, we have the reciprocal of the vertex IIB, which is AHIBA^
giving the stresses on the pieces which meet at that vertex.
The direction of HI is known, and we see by following the
1
Fig.
30.
strain polygon that IB^ BA and AH all act towards the vertex.
The stresses on the corresponding pieces are therefore com-
pressions.
For the vertex AC we have in the strain diagram NO^ OA
5^ MECHANICS OF SOLIDS.
and AB, Draw BC and NC parallel to the corresponding lines
in the frame, and we have NOABCN ior the reciprocal of the
vertex AC. BC is compression and CN tension.
Taking the remaining vertices in succession, we have:
¥or B/y the polygon CBIJDC] JD being compression and
Z>C tension.
For iV^^, the polygon MNCDEM \ DE and EM being ten-
sions.
For DK^ the polygon EDJKFE\ KF being compression and
FE tension.
For EG, the polygon MEFGM \ FG and GM being tensions.
For FL, the polygon GFKLG\ LG being compression.
For LM^ the polygon LMG.
From which all the stresses and their characters are com-
pletely determined.
(2) The Simple Warren Truss, having equal loads at the lower
vertices (Fig. 31).
Construct the force polygon of the applied forces IJKLM,
Fig. 32. Their resultant is /J/", which evidently acts through
the middle point of the truss. The reactions at the points of
support are each equal to half the total load, and are MA and AI.
For the vertex AI we have the polygon A IB A ; IB being
tension and BA compression.
For the vertex IC we have BIJCB\ JC being tension and
BC compression.
For the vertex AC ^t. have ABCDA\ CD being tension and
DA compression.
For the vertex Df we have DCJKED\ and for the vertex
AE we hsLVQ ADEFA. In the last two polygons, since E and Z>,
and ^ and Fare coincident, the parts DE and EF support no
stress.
The stresses on the remaining parts may be found by pro-
ceeding with this construction through the vertices H^F, AG and
ZZT, or by a construction similar to the above, beginning at the
vertex AM.
The upper chord is subjected to a stress of compression and
GRAPHICAL STATICS.
57
the lower chord to one of tension, these stresses being greatest
at the middle of the truss, while the stresses on the diagonals
are greatest at the ends of the truss.
(3) The Simple Warren Truss, loaded unequally at the lower
vertices.
Let the frame be the same as in the preceding example, Fig.
Fig. 31.
Fig. 32.
Fig. 33.
31, and let the two right-hand loads be each one half as great as
the others. The force polygon of the applied forces is IJKLM
{Fig. 33). The intensity of their resultant is IM. To find its
action-line and the reactions at the points of support, assume
the pole O and construct the polar polygon 6123456. -ffis the
58
MECHANICS OF SOLIDS.
action-line of the resultant, and drawing OA parallel to 5 6, we
have MA and AI as the intensities of the reactions.
The determination of the strains on the parts is sufficiently-
indicated by the nomenclature in the diagram.
Fig. 34.
(4) Figs. 34, 35, 36 and 37 represent loaded trusses with their
strain diagrams. As an exercise the student should supply the
nomenclature and determine the character of the strains. It
will be observed that at one vertex of the frame in Fig. 37 there
are three forces to determine, and the problem therefore requires
the application of proposition 4, Art. 62.
GRAPHICAL STATICS.
59
6o
MECHANICS OF SOLIDS.
.
/}
\\
■<
'),. ,
<^'
2
h
7
^"^"""^^ #
iv^
■ .
^^~-~>^^
Fig. 37.
Work and Energy.
64. Work is said to be done by a force when its point of ap-
plication has any motion in the direction of its action-line. The
unit of work is the quantity of work done by a force of unit
intensity while its point of application moves over the distance
WORK AND ENERGY, .6l
unity in the direction of the action-line of the force. The unit
of work used throughout the text is called the foot-pound^ the
unit of distance being one foot and the unit of intensity one
pound. The unit of work of the C. G. S. system is the work
done by a dyne over a centimeter, and is called the Erg,
The simplest illustration of work is that of lifting a weight
through a vertical height. Thus, it requires the expenditure of
one foot-pound of work to lift one pound through a height of
one foot. Hence, the work done in lifting a weight through
any height is equal to the product of the weigiit and height;
and, in general, the work done by any constant force is found by
multiplying its intensity by the path of its point of application,
estimated in the direction of its action-line. If the force be
variable it may be regarded as constant while its point of appli-
cation describes a path dp, estimated along its action-line, and the
elementary quantity of work is
dW=Idp (86)
The summation of all the elementary quantities of work gives
the total quantity done by this force, and we have
W^^dW=:2Idp (87)
Where the principles of the calculus can be applied, we have
W=fdW=Jldp (88)
Hence, whenever such a relation can be established between the
variable intensity /and the path / so that the second member
can be expressed as a known integrable function of a single va-
riable, the total quantity of work can be determined by integra-
tion.
62
MECHANICS OF SOLIDS.
65. The symbol expressing work, / Idp, is analogous to the
symbol j ydx in the calculus. The latter is the representative
of the quadrature of a curve whose varying ordinates are j' and
abscissas x, both expressed in the same unit. From this analogy
we may graphically represent work by the area contained be-
tween the axis of p and a curve whose ordinates measure the
varying intensity of the working force at the different points of
the path p. The unit of work is graphically represented by a
square whose side is the unit of the scale from which the inten-
sity and path are taken.
When the expression / Idp is not integrable, the quantity of
work can be determined approximately
by the usual methods for the estima-
tion of the area included between the
curve, the extreme ordinates, and the
path, as in mensura tion. Thus let the
ordinates of the curve j^', Fig. 38, repre-
sent the varying intensity of the force
while its point of application passes
over the path pp\ Poncelet's formula for the approximate
area is
^
V-
"^
I1
u
I3
I4
I7
^^-^
£-
Fig. 38.
Q = d[2{/, + /, + /. + .../.)+ i(/_ +/.+,)_ i(/.+/«)],(89)
in which d is the distance between the consecutive ordinates 7^,
/„ /g, etc., when the whole path,^', is divided into any even
number, n, of equal parts.
If the varying values of / be known only at certain points of
the path, the extremities of these may be joined by right lines,
thus forming trapezoids whose aggregate area ox\\y approximately
represents the quantity of work.
66. Energy, — Force and matter are inseparably connected.
Any system of masses is accompanied by forces, and these forces
WORK AND ENERGY. 63
perform work during any change in the configuration of the sys-
tem. Energy is the capacity for doing worky and it is measured
in units of work. Taking a single molecule of the system mov-
ing under the action of the resultant of all the forces applied to
it, we have
Idp = m-^ds\ ....... {90)
and for the whole system,
. 2n^p=^-^^<}i-^i!^ (9.)
t/a 2 2
When the molecule is at the position i, the quantity of work
represented by the first member of Eq. (91) is called potential
energy, since it measures the capacity of the force to do work
while the molecule passes from i to 2. It is simply energy of
position; that is, by virtue of the position of the molecule in the
system the forces acting upon it have a certain power to do work
while its position is changing. When the molecule arrives at
the position 2, Xh^ potential energy represented by / Idp has been
converted into energy of motion, called kinetic energy, which is
measured by ^ ^' ■ , the kinetic energy of the molecule
being ^ ^' at i and ^ "^ at 2. '
2 2
HQnce, potential energy is defined to be that part of the energy
of a system which it possesses by virtue of the relative positions
of its different masses, and kinetic energy to be the energy which
the system possesses by virtue of the motions of its different
64 MECHANICS OF SOLIDS.
masses. The term work is applied to the change of energy from
one form or body to another.
For example, let the system be composed of a unit of 7nass
and the earth, and let the limits be determined by two horizontal
planes separated by a distance of lo feet. Taking the body at
the upper limit, we have
/>
32.2 X 10 = 322 ft.-lbs.
If the body start from rest and fall freely in vacuo to the lower
plane, we shall have this potential energy converted into kinetic
energy, or ^
mv V
— =- = 322:
2 2
r
while at any intermediate point, part of the energy is potential
and part kinetic. Thus, when the body has fallen to a point
midway between the two limits, its potential energy with respect
to the lower plane is 161 tt.-lbs., and its kinetic energy is also
161 ft.-lbs. Each form of energy is measured in units of work,
but no work is done unless there be a transformation of energy. This
illustrates what is meant h^ energy of position and energy of motion.
To illustrate further, the muscular potential energy in a man's-
arm may be changed into potential energy of elasticity in a bent
bow, and the potential energy of the bow may be changed into-
kinetic energy of a moving arrow, work being done in both cases.
Kinetic energy cannot, however, be transferred from one body
to another without passing through the potential form.
67. The Law of the Conservation of Energy. — Scientific investi-
'gation points to the conclusion that the total quantity of energy
in the universe, as well as the total quantity of matter, is invari-
able; that is, that neither matter nor energy can be created or
destroyed by any known means. Accepting this as a scientific
truth, we must admit that the energy gained or lost in any
WORK AND ENERGY. 65
limited system of masses in which the energy varies must have
been obtained from other masses or transferred to them.
A conservative system is one containing a certain definite
amount of energy. It. consists of limited masses subjected to
the action of definite forces. The law of energy for such a sys-
tem is
n-^rK=C, (93)
in which TL represents the potential and K the kinetic energy at
any time, and C the constant quantity of energy in the system.
II and K may both vary with the time, but C is constant ; and if
any change occur in the potential energy we shall have a cor-
responding and equal but opposite change in the kinetic energy;
thus,
Ut. ~-Ut» = Kt>f-Kt', (94)
each member representing the change in the corresponding en-
ergy during the time /" — /'. During this time the forces of the
system act upon the masses and cause them to change their con-
ditions of motion and relative positions, the change in the po-
tential energy being
'fj'P (95)
:2
The change in the kinetic energy of a single molecule of the
system during this interval is
£,'4?'^'' (96)
and for the whole system,
^S!..'"%^'- (97)
The change in the potential energy in the time dt is evi-
dently
:2idp (98)
5
66
MECHANICS OF SOLIDS.
and the change in the kinetic energy during the same time is
-^^^^^•^ (99)
Since these two quantities are always equal, we have
:EIdp = :2m^^,ds; (E)
an equation which expresses the Law of the Conservation of Energy.
This law may be stated as follows:
The total energy of any conservative system is a quantity which can-
not be increased or diminished by any mutual action of the bodies of the
system, and any change of either potential or kinetic energy must always
be accompanied by an equal change in the other*
It is evident from this statement that the universe is the only
rigidly conservative system. But many limited systems are so
remote from all other bodies that the effect of these latter* is
insignificant when considering the relative motions of the former.
Eq. (E) is the fundamental equation of mechanics, and it
involves all relative changes in the configuration and motion of
any conservative system.
68. The Principle of Virtual Velocities. — If no change of state
occur in any of the molecules, the factors —^ will each become
zero, and the equation reduces to
:SIdp = o', (S)
or, the total quantity of work done by the forces upon the system
of masses is zero. Any one of the elementary quantities of
work represented by the type-symbol Idp is exactly equal in
amount, but of a contrary sign, to the aggregate quantity of
work of all the other forces represented by ^/V/. Such a
system of forces is said to be in equilibrio, and the masses in
equilibrium. If the latter be in motion, this motion must be
WORK AND ENERGY. 67
uniform. Regarding the intensities of forces as always positive,
the sign of the products Idp depends on the
sign of dp. The sign of dp is taken positive fny-^^-^m'
when it falls on the action-line of the force, « — ^ — ' E »
and negative when it falls on the action-line
produced (Fig. 39).
The elementary paths whose projections are dp are called
virtual velocities. Being the actual paths described in the time dt,
they have the same ratio to each other as the velocities of the
points of application at the instant considered.
The products Idp^ J'dp\ etc., are called virtual moments; they
are the elementary quantities of work done by the forces while
their points of application move over the distances whose pro-
jections on the action-lines are dp^ dp\ etc.
Equation (S) is the form taken by the fundamental equation
in Statics, and is the mathematical statement of the principle of
virtual velocities; that is, when any system of forces is in equilibrio
the algebraic sum of their virtual moments is equal to zero. In such
a system the potential energy is constant, none being trans-
formed into kinetic energy.
The converse of this principle is also true; that is, when the
algebraic sum of the virtual moments of any system of forces is
equal to zero the forces are in equilibrio.
69. Equation (E) referred to Rectangular Co-ordinate Axes. —
Let a, b, c be the angles made by the virtual velocity of the
point of application of a force with the axes, and d the angle
between this virtual velocity and the action-line of the force.
Then we have
cos d = cos a cos a -j- cos P cos b -{- cos y cos c\ , (100)
and multiplying by ds^
ds cos d = cos ads cos a -\- cos fids cos b ■\- coc y^^ cos ^,(101)
or
dp = cos adx -\- cos /3dy + cos ydz (102)
68 MECHANICS OF SOLIDS.
Multiplying both members of this equation by the intensity
of the force, we have
Idp = / cos adx + / cos ^dy -\- 1 cos ydz. . . (103)
That is, f/ie virtual moment of any force is equal to the sum of the
virtual moments of its rectangular components.
For the whole system we have
'^Idp = ^/ cos adx + -2"/ cos ^dy + ^I cos ydz. (104)
70. Let «', b'^ c' be the angles made by the elementary path
of any molecule with the axes, and we have
I = cos'^ «' + cos'^ b' + cos'' c'y . , , , (105)
and, multiplying by m—=-ds^
d^s . d'^s , a - , d^s . » ,, , d^s J » . , ..
^n-zpds = ^;^^-f cos" a' + ^-rpf^^ cos' h' + ^-j^ds cos' r . (106)
But
ds cos a' = dx; ds cos b' = dy; ds cos c' = dz\ ) / ^\
^V cos a* - d'x\ d^s cos b' = d''y\ d'^s cos .;' = d'^z. ) ^ ^'
Hence
d'^s , d^x . . d^y , , d^z. , ..
