical terms. Let
c - a be the radius of curvature of the surface of revolution at the point of contact
of the sphere in stoady motion, so that p differs from c only by small quantities,
and may be put equal to c in the small terms. Also we have »• - 6 + c cos a . 0'.
t :'
i
fixed sphere
n about the
will be stable
radius of the
axis verticaL
I it so that its
>ing disturbed,
ns adopt the
lall put for Ar»
)nstant. Let
3 M=0, V=bf*,
lation between
md to different
have opposite
;h the point of
le vertical less
limum when
lin all the con-
rical terms. Let
point of contact
aall quantities,
c cos a . 0'.
MOTION OF A SPHEIIE.
Now by equations (iv) and (v) of Art. 581 we have
4G0
rfwj _dd d\f^ p Bin 0-r _dff c sin a - &
lU ~ dt dt a ~ dt ^ a
c sin a - 6
W3 = /t
-- — O' + n,
de .
¥''-dr^'
where n is the whole of the constant part of u^.
Again, from equation (ii), we have
adt\ dlj a dt dt o'+
, H dff 5dY ccosoAid^' 2 d9 ^
'•-r''''''l^--a-dt^ ^dr + 7"d*=^'
integrating we have
(2 _ 2nc cos a\ ^ _h d\f/
7" a J ~a~dt'
the constant being put zero because ^ and ^' only contain trigonometrical terms.
Thirdly, from equation (i), we have
Id f de\ r fd^y „ 2 . ^d^L 5 a .
- :i; ( P j7 ) - - 1 j7 ) cos 0+s w«sin 0-J- = ^ - sin tf ;
adtydtj a\dtj 7 * dt 7 a '
cd?e' b + ecoaae' . ^/. „ d\l/\
••• aW^ a («o««-Bma<0(/*'' + 2M.J-)
+|(sma+C03a<»')(M+^)(» + /i^^^^(?') = |f(sma + C03atf').
This expression must be expanded and expressed in the form
Jn this case, smce 0' contains only trigonometrical expressions, we must have ^=0.
Putting ^=0 in the above expression, we find the same value for n as in steady
motion. After expanding the preceding equation we find
A=ii^(- cob' o+I sin'a^ + /t« -I— ( 1
V 7 /cBma\
2oos'o+ssin'o)
c Bin a V 7 /
25o'sino 10 o . 10 o
+^—r. -=- |smocosa+-=-^coso.
In order that the motion may be steady, it *3 sufficient and necessary that this
2t
value of A should be positive. And the time of oscillation is then -;= .
s/A
It is to be observed that this investigation does not apply if a and therefore b be
small, for some terms which have been rejected have b in their denominators, and
may become important.
684. The general equations of the motion of a sphere on an imperfectly rough
surface may be obtained on principles similar to those adopted in Art. 306. The
difference in the theory will be made clear by the following example, in which a
method of proceeding is explained which is generally applicable, whenever the
integrations can be effected.
■
n
'^1
liii
n'-
r -
m
1 II
470
MOTION UNDER ANY FORCES.
it r
f I
1 1
Hi
\
685. .4 homogeneous sphere moves on an imperfectly rough inclined plane with
any initial conditions, jind the direction oj the motion and the velocity of its centre
at any time.
Let O be the centre of gravity of tlie sphere. Let the axes of reference GA, GD,
GC have their dii-cctions fixed in space, the first being directed down the iucUned
plane and the last normal to the plane. Let u, v, w be the velocities of resolved
parallel to these axes, and w^, wa, u^ the angidar velocities of the body about these
axes. Let F, F be the resolved parts of the frictions of the plane on the iphere
parallel to the axes QA, GB, but taken negatively in those directions. Let k be tlio
radius of gyration of the sphere about a diameter, a its radius, and let the mass be
unity.
Let a be the inclination of the plane to the horizon. The equations of motion
will then be
h^'^-^=-F'a]
at
k-%*=Fa
at
.(1).
du
di
dv
di
= -F+gs'ma]
= -F
.(2).
Eliminating F and F from these equations and intcgrafcuig wo have
U+-5 awa = l7o+5' sin at
.(3).
where Uq and Vq are two constants determined by the initial values of «, v, w^, Wj.
The meaning of these equations may be found as follows. Let P be the point
of contact of the sphere and plane, let Q be a point within the sphere on the normal
at P so that PQ= , so that Q is the centre of oscillation of the sphere when
suspended from P. It is clear that the left-hand sides of the equations (3) express
the components of the velocity of ^ parallel to the axes. The equations assert that
the frictional impulses at P cannot affect the motion of Q, and this readily follows
from Art. 119, because Q is in the axis of spontaneous rotation for a blow at P.
586. The friction at the point of contact P always acts opposite to the direction
of sliding and tends to reduce this point to rest. When sliding ceases the friction
(see Art. 148) also ceases to be Umiting friction and becomes only of sufficient mag-
nitude to keep the point of contact at rest. If sliding ever does cease, we then have
u - awj — 0, v + awi = (4).
The equations (3) and (4) suffice to determine these final values of u, v, w, and
W.J. Thus tho direction of the motion and the velocity of the centre of gravity after
sliding has ceased have been found in terms of the time. It appears that both these
elements are independent of the friction.
If the equations (4) hold initially the sphere will begin to move \fithout sliding
if the friction found from the equations (1), (2) and (4) is less than tho limiting
friction. As in Art. 147, this requires that the coefficient of friction /«> -^ — r;, tan a.
Supposing thia inequality to hold, the friction called into play will be always loss
than the limiting friction and therefore equations (3) and (4) give the whole motion.
MOTION OF A SPHERE.
471
plane with
f its centre
e GA, GB,
lie inclined
resolved
ibout these
the Lphere
et k be tlio
he mass be
; of motion
.(2).
.(3).
t, V, Wi, Wj.
le the point
X the normal
iphere vrhen
(3) express
assert that
idily follows
)w at P.
ilie direction
the friction
fficient mag-
re then have
(4).
u, V, Ui and
gravity after
vt both these
hont sliding
the limiting
always loss
hole motion.
587. If the equations (4) do not hold initially or if the ineqtiality jnst men-
tioned is not satisfied, let S be the velocity of sliding and let be the angle the
direction of sliding makes with OA. To fix the signs we shall take S to be positive
while $ may have any value from - b- to jr. Then
S cos d=u-au„ 3 Bin d=v + auy
.(5).
The friction is eqnal to fig cos a and acts in the direction opposite to sliding,
hence
F=ngeoBaco39, F" = fig eoa a Bin 9.
The equations (1), (2) and (5) therefore give
d (S cos
, — =-( l + pj/tr^cosacostf+flrsmo
d(5sine) /, a«\ . .
.(6).
Expanding we find
dt v^ky
fig cos a+9 sin a cos
,de
.(7).
S -TT = - fl sin a sin
at
If be not constant, we may eliminate t and integrate with regard to 9, this
gives
Ssin9=2A (tan^V (8),
where n = f 1 + p j ju cot a, and A is the constant of integration. If Sq and ^o ^^ the
initial values of S and determined by equations (5), we have
2A=SaBin0,
(cot|
.(9).
Substituting the value of S given by (8) in the second of equations (7) and inte-
grating we find
«-l n+l n-1 n+1
A
t.
.(10),
the constant of integration being determined from the condition that tf = ^oWhen
t=0. The equations (8), (9) and (10) give S and in terms of t. The equations
(3) and (5) then give u,v,u^ and Wj in terms of t.
d9
The second of equations (7) shows that -:- has an opposite sign to 0, hence be-
ginning at any initial value except ijr continually approaches zero. It follows
that, unless a is zero, will be constant only when 0q=O as ± tt.
If n > 1, i.e. fi > — — Tj tan a, we see from (8) that sliding will cease when
Cb -J' fC
vanishes. Tliis, by (10) will occur when
g sin a ^n - 1 Ji + 1 '
{: :
«1
it it
ill
m I
t .
TiiO subsequent, motion has already been found.
:>).
! I
472
MOTION UNDER ANY FORCES.
If n < 1 we see by (8) that S increases as $ decreases, so that sli'ling will never
cease. It also follows from (10) that vanishes only at the end of an infinite time.
li Sq=0, sliding will never begin if n > 1, but will immediately begin and never
cease if n < 1.
588. The theory of the motion of a sphere on an imperfectly rough horizontal
plane is so much simpler than when the plane is inclined or when the sphere rolls
on any other surface, that it seems unnecessary to consider this case in detail. At
the same time the game of biUiards supplies many problems wliich it would be
unsatisfactory to pass over in silence. The following examples have been arranged
eo as both to indicate the mode of proof to be adopted and to supply some results
which may be submitted to experiment.
The result given in Ex. 1, was first obtained by J. A. Euler the son of the cele-
brated Euler, and published in theJlf^ni. de. I'Acad. de Berlin, 1758. Most, possibly
all, of the other results may be found in the Jeu de Billard par 0. Coriolis, pub-
lished at Paris in 1835.
Ex. 1. A billiard-ball is set in motion on an imperfectly rough horizontal
plane, show that the direction and magnitude of the friction are constant through-
out the motion. The path of the centre of gravity is therefore an arc of a parabola
while sliding continues, and finally a straight line. The parabola is described with
the given initial motion of the centre of gravity under an acceleration equal to fig
tending in a diiectiou opposite to the initial direction of sliding.
Ex 2. If Sq be the initial velocity of sliding prove that the parabolic path lasts
1
5
If the initial velocity of sliding be one foot per second, the parabolic path lasts
therefore less than a twentieth part of a second.
Ex. 8. If P be the point of contact in any position and Q the centre of oscilla-
tion with regard to P, prove that the velocity of Q is always the same in direction
and magnitude. Thence show that the final rectilinear path of the centre of gravity
is parallel to the initial direction of the motion of Q and the final velocity of the
centre of gravity is = of the initial velocity of Q. If PP be the initial direction of
motion and V the initial velocity of the c autre of gravity and 1 1L3 time given by
Ex. 2, prove that the final rectilinear path of the centre of gravity intersects PP" in
a point P' so that PP=l Vt.
Ex. 4. A billiard-ball, at rest on an imperfectly rough horizontal table, is straok
by a cue in a horizontal direction at any point whose altitude above the table is h,
and the cue is withdrawn as soon as it has delivered its blow. Supposing the cue
to be sufficiently rough to prevent sliding, show that the centre of the boll will
move in the direction of the blow and that its velocity will become nniform and
equal to = - 5 after a time — ^ where B is the ratio of the blow to the mass
'■la 7a fig
2 S
for a time = — . From some experiments of Coriolis it appears that fi=^ nearly.
of the sphere and a is the radius.
In order that there should be no sliding the distance of the cue from the centre
of the ball must be loss than a sin e where tan c is the coefficient of friction between
the cue and ball. ^
; will never
finite time.
1 and never
1 horizontal
pbere rolls
detail. At
t would be
n arranged
)ine results
of the cele-
i?t, possibly
>riolis, pub-
horizontal
at through-
' a parabola
cribed with
equal to ng
path lasts
=■= nearly.
o
path lasts
'6 of oscilla-
in direction
re of gravity
}city of the
direction of
me given by
sects PP" in
ble,isstniok
e table is h,
ing the cue
he boll will
oniform and
to the mass
n the centre
tiou between
MOTION OF A SOLID BODY ON A PLANE.
473
Ex. 5. A billiard-ball, initially at rest and touching the table at a point P, is
etioick by a cue making an angle /3 with the horizon. Show that the final recti-
linear motion of the centre of gravity is parallel to the straight line PS joining P
to the point S where the direction of the blow meets the table, and the final velocity
5 PS
of the centre of gravity is - — ^ sin /3 in the direction of the projection of the blow
on the horizon.
It will be noticed that these results are independent of the friction.
Ex. 6.
Measure Sr=^ccot/3 along the projection of the blow on the horizon-
tal table, then TS measures the horizontal component of the blow referred to a
unit of mass, on the same scale that PS measures the final velocity of the centre of
gravity. Prove that during the impact and the whole of the subsequent motion the
friction acts along PT and that the whole friction called into play will be measured
by PT on the scale just mentioned.
5 PT
Thence show that unless /* < = — the paraboho
arc of the path will be suppressed. Show also that PT is the direction in which
the lowest point of the ball would begin to move if the horizontal plane were smooth
and the ball were acted on by the same blow as before.
Motion of a Solid Body on a plane.
589. A solid of revolution rolls on a perfectly rough horizontal plane under the
action of gravity. To find the steady motion and the small oscillations.
Let be the centre of gravity of the body, OC the axis of figure, P the point of
contact. Let OA be that principal axis which lies in the plane POC and GB the
axis at right angles to GA,GC. Let GM bs a perpendicular from G on the hori-
Hi
;1
r. {
i i
I;
■I
474 MOTION UNDER ANY FORCES.
zontal plane, and PN a perpendicular from P on GC. Let be the angle QCtaskdi
with the vertical, and ^ the angle MP makes with any fixed line in the horizontal
plane. Let R be the normal reaction at P; F, F' the resolved parts of the frictions
respectively in and perpendicular to the plane PQC. Let the mass of the body be
unity.
Let us take moments about the moving axes QA, OB, QC according to Art. 253.
As in the second case of Art. 254, we put B^^w^, &t=w, and &a= -^cos 9. Bemem-
bering that h^=Awi, h^=Au^, h^=Cu^ we have
A^-AwJ^^GOBe+Cu,i»t=-F.GN (1).
A^- CwgWi + ^Wj^cos tf= -F. OM-R. MP (2).
Q/% at
C^^F.PN (3).
The geometrical equations are
dt="« (^)- ""<'dF=-"i <«)•
Let u and v be the velocities of the centre of gravity respectively along and per<
pendicular to MP, both being parallel to the horizontal plane. The accelerations
of the centre of gravity along these moving axes will be
Tt-''dt=^ <^)'
Tt^'^Tt-^ ^^^'
And if 2 be the altitude of above the horizontal plane, we have
5^=-^+^ («)•
Also since the point P is at rest, we have
u-GMu,=0 (9),
v + PNus-GN'-^i=0 (10),
2=-(?iVcos<;+PiVsintf (11).
These are the general equations of motion of a solid of revolution moving on a
perfectly rough horizontal plane. If the plane is not perfectly rough the first eight
equations will still hold, but the remaining three must be modified in the manner
explained in the next proposition.
When the motion is steady, we have the surface of revolution rolling on the
plane so that its axis makes a constant angle with the vertical. In this btate of
motion, let tf=o, ^=/*, W8=n, GM=p, MP=q, GN=^, NP=ri, and let /5 be the
radius of curvature of the rolling body at P. Then the relations between these
quantities may be found by substitution in the above equations.
Suppose it were required to find the conditions that the surface may roll with a
given angular velocity n with its axis of figure making a given angle with the verti-
cal. Here n and a arc given, and p, q, f, ij, p may be found from the equations to
MOTION OF A SOLID BODY ON A PLANE.
475
(1).
(2).
(3).
(6).
1 pel
itioi
..(6),
(7).
..(8).
..(9),
(10),
(11).
the snrface. We have to find /x, Wj , w^ , u, v and the radius of the circle deacribcd
by in space. Then eliminating F, F', li, we get
/tt' sin a {A cob a-p^)- u/jl (C sin a + pri)-gq= 0,
Wi=-^sino, (i>2=0,
M=0, « = - rnj - f/t sin o.
Let r be the radius of the circle described by as the surface rolls on the plane.
Since Q describes its circle with angular velocity /jl, we have r/i=v, and hence
r= - — -f smo.
/«
Eliminating n we have
H^ {^ i; sin a cos a + C| sin' a + r (C sin a +pri)] = ffiV-
For every value of n and a there are two values of /*, which however correspond
to different initial conditions. In order that a steady motion may be possible, it
is necessary that the roots of this quadratic should be real. This gives
(C7 sin a +priy «' + ^02 sin o {A. cos a-p^)=a, positive quantity.
If the angular velocity n be very great, one of these values of fi is very great
and the other small. If the angular velocity be communicated to the body by
unwinding a string, as in a top, the initial value of Wx will be small. In this case
the body will assume the smaller value of (j^, and we have approximately
u= Ql .
•^ n(CBma+pr})
To find the small oscillation, we put $ = a-\-d', 37 = /*+ -^t wj-n + Wg'. Then
we have by geometry,
t=0M=p + q9', PM-=q + (^-p)e',
GN=^+p9'('ma, PN='n+p8^eoBa,
and substituting in (6), ['J), '\0), (6), (7) respectively, we find
Ui=i - fi sin a - n cos a^ - sin a
dff
d£
dt '
d^|/
V = - /i sin o{ - nij - (/* cos of + up sin' a + np cos o) ^ - sin a^ —~ i;wa'.
dt
d^B' „ . d-i/ d^'
+ /i (^ cos of + up sin' o + np cos o) 6' + i;ai«8',
dd' . d'-J/ do},'
F'x: - (/i cos a^-p/x + ixp sin' a+npMBa)-r - sin of -j.^- - 17 -j- .
Substituting these in equation (3) and integrating, we have
d\f/'
(C + 1;') w'j = (PM - Mf cos a -up sin' a - up cos o) r]9' - ij sin of -;- .
• (A),
the constant being omitted because «, a and /* are supposed to contain all the
,d^
i
' n
pit'
■ 1
■';!
m
constaut parts of w^, 9, and
df
i
476 MOTION UNDER ANY FORCES.
Again substituting in (1) and integrating, we have
{Cn - 2Afi cos o + { (pn - /« cos of - /* sin'o/) - np cos a)}0'-{A+ ^) sin a J =ifiu^'(ii).
Also substituting in (2), wo have
{A+p''+q*)-j-^+ff{Afi'(sia^a-coB,^a) + CnfiCOBa+{p-p)g^
+ /t'sina fj + nuriq + /jflcoa o^ + nftfrp cos o + m' sin' "■fP )
+ --^{ -?J^sino coso + C»sina + 2^^sina+npi>} '~ " '*
tic
+ u^'{Cfi sin a+fipri)
+ i - il sin a COB o/t* + Cn/t sin a 4 gq + sin aft'^p^ + nupr} }
d\f/
The last term 6f this equation must vanish since 0', -~ , ug'only contain periodic
terms. It is the equation thus formed which determines the steady motion and
gives us the value of /x.
To solve these equations we may put
e'=LBia{\t+f), ^ =MBm{\t+f), ua'=NBin{\t+f).
If we substitute these in (A), (B), (C) we shall get three equations to eliminate
the ratios L:M:N. Before substitution it will be found convenient to simplify
the equations first by multiplying (A) by f and (B) by 17 and subtracting the latter
result from the former, and secondly by multiplying (A) by — and adding the re-
suit to (C). We then obtain the following determinant,
-(A+p^ + q')\H{p-p)9
+ ft? {p^- A COB 2a -qr)
+ n/iC cos a
Afi sin a COB a
n
Cn{ri Bi a-p)
Cn - 2A/1 cos a
^ sina
Cf
{p-^eoBa-p sin^a) /j,
- pn cos a
{sin a
-(C+v')
=0.
590. Ex, 1. To find the least angular velocity which will make a hoop roll
in a straight line.
In this case r is infinite and therefore /x must be zero. It follows from the
equation of steady motion that 9=0, or the hoop must be upright. We have
p=a, 2=0, {=0, 71= a, n=0, aaH C=2A. The determinant becomes
^,_ 2tfi{2A + a'')-ag
^ aTTT* '
so that the least angular velocity which will make X a real quantity is given by
2((7+a«)'
=fW(B)-
.=0...(C).
bin periodic
notion and
) eliminate
to simplify
g the latter
ling the re<
=0.
I a hoop roll
wa from the
. We have
iven by
MOTION OF A SOLin BODY ON A PLANE.
477
Let the hoop ba an arc, we have C=a\ and if 7 be the least velocity of the
Let the hoop be e. disc, then
centre of gravity, this equation gives V> g >/"//•
C= g , and we have V>
/as
V ¥•
Ex. 2. A circular disc is placed with its rim resting on a perfectly rough
horizontal table and is tpun with an angular velocity Q about the diameter through
the point of contact. Prove that in steady motion the centre is at rest at an
altitude - above the horizontal plane, where I is the radius of gyration about a
diameter ; and, if a be the inclination of the plane to the horizon, the point of
If the disc bo slightly
2ir
contact has made a complete circuit in the tiiie ^ sin a.
disturbed from this state of steady motion, show that tlie time of a small oscillation
i«a| •
{ga 3*''co8"''a + tt*sin«
Ex. 3. An infinitely thin circular dit . moves on a perfectly rough horizontal
plane in such a manner as to preserve a constant inclination a to the horizon.
Find the condition that the motion may be steady and the time of a small oscillation.
Let the radius of the disc be a, and the radius of gyration about a diameter k.
Let Wj be the angular velocity about the axis, /i the angular velocity of the centre
of gravity about the centre of the circle described by it, r the radius of this circle,
then in steady motion
(2i» + o') «a = iV cos o - — cot a, (2i' + d')r = - Ic'a cos a + ^ cot o.
/* fit
If 2* be the time of a small oscillation
^^ y(*« + a») = m' { fc''( 1 + 2 cos^a) + a" sin'a } - n/* cosa(6A'' + o") + 2n«{2P + o') - ^a sino.
Ex. 4. A heavy body is attached to the plane face of a hemisphere so as to form
a solid of revolution, the radius of the hemisphere being a and the distance of the
centre of gravity of the whole body from the centre of the hemisphere being h. The
body is placed with its spherical surface resting on a horizontal plane, and is set
in motion in any manner. Show that one integral of the equations of motion is
il sin*^ --^+C7w3 [ cos<' + -] =constant whether the plane be smooth, imperfectly
rough, or perfectly rough.
It is clear that the first two terms on the left-hand side of this equation is the
angular momentum about the vertical through 0. Let this bo called /. Since wo
may take moments about any axis through G as if (? were fixed in space, we have
dl
dt
=F'.P3f. But PM= -PN.-, hence eliminating F' by equation (3) and in-
tegrating, we get the required result.
Ex. 6. A surface oi revolution rolls on another perfectly rough surface of
revolution with its axis vertical. The centre of gravity of the rolliug surface lies
in its axis. Find the cases of steady motion in which it is possible for the axes of
both the surfaces to lie in a vertical plane throughout the motion.
Ltt tf be the inclination of the axes of the two surfaces, P the point of
contact, GM a perpendicular on the tangent plane at P, PN a perpen-
dicular on the axis GC of the rolling body; F the friction, E the reaction at P;
[X
;
h
■ ,
i
..i
I
^
i
u
t. ;
^
'
;:l
..; x^
■!l
ii
478
MOTION UNDER ANY FORCES.
n the anfi^lar velocity of the rolling body about itfl axis GC, ix the angiilar rate at
which Q deBcribea its oiroulor path in space, r the radius of this circle. Then in
steady motion
Mn sin e(Cn- An COB e)=- F .QM-R. MP,
R= ~ Mr/*' sin a + Mg cos a,
F= -Mrft!' cos a -Mgeia a,
n.PN+fABinO. QN= ~rn,
where M is the mass of the body.
691. A surface of any form rollt on a fixed horizontal p^ane under the action of
gravity. To form the equations of motion*.
* The motion of a heavy body of any form on a horizontal plane seems to have
been studied first by Foisson. The body is supposed to be either bounded by a
continuous surface which touches the plane in a single point or to be terminated
by an apex as in a top, whUe the plane is regarded as perfectly smooth. Poisson
nses Eoler's equations to find the rotations about the principal axes, and refers
these axes to others fixed in space by means of the formula; of Art. 235. He finds
one integral by the principal of vis viva and another by that of angular momentum
about the vertical straight line through the centre of gravity. These equations are
then applied to find how the motion of a vertical top is disturbed by a slow move-
ment of the smooth plane on which it rests. See the Trait6 de MScanique.
In three papers in the fifth and eighth volumes of Crelle'a Journal (1830 and
1832) M. Coumot repeated Poisson's equations, and expressed the corresponding
geometrical conditions when the body rests on more than one point or rolls on a*>
edge such as the base of a cylinder. He aJso considers the two cases in which the
plane is (1) perfectly rough, and (2) imperfectly rough. He proceeds on the same
general plan as Poisson, having two sets of rectangiilar axes, one fixed in the body
and the other in space connected together by the formulas usually given for
transformation of co-ordinates. As may be supposed, the equations obtained are
extremely complicated. M. Goiunot also forms the corresponding equations for
impulsive forces. Those however which include the effects of friction do not
agree with the equations given in this treatise. •
In the thirteenth and seventeenth volumes of Liouviiys Journal (1848 and
1852) there will be found two papers by M. Piiiseux. In the first he repeats
Poisson's equations and applies them to the case of a solid of revolution on a
smooth plane. He shows that whatever angle the axis initially makes with the
vertical, this angle will remain very nearly constant if a sufficiently great angular
velocity be communicated to the body about the axis. An inferior limit to this
angular velocity is foimd only in the case in which the axis is vertical. In the
second memoir he applies Poisson's equations to determine the conditions of
stability of a solid of any form placed on a smooth plane with a principal axis at
its centre of gravity vertical and rotating about that axis. He also determines
the small oscillations of a body resting on a smooth plane about a position of
equilibrium.
In the fourth volume of the Quarterly Journal of Mathematics, 1861, Mr G. M.
Slesser forms the equations of motion of a body on a perfectly rough horizontal
plane and applies them to the problem considered at the end of Art. 597. He uses
moving axes, and his analysis is almost exactly the same as that which the author
had adopted.
MOTION OF A SOLID BODY ON A PLANE.
479
dar rate at
, Then in
he action of
sms to have
anded by a
terminated
h. Poisson
, and refers
;. He finds
laomentum
^nations are
, slow move-
al (1830 and
^responding
rolls on a*^
in which the
on the same
I in the body
ly given for
obtained are
quations for
etion do not
il (1848 and
it he repeats
olation on a
akes with the
great angular
r limit to this
tical. In the
conditions of
icipal axis at
so determines
a position of
361, Mr G. M.
gh horizontal
597. Ho uses
Lch the author
Let GA, OB, OC, the principal axes at the centre of gravity, bo the axes of
reference and lot the mass be unity. Lot ^ ((, 17, ^) = be the equation to the
bounding surface, (f, 1;, f) the co-ordinates of the point P of contact. Let (p, 7, r)
be the direction-cosines of the outward direction of the normal to the surface at
the point f , tj, f, tlien
a^ dift d<ft'
«/( dv d}
Firstly, let the plane be perfectly rough. Let X, Y, Z be the resolved j-arts
along the axes of the normal reaction and the two frictions at the point {, 17, f, and
let the mass of the body bo unity. By Euler's equations we have
C-^^''-{A-B)w,u,,=^Y-vX
Also the equations of motion of the centre of gravity are by Art. 24' i
du
dv
^^-wu}i + uu^=gq+Y
dw
Also since the line {p, q, »•) remains always vertical,
dp ,
dq
dr
and since the point (f, 1;, f) is at rest we have
U=u-'
(1).