That is, the increment of the kinetic energy of any molecule is equal
to the sum of the increments estitnated in any three rectangular direc-
tions.
WORK AND ENERGY.
69
For the whole system we have
'2m— ^ds = 2m—pdx + 2m-^^dy + 2m— ^dz. . (109)
Substituting in Eq. (E), we have
21 cos adx 4" -2"/ cos /3dy -\- 21 cos ydz =
~dF + ^p"^ '^^^^^dP ' '
(110)
(^.
/
x^
"7\
This transformation has not in any way affected the gener-
ality of Eq. (E) which, in its new form, still embodies all the cir-
cumstances of motion of the molecules of a body, or of a sys-
tem of bodies, under the action of any system of extraneous
forces whatever.
O^' 71. Application of Equation E to the Motion of a Rigid Solid. —
A rigid solid is a body whose molecules are supposed to preserve
unchanged their relative distances from each other. The most
general motion that can be im-
agined .for such a hypothetical 2 z'
solid is one compounded of a
motion of translation and rota-
tion. Its motion of translation ,/ I
may be defined by that of one
of its molecules, and its motion
of rotation by that of the body
about this molecule. In Fig.
40 let O be any fixed origin, O'
the position at any instant of Fig. 40.
the particular molecule which
determines the motion of translation, and ;// the position of any
other molecule at the same instant. Let ic, y, z be the co-
ordinates of ;;/ referred to the fixed origin; x^^y^, z^, the co-ordi-
nates of the movable origin 6>' referred to the fixed, and x\y', z'
the co-ordinates of m referred to the movable origin.
X'
4/
70
MECHANICS OF SOLIDS.
Then, supposing the axes at the movable origin O' to be
always parallel to the fixed axes at 6>, we have
x = x^^rx'\ y=yo+y'; ^ = -^0 + ^'; • (m)
dx = dx^ + dx'; dy = dy^-\- dy*\ dz — dz^ + dz\ . (112)
Measuring the angles about O* as indicated in the figure to
conform to Art. 35, we have, for the increments of x\y\ z\ due
to rotation about the axes X\ K', Z',
dd
X\
dd=o\
^dS = - add sin ^ = - z'dd-,
du
dO = add cos d = fdO.
djl
dd
dx'
— dtp=. cdi/) cos tp = z^dtp ;
dz'
— dtp=— cdip sin = x'dcp;
dcf)
d^
("3)
(114)
("5)
Hence we have, for the total differentials,
dx' = z'd^-/d(p',
d/ = x'dtp- z'dd;
dz' =yde - x'dip.
(116)
WORK AND ENERGY.
71
Substituting these values in Eqs. (112), we have
dx = dx^ + z'd^ —y'd)
+ 21 cos P(dy^ + x'd(t) - z'dd)
+ 21 cos y{dz^ -\-yde - x'dip)
^2m—-,(dx,^z'd^--yd)
J^2f/-^ldy,^x^d^z'de)
+ ^tn^^Xdz, -VydB - x^dtp).
(118)
But dx^^ dy^^ dz^ relate to the movable origin, and dd, dtpy d(j>
are independent of the position of m because the body is rigid ;
each of these differentials is therefore a common factor of the
terms which it enters, and Eq. (118) may be written
{21 cos a - ^ffi^dx, + {21 cos /? - ^^5^)^A
+ [21 cos y - 2m-^^dz,
+ [^/(^' cos /? -/ cos a) - 2m'^^^^^^^yil> \ (i 19)
I V^Tf f f \ ^ z^d^x — x'd'z'l ,,
-\- 2il(^z' cos a — x' cos y) — 2m -— \dip
+ [:^/(/cos y-z^ cos /?) - 2m^ ?-/'^'y ']de = o.
72
MECHANICS OF SOLIDS.
72. Application of Eq. (E) to a Rigid Solid ^ perfectly free to
Move. — The only restriction which has thus far been imposed is
that this equation, which has been derived from Eq. (E), shall
apply 'to. a -single rigid solid acted upon by any extraneous
forces. The body may be subjected to any conditions whatever
as to its possible motion under the action of these forces, and
the values of dx^, dy^^ dz^, dd, dij;, dcf) will depend on these con-
ditions. If no conditions be imposed, that is, if the body be
free, then dx^, dy^, dz^, dd, dip, dcf), will be entirely arbitrary and
independent of each other. Hence we have, by the principle of
indeterminate coefficients,
X = 21 cos a =: 29
dy
dt''
Z = 2/ cos y = 2m
x^dy —y^d^x
Y= 21 cos /3 = 2m -f-,; (-
d\^
dt^''
2J(x' cos fi — y* cos a) ■=• 2
2I(z' cos a — x' cos y) = 277i
21{y' cos y — z' cos ft) = 2
df
z'd'^x — x' d'^z
_ y'd'^z — z'dy
dt''
(T)
(R)
73. Interpretation of Equations (T) and (R). — These six condi-
tions, applicable to the case of a free rigid solid, having been
derived from the general equation of energy, embody all the
circumstances of motion of its molecular masses, caused by the
action of extraneous forces. Considering Eqs. (T), we see that
their rniddle members are the sums of the component intensities
of the extraneous forces in the directions of the rectangular axes,
and the last members are the sums of the products of the
molecular masses of the body by their accelerations in the cor-
responding directions. But these products are the type-symbols
WORK AND ENERGY. 73
of the intensities of extraneous forces acting on the molecular
masses. Hence in a free rigid solid any system of extraneous
forces may be replaced by an equal set whose points of applica-
tion are the molecules of the body, and the circumstances of the >.
motion of translation of the latter will not be changed. We see
also that whether there be one or many extraneous forces acting
on the solid, the connections which unite its molecules together
cause the effect of these forces to be distributed throughout the
whole body. Eqs. (T) therefore refer to motion of translation, and
express the fact that the algebraic sum of the component intensities of
the extraneous forces, estimated in any direction, is measured by the sum
of the products of the mass of each molecule by its acceleration in that
direction.
Referring now to the second members of (R), and consider-
ing the first of these, we see that it is the summation of terms of
the form of
, .d'^y , ,d*x . .
But ^-^'—f-i is the measure of the intensity of the force which for
the instant dt acts on m' in the direction of the axisjj'', and x* be-
7 2
ing the co-ordinate of ;;/' referred to O' , the product m'x* , is
the moment of that force with respect to the axis z* . Similarly
d'^x . , . . . ,d^x
-j-j- IS the moment of the force ^^t—-^
d'^x d^x
m'y'-j-^ is the moment of the force w'— jy- with respect to the
force acting on ;// at that instant with respect to the axis z\
We see, therefore, that each molecule may be regarded as being
subjected to a force of certain intensity, and the algebraic sum of
the moments of these forces at any ifistant, with respect to any axis, is
exactly equal to that of the extraneous forces at the same instant with
respect to the same axis ; and this is what is expressed by Eqs. (R).
74. If the solid be not free, tlie conditions to be satisfied are
less than six in number. For example, if one point of the body
74 MECHANICS OF SOLIDS.
be assumed as fixed, this point may be taken as the origin 0\
and we shall have
dx^ = dy^ = dz^ = o.
The first three terms of Eq. (119) then reduce to zero; and
since dd^ dtp, d(f) are still arbitrary and independent, this equa-
tion is satisfied when the three conditions (R) are satisfied.
If two points be fixed, the right line joining them will be a
fixed axis, and may be taken as the axis Y'. We shall then have
dx^ =. dy^=. dz^ = o, and d^ = ^^ = o; and since dtp is not neces-
sarily zero, Eq. (119) is satisfied when the second of Eqs. (R) is
satisfied; that is, the single condition of rotation about the fixed
axis ; and similarly for other conditions of constraint.
75. If the molecules of the solid be in uniform motion or at
rest, the forces and moments are balanced, and the body is in
equilibrium both as to translation and rotation. Eqs. (T) and
(R) then become
X ==■ 2/ cos £x = o;\
V=::S/cos/3 = o;y (T')
Z = 2/ cos y =z o;)
Vx — Xy = 2/{x^ cos /3 — y' cos or) = o; )
Xz — Zx = '2l{z' cos a — x' cos ;/) = o; )■ . , (R')
Zy — Yz = 2/{y' cos y — z^ cos /?) = o; )
which are the six conditions of equilibrium.
76. Analytical Mechanics consists essentially in the application
of Equations (E), (T), (R), (T') and (R') to conservative systems
of masses and forces. The object of the discussion is to ascer-
tain the position of any and all molecules at any time, the nature
and direction of their motions, and the configuration of the
bodies of which they are the elements. The theory of the in-
vestigation is simple, but the practical application is limited by
our mathematical knowledge and skill.
work: and energy.
75
77. Equations (T) and (R) referred to the Centre of Mass as a
Movable Origin. — Since there are as many terms in the last mem-
bers of Eqs. (T) and (R) as there are molecules in the body,
these equations are not in a convenient form for discussion. By
taking the movable origin at the centre of mass, the resulting
equations are no less general than before, while the solution of
practical problems is much simplified.
78. Equations of Translation. — In the first of Eqs. (T),
d^'x
X = J^/ cos a = ^^-77-% '
(12.)
Substitute for d*x its value obtained by differentiating Eq. (112),
and we have
d^x d'^x d'^x'
X = J?/ cos a = ^m-j^ = 2m-^ -f .^;//— 5-. . (122)
dt'
dt'
But d*x^ is a common factor in the term which it enters, and
from the principle of the centre of mass, Eqs. (64), we have
2md^x' z^ Md*x, (123)
and Eq. (122) reduces to
d^x d^x
X=:SIcos a = m"^ + M~.
at at
("4)
_ Taking the movable origin at the centre of mass, we have
^ = o, and Eqs. (T) become
Ar=^/cosar = i^-'^'"^» ^
dt^
Y=2I cos fi = M-^
Z = ^I cos y = M-
dt^
d\
dr
(T.)
76
MECHANICS OF SOLIDS.
from which the motion of translation of the centre of mass of a
free rigid solid, under the action of incessant forces, can be found.
If the forces be impulsions^ then Eqs. (T) become . ,
X = 2/^ cos a = ^ni—-'
y= :2I^ cos /3 = 2m^-;
' Z = ^I. cos y = ^m-r-\
at
which, when the movable origin is the centre of mass, become
dx
X= 21, COS a = AI-^ = MFy,
Y= 21, COS /? = m"^ = MVy\
Z = 2/, cos r = M^ = MV, ;
dt
(T.')
from which the motion of translation of the centre of mass of a
free rigid solid, under the action of impulsions, can be found.
From Eqs. (T^,) and (Tq/) we draw the following conclu-
sions:
(i) That the motion of the centre of mass of a free rigid solid.,
under the action of extraneoics forces^ is entirely independent of the
relative positions of the molecular masses, since their co-ordinates have
disappeared from the equations of motion.
(2) That the motion of the centre of mass depe7ids only upon the
mass of the body and the intensities and directions of the extraneous
forces, and is i7idependent of the points of application of the forces.
(3) That the motion of the centre of mass will be precisely the same
as that of a material point whose mass is equal to that of the body., sub-
WOJ^K AND ENERGY.
77
----■^R'
jected to the action of forces equal to the given forces in intensity and
having the same direction.
79. To illustrate tliese conclusions, let us consider the motion
of translation of the centre of mass of the free rigid solid ^4, Fig. 41,
first, under the action of the
several incessant forces /^^jP",
P'" and /*'", and, second, under
the action of the impulsions
^/, Pn^ ^iu and P,,.
Let O be the centre of
mass, and let forces equal to
the extraneous forces in inten-
sity and direction be supposed
applied to it. Let R' be the
resultant of those forces which
have O as their common point of application; then from the above
principles it can be asserted that under the action of the given
incessant forces the centre of mass will move along the right line
R*
OR* with a constant acceleration equal to -7>; and if the forces be
impulsions, that the centre of mass will move along the right
R
line OR^ with a constant velocity equal to -~=.
80. Equations of Rotation. — When the forces are incessant and
the movable origin is the centre of mass, Eqs. (R) readily-
red uce to
Fig
L- Yx - Xy=:SI{x' cos /?-/ cos a)='Sf
M=Xz — Zx=i^I{z' cos a— x' cosy)— '2m
iV= Zy-Yz = ^/(/cos y- z' cos fi) = 2m
x'dy-yd'x' ^
df
z'd'x'-x'd'z'
dt'
yd\'-z'dy
dt^
; KR«)
To show this, let us reduce the second member of the first of
Eqs. (R); we have
78
MECHANICS OF SOLIDS.
^ x'd'y-fd'x „ x'd'y.-Y x'dy - Vd'x- y'(fx'
^"^ dt' = ^'" Te
__ d'^y^^vix' — d'^x^'^my' -}- ^mx'd^y' — ^my'd^x'
x'dW - y'd'^x'
= ^^ le '
since, by the principle of the centre of mass, ^mx' = 2my' = o.
Similarly, w/ieu the forces are wipulsions^ Eqs. (R) become V
Z,=: Yx-Xy^ :2l,(x' cos y^-/ cos a) = '^m'^^jf—' ^
M^=Xz—Zx=:2lXz' co^ a— x' cosy) = :2m ; }{^2^)
N,=Zy - Vz==2I^{y cos y - z' cos/3) = 2^ /"^' ~ ^'^-^
which, when referred to the centre of mass as a centre of rota-
tion, become
L- Yx-Xy^2I,{x' cos /?-/ cos a) = ^n T ^-^ / ^^ ;
M^^^Xz —Zx=2I^(z' cos a —x' cos y) = 2?
N^=Zy — Yz=2/,{y' cos y—z* cos /3) = 2
dt
'dx'-x'dz' ,
d~t '
_ /dz'-z'd/
MR-')
dt
8l. If one point of the body be fixed, we have, by taking it
as the origin 0\
^»^ = ^V; dy = dy; d'zz=dV;
and Eqs. (R) reduce to the form of Eqs. (RJ) or (Rm')> independ-
ently of the principle of the centre of mass.
Also, if an axis be fixed, that one of Eqs. (R) which applies
GENERAL THEOREM OF ENERGY. 79
to the axis in question reduces to the corresponding one of Eqs.
(Rjn) or (Rni')' according as the forces are incessant or impulsive.
82. In Eqs. (Rm) and (Rm') the co-ordinates of the centre of
mass do not appear; therefore the motion of rotation of the body
about the centre of mass is independent of the position of this
centre, and will be the same whetlier it be considered at rest or
in the state of its actual motion. This exhibits the complete
independence of the motions of translation and of rotation, and
permits the investigation of either as if the other did not exist.
By means of Eqs. (Tm) the position, velocity and acceleration of
the centre of mass can be theoretically determined at any time,
and by Eqs. (Rm) the corresponding positions, angular velocities
and accelerations of every molecule with respect to the centre of
mass; and thus the configuration of the w^hole body about this
point can be determined for the same instant. The mathematical
difficulties, however, due to integration, limit their application to
but a few simple cases. It is to be noted that when the problem
involves incessant forces, Eqs. (T^) and (Rm), and when impul-
sions alone, Eqs. (T^') and (Rm')> ^^^ ^^ ^^ used.
General Theorem of Energy applied to a Free Rigid
Body whose Centre of Mass is referred to a Fixed
Point in Space.
83. Translation under Incessafit Forces. — Multiply Eqs. (T^) by
dx, dy, dz, respectively, add the results and integrate, and we
have
fiX^. + Ydy + Zdi) = j^,f^J^^:f+^y±J^
= f(^|:±^)+C=^Vc,(.6)
in which x^y, z are the co-ordinates of the centre of mass referred
to the fixed origin, V is the variable velocity with respect to the
sar.ie point, and Af is the mass of the body. The first member,
80 MECHANICS OF SOLIDS.
J (Xdx ■\- Ydy -\- Zdz), (127)
is, Art. (i(i, an expression for the quantity of work done by the
extraneous forces, or the total quantity of potential energy ex-
pended by them, and whose particular value in any case will be
determined when the limits of the integration are fixed. The
constant C of the second member is evidently the kinetic energy
which existed in the body at the instant the extraneous forces
began to act; for at that instant, say /j, their work is zero, and
if Fj be the corresponding velocity, then
C^-'JLLi. (i23)
2 ^ '
The first term of the second member, , is the total ki-
2
netic energy at any time, and hence the whole second member,
—- -^ (129)
being the difference between that possessed by the body at any
time and that when the extraneous forces began to act on the
body, is the exact equivalent of the potential energy represented
by the first member. Here we have, as should have been ex-
pected, the general law of the transformation of energy.
84. The first member may be integrated when the compo-
nent forces in the directions of the co-ordinate axes are constant,
and when, if these forces be variable, it becomes a known differ-
ential function of the three co-ordinates .r, 7, z. In the first case,
let R be the constant intensity of the resultant of the forces, and
a, b^ c the angles which its action-line makes with the co-ordi-
nate axes. Then we have between the limits 1 and 2
R(x cos a +/ cosb-\-z cos ^),''= F'(x,y, z)^=M^—^ ^—', (130)
GENERAL THEOREM OF ENERGY. 8 1
In the second case, in order that integration may be possible,
the intensities of the forces must be functions of x,y, 2, and we
have
r\x,y,z): = M^-^^:^ (131)
Hence we may write Eq. (126) in either case
F(.^,y,i); = F{x^.\)-F(xj.^:^ = M^^^-^, (132)
as the general law of energy when forces act on a free rigid
solid to give it motion of translation, under the conditions im-
posed above. From this equation we conclude that the velocity
generated in a free rigid solid by constant forces^ or by variable
forces whose intensities are functions of the co-ordinates of the centre
of mass ^ varies only with the values of the co-ordinates; and that,
should the centre of mass ever return to the same position in
space, its velocity will be the same as before, whether the path
by which it reaches this point be the same or not.
If the forces be in equilibrio, then, Art. 68,
Xdx^Ydy^Zdz:=.o, (133)
and we have
= a constant; (^34)
that is, the velocity is constant.
85. Rotation under Incessant Forces. — If we multiply Eqs. (Rm)
by dcf), dtp and dd^ respectively, add the results and reduce by
Eqs. (116), we obtain
21 cos adx' + 21 cos fidy' -\- 21 cos ydz'
^ (dx'd\x' + dy'dy + dz'd^z'
), (135)
82 MECHANICS OF SOLIDS.
which, by integration, becomes
J:2Idp =jRdr = i2m[^^^^±^±^)+C'=i2mv'+C', (136)
when, Eq. (104), we replace the sum of the virtual moments of
the extraneous forces, 2Idp, by the virtual moment of the result-
ant, J^dr, But dr, being the elementary path of the point of ap-
plication of the resultant projected on the action-line of the re-
sultant, is equal to the product of the path described by a point at
a unit's distance from the centre of mass by the lever arm of i?.
Let k be this lever arm, and ds the elementary arc at a unit's
distance, and we have
fj^Ms = i2mv' + C (137)
The first member is the general expression for the work of
rotation done by the resultant of the system, or the potential
energy transformed; ^^mv^ is the kinetic energy of rotation of
the body with respect to the centre of mass; C is the kinetic
energy of rotation in the body before any has been transferred
to it by the extraneous forces, and is equal to — \^mv^\ there-
fore we hflve'
CRkds = ^^mv" - i'Smv* ... . (138)
for the total kinetic energy of rotation put into the body by
the expenditure of the equivalent amount of potential energy
likds.
s
86. Adding together Eqs. (132) and (138), we have
£\xdx + ydy + Zdz) -^-fjjRkds
= M^ -^- -f i:Sm(v,' - v:), (139)
GENERAL THEOREM OF ENERGY. 83
in which the first member expresses the total expenditure of
potential energy of the extraneous forces in producing motion
of translation and of rotation, and the second member the equiv-
alent quantity of kinetic energy which has resulted therefrom.
If the action-line of the resultant of the extraneous forces pass
through the centre of mass, the second term of the first member
and the second term of the second member become zero.
Thus we see that the poi7it of application is of importance in
determining the effect of a force, since by supposing it to be
changeable the resulting kinetic energy of rotation imparted to
the body will be correspondingly varied.
87. Translation under Impulsive Forces. — Squaring Eqs. (T^')
and adding, we have
X« -f F» + Z' = i?/ = M\ VJ -f F/ + F,') = M' V\ (140)
whence
R,-MV, (141)
Hence, when a free rigid solid has been subjected only to a
system of .impulsions, its centre of mass will move with a con-
stant velocity, -7^.
88. Motion of Translation. — In the discussion of the motion of
translation of bodies two classes of problems arise. In the first,
which are called direct^ we have given the mass of the body and
the forces acting upon it; and it is required to find the path of
the centre of mass and all the circumstances of its motion. In
those of the second class, which are called inverse problems, the
path of the centre of mass is given, and it is required to find
the forces which will cause the body to follow that path.
89. The Direct Proble7n. — To solve the direct problem we sub-
stitute in Eqs. (Tn,) the mass of the body and the component in-
tensities of the forces, and obtain
84
MECHANICS OF SOLIDS.
d'x
X
d^y
V
d'z
Z
de "
~ M'
df
M'
de
~ M'
• (142)
in which the accelerations are constant or variable according as
the forces are constant or variable. Integrating, we have
(■43)
The arbitrary constants in these equations are the values of
the component velocities when / = o; that is, at the epoch or in-
stant from which / is estimated.
Integrating again, we have
(144)
D, D* and D*' being the co-ordinates of the centre of mass at
the epoch.
The values of C, C, C", Z>, D* and D" are called the initial
conditions^ since they are the component velocities and the co-
ordinates of the centre of mass, when the forces began their
action.
The integrals in Eqs. (144) can readily be found when the
^iven forces X^ Y, Z are constants, or, if variable, when they can
be expressed in such terms of / as to make them known integrable
GENERAL THEOREM OF ENERGY. 8$
expressions. Then by eliminating t from Eqs. (144) we obtain
two equations containing x^ y, z, and constants, which are the
equations of the path. The problem is then completely solved,
since we have the position, velocity and acceleration at any time,
and the entire path of the body.
90. The Inverse Problem. — Since the centre of mass may de-
scribe the same path under different conditions of velocity and
acceleration, the inverse problem is indeterminate. It may, how-
ever, be made determinate by assuming the initial conditions
and one component velocity or one component acceleration.
Let the equations of the path be
By differentiating and dividing, by dt, we obtain two equa-
tions involving three component velocities, and by a second dif-
ferentiation and division we get two equations containing three
component accelerations. To obtain an equation connecting
the velocity with the component accelerations, differentiate Eq.
(126) and divide by the differential of one of the variables, as
dx. We thus have
or
1 d{V') _d*x dy dy d'z dz
2 dx ~ dt'''^ dt^'d'x'^ df^'dx ' ' ' ^^"^^^
Now if one of the component velocities be assumed, we may
obtain the value of Ffrom the equations obtained by the first
differentiation of the equations of the path, since
---(ir+dr +(§)■•
86 MECHANICS OF SOLIDS.
Then we have three equations in which the three component
accelerations are the only unknown quantities, and the problem
may be completely solved.
If one component acceleration be assumed, we may find V
from Eq. (147) by integration, and this value, together with the
equations involving the component velocities, makes the solu-
tion possible. Thus, in either case the problem is determinate,
and all the circumstances of the motion may be found as in the
direct problem.
It is also evident that the problem may be solved by assum-
ing any new condition connecting the six unknown quantities,
since we already have five equations containing them.
91, Examples of the Direct Problem.
(i) Constant Forces. — Integrating Eqs. (142) twice, we have
dt-'M^^^' Tt-M^^^^ di-^M^+^ ^ ('48)
(149)
Let us suppose that the centre of mass at the epoch is at rest
at the origin of co-ordinates; then Eqs. (148) and (149) become
• • • (ISO)
• . . (iSi)
Eliminating from either pair of equations pertaining to the
dx X
dt ~ ~M '
^y - y ,.
dt ~ M '
dz z
dt ~~ M '
X t'
^"-J/2'
^~ M2 '
Zt""
^~ M2'
GENERAL THEOREM OF ENERGY, 8/
same axis, as x for example, the factor -7^, and indicating the ve-
locity in that direction by the subscript, we have
(152)
Laws of Constant Forces, — Eqs. (150), (151) and (152) express
the laws of constant forces, which are:
I St. The velocity of the centre of mass in any direction varies
directly with the time, and at any instant is equal to the product of the
acceleration in that direction multiplied by the time.
2d. The space passed over in any direction varies directly as the
square of the time, and at any instant is equal to the acceleration in that
direction multiplied by half the square of the time.
3d. The space described in any direction is equal to the component
velocity in that direction at the time considered, multiplied by half the
time since the epoch ; hence the space described in the first unit of time is
equal to half the acceleration.
92. (2) Motion due to Gravity. — Let the weight of the body,
wliich is the only force acting, be supposed constant. Take the
axis of z vertical and positive downward, and let a, fi, y be the
angles which the weight Mg makes with the axes x^y^ 2, respec-
tively. Then Eqs. (148) become
d'^x MfT cos a
d\ Mg cos B _
d^z Mg cos y
('53)
Under the general supposition that the centre of mass at the
epoch is in motion and not at the origin of co-ordinates, these
equations, by integration, give
88 MECHANICS OF SOLIDS.
^=^ 1=-'' %=^'^C', . . . (.54)
and
x=^Ct-\-D, y=:Ct^D\ z:=^\gt'^C^t^D*\ (155)
Hence, Eqs. (153), the accelerations in the directions of x and
y are zero, and if there be any motion in a horizontal direction
it must be uniform; this is also shown by the first two of Eqs.
(154). The acceleration g in the vertical direction is that due to
gravity; and the velocity in this direction must increase alge-
braically, as shown by the last of Eqs. (154). From Eqs. (155)
we see that the distances passed over in the directions of x and j
vary directly with the time; also that the distance of the centre
of mass from the origin, estimated in the vertical direction, is
composed of three parts, viz., the initial co-ordinate D" , the
space due to the initial velocity C", and that due to the constant
effect of gravity.
Confining the discussion to motion in a vertical direction,
and omitting the accents from the constants, the equations
become
Jf=^; Tt=St^C; z = ^r + Ct + D. . (156)
If the body start from rest at the origin, C and Z> will both
be zero; and letting v represent the velocity and /i the height
fallen through, we have
v = gt; h = igf (157)
Eliminating /, we have
V* = 2gh (158)
GENERAL THEOREM OF ENERGY. 89
This relation between v and h is of frequent use in problems of
motion; h is called the height due to the velocity Vy and v the velocity
due to the height h\ either may be found in terms of the other
when g is known.
This relation may also be obtained from the general law of
energy, since we have
wh = mgh = , (159)
or
v" = 2gh. (160)
If the body be projected vertically upward from the origin
with an initial velocity C, then, to find the duration of its ascent
and the height to which it will rise, we have, Eqs. (156),
^ o=gt—Cy or / = -, (161)
and
h = igt'-Ct=-— (162)
Gravity therefore abstracts g units from the initial velocity
every second until the body comes to rest at the altitude //, after
which it will restore g units of velocity each second, and the
body will reach the origin with its initial velocity.