.(2).
(3),
U=u - 170)3 -ffw2=0)
ir=w-fw„ + Ma>, =0)
(4).
where U, V, IF are the resolved parts of the velocity of the point of contact P in
the positive directions of the axes.
:»■•
1^
I
f) I
III I
il
\V
4S0
MOTION UNDER ANY FORCES.
592. SfcomVy, let the plane be perfectly smooth. The oqnations (1), (2), (.1>,
apply equnliy to thin cnne, but cqnationH (1) are not true. Since the rcaultaut of
X, Y, Z ia & reaction 72 normal to the fixed plane, wo have
X=-pR, Y=t-qR, Z=-rR (5).
The negative sign is prefixed to R bccanso (p, q, r) are the directlon-coHines of
the outward direction of the normal, and it ia clear that when these are taken posi'
tively, the components of R are all negative. If at any moment R vanishes and
changes sign the body will leave the plane.
Siuce the velocity of parallel to the fixed plane is constant in direction and
magnitude, it will usually be more convenient to replace the equations (2) by the
following single equation. Let OM be the perpendicular on the fixed plane and lot
MQ=z, then
%--o^^ («)•
It is necessary that the velocity of the point of contact resolved normal to the
plane should be zero, this condition may bo written in either of the equivalent
forms
UpJt-rq^-Wr^Q
dz
di
+ (v-ia - fw*) P + (f w, - f Wa) 1 + (I", - i^Wj) r =
r=o|
(7).
693. Thirdly, let the body slide on an imperfectly rough plane. The equa-
tions (1), (2), (3) and (7) hold as before. If /t be the coefficient of friction tho
resultant of the forcos X, Y, Z must make an angle tan~' n with the normal at tho
point of contact, hence
{Xp+Yq + Zr)^_ 1_
~1 + M»
.(8).
X-'+Y-'+Z'
Also since the resultawt of {X, Y, Z), the normal at P and tho direction of slid-
ing must lie in one plane, we have the determiuantal equation
X(qW-rV)+Y(rU-pW) + Z{pV-qU)=0 (9).
Siuce the friction must act opposite to the direction of sliding, we must have
XU+ YV+ZW negative. When this vanishes and changes sign, the point of con-
tact ceases to slide.
If the body start from rest we must use the method explained in Art. 146 to
determine whether the point of contact will begin to slide or not. Assume X, Y, Z
to be tho forces necessary to prevent sliding. Then since m, v, w, Wj , Wj , Wg are all
initially zero, we have by differentiating (4) and eliminating the differential coeffi-
cients of u, V, w, Wp Wjj, Wj three linear equations to find X, Y, Z, in terms of the
known initial values of {p, q, r) and {^, v, f). The point of contact will slide or not
according as these values make the left-hand side of equation (8) less or greater
than tho right-hand side.
The equations to find X, Y, Z may be obtained by treating the forces as if they
were indefinitely small impulses. In the time dt, we may regard the body as acted
on by an impulse r/dt at and a blow whose components are Xdt, Ydt, Zdt at P.
By Art. 296 we may consider these in succession. The effect of tho fir^t is to com-
municate to P a velocity gdt in a direction normal to the fixed plane and outwards.
If P does not slide, the effect of tho blow at P must be to destroy this velocity.
Hence X, Y, Z may be found from the equations of Art. 304 if we write Ui=pff,
MOTION OF A SOLID BODY ON A PLANE.
481
:i), (2), (3^.
eaultaut of
(5).
ii-coflines of
taken posi.
auiobes and
irection and
18 (2) by the
lano and lot
(6).
lormnl to the
10 equivalent
(7).
The cqua-
f friction the
normal at tbo
(8).
rection of slid-
(0).
we must have
e point of con-
i in Art. 146 to
tssnme X, 1', 2
,, Wa, wgare all
fferential coeffi-
n terms of the
will slide or not
) less or greater
forces as if they
le body as acted
, YiU, Zdt at P.
3 first is to cora-
and outwards.
)y this velocity,
ye write Mi=iV»
Vi'^qy, u>i = rtf and u„ r,, tr, all equal to zero on the left-hand sides and (to suit the
notation of this article) change p, q, r on the right-hand sides into {, n, f.
Oeomotrically the point of contact will not slide if the diametral line of the fixed
plane with regard to the ellipsoid called E in Art. S04 makes a less angle with the
normal than tan~i ft.
In any of these cases when p, q, r have been found, the inclinations of the prin-
cipal axes to the vertical are known. Their motion round the vortical may thou be
deduced by the rule given in Art. 249, When u, r, w and the motions of the axes
have been found, the velocity of the centre of gravity resolved along any straight
line fixed in space may be found by resolution.
594. Some integrals of these equations are supplied by the principles of angular
momentum and Vis Viva. If the plane is perfectly smooth we have
A Wjp -t- Bu^q + Cuar = a,
Au>,*+ Bw,« -t- Cwa' + (f|)* = /9 - 2^7^.
where a and /9 are two constants.
If the plane is perfectly rough we have
Aui' + Bu.j^ + Cua^ + u'' + v^ + w*=p-2gz.
595. Ex. 1. A body rests with a plane face on an imperfectly rough horizon*
tal plane whose coefficient of friction is /x. The centre of gravity of the body is
vertically over the centre of gravity of the face, and the form of the face is such
that the radius of gyration of the face about any straight lino in its plane thropsh
its centre of gravity is y. The body is now projected along the plane so that the
initial velocity of its centre of gravity is Vq and the initial rotation about a vertical
axis through its centre of gravity is Wo. If Uo be very small, prove that the centre
of gravity moves in a straight line and its velocity at the end of any time t is v^ - fxgt.
Show also that the anguloi: velocity at the same time is w,, [ 1 - — 1 , where Jc
the radius of gyration of the body about a vertical through the centre of gravity.
[Poisson, Tmiti de Mecanique,]
Ex. 2. A body of any form rests with a plane face in contact with a smooth
fixed plane so that the perpendicular from the centre of gravity on the plane falls
within the face. If the body is then struck by a blow which passes through or
begins to move from rest under the action of any finite forces whose resultant
passes through 0, prove that it will not turn over, but will begin to slide along the
plane, even if the line of action of the force cuts the plane outside the base.
[CoumoW]
696. Wliatever the shape of a body may be we may suppose it to be set in
rotation about the normal at the point of contact with an angular velocity n.
If this angular velocity be not zero, the normal must be a principal axis at the
point of contact, and yet it must pass through the centre of gravity. This cannot
be unless the normal be a principal axis at the centre of gravity. If however n=0,
this condition is not necessary. There are therefore two cases to be considered.
Case 1. A body of any form is placed in equilibrium resting tcith the point C on
a rough horizontal plane, with a principal axis at the centre of gravity vertical, and
R. D. 31
13
• '
K: „■
482
MOTION UNDER ANY FORCES.
it then set in rotation with an angular velocity rx about GO. A small disturbance
being given to the body, it is reqtiired to find tlie motion.
Case 2. A body of any form is placed in equilibrium on a rough horizontal plane
with the centre of gravity over the point of contact, A small disturbance being given
to the body, to find the motion.
i !
§ i^
I' i
697. Case 1. Supposing the body not to depart far from its initial position,
we have p, q, «, v, w, Wj, w, all small quantities and r=l nearly. Hence by (2),
when we neglect the squares of small quantities, we see that X, Y are also small,
and Z = -g nearly. It follows by (1) that Wg is constant and . . =n. Also | and i)
are small and f = h nearly, where h is the altitude of the centre of gravity above the
horizontal plane before the motion was disturbed. The equation to the surface
may, by Taylor's theorem, be written in the form
^-h V^ + 2|« ^n
where (a, h, c) are some constants depending on the curvatures of the principal
sections of the body at the point C,
The squares of all small quantities being neglected, the preceding equations
become
A~^-{B-C)nu^=-gv-hY
,dw<
B^-{C-A)nw^ = hX+gi
da
di'
dt
nv=gp + X,
dv
di
+ nu=gq + Y,
dp
Tli
= nq~Wfi,
da
« - Jiij + // W.J = 0, r - /( Wj + «? = 0,
P
abbe
. = 1.
Eliminating X, Y, u, v, Wj, Wj from these equations, we get
d^q
dp
(A + h^)-^^ + {A+B + 2¥ -C)n f-{{B-C) n^ + hg + hhi^ \q=-{9 + hn'')v + hn
' df^
dt
-(fi + 7»«)|^ + (J+J5 + 27t2-(7) n ^ + {{A-CW + hg + hhi^}p = {g + hn^)^ + hn^.
It will be found convenient to express ^, rj in terms of p, q. The right-hand
sides of each of these equations will then take the form
Lp + Mq+L'^ + M''^,
at dt
To solve these equationH, we must then assume^;, q to be of the form
p = P„ cos \t + Pj ti'in \t )
q = Po cos \t + Q, sin X( ) '
MOTION OF A SOLID BODY ON A PLANE.
483
, disturiance
izontal plane
e being given
itial position,
Hence by. (2),
re also small,
Also J and ij
riiy above the
;o the surface
the principal
ling equations
The right-hand
e form
If the tangents to the lines of ourvature of the moving body at C be parallel to
the principal axes at the centre of gravity, these equations admit of corsidorable
simplification. In that case the equation to the surface may be written in the form
t--KM)-
where a and c are the radii of curvature of the lines of curvature. The right-hand
sides of the equations then become respectively
- (g + An") cq + hna ~ and (g + hn^ ap + hnc -^ .
To satisfy the equations, it will be sufficient to put
p=Fco3(\t+f), q = Bin (\t+f).
P
This simpUfication is possible, because we can see beforehand that ^ = ~^ .
P
Substituting and eliminating the ratio - , we get the following quadratic to de-
Cr
termine \'.
{{A-\-h'')\'' + {B-C-{-h{h-c)\n''+g{h-cMB-\-hP)\^ + {A~C+h{h-a)W-{-g{h-a)]
=\H^{A-\rB + 2h'^-C-ha}{A-\-B-v21i^-C-hc).
If \, \ be the roots of this equation, the motion is represented by the
equations
p=Pi cos {\t +/i) + Fj cos (\t -f-Za)
g = (?i sin (\t +/i) + G, sm {\t +f^)
f (i
where tt « ^ ^™ known functions of X^, \ respectively, and Fy, F^, f^, /j are
constants to be determined by the initial values of », g, ^ , -p .
at at
In order that the motion may be stable, it is necessary that the roots of this
quadratic should be real and positive. These conditions may be easily expressed.
Ex. 1 . A solid of revolution is placed with its axis vertical on a perfectly rouf,'h
horizontal plane and is set in rotation about its axis with an angular velocity n.
If c be the radius of curvature at the vertex, h the altitude of the centre of
gravity, k the radius of gyration about the axis, k' that about an axis through tho
vertex perpendicular to the axis of figure, show that the position of the body will bo
stable if n > 2 — ir-h — - '
k^ + hc
Ex. 2. An ellipsoid is placed with one of its vertices in contact with a smooth
horizontal plane. What angular velocity of rotation must it have about the vertical
axis in order that the equilibrium may be stable ?
Eesult. Let a, h, c be the semi-axes, c the vertical axis, then the angular
(Hg Jc*-a* + Jc*-b*
velocity must be greater than
/5g
V c
a^ + b'^i
[Puiseux.]
Ex. 3. A solid of any form is placed in equilibrium with tho poinl C on a
smooth horizontal piano, a principal axis GC at the centre of gravity being vertical,
and an angular velocity n is then communicated to it about GC. A small disturb-
31—2
■4
■
■
111
lii \
fm
! &-
wn tf W M iw'r-' -.m"
R r?f
484 MOTION UNDER ANY FORCES.
ance being given, show that the hanuonic periods may be deduced from the qaad<
ratio
{A\^ + E)(B\'+F) = {A +B-C)n^\^+g^p'-p)^Bin^d COB.* S,
where
E = {B-C)n*+g{{h-p)sm^8 + {h-p')coa'*S\,
F={A-C]n'^+g{{h-p)eoa'S+{h-p')Bin?d].
Also h is the altitude of the centre of gravity, p, p' are the principal radii of
earvature at the vertex, and S is the angle the principal axis GA makes with the
plane of the section whose radius of curvature is p. [Puiseux.]
698. Case 2. Supposing the disturbance to be small, we have 0^, u^, w^,
u, V, w all small quantities. Hence when we neglect the squares of small quantities
the equations (1) and (2) become respectively,
^|.=,z-!-F, B'^'=tx-cz. o^'=sr-,x (■).
dtt ,, dv „ dw _ ....
^^=9P + X, ai=!^i+Y, -arOr+Z (u).
Let fo» %> to ^^ *^^ co-ordinates of the point of contact in the position of equili-
brium, and let f=fo + f'i V=Vo + v'i f=fo + f' Then in the small terms of
equation (4) we may write f^, %, fo for f, 17, f. Hence differentiating these and
eliminating X, Y, Z, u, v, w by help of equations (i) and (ii), we get
(^ + V + fo'')^^-fo'?o'^^-Wo-~^=-ff('?r-f2) (xii),
and two similar equations.
Let Pq, go, Vq be the values of p, q, r ia the position of equilibrium. Then
^ = — = i^=p where p is the radius vector from to the point of contact. Now
Po <lo ^a
in the small terms of equations (3) we may write p^, Qq, r^ for p, q, r. Hence equa-
tions (iii) "become by substitution
. dw, d^r ^ d\ , . .
^d<='''^dra-f«''dt«-^('^-^«) (^^)'
and two similar equations. At the time I l^ip^p^^+p', q=q(, + q', and r=ro + /.
Then since (Pq -^p'f + {q^ + 3')' + (^o + rf=l, we have Pop' + q^q' + Tot' = 0. The
form of the surface being known we can find p', q', / in terms of f, rj', f , and thus
express »;r - fj, ip - fr, ^q- tip in the form - g (rjr - tq) = I^' + Mq'.
The equations (iv) now become
, du, d^r' dy , , „ ,
^-dr=''«^d^-^<"'dr«+^^'+^3'. (V),
and two similar equations.
Differentiating equations (3), and substituting for -^^ , -J^", -^, r' and -^ ,
(It Ctt lit (It
we get equations of the form
dV, ^d»g'_
MOTION OP A ROD.
tlie quad-
)1 radii of
8 with the
■»1. Ws» '^S'
I quantities
(i).
....(ii).
on of equili-
II terms of
ig these and
(iii).
rium. Then
ontact. Now
Hence equa-
(iv),
lr=ro+r'.
ror'=0. The
f', f , and thus
■(V).
^\r'andg',
485
To solve these we put p' = P cob (\t+f), 2'=Qcos(X«+/), substituting and
elin> mating the ratios ^ , we have '.he quadratic
to determinf X'.
=0.
(vi).
Thus by virtue of the relation existing between p', q', j-*, each of these may be
represented by an expression of the form
Pi cos (Xjf +/i) + P, cos (Xsj« +/j).
Substituting these values in equations (v) we see that w^, w^, u^ can each bo
represented by an expression
Oj + JSj cos (\t +/,) + E^ cos (\t+f.,),
where E^, E^ are known functions of P^ P^ ... and \, \, but Q^, n^, O3 are »niall
arbitrary quantities. By substituting in equations (3) and equating the coefficients
of cos (\t+fi) and cos {\t+f.^), we may find the values of E^ and E^ without diffi-
culty. And we also see that we must have
Po 5* J-o '
so that, of the three 0„ Q^, Oj, only one is really arbitrary. We have therefore
but five arbitrary constants, viz. Pj, P^, f^, f.^, and Oj. These are determined by
the initial values ot Wj, Wj, W3, p' and q'.
To find the motion of the principal axes round the vertical, let <f> be the angle
the plane containing GC and the vertical makes with the plane of AC. Then by
drawing a figure for the standard case in which p, q, r are all positive, it will be
seen that if /* be the rate at which OG goes round the vertical,
/ji>Jl-j
. ^ i'o<^i+9oWi
: Wj COS ^+ Wj Sin ^ = /T^^T •
Substituting for u^, ci>2> ^^^ takes the form
At = Wj + A^i cos (Xi« +/i) + N^ cos (Xj,< +/j),
where n,, N-^, N^ are all known constants.
In order that the equilibrium may be stable it is necessary that the roots of
the quadratic (vi) shoiUd both be real and positive. These conditions may easily
be expressed.
These conditions being siipposed satisfied, the expressions for p', q', r' will only
contain periodical terms, and thus the inclinations of the principal axes to the
vertical will not be sensibly altered. But the expressions for Wj, Wj, Wj may each
contain a non-periodical term, and if so the rate at which the principal axes will
go round the vertical will also contain non- periodical terms. The body therefore
may gradually turn with a slow motion round the normal at the point of contact.
The expressions for «, v, lo will contain only periodic terms, so that the body will
have no motion of translation in space.
Motion of a Rod,
599. Wlicn the body whose motion is to be determined is a rod, it is often
more convenient to recur to the original equations of motion supplioil by
D'AIombert's Principle. The equations of Lagrange may also bo used with
advantage. These methods will be illustrated by the following problem,
Hit
if!?
'4
: 'I
■ 1 i
i
f ■
M^yiM.-rW.
486
MOTION UNDER ANT FORCES.
A uniform heavy rod, suspended from a fixed point by a string, ma^ws small
oscillations about the vertical. Determine the motion.
Let be taken as origin, and let the axis of z be measured vertically downwards ;
let 2a bo the length of the rod, 6 the length of the string. Let (I, m, n) (p, q, r)
be the dii-ection-coBiues of the string and rod. Then I, m, p, q are small quantities
whose squares are to be neglected; and we may put n and r each equal to unity.
Let w be the distance of any element du of the rod from that extremity A of the
rod to which the string is attached. Let {x, y, z) be the co-ordinates of the element
du, then we have
»=hl + vp, y=ibm-\-uq, 8=6 + M (1).
Let M be the mass of the rod, MT the tension of tho string, the equations of
motion oi the centre of gravity will be
dH dJ'p^
d^m d^q
0=g-T
By D'Alembert's Principle the equation of moments round x will be
Zdu (y^^,-z'^f^=2du{yZ-2r)=:ldu(j!,o).
By equations (1) this reduces to
£"du^- (b + u) (b^-^^+u^^^,y^=2ay(bm + aq).
Integrating, we get
which by equations (2) reduces to
.(2).
, d^m . - „ ,
4 d'q
-a — --
a dt^
Therefore the four equations of motion are
. dH d^p
cPl i d^p
dt« ■*■ 3 " dt '
■9P-
■(3),
and two similar equations for m, q. These equations do not contain m or q, and
on the other hand the equations to find in and q do not contain I or p. This shows
that the oscillations in the plane xz are not affected by those in the perpendicular
plane yz. 6ee Art. 450.
To solve these equations , put l=Fain{M + a), p = G sin (\t + a) ,
we get
b\^F + a\^G = .jP,
b\^F+-aVG=gGi
^. ia + 3b ^. 3(;« „
... \4 — —o\^+ 4=0,
ab ab '
and the values of X may be found from this equation.
i';i'(( small
ivmwards ;
n) ip, q, r)
quantities
to unity.
A of the
le element
(!)•
[uations of
.(2).
EXAMPLES.
487
In order to make a comparison of different methods, let na deduce the motion
from Lagrange's equations. In this case we must determine the semi vis viva T
true to the squares of the small quantities p, q, I, m, we cannot therefore put r=l,
11=1. Sincej!)« + g''' + r* = l, i' + m* + »"=!, we have
r = l
p^ + ^
n = l-
I»+m"
2 ' 2
we must therefore replace the third of equations (1) by
z = bn + ur=b + u-b —^ u*--~.
If accents denote differential coefficients with regard to t, as in Lagrange's
equations we have
Sffix'3=2m(6«n + 26jyM+j>'«uS) = iH (m'' + 2hiya+'^ pA .
The value of Sm/^ may be found in a similar manner. The value of Sm/' is of
the fourth order and may be neglected. Hence we have
2r=6» {l'f + m'^) + 2ab{l'p'+m'fi')+ ~ (p'^+q'%
6 — ^ +a ^ „ j + constant.
_,, ^. d dT dT dU , , ,,, , ,,
The equation -n t;; - -r. = tt becomes oi" +.--»"= -gl;
at dl dl dl to'
similarly we get
bl" + ^p"=-gp.
These are the same equations which we deduced from D'Alembert's Principle,
and the solution may be continued as before.
EXAMPLES'.
i It
II
ill!
(3),
ji m or g, and
This shows
perpendicular
1. A uniform rod, moveable about one extremity, moves in such a manner as
to make always nearly the same angle a with the vertical ; show that the time of a
small oscillation is 2ir
/2a
V 3« • i:
cos a
3ff 1 + 3 cos'' a
, a being the length of the rod.
2. If a rough plane inclined at an angle a to the horizon be made to revolve
with unifoim angular velocity n about a normal Oz and a sphere be placed at rest
upon it, show that the path in b]iace of the centre will be a prolate, a common, or a
curtate cycloid, according as the polat at which the sphere is initially placed is with-
out, upon, or within the circle whoso equation is a?+y^= — ^-— — x, the axis Oy
being horizontal.
When the sphere is placed at rest on the moving plane, it should be noticed
that a velocity is suddenly given to it by the impulsive frictions.
■1 1 f
* These Examples are taken from the Examination Papers which have been
net in the University and in the Colleges.
*ii-i:
mummmmmm^
■»«?
.--^JU»L... II I L.I H—i
488
MOTION UNDER ANY FORCES.
8. A circular disc capable of motion about a vertical axis through its centre
perpendicular to its plane is set in motion with angular velocity O. A rough
uniform sphere is gently placed on any point of the disc, not the centre, prove that
the sphere will describe a circle on the disc, and that the disc will revolve with
angular velocity =-.iri- a — ;, Of where MP is the moment of inertia of the disc
about its centre, m is the mass of the sphere and r the radius of the circle traced
out.
4. A sphere is pressed between two perfectly rough parallel boards which are
made to revolve with the uniform angular velocities Q and Q' about fixed axep per-
pendicular to their planes. Prove that the centre of the sphere describes a circle
a'uout an axis which is in the same plane as the axes of revolution of the boards and
whose distances from these axes are inversely proportional to the angular velocities
about them.
Show that when the boards revolve about the same axis, their points of contact
will trace on the sphere small circles^ the tangents of whoso angular radii will be
c CI' "
a 'W+U
centre.
; , a being the radius of the sphere and c that of the circle described by its
5. A perfectly rough circular cylinder is fixed with its axis horizontal. A
sphere being placed on it in a position of unstable equilibrium is so projected
that the centre begins to move with a velocity F parallel to the axis of the cylinder.
It is then slightly disturbed in a direction perpendicular to the axis. If d be
the angle the radius through the point of cor+act makes with the vertical, prove
'2
that the velocity of the centre parallel to the axis at any time ( is Fcos */ -
10
e
and that the sphere will leave the cylinder when cos fi-
ll'
6. A uniform sphere is placed in contact with the exterior surface of a perfectly
rough cone. Its centre is acted on by a force the direction of which always meets
the axis of the cone at right angles and the intensity of which varies inversely as
the cube of the distance from that axis. Prove that if the sphere be properly
started the path described by its ceniie will meet every generating line of the cone
on which it lies in the same angle.
See the SohUions of Cambridge hrohlcmi for 1860, page 92.
7. Every particle of a sphere of radius a, which is placed on a perfectly rough
sphere of radius c, is attracted to a centre of force on the surface of the fixed sphere
wi th a force varying inversely as the square of the distance ; if it be placed at the
extremity of the diameter through the centre of force and be set ri tating about that
diameter and then slightly displaced, determine its motion ; and show that when it
leaves the fixed sphere the distance of its centre from the centre of force is a root of
the equation 20x'< - 13 (2c + a) jc* + 7rt (2c + o)" = 0.
8. A perfectly rough plane revolves uniformly about a vertical axis in its own
plane with an angular velocity n, a sphere being placed in contact with the plane
rolls on it under the action of gravity, find the motion.
Take the axis of revolution as axis of z, and let the axis of x be fixed in the
plane. Let a be the radiuH, m the mass of the sphere ; F, F the frictions resolved
EXAMPLES.
489
parallel to the axes of x and z and R the normal reaction. The equations of
d'x F
motion are therefore by *. 1. 179 tt:. - «'*=
df*
m
The equations of rotation by Art. 255 are
, „ dx R ,rf«z F'
-an' + 2n j, = - and -r-= -g-^~.
dt m iil^ " m
(?», Fa dwu
du,
'di '
Fa
Since the point of contact has the same motion as the plane the
geometrical equations by Art. 244 are — - on + awg=0, j -aug=0. Solving these
equations we find that the sphere will not fall down. If the sphere sta^^ from
relative rest at a point in the axis of x, we have z = - -j tan^ i { 1 - cos (nt cos i) )
where sin i = a/ -. The sphere will therefore never descend more than —^ below
its original position.
9. A perfectly rough vertical plane revolves with a uniform angular velocity n
about an axis perpendicular to itself, and also with a uniform angular velocity Q
about a vertical axis in its own plane which meets the former axis. A heavy uni-
form sphere of radius c is placed in contact with the plane ; prove that the position
of its centre at any time t, will be determined by the equations
i§-6n^^-2/£=o,
_ d^
df' ' "■■ dt ' ''" \dt^
+ QH
)=o,
z denoting the distance of the centre from the horizontal plane through the hori-
zontal axis of revolution, and ^ that from the plane through the two axes.
Prove also that 7u=7ca + 2ixb, 7v + 2)ua = 0, if a and 6 be the initial values of $
and z, u and v those of 3? and -j- .
at dt
10. A hoop AGBF revolves about AB its diameter as a fixed vertical axis. GF
is a horizontal diameter of the same circle which is without mass and which ia
rigidly connected to the circle; DQis a smaller concentric hoop wlxich can turn
freely about OF as diameter. If 0, fi', w, w', be the greatest and least angular
velocities about AB, GF respectively, prove that . fi'=u'' - w'*.
11. OA, OB, 00 are the principal axes of a rigid body which is in motion
about a fixed point 0. The axis 00 has a constant inclination a to a line OZ
fixed in space, and revolves with uniform angular velocity fl round it, and the
axis OA always lies in the plane ZOO. Prove that the constraining couple has its
axis coincident with OB, and that its moment is -(A-C) 0^ sin a cos a.
"I
: 1
!l.
I
>;F'
i'V ■ '
•••"
CHAPTER XI.
PRECESSION AND NUTATION,
&C. &C.
On the Potential,
COO. To find the potential of a hody of any form at any
external distant point.
Let the centre of gravity O of the body be taken as the origin
of co-ordinates and let the axis of x pass through 8 the external
point. Let the distance GS= p. Let {x, y, z) be the co-ordinates
of any element dm of the body situated at any point P and let
QP = r, then P/S» = p" + r' - 2pa?. The potential of the body is
^ PS'
1 2/30;- r" 3 /2px—r
?si
dm (
P
5 /2/:
arranging these terras in descending powers of p, we get
dtn
U +
P
F=S
a; 3a;' - r" 5x' - Sxr^ 35.e* - 3 0a; V -H 3/
^^"■^"V^'^ 2p' ■•■ 8p* +
...}
Let ilf be the mass of the body, then %dm = M. Also since the
origin is at the centre of gravity, we have Xxdm = 0.
Let A, B, G be the principal moments of inertia at the centre
of gravity, / the moment of inertia about the axis of x, wliich in
our case is the line joining the centre of gravity of the body to
the attracted point. Then
Xdmr'^^iA+B+C),
-^dmx'^^dm {r' -f-z'')=\{A ^ B + C) - I.
•m at any
the origin
le external
o-ordiuates
; P and let
body is
so since the
,t the centre
X, which in
the body to
ON THE POTENTIAL.
491
Let I be any linear dimension of the body, then if p be so
great compared with Z that we may neglect the fraction f- J of
the potential, we have
^r_M ^ A + B + C-SI
If we wish to make a nearer approximation to the value of V,
we must take account of the next terms, viz.
5X mx^ — 32ma;r'
Let (I, t), ^ be the co-ordinates of m referred to any fixed
rectangular axes having the origin at 0, and let (a, /8, 7) be the
angles 08 makes with these axes. Then
a; = f cos a + 17 cos /3 + (fcos 7 ;
.'. "Zmx^ = cos' a Swi|' + 3 cos'a cos /3 "Zm^if +
If the body be symmetrical about any set of rectangular axes
meeting at G, we have Sm|' = 0, Xm^rj = 0, &c. = 0, so that this
next term in the expression for the potential vanishes altogether.
Thus the error of the preceding expression for V is comparable
to only the fraction (-) of the potential. This is the case with
the earth, the form and structure of which are very nearly sym-
metrical about the principal axes at its centre of gravity.
This theorem is due to Poisson, but it was put into the con-
venient form just given by Prof. MacCuUagh. See Boyal Irish
Transactions for 1855, page 387.
001. In the invest'gation of this value for the potential, S
has been supposed to be at a very great distance. But the ex-
pression is also very nearly correct wherever the point 8 be
situated, provided the body be an ellipsoid whose strata of equal
density are concentric ellipsoids of small ellipticity.
To prove this, we may use a theorem in attractions due to
Maclaurin, viz. The potentials of confocal ellipsoids at any ex-
ternal point are proportional to their masses. Let us first con-
sider the case of a solid homogeneous ellipsoid. Describe an
internal confocal ellipsoid of very small dimeniions and let a', h', d
be its semi-axes. Then because the elliptlity is very small, we
can take a', h', c so small that 8 may be regarded as a distant
point with regard to the internal ellipsoid. Hence the potential
due to the internal ellipsoid is
^„ M' . A'^B'+C'-M'
u'\\
'^■\
U
'i;
492
PRECESSION AND NUTATION.
where accented letters have the same meaning relatively to the
internal ellipsoid that imaccented letters have with regard to the
given ellipsoid. The error made in this expression is of the
/a'\*
order ( - J V. Hence, by Maclaurin's theorem, the potential V
of the given ellipsoid is
y_M M A' ^-B'+C'-^ir
and the error is of the order
a* -
If a, h, c be the semi-axes of the given ellipsoid, we have
Similarly, B=^,B+^^ M\\ 0= ^, C + \ M\\
Also if (ot, /3, 7) be the direction-angles of the line GS with
reference to the principal axes at G, we have
/= J cos' a 4- 5 cos" ^+G cos' 7 = 177 / + ^ M\^.
Hence, substituting, we have
V=
Jf . J+5+C-3J
P 2p»
If a, 6, c be arranged in descending order of magnitude, we
can by diminishing the size of the internal ellipsoid make c as
small as we please. In this case we have ultimately a = Va"* — c\
Let 6 be the ellipticity of the section containing a and c the
greatest and least semi-axis. Then a' = ' V2e, and the error of
the above expression for V is of the order 4 (- j e'F.
The theorem being true for any solid homogeneous ellipsoid
is also true for any homogeneous shell bounded by concentric
ellipsoids of small ellipticity. For the potential of such a shell
may be found by subtracting the potentials of the bounding
ellipsoids, A^-B+ C hy Art. 5 being independent of the direc-
tions of the axes.
Lastly, suppose the body to be an ellipsoid whose strata of
equal density are concentric ellipsoids of small ellipticity, the
external boundary being homogeneous. Then the proposition
being true for each stratum, is also true for the whole body.
ne GS with
I make c as
3f the direc-
ON THE POTENTIAL.
403
This theorem was first given by Prof. MacCullagh as a
pro])lcm, and was pubHshed in the Dublin University Calendar for
1834, page 268. Some years after, about 184G, he gave his proof
of the theorem in his lectures, which is substantially the same
as that given in this Article. See the Transactions of the Royal
Irish Academy, Vol. xxii,. Parts i. and ii., Science.
602. Tlie following geometrical interpretation of the formula of Art. 600 is
also due to Prof, MacCullagh. His demonstration and another by the Rev. R.
Townsend may be found in the Irish Tramactiom for 1855.
A system of material points attracts a point S whose distance from the centre
of gravity G of the attracting mass is very great compared with the mutual
distances of the particles. If a tangent plane be drawn to the ellipsoid of gyration
perpendicular to GS, touching the ellipsoid in T and cutting GS in U, then the
resultant attraction on S lies in the plane SOT. The coinponent of the attraction
on S in the Erection TU= — ^ GU.UT. The component of the attraction on
c . r ^. .. Tjr, ^I 3A + B + C-3I
S in the direction UG= ^ + s ,, .
These theorems are also true if we replace the ellipsoid of gyration by any
confocal ellipsoid. Let a, h, c be the semi-axes of tliis confocal, and lot p be the
perpendicular GU on the tangent plane. Since by Art. 26, A = Ma^ + X, £=Mb'' + \,
c ^, ^ • <. L u IT ^J- M(a^ + b^ + c^-3p'')
&e. where \ is some constant, we have V— — -\ ^ --^i ^—^ .
P 2/)'*
To prove that the resultant force on S lies in the plane SGT, let us displace
Sio S' where SS' is perpendicular to this plane and is equal to pd'^. By Art. 326
\dV
the force on S in the direction SS' is - — - .
pdyp
But after this displacement the tan-
gent plane perpendicular to GS intersects along TU the former tangent plane, hence
dp
df
To find the force P acting at S in the direction TU, let us displace S to S" where
dV
:0, and .\-r-r=0.
d\y
SS" is parallel to TU and is equal to pd^. Since OU is perpendicular to UT we
have, exactly as in the Differential Calculus, TiZ^ 7^. Hence
d^f/
pdtp p* ^
Lastly, to find the force S in the direction SO we have by Art. 326
_ dV M BA+B + C-3!
■»= --3- = rsT ;
dp
2
Ex. Show that the product GU. TUia the same for all confocals.
I'M
n i
i^.
494
PRECESSION AND NUTATION,
■ !
608. Ex. If QP bo a straight line through tho oentro of prrnvity Btich that
tho moment of inertia about it is equal to tho mean of the three principal momontfl
of inertia at O, t}ien tho resolved attrnctiou of tho body on any point S in tho
direction <S'^ is more nearly tho same as if tho body wore collected into its centre of
gravity when iSf lies in OP, than when S lies in any other straight line through Q.
Show also that the moment of inertia about GP is equal to the mean of tho
moments of inertia about all straight lines passing through 0,
If two of the principal moments of inertia aro equal, prove that QP makes with
1
the axis of unequal moment on angle equal to cos-i -/-•
C04. Ex. 1. If two bodies exert equal attractions on all external points, prove
that their centres of gravity must coincide and their mnHSos must bo equal. Tho
principal axes at their common centre of gravity must bo coincident in direction,
and the diilerence of thoir moments of inertia about any straight lino constant.
Ex, 2. Thence show that two Chaslesian shells of the same body have the
same principal axes at their common centre of gravity and tho di£forcuce of their
moments of inertia about any straight lino constant.
Ex. ."5. If tho attraction of a body on every external point be the same as that
of a single particle placed a some point, then the mass of the particle is equal to
the muss of the body, the point is the centre of gravity, and unless the law of
attraction be as the direct distance, every axis through the centre of gravity is a
principal axis at the centre of gravity. See tho Quarterly Mathematical Journal,
1867, Vol. II. page 136.
These results follow readily from Ex. 1.
Ex. 4, Let an ellipsoid be described having its semi-axes a, b, e such that
2 . . . „2„ „ . _ . „2
i./|6».
M'^c*=A + B-C + \, where \ is at
C+A-B-\-\
our disposal, and may be any quantity positive or negative which does not make
a, b, e imaginary. Let an indefinitely thin shell of mass M be constructed
bounded by similar ellipsoids and having this ellipsoid for one bounding surface.
Then the attractions of the given body and this shell on any distant external point
are the same in direction and magnitude.
The attraction of such a shell on any externa^ ' it is normal to the confocal
through that point and is equal to -rr/ p', where a', b', c' are the semi-axes of the
confocal and p' the perpendicular on the tangent plane at the attracted point. See
the Quarterly Journal of Pure and Applied Mathematics, 1867, Vol. viii. page 322.
Ex. 5. The attraction of a body two of whose principal moments at the centre
of gravity A and £ are equal and greater than tho third attracts a distant point as
/ A -C
if its mass were equally distributed over a straight line 2a/ 3- placed per-
pendicular to tho plane oi A, B with its middle point at tho centre of gravity.
This proposition is accm-ately true if the body be an indefinitely thin shell bounded
by similar prolate spheroids. In any case it is necessary that the equal moments
A, B should be greater than the third moment of inertia C.
Ex. 6. Whatever be the relative magnitudes of the three principal moments
of inertia, the attraction on a distant point is the same as if the mass was distributed
ity sncli tlint
jipal momontB
oiut S in tlio
o itR oentro of
) tbroitKh 0.
le mean of tbo
\p makes with
I pointR, prove
)0 eqiml. Tho
it iu iliroction,
constant.
body have the
erence of their
be same as that
iolo is equal to
iless the law of
> of gravity is a
natical Journal,
h, e such that
where X is at
does not make
be constructed
)unding surface,
external point
to the confocal
emi-axes of the
ited point. See
VIII. page 322.
is at the centre
distant point as
-C
, placed per-
M
ntre of gravity.
■in shell bounded
equal moments
ncipal moments
1 was distributed
ON THE POTENTIAL.
495
over tho focal conic of tho ellipsoid described in (4) so that tho density at any point
, whore AB is tho diameter throngh P.
P is proportional to - -z=
^AP . PJi
Ex. 7. The attraction of any body of mass M on a distant particle may bo
found iu the following manner. Lot an indefinitely tiiin shell of mass ZM bo
constructed bounded by similar ellipsoids and having the ellipsoid of gyration at
tho centre of gravity for one bounding surface. Also lot a particle of mass 4.1/ bo
collected at the centre of gravity. Then tho attraction of M on any distant
particle is tho same in direction and magnitude as if 4i>/ attracted it and 3If
repelled it.
605. Ex. If the law of attraction had been - 4> (dist.) instead of tho inverse
square, tho potential of a body on any external point H would have been represented
by iiH^i (i'<Si), where (p(p) is the differential coeflicicnt of 0,(/»). In this case, by
reasoning in tho same v/ay as in Art. 600^ we got
whore A, B, C and / have the same meanings as before.
If («', y', z") be the co-ordinates of 8 referred to tho principal axes at 0, tho
moment of the attraction of S about tho axis of m is = - -r ^^^' , (C-A)x'z'. See
pdp p ^
Art. 326.
606. Ex. An indefinitely thin stratum is placed on a sphere and the density at any
point Pis equal to at* + ?>m* + w*, where I, m, n are the direction-cosines of tho
radius through P referred to any rectangular axes. Show that the potential of the
stratum at any external point is equal to
Eja + b + c) Ef ■ 6{al^ + bm^+ ck") -a ~b- c
- pf +5 p^ -'
where / is the radius of the sphere and E its volume.
007. To find the Force-function dm to the attraction of any
body on any oilier distant body.
Let G, O' be the centres of gravity of the two bodies, and let
OQ' = B, Let A, B, C; A', B, C be the principal moments of
inertia of the two bodies at G and G' respectively ; I, T the
moments of inertia about GG\ and let JLT, M' be the masses of
the two bodies.
Let m be any clement of the body If situated at the point 8,
and let GS = p. Then the potential of the body 31 at m' is
m < — H-5 \> where /, if, the moment of mertia of
IP V )
the body M about GS. We have new to sum this expression for
all values of m. This gives
P -P
\
!
!'
I
I
\\
•1'
i
I i i
If
tri:
:!i;!
;!-:
'msmssa
'< 4^^
496 PRECESSION AND NUTATION.
The first term by the same reasoning as before gives
+ M —
R
2R'
In the second term, let x, y , z be the co-ordinates of m'
Then
referred to Q' as origin
j[) = ^fl+-5-|- squares of oj', y , z'j ,
7j = /(I + ace + ^y + yz' + squares),
where a, fi, 7 are some constants. Substituting these, and re-
membering that 'Zm'x = 0, Xmy = 0, Xm'z' = 0, we get
Jif .
A + B+G-SI
21^
( /terms depending on the\)
( V squares of x', y, z J)
Hence the required force-function is
F=
MM
B
M
A' + E+C
2W
37' . ^,,A + B+C-M
V M —
2R^
•11'^""
(II \
712) ^> where I, V
are any linear dimensions of the two bodies respectively.
608. To find the moment of the attraction of the sun and
moon about one of the principal axes of the earth at its centre of
gravity.
Let the principal axes of the earth at its centre of gravity be
taken as the axes of reference, and let a, /S, 7 be the direction-
angles of the centre of gravity G' of the sun. Then if Fbe the
potential of the sun or moon on the earth, we have
F=^+j,f4>^'+^'
R
2R'
37' , „,^ + 5+C-37
h if —
2R'
where unaccented letters refer to the earth, and accented letters
to the sun or moon. Let 6 be the angle the plane through the
sun and the axis of y makes with the plane of wy, then ~rs is the
required moment in the direction in which we must turn the body
to increase 0. From the above expression, since 6 enters only
through 7, we have
dV_SM[dI
de ~ 2R' dd'
ON THE POTENTIAL.
497
lies of w'
le, and re-
11-
1-M
7", where I, I'
the sun and
its centre of
Df gravity be
he direction-
lif F be the
7-37
3ented letters
through the
en -ja IS the
turn the body
9 enters only
Now I = A cos'a + B cos'yS + C cos^, and by Spherical Trigo-
nometry, we have
cos 7 = sin yS sin $
cos a = sin yS cos 6
h
dl
.'.'^ = -2{A-C) sin*)8 sin ^ cos ^ ;
.'.the moment required) « -^7' ^ ..
about the axis of y | = - 3 -^t ( C - ^) cos a cos 7.
In this expression the mass of the attracting body is measured
in astronomical units. We may eliminate this unit in the fol-
lowing manner. Let n be the mean angular velocity of the sun
about the earth, R^ its mean distance, so that if M be the mass
of the earth, we have
M' + 3I
^0'
=?= n-
Now M is very small com-
iM
pared with M', so small that jp is of the order of terms already
31'
neglected. Hence we may in the same terms put ^^ = n'^, and
therefore
the moment of the sun's at-
traction about the axis of
H = -Sn''{G- A) cos a cos 7 (^J.
Let n" be the mean angular velocity of the moon about the
earth, so that, if M" be the mass of the moon, B' the mean dis-
M"+M
tance, we have — ^7-3 — = n"^ Let v be the ratio of the mass of
the earth to that of the moon, then we have 57-3 — - = w"", and
• TV . °
therefore if it be the distance of the moon
the moment of the moon'
attraction about the axis
on's ] Sir ,^ ., (R'y
isofyj l + v^ ' ' \E J
In the same way the moments about the other axes may be
found. Putting k for the coefficient, we have
moment about axis of a? = — S/c (J? — C) cos ^ cos 7,
moment about axis oi z =: — Sk {A ^ B) cos a cos /3.
609. Ex. 1. A body free to move about its centre of gravity is acted on by any
number of attracting particles arranged in any way at a constant distance p from
the centre of gravity. If ^,, By, Cp D^, E^, F^ be the moments and products of
inertia of the body referred to any rectangular axes meeting in tlie centre of gravity,
R, T). 32
'11
ill
il
:
vf
■^
Mmm
498
PRECESSION AND NUTATION.
r=
and if accented letters represent corresponding quantities for the particles referred
to the same axes, prove that the mutual potential of the body and the particles is
MM> 3(AiAi'+BiBi'+CiCj;+2FyFj^'+2DiD^'+2EiEi')-{Ai + B^ + C^){A^' + Bi' + Ci')
P "^ V
where HT is the mass of all the particles. If the axes of reference be principal
axes for either body, this result admits of considerable simplification.
Show that the numerator of the second term may be expressed in terms of the
invariants of the momental eUipsoids of the body and of the system of particles.
Ex. 2. The force function between a body of any form and a uniform circular
ring whose centre is at the centre of gravity of the body and whose mass is Jl ' is
^ MM^ ^,A + B + C-SJ
where / is the moment of inertia of the body about an axis thi-ough its centre of
gravity perpendicular to the plane of the ring, and A, B, Q are the principal
moments of inertia at the centre of gravity.
This follows from Ex. 1.
Ex. 3. Thence show iJiat Saturn's ring supposed uniform will have the same
moments to turn Saturn about its centre of gravity as if half the whole mass were
collected into a particle and placed in the axis of the ring at the same distance
from Saturn, provided the particle repelled instead of attracted Saturn.
Ex. 4. If the earth be formed of concentric spheroidal strata of small but
different ellipticitieB and of different densities, show that
rd{a^e).
C-A
fi
da>
where e is the ellipticity and p the density of a stratum, the major-axis of
which is a ; the square of « being neglected. It follows that if e be constant
C-A *
- is independent of the law of density.
If we assume the law of density and the law of ellipticity given in the Figure of
C-A
the Earth, this formula gives -^=-00313593. See Pratt's Figure of the Earth.
Ex. 5. A body free to turn about a fixed straight line passing through the
centre of gravity is in equilibrium under the attraction of a distant fixed particle.
Show that the time of a small oscillation is 2^^^^^^^^—^-—^^, where the
fixed straight line is the axis of y, the plane of xy in equihbrium passes through the
attracting particle, and f, -q are the co-ordinates of the particle. Also^, B, C,D, E,F
arc the moments and products of inertia of the body about the axes.
If the straight line did not pass through the centre of gravity show that the
time would be proportional to p.
es referred
irticles is
e principal
rms of the
pai'ticles.
)rm circular
EisB is ilf' is
its centre of
ihe principal
ave the same
le mass were
ame distance
•n.
of small hixt
major-axis of
6 be constant,
n the Fignre of
•e 0/ the Earth.
ig through the
fixed particle.
- I , where the
ises through the
)A,B,C,D,E,F
show that the
MOTION OF THE EARTH ABOUT ITS CENTRE OF GRAVITY. 499
Motion of the Earth about its Centre of Gravity.
610. To find the motion of the pole of the earth about its
centre of gravity when disturbed by the attraction of the sun and
moon, the figure of the earth being taken to be one of revolution.
Let us consider the effect of these two bodies separately.
Tlien, provided we neglect terms depending on the square of
the disturbing force, we can by addition determine their joint
effect.
The sun attracts the parts of the earth nearer to it with a
force slightly greater than that with which it attracts the parts
more remote, and thus produces a small couple which tends
to turn the earth about an axis lying in the plane of the equator
and perpendicular to the line joining the centre of the earth
to the centre of the sun. It is the effect of this couple which
we have now to determine. It clearly produces small angular
velocities about axes perpendicular to the axis of figure. We
shall also suppose that the initial axis of rotation so nearly coin-
cides with the axis of figure, that we may regard the angular
velocities about axes lying in the plane of the equator to be small
compared with the angular velocity about the axis of figure.
Let us take as axes of reference in the earth, GG the axis
of figure, GA and GB moving in the earth with an angular
velocity 6^ round GG. Then following the notation of Art. 252,
we have
h^=A(a^,
h' = A(o., h'-G<o.,
^t =
a>
i»
^a= «2-
The equations of motion are therefore
d(o.
A~'-G<o,(o, + A(o,0,=M
^St
=
(1).
Let
The last of these equations shows that oj, is constant,
this constant be denoted by n.
The other two angular velocities are to be found by solving
the other two equations. This solution must be conducted by
the method of continued approximation, a)^ and m.^ being regarded
as small compared with n.
32—2
I
:i^ I
I
\'i
m
MMHpi
mimmm
mmm
600
PRECESSION AND NUTATION.
In the first instance let us suppose the orbit of the dis-
turbing body to be fixed in space. This is very nearly true
in the case of the sun, less nearly so for the moon. This limi-
tation of the problem proposed will be found greatly to simplify
the solution. We can now choose as our axes of reference in
space two straight lines OX, GY at right angles to each other
in the plane of the orbit and a third axis GZ normal to the
plane.
! If
611. In these Equations of motion the quantity 0^ is at
our choice, let it be so chosen* that the plane containing the
* We might also very conveniently have chosen as axes of reference, GC the
axis of figure and axes GA ', QB' moving on the earth so that GB' is the axis of
the resultant couple produced hy the action of the disturbing body on the earth.
In this case the plane OA' moves so as always to contain the disturbing body S,
BO that ^8 is the angular velocity of CS round C and is therefore a small quantity of
the order n'. We shall therefore reject the small terms u^O^ and u-^d^ in equations
(1). The equations now become
A^^^^Cn.,^0
A -j^ - Cnt)i = M= - 3(«(C- il)cos o cos 7
where the value of M is at once obtained from Art. 608, and in onr case a= -- 7.
Eliminating w, we have
- -1 ^ /C'A'
Cn
M.
Since the angular distance 7 of the disturbing body from the pole of the earth
MOTION OF THE EARTH ABOUT ITS CENTRE OF GRAVITY. 501
axes GC, OA also containr" GZ. Then ^3 is the angular velocity
of the plane ZGC round GG. If Wj and a>^ were zero, and
the -arth merely turned round its axis GC, it is clear that
GC and therefore also the plane ZGC would be fixed in space.
Hence 6^ is a small quantity of the same order at least as (o^
or «Dj, For a first approximation we neglect the squares of the
small quantities to be found. We therefore reject the small
terms a).^d^, w^d^ in the equations (1). The equations now become
(2).
Following the usual notation let 6 be t^^i angle ZG and
^ the angle the plane ZC makes with the fixed plane ZX. We
have then the two geomv-trical equations
ft).
sin^
dt
d0
'"»=dr
,(3).
These follow at once from a mere inspection of the figure, or
we may deduce them from Art. 235, by putting ^ = 0.
We have now to find the magnitudes of L and M. Let S
be the disturbing body and let it move in the direction X to Y.
According to the usual rule in Astronomy, we shall suppose
the longitude I of /S to be measured in the direction of motion
Ui
I.
e of the earth
varies very slowly, the term on the right-hand side is very nearly constant. If
this be regarded as a suiHcieut approximation we have
' wi= -2^-^y-sm27, and Wj=0.
But in fact these are nearly true when we take account of the periodical term
provided only S moves slowly. For suppose
^ = 7lfo + 2Psin(/)« + 0,
where p is small; we have in that case
M'J::
Mt _. CnP . , . , ^v
M
neglecting the small term p^ in the denominator we have as before Wj = - -y .
The motion of the axis O in space is therefore simply that duo to an angular
velocity Wj about the axis A'. Since the plane 4'C moves so as always to contain
the disturbing body <S, the axis of figure GC is at any insta.. •^^ moving perpendicular
to the plane containing it and the disturbing body (i.e. in the figure C is always
3k C— a
moving perpendicular to SC) with an angular velocity equal to —ji— sin 27. If
we resolve this in the direction along and perpendicular to ZC we easily deduce the
equations (7) in the text and the solution may be continued as above.
Ill
I ■; II::
i<«l
602
PRECESSION AND NUTATION.
from the point on the sphere opposite to B. This point is
usually called the firsi point of Aries. Then
TT
BS = 'ir -I and SN=l-ir.
it
By Art. 608 we have
L = -^k{B-C) cos )S cos 7 = - 3« (^ - C) sin >SfiVcos 8N sin Q
= ^ « ( J. — C) sin ^ sin
2Z.
■(4),
if=-3«:(C'--u4)cosoco37 = -3/c(C— -4} cos' /SiVsin ^ cos ^
= --^-/c(C-^)3in^cos^(l-cos2Z).
.(5).
Since the motion of the disturbing body is very slow com-
pared with the angular velocity of the earth about its axis,
I and therefore L and M are very nearly constant. If this be
reg rded as a sufficiently near approximation we have at once
by (2)
M L
"^^-C^'
""^^Cn
.(6).