From Eq. (162), as also from Eq. (158), we see that the height
which the body will attain is equal to tlie square of the initial
velocity divided by twice the acceleration due to gravity.
93- (3) ^^^ Trajectory in Vacuo. — The path described by the
centre of mass of a body is called the trajectory; if the body be
given an initial velocity, it is called a projectile; but the term
trajectory is usually limited to the paths of projectiles intended
to be thrown from guns by means of some explosive — generally
gunpowder.
90 MECHANICS OF SOLIDS.
The discussion of the trajectory in vacuo limits the forces
acting to the weights alone; and since for short distances on its
surface the radii of the earth are sensibly parallel, the weight at
all points of the trajectory may be considered as acting parallel
to its direction at the origin. Let the projectile start from the
origin, and take the axis z vertical and positive upward; the
trajectory will lie in the plane of this axis and the initial direc-
tion of motion, since there is no force acting oblique to this
plane. If we take the axis of x in this plane, the differential
equations of motion become
d^z Z Mg cos y , , ^
d^x X M^ cos a , ^ ^
ir = M= M ^SCosa = o; . . . (164)
since y = i8o° and a = 90°.
These equations give, by two integrations,
X
= a; 2 = - igt' -i- C/. ...... (166)
Let V be the initial velocity, and 6 the angle which the tra-
jectory at the origin makes with the axis at x. Then we have
C= Fcos 6; C"= Ksin 6.
Substituting these values of Eqs. (165) and (166), we have
^-=^Vcos6', g=-^/+Fsin^; . . . (167)
x=Fcosef; 0= ~i^/»+ Tsin ^/. . . (168)
Eliminating / from Eqs. (168), we find the equation of the
GENERAL THEOREM OF ENERGY. 9 1
trajectory to be
«=*""^-*?F^^i^ (169)
or
x"
ZZ^X\.2XiQ 7 j-^, (170)
4^ COS Q \ I /
when for V^ we substitute 2gk.
To find the co-ordinates of the highest point, we have
dz ^ 2X , .
~ = o — tan 6 7 ^~2j, .... (171)
dx 4/1 cos 6 \ I /
or
X =. 2k tan 6 cos' ^ = -^ sin 2^. . . . . (172)
This value substituted in Eq. {170) gives
, . n /I ^' sin" 20
z •=. h sin 2u tan u ; —-p,
4/1 cos u
= 2/4 sin' e-/i sin' 6 = /i sin' 6. , . . (173)
Transfer the origin to the highest point, without changing
the directions of the axes, and we have (H 1 1
, 7 • 9 n / , r . /i\ /I {x -\- h sin 26^)' , .
2 4- >4 sin' ^ = (a: + y^ sin 26) tan ^ — ^— ^ 7-5-^, (174)
^ ^ 4^ cos' 6 ' ^ ' ^/
or
4^ cos' ^js:+ 4^' sin' ^ cos' 6^ = 4^ cos' ^ tan ^j;
-j-4^'cos' ^sin 2^ tan ^—^'—2^ sin 2^;t— ^'sin* 2O, (175)
But
4^' sin' 6 cos' (9 = 4>4' cos' d sin 2(9 tan 6 ^ h* sin' 2^, (176)
92 MECHANICS OF SOLIDS.
and
4^ cos' d tan 6 •=. ih sin 2^, {i77)
and Eq. (175) reduces to
x^ — ^ ^h cos'' Qz^ . . . . . . (178)
the equation of a parabola whose axis is the vertical through the
highest point.
The range, which is that portion of the original axis of x
between the two branches of the curve, is seen from Eq. (172) to
be 2/1 sin 26, and its maximum value for any given value of V
is obtained when the angle of projection is 45°; and since this
value is 2h, the maximum range is equal to twice the height due
to the initial velocity. The corresponding value of z is ^//, or |-
of the maximum range.
The time required to describe any portion of the curve is evi-
dently
The time from the origin of motion to the highest point is
also given by the second of Eqs. (167),
~=o^-gt+Vs\ne, (180)
or
t=^^^^. (.80
g
The value for the velocity at any point, obtained from Eqs.
(167), is
Z;« = ^^' + ^< =: F'+/^'-2F^/sin^, . . (182)
GENERAL THEOREM OF ENERGY. 93
or, eliminating thy means of Eq. (179),
V = ^ V^ - 2g tan ex + j,ff^^. ff . . . (183)
From the symmetry of the curve with respect to the vertical
through the highest point, it is evident that
(i) The two branches are described in equal times.
(2) For points at the same height the angle of fall is the sup-
plement of the angle of rise.
(3) For points at tlie same height the velocities are equal,
since the horizontal velocity is constant and the vertical veloci-
ties are numerically equal for equal values of z.
In Eq. (170) substitute for cos 6 its value in terms of tan 6,
and we have
„ ^' + tan' dx" ...
z = tan 6x ' — -r , .... (184)
or
tan e = (185)
The point (x, z) can therefore be reached by one or by two
angles of projection, according as the quantity under the radical
sign is zero or positive, and cannot be reached when this quan-
tity is negative.
It is evident that if the parabola whose equation is
4>^' - 4/^0 - ^» = o (186)
be revolved about its axis 2, it will generate a surface which will
be the locus of all points which can be reached by but a single
angle of projection, and beyond which the projectile cannot be
thrown by the initial velocity due to h. Any point within this
surface can be reached by two angles of projection. This limit-
94 MECHANICS OF SOLIDS.
ing surface is a paraboloid of revolution, whose vertex is on the
axis of z at a. height /i from the origin, and the radius of whose
circular section in the plane normal to the axis through the
origin is 2^,
94. (4) T/ie Trajectory in Air. — The air resists the passage of
a projectile through it, and thus abstracts a part of its kinetic
energy. The resistance of the air is chiefly due (i) to the dis-
placement of the air particles by the forward motion of the pro-
jectile; (2) to the excess of air pressure in front; (3) to the cohe-
sion of the air and its friction on the surface of the projectile.
The resultant of these forces, called the resistance of the air, there-
fore varies with the velocity and. form of the projectile, the nature
of its motion, and the condition of the atmosphere. It is a force of
variable intensity, and its law of variation is not accurately
known; hence the theorem of energy Eq. (126) cannot be directly
applied, since X, V, Z are unknown.
When the velocity of any particular projectile is known at
certain points of its trajectory, the loss of energy between any
two of these points, making due allowance for the effect of known
forces, will be iM( V^ — V^), which, being divided by the dis-
tance between the two points, will give the mean resistance. An
approximate law of resistance may, be obtained by taking these
points sufficiently near together, and then varying the initial
velocity so as to include all service velocities. The details of
this method being given in the course of Ordnance and Gunnery,
only a brief statement of the mechanical principles is here given.
Consider the projectile to have motion of translation only,
and the acting forces to be the weight and the resistance of the
air. Let W be the weight in pounds,^ the acceleration due to
gravity, r the acceleration due to the resistance of the air, Fthe
velocity of the centre of mass at any point of the trajectory, and
the angle which the trajectory makes with the axis of x. Since
the resistance of the air acts along the tangent, the trajectory
will lie in a vertical plane, and the axes of co-ordinates may
therefore be assumed as in Art. 93.
The total acceleration at any point will be the resultant of
GENERAL THEOREM OF ENERGY. 95
two accelerations, one in the direction of the tangent to the tra-
jectory and the other in the direction of the radius of curvature,
Eq. (8), Art. 16. These are
•^ = -r-'gsin, (191)
whence we have
du
dt= -; (192)
r cos
V^ ,,ds I ..ds dcf) ..d(t> . i ^\
and therefore
^=-^^f = ^. (.97)
From Eqs. (192), (193), (195), (197), we have
du
cos 0'
udu
cos
2^ tan (})du
r cos
ngdu_
J Vr
(198)
These integrations depend upon the variables u^ and r, and
as there is no known relation connecting these quantities, the
direct solution, which requires x, z^ 0and / to be known through-
out the entire trajectory, is impossible. The methods of ap-
proximation used in the solution of practical problems of ballis-
tics will be found in the course of Ordnance and Gunnery.
MOTION OF ROTATION, 97
Motion of Rotation.
95, Moments of Inertia. — When a body rotates about an axis,
the velocity of any molecule is (Art. 17)
v—oor, (199)
r being its distance from the axis. Its kinetic energy of rotation
is therefore
^mv^ = \moo^r^^ (200)
and that of the whole body is
^'2mv^ = ioo*2m.r*j (201)
or one half the product of the square of the angular velocity by
2mr*. The latter is called the moment of inertia of the body with
respect to the axis, and is the sum of the products obtained by multi-
plying the mass of each molecule by the square of its distance from the
axis.
Since for a given angular velocity of the body about different
axes the kinetic energy of rotation is directly proportional to
'2mr^^ the moment of inertia of a body measures the capacity of the
.body to store up kinetic energy during a motion of rotation about the
axis with respect to which the moment of inertia is taken.
The angular velocity being the actual velocity at a unit's dis-
tance from the axis, we may write
or
\M ,QD^ =^ \Q!)^'2mr\ (202)
M^ = J2/«r' (203)
Hence, ^wr' measures the mass which would, if concentrated at a
unifs distance from the axis, have the same ?fwment of inertia as the
body with respect to that axis.
7
98
MECHANICS OF SOLIDS.
96. Radius and Centre of Gyration. — Let M be the mass of the
body, and write
^' = '2mr'^ (204)
Solving with respect to ^, we have
k
= V'
^mr"^
M
(205)
The distance k is called a radius of gyration^ and its extremity
not on the axis, a centre of gyration. When the axis passes
through the centre of mass the radius and centre of gyration are
coWtd principal, and such a radius is generally denoted by k^.
From Eq. (204) we see that // the whole mass of the body be con-
centrated at the centre of gyration the moment of inertia of the body
with respect to the axis will not be changed. The radius of gyration
may therefore be defined to be the distance from the axis at which
the whole mass of the body may be concentrated without changing its
moment of inertia.
Since the kinetic energy of rotation of a body depends only
on the angular velocity and moment of inertia, we see that for a
given value of qd the kinetic energy of rotation is the same as if
a mass equal to 2mr^ were concentrated at a unit's distance from
the axis, or the whole mass of the body at the distance k from
the axis.
97. The Momental Ellipsoid. — Let it be required to find the
relations existing between the moments of
inertia of a body with respect to all right
^ lines passing through a single point. Let
the assumed point be taken as the origin
(Fig. 42), and let a, /3, y be the type-sym-
3 bols of the angles made by the right lines
- with the co-ordinate axes. Let m be the
mass of one of the molecules of the body,
and we have for its moment of inertia, with
respect to OR,
Fig. 42.
mr"^ = m[x^ -\- y^ -\- z^ — (x cos a ^^^^ cos /3 -\- z cos yY], (206)
MOTION OF ROTATION.
99
Summing the moments of inertia for all the molecules, we
have, for the moment of inertia of the body with respect to OR^
'Smr^ — ^m^x^ + / + z^ — (^^ cos or + / cos /3 + 2 cos yY\
= 2m{{x'^-\-y'^-\- z'^){cos^a + cos^ /?-j- cos^ r)—{x cos a -\-y cos /?+2 cos yf^
- 2m{f + z'') cos' a + 2m{x'' + z^) cos^ /3 + 2m{x'' +/) cos' r
— 2^myz cos ^ cos y — 22mxz cos « cos >^ — 22mxy cos a cos /?. . (207)
But 2m{y' + 2'), :S;//(^' + z') and ^/^^(jt:' +/) are the mo-
ments of inertia of the body with respect to the axes x, y, Zy
respectively. Representing these moments of inertia by A, B, C,
we have
2mr^ = A cos' a -{• B cos'' /3 -\- C cos' y — 22myz cos /3 cos y
— 22mxz cos a cos ;^ — 22mxy cos of cos jS. (208)
Lay off on OR a distance from 6> equal to
f^
:, and let
mr
x', y, 0' be the co-ordinates of the point thus determined. Then
we have
or
*' =
cos a
V2mr''
/=
cosyS
z' =
cos y
cos a =
x' V2mr'
cos/? =
y V:Emr^
(209)
cos y =^ z^ 4/^
(210)
Substituting these values of the cosines in Eq. (208), we have
. (211)
I = Ax'' + By'' + a" - 2{^2myz)yz' - 2{2mxz)x'z'
— 2(2mxy)x'y'
c
ICXD MECHANICS OF SOLIDS.
which is the equation of the locus of all points that are at a dis-
tance from the origin equal to the reciprocal of the square root
of the moment of inertia of the body with respect to the line
upon which this distance is laid off. We see from the equation
that this locus is a surface of the second order; and, since the
radius-vector is always finite, it is an ellipsoid.
This ellipsoid is called the momental ellipsoid of inertia^ for the
reason that the square of the reciprocal of any one of its semi-
diameters is the moment of inertia of the body with respect to
the coincident right line. It presents a geometrical image of
the values of the moments of inertia of the body with respect to
all lines radiating from the assumed point.
The greatest moment of inertia is that with respect to the
shortest diameter, and the least is that with respect to the great-
est diameter. As all semi-diameters of the cyclic sections are
equal to the mean semi-axis of the ellipsoid, the moments of
inertia with respect to these lines are equal to each other. The
origin O having been assumed at pleasure, it is evident that
there is a momental ellipsoid of the body for each point in space.
98. Principal Axes. — Since the equation of an ellipsoid when
referred to its centre and axes takes the form of
Ax*"" + Bf^ 4- a'" = I, (2I2>
we see that for at least one set of rectangular co-ordinate axes
through any point in space we must have the conditions
E^l
2mxy = o;
2mxz = o; J- (213)
2myz
The axes of figure of the momental ellipsoid are called prin-
cipal axes at the point considered; and since for such axes we
have the conditions expressed by Eqs. (213), the latter are called
l/ie conditions for principal axes.