That these are the integrals of equations (2) when we take
some account of the variability of L and M may be shown by
substitution in those equations. We see that they are satisfied
if we may neglect such a term as
^--L(JS-(7)Icos(9sin2;^ + 2sin(9cos24!l.
dt
dt
dt
dd
Since k(B — C) and -77 are both small quantities of the order
ft), or Wg, the first of these terms is of the order o)^ and such
terms we have already agreed to neglect. The last term is of the
n
order — &>„, where n is the mean angular velocity of the disturb-
n
ing be 'v about the earth. Rejecting these terms also, we have by
(3), f4) and (5),
d0 ^kC-A . . . „.
-77 — — o p.— sni 6 sm zl
dt zn G
dyfr SkC-A .... „,.
■in
612. To find the motion of the pole of the earth in space
referred to the pole of the orbit of the disturbing body as
origin, wc have merely to iutograte the equations (7). For a
point i
IS
N&ind
....(4),
(5).
low com-
its axis,
f this be
J at once
(6).
I we take
shown by
e satisfied
ity
the order
and such
m is of the
le disturb-
ve have by
.(7).
,h in space
body as
7). For a
MOTION OF THE EARTH ABOUT ITS CENTRE OF GRAVITY. 503
first approximation, in which we reject the squares of the small
quantities to be found, we may regard 6 on the right-hand side
as constant and equal to its mean value. If we write for I its
approximate value
l = nt + e,
wc find by integration
.(8).
= const. + ~. — > — Pi — sin 6 cos 21
^nn U
ifr = const. — ^ — -, — 7=— cos (l — ^ sin 21)
^ 2nn C ^ 2 '.
613. We may also solve equations (2) in the following manner. Since wo
reject the squares of the small quantities to be found, we may in calculating the
values of L and ilf to a first approximation suppose d to be constant and I to be
measured from a fixed point in space. We then have by the theory of elliptio
motion
l=n't + €' + Pi sin {pjt + gi) + i*2 sin (i^a* + (Za) + &c.,
where the coefiBcients of the trigonMnetrieal terms are all known small quantities,
and all the coeflScients of t are very small compared with n. In the case of the
sun the coefficient of t in the greatest of the trigonometrical terms is ^^ n and in
the case of the moon ^ n.
We may also include in thds formula the secular inequalities in the value of I.
For, we shall presently find that d has no secular inequalities, and that the first
point of Aries from which I is laeasured has a very slow motion which is very
nearly uniform on the plane of the orbit of the disturlnug body. This slow motion
m^ obviously be included in the n'.
If we eliminate u^ between equations (2) we hr.ve
d»wi CV 1 dL Cn^
The first term on the right-hand side we have already agreed to neglect. Sub-
stituting in the expression for M given in (5) the value of I, suppose we have
if=SFcos(\«+/>,
where the constant part of M is given by X - and all the other values of X are
very small. Then solving, we find
FCn ,», , ,v
w,=
Since P and X" are both very small we may reject the small term X' in the
denominator, we then have
1 M
.,,= --Si?cos(Xt+/)=-^^.
This result is strictly true for the constant term and very nearly true for the
periodical terms. In the same way we may prove that w^= -,- .
U
5 ;
■* 'J.
■WW
P<HMII«va
fn
ft ^*
504
PRECESSION AND NUTATION.
When we proceed to find 9 and \f/ from the values of Wj and w, by the help of
equations (3), it will be seen that no term will rise on integration in which \ is not
^mall. These rejected terms will not therefore afterwards become important.
614. The integration of equation (7) may be effected without neglecting the
terms containing the powers of e' in the expression for I. By the theory of
elliptic motion we have
R^ ^l = constant = Ro^'jr^*,
at
where a very small term has been rejected on the left-hand side depending on the
motion of Aries. Substitutiug for k its value given in Art. COS we find
dl'
3n'
i?o
1 C-A
■2nl + ,. C Jiji
■ sin sin 21
df Bn' 1 C-A Ro ^,- „„
-3, = -rr- ■, 7T- — p^=— cos 0(1- cos 20
where v is to be put equal to zero when the disturbing body is the sun. From
the equation to the ellipse, we have
^jP^^ = l + e'fOB(l-L).
It
If this value of R be substituted in the equations, the integrations can be effected
without difficulty. But it is clear that all the tenuis which contain e' are periodic
and do not rise on integration so as to become equally important with the others.
Since then e' is small, being equal in the case of the sun to about ^., it will be
needless to calculate these terms.
615. Let us now examine the geometrical meaning of the
equations (8). For the sake of brevity, let us put 8= ^ — > — ^— ,
so that by Art. 608 o = s — t^ or o = -^ — 7=j- v — ac-
•' 2 G n 2 C n 1 + v
cording as the sun or moon is the disturbing body, the orbit of the
disturbing body being in both cases regarded as circular.
Let us consider first the term —S cos 6 1 in the value
of ■\|r. Let a point C^ describe a small circle round Z the pole
of the orbit of the disturbing planet, the distance CZ being
constant and equal to the mean value of 6. Let the velocity
be uniform and equal to Sn' cos sin 0, and let the direction of
motion be opposite to that of the disturbing bouy. Then 0^
represents the motion of the pole of the earth so far as this
term is concerned. This uniform motion is called Precession.
Next let us consider the two terms
Bd=^ Ssm e cos 21, 5^ = I 5' cos 6 sin 21
I !l
the help of
ch \ is not
tant.
[lecting the
theory of
ding on the
sun. From
be effected
' are periodic
L the others.
1
, it will be
ng of the
\k G-A
m
c
ac-
l + v
rbit of the
the value
Z the pole
GZ being
le velocity
irection of
Then C;
ar as this
ission.
MOTION OP THE EARTH ABOUT ITS CENTRE OF GRAVITY. 505
If we put a; = sin ^ Si/r, y = W, we have
a ^ /I \l *■>
^Sfcos^sin^Y fi/Ssin^)
which is the equation to an ellipse.
Let us then describe round G^ as centre an ellipse whose
semi-axes are „ S cos 6 sin 6 and ^ S&mO respectively perpen-
dicular to and along ZG\ and let a point C, describe this
ellipse in a period equal to half the periodic time of the dis-
turbing body. Also let the velocity of G^ be the same as if
it were a material point attracted by a centre of force in the
centre varying as the distance. Then 0^ represents the motion
of the pole of the earth as affected both by Precession and the
principal parts of Nutation.
If we had chosen to include in our approximate values of
B and ^/r any small term of higher order, we might have re-
presented its effect by the motion of a point G^ describing an-
other small ellipse having C^ for centre. And in a similar manner
by drawing successive ellipses we could represent geometrically all
the terms of Q and ■^.
616. In this solution we have not yet considered the Com-
plementary Functions. To find these we must solve
^^t+^^ 0, ^^-^-Ono) =0.
at ^ ^* ^
dt
We easily find (o^ = Hsm(-j-t+ k\ a)^ = — Hcos(^t + Kj.
The quantities H and K depend on the initial values of ot^ w,.
As these initial values are unknown H and K must be de-
termined by observation. If H had any sensible value it would
be discovered by the variations produced by it in the position
in space of the pole of the earth. The period of these would
be — >, , as -4 and G are nearly equal in the case of the earth,
this period is nearly equal to a day. No such inequalities have
been found. If however any such inequality existed we might
consider these two terms together as a separate inequality to
be afterwards added to that produced by the other terms of a^ w^
whose period is half a year.
The effect of the complementary function on the motion
of the pole of the earth has been already considered. The
motion is the same as if the earth were at aay instant set in
n
». i
ill
if
,i
>00
PRECESSION AND NUTATION.
'•/ I
rotation about an axis whose direction-cosines are proportional
to Ha'm(--7-t + K], — Hcoa [— j t + Kj and n and then left
to itself. The instantaneous axis will describe a right cone of
small angle round the axis of figure and also a right cone of
small angle in space. Hence from this cause there can be no
permanent change in the position in space of the axis of the
earth. See Art. 522.
617. The preceding investigations are of course approxima-
tions. In the first instance we neglected in the differential equa-
tions the squares of the ratios of (o^ and (o^ to n, and afterwards
some periodical terms which are an — th of those retained. We
see by equations (3) and (8) that the second set of terms rejected
is much gi-eater than the first, and yet when the sun is the dis-
turbing body these terms are only about -^— ^ th part of those
retained, and when the moon is the disturbing body these
are only ^ th part of terms which themselves are imperceptibte.
We have also regarded the earth as a solid of revolution so
that A — B may be taken zero, a supposition which cannot be
strictly correct.
3 C—A n
618. In the ca.-^o of the sun we have S=^ — /y , so that
2 C n
the precession in one year is ^
SO- An'
cos 9 27r. It is shown in
2 C n
treatises on the Figure of the Earth that there is reason to put
"^ = -0031. Also we have - = -~ , and ^ = 23°. 8'. This
C n 36o
gives a precession of about 15"'42 per annum. Similarly the
coefficients of Solar Nutation in -^ and 6 are respectively found
to be 1"*23 and 0"*53. If we supposed the moon's orbit to be
fixed, we could find in a similar manner the motion of the pole
produced by the moon referred to the pole of the moon's orbit.
In this case 8=^ — p.- — t; . - The value of varies be-
2 G n \+v
11 1
tween the limits 23" ±5". Putting - = ^^ , j; = 80, ^ = 23°, we
find a precession in one year a little more than double that pro-
duced by the sun. But the coefficients of what would be the
nutations are about one-sixth of those produced by the sun.
619. We have hitherto considered the orbit of the disturbing
body to be fixed in space. If it be not fixed, we must take the
MOTION OF THE EARTH ABOUT ITS CENTRE OF GRAVITY. 507
)portional
then left
t cone of
b cone of
in be no
is of the
>proxima-
tial equa-
bfterwards
aed. We
s rejected
s the dis-
; of those
)dy these
ceptiblie.
olution so
cannot be
— , so that
n
. shown in
son to put
8'. This
ilarly the
irely found
rbit to be
f the pole
on's orbit.
varies be-
= 23°, we
that pro-
ild be the
3un.
disturbing
5t take tlie
plane CA perpendicular to its instantaneous position at the
moment under consideration. The quantity 0,, will not be the
same as before*, but if the motion of the orbit in space be very
slow, Oj will still be very small. We may therefore neglect the
small terms O^m^ and 0^(o^ as before. The dynamical equations
will not therefore be materially altered. With regard to the
geometrical equations (8) it is clear that w,, m, will continue to
express the resolved parts of the velocity of C in space along and
perpendicular to the instantaneous position of ZC. To this degree
of approximation therefore, all the change that will be necessary is
to refer the velocities as given by equations (7) to axes fixed in
space and then by integration we shall find the motion of C. This
is the course we shall pursue in the case of the moon.
The attractions of the planets on the earth and sun slightly
alter the plane of the earth's motion round the sun, so that the
position of the ecliptic in space varies slowly. It can oscillate
nearly five degrees on each side of its mean position. If the earth
were spherical there would be no precession caused by the at-
tractions of the sun and moon. The direction of the plane of the
equator would then be fixed in space, and the changes of its
obliquity to the ecliptic would be wholly caused by the motion of
the latter, and would be very coi.>siderable. But, as Laplace re-
marks, the attractions of the sur and moon on the terrestrial
spheroid cause the plane of the equator to vary along with the
ecliptic so that the possible change of the obliquity is reduced
to about one and a third degrees which is about one-quarter of
what it would have been without those actions.
At present the obliquity is decreasing at the rate of about
48" per century. After an immense number of years, it will begin
to increase and will oscillate about its mean value. These in-
equalities we do not propose to discuss in this treatise. We must
refer the reader to the second volume of the Mecanique Celeste,
livre cinquifeme. He may also consult the Connaissance des Temps
for 1827, page 234.
620. Ex. 1. If the earth were a homogeneous shell bounded by similar elhpsoids,
the interior being empty, the precession would be the same as if the earth were
solid throughout.
* The value of ^3 may be found in the foUov ng manner. The orbit at any
inst;\nt is turning about the radius vector of the planet as an instantaneous axis.
Let u be this angular velocity which we shall suppose known. Let Z, Z'; B, E bo
two successive positions of the pole of the orbit and the extremity of the axis of B
respectively. Then ZB=a right angle =Z'i5'. Hence the projections of ZZ', liB',
on ZJ are equal. This gives, since ZB is at right angles to both CZ and SB,
BSB' am BS=ZVZ' sin ZC. Now the angle ZCZ'- - 5^;, and the angle BSB'=u,
hence 8^3 . sin 9= -n sin I. The value of 5^3 must bo added to the former value of 0^,
,4f«MM
508
PRECESSION AND NUTATION.
Ex. 2. If tho earth wero a homogeneous shell bounded externally by a spheroid
and internally by a concentric sphere, the interior being filled with a perfect fluid
of the Hame density as tho earth, show that tho preceshiiou would be greater than if
the earth were solid throughout.
Let (a, a, e) be the semi-axes of the spheroid, r the radius of tho sphere. Then
C - A
since the precession varies as — - by Art. 615, the precession is increased in tho
ratio a*c : a*c — r".
Ex. 3. If the sun wero removed to twice its present distance show that the
solar precession per unit of time would be reduced to one-eighth of its present
value; and the precession per year to about one- third of its present value.
Ex. 4. A body turning about a fixed point is acted on by forces which tend to
produce rotation about an axis at right angles to the instantaneous axis, show that
the angular velocity cannot be uniform unless the momentiJ ellipsoid at the fixed
point is a spheroid.
The axis about which tho forces tend to produce rotation is that axis about
which it would begin to turn if the body were placed at rest.
Ex, 5. A body free to turn about its centre of gravity is in stable equilibrium
under the attraction of a distant fixed particle. Show that the axis of least
moment is turned toward tho particle. Show also that the times of tho
principal oscillations are respectively 2t lonfn^i^ju 8"^ ^tt ja-i/'/irr-iU •
If the body be the earth and M' be the sun, show that the smaller of these two
periods is about ten years.
621. To giv6 a general explanation of the manner in luhich
the attraction of the Sun causes Precession and Nutation.
If a body be set in rotation about a fixed point under the
action of no forces, we know that the momenta of all the particles
are together equivalent to a couple which we shall represent by G
about an axis called the invariable line. Let T be the Vis Viva
of the body. If a plane be drawn perpendicular to the axis of G
at a distance — jj- e* from the fixed point, then the whole motion
is represented by making the momental ellipsoid whose parameter
is e roll on this plane. In the case of the earth, the axis 01 of
instantaneous rotation so nearly coincides with OG the axis of
figure that the fixed plane on which the ellipsoid rolls is very
nearly a tangent plane at the extremity of the axis of figure.
This is so very naarly the case that we shall neglect the squares
of all small terms depending on the resolved part of the angular
velocity about any axis of the earth perpendicular to the axis of
figure.
Let us now consider how this motion is disturbed by the action
of the sun. The sun attracts the parts of the earth nearer to it
with a slightly greater force than it attracts those more remote.
by a sphoroiil
I perfect fluid
;reater than if
iphero. Then
lOreased in tbo
show tliftt the
of itn preueut
value.
1 which tend to
ixis, show that
)id at the fixed
that axis about
hie equilihrinm
3 axis of least
times of tho
{'(B-A)) '
er of these two
,er in which
n.
under the
the particles
(resent by (^
he Vis Viva
le axis of G
i^hole motion
e parameter
\e axis 01 of
the axis of
rolls is very
is of figure,
the squares
the angular
the axis of
)y the action
nearer to it
nore remote.
MOTION OF THE EARTH ABOUT ITS CENTRE OF GRAVITY. 509
Hence when the sun is either north or south of the equator its
attraction will produce a couple tending to turn tho earth about
that axis in the plane of the equator which is perpendicular to
the line joining the centre of the earth to the centre of the sun.
Let the magnitude of this couple be represented by a, and let us
suppose that it acts impulsively at intervals of time dt.
At any one instant this couple will generate a new momentum
adt about the axis of the couple a. This has to be compounded
with the existing momentum G, to form a resultant couple G'.
If the axis of a were exactly perpendicular to that of we should
have G' = \/'W^'\oLdtf = G ultimately.
Let be the angle that the axis of G makes with OG, then
^ is a quantity of that order of small quantities whose square is
to be neglected. Taking the case when OG, DC and the axis of a
are in one plane, for this is the case in which G' will most difter
from G, wo have
G" = (G cos eY + {G sin e + adt)'
= G^+2Gxainddt (1).
Then a and being of the same order of small quantities, the
term a sin is to be neglected He'ace we have G' = G. But the
axis of G is altered in space by an angle — ^ in a plane passing
through OG and the axis of a.
Next let us consider how the Vis Viva T is altered. If T' be
the new Vis Viva we have
T' — T = twice the work done by the couple a
= 2a (ft) cos /3) c?« (2),
where to cob ^ is the resolved part of the angular velocity about
the axis of a. For the same reason as before the product of this
angular velocity and a is to be neglected. Hence we have T' = T.
It follows from these results that the distance — ^— e^ of the fixed
G
plane from the fixed point is unaltered by the action of a.
Thus the fixed plane on which the ellipsoid rolls keeps at the
same distance from the fixed point, so that the three lines OG,
01, OG being initially very near each other will always remain
very close to each other. But the normal OG to this plane has
a motion in space, hence the others must accompany it. This
motion is what we call Precessiou and Nutation.
Lastly these small terms which have been neglected will not
continually accumulate so as to produce any sensible effect. As
the earth turns round in one day, the axis OG will describe
' i
4
it ■•
I'i^
'■:,
I!
wammn
510
PRECESSION AND NUTATION.
a cone of small angle 6 round OG. The axis about which the sun
generates the angular velocity a is always at right angles to the
plane containing the sun and OC. Hence, regarding the sun as
fixed for a day, the angle 6 in equation (1) changes its sign every
half day. Thus 0' is alternately greater and less than 0. Simi-
larly since the instantaneous axis describes a cone about OG it
may be shown that T' is alternately greater and less than T,
622. Let us trace the motion of the axis OG through a whole
year. Describe a sphere whose centre is at and let us refer the
motion to the surface of this sphere. Let K be the pole of the
ecliptic and let the sun 8 describe the circle DEFH of which K
is the pole. Let DF be a great circle perpendicular to KG, then
since OG and the axis of figure of the earth are so close that we
may treat them as coincident, D and i^'will be the intersections of
the equator and ecliptic. When the sun is north or south of the
equator, its attraction generates the couple a, which will be
positive or negative according as the sun is on one side or the
other. This couple vanishes when the sun ir passing through the
equator at D or F. If the sun be anywhere in DEF, i.e. north
of the equator, G is moved in a direction perpendicular to the
arc 08 towards D. If the sun be anywhere in FHD, a has the
opposite sign and hence G is again moved perpendicular to the
instantaneous position of G8 but still towards D, Considering
the whole effect produced in one year while the sun describes the
circle DEFH, we see that G will be moved a veiy small space
towards D, i.e. in the direction opposite to the sun's motion.
Resolving this along the tangent to the circle centre K and radius
KG, we see that the motion of G is made up of (t) a uniform
MOTION OF THE EARTH ABOUT ITS CENTRE OF GRAVITY. 511
ch the sun
gles to the
bhe sun as
sign every
0. Simi-
out Oa it
lan T.
Lgh a whole
as refer the
e pole of the
of which K
to KG, then
Iclose that we
Itersections of
south of the
hich will be
|e side or the
through the
2F, i.e. north
ticular to the
ED, OL has the
[licular to the
Considering
describes the
small space
[sun's motion.
\K and radius
(1) a uniform
motion of G along this circle backwards, which is called Preces-
sion and (2) an inequality in this uniform motion which is one
part of Solar Nutation. Again as the sun moves from D to E, G
is moved inwards so that the distance KG is diminished, but as
the sun moves from E to F, KG is as much increased. So that
on the whole the distance KG is unaltered, but it has an in-
equality which is the other part of Solar Nutation.
It is evident that each of these inequalities goes through its
period in half a year.
623. To explain the cause of Lunar Nutation.
The attraction of the sun on the protuberant parts at the
earth's equator causes the pole C of the earth to describe a small
circle with uniform velocity round K the pole of the ecliptic with
two inequalities, one in latitude and one in longitude, whose period
is half a year. These two inequalities are called Solar Nutations.
In the same way the attraction of the moon causes the pole of the
earth to describe a small circle round M the pole of the lunar
orbit with two inequalities. These inequalities are very small
and of short period, viz. a fortnight, and are therefore generally
neglected. All that is taken account of is the uniform motion
of C round M. Now K is the origin of reference, hence if M
were fixed the motion of G round M would be represented by a
slow uniform motion of G round K together with two inequalities
whose magnitude would be equal to the arc MK, or 5 degrees, and
whose period would be very long, viz. equal to that of G round K
produced by the uniform motion. But we know by Lunar Theory
that M describes a circle round K as centre with a velocity much
more rapid than that of G. Hence the motion of G will be repre-
sented by a slow uniform motion round K, together with two
inequalities which will be the smaller the greater the velocity
of M round K, and whose period will be nearly equal to that
of M round K. This period we know to be about 19 years.
These two inequalities are called the Lunar Nutations. It will
be perceived that their origin is different from that of Solar
Nutation.
624. To calculate the Lunar Precession and Nutation.
Let K be the pole of the ecliptic, 3f that of the lunar orbit,
G the pole of the earth. Let KX be any fixed arc, KG= 0,
XKG=yfr, then we have to find 6 and yjr in terms of t. By
Art. 615 the velocity of G in space is at any instant in a direction
perpendicular to MG, and equal to
SuT G- A _1_
2n C' l+u
cos MC am MG.
11, a
1 .;
V:>,!!
r\
If!
;:i
1 ■ f
[
■i
B H-i\
ill
|!
M2
PRECESSION AND NtJTATION.
For the sake of brevity let the coefficient of cos MG sin MG
be represented by P. Then resolving this velocity along and
perpendicular to KG, we have
^ = - P sin If C cos ilf C sin ^Cif 1
sin ^ § = - P sin MG cos MG cos KGM
at
By Lunar theory we know that M regredes round K uniformly,
the distance KM remaining unaltered. Let then KM=i, and
the angle XKM= — mt + a.
Now by spherical trigonometry,
cos MG = cos t cos 6 + sin « sin cos MKG,
• Ti*-/-/ TjryyTir cost — cos iJfC cos ^
sm MG cos KGM= r—^
sm a
= cos t'sin — sin i cos cos MKG,
s\ii MG. sin KGM == sin t sin MKG.
Substituting these we have
^ = _ p jsin t cos i cos sin J/ZC + | sin't sin ^ sin 2MKg\ ,
sin ^ -^ = — P -Isin ^ cos f cos"* t — ^ sin'i j
— sin tcos I cos 2^ cos MKG— ^ sin'isin ^cos ^cos 2MKG[ .
For a first approximation we may neglect the variations of
d0
and -^ when multiplied by the small quantity P. Hence -jr
contains only periodic terms, and the inclination has no per-
manent alteration. But -^ contains a term independent of
MKG ; considering only this term, we have
^ = constant — Pcos ^ [cos'* — ^ sin' ijt.
This equation expresses the precessional motion cf the pole
due to the attraction of the moon. We may write thib liquation
in the form ■'/r = ^^ —jft.
To find the nutations, we must substitute for MKG its r .pproxi-
mate value
MKG= {-m+p) t+a-'>lr^.
V sin MG
dong and
uniformly,
M=i, and
MKC,
IMKC\-
mkg\.
•iations of 6
dd
Hence ^
las no per-
pendent of
f the pole
lib equation
its f.pproxi-
MOTION OF THE EARTH ABOUT ITS CENTUE OF GRAVITY. 513
We then have after integration
/, . Psini cost cos ^ irr'/^ Psin'e'sin^ c^ttr^n
^ = const.-— C03 MKC — , ,- co3 2iI/AC7.
m — p 4 {m — p)
The second of these two periodic terms being about one-
fiftieth part of the first, which is itself very small, is usually
neglected. Also p is very small compared with m, hence we have
- - Psin icosi cos ^ ,,rT^
— da cos MKC.
This terra expresses the Lunar Nutation in the obliquity.
In the same way by integrating the expression for ^, and
neglecting the very small terms, we have
I I D a f 2 • 1 • a A ^ n sm 2i C(
>Ir = llr — P cos ^ I cos" I — rx Slli't ]t — F —r — . -
^ " \ 2 / ziii s
sin 2i cos 2$
siu^
sm
MKC.
The angle MKC is the longitude of the moon's descending
node, and the line of nodes is known to complete a revolution
in about 18 years and 7 months. If we represent this period by
27r
T we have MKG= — „ / + constant.
The pole M of the lunar orbit moves round the point of re*
ference K with an angular velocity which is rapid compared with;?,
but yet is sufficiently small to make the Lunar Nutations greater
than the Solar. We may also notice that if M had moved round
K with an angular velocity more nearly equal to p the Nutations
would have been still larger. This may explain vhy a slow motion
of the ecliptic in space may produce some corresj ending nutations
of very long pciiod and of considerable magnitude.
R. D.
nf]
•h
■ 1
vl
n
mw
m!
ii
j
514
PBECESSION AND NUTATION.
Motion of the Moon about its centre of gravity.
625. In discussing the precession and nutation of the equinoxes, the earth has
been regarded as a rigid body two of whose principal moments at the centre of
gravity are equal to each other. One cons -r^aence of this supposition was that the
rotation about the axis of unequal moment s not directly altered by the attraction
of tho disturbing bodies. As an examplo of the ^ilect of these forces on the
rotation when all the three principal moments are unequal, we shall now consider
the case of the moon as disturbed by the attraction of the earth. As our object is
to examine the mode in which the forces alter the several motions of tho moon
about its centre of gravity rather than to obtain arithmetical results of the greatest
possible accuracy, we shall separate the problem into two. In the first place we
shall suppose the moon to describe an orbit which is very nearly circular in a plane
which is one of the principal planes at its centre of gravity. In the second case we
shall remove tho latter restriction and examine the effects of the obliqiuty of the
moon's orbit to the moon's equator.
626. The moon describes an orbit ahont *lu! centre of the earth which is very
nearly circular. Supposing the plane of the o bit to be one of the principal planes
of the moon at its centre of gravity, find the motion of the moon about its centre of
gravity.