\i/K^\ vv*
Qi i v^Y
MOTION- OF ROTATION. lOI
The quantities 2mxy, 2mxZy 2myz reduce to zero for princi-
pal axes because the sum of the positive and negative products
arising from the signs of the co-ordinates x, y, z are numerically
equal to each other. The moments of inertia for such axes are
called principal moments of inertia; they evidently include the
greatest and least moments of inertia at the point.
The value of JS'/z/r^ for any axis, Eq. (211), in terms of the
principal moments of inertia at the point considered, becomes
^wr' = A cos' a-\- B cos' ^ -\- C cos' y\ . . (214)
that is, the moment of inertia of any body with respect to any line what-
€7!er is equal to the sum of the products obtained by multiplying the
principal moments of inertia at any point of the line^ respectively^ by the
squares of the cosines of the angles which the line makes with the prin-
cipal axes at the point.
99. It is readily seen, Eq. (214), that v^rhen the principal mo-
ments of inertia of a body are known at any point, all its other
moments of inertia with respect to that point may be deter-
mined; and it will now be shown that if the principal moments
of inertia be known at the centre of mass, the moments of inertia
with respect to all lines whatever can be readily computed.
Let any right line be taken as the axis of z\ then the moment of
inertia with respect to this line is
2/«r' = ^/«(^'+/) (215)
Let ^o>^o ^^ the co-ordinates of the centre of mass referred
to the assumed axes, and x\y* the co-ordinates of the molecules
of the body referred to the centre of mass; then we have
x=zx^-\-x*\ y=y,+y';
which, substituted in Eq. (215), give
Smr' = S>n{x' +/) = 2m[{x, + x')' + {y, +/)']
= 2m{x,'+y;) + 2m{x"+/')+2x,2mx'+2y,2m/. (216)
I02 MECHANICS OF SOLIDS.
Placing x^ -\- y^ = ^', and remembering that by the princi-
ple of the centre of mass
^mx' = ^my' = o,
we have
2mr'=:Md'^:2m{x''' -{■/') (217)
Therefore, the moment of inertia of the body with respect to any
line in space is equal to its moment of inertia with respect to a parallel
line through the centre of mass, increased by the product of the mass of
the body by the square of the perpendicular distance from the centre of
mass to the given line. The least principal moment of inertia at
the centre of mass is therefore the least of all the moments of in-
ertia of the body.
100. Discussion of the Momental Ellipsoids of a Body. — Let Ay
£, C be the principal moments of inertia at the centre of mass.
(i) Suppose A =^ B ^^ C. Then the central ellipsoid is a
sphere, and therefore all moments of inertia at the centre of
mass are equal, and all axes through it are principal. For every
other point in space the ellipsoid is a prolate spheroid whose
axis passes through the centre of mass; for, the moment of in-
ertia with respect to this line is the same as the central mo-
ments of inertia, while those with respect to all lines perpendicu-
lar to this are greater than the central moments of inertia and
equal to each other; and these lines are principal axes since the
moment of inertia with respect to the line through the centre
of mass is the least of all the moments of inertia at the point in
question.
(2) Suppose A y B and B =. C. The central ellipsoid is an
oblate spheroid whose axis is that of the greatest moment of
inertia. There are two points on the axis of the spheroid at
which the ellipsoid is a sphere, and they are found thus: At
these points all moments of inertia must be equal to A. Then,
denoting by x the distance of these points from the centre, we
have
MOTION OF ROTATION. IO3
A=iB-^Mx^^C^Mx^', .... (218)
whence
lA-B
(219)
It is evident that the ellipsoid can be a sphere at no other
point.
(3) Suppose A = B and B > C. The central ellipsoid is a 'Tlt~"
prolate spheroid whose axis is that of C. There is no point at
which the ellipsoid can be a sphere.
(4) When A > B > Cy the central ellipsoid is one of three ^ /^ ^ ^
unequal axes at the centre of mass, and cannot be a sphere at ^.^
any point in space. )^'\^.
10 1. Determination of the Moment of Inertia. — The moment of ^^ 1,^^
inertia may sometimes be found by the summation of the sepa- '
rate values of wr*.
Whenever the body is one whose density and boundary vary
by some law of continuity, we may write
m-= dAf =■ 6dV (220)
and
2mr^=rr^dM=rr''6dV', (221)
from which the moment of inertia can be found whenever the
expression can be integrated between the limits that determine
its volume.
Having found the moment of inertia of a body with respect
to any line, that with respect to a parallel line may be found by
Eq. (217); and having found the principal moments of inertia at
any point, that with respect to any other line through the point
can be found from Eq. (214).
In the following examples let S represent the density, go the
area of cross-section of a material line, and / the thickness of a
material surface.
104 MECHANICS OF SOLIDS.
Ex. I. A Uniform Straight Rod. — Let the axis be perpen-
dicular to the rod at its middle point, and represent the length
of the rod by 2a\ then we have
x^dx = 2a6oo— .
3
Whence, since M = 2adGo, we have
Mk; = M- and k/ = - (222)
3 '3 ^
The centre of gyration is therefore at a distance from the
centre of mass equal to = .577^:, nearly.
1.732 -I- *^" '
For an axis perpendicular to the rod at any distance d from
the centre we have
)mr
= Ml+'^')= ("3)
which becomes, for the perpendicular axis at either extremity.
M±a\
Ex. 2. A Circular Arc^ subtending an angle 2^ at the centre
and whose radius is a,
I. Axis perpendicular to the plane of the arc through its
centre.
■:2mr'' = doo f^a'dO = 26000,' 6 = Ma*)
MOTION OF ROTATION. IO5
and, for the whole circumference, '
^wr' = 2716000" = Ma* (224)
2. Axis in the plane of the arc through its centre and middle
point.
Smr" = doo C^ a' sin' Odd
= dooa\d — sin 6 cos 0)\
and, for the whole circumference,
Smr" = TtSaoa* = J/ -. (225)
From Eqs. (224) and (225) 2mr* can be found for any right
line passing through the centre by the application of Eq. (214).
.Ex. 3. A Rectangular Plate whose sides are 2a and 2b. Take
the centre of the plate as the origin, and the axes parallel to
its sides; then for the axis x^ perpendicular to 2^, we have
dM = 2adtdy^ and therefore
2mrJ = 2a6t f^ y'dy = '^6tab'^M^^\ . . (226)
J-b Z 3
and similarly, for the axis y^
2mry = M" (227)
For the axis z we have
2m(x* +/) = 2m(y' + z") + 2m{a^ + 2"),
io6
MECHANICS OF SOLIDS.
since z = o; hence
2Mr/=.M^^±^ (228)
3 ^ '
If the plate be square^ a = b, and we have
'2mrJ = SmrJ' = M
:^mr^ = M
y
2a'
(229)
Ex. 4. A Triangular Area about an Axis through a Vertex. —
V ^ B Let AjBC, Fig. 43, be the triangle, and take
the axes x and y through the vertex in its
plane; let /3 and yS' be the distances of the
vertex B from y and x respectively, and y
and ;/' those of the vertex C from the same
axes, and let AZ> = /; then we have for the
triangle ABD
Fig. 43.
dM = dtPQdx - dtl^—^dx.
Similarly we have, for the triangle ABC,
:Emry' = dtl r(i - -\x''dx = (J//^-. . . (231)
But ^mry of ABC is equal to the difference betw-en that of
ABB and ACD\ therefore we have, for ABC,
MOTION OF ROTATION. lO/
:2Mr; = Stl^^ (,3,)
The mass of ABC is evidently dtl— ^; hence
2
:2mr; = ~l,P' ^ §Y + Y') (^33)
which is a general formula for the moment of inertia of any tri-
angle with reference to an axis in its plane, passing through its
vertex and being wholly without the triangle. Similarly we have
:S«r,' = ^(/J" + /?>' + /'),. . . . (234)
and, for the axis z at A,
2mr.' = ^(/J' + fiy + y' + P" + /?>' + y"). . (23s)
Ex. 5. A Triangular Area about any Axis whatever. — At the
middle point of each side let one third the mass of the triangle
be concentrated; then the centre of gravity of the three material
points coincides with that of the triangle.
The moment of inertia of the material points with reference
to any line Ay drawn through A is
and with reference to a line through A perpendicular to the
plane of the triangle is
108 MECHANICS OF SOLIDS.
M
3
Uf+f)+(?+?)+('^+'^^)(
= -^W + Py + f + r + P'y' + r"); (^'s?)
and these moments of inertia are the same as those of the tri-
angle with respect to the same lines.
The moments of inertia expressed by Eqs. (235) and (237)
are evidently the greatest moments of inertia at the point, and
they are therefore principal moments of inertia; hence two of
the principal axes lie in the plane of the triangle. But, Eqs.
(233) and (236), the sections by this plane of the ellipsoids for
the point A of the triangle and system of points coincide
throughout at least half their length; therefore the principal
axes in this plane are also equal and coincident, and the ellip-
soids coincide throughout. Hence the ellipsoids at the common
centre of mass are one and the same ellipsoid. The triangle and
system of material points, having then the same central ellipsoid,
have equal moments of inertia with respect to all lines in space.
Therefore, to determine the moment of inertia of a triangular
M
area, find that of three masses each equal to — at the middle of
its sides with respect to the given line, and it will be the required
moment of inertia of the triangle.
Ex. 6. An Elliptical Area. — Let the equation of the ellipse be
ay + ^V = a^V"',
then, for a line through its centre coincident with its major axis,
we have
]mrx ^ =■ 461 j j y^dxdy
t/o t/o
= 7tabdt-=M--
4 4
'. . (238)
'^^tfCiT'-^;:
MOTION OF ROTATION. IO9
after substituting -(«' — x"^ iox- y after the first integration.
Oi
Similarly, for the minor axis, we have
2mry = ^6t I j x^dydx
t/o t/o I . .
«» r • • • ^'^9)
4 4 J
and by combination we have
2».r/ = ^^^l±^) (240>
For an axis in the plane of the ellipse coincident with any ra-
dius vector r we have, Eq. (214), since y = r' sin' a and x^ = r*
cos' a,
2mrr^ = M ("' ^^"' ^ + ^' '" "^^ = M^. . (244
4 4r' ^ ^ '
For a circular area we have
4 2
Ex. 7. ^« Ellipsoid. — Let the equation of the ellipsoid be
then the area of any section perpendicular to the axis of x is
a a a ^ '
and therefore
no
MECHANICS OF SOLIDS.
dM = d^{a' - x^)dx.
From Eq. (240) we see that the square of the radius of gyra-
tion for an ellipse with respect to the normal through its centre
is equal to the sum of the squares of its semi-axes divided by 4;
therefore we have
•'^ nbc
./' + ^%..
:2mr,' = S r "^{a^ - xy-—^-(a' - x*)dx
^^jc(^-^n r,^.
Sn ^^^ \' l r (a' ~ xy.
-u Tiauc
3
dx
—dnabc
5
= M
b' + c'
In the same way we readily obtain
•" 5 '
^mr^ = M ' — .
(242)
(243)
For a spheroid whose axis is a, and b =■ c, vjq have
^pirj" = M
2a'
'!2mry = "^mrz = M
a^-\-b''
(244)
(245)
For a sphere^
2ar
:SmrJ = 2mry^ = 2mr/ =M — .
(246)
MOTION OF ROTATION.
Ill
Ex. 8. A Rectangular Par allelopipedon, — Assume the origin at
one of the angles of the solid; let the co-ordinate axes coincide
with its edges, and let a, b, c be their lengths along the axes x^
y, Zf respectively; then
A= r f^ r p{f + z^)dxdydz
«/o t/o t/o
__ pabcip' -f c*) __ ^^ ^' + c\
3 3
^ ^ pabc{a' -\- c') ^ ^^l+f\
3 3 '
^ ^ pabcia^-^b-^) ^ j^a^jV_b\,
3 3 '
and for principal axes at the centre of mass, (Art. 99),
A^m'L±S\ B=.M^l±fl C=M^1±^:,
(247)
12
12
12
(248)
For the cube^ since a=z b = c.vjq have for the edges
2«'
A=B=C= M^
and for principal axes at the centre of mass
\ -i 2 / 6
(249)
Hence the momental ellipsoid at the centre of the cube is
a sphere, and all moments of inertia are equal.
It might appear that the edges are principal axes at the angle
of the cube from the equality of their moments of inertia. But
the line joining the centre and angle is a principal axis at the
angle, since the moment of inertia with respect to it is less than
that with respect to any other line; hence the principal axes at
112
MECHANICS OF SOLIDS.
the angle are the diagonal of the cube and any pair of perpen-
dicular right lines in the normal plane to the diagonal at the
angle. The moment of inertia with respect to every right line
in this plane passing through the angle of the cube is readily
seen to be
M
(6+t) = ^— (^5°)
102. The moments of inertia with respect to central principal
axes are here tabulated for convenient reference.
Mass.
Dimensions.
X.
Y.
1. Rod or Cylinder
2. Circular Rim
3. Rectangular Plate
4. Elliptical Area
5. Circular Area
6. Ellipsoid
7. Spheroid
8. Sphere
9. Rectangular Paral
lelopipedon
10. Cube
length = 2a,
radius = r
radius = a
sides=aand^
axes=«and3
radius = a
axes = a, d, c
d = c
radius = a
edges = a,d,c
edge = a
M -
3
Ma"
4 3
AI
b''
M-
M
M
M'
4
5
5
2^2
h'^
■M
4 3
-M —
2 3
M —
3
Md"
M-
4 3
.a''
2
M-
4
a'-
4
5
5
M —
5
-M ■
4 3
-M-
2 3
M-
2
M-
2
2
5
4 3
-M —
2 3
From the moments of inertia above we can readily derive the
corresponding radii of gyration by dividing by the mass of the
body and taking the square root of the quotient.
INSTANTANEOUS AXIS. II3
If the body be irregular in form and be not homogeneous,
the principles of the calculus cannot be applied to find its mo-
ment of inertia. In such cases the moment of inertia can be
experimentally found by means of the principles of the com-
pound pendulum, a method which will be explained subse-
quently.
Instantaneous Axis.
1O3. Whatever may be the component angular velocities of
a body when rotating about a centre, the resultant angular ve-
locity and the axis of rotation may be found by the application
of the principle of the parallelopipedon of angular velocities. 4?
When the centre of mass is taken as the centre of rotation, these
are called instantaneous angular velocity and instantaneous axis. At
any instant the path of each molecule of the body is in a plane
perpendicular to the instantaneous axis, and all points on this
axis have no motion with respect to the centre of mass.
The component velocities of any molecule with respect to
the centre of mass are obtained by dividing Eqs. (116) by d?"/;
thus we have - ^
dx' ,dtb , d(b , ,
dt ~~ dt -" dt "^ -^ ''
d/ /0 ,dd ,
■^="-^-^^="^^-^^-
dz' ,dd Jtb
dt ^ dt dt ^ "" •^'
(251)
when we substitute the symbols oox^ ooy^ oo^ for the component
, . . dd dtp d(l) ^
angular velocities — , -^, -j- about the co-ordinate axes x\yy
z', respectively. If in these equations we make
dx^__^ _ dz^_
dt ^ dt ~ dt "^ °'
114 MECHANICS OF SOLIDS.
we shall obtain the equations of the locus of all those points
which are at rest with respect to the centre of mass; and, since
these points lie on the instantaneous axis, we have
z'ooy — y'coz = o, )
X^GOz — Z^GDje= 0,y (252)
y'GJj; — x'ooy — o,)
for the equations of the instantaneous axis. All molecules of
the body not on this right line will have, at the assumed instant,
a motion with respect to the centre of mass, and will therefore
rotate about it. Since this axis passes through the centre of
mass, the position of the axis in space depends only on the val-
ues of the angular velocities gDjc, cOy, gOz; and as these values
generally remain constant only for the instant df, the instanta-
neous axis describes the surface of a cone whose vertex is the
centre of mass.
104. Let a, /3f y he the angles which the instantaneous axis
makes with the co-ordinate axes, and co the instantaneous angu-
lar velocity; then, by the principle of the parallelopipedon of an-
gular velocities, we have
COS a = — , cosp = -^, COS V = — , . (253)
00 ' 00 ' 00
and
co' = G?^' -f (W^ ' 4- ffi>^ » (254)
Hence it is necessary to find values for the component angu-
lar velocities before the position of the axis and the resultant
angular velocity of the body can be determined. The two cases
to consider are (i) rotation due to the action of incessant forces ^ and
(2) rotation due to impulsions. The latter, being the simpler case,
will be discussed first.
ROTATION DUE TO IMPULSIVE FORCES. II5
Rotation of a Rigid Solid due to Impulsive Forces.
105. When the centre of mass is the movable origin, the mo-
tion of translation of this point and the motion of rotation of the
body about it have been shown to be wholly independent of
each other. Hence we may regard the centre of mass as a fixed
point in space, and consider only the rotation of the body about
it. The body is supposed to have been subjected to a system
of impulsions whose effect is completed in a very short time, and
the body is then abandoned to itself, free from the action of any
extraneous force whatever. It is required to find its subsequent
motion of rotation in all of its particulars.
When the moments of the several impulsions may be com-
pounded into a single resultant moment, Rk^ having the centre
of mass as the centre of moments, the moment axis of R is call-
ed the resultant or invariable axis, and the plane of R and k is
called the resultant or invariable plane. They cannot change their
direction in space unless other forces be introduced, which is
not supposed.
106. Assuming Eqs. Rm', which are here applicable, / v / / i>
21 Xx' cos fi^y COS a) = :2v{^-^^^^^-^ = Z„
2/,(z' COS a^x' cos y) = 2m?^^^^^-^^f^ = M,, )- {RJ)
2lXy cos y^z' cos /?) = :2m^l^^-^I-^^ = N,,
and substituting the values of dx\ dy\ dz' given in Eqs. (251),
we have
:2m ''''^^'~j'^''' =^:2m(x''^y')cso,---:2mx'z'oD^--:2myz'GOy',
z'dx* x'dz'
'2m — = 2m(x'*-\- z'^)G0y'-'2my^ z' GDs—'2mx'y' GDjc'y K^SS)
2m- — — = 2m{y^-\- 2')<»^ —2mx'y'G0y—2mx'z'G0g,
ii6
MECHANICS OF SOLIDS.
If the axes be principal, these equations reduce to
/v^' ^'^«'
2V;c'
^/
= ^;??(y' + z'^)G9y = ^GJy = M,\
2/w^i^^^^' = :^/«(/» + 2'»)a?, = ^(i?^ = iV^^.
(256)
Hence
Z, i^, ^. , ,
That is, //z^ angular velocity due to an impulsion about a principal axis
is equal to the component moment of the impulsion divided by the moment
of inertia of the body^ both taken with respect to that axis.
This principle is true also for the instantaneous axis; for if z
be the instantaneous axis, we have
Cb?^ z= Ce?^ = o, and 00^ = Ce?.
And substituting these values in the first of Eqs. (255), we have
:2m'^^-^— = :^m(x'' +/')o^. = Cg? = Z,. . (258)
Since Eqs. (Rm') apply to rotation about a fixed point or a
fixed axis, this principle is likewise applicable to both of these
cases.
107. Let "^mr"^ be the moment of inertia with respect to the
instantaneous axis, and the angle between this axis and the
invariable axis. Then we have
00 =
Rk cos
-v--
(259)
ROTATION DUE TO IMPULSIVE FORCES. 11/
hence
, . '2mv'^ = G0^'2mr^ = — = Ekoo cos 0, . . . (260)
d being that semi-diameter of the central ellipsoid which coin-
cides with the instantaneous axis.
Squaring Eqs. (256) and adding, we have
A^QD:c' + B'ooy' + c'ci?^' = l; -f m; + n; = R^k*-, (261)
-ff-^ being the resultant moment of the system of impulsions with
respect to the centre of mass.
From Eq. (260) we conclude, since 2mv^ is constant,
(i) The instantaneous angular velocity varies directly with
the length of the semi-diameter of the ellipsoid which coincides
with the instantaneous axis, or inversely as the square root of
the moment of inertia with respect to this axis.
(2) The angular velocity about the invariable axis, an cos 0,
is constant; therefore, as increases or cos diminishes, go in-
creases; that is, as the instantaneous axis increases its inclina-
tion to the invariable axis, the instantaneous angular velocity
increases.
Eqs. (260) and (261), together with that of the central ellip-
soid, give the circumstances of the rotary motion of a free rigid
solid under the action of impulsive forces whenever we can find
the value of Rk.
108. The equation of the invariable plane is
Call the point in which the instantaneous axis pierces the
central ellipsoid the instantaneous pole,, and let x\y', z* be its co-
ordinates. Then the equation of the tangent plane to the ellip-
Il8 MECHANICS OF SOLIDS.
sold at the instantaneous pole is
Axx' + Byy* + Czz' = i. . . . . . (263)
From Eqs. (252) we liave
QD:c OOy CDz GO / , v
^ = y=ir=d=^' • • • • (^^4)
which reduces Eq. (263) to
AoOxX + Booyy + CoOzZ = €, . . . . (265)
and this, by Eqs. (256), becomes
N,x + M,y + Z,z=€ (266)
Dividing both members by i?/^, we have
The sum of the squares of the coefficients of the variables be-
ing unity, these coefficients are the cosines of the angles which
the normal to the plane makes with the co-ordinate axes, and
— is the perpendicular distance from the centre of mass to the
plane. We therefore see that the tangent plane at the instanta-
neous pole is parallel to the invariable plane, and that these two
planes are separated by the constant perpendicular distance, /.
Hence the ellipsoid ro//s, without sliding^ on a tangent plane
parallel to the plane of resultant rotation of the system, and at a
fixed distance/ from the centre of the ellipsoid. As different
points of the ellipsoid come successively into the tangent plane>
ROTATION DUE TO IMPULSIVE FORCES. IIQ
the semi-diameters which join them with the centre become in
turn the instantaneous axis. The locus of the tangent points on the
ellipsoid is called the Polhode^ and in the general case is a curve
of double curvature. The locus of the points of contact on the
tangent plane is necessarily a plane curve, and is called the Her-
polhode. If we imagine all points of the polhode joined with the
centre of the ellipsoid by the various semi-diameters, and all
points of the herpolhode with the same point by the various in-
stantaneous axes, we will have two cones; the former called the
rolling cone, described about a principal axis of the ellipsoid, and
the latter called the directing cone, about the invariable axis; at
any instant they are tangent to each other along the instanta-
neous axis.
109. The Rolling Cone. — Dividing both members of Eq. (261)
by e', we have
^•^- + ^'^ + C'^:-=j-... . . (^68)
But, Eqs. {264),
GoJ „ GoJ ,, 00 *
€'
^=/»; ^=Z'\ . . . (269)
and Eq. (268) becomes
^V" + ^y + C V» = i, .... (270)
which is an equation of condition for points of the polhode.
Since the instantaneous pole is on the ellipsoid, we have also
the condition, Eq. (212),
Ax'' + B/' + a" = I, (27i>
I20 MECHANICS OF SOLIDS,
and dividing by/^ we get
>
-Ti-+-^ + -ir-=T, (272)
/ / / /
Subtracting Eq. (270) from Eq. (272) and omitting accents,
we have
which expresses the relations existing between the co-ordinates
of any point of the polhode. But we see that this equation is
satisfied by the co-ordinates of the origin, and also by any set of
values which bear a constant ratio to the corresponding co-
ordinates of any point of the polhode; that is, the equation Is
satisfied by the co-ordinates of all points of the instantaneous
axis in all of its positions. It is therefore the equation of the
rolling cone.
By replacing A, B, C within the parentheses by -5, j,, -5,
respectively, a^ b^ c being the semi-axes of the ellipsoid, the
equation of the rolling cone becomes
Aj - i\'' + ^(7 - -^y + <7 - 7>' = °- (^7-^)
The position and character of this cone depend on the values
of the constant/.
110. Discussion of the Rolling Cone.
(i) Let/ = a. Eq. (274) becomes
ROTATION DUE TO IMPULSIVE FORCES. 121
which is satisfied only by
jt: = — y = o and z -=■ o,
o
Hence the axis of x is the rolling cone, the directing cone, the
invariable axis, the instantaneous axis, and the longest principal
axis of the body. The body rotates uniformly about this axis.
(2) Let ay py b. The second and third terms of Eq. (274)
then have the same sign; x is then the axis of the cone, and the
sections of the cone normal to x are ellipses.
(3) Let/ = b. Eq. (274) becomes
and we have
which are the equations of two planes equally inclined to the
principal plane ab of the ellipsoid and intersecting in the mean
principal axis. They cut from the ellipsoid two equal ellipses,
which are called the critical ellipses, or separating polhodes, of the
central ellipsoid. The semi-axes of the critical ellipses are
sT.
a c
a^ -\-c'' TT and ^ (278)
(4) Let by- py c. Then the first and second terms of Eq.
(274) have the same sign; z is then the axis of the cone, and the
sections of the cone normal to z are ellipses.
(5) Let/ = c. Then
o
122
MECHANICS OF SOLIDS.
and the axis of z is the rolling cone, the directing cone, the in«
variable axis, the instantaneous axis, and the shortest principal
axis of the body. The bod}- rotates uniformly about this axis.
III. The Polhode and Herpolhode. — In the ist and 5th cases
the polhode and herpolhode are points.
In the 2d and 4th cases the polhode is in general a curve of
Fig. 44.
double curvature, and the herpolhode is a wavy curve, as indi-
cated in Fig. 44.
In Fig. 44 let « >/ > b, take the tangent plane to be the
horizontal plane, and assume the vertical plane parallel to the
plane of the longest and shortest principal axes. Then the long-
est, shortest and mean axes are projected equal to themselves
in aa'\ CC and bb\ respectively; //'/" is the vertical projection
ROTATION DUE TO IMPULSIVE FORCES. 1 23
of the polhode, and hh'h"h"\ etc., is the herpolhode; the invari-
able axis is projected in 00" and C?', and the instantaneous axis
in Op and O'h', OE' and (9^ are traces of the cyclic planes, and
OE" and OE"* are traces of the planes of the critical ellipses.
The maxima and minima values of d recur when the vertices
of the polhode come into the tangent plane. These vertices are
the intersections of the polhode by the principal planes of the
ellipsoid. The maxima and minima values of the radius vector of
the herpolhode correspond to those of d, and hence this curve
will lie between the circumferences of two circles whose common
centre is (9', and whose radii are O'h and 0'h\ corresponding to
the greatest and least values of S. If the angle included between
two consecutive maximum radii-vectores of the herpolhode be
commensurable with a right angle, the curve will be retraced
after a certain number of complete turns, and the herpolhode
will be a closed curve. If this angle be incommensurable with
a right angle, the instantaneous pole will never retrace its former
path on the tangent plane.
The general value of the radius vector of the herpolhode is
given by the equation
p" = Z' = o;
(286)
and hence, by the principle of the parallelopipedon of angular
velocities, we have
C0x= ~77^^^^ '^ dt ^^" ^ ^^" '
de . ^ , dip . . n
Wy = — — - sin + -^ cos sin C^;
. (287)
117. T/te Gyroscope, — The problem of the gyroscope illustrates
this subject. It may be stated thus:
Find the circumstances of motion of a solid of revolution'
about a fixed point on its axis, it having been given an initial
ROTATION DUE TO INCESSANT FORCES.