Tict uA. GB, GC be the principal axes at G the centre of gravity of the moon,
and let GC be the axis pei'pendicular to the plane in which G moves. Let A, B, C
be the moments of inertia about GA, GB, GC respectively, and let M be the mass
of the moon, and let accented letters denote corresponding quantities for the
earth.
Let be the centre of the earth, and let Ox bo the initial line. Let OG=r,
GOx = 6. Let us suppose the moon turns round its axis GC in the same direction
that the centre of gravity describes its orbit about 0, and let the angle OGA = <p.
The mutual potential of the earth and moon is by Art. 607
v^^j^^M^.:±i±P^^M'^^:ji^,
r 2r^ 2r'
Here I=A cos'0 + i?sin'^ and therefore the moment of the forces tending to
turn tho moon round GC ifl
dV
d<t)
2-s(Z?-.l)8m20
(1).
MOTION OF THE MOON ABOUT ITS CENTRE OF GRAVITY. 515
! earth has
B centre of
as that the
attraction
les on the
iw consider
ir object ia
' the moon
he greatest
it place we
r in a plane
ond case we
iiity of the
hich is very
cipal planes
its centre of
f the moon,
Let^, B, C
be the mass
ties for the
Let OG=r,
me direction
es tending to
(1).
Since &+</> is the angle which GA, a line fixed hi the body, makes with Ox, a
line fixed in space, the equation of the motion of the moon roand GC is
d^9 dV sy'B-A
dt.^ ■*■ dt
2
sin 2<p.
.(2).
The motion of the centre of gravity of the moon referred to the centre of the
earth as a fixed point is found in the Lunar Theory. It is there dhowu that r and
may bo expressed in the form
r= c{ 1 + i cos (p* + o) + &c. f ,
, —n+pt\-Mncoa{pt + a) + Sce.,
where pt is a very smali term which represents a secular change in the moon's
angular velocity about the earth, and is really the first term of the expansion of a
trigonometrical expression.
If we substitute the value of — ir. equation (2) we have the following equation
to determine 0,
^ I = - 2 (Z' sin 2v'> - /J + npM sin {pt + a) + &c.
.(3),
3 B — A
where for the sake of brevity we have put n' ^ — -^
■ 2'
Now we know by observation that the moon always turns the same face towards
the earth, so that amongst the various motions which may result from different
initial conditions, the one which we wish to examine is characterized by ^ being
nearly constant. Let us then introduce into this equation the assiimp^-ion that
is nearly constant; we may then deduce from the integral how far this assumption
is compatible with any given initial conditions which we may suppose to have been
imposed on the moon. Tutting 0=0o + ^'> where ^^is supposed to contain all the
constant part of ^, we easily find
(4).
2<j«6in2^o=-/3
Ma.'
"; i + 3* ^'^^ 2^u^' = npM sin {ft + o) + &c. I
etc ^
The numerical value of q depends on the structure of the moon and can there-
fore only be found by comparing the results of tliis investigation or some otlier
results with observation. T)\e first of equations ^4) shows that 2/3 must be less
than g". But for various reasons, though 5 is very small, we must yet suppose that
- is also extremely small. Assuming this, we see that (^„ must also be very small.
It follows also that we may write 20„ for sin 2(/>o and unity for cos 2^,, in these equa-
tions. Solving the second equation, we find,
0=7/ sin {qt + A') - ^^ + 3/ ^"^^ sin (pt + a) + &o.
.(5),
where II and K are two arbitrary constants whose values depend on the initial con-
ditions. The angular velocity of the moon about its axis is therefore given by the
formula
d9 d(f>
dt "*■ (it
M7"
= n-\-pt + IIqBm(qt + R) + M ^"'^^jnin{pt + a) + &o (6).
S3— 2
II
l::;1 I
*:|
^' ''1 .
w
ml
int
, II
516
PRECESSION AND NUTATION.
If 5' were negative or zero, the character of the sohition of (3) would be altered.
In th 3 former case the expression for tf» would contain real exponentials. If the
initial conditions were so nicely adjusted that the coefficient of the term containing
the positive exponent were zero, the value of would still be always small. But
this motion would be unstable, the smallest disturbance would alter the values of
the arbitrary constants and then ^ would become large. If we also examine the
solution when q'=0, we easily see that ip could not remain small. We therefore
infer that of the axes 6A, oJ of the moon, the axis of least moment is turned
towards the earth and that these two principal momeutb are not equal.
In order that the expression ^5) for ip may represent the actual motion it if.
necessary and sufficient that H when found from the initial conditions should
je small. Wo see, by differentiation, that £fq is of the same order of small
quantities as ^. Hence B will be small if at any instant the angular velocity,
viz. TT + -17 , of the moon about GO were so nearly equal to the angular velocity,
(it itt
do
viz. — , of its centre of gravity round the earth, that the rt.tio of the difference to
q is very small.
If therefore we suppose the moon at any instant to be moving with its axis of
least moment pointed towards the earth and its angular velocity about its axis of
rotation to be nearly equal to that of the moon round the earth, then the axis of
least moment will continue always to point very nearly to the earth. The mean
angular velocity of the moon about its axis will immediately become equal to that
of the moon about the earth and will partake of all its secular changes. This is
Laplace's theorem. It shows that the present state of motion of the moon is
stable, rather than explains how the angular velocity about the axis came to be so
nearly equal to the angular velocity about the earth.
627. By comparing the value of the angular velocity of the moon about its
axis obtained by theory with the results of observation, wo may hope to obtain
some indications of the value of q^ and thence of
V-A
C
. If the term Ilq sin {qt + K)
B- A
could be detected by observation, we should deduce the value of — ^— .*fom its
period.
Among the other terms of the expression for the angular velocity of the moon
about its axis, those will be beat suited to discover the value of q which have the
largest coefficients, that is ihose in which either the numerator M is the greatest
or the denominator 2' -p"^ the least possible. By examining the numerical value of
B- A
their coefficients Laplace has shown that if — ^ were as great as "03 the elliptic
inequality could be recognized by observation, and if it were between •0011 and -003
the annual equation could be observed.
628. We may also calculate by the help of Art. 326 the radial and transverse
forces which act on the centre of gravity of the moon due to the mutual
attractions of the earth and moon. Since the principal moments of the moon
are nearly equal and its linear size small compared with its distance from
the earth, these forces are very nearly the same as if the moon were collected
MOTION OF THE MOON ABOUT ITS CENTRE OF GRAVITY. 517
be altered,
lis. If the
containing
mall. But
e values of
xamino the
e therefore
t is turned
action it if,
ins should
ir of small
lar velocity,
lar velocity,
iifferenoe to
;h its axis of
it its axis of
the axis of
The mean
c[ual to that
res. This is
the moon is
ime to be so
on about its
pe to obtain
^2 sin (3* + A')
-A
from its
of tho moon
lich have the
the greatest
■rical value of
)3 the elliptic
3014 and -003
nd transverse
the mutual
of the moon
listance from
were v^oUccted
into its centre of gravity. The effect of the small forces neglected by
this assumption will be insignificant compared with the other forces which act ou
the centre of gravity of the moon. The motion of the centre of gravity of the
moon is therefore very nearly the same as if the Vviiole mass were collected into its
centre of gravity.
Since however there are no other forces which have a moment round GC besides
those found above, the effect of these may be perceptible. The effects of tidal
friction on the rotation of the moon may be omitted, at least at the present time.
Ex. The centre of gravity (? of a rigid body describes an orbit which is
nearly circular about a very distant fixed centre of force attracting according
to the Newtonian law and situated in one of the principal planes through 0. If
r=c(l + p), 6=nt + v^ ba the polar co-ordinates of G referred to 0, show that the
equations of motion are
3
'2''
3«V - Sh" '^ = - j «'7' - I n'7 cos 2^ '
dt
2 T + "-iiTf = 5 "> sill 2^
d«0 dV
where 7 =
7 =
2C-
A-B
+ ji
dt^
= - ^ sin 20
•6Mc^
We may notice that the values of 7 and 7' are much smaller than 3' and might
therefore be rejected in a first approximation.
If the body always turns the same face to the centre of force so that <f> is
nearly constant and in small, show that there will bo two small inequalities in tho
value of of the form 1 sin {pt + a), whore p ia given by
(i>2 - n") (p"^ - 2") - 3/i'7 (p" + 3?i=) = 0,
one of these periods being nearly the same as that of the body round the centre
of force and the other being very long.
If the body turns very nearly uniformly round its axis GC, so that = n't + e'
nearly, show that there will be two small inequahties in the value of <p, one iu
which 2>=w aud another in which p = 2n'.
629. E". 1. Show that the moon always turns the same face very nearly to that
focus of her orbit in which the earth is not situated. [Smith's Prize.]
Ex. 'z. If the centre of gravity G of the moon were constrained to describe a
circle with a uniform angular velocity n about a fixed centre of force attracting
according to the Newtonian law ; show that the axis QA of the moon will oscillate
on each side of GO or will make complete revolutions relatively to GO according
as the angular velocity of the moon about its axis at the moment when GA and GO
coincide in direction is less or /jreater than n + q. Find also the exteii' of the
oscillations.
Ex. 3. A particle m moves without pressure along a smooth circular wire of
mass M with uniform velocity under the action of a central force bituated in the
centre of the wire attracting according to the law of nature. Show that this system
of motion is stable if ,, > r^^-- . The disturbance is supposed to be given
M 2o
to tho particle or tho wire, the contre of force remaining fixed in space.
i
I!
ilif
5 .<;
■
f
•ill
T
lli
i !
518
PRECESSION AND NUTATION.
Ex. 4. A uniform ring of mass M and of very small Bection is loaded with a
heavy particle of mass m at a point on its circumference, and the whole is in
uniform motion about a centre of force attracting according to the law of nature.
m
Show that the motion cannot be stable unless ,, lies between
M + 111
•8279.
•815865 and
This example shows (1) that if a ring, such as Saturn's ring, be in motion
about a centre of force, its position cannot be stable, if the ring be uniform ; and
(2) that if, to render the motion stable, the ring be weighted, a most delicate
adjustment of weights is necessary. A very small change in the distribution of
the weights would change a stable combination to one that is unstable. This
example is taken from Prof. Maxwell's Eatay on Saturn's Sings.
Ex. 6. The centre of gi-avity of a body of mass 3f , sjmmetrical about the plane of
xy, is ', and is a point such that the resultant attraction of the body on is
along the line GO. Then if the body be placed with coinciding with a fixed
centre of force S, and be set in rotation about an axis through perpendicular to
the plane of xy with an angular velocity w, G will, if undisturbed, revolve uniformly
in a circle, always tiurning the same face towards O, provided Mau^ is equal to the
resultant attraction along GO, where a is the distance GO. It is required to
determine the conditions that this motion should be stable.
The motion being disturbed, will no longer coincide with the centre of force
S. Let two straight lines at right angles revolving uniformly round fi^ as origin
with an angular velocity u be chosen as co-ordinate axes, and let x be initially
parallel to OG. Let {x, y) be the co-ordinates of 0, ^ the angle OG makes with
the axis of x, then x, y, <f> are all small. Let V be the potential of the body at 0,
cPV dT ^
™„. ^' dy^=^'
of force.
and let dj^^"
,«-,-. Let S be the amount of matter in the centre
dxdy ay*
Then the equations of motion of G, Art. 179, will reduce to
and the equation of angular momentum about S will lead to
2uax+aj^y+ (a^+k^) j^<t>=0,
where k is the radius of gyration of the body about 0. Combining these equations
as a determinant and reducing we &id that the differential equation in |, rj, or (ft
is of the form
The condition of stability is that the roots of this equation should be real and
negative. Hence A, B, O must be of the same sign and B'^>'iAC. This pro-
position is due to Sir W. Thomson and is given in Prof, Maxwell's Essay on Saturn's
Rings.
630. The motion of a rigid body about a distant centre of force has been
investigated on the supposition that the motion takes place entirely in one
plane. We see by equation (2) of Art. 62C that the case in which the centre
II
ded with a
rhole is in
of nature.
115865 and
in motion
form; and
)st delicate
ribution of
ible. This
;he plane of
idy on ia
ith a fixed
indicular to
3 uniformly
squal to the
required to
ire of force
5 as oriRin
be initially
nakes with
body at 0,
1 the centre
36 equations
in $, i;, or <t>
be real and
This pro-
/ on Saturn's
ae has been
xely in one
1 the centre
MOTION OP THE MOON ABOUT ITS OfNTRE OF GBAVITY. 619
of gravity describes a circular orbit, and the rigid body always turns the axis
of least moment towards the centre of force, is one of steady motion. The
preceding investigation also shows that this motion is stable for all disturbances
which do not alter the plane of motion. It remains now to determine the effect of
these disturbances in the more general case when the motion takes place in three
dimensions.
Tho whole attraction of the centre of force on the body is equivalent to a single
force acting at the centre of gravity, and a couple. If the size of the body be small
compared with its distance from the centre of force we may neglect the effect of the
motion of the body about its centre of gravity in modifying the resultant force.
The motion of the centre of gravity will then be the same as if the whole were
collected into a single particle. The problem is therefore reduced to the following.
A rigid body turns about its centre of gravity 0, and is acted on by a centre of
force E which moves in a given manner. In the case in which the rigid body is
the moon, this centre of force, i.e. the earth, moves in a nearly circular orbit in a
plane which itself also has a slow motion in space. This motion is such that a
normal GM to the instantaneous orbit describes a cone of small angle about a
normal OK to the ecliptic. The two normals maintain a nearly constant Jn-
clination of about 5". 8'; and the motion of the normal to the instantaneous orbit is
nearly uniform.
631. It will clearly be convenient to refer the motion to axes OX, OY, GZ
fixed in space such that OZ is normal to tho ecliptic. Let GA, GB, GC be the
principal axes of the moon at the centre of gravity G. Let (p, q, r) be the direction-
cosines of OZ referred to the co-ordinate axes GA, GB, GC. Then we have, since
GZ is fixed in space,
do ft
.(I).
■^ - Ujr+ wgp =
df
^-WjlJ + Wi^^O
Now our object is to find the small oscillations about the state of steady motion
in which OZ, GC, GM all coincide. We shall therefore havep, q, Wj, Wj all small,
and r very nearly equal to unity. The equations (I) will therefore become
dp
dq
dt
-Wi + ?ip =
where n is the moan value of u^.
Let X, ft, V be the direction-cosines of the centre of force E as seen from Q.
Then we have by Euler's equations and Art. 608,
dt
-(B-C) wa«a= -3n''>(/?- C7)/«i»
C-^^-{A-B) wi«j= - 8rt'»(4 - B}\n
(11).
11 i
tit
i
520
PIIECESSION AND NUTATIOf,
Id the case of stead/ motion, the rigid body ulways turns the axis {GA) ot lenst
momeut towards the centre of force, and w^=n'. We liave then both fi and i> small
quantities, uo that in the first equation we may neglect their product /uf, and in
the second equation we may put v\=v. Also, we may pat W3=Tt=n' in the small
terms.
If I be the latitude of the eoi-th as seen from the moon, we have
Bin l=coa ZE=p\+ qn + rv~p + v nea,T\y.
Hence the two first of Eoler's equations f^ake the form
dt
(C-A)nwi= -3n^{C-A)(-p+Bml)
.(III).
If the earth, as seen from the moon, be supposed to move in a circular orbit in
a plane making a constant inclination tan~^ k with the ecliptic, and the longitude
of whose node is -gt + /3, we shall have
Bin I =kBia (n't + gt - p).
In this expression g measures the rate at wliich the node regredes, and Is abont
the two hundred and fiftieth part of n. We shall therefore regard - as a small
n
quantity.
To solve these equations, it will be found convcnieDt to substitute for Wj, w,
their values in terms of p, q. We then have
d*q
.^P
A-j^l+iA + B-OnJ^-n^B-Oq^O
^%-i^+^-<^)n^f+^n^{C-A)p = Sn^{C-A)sml
1) of lenst
nd V small
fiv, and in
the small
.(III).
liar orbit in
B longitude
md is about
as a small
ite for Ui, Wj
MOTION OF THE MOON ABOUT ITS CENTRE OF GRAVITY. 521
To find p, q, let us put
p = P sin {{n'+g)t-p], q-=Qcoa {{n' +g)t-p},
where P, Q are some constants to be determined by substHution in the equation.
We have
Q{A{n+g)'\- (B - On'] =P(A + B-C)n{,Hg) )
P{£(n+g)'-i(C-A)n'\-Q{A + B-C)n{n+g)=-3n*k{C-A)S'
We may solve these equations to Gud P and Q accurately. In the case of the
moon the ratios — —— , — - — , — -— and - are all small. If then we neglect the
C A H 71 °
products of these small quantities, the first equation gives us p=l- ^. The
second equation will then give
n
P=
Snh{C-A)
Sn{V-A)-2Dg'
As g is very small compared with n, we may regard P and Q as equal.
632. The complementary functions may be found in the usual manner by
assuming
p=FBm(8t + II), q = Q COB (st+ IT),
on substituting we have the quadratic
AB8*-{{A+P-C)'-B{B-C)-U{A~C)]nU'' + i{A-q{B-C)n*=0,
to find «>, and
G _ (A + B-C)n8
to find the ratio of the coefficients of corresponding terms in p and q. If the roots
of this equation were negative p and q would be represented by exponential values
of t, and thus they would in time cease to be small. It is therefore necessary for
stability that the coefficient of a' should be negative and the product (A - C)(£ - C)
positive. Both these conditions are probably satisfied in the case of the moon.
For since B-C and A-C are both small, the term (A + B- C)' is much greater
than the two other terms in the coefficient of s^. Also, since the moon is flattened
at its poles, we shall probably have A and B both less than C.
633. Let 31 be the pole of the moon's orbit, which is the same as that of the
earth's orbit as seen from the centre of the moon. Then M is the pole of the
dotted line in the figure of Art. 631. Therefore the angle EZM measured by
turning ZE in the positive direction round Z until it comes into coincidence with
Z}f,iB= -^-{{n + g)t~p]. Again, if the angle J?ZC be measured in the same
direction, we have
cos EC -COB CZ cos ZE _ v-r{ p\ + qn+r¥)
sin CZaiixZE ~ J^T^emZE
Hence we easily find BxaEZC= f iT~^'
But BinCZM^sin EZM ooB EZV- cos EZM Bin EZC
_cos{{ii + g)t ■ p\p-Bin{ (n+g)t- p\q
cos EZC=
-P
v'p'+a'
, nearly.
^^M
m
^: fi
,;:.;
522
PRECESSION AND NUTATION.
.1
Ml;
If now wo substitute for p and q their valnen, it is clear that the terras in p and
q, whose argument is n + g, disapijear. So that if F and were zero, the Hiue of
the angle VZM would be absolutely zero. In this case the tlirce polos C, Z, M
must lie in an arc of a great circle, or, which is the Hame thing, the moon't equator,
the moon's orbit, and the ecliptic inunt cut each other in the same line of node$.
It however F and G bo not zero, but only very small, wo have
SF'sin (»'<+//')
Bin CZM=
^//'=' + 20'" Bin («'« + //')'
where F, G' contain either i!" or as a factor, and are therefore small. If then F
and O be both small compared with P, llio angle VZM will remain either alwuyu
small or always nearly equal to tt.
The intersection of the moon's equator with the ecliptic will then oRcillato about
the intersection of the moon's orbit with the ecliptic as its mean position. Since
these oscillations are inHcnsible, it follows that in the case of nature, the com-
plementary functions must be extremely small compared with the terms depending
directly on the disturbing force.
634. If we disregard the complcmontary functions we have p = P sirup,
q=P coa 4>, where = (n' + <;) t - /3. Now Hin' CZ ssp" + q''; therefore CZ= -P very
nearly. The value of CZ, the inclination of the lunar equator to the ecliptic, is
known to be about 1"'.28'. Hence, since - =-004, we may deduce from the ex-
n
C-A
pression for P at the end of Art. 631 an approximation to the value of - „
C — A
In tlxia manner Laplace finds —jr- = -000599.
it' '■.'
IS in p and
bo Hiue of
m a, Z, M
i's equator,
noilc$.
If then F
her alwttya
illato about
ion. Since
3, tbe com-
s depending
jj=P8in^,
:=-P very
e ecliptic, is
rom tbo ex-
, C-A
ae of - D .
CHAPTER XII.
MOTION OF A STUING OR CHAIN.
The Equations of Motion.
635. Prop. To determine the generil ejuations of motion
of a stnng under the action of any forces.
First. Let the string he inextensihle.
Let Ox, Oji, Oz be any axes fixed ia space. Let Xmds,
Ymds, Zmds bo the impressed forces that act on any element
ds of the string whose mass is mds. Let u, v, to be the resolved
parts of the velocities of tliis element parallel to the axes. Then,
by D'Alembert's principle, the element ds of the string is iu
equilibrium under the action of the forces
»"* (^ - ^) ' "^^^ (^-f ) ' ^^* (^- ft) •
and the tensions at its two ends.
,da}
Let T be the tension at the point (a?, y, z), then T-..- , T
dy
,dz
ds' ^ ds'
T^ are its resolved parts parallel to the axes. The resolved parts
of the tensions at the other end of the element will be
j,dx^d
ds
and two similar quantities with y and z written for x.
Hence the equations of motion are
du
di
dv
m
m
m
dt ds
dw
dt
t) + "^^
.(1).
y
m
ill
I t
■■^ I ;
5
)
i
f ( iii
I
W
i 'i
524 MOTION OF A STKING.
In these equations the variables s and t are independent. For
any the same element of the string, s is always constant, and its
path is traced out by variation of t. On the other hand, the
curve in which the string hangs at any proposed time is given by
variations of s, t being constant. In this investigation s is
measured from any arbitrary point, fixed in the string, to the
element under consideration.
To find the geometrical equations. We have
(^)'-(i)'-(iy--
Differentiating this with respect to t, we get
dx du dy dv dz dw „ ,c>\
1 ><. — I = (2).
ds ds ds ds ds ds
The equations (1) and (2) are sufficient to determine w, y, z,
and T, in terms of s and t
Ex. If V be the Vis Viva of any arc AB ol the chain ; T^, T^ the tensions at
the extremities of this arc ; »/, tf,' the velocities of the extremities resolved along
the tangents at those extremities, prove that
Y-^= T^Ui - ^iWi' + f(Xu +Yv + Zio) mds,
the integration extending oyer the whole arc.
636. The equations of motion may be put under another
form. Let <^, -v/r, ^ be the angles made by the tangent at x, y, z,
with the axes of co-ordinates. Then the equations (1) become
^f = |,(2^cos<^) + m.Y (3),
with similar equations for v and w.
dx
To find the geometrical equations, differentiate cos = -^ with
respect to t ;
, ddi du ...
'''-'"'''^dt^ds- (^)-
Similarly, by differentiating cos "^ = ^ - and cos ;j^ = ^- , we
get two other similar equations for -^jt and ;^. Taking these six
equations in conjunction with the following
cos" + cos''i^+ cos';j^ = l (5),
we have seven equations to determine u, v, w, ^, >/r, -^ and T.
Wll
THE EQUATIONS OF MOTION.
525
nt. For
t, and its
land, the
given by
;ion s is
<y, to the
.(2).
le X, y, z,
3 tensions at
solved aloug
r another
i at X, y, z,
)ecoine
.(3),
I = -^ with
(4).
dz
Iff these six
ind T.
•(5),
If the motion takes place in one plane, these reduce to the
four following equations :
m
du d
= -J- (T cos ^) + mX
dt ds
dv
dt ds
»^:l = ;77(^s^^*/')+*'*^
(fi).
— sin^
d(f>
dt
du^
ds
(7).
. dd> dv
''''^Tt=ds
The arbitrary constants and functions which enter into the
solutions of these equations must be determined from the peculiar
circumstances of each problem.
637. Secondly. Let the string be elastic.
Let ff be the unstretched length of the arc s, and let tndff be the mass of an
element da of unstretched length or ds of stretched length. Then by the same
reasoning as before, the equations of motion become
m
du d /^dz\ ,,
i[t=d.Vdir'^'^ (')•
and two similar equations for v and lo. To find the geometrical equations we must
differentiate
the independent variables being now <r and t. Differentiating with regard to t we
have
dx du dv dv dz dw
d(7 dff d<s da da da
_ ds d fds \
da dt \da) '
But if \ be the modulus of elasticity of the string, we have
— = 1 ^
da \ '
(ii).
Substituting we have
dx du dy dv dz dw
da da da da da da
<^4)l
dT^
\ dt '
.(iii).
The two equations (ii) and (iii) together with the three equations (i) will suffice
for the determination of u, v, w, s and T in terms of a and t.
If we wish to use the equations of motion in the forms corresponding to (3) or
(C), the dynamical equations become
VI
du _ d
dt " da
with similar equations for v and w.
(7'cos</)) + niJ,
:^
I -
I
[
i:
i
i
j;
i
n
t
?li
■It-
m •',
fit
626
MOTION OF A STRING.
The geometrical eqaations corresponding to (4) or (7) may be found tlius. We
have
d.='''"f'd?="'''f[^'-\)'
Differentiating, we have
du . dd) 1 d ,- .
with similar expressions for v and w.
G38. When the motion of the string takes place in one plane,
is often convenient to
and normal to the curve.
it is often convenient to resolve the velocities along the tangent
Let u', V be the resolved parts of the velocity of the element
ds along the tangent and normal to the curve at that element.
Let (j) be the angle the tangent to the element ds makes with
the axis of x. Then by Art. 179 or 252, the equations of motion
are
du' .d^_y,dT-
dt dt mds
dv , d(f> _
'dt'^'' di-
T
mp
.(1).
The geometrical equations may bo obtained as follows. We
have
u = u cos <f) — r ' sin <f>.
Differentiating with respect to s, we have by Art. 636,
d<f>
- dt ^^"
dv u'\
, (du v\ , fdv ii\ . ,
ds
where p is the radius of curvature, and is equal to -v^ . Since
the axis of x is arbitrary in position, take it so that the tangent
(luring its motion is parallel to it at tlie instant under considera-
tion ; then ^ = and we have
=
du v
(2).
ds p
Similarly, by taking the axis of x parallel to the normal,
d^ _ dv iL
dt ds p
These four equations arc suIBcient to determine u', v, (j> and
T in terms of s and t.
•(3).
THE EQUATIONS OF MOTION.