129
rotation about its axis and then left to the action of its own
weight.
Let (9, Fig. 46, be the fixed point,
Ox^ Oy, Oz the fixed axes, and Ox', Oy\
Oz' the principal axes at O, Oz being
taken vertical and positive upward. Oz'
is the axis of revolution of the body.
Let h be the distance from O to the
centre of gravity, and assume A =■ B
and B (289)
Z = Y'x'- X'y'=o. )
Substituting these values in Euler's Eqs. (283), we have
dt
dOOy
~dt
— {A — C)GOyGDz = iV= mgh sin 6 cos 0;
-\- {A — C)gDxOOz = M = — mgh sin 6 sin 0;
^(290)
Integrating the last equation, we have
Cl»2 = a constant =■ n (291)
130 MECHANICS OF SOLIDS.
Multiplying the first of Eqs. (290) by (y^and the second by
GOy^ we have by addition, Eqs. (287),
A\QOx-Tr + 00y—zj\ = mgh sm uyoo^ cos -- CsJ^sin 0)
= mgh sin c'— .
(292)
Integrating this equation, we have
iA{GoJ + gd/) = ^«^y^ (cos ^0 — cos 6), , . (293)
in which 6^ is the initial angle zOz\
Adding the initial kinetic energy of rotation iCn^, we have
iAiGoJ" + gd/) + iCfi" = mgh (cos B^ - cos 6) + iC•«^ (294)
Since mgh (cos ^^ — cos 6) is the work of the weight while G
falls over the distance h (cos Q^ — cos ^), we see that Eq. (294)
expresses the theorem of kinetic energy of rotation.
118. From Eqs. (287) we readily get
<»/ + <»/ = 5 + sin»^fl;. . . . (295)
which substituted in Eq. (294) gives
A-j-^ + A sin'' Q-j-i, — 2mgh{cos 6^ — cos 6). . (296)
at at
Multiplying the first and second of Eqs. (290) by sin and
cos respectively, we have, by addition,
ROTATION DUE TO INCESSANT FORCES I3I
^[sin 0—^+cos0-^j+«(C— -(4) [cousin 0—0?;^ cos 0] =0,(297)
and, since Eqs. (287) give
ooy sin — fl?^ cos = — ^, . . . , (298)
Eq. (297) reduces to
^^sin 0-^ + cos 0-^j + nA-j^ - «C-^ = o. (299)
We have also, from Eqs. (297),
cOx sin + (Wy cos = sin ^-^. . , . (300)
Differentiating this last, dividing by dt and reducing by the
relations of Eqs. (287), we have
sin 0^^ +COS 0-^ = cos ^ ^ + sin ^
{gOx cos — G7y sin 0)
^0
Substituting this value in Eq. (299), we have
./ jedil,\ . uPtp\ ^dd , .
132
MECHANICS OF SOLIDS,
which, after multiplying by sin d and integrating, gives
A sin' QJ- + CV/ cos ^ = a constant = ^'; . . (303)
or between limits.
A sin' e^^ = C«(cos B^ - cos 6),
(304)
From the last of Eqs. (287) we also have, since aOg = n.
dcf) dtp
(305)
Eqs. (296), (304) and (305), viz.,
A-—^ + A sin' 6-^^ = 2mgh (cos Q^ — cos 6),
A sin' 0-^ = Cn (cos d^ — cos ^),
d(f) dip ^
(306)
are the differential equations of motion of the gyroscope. From
them the values of 6, tp and i, and hence corresponds
to no angle. Thus we see that the body falls from its initial
position until 6 = 6^, then rises until 6 = 6^, and continues to
oscillate between these two values. The integral of dt (Eq. 307)
between the limits 0^ and 6^ evidently gives half the time of a
complete nutational oscillation.
The precessional velocity given by the second of Eqs. (306) is
d(p Cn{cos 6^ — cos 6) , .
-di = ASTxTe : "°9)
and it is zero when 6 = 6^, and a maximum when 6 = 6^. It is
direct or retrograde according as n is positive or negative.
Combining the motions in nutation and precession, we find
that the horizontal projection of the path of the centre of gravity
134 MECHANICS OF SOLIDS.
lies between two concentric circumferences whose radii are
h sin d^ and h sin ^„ and is tangent to the outer and normal to
the inner circumference.
From Eq. (308) we have
• a r- i/cos^. — cos^ . .
s.n e = C« f — ^-p^^— (3,0)
from which we see that B^ — 6^ may, by increasing the value of
«, be made less than any assignable quantity. In the common
gyroscope we can give n such a value that the eye can detect
neither the vertical motion nor the variation in the precessional
velocity.
This discussion gives only the general character of the mo-
tion. The complete solution of the problem requires the inteorra-
tion of Eqs. {306), and this involves methods not given in the
course of mathematics at the Military Academy.
Impact.
120. When two bodies collide there is a transfer of energy
from one to the other, by which changes in the velocities of both
bodies, and in their form and volume, are effected. The impact,
though ordinarily said to be instantaneous, requires a finite time
for its completion. When after collision we examine the sur-
faces of two ivory balls which have been previously oiled, we
notice that their areas of contact during collision must have
been very much greater than when they simply rest against each
other. Hence the distance between their centres of mass during
impact must be less than the sum of the radii of the spheres, and
the intensity of the mutual pressure of the colliding bodies evi-
dently varies by continuity from zero to a maximum and then to
zero again. The instant of nearest approach of their centres
separates the period of compression from that of restitution^ and at
that instant their centres have the same velocity.
IMPACT. 135
Actual solids possess a certain degree of elasticity of form
and volume, by which they regain approximately their original
form and volume when the impact has ended. If C be the in-
tensity of the impulsion producing compression, R that which
restores the form and volume, and e their ratio, we have
R^eC', (311)
e is called the coefficient of restitution^ and in actual bodies is less
than unity and greater than zero.
121. Direct and Central Impact. — Let a spherical mass w, mov-
ing with a velocity z;, collide with a similar mass m' , whose ve-
locity is u in the same direction, and let w be their common
velocity at the instant of nearest approach of their centres; then,
taking velocities in opposite directions to have opposite signs,
we have
C= tn{v — w) = m'{w — «), .... (312)
whence
and
C= — i — }{v—u) (313)
i? = a=-^^(z;-«y (314)
Let Fand C/'be the final velocities of m and m\ and we have
je + c = c(i + .) = ^^^(t, - «)(i + .)
= m{v-V)=m\U-u) (315)
I^nce
136
MECHANICS OF SOLIDS.
V=V
m'
m-\- m
-,(v- «)(i +^);
U:=zu-\-
-,(v - u)(T^e\
(316)
7?i -j- nv
The limits of these values, found by making ^ = i and e — o,
are
and
2m'
V = v —
-,{^ - ^)f
U^=u-{-
2m
m-\-m
li^ - «)»
y = y [p ^U)= ; J-,
" m-\-m
-(z; - «) =
m -\- m' ^
(317)
(318)
In actual cases ^ is a constant to be determined by experi-
ment. Whatever its value may be, it is readily seen from Eqs.
(316) that the sum of the momenta of the two bodies after colli-
sion is equal to that before collision; therefore no momentum is
destroyed by the impact.
The sum of the kinetic energies of the masses before and
after impact are respectively
i(wz/'4-wV) (319)
and
mm
i(mi^ + m'u')-i-^^-^iv-uY(i-e'). . . (320)
Whence we see that a loss of kinetic energy always accompanies
the impact of actual masses.
IMPACT.
137
122. Oblique Impact. — If the paths of the colliding bodies be
oblique to each other, their velocities may be resolved into com-
ponents in the directions of the common tangent and normal at
the point of impact. C and R will depend only on the normal
components, the tangential components having no effect on the
impact. The final velocity of each body will therefore be given
by the resultant of the changed normal component and the un-
changed tangential component.
123. If m' be very great with respect to m and at rest we
have the case of impact against a fixed obstacle. -;
Then if be the angle of incidence, Fig. 47, or that | —
which the direction of the path of m makes with
the normal to the deviating surface at the point oifn'l
impact, we have for the component impulsions of
compression
mv sin (p and mv cos 0,
Fig. 47.
and for those of restitution
mv sin and — mev cos 0;
the resultant of the latter being
mv 4^sin'0H-^' cos'0.
If the angle of reflection is 0' we have
sin tan
tan 0' =
e cos
(3")
which varies between — tan 0for ^ = i, and 00 for ^ = o. Hence
in all actual cases of impact the path of the reflected body will
make an angle with the normal greater than that of incidence
and less than 90°, depending on the value of e for the bodies
considered.
138
MECHANICS OF SOLIDS.
Axis of Spontaneous Rotation.
124. A spontaneous axis is a right line fixed in space^ about
which a free body rotates during impact while the centre of
mass of the body is in motion. Its position and the necessary
conditions for its development are derived from Eqs. (117).
Dividing these by dt, and replacing the component angular
velocities about the centre of mass by their symbols, we have
dx dx. , - -
dz dz . ,
(322)
in which the first members are the component velocities of any
molecule of the body with reference to a set of axes fixed in
space, and the first terms of the second member are the compo-
nent velocities of the centre of mass with respect to the fixed
origin, when the centre of mass is taken as the movable origin.
From Eqs. (Tm') we have
'dt~^''~ M'
dt~ ''~ M'
^-v - ^.
dt ~ '~ M'
' (323)
in which X^, Y^, Z^ are the component intensities of the result-
ant impulsion in the direction of the co-ordinate axes.
AXIS OF SPONTANEOUS ROTATION.
139
Substituting these values in Eqs. (322), we have
dt M ^
y oD^
dt M
+ x'ooz — z'0Ojc\
dz Z. . . ,
(324)
For all points at rest with respect to the fixed origin we have
the conditions
dx _ dy _dz _
~dt~~dt~~dt~^'
(325)
and Eqs. (324) become
M
*--\- z'ooy —y'GO^ = o;
(326)
which will be the equations of a right line fixed in space when
X,(w^ + K,(»y + Z.tt?, = o (327)
Dividing Eq. (327) by R,qo, we have
R. oo~^ R. 00^ R. GO *
(328)
which expresses the condition that the action-line of the result-
ant impulsion is perpendicular to the instantaneous axis; hence
I40 MECHANICS OF SOLIDS.
we conclude that a spontaneous axis will be developed when a body is
so struck as to make the instantaneous axis perpendicular to the result-
ant impulsion; otherwise Eqs. (326) are the equations of a single
point whicli alone is at rest for the instant. We see also by
comparing Eqs. (326) with Eqs. (252) that the spontaneous axis is
always parallel to the instantaneous axis.
125. It is readily seen from Eq. (327) that the required con-
dition can be satisfied only when at least one factor of each term
is zero; that is, either when the line of impact lies in a central prin-
dpal plane or when it is parallel to a central principal axis. The dis-
cussion is the same for all cases.
126. Let the line of resultant impact, Fig. 48, be parallel to
the principal axis y' and lie in the prin-
cipal plane x'y', and let R^ = J/Fbe the
intensity of the resultant impulsion; then
we have
X, = o; Y^ = MV- Z, = o;
MVh )- (329)
GOr = 0: G?v = O; ODz =
Fig. 48.
h being the lever arm of the impulsion
with respect to the axis z'\ Eq. (327) is satisfied, and the equa-
tions of the spontaneous axis developed by the impact are
, o , C Mk; k; . ,
which are those of a right line parallel to the principal axis z\
intersecting the axis x' at a distance from the centre of mass
k '
■equal to j-.
Let / be the perpendicular distance between the line of im-
pact and the spontaneous axis; then we have
/ = '5 + ^; (331)
AXIS OF SPONTANEOUS ROTATION. I4I
whence
{i-K)h = k: (332)
Since /&/, the square of the principal radius of gyration of
the body with respect to the axis 2', is constant, we see that the
two distances, viz., // from the centre of mass to the line of im-
pact, and I — h from the centre of mass to the spontaneous axis,,
are reciprocally proportional; hence, as the line of impact re-
cedes from the centre of mass, the spontaneous axis approaches
that point, and conversely. When the line of impact passes
through the centre of mass the spontaneous axis is at an infinite
distance, as it should be, since in this case the body will have
motion of translation only.
127. When a spontaneous axis is developed the action-line of
the corresponding impulsion is called an axis of percussion^ and
each of its points a centre of percussion; the latter term, however,,
being generally applied to the point in which the axis of per-
cussion intersects the line h. The corresponding point of the
spontaneous axis is called a centre of spontaneous rotation. The
axis of percussion and the spontaneous axis are conjugate lines,
each of which implies the other; the positions of both are con-
nected and determined by Eq. (332). For example, the spon-
taneous axis of a straight rod struck at its extremity in a direc-
tion perpendicular to its length is, Eq. (222), given by
(/-/i)yi= (/-«)« = !'; (333).
whence
/ — flj = J^, or / = |. 2«, (334).
or is at a distance two thirds of the length of the rod from the
line of impact. All elements of the rod beyond the spontaneous
axis will have a motion in a direction opposite to that of the
impact, and those between the axis and line of impact a motion
142 MECHANICS OF SOLIDS.
in the same direction as the impact. This explains the cause of
the physical shock experienced when, in striking a ball with a
bat or in chopping with an axe, the part held by the hand does
not conform to the position of the spontaneous axis correspond-
ing to the line of impact.
128. In Eq. (332) substitute I — h ior h, and we have
[i-{i-h)-\(i-h) = h{i-k) = k;. . . . (335)
Hence, for parallel impacts, the centre of percussion and the
centre of spontaneous rotation are reciprocal and convertible;
that is, if the centre of spontaneous rotation become a new cen-
tre of percussion, the old centre of percussion will become the
new centre of spontaneous rotation.