627
thus. We
If the string he extonsihle, the dynamical equations become
me plane,
e tangent
> element
element.
ikes with
of motion
(1).
3WS. We
6,
- . Since
tangent
consitiera-
,...(2).
....(3).
', V, ^ and
dt dt mdff
at at mp da
To find the geometrical equations, we may differentiate
M=u'co8 ^-ti'sin <f>
with regard to a. This gives by Art. 637
. ^d4>
-^^(7'cos^)=(^^^----Jcos^-(^- + --JBm0.
rt. C38, this reduces
_ du' v' / T\
- dcr pV X/'
By the same reasoning as in Art. 638, this reduces to
IdT
X dt
dj>
dt
('-D-S'-K'-D
639. The equations (2) and (3) may also be obtained in the
following manner. The motion of the point P of the string being
represented by velocities w' and v' along the tangent PA and
normal PO at P, the motion of a consecutive point Q will be
represented by velocities u' + du and v' + dv' along the tangent
QB, and normal QO at Q, Let the arc PQ = ds, and let ^i^be
a perpendicular on PA. Since the string is inextensible, the
resultant velocity of Q resolved along the tangent at P must be
ultimately the same as the resolved part of the velocity of P in
the same direction. Hence
(u + du') cos d^ — (v + dv) sin dj> = u,
or, proceeding to the limit,
nil
du — vd(^ = ; .*. -v-
^- = 0.
<70
Again, .^ is the angular velocity of PQ round P.
Hence
the difference of the velocities of P and Q resolved in any direc-
tion which is ultimately perpendicular to PQ must be equal to
.'. (u + du) sin d(f> + (v + dv) cos d(f> — v'= ds -^ ,
or in the limit
dj) _ dv u'
dt ds p '
640. Ex. 1. An elastic ring without weight, wlioso length when unstretched is
given, is stretched round a circular cylinder. The cylinder is suddenly annihilated,
m
i I !
h
628
MOTION OF A STRING.
show that ' the time which the ring will take to collapse to its natiiral length ia
/ Mav
the natural radius.
- , where M is the mass of the string, X its modulus of elasticity, and a is
Ex. 2. A homogeneous light inextensible string is attached at its extremities
to two fixed points, and turns about the straight line joining those points with uni-
form angular velocity. Find the form of the string, supposing its figure per-
manent.
Hcsult. Let the straight line joining the fixed points bo the axis of x, then the
form of the string is a plane curve whoso equation is 1 + ( -^ j = ( ' J , where a
and h are two constants.
On Steady Motion.
04)1. Def. When the motion of a string is such that the
curve which it forms in space is always equal, similar, and siirii-
larly situated to that which it formed in its initial position, that
motion may be called steady.
642. Pkop. To investigate the steady motion of an inexten-
sible string.
It is obvious that every element of the string is animated with
two velocities, one due to the motion of the curve in space, and
the other to the motion of the string along the curve which it
forms in space. Let a and h be the resolved parts along the axes
of the velocity of the curve at the time t, and let c be the velocity
of the string along its curve.
Then, following the usual notation, we have
w = a -f c cos 0|
v= 6 +csin0j
dtL
(1).
d<b
Now a, h, c are functions of t only, hence -,- = — c sin ^ .
Therefore by equation (7) of Art. C30 we have
(2).
d<f> d<f)
dt ' ds
Substituting the values of u and v in the equations of motion,
Alt. G35, we get
da do , . , d<b
^-f^^cos0-csm<^-J
db dc .
Tt-^dt'"''
.. d [T
A -}- -, - COS <A
ds \in ^
A -f- c COS (A ; = r -f -7- ( - sin 6 )
^ dt as \m ^ ) i
ON STEADY MOTION.
529
length is
r, and a is
xtromitioB
1 with uni-
igure per-
5, then the
) .where a
that the
md simi-
tion, that
I inexten-
ateJ with
lace, and
which it
the axes
e velocity
..(I)-
. ,dib
(2).
of motion,
,(3).
Substituting for ~ , these equations reduce to
The form of the curve is to bt independent of t; hence, oi:
eliminating T, the resulting equation must not contain t. This
will not generally be the case unless t- , -r. i -ji are all con-
stants. In any case their values will be determined by the known
circumstances of the Problem. The above equations must then
be solved, s being supposed to be the only independent variable,
and t being constant.
643. If the string move uniforn^Iy in space, and the elements
of the string glide uniformly along the string, -77 = 0, tv— 0,
dc
dt
dt
-Tf = ^' It will then follow from the above equations, that the
form of the string will be the same as if it was at rest, but the
tension will exceed the stationary tension by mc^.
644. Ex. 1. Let an electric cable he deposited at the bottom of a sea of uniform
depth from a ship moving icith uniform velocity in a straight line, and let the cable
be delivered with a velocity c equal to that of the ship. Find the equation to the
curve in which the string hangs.
The motion may be considered steady, and the form of the curve of the string
will be always the same.
If the friction of the water on the string be neglected, gravity diminished by the
buoyancy of the water will be the only force acting on the string, let this be repre-
sented by g'. Hence the form of the travelling curve will be the common catenary,
and the tension at any point will exceed the tension in the catenary by the weight
of a length of string equal t - -, .
Next let the friction of the water on any element of the cable be supposed to
vary as the velocity of the element, and to act in a direction opposite to the direc-
tion of motion of the element*. Let fi be the coefficient of friction.
Lot the axis of x be horizontal, and let x' be the abscissa of any point of the
cable measured from the place where the cable touches the ground, in the direction
* Each element of the string has a motion both along the cable and trans-
versely to it. The coefficients of these frictions are probably not the same, but
they have been taken equal in the above investigation.
R. n. 84
m
^i(
530
MOTION OF A 8TBIN0.
s
l!
!',!
Ih
of the ship's motion. Also let a' be the length of the curve measured from the same
point. Then x=x' + ct, aai 8=8' + ct.
Following the same notation as before, we have
X=-nu, Y= -g'-(iv.
But u=c-ccos0, v=-csin^.
Hence the equations (3) )jecome
0= -/xc+fju: COB ip + -r \( — c' jcos^j
0= - Ijr' + /*« sin + J- I ( — c' J sin ^
Hence,
g'A= ~fict+ncx+( — c" J cos^
To integrate these put ein </>=■—-, cos 0= t-
+ g-c«)8in0J
■a).
g'B= -g'8 + ficy
where A and B are two arbitrary constants.
At the point where the cable meets the ground, we must have either T=0 or
^=0. For if be not zero, the tangents at the extremities of an infinitely smaU
portion of the string make a finite angle with each other. Then, if T be not zero,
resolving the tensions at the two ends in any itircction, we have an infinitely small
mass acted on by a finite force. Hence the element will in that case niter its posi-
tion with an infinite velocity. First, let us suppose that ^=0. Also at the same
point, y=0 and »' = 0. Hence B= -ct.
IXC
Putting S-=«i we get by division
dy
dxf'
ey
A-cuf+ea'
This is the differential equation to the curve in which the cable hangs.
To solve this equation*, let us find a' in terms of the other quantities,
A^,-e^%-^eu
dx dx
.(2).
8 =
Differentiating, we have
s/^^m-
^^^.(A-cx'^ehj)
0-'^)'
f.
* The problem of the mechanical conditions of the deposit of a submarine cable
lias been ronsidered by the Astronomer Royal in the Phil. Mag, July 1858. His
solution is different from that given above, but his method of integrating the differ-
ential equation (2) has been follo\Yed.
tl
c
HI
Ki-
ln
mm
■^i^
ON STEADY MOTION.
531
Put p for y '.rbere convenient, and put v for A -ex'+e^i/; the equation then
becomes . .
dp
1 dv
-c
dx'
vdm^ {l-ep),Jl+p*
in which the variables are separated, and the integrations can be effected. The
equation can be integrated a second time, but the result is very long. The arbitrary
constant A may have any value, depending on the length of the cable hanging from
the ship at the time «=0.
The curve in its lower part resembles a circular arc or the lower part of a com-
mon catenary. But in its upper part the curve does not tend to become vertical,
but tends to approach an asymptote making an angle cot~ie with the horizon. The
as3'mptote does not pass through the point where the cable touches the ground but
A
below it, its smallest distance being — ; ^=^ ; the asymptote also passes below the
ship.
If the conditions of the question be such that the tension at the lowest point of
the cable is equal to nothing, the tangent to the curve at that point will not neces-
sarily be horizontal. Let \ be the angle this tangent makes with the horizon,
Beferring to equations (1) we have simultaneously
Hence
=0, 2/=0, «'=0, r=0, and ^-X.
A= --. cos X, i/= - , sin X - ct.
9 9
Tlie diJTerential equation to the curve will now become
11
dy
dx''
— ; sin X + s' - ey
(I
■■■^^"" *
;COSX + e«'-f»'
9
.(3),
which can be integrated in the same manner as before. One case deserves notice;
viz. when e=cotX. The equation is then evidently satisfied by y=-x'. The two
constants in the integral of (3) are to be determined by the condition that when
a;'=0, y = 0, then -y^,=tanX. Both these conditions are satisfied by the relation
y=-a;'. Hence this is the required integral. The form of the cable is therefore a
straight line, inclined to the horizon at an angle X=cot~^£; and the tension may be
found from the formula 7= , — ^"Jl— .
1 + cos X
Ex. 2. Let a cable be delivered with velocity c' from a ship moving with uni-
form velocity c in a straight line on the surface of a sea of uniform depth. It the
resistance of the water to the cable be proportional to the square of the velocity,
the coelTicient B, of resistance for longitudinal motion being different from the
coefficient A , for lateral motion, prove that the cable may take the form of a
straight line making an angle X with the horizon, such that coi^\= sJ<i* + \-h
where c is the ratio of the speed of the ship to the terminal velocity of a length of
34—2
1 1
St
532
MOTION OF A STRING,
cable falling laterally in water. Prove also that the tension will be found from the
equation
r = jj,-f e. (^-cosx)'^-^jm/. [PhU. Mag.]
ir
On Initial Motions.
645. A string, under the action of any forces in one plane,
begins to move from a state of rest in the form of any given cui've.
To find the initial tension at any given point.
Let mPds, mQds be the resolved parts of the forces respectively
along the tang nt anr* ^ ormal to any element ds. The force P is
taken positive! v> it acts in the direction in which s is mea-
sured, and Q L . uritiv" when it acts in the direction in which
p is measured ak .< ^ lie rormal, viz. inwards. Let m be the mass
of a unit of length.
Let u, V be the velocities of the element along the tangent
and normal. Then the equations of motion are by Art. 638
du d(b -n 1 dT ,_.
-jT-W j7 =P + --7- (1),
dt dt m da ^ "
|+„#=«+i?: (2),
dt dt m p ^ '
where T is the tension, p the radius of curvature, and (fy the angle
the tangent makes with any fixed straight line. The geometrical
equations are
l-r" f^)' ^p'4! <*'•
Differentiating (1) and multiplying (2) by - , we get
r
d'u dJ'i^ dvd<f>^dP 1 ffr-i
dsdt dsdt dsdt ds m ds* \ ,^>
Idv^^nd^^Q^ll^ I ^''^'
p dt p dt p m p* i
But by differentiating (3) we have, since - — -j,
d^u ^ d^<}> 1 ^» ^ A /px
dsdt dsdt p dt A^'^"
Hence, subtracting the second of equations (5) from the first,
we have by (4) and (6)
m \ds p*/ ds p \dt J '
ON INITIAL MOTIONS.
633
rom the
plane,
n curve.
ectively
■ce P is
is mea-
n which
he mass
tangent
^8
...(1),
...(2),
he angle
Dmetrical
....(4).
(•^).
....(0).
the first,
In the beginning of the motion just after the string has been
cut we may reject the squares of small quantities, hence (-^)
may be rejected. Hence we have
d'T T dP Q
— 3= — m-T-+m —
ds'
ds
(7).
p as p
This is the general equation to determine the tension of a
string just after it has been cut.
The two arbitrary constants introduced in the solution of this
equation are to be determined by the circumstances of the case.
If both ends of the string are free, we must have T= at both
ends.
Since the string begins to move from a state of rest we have
flit fll)
initially u = 0, v = 0. At the end of a time dt, -,, dt and -^ i/t
will be the velocities of any element of the string. Hence if yfr
be the angle the initial direction of motion of anj el .or* of the
string makes with the tangent to the element, we hn e : equa-
tions (1) and (2)
1 T
tan'^ =
m p
m ds
(8;.
It must be remembered that the constants of integration are
necessarily constant only throughout the length of the string at
the time ^ = 0. They may be functions of t and may be either
continuous or discontinuous. For example, if a point of the string
be absolutely fixed in space, the transverse action of the fixed
point on the string may cause the constants to become discon-
tinuous at that point. In this case equation (8) is not necessarily
true in the immediate neighbourhood of the fixed point.
646. If the string be heterogeneous we may easily show in
the same way, that the initial tension is given by
±(\dT\_\T ^_dP Q
ds\mdsj m p^ ds p '
647. A string is in equilibrium, under the action of forces
in one plane. Supposing the string to he cut at any given point,
find the instantaneous change of tensic n.
Let Tq be the tension at any point (ic, y) just before the
string was cut. Then the forces P, Q satisfy the equations of
equilibrium
= P+
m ds
0=^ +
ir„
I
?
m
VI
534
MOTION OP A STIIING.
Hence
ds p m ds* m p*
If T'bo the instantaneous change of tension, wo have T'=T-T^,
The equation of the last article therefore becomes
da' p' ~
'111
648. Ex. 1. A strin(j is in equilihrlum in the form of a circle about a centre of
repulsive force in the centre. If the string be now cut at any point A, prove tliat the
tension at any point P is instantaneously changed in the ratio of
o'^ + e"''
where is the angle subtended at the centre by the arc AP.
Let Fho the central force, then P=0, and mQ= - F. Let a bo the radius of tbe
circle. Then the equation of Art. G45 to determine T becomea
ds^ n'^ a *
Let 8 be measured from the point A towards P, then s-=ad\ also F is independ-
ent of s. Hence we have
T=FaJrA€^ + Be-^.
To determine the arbitrary constants A and B we have the condition r=0 when
^=Oand^=27r;
T=^Fa.{l-
e^ + e"
But just before the string was cut T-Fa.
tiou follows.
Hence the result given in the onunoia-
Ex. 2. A string is wound round the under part of a vertical circle and is just
supported in equilibrium at the ends of a horizontal diameter by two fore a. The
circle being suddenly removed, prove that the tension at the lowest point is
instantly decreased in the ratio 4 ;e^+e ^.
Ex. 3. The extreme links of a uuiform chain can slide freely on two given
curves in a vertical plane, and the whole is in equilibrium under the action of
gravity. Supposing the chain to break at any point, prove that the initial tension
at any point is r=y (A(jy+B), where y ia the altitude of the point above the direc-
trix of the catenary, fj> the angle the tangent makes with the horizon, and A, B two
arbitrary constants. Explain how the constants are to be determined,
Ex. 4. A string rests on a smooth table in the form of an arc of an equiangular
spiral and begins to move from rest under the action of a central force F which
tends from the pole and varies as the 7i"' power of the distance, show that the initial
tension is given by
It coB^ a + sin'' a
T=-rF —
« (H + l)cos*a-sin'a
'rArP+ Hi'',
ON INITIAL MOTIONS.
535
wbore a is the angle of the spiral, p and q are the roots of the qnadratio
a; (a; - 1) = tan* a.
Show that the solution changes its form when a is such that the first term h
infinite, and find the new form.
649. A string rests on a smooth horizontal table and is acted
on at one extremity by an impulsive tension, find the impulsive
tension at any point and the initial motion.
Let T be the impulsive tension at any point P,T -\-dT i\\Q
tension at a consecutive point Q, then the element P^ is acted on
by the tensions T and T+dT at the extremities. Let <^ be the
angle the tangent at P to the string makes with any fixed line ;
II, V the initial velocities of the element resolved respectively
along the tangent and normal at P to the string. Then, resolving
along the tangent and normal, we have
muds == (T-hdT) cos d<j>-T
mvds = {T + dT) sin d(l>
therefore proceeding to the limit
1 dT
m as
1 T
v= .
m p
But by Art. 639, we have ~^ = - - Hence the equation to find
T becomes
ds' ~p'~^'
This, as might have been expected from mechanical consi-
derations, is the same as the equation in Art. 647.
If the chain be heterogeneous we easily find in the same way
d ndT^ ^_
Is \m ds j VI p''* '
ds \m ds
The two results in this article appear to have been first given
in College Examination Papers.
650. Ex. If Ti, T„ bo the imp'olsive tensions at the extremities of any arc of
the chain, wi, Mj the initial velocities at the extremities resolved along the tan^
gents at the extremities, prove that the initial kinetic energy of the whole arc is
This readily follows by integrating m {u' + v^)ds along the whole length of the
arc. But it also follows at once from Art. 331, for the work done at either extre-
mity is the product of the tension into halt the initial tangential velocity.
1"
•'
i
536
MOTION OF A STRING.
Small Oscillations of a loose chain.
651. A heavy heterogeneous chain is suspended hy one ex-
tremity and hangs in a straight line under the action of qravity.
A small disturbance being given to the chain in a vertical plane,
it is required to find the equations of motion*.
Let be the point of support, let the axis Ox be measured
vertically downwards and Oy horizontally in the plane of disturb-
ance. Let mda be the mass of any elementary arc whose length
PQ is ds, and let T be the tension at P. Let I be the length of
the string, and let us suppose that a weight Mg is attached to the
lower extremity.
The equations of motion as in Art. G35 will be
df ~mds\ ds)'^^
df m ds \ ds)
.(1).
Since the motion is very small, the point P will oscillate in a
very small arc, the tangent at the middle point being horizontal.
d'jtj
Hence we may put -ji = ^' For a similar reason we may put
dx = ds. We therefore have by integrating the first of equa-
tions (1)
T= constant —g jmdx.
But T= Mg when x = l, hence we find
T=Mg + gj mdx.
(2).
f.«V
• In the Seventh Yulume of the Journal Poly technique, Poissc i discusses the
oscillations of a heavy homogeneous chain suspended by one extremity. Putting
U,-x)^i:„gh equal to « or «' according as the upper or lower sign is taken, and
dy
y
He obtains
•/' = y{l- x)i , he reduces the equation to the form , , , _ - -7 -; ^.
i> J^ ' > "* dsda' 4 (8 + 8')3
the integral by means of two definite integrals and two infinite series. After a
rather long discussion of the forms of the arbitrary functions which occur in the
integral, he finds that a solitary wave will travel up the chain with a uniform
acceleration and down with a uniform retardation each equal to half that of
gravity.
' one ex-
f pravity.
',al planOf
measured
f disturb-
)se length
length of
led to the
.(1).
illate in a
liorizonttil.
may put
of equa-
(2).
iscusses the
ty. Patting
taken, and
He ottains
es. After a
)ccur in the
a uniform
lalf that of
I
SMALL OSCILLATIONS OF A LOOSE CHAIN.
537
Wlien the chain is homogeneous, thia equation takes the simple
form
T=Mf; + mff{l-x) (3).
It may bo noticed that this expression is independent of the
time ; the tension at any point of the chain is ccjual to the total
weight of matter below that point.
The secoT'.J equation may be written in either of the forms
df m dx \ dxj
= lr^ + i^^2/|
in dx^ m dx dx J
where T is a function of x given by the equations (2) or (3).
(4),
652. Let us suppose that the displacements of the particles
forming any finite portion of the chain during a finite time, are
represented by ;/ = <^ {x, t), where is a continuous function of x
and t Let P be a geometrical point within this portion of the
di/
chain which moves so that the particle- velocity at P, i. e. -4^ is
.always equal to some constant quantity A, Let v be the velocity
with which P moves, then following in our mind the motion of P,
we have
d^y . d^y „
dt' dxdt
.(5).
Let Q be a point also within the portion, such that the tangent
to the chain at Q makes with the vertical an angle whose tangent,
i. e. -r- , is yp> where B is some constant quantity.
Let v' be the velocity with which Q moves, then
(^S)^-"
dxdt dx \
,(G).
Eliminating the second dififerential coefficients of y from equa-
tions (4), (5) and (6), we easily deduce that if P and Q coincide
at any instant,
vv' = - (7).
m
This reasonipg requires that all the second differential coeffi-
cients should be finite, and that y should be a continuous function
of x and t. It would not apply to any point P, if the discontinuous
extremities of two waves were passing over P in opposite direc-
tions. But the consideration of these exceptions is uunecc .ary
for our present purpose.
r)38
MOTION OP A STFINQ.
i
t
f
, I
m
Let AB be a disturbed portion of the chain travelling in the
direction ^5 on a chain otherwise in equilibrium. At the con-
fines of the disturbance the two portions of the string must not
make a finite angle with each other. If they did, an element of
the string would be acted on by a finite moving force, which is the
resultant of the two finite tensions at its extremities. In such
a case the disturbance would instantly extend itself further along
the chain and take up some new form. Supposing we exclude
any such case as this, we must have, as long as the motion is
finite, both h{ — 0, and ^- = 0, at both the upper and lower ex-
dt
dx
tremity of the disturbance. If then P be a point at which -^^ = 0,
and Q a point at which -j- = Q, P and Q may be considered as
taken just within the boundary of the v/ave ; P and Q Avill there-
fore each travel with the velocity of that boundary. Hence
putting V = v , we find for the velocity of either point
,,2^
T
m
,(8).
It appears therefore that if a solitary wave travel up the chain,
the velocity increases as the wave approaches the upper extremity.
The upper end of the wave will travel a little quicker than the
lower end, because the tension at the upper end exceeds that at
the lower; thus the length of the wave will gradually increase.
When the wave travels down the chain, the velocity for the same
reason decreases.
C53. Ex. 1. If the chain be homogeneous, show that the boundaries of a
solitary wave will travel up the chain with an acceleration equal to half that of
gravity, and down the chain with a retardation of the same numerical amount.
Ex.2. Let the law of density be m=A(l + l' -x)~i where I is the length of
the chain and A, I' two constants. Also let a weight equal to 2Ag\/l' be fastened
to the lower extremity, prove that
This integration may be effected by writing 0=:(l + l')i -(l + l'-x)K The equation
of moticu then takes the form ^-| = ^ '- .^ , which can be solved in the usual manner.
Ex. 3. The chain is said to sound an harmonic note when its motion can bo
represented by an expression of the form y = <p(x) sin {kI + a); so tha*: the motion of
every element repeats itself at the same constant interval. Show that the harmonic
periods of the chain and weight are given by
KZ'4auK{(« + /')i-rij--.l (1).
!
ling in the
t the con-
T must not
element of
hich is the
. In such
rther along
ve exclude
motion is
I lower ex-
,ieh|=0,
nsidered as
will there-
y^. Hence
,(8).
3 the chain,
■ extremity,
ir than the
seds that at
ly increase,
r the same
lundaries of a
to half that of
,1 amount.
1 the length of
'I' be fastened
The equation
asual manner.
motion can bo
the motion of
the harmonic
(1).
SMALL OSCILLATIONS OP A LOOSE CHAIN.
539
To prove this, we substitute y=/(e) 8in(K« + o) in the differential equation
obtained in the last Example; we thus find/(^) to be trigonometrical. Since y =0
when a;=0 for all values of t, the expression for y reduces to
7j = %\nKB\A^smKt\jA ■\-B^eosKt\^A \ (2),
where Ak and Bk are two arbitrary constants. But when x=l, y must satisfy the
equation of motion of the weight, viz. -ifi~ ~g y . Whence the result follows by
Eubstitution.
Ex. 4. If the initial motion of the chain and weight be given by the equations
y=.f(x), —=F{x) when t=0, tlien y can bo expanded in a series, the general term
of which is expressed by equation (2) of the last example. Find the values of
Ak and Bk.
We notice that equation (1) of the last example may be written in the form
cos K^j = kJi' sin Kdi,
whci'6 ^1 is the value of when x = l. We then easily find that
/ sin k6 sin K'6dd = - \/l' sin k0i sin «'<?,,
rOi 1 1
/ sin" KOdO = ^6,- ;, yjV sin' kO^.
These results may be obtained by integi-ating the left-hand sides and substi-
tuting for cos K^i and cos K'd^ their values in terms of sin kB-^ and sin k'O-^.
If we now multiply both sides of equation (2) by sin kO and integrate from
^=0 to 5=^1, we find by the use of these two results
5 Bk (^1 + sjV sin2 K^i) = / y sin kQM +/(?) sjl' sin k^^.
Z •'0
Differentiating (2) and performing the same process, w/ have
Co-t. An inelastic heterogeneous chain is suspended from two
fixed points under the action of gravity. Any small disturbance
being given in its oiun plane, it is required to find the small oscil-
lations.
Let the axis of x be horizontal and that of y vertical. Lot C
be any point on the chain when hanging in equilibrium, and let
the arc s be measured from C. Let (.»•, ?/) be the co-ordinates of
any point P determined by CP-s. Let T be the tension at P,
mgds the weight of an element ds situated at P. The equations,
of equilibiium are
ds
:^)-. l(^l)-"^-'-
!l
■■'■.: .
Hi
I
!■
,
■I'
540
MOTION OF A STRING.
Let a be the angle the tangent at P makes with the axis of x,
then we easily find
T:
wg_
cos a'
m — w
<?tan
a
da
(1).
where w is an undetermined constant.
When the chain is in motion, let (a? + 1, y + v) he the co-
ordinates of the position of the particle P at the time t, and let
the tension at that point be T' = T+ U. The equations of motion
will be
cP^ _'i^d_{rp' (dx , d^\
d^~ mds\ \i
^V^}_ d U, (dif d^\\ _
df m~ds\ \ds dsj\ ^'
which, by subtracting the equations of equilibrium, reduce to
dt in as \ as dsj \ • ,_.
dt^ m ds \ ds ds) J
when the squares of small quantities are neglected.
Since the string is inelastic, we have
{dx + d^Y + [dy + drjf = {dsf.
ExDanding and rejecting the squares of small quantities, this
becomes
fj f nS rrii rim
(S).
dx d^ dy drj _
ds ds ds ds
We have thus three equations to find |, rj and U as functions
of s and t
655. To find the velocity with which a solitary ivave will
travel along the chain.
If we suppose a small disturbance to travel along this chain,
so that there is no abrupt change of direction of the chain at the
d^
boundaries of the wave, we must have at those points -—=0,
^ = 0, ^ = 0, i]-0, and f/=0. Let v be the velocity
ds "' dt ^' dt
with
which one boundary of this wave travels along the chain, then,
following that boundary in our mind, wo have
d'^ . d'^
- -f- u -^
di' dsdt
i"f.+/''f = o
I axis of X,
(1)»
36 the co-
3 t, and let
s of motion
uce to
(2).
.ntities, this
(3).
IS functions
/ ivave tuill
this chain,
lain at tlie
ints ~=\j,
as
elocity with
chain, then,
I
SMALL OSCILLATIONS OF A LOOSE CHAIN.
and therefore
541
with a similar equation for rj. Thus the dynamical equations be-
come at the boundary
( 2_^\^^]^dUdx^
\ m) ds^ ~ m ds ds
f._T\dS,^}^dUdy
\ m) ds^ m ds ds
and the geometrical equation becomes
<^f dx _ d^n dy
ds* ds " ds^ ds'
T
From these we easily get ?;' = — . Substituting for Tand m their
values, we have if p be the radius of curvature at P,
v = ^{gpeosa) (4),
so that the velocity of either boundary of the wave is that due to one
quarter of the vertical chord of curvature at that point.
Ex. 1. A chain is in equilibrium under the action of any forces which are
functions only of the position in space of the element acted on. Show that the
velocity of either boundary of a solitary wave is that due to one quarter of the chord
of curvature in the direction of the resultant force at that boundary.
656. To solve at far as possible the equations of motion of a heavy slack
heterogeneous chain.
It will be convenient to express the unknown quantities f, ?;, Uin terms of
some one function </>,
Let a + be the angle the tangent at P makes with the horizon at the time t.
Then
coa(a + ^-)=-~^^»
ds'
sin (a + 0) =
dy + dy
ds '
sin a =
' da.
= - p(f> sin o,
dn
A COS a = — .
ds
d-n ^
^= - jp^ Bin ada-{- A, r) =J pip coa ada + B
where A and B are two undetermined functions of t.
The equations (2) now become
1-3 — fl = T" ( - iir<^ tan a + - cos o
dt» cos' a da, \ -'^ w J
d'n 1
(5);
(C),
(7).
df co8*o da
d / U .
{9<f> + ~ sin
')
■(«).
w
ii
'I
1!'
il
i'f r
i
If
ft
I'll
I
8
542
MOTION OF A STRTNG.
For C 3 sako of brevity let accents denote difierentiationb th i;\;>i ' to t.
Expanling Uo differentiations on the right-hand side, these eclua'.iol^! n.ay bo
written in the form
■f'sina + 17' coso-^( 0sina+ j- cosa\-U }
.,, ,, . , dU cos^a ['
f cos a + 1; am + (7^ cos a = j_ ~;r~J
_dU co8°(
~ da, w
Differentiating the first with regard to o and adding the result to the second,
we obtain
COS(
P" _ d^<f> d ni con a \
3 a da'' "" da \ w J '
Differentiating the second and subtracting the first irom the result, we obtain
2/7
d<f>
d" /f/cosa>
w
da da" \ w J '
These equations evidently give
Uco3a=u-g ( 2 /Jida-l-Co + i^ j.
•(9J,
dV
-dt^=^
cos
p
-■■(g+4* + 2^) (10),
where C and D are two undetermined functions of t. These are the general
equations to determine the small oscillations of a slack chain.
The undisturbed form of the curve being givon, p is known as a funcsion of o.
We may then use the equation (10) to find (p as " function of a and t. T'i'j tension
is then found from the equation (9), and the disjilacements f, 7; of any point of the
chain by equations (7).
657. The determination of the whole motion depeads therefore on the solution
of a single equation. Supposing the integration to have beeu otTi;rl,f>d, the ex-
pression for (p will contain two new arbitrary functions of a and t. ll'hcse wo may
represent by \I/(P) andx((?) where i/'and x are arbitrary iunctions of two determinate
combinations P and Q of the variables, Tbo arbitrary fimctions A and U are not
independent of C and B, and the ro' ., . 'letween them uiay be found by substi-
tuting in equations (8).
We have thus four arbitrary functions whoso values have to be determined from
the conditions of the question. Let a^, Oj, be the values of a which correspond to
the two extremities of the string. Then the values of and J^ are given by the
etc
question when ( = for all values of a from a-a,^ to o-a,; also the initial values
of yl and /? are given. Thus the values of \j/(P) and x(Q) arc determined for all
values of P and Q between the two limits which correspond to a= a,„ t = and a = Oj,
t = 0. The forms of tp and x for values of P and Q exterior to these limits, and the
values of A and li when t is not zero, ai'e to be found from the conditions at the
extremities of the chain. If the extremities be fixed, we have both ^ and r/ equal to
zero for all values of t when a^a^ and a^a^. It may thus happen that tlio
ciVitrary functions A, H,\(/ and x are discontinuous.
In many cases the circumstances of the problem will enable us to determine at
once the form of C. Thus, suppose the string when in equilibrium to be
symm'hical about t'. vertical line, say the axis of y, and lot the points of support be
wher
terms
of br
for sii
? Hi ay be
le second,
e obtain
(9).
(10),
Ike general
if'tion of a.
Tli'j tenBion
point of the
the solution
U'd, tho ox-
lese wo may
determinate
d B are wot
il liy substi-
t-miucd from
orrespond to
^vcn by the
nitial values
lined for all
and a = ai,
nits, and the
itions at the
lid ri equal to
pen that tho
dotermiuo at
brium to be
of support bo
I
SMALL OSCILLATIONS OF A LOOSE CHAIN.
543
xi::ud m tiie same horizontal line. Then if the initial motion bo also symmetrical
about the axis of y, tho whole Bul^sequent motion will Le lym metrical. Thus ^
must be a function of o, contaming when expanded only odd powers of o. Sub-
Btitnting such a series in equation (10) wo see that C must be zero.
658. There are several cases in which the equation to find the small motions
of a chain may be more or less completely integrated. One of the most interesting
of these is that in which the chain hangs in equiUbrium in the form of a cycloid.
In this case we have, if b be the radius of the generating circle, p=ib cos a. Tho
density of the chain at any point is given by vi= jr 3-, so that all the lower
part of the chain is of nearly uniform density, but the density increaaes rapidly
hi, ler up the chain and is infinite at the cusp.
The equation to find the oscillations now takes the simple form
d^<f> _ {I { d"<t>
^-&i^-^*-H '")■
in which all the coefficients are constants.
There are two cases of motion to be discussed, (1) when the chain swings up
and down, and (2) when it swings from side to side. The results are indicated in
tho two following examples.
Ex. 1. A heavy chain suspended from two points in the same horizontal line
hangs under gravity in the form of a cycloid. Find the symmetrical oscillations
of the chain, when the lowest point moves only vp and down.
In this case we have C7=0. To find the nature and time of a small oscillation,
we put
0= S/2 sin Kt + SiJ' cos Kt,
where ':'■ implies summation for all values of k, and B, R' arc lanotions of a only.
Substituting, we have
with a similar equation to find If. Therefore
i?=Lsin2 ^f l + -^ja,
where L is an arbitrary constant, the other constant being determined by tlie
consideration that tho motion is symmetri'ial about tho axis oi y. F ihe sake of
brevity, put X = 2. /(l+— j. Substituting in (7), wo find that the . jrms derived
from II become
| = SL
2b
\'-4.
{ \ cos \a sin 2a - 2 sin \a cos 2a } sin Kt,
,, = 2
[-
26 2b
L^„ — j{\cosXocos2a + 2smXasm2a} -L -r-cosX
A^ - 4 A
a + J/jsi
sin Kt,
where If is a constant depending on tho position of the points of support. Tho
terms derived from li' n.ust bo added to these, but havo been omitted for tho sako
of brevity. They may bo derived from those just written down by writing cos Kt
for sin Kt and changing ihe constants L, II into two otlier constants L', 11'.
:j
(I ^"#^ %
!--.f
Mi
544
MOTION OF A STRING.
Let the length of the chain be 21, then at either end 8inao= tt. At both
extremities we must have f =0, i;=0. All these four conditions can he satisfied if
tanXag tan2a(,
This equation therefore determines the possible times of symmetrical vibration
of a heterogeneous chain hanging in the form of a cycloid.
659. If a be not very large, the oscillations are nearly the same as those of a
uniform cliain*. In this case since Oq is small but Xao is not necessarily small,
the equation to determine X is approximately
tan Xag=Xao,
Sir
The least value of Xa which can be taken is a little less than
y
Hence X
now
is great, and therefore k = a/( ji) ^ nearly. The expressions for f and 17
tnke the simple forms
f = Si T-j |XoCOsXa-sinXa} sin ( a /h At + e)
i7=SZi — {cos Xoj - cos Xa} sin ( » /^^ Xf + e )
Th(! terms depending on cos Kt have been included in these expressions for f and
r) by introducing e into the trigonometrical factor.
The roots of the equation tan Xoo=Xao may be found by continued approxi-
mation. The first is zero, but since X occurs in the denominator of some of the
small terms, this value is inadmissible. The others may be expressed by the
formula Xao=(2i + l) 5 -^, where 6 is not very large
vibration nearly equal to
are all short.
2i + l' ^igb'
This makes the time of
Thus the times of vibration of the chain
This result will explain why the marching of troops in time along a suspension
bridge may cause oscillations which are so great as to bo dangerous to the bridge.
It is clearly possible that the " marching time" may be equal to, or very nearly
equal to some '^no of the times of vibrations of the bridge. If this should oocur
it follows from Arts. 433 and 503 that the stability of the bridge may be severely
strained. "
* The rc-der ..ho may wish to see another method of discussing the small
oscillations of -i uif (■ 'nsion chain may consult a memoir by Mr Rolirs in the ninth
volume of the Caml>]ldge Tramactions. Mr Eohrs considers the chain to be homo-
geneous, ByrainrtriciiJ iibout the vertical, and nearly horizontal from the beginning
of tlie process. In the necond edition of this treatise the small oscillations wore
iilfo treated lO thj same hypotliesis, but in a different manner. That method,
Luwover, is not nexrly so siiaple as tlie one here given in which the approximate
oscillations for a catenary are deduced from the accurate ones for a cycloid.
'"""net
SMALL OSCILLATIONS OF A LOOSE CHAIN.
54^
It sliould be noticed that the terms in the exprjssion for f have the square of \
in the denominator, whilo those in the expiission for ij have the first power of X.
Since \ is great we might as a first approximation reject the values of f altogether,
and regard each element of the chain as simply moving up and down.
of the chain
660. Ex. 2. A heavy chain suspended from two points hangs under gravity in
the form of a cycloid. If it swings from side to side in its own plane so thut the
middle point has only a lateral motion without any perceptible vertical motion,
find the times of oscillation.
As in the last example, we put
0= 2J? sin Kt + S/J' cos id,
where R and R' are functions of a only. Substituting in equation {11) we see that
2C=lhfim Kt + l^k Bin Kt where h and k are arbitrary coubtauts. The equation to
find R becomes
d^R
da*
+ 4
(6(('\ h
1 + - I as before, we find iJ- - — + Z sin (\o+ Af).
Thence taking the term of (f> which contains sin Kt,
f h'-hb C0B2a . 26 ,, ,, .„ • «
^T^-^= j^ + Zj-^--^{\cos(\a + ilf)sm2a-2 8:n(Xa + .U)cos2a[,
where h' is an arbitrary constant introduced on integration. Substituting in
equation (8), we find h'= - ^ ( 6 + i ) • Also, we havo iu the same way
Bin
26 26
-I. 5— j-{\cos{\tt + Jf)cos2a + 2sin(\a + ilf)sin2a} -L - aos{\a + M) + H.
A — 4 A
If we suppose the two supports to be on the same horizontal line, we must havo
1=0 aud 7]=^} when a=±0o. These conditions may bo satisfied if we take
M=^, H=0, for then ^ becomes an even and 17 an odd function of a. In this case
»j=0 at the lowest point of the chain. We have then two equations to find — ,
h
equating these values, we have
2 tan 2ao - \ tan Xao
tan Xoo X' - 4
cos 2a X
2a^ + bin '2ao
X tan Xao '■^.n 2aQ + 2
4,
2cos''ao+j^3_^
661. If oj be small, this equation is very nearly satisfied by Xag-iir where
i is any integer. In this case the complete expressions for ^ and 7) take the simple
forms
^='SL r^(cos XOfl-cos Xa-Xa sin Xa)siu( . / jy X/ +« 1 j
»;=SZ— sinXa sinf . / jr Xt + e j
R. I).
35
54G MOTION OF A STRING.
662. Ex. 1. If we clmngo ti.o vfiriablos from a, t to p, q whoro
P = t+ r. / — " — da, q= -t+ A / — ^— da,
J\/ gcoBa ' '■ JV gcoaa
show that tho general equation (10) of small oscillations takes tlio form
, . ff cos a , . .,
where /*■• = and <f>=fi<p'.
P
Show also that the coefficient of (/>' is a function of p + q, tho form of tho
function depending on tho law of density of tho chain.
This transformation may be usi^ful, hecauso it follows from Art. 055 that p is
constant for the boundaries of a solitary wave travelling in one direction, and q for
a wave travelling in the other direction.
Ex. 2. A heavy string lianga in equilibrium under gravity in such a form that
its intrinsic > quation is = -sin''(2a + c) where h and c are any constants.
P 9
h^ sin'' (2a 4- c)
Show that its law of density is given by m=w - '- — -^ . If such a chain be
n COS' ci
set in motion in any symmetrical manner, prove that its motion is given by
Ex. 3. If in addition to gravity, each element of the chain bo acted on by a
small normal force whose magnitude is Fg, prove that the equation of motion
of the chain is
! f ~-da.
J cos a
y cos a dt^ da? cos a da
ii!
If tJio chain is nearly horizontal, so that a is very small, and if F—f^va. (at — ca),
prove *.hat the denominator of the corresponding term in tho expression for </> is
g{c''~\)-pa\
Ex. 4. A heavy chain of length 11 is suspended from two points A, B\\\ tho
same horizontal line whose distance apart is not very different from 2/. Each
particle of tlie chain is slightly disturbed from its position of rest in a direction
perpendicular to the vertical plane through AB. Find the small oscillations of tho
chain.
Ex. 5. A heavy string is suspendod from two fixed points A and B and rests
in equilibrium in the form of a catenary wlioso parameter is c. Lot tlic string
be initially displaced, the points of support A , B being also moved, so tliat
^ = o-(l + cos 2a) + 0-' sin 2a,
where a and a' are two small quantities and tho other It'ttors liavo tho sanio
meaning as in Art. OSfi. If the string he placed at rest in this new position, prove?
that it will always remain at rest.
.. ^ U - Jt- i t
» ■ „«.. ' , uimt^ '
SMALL OSCILLATIONS OF A TIGHT STRIN(J.
547
Small Oscillatiom of a tight ttring.
663. An elastic string whose xoeight mag he neglected and whose unstretched
length is 1 has its extremities fixed at two points whose distance apart is V, The
string beirj disturbed so that each particle is moved along the length of the string,
find the equations of motion.
Let A bo ono of the fixed pointa, and let AB be tlio string when unstretched
and placed in a straight line. Let the extremity B be pulled until it reaches the
other fixed point B'. Let PQ bo any c^imont of tho unstretched string, P'Q' the
same element at the time t. Let AP=oi, and let the abscissa AP' be x'. Let T and
T+ dT be the tensions a* P' and Q'. Lot 31 bo the mass of the whole string, m the
mass of a unit of length of unstretched string. Then, as in Art. 637, the equation
of motion is
dV dT ,,,
"'d^^Tx <^)-
If E be the modulus of elasticity, we have by Hooke's law
d^-^^E ^^^'
Eliminating T, we have
fPx^ _ E d'x' ,3.
d«3-„i dxi '•
If we put E=ma*, the integral of this equation is
x'=f[at-x) + F(at-{x),
where /and F are two arbitrarj' functionH.
Tho discussion of this e>^uation may bo found in any treatise on Sound. Tho
result is, that a function of the form (p {at - x) represents a wave which travels with
a velocity equal to a. In tho case therefore of the string, the motion will be repre-
sented by a series of waves travelling both ways along the string with the samo
velocity. This velocity is sncli tliat the time of traversing a length I of unstretched
string or a length V of stretched string is I a/ — . It should be noticed that tliia
time is independent of the nature of the disturbance, and is the same whether the
string be originally stretched or not.
It should also be noticed, that assuming as usual tlio truth of Hooke's law, the
equation (3) and these results are not merely approximations, but are strictly
accurate.
It is often move convtsnient to select some particular state of the string as a
standard of reference and to express the actual position of any particle at the time
t by its displacement from its position in this standard. Thus if the unstretched
state AB ot the string bo chosen as tho standard of reference, we put x'=x+^, so
that ^ is the displacement of the particle whose abscissa in the unstretched state
is X. Tho equation of motion now tnkes the form
f/_2^ E d'i
}tt'^ ^ m dx^ '
apd tlie integral may be obtained as before.
3.-)— 2
i,-1
548 MOTION OF A STRING.
6Ci. An elastic string being stretched as in the last proposition is slightly dis-
turbed in any manner, find the equations of motion.
Following the same notation as before, lot {x', j/*, z') bo the co-orJinatos of P'.
Then, as in Art. 637, the equations of motion are
»f4(^£) ■ <».
»^^^(^l) <■"■
-f=.4(^S) '".
where ds' is the length of the element FQ'. If E bo tlio modulus of elasticity wo
have by Hooke's law
ds' . T ...
dx=l + ^ <*>•
Since the disturbance is very small — and -;- are very small and ," > is very
da di da
nearly equal to unity. Heuco the first equation takes the form
d V _ dT
"^ dt^~dx'
and Hooke's equation takes the form
dx~ ^ E'
which are the same equations as in the last proposition, so that when the disturb-
ance is small the longitudinal motion is independent of the motion transverse to
the string.
In the second equation we may regard T as constant, its small variations being
multiplied by the small quantity -^ . Hence we may put T= Tq wh(!re To=E -— .
cts c
ds' I'
This gives by equation (4) -^ =- . The equation of motion therefore becomes
ax t
dhf ^TpldY
dt'^ ~ vi V d«« ■
The third equation may be treated in the same way.
The velocity of a transverse vibration measured in units of length of unstretched
lYi
string is therefore */ ~y • ^^° ^^^^ °^ traversing a length I of unstretched string
or I' of stretched string is */ -^ . This velocity is independent of the nature of
the disturban"'^ ■ ut depends on the tightness or tension of the string.
If the string be very slightly elastic we may, in this last formula, put l'=l. In
this case we obtain the results given in all treatises on Sound.
605. There are two modes of applying the equations of motion to actual cases.
We shall first illustrate these by solving a simple example by both methods, and we
shall then make some remarks on the results.
••»■■-•»»—•-
SMALL OSCILLATIONS OP A TIGHT STRINQ.
549
hthj dii-
!8 of P'.
(1).
(2).
(3),
sticity vro
(4).
f'
, is vory
it
le disturb-
ansvcrse to
tions being
I' -I
0— ^ I
ecomes
unstretched
tolled string
le nature of
nt l'=l. In
actual cases.
lodS; and wo
An elastic string whose unstretched length it I rests on a perfectly smooth table
and has its extremities fixed nt two yints A, B' tehose dintance apart it 1', where V is
greater than 1. The extremity B' is suddenly released, find the motion.
Following the samo notation as in Art. GG3, tho motion is given by the equation
^=--f(at-x)+F{at + x),
where { is the displacement of the particle whose abscissa in tho unstretched string
is X. The conditions to determine / and F are as follows.
1. When x=0, {=0 for all values of t.
2. When x= I, T=0 and /. ^ =0 for all values of t.
dx
3. When«=0, f=rxfrom«=Otox=Z, wheroI'={r+l) J.
4. When t=0, -f=0 from a:=-0 to x=?.
at
From the first condition it follows that the functions F and / are the samo with
opposite signs. From tho second condition we have /' (at + l)= -/' (at - 1), so that
the values of the function /' recur w'th opposite signs when tho variable is in-
creased by 21. If then we know the values of/' (z) for all values of z from 2=2^ to
z=Zq + 21 where Zq has any value, then the form of the function is altogether known.
Now tho third condition gives / ( - x) - / (x) = ra; and tho fourth gives f'{-x) =/' (as)
from a:=0 to x=l. Hence f'(x)=-~ from x~-l to x~l.
It follows that
f (a) = - - from 2 = - 1 to ?, /' (z) = ^ from z = l to SI and so on changing sign every
time the variable passes the values I, 31, 51, &o. Lot us consider the motion of any
point P of the string whose unstretched abscissa is x. Its velocity is given by the
formula -=/'(a«-a;)-/'(at + x). Since x<l wo have -=-- + -=0; hence the
particle does not move untU at + » = L The second function then changes sign and
V T T
we have -= -a~ n~ -'''• '^^° particle continues to move with this velocity until
at~x=l, when the first function changes sign and so on. Let ABhe the unstretched
string, and let a point R starting from B move continually along the string and
back again with velocity a. Then it is easy to see that ^7llcn R is on the same side
of P as the loose end of the string, P will be at rest, and when R is on the samo
side of P as the fixed end, P will be moving with a velocity alternately equal to
dbra. The general character of the motion is; the equilibrium cf the string being
disturbed at B, a wave of length il travels along the string, so that P cloes not
begin to move until the wave reaches it. This wave is reflected at A and returns.
666. The second method of conducting the solution is as follows. Taking as
before the expression
i=Mf[at-x)-\-F{at-\-x\
let us expand each function in a series of sines and cosines, so that we have
{=S[4 sin {7i(a«-x) + a} + Ssin{n(a«+a;) + j3}],
where S imphes summation for all values of n, and A, B, a and ^ are constants
which are different in every term and may conveniently be regarded aa functions
of n.
•--r^Md
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'/J
550
MOTION OF A STRING.
Since the motion is osoillatovy, we may suppose that all the values of n are real,
and it is clear that without loss ot generality we may restrict n to be positive. We
do not propose to discuss the circumstances under which these suppositions may be
correctly made. For these we must refer the rej,der to Fourier's theorem. We
may here regard the assumptions as justified by the result, because we con thus
satisfy all the data of the question.
The four conditions of the problem enable us to determine the constants. From
the first condition we have /3=o+K7r, -B=(-l)*'*'^^ where k is any integer. It
easily follows, by expanding, that f may be written in the form
f = S (C sin nat + D cos nat) sin not,
where and D are to be regarded as functions of n. From the second condition
we have cos nl=0, hence nl={2i + 1) 5- where i is any positive integer.
The possible
harmonic periods (see Arts. 412 and 450) of the string, with proper initial dis-
turbances, one end being fixed and the other loose, are therefore included in the
form 7-r-. — 7-—.
(2i + l)a
The initial disturbance is given by the third and fourth conditions. We have
2DBmnx=rx, 2GnBuxnx=0.
To find the value of 2) in any term we multiply the first equation by the coefficient
of D in that term and integrate throughout the length of the string, i. e. from
x-Otox=l. This gives
In. , sinnl
0=''/ xam7ixdx=r
71"
The other terms all vanish since /em nx Bmn'xdx=0, when n and n'are numerically
unequal.
Treating the second equation in the some way, we find C=0. Hence the
motion is given by
, _, 2r sin nl ^ ,
• i= 2, -, — 5— cosnatsmnas.
Writing for i its values 1, 2, S, &c. successively, this equation becomes .irhcn
written at length
^ 8rl ( vat . itx 1 Swat . 3irx 1 5irat . Birat ,
^= ^ r^ 2r "" 2f - 3^"^^ -2T '"" •2r + 5«*'°' Tr '"" -2T -*"•
This is a couvergout series for £ and it may be a sufficient approximation to the
motion to take only the first few terms. For example, suppose we reject all beyond
the first two terms, and in order to compare the result with that obtained in the
first solution let us put at=^l.
If wc trace the curve whose ordinate is - j|
and
abscissa x, we find that it resembles ^=0 for small values of x, then rises with a
point of contrary ficxure and becomes nearly horizontal as x approaches I. This
agrees very well with the former result.
667. If' wo examine these solutions, we shall see that we have two kinds of
conditions to determine the arbitrary functions ; (1) There are the conditions at
the two extremities of the string. The peculiarity of these is, that they hold for all
values of t. (2) There are the initial conditions of motion. The peculiarity of
mm
<mam
SMALL OSCILLATIONS OF A TIGHT STRING.
551
are real,
ive. We
IS may be
em. We
con thus
s. From
teger. It
condition
le possible
initial dis-
ied in the
We have
3 coefficient
g, i. e. from
numerically
Hence the
comes .fhcn
-&c
)
aation to the
ct all beyond
tained in the
„ . § ..d
I rises with a
shes I. This
two kinds of
sonditions at
y hold for all
peculiarity of
•those is, that they do not hold for all values of x, but only for all values within a
certain range limited by the length of the string. The first set of conditions is
used to determine the mode in which the values of the functions recur, so that
when their values are known through a certain limited range, they will become
known for all those values of the variable which occur in the problem. The second
set of iionditions is used to determine their values during this limited range.
The functions were found to be discontinuous. It may bo objected that no
notice was token of any possible discontinuity in forming the equations of motion ;
and that therefore these equations cannot be applied, without further examination,
to any cases which require the arbitrary functions introduced into the solution to be
discontinuous. This question has been much discussed, but we have not space here
to enter into it. We must refer the reader to De Morgan's Differential Calcuhu,
Chap. XXI. Art. 92, where both a short history of the dispute between Lagrange
and D'Alembert and a discussion of the difficulty may be found. See also the
Mecanique Analytique, Seconde Partie, Sect. vi. § iv.
In the second form of the solution we replace the arbitrary functions by a
convergent series of harmonic vibrations. Taking a finite number of terms aa an
approximation, we have a perfectly cor ;U0U8 solution whose initial conditions
differ but slightly from those of the proposed problem. This difference is less and
less, the more terms of the series are included in the solution.
In comparing the two results, we see that each form has its advantages. The
first determines the motion by a simple formula. The second is more convenient
when the harmonic periods are required.