Constrained Motion.
129. When a rigid surface or curve deflects a body from the
free path which any given system of forces would cause it to
take, the motion is said to be constrained. Let the motion of its
centre of mass determine the translation of the body, and, by the
principles in Art. 82, we may omit the present consideration of
its motion of rotation about that point. Eq. (119) then becomes
(^-^S"'-^ + ( ^-^S)'^^ + (^-^S)'^^ = °- (336)
130. Equations of Constraint. — Let
L^f(x,y,z) = o (337)
be the equation of the surface upon which the centre of mass is
constrained to move. Differentiating this equation, we have
g^^ + ^^_, + ^^, = o (338)
CONS TRA I NED MO TION.
143
Since the path of- the centre of mass lies in the given surface,
Eqs. {zz^) ^"d {ZZ^) maybe combined by considering only those
values of dXy dy and dz which are common to the path and sur-
face. To make the terms of these equations quantities of the
same kind, multiply the differential equation of the surface by
an intensity /. Add the resulting equation to Eq. (336), and we
have
('----S:+'S)*
Now, if
.dL ^dL . .dL . ^
^^' ^^ ""^ ^^' (340)
be the rectangular components of a force which, together with
the given extraneous forces, will cause the body to remain con-
tinually on the geometrical surface whose equation is that of the
rigid surface (337), the latter may be supposed removed and the
body will be a free body subjected to the action of the com-
ponent extraneous forces X -{- 1-^, etc.; hence, by Eqs. (Tn,), we
will have
dx
dL .^d^x
dz
dt
• • •
(341)
Eliminating / from these equations, we have
144
MECHANICS OF SOLIDS.
(.-
M
dy
•\dL _
dt^'Jdz
dt'ldx ~
>.
(342)
which are called the differential Equations of Constraint^ and from
which, with the equation of the surface, the path of the centre
of mass and its position at any time may be determined.
131. The Normal Reaction. — Let N represent the intensity of
the resultant whose component forces are
-dL dL dJL
dx^ dy dz^
and Qjc-> ^y^ ^z the angles which N makes with the co-ordinate
axes; then
„ I dL
cos d:c = -T-T-J- =
N dx
dL
dx
cos dy
^ dx" '^ df'^ dz"
dL
L dL dy
"T ^,.3 ~r j„i
cos dz = T-T-r
N dy /Jn
^ dx" ' df ' dz
dL
dz
N dz JdV_ d£ djy
^ dx^ "^ df "^ dz" J
• (344)'
CONSTRAINED MOTION.
145
Hence N acts always in. the direction of the normal to the devi-
ating surface.
Substituting in Eqs. (341) for /^r-, etc., their equals N cos 0^^
etc., we have, after transposing,
Y- M~^-^= -iV^cos ey\
(345)
Squaring and adding, and extracting the square root of the
resulting equation, we have
Representing the first member by P and dividing each of
Eqs. (345) by (346), we have
X- M
y-M^^
= — cos dx\
— cos 6
cos Sz,
(347)
The first members are the cosines of the angles which the
resultant P makes with the co-ordinate axes, and we see there-
10
146 MECHANICS OF SOLIDS.
fore that P acts in a direction opposite to the normal of the
deviating surface. We also see that P is the resultant of that
part of the extraneous forces which generates no momentum,
and as its action-line makes an angle of 180° with the normal to
the constraining surface, it is the measure of the direct pressure
on the surface; the equal intensity N is the equivalent normal
reaction of the surface. We therefore see that the force N,
which we have apparently introduced, already existed in the re-
action of the rigid surface. Hence we conclude that if a body
be acted oil by any system of extraneous forces and constrained
to move upon a rigid surface, the circumstances of motion of
translation will be precisely the same as if it were a free body
acted upon by a system consisting of the given forces and one
whose intensity and direction are those of the normal reaction
of the surface. Equations (345) may therefore be employed in
problems of constraint, just as Eqs. (Tj^) are employed in prob-
lems of free motion.
132. Transposing the second terms of the first members of
Eqs. (345) to the second members, and multiplying the resulting
equations by dx, dy^ dz, respectively, we have, by addition,
Xdx ■\-Ydy-\-Zdz-^NY-^- cos Qx-\-~ cos By + ~ cos e^ j- ds
,^dxd^x -^ dydy-{-dzd^z . „.
=^ ^^ ■• • • • (348)
which, since
dx Q ^ dy /J , ^^
-r cos Bx-\--j- cos By-\--r
ds ^ ds ^ ^ ds
^ cos 6^ -\- -^- cos By -\- —cos 6^ = o . . . (349)
is the cosine of the angle which the normal reaction makes with
the tangent to the surface, reduces by integration to
J {Xdx + Ydy + Zdz) = ^^ + C, . . . (350)
CONSTRAINED MOTION.
147
the equation of energy (126) of a free body under the action of
the same forces. Hence the conclusions derived from the
theorem of energy in free motion of translation are true in con-
strained motion also (Arts. 83 and 84); the constraint being
supposed to be without friction.
133. To find the value of N^ eliminate dt from Eqs. (345) by
the relation
(351)
and we have
as
NCOS By =MV'^- V;
JVcos 0, = MV'^ - Z.
as
(352)
Squaring and adding, we have
yV^'(cos' e.+ cos' e,+ cos' 0.)=M' F' \ (5) + {ff)'+ iff)' \
^x' -i- F'4-z^
(353)
Let p be the radius of curvature of the surface at any point,
and the angle which the resultant -ff makes with it; then sub-
stituting in Eq. (353) the following values:
cos' 6jc -f cos" 6y + cos' 6j = 1; . . . (354)
^»+ K» + Z' = i?»; (355)
i?sin0 = ^^;f = ^F'g;
. (356)
148
MECHANICS OF SOLIDS.
4-
ds
d'x
dx d's
ds
~ ds'
ds' ds"
4
ds
ds
_dy
~ ds'
dy d's^
ds' ds'''
4
ds
ds
d'z
~ ds'
dz d's
ds' ds''
the latter being obtained from Eq. (7), — we have
, dx dy
dz
, I ds dz d's\
"^ V ^^ '^ ds' ds'/
= H* sin' +
M'V
dx dy^ ,d_z_ ^
,^,,,^X "^Ts , Y "^ ds Z "^ ds\R
ds' {Rds Rds Rds S
+ R' cos" 0.
(357)
(358)
-VR'
(359)
But since
and
CONSTRAINED MOTION. ^49 ,
dx dy dz
X ds ^ Y ds ^ Z ds , i r \
X dx ^ Y dy , Z dz . , y^v
we have finally
^« ^ .^^' _ £^Z!^.£2i^ + j?.cos'0+ ^^'sin',^- 2^'sin'0
="—. -^ r+ie^'cos'^; (362)
whence
and
N •=. -^ cos (363)
iV=^cos0 (364)
I Therefore the normal reaction of the surface at any point is
equal to the difference between the normal component of the
resultant of the extraneous forces and .
9
134. When the body is in motion on the concave side of the
surface the value of N is given by Eq. (363), and when on the
convex side by Eq. (364). In the first case we see then that
when the action-line of R lies outside of the tangent to the
curved path of the body the intensity of the normal reaction is
equal to the sum of and R cos 0, and when it lies within
the tangent, to the excess of over R cos 0. When, in the
ISO
MECHANICS OF SOLIDS.
latter case, R cos becomes greater than , the body will
leave £he° concave surface and describe a path of greater curva-
ture, since iVcan never become negative.
In the second case Eq. (364) shows that the body can only
remain on the given surface as long as R cos is positive and
numerically greater than — —-\ when this condition is not ful-
Fig. 49.
filled the body will leave the surface and describe a path of ^ess
curvature. Fig. 49 illustrates the two cases.
135. Centrifugal Force. — The force whose intensity is meas-
MV
ured by was formerly supposed to be exerted by the body
itself and to act from the centre of curvature outward. It was there-
fore called centrifugal force. This name is still in general use,
but it is evidently a misnomer arising from erroneous conclu-
sions from well-known phenomena. We have seen, Art. 16, that
when the path is a curve the total acceleration is the resultant
of two rectangular components, one, -3-^, in the direction of the
v^
tangent, and theother, — , along the radius of curvature towards
P
CONSTRAINED MOTION, 15I
the centre. The intensities of the corresponding forces are there-
fore J/ -7-5 and M — .
In the discussion above, we find the latter force to be that
part of the normal reaction of rigid curves or surfaces which is
called into play by the equal direct pressure due to the change
in the direction of motion of the bodv. Hence is the in-
tensity of that force which actually deflects the body from its
rectilinear path. It varies directly as the square of the velocity
of the body, and inversely as the radius of curvature of its path;
it is zero when p is 00 , that is, when the path is a right line, and
infinite when p is zero, that is, no finite force can abruptly change
tlie direction of motion of a body.
Let a body B, Fig. 50, be whirled about a centre by means
of a cord dB held by the hand at d, tlie latter describing the
small circumference. The pull on the cord,
represented by Ba^ has two components, Bb
and Bc\ the former accelerates the motion of
B, while the latter deflects it from the rectilin-
ear path which it would follow due to its
acquired velocity and the component Bb. If
the motion be accelerated until , equal to
P
Ba cos 0, is greater than can be applied through
the medium of the cord, the latter will break
and the body will continue to move in the
direction of its motion at the instant of rup-
ture.
Due to the great angular velocities with
which fly-wheels, grindstones, etc., are often
made to rotate, thev sometimes break in pieces;
f ' YiG, 50.
this occurs when the cohesion at the sections
of rupture is less than the pull required to cause the parts of the,
body to describe their circular paths with the required velocity.
"When the areas of section of such bodies and the values of thg.
152 MECHANICS OF SOLIDS.
cohesion per unit of area are known, the safe values of V can
readily be determined from the expression .
136. Referring to Art. 44, we can now see why the rotation
of the earth on its axis diminishes the weight of bodies, and
makes the apparent less than the actual weight.
Since the mass m^ Fig. 51, is constrained to re-
main on the convex side of its parallel of latitude,
the normal reaction is
JV = mg cos A
Fig SI. = cos :i(^ — Qd'R)m, . . (365)
I
in which oo is the angular velocity of the earth, X the latitude of
m, and p and R the radii of the parallel and of the equator
respectively. At the equator the acceleration due to gravity is
diminished by the value of co^R^ or
4;r'3962.72 X 5280 ^ . , ,,^
which, as was previously stated, is about ^^ that due to gravity;
hence if the angular velocity of the earth were seventeen times
as great as it now is, bodies at the equator would have no appar-
ent weight.
137. Problems in Constrained Motion. — To solve these problems
we may either make use of the equations of constraint (342),
together with the equation of the surface, or by means of Eqs.
(345) treat them as oases of free motion. By the first method
we find the partial differential coefficients from the equation of
the surface and substitute them, with the component intensities
of the extraneous forces, in the equations of constraint. We
will have then two equations involving three second differential
coefficients, and a third equation can be obtained by differentiat-
ing the equation of the surface; hence we can thus find a single
CONSTRAINED MOTION.
153
equation by their combination, involving only a single second
differential coefficient, its corresponding variable and constants.
If integration be possible, the solution of the problem can then
be accomplished. By the second method the substitution of the
normal reaction and the component intensities of the extraneous
forces in Eqs. (345) gives three equations, each involving the cor-
responding component acceleration; the steps are then those
employed in cases of free motion.
138. On an Inclined Plane. — Let the forces be friction and the
weight of the body. Assume that the friction is constant and
directly opposed to the motion, and let F be its intensity. Let
Mg be the weight of the body, and take the axis of y in the in-
clined plane, the latter making the angle a with the axis of x.
Let the body start from rest at the origin; the motion will then
be in the plane xz, and we have for the equation of the path
L-=. z -\- X tan a = o.
(367)
Assume the second of Eqs. (342), and substitute in it the
following values:
dL
'dz ~
i;
^
dL
d'x ~
tan a\
d'x =
d'z
tan Of'
X=:
— i^cos a\
z =
^Mg-\-F
sin a, -
(368)
This gives us ^ Fig. 52.
d^z _ i^ sin a-\- F sin a tan' a — Mg tan' a
dt* ~ M(\-^ tan' a)
^ ( 77 ' n^ tan' a\ .IF . \
= M[^^^^oc^Mg^-^ = sina[-^gsma).
(369)
154 MECHANICS OF SOLIDS.
Multiplying by 2dz and integrating, we get
dt
{F,y = 2 sin «r^— - g sin ajz, . . . (370)
or, solving with respect to ^—--\-Mg--. . . (383)
By differentiating the equation of the curve, we get
dx dz ds
(384)
and since the radius of curvature is
p = \Za(2a - ^)]i, (385)
these values give, for W,
^ - FF-7 m + ^S- i u = ^SV-T Tu . (386)
If the body start from rest at the highest point of the curve,
then will V^ — 2g/i = 4^^ when it reaches the lowest point ; then
z = o, and we have
JV=Mg + Mg=2Mg, (387)
or double the weight ; therefore the direct pressure due to the
velocity generated by the weight in falling from the highest to the
lowest point is equal to the weight of the body.
158 MECHANICS OF SOLIDS.
Let the body start from any point, as P\ then OP = j, and
we have, Eqs. (382) and (384),
\ pp ds _ fay M dz
{2g)^Jk W^'^)^~^§l Jo (hz - zy
=(-;ri-'"-Ti:-(ir= • • • • • (388)
that is, the time of descent from any point on the curve to the
lowest point is independent of its height h and will be the same
no matter from what point the body starts. The velocity with
which it passes the lowest point will depend on the vertical
height of fall, and is equal to ^ 2gh : due to this velocity