668. Ex. A heavy elastic string AB whose unstretched length is I is suspended
from a point A under the action of gravity. If ^ be the vertical displacement of
any point whose distance from A is x when the string is unstretched, and if a be
the velocity of a wave measured in units of unstretched length, prove that
where/ (2) recurs with an opposite sign when z is increased by 21. If the string
is initially unstretched and at rest, prove that
•'W =^ 4^2 + 2a'''
the upper sign being taken when z lies between - 1 and 0, and the lower when 2
lies between and I. Thence show that the whole length oscillates between
a'
Taking the other form of solution, show that the harmonic periods are
where i is any integer.
(2i + l)a
Show also that
i=-
gx'
2ai
. /2i + l irx\ [
glx 16^»5;n_Jil2!!!i
2i + l irat\
'2' I )
the summation extending from t = to i-(x> ,
669. Three elastic strings AB, BC, CD of different materials are attached to
each other at B and C and stretched in a straight line hetiveen two Jixed points A, D.
n
552
MOTION OF A STRING.
If the particles of the string receive any longitudinal displacements and start from
rest, find the subsequent motion.
Let A be the origin, AD the direction in which x is measured. Let the nn-
stretnhef'i lengths of AB, £C, CD be 2j, 2„ ^. Let B'l, E^, £, be their respective
coeffici'ints of elasticity, m^, m^ m, the masses of a unit of length of each string.
For the sake of brevity let £i=miai', E^=m^a^*, E^=.m^a^. Let the rest of the
notation be the same as before.
When the string is stretched in equilibrium between the two fixed points A and
D, let To be the tension of the string. In this position the displacements of the
elements of each string from their positions when nnstretched may be written
At the time t after the eqnilibrinm has been disturbed, let these displacements
be respectively f j + f i', fa + f j', f g + f g'. We then have
fi' = SXj sin (ni* + JfJ cos n^a^t,
fa' = 2La sin {n^(x-lj)+ M^] cos n^a^t,
fa = Sij sin {rij {x-l^- 1^) + M^] cos n^a^t,
where 2 implies summation for all the harmonics. In order to compare the coeffi-
2t
cients of the same harmonic we must suppose niai=n^a^=n^a^= — , where p is the
period of the harmonic.
To find the constants we have the conditions
when x=0,
X=li,
X = li + l„
X = ly + l^ + l^,
f,'=o,
li'={,'.
I,'=f3'.
f3' = 0,
/
-.t'-.1^'.
These give
IiaBin3fa=LiSin(niii + Jlfi) /
EjnjL J cos itf a = ^iTJi £i cos (nrj^ + M^)\
L3 sin ilfg = ia sin ("a^j + -^i) \
E.^n^L^ cos ifg = £ jMa L, cos {n^^ + Ma) > '
Those give the following equations to find the ill's ;
tan M^ _ tan(n i? i + itfi) i&nM^ _ ioxi {n ^\ + M^ . _ tan {nJ,^-\-M .^
iJjtta E{ni
Solving these we find
£3^3
£3^8
■BsMg
tanrij?! tan »,?, tanngJg j tannj?! tan Hj/j tannj/j
iii«i
SMALL OSCILLATIONS O' A TIGHT STRING.
553
art from
t the nn-
espeotive
ih string.
3st of the
its A and
its of the
,tten
laoements
the coeffi-
jre p is the
'j. + 'j + '3 >
0,
■BsWa
Substituting for n^, n,, n, in terms of p we have an equation to find the har-
monics.
The Talnes of p being known, it is clear that the preceding equations determine
all the constants except L^. We have therefore one constant undetermined for
each harmonic. To find these we must have recourse to the initial conditions.
The equations may be written in the forms
{/=SP,, cos na«, f,'=SQ„ cos na*, {8'=2;i2„ cos na«,
cPP
where P„, Q„ and R^ satisfy the equation -^ = - w'P. We have therefore, after
integration by parts.
Similar theorems apply to Q„ and i2„.
We also have the conditions
when 06=0, x=li,
P=0, P=Q,
dx
dx'
dQ dR
X — ^1 T frg X frjji
R=0,
whatever the suffixes may be, provided they are the same in each equation. If
then we put
4>(m, n) =. f^' E.P^P^dx +f^'^^' E,Q^Q^dx+ f^''^^''^^ E^R^R^dx,
we have mV (»». ») =nV (m, n), and therefore each is necessarily zero when m and n
are difierent. A precisely similar theorem would apply if one or both ends of the
string were loose, or if the string were vibrating transversely instead of longitudinally.
Suppose now that we have initially ^i'=/i (x), l»'=/a {«). ^3=/^ («). We easily
find
/ jBi/i {«) sm {njpe +Mj)dx+ f ^ E^f^ («) sin {n^ (« - U) + if,} dx
•'0 «'?.
+ (, It " "-^a/s (*) Bin {ng {x-l^- 1,) + M^\ dx
= EJj, f ' Bin2 (ni« + M.) dx + E^L^ f ' ' sin'' {n^ (x - 1^) + M^} da
+ E^L, fj^'^jj'^^' sin" {», (x-l,- 1,) + M.,] dx,
these integrations may bo easily effected and give an additional equation to find the
L, which corresponds to any value of p.
If the strings did not start from rest, we should merely have to add to the
expressions for |/, {,', ^3' similar functions of x but with sinnat written for cos naf.
670. Ex. 1. If the three strings vibrato transversely, and a^, CTj, a^ be the
velocities of a wave along them measured in units of length of unstretched string,
prove that the periods of the notes are given by the equation
tan tijl^ tan nj/g tan nj/j _ ^ tan nJi tan n^l^ tan n^l^
. «l »• "s "i "« "3
554
MOTION OF A STRINQ.
T
2ir
where riia^ = n^a^ = n-jO-^ = — . If tlio initial disturbance ia given show bow to find
the subsequent motion.
Ex. 2. Two heavy strings A B, BC of different materials are attached together
at B and suspended under gravity from a fixed point A. Prove that the periods of
the vertical oscillatio-is ore given by the equation
tan2'^^tan?-'l'' = |lfL«,
the notation being the same as before. If the two strings be initially unstretched,
find their lengths at any time.
671. An elastic string is stretched between two fixed points A and B' and is set
in vibration, it is required to find the energy.
Let the notatio** be the same as that used in Arts. GC3 and 664.
First let the vibrations be longitudinal. The equation of motion is
dt^ c/x3*
Hence we have
t -I
1= -j-x + :2[Asin{n(at~x) + a)+BBm{n(at + x) + p}].
Since ^ must vanish when x^O and be equal to I'-l when «=! we find, as
in Art. 666,
I' -I
i=— J- 05+ 20 sin Mac sin (nat + y),
where nl=iir and S implies summation for all positive integer values of f. The
letters C and y are constants which may be different in every term and which de-
pend on the initial disturbance.
The kinetic energy of the whole string is
rii , /rfA" r^i
=zj - max I.I = / 2 mdx {ZCna sin nx cos (nat + y)\*.
rl
Now / sin nx sin n'xdx=0 when n and n' are numerically unequal since nl and
«7 are both integer multiples of ir. Hence, when the square of the series is ex-
panded, the integral of the product of any two terms is zero.
rl 1
Also / Bio!' nxdx=„ I, hence the kinetic energy becomes
= 2 mia' 2C^n' cos^ {na« + 7).
To find the potential energy; we notice that the work done in stretching an
element from its unstretched length dx to its length dx + d^ ia, by Art. 327, equal
tu
1 /(/'\'
- £ ( y J dx. Hence the whole work done in stretching the string is
=f^^lEdx(^^' = f^lEdx\^^ + ^Cncosnxshxinat + y)\\ .
ri 1
Now / cos nx cos n'xdx-0 or ^ I according as n and n' are numerically unequal
Jo i
SMALL OSCILLATIONS OF A TIGHT STRINQ,
555
or equal to cacb other ; also T coanxdx=0. Hence as before, the integral becomes
2^^
^ + ^ EiSCV sin" (nat + 7).
The first term is the work done in stretching the string from the unstretched
length I to the stretclied length V. If we refer the potential energy to the position
of the string when stretched in eciuilibrium between the extreme point? A and B'
as the standard position, we retain the latter term only.
The energy is the sum of the kinetic and potential energies. Bino E^ma',
this becomes
energy = J jnZa'SCV.
This result might have been deduced more simply from Art. 458, where it
is shown that the energy of a compound vibration is the sum of the energies of the
simple vibrations into which it may be resolved. See also Art. 451. The kinetic
energy of any single harmonic is easily Been by integration to be
Hence the whole energy is
■mla^I,Cr'n^.
We may also notice that, as in Art. 457, the mean kinetic energy is equal to the
mean potential energy, the means being taken for any very long period.
672. Next, let the vibrations be transversal.
Following the notation of Art. 664, the motion is given, as before, by
3/' =2(7 sin nx sin (nat + 7),
where nl=iir and 2 implies summation for all positive integer values of t.
The kinetic energy by fhe same reasoning as in Art. C71 is equal to
jmla^SCPn^coa'inat + y).
To find the potential energy, we notice that the work done in stretching an
clement from its unstretched length dx to its stretched length ds' is by Art. 327
1 /<?s' \* "
equal to 5£K-lj dx. Now
(dsy = (dxT + (dyy = (^'- dxj + dy'\
s' l'\, ll^/dy'Y) ,
ds'
dx
,l'-l
liemcmbcring that, by Art. 664, ma^=E—jj- ; wo find that the whole work done
iu stretching the string is
./;>|.(-)V^.(g)'i.
Substituting for y' and integi'ating wo find that the work is equal to
I E ^^—Jt + ^»i/<i''SC«H'Biu« {nat + i).
556
MOTION OP A STRINQ.
If we take the position of equilibrium of the string when Btretchcd between tho
extreme points A and B' as the position of reference, we find that the
energy = -. mla'2 (7*n*.
This we may call the energy of the disturbance.
Prof. Donkin in his treatise on Acoustics, page 128, has found the energy of a
string vibrating transversely, by an ingenious application of the method of sub-
tractions.
Ex. 1. An elastic rod AB has the end A fixed and B free. Being placed on a
perfectly smooth table, it vibrates longitudinally. Show that the energy of a disturb-
ance represented by ( = 2 C sin nx sin {^% + 7) where n{ = (2i -f- 1) ^ is ^ m2a*Z £7'n'.
.,
I
)etween tho
energy of a
biod of sub-
placed on a
of a disturb-
ftio«2C«n».
NOTES.
On D'Alemhert's Principle, by Sir G. B. Airy.
I HAVE seen some statements of or remarks on this principle which
appear to me to be erroneous. The principle itself is not a new physical
l)rinciple, nor any addition to existing physical principles ; but is a con-
venient principle of combination of mechanical considerations, which
results in a comprehensive process of great elegance.
The tacit idea, which dominates through the investigation, is this : —
That every mass of matter in any complex mechanical combination may
be conceived as containing in itself two distinct properties :— one that of
connexion in itself, of susceptibility to pressure-force, and of connexion
with other such masses, but not of inertia nor of impressions of momen-
tum: — the other that of discrete molecules of matter, held in their places
by the connexion-frame, susceptible to externally impressed momentum,
and •possessing inertia. The union produces an imponderable skeleton,
carrying ponderable particles of matter.
Now the action of external momentum-forces on any one particle
tends to produce a certain momentum-acceleration in that particle,
which (generally) is not allowed to produce its full effect. And what
prevents it from producing its full efibct ] It is the pressure of the
skeleton-frame, which pressure will be measured by the difference be-
tween the impressed momentum-acceleration and the actual momentum-
acceleration for the same. Thus every part of the skeleton sustains a
pressure-force depending on that difference of momenta. And the whole
mechanical system, however complicated, may now be conceived as a
system of skeletons, each sustaining pressure-forces, and (by virtue of
their combination) each impressing forces on the others.
And what will be the laws of movement I'esulting from this connexion?
The forces are pressure-forces, acting on imponderable skeletons, and
they must balance according to the laws of statical equilibrium. For if
they did not, there would be instantaneous change from the understood
motion, which change would be accompanied with instantaneous change
of momentum-acceleratica of the molecules, that would produce different
pressures corresponding to equilibrium. (It is to be remarked that
momentum cannot be changed instantaneously, but momentum-accelera-
tion can be changed instantaneously.)
558
NOTES.
>1
,1'
Wo como thus to tho concluaion, tliat, taking for ovciy molecule tlio
difference between tho impressed momentum-acceleration an<l the actual
momentum-acceleration, those differences throiigli the (uitiro machine
will statically balance. And — combining in one group all tho impressed
momentum-accelerations, and in another group all the actual momentum-
accelerations — it is the same thing as saying that the impressed momen-
tum-accelerations through tho entire machine will balance the actual
momentum-accelerations through tho entire mnchino. This is the usual
expression of D'Alembert's jninciple.
Elders Geometrical Equations,
Art. 235. It is sometimes necessary to express the angular veloci-
ties of the body aboiit the fixed axes OX, OY, OZ in terms of ^, <^, i/r.
Tliis may be effected in the following manner. Let W;,, w,^, w^ be the
angular velocities about the fixed axes, 12 the resultant any velocity. If
we impress on space and also on the body in addition to its existing
motion, an angular velocity equal to — O about the resultant axis of
rotation, the axes OA, OB, OC will become fixed, and the axes OX, OY,
OZ yiiM move with angular velocities -w^j, — <o,^, -w,. Hence, in the
formulae of tho text, if we change
</) into - 1/^
$ ... ~e
^ ... -<t>
!•
oi| will become - to.
*"2
0)J
Thus, we have
de
w.
dt^
-J sin i/r + "".y sin d cos \^,
dO , dtt> . . . ,
cos i/' + -77 sin sin ip,
dt
dt
d<i> . dtp
di dt
Sometimes it will be more convenient to measure tho angular co-
ordinates $, <ji, xf/ in a different manner. Suppose, for example, v/e wish
to refer the axes fixed in space to the axes fixed in the body as co-ordi-
nate axes. To obtain the standard figure corresponding to this 3ase, we
must in the figure of Art. 235 interchange the letters X, Y, Z with
A, B, C each with each. The angles 6, ^, ^ being measured as indicated
in the figure after this change, the relations connecting them with the
angular velocities about the axes fixed in space, are obtained from those
in the text by simply changing <a^, w„, Wg into —u}x, — Wy' ~'^^' ^^ ^^
choose to measure 6 in the opposite direction to that indicated in the
figiire, the expressions for ta^, w,^, become identical with those for to,, w^,
in the text.
1
oculo tho
ho actual
inachiiH'
impressetl
>mentum-
1 momen-
he actunl
tlie usual
liar velocl-
of e, ^, yi>.
0)^ be tlio
locity. If
;8 existing
iiit axis of
^OX,OY,
nee, in the
mgular co-
le, v/e wish
T as co-ordi-
iis 3ase, we
7, Z with
IS indicated
im with the
from those
0) . If we
ated in the
for w, , W-,
NOTKS.
"..')()
0)1 the Impact of Bodies.
Arts. 156 and 30.5. The pi-oblem of the impact of two smooth
inelastic bodies is considered by PoLssoii in his Traitc de Mecanltjur,
Seconde Edition, 1833. The motion of each body just before impact
being supposed given, he forms six equations of motion for each body to
determine the motion just after impact. These contain thii'teen un-
known quantities, viz. the resolved velocities of the centre of gravity of
each body along three rectangular axes, the three i-esolved angular
velocities of each body about the same axes, and lastly the mutual
reaction of the two bodies. Thus the equations are insufficient to
determine the motion. A thirteenth equation is then obtained from the
principle that the impact terminates at the moment of greatest compres-
sion, i. e. at the moment when the normal velocities of the points of con-
tact of the two bodies which impinge, are equal.
When the bodies are elastic, Poisson divides the impact into two
periods. The fii'st begins at the first contact of the bodies and termi-
nates at the moment of gi-eatest compression. The second begins at the
moment of greatest compression and terminates when the bodies separate.
The motion at the end of the first period is found exactly as if the bodies
were inelastic. The motion at the end of the secud period is found
from the principle that the whole momentum communicated by one body
to the other during the second period, bears a constant ratio to that com-
municated during the fii-st period of the impact. This ratio depends on
the elasticity of the two bodies and can be found only by experiments
made on some bodies of the same material in some simple cases of
impact.
"When the bodies are rough and slide on each other during the impact,
Poisson remarks that thei'e will also be a fiictional impulse. This is to
be found from the j^rinciple that the magnitude of the friction at each
instant must bear a constant ratio to the normal pressure and the direc-
tion must be opposite to that of the rehitive motion of the points in
contact. He applies this to the case of a sphere, either inelastic or
perfectly elastic, impinging on a rough plane, the sphere tui-ning before
the impact about a horizontal axis perpendicular to the direction of
motion of the centre of gravity. He points out that there are several
cases to be considered; (1) when the sliding is the same in direction
during the whole of the impact and does not vanish, (2) when the sliding
vanishes during the impact and remains zero, (3) when the sliding
vanishes and changes sign. This third case, howovc r, contains an un-
known quantity and his formulae therefore fail to determine the motion.
Poisson points out that the problem would be vei'y complicated if the
sphere had an initial rotation about an axis not perpendicular to the
vertical plane in which the centre of gravity moves. This case he does
not attempt to .solve, but jiasses on to discuss at greater length the im-
pact of smooth bodies.
M. Coriolis in his Jeu de Billard (1835) considers the impact of two
rmt{}h spheres sliding on each other during the whole of the impact. He
obtains the restilt given in Ai't. 312, Ev. 3.
i)60
NOTES.
1^
mk
H '
i
1
;fl| i
i|||j
W
'il| i >
iHii
^■l^^^l^^l
M. Ed. Phillips in tlio fourteenth volume of Liouville' a Journal, IS-ID,
considers the problem of the impact of two rouyh inelaatic hodios of any
form when the direction of the friction is not necessarily the same
throughout the impact, provided the sliding does not vanish during the
impact. He divides the period of impact into elementary portions and
applies Poisson's rule for the magnitude and direction of the friction to
each elementary period. He points out how the solution of the equa-
tions may be effected, and in particular he discusses the case in which the
two bodies have their principal axes at the point of contact parallel each
to each and also each body has its centre of gravity on the common
normal at the j)oint of contact. Ho deduces from this the two results,
given in Art. 312, Ex. 4 and 5.
M. Phillips does not examine in detail the impact of elastic bodies,
though he remarks that the period of impact must be divided into two
portions which must be considered separately. These however, he con-
siders, do not present any further peculiarities.
The case in which the sliding vanishes and the friction becomes
discontinuous, does not appear to have beeu examined by him.
Sir W. R. HamiltorCa Equations.
Art 378. The demonstration as given by Sir W. R. Hamilton
requires that T should be a homogeneous quadratic function of the
accented letters and this is generally the case in dynamics. The exten-
sion to the case in which the geometrical eqiiations do not contain the
time explicitly is due to Prof. Donkin. Prof. Donkin has made a
further extension of this theorem which is sometimes useful. If T be'
a function of any other letter, say f, as well as $, <f), &c., then we sliall
dT dT
have -j^ = — ~ , the differentiation with respect to ^ being in each
case performed only so far as f appears explicitly. The theorem may be
demonstrated as in the note to page 374.
On the Principle of Least Aotion.
The argument in Art. 394 shows that 8 / Tdt = under certain
conditions. According to the usual phraseology it is asserted that
Tdt is either a maximum or a minimum. But this is not stric: >y
/
correct. It seems clear that since the Via Viva cannot be negative, there
must be some mode of motion from one given position to another, for
which the action is the least possible. When, therefore, the equations
supplied by the Calculus of Variations lead to but one possible motion,
that motion must make / Tdt a minimum. But v/hen there are several
NOTES,
561
])OH8iblo modes of motion, though none can bo a maximum for tho
reaHon given in the text, some of these may be neither maxima nor
minima.
To dotermino whether tho integral is a maximum or a minimum or
neither, we must examine tho terms of the second order in the variation
of the integral to ascertain if their sum keeps one sign or not for all
variations of the independent variables. This is a very troublesome
prooes!*, and we do not propose to discuss it. It will be sufficient to call
the reader's attention to some remarks of Jacobi, given in the seven-
teenth volume of Crelle'a Journal, 1837, and translated in Mr Tod-
hunter's History of the Calculua of Vernations, page 250.
Suppose a dynamical system to start fi'om any given position which
we shall call A, and to arrive at lome position B. If the time be
given, the motion is found by making 8 / Ldt = ; if the energy be given,
by making 8 / Tdt = 0. The constants which occur in integrating the
differential equations supplied by the Calculus of Variations are to be
determined by means of the given limiting values ; but as this involves
the solution of equations tliere will in general be several systems of
values for the arbitrary constants, so that several possible modes of
motion from ^ to ^ may be found which satisfy the same differential
equation and the same limiting conditions. Now let one of these modes
of motion be chosen, and let the position B approach j4, so as to be
always on this chosen mode of motion. Suppose that when B reaches
the position G another possible mode of motion from A to B in indefi-
nitely near to the chosen motion. Then C determines the boundary up
to which or beyond which the integration must not extend if the inte-
gral is to be a minimum.
The reason seems to be as follows. If U be equal to the integral
under consideration, we have along each of the motions from A to B
81/ =0. Hence when B coincides with C, we have both 817=0 and
8{U+8U) = 0. It easily follows that the terms of the second order can
be made to vanish by a proper variation. When the limits of integra-
tion are more extended than AC, it is not difficult to show that the
terms of the second order can be made not merely to vanish, but to
change sign.
Jacobi illustrates his rule by considering the principle of least action
in the elliptic motion of a planet. Let S be the sun, and let the particle
start from A towards aphelion to arrive at a point B. The path is
known to be an ellipse with aS" for focus. Since we use the principle of
least action, the energy of the motion is given : hence the major axis of
the ellipse is known, let this be 2a. The other focus H of the ellipse is
the intersection of two circles described with centres A and B and radii
2a — SA,2a — SB respectively. The two intersections give two solutions
which only coincide when the circles touch, that is when the line AB
passes through the focus H. Thus if we draw a chord AC through H
to cut the ellipse described by the particle in C, then the terminal posi-
tion B must fall between A and C if the integral which occurs in the
principle of least action is really to be a minimum for this ellipse. If ^
R. D.
36
562
NOTES.
coincide with (7, then the second variation cannot become negative, but
it can become zero, so that the variation of the integral is then of the
third order, and may therefore be either positive or negative. If B be
beyond C the second vaiiation itself can become negative.
If the particle start from A towards perihelion, then the extreme
point G is determined by drawing a chord AC through the focus S to
cat the ellipse in C. For if A and C are the limits we can obtain an
infinite number of solutions by the revolution of the ellipse round AC.
If then in the last case the second limit B fall beyond G there will be a
curve of double curvature between the two given points for which
i Tdt is leas than it is for the ellipse.
On Sphero- Conies.
The following properties of a sphero-conic will be foimd useful in
connexion with the theorems of Art. 527. They appear to be new. The
curve is represented by the line DED'E'. As in the text, the eye is
supposed to be situated in the radius through /., viewing the sphere
from a considerable distance. The three principal planes of the cone
intersect the sphere in the three quadrants AB, BC, CA, and any one of
the three points A, B, C might be called the centre. The arcs AB and
AE are represented by a and b.
1. Equation to the conic. Draw the arc PN perpendicular to AD
and let PN=y, AN-x, Let NP produced cut the small circle de-
scribed on Diy as diameter in F, let NP' be called the eccentric
ordinate and be represented by y'. We then have
tany ^ . tanJi
= constants
tan?/'
cos a = cos y
, tanoj
tan a> •
' cos X f
NOTES.
565
ative, but
en of the
If 5 be
) extreme
jcus S to
obtain an
otind AG.
I will be a
for which
useful in
lew. The
the eye is
the sphere
: the cone
Bmy one of
3 AD and
ular ix) AD
[ circle de-
e eccentric
i
2. The projection of the normal PG on the focal radius vector SP,
i. e. PL, is constant and equal to half the latus rectum.
If 21 be the latus rectum, then tan I =
tan* 6
tana '
Also
tan GL
sinPiV
= constant.
3. IS QAF be an arc cutting PG at right angles, QA may be called
the semi-conjugate of ^ P. Then
tsinPG.tsaiPF=t&n'b.
4. The length PK cut off the focal radius vector by the conjugpte
diameter is constant and eqiial to a. This follows from (2) and (3).
8111 &
5. If 1 -e' = -T-j— , e may be called the eccentricity of the sphero-
conic. Then
tan -4^ = e* tan ^iV.
6. Also S being a focus
tsi3i(SP-a)=:etanAN.
7. Polar equation to the conic
tan 2
= 1-
e A
cos PSA.
tsinSP ^ cos* 6
8. If p be the i-adius of curvature at P, then
tan'w
9. Regarding AP, AQ aa conjugate semi-diametera,
sin' AP + ain'AQ = sin* a + sin*6
ainAQ . sin PF= sin a . sin 6
}•
10. If ^ be the perpendicular from the centre A on the tangent
atP,
tan' a tan' 6 . . . g . x a ^ «
= tan* a + tan* o - tan' AP.
11. Also
12.
Cor.
tan'^
tan» PG - tan' Z = -An sin' P^-
cos
sin' a - sin' AP
= sin* J^- sin' ft
tan' b
/ 1-,
,sin'Pir.
tan' PG =
cos' b sin* a
(coa'AP- cos' a cos* b).
564
NOTES.
If sin AM =Bm. AM = -. — , the planes of the arcs BM and SM' are
sin a
parallel to the circular sections of the cone . Some of the properties of these
arcs resemble those of asymptotes when £ is regarded as the centre of
the conic. The properties which connect the sphero-conic with the arcs
BM and BM' will be found in Dr Salmon's Solid Geometry.
Many other properties of sphero-conics will also be found in Mr Frost's
Solid Geometry.
MiscellaneoiLs Notes.
Art. 3. The term moment of inertia with regard to a plane seems to
have been first used by M. Binet in the Journal Folytechniqiie, 1813.
Arts. 19 and 182. So much has been written on the ellipsoids of
inertia and on the kinematics of a solid body that it is diflScult to
determine what is due to each of the various authors. The reader will
find much information on this point in Prof Cayley's report to the
British Association on the Special Problems of Dynamics, 1862.
CAHBBIDOE: PniNTED BY C. J. CLAY, M.A. AT THE UNIVERSITY I'REBS
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5, 1813.
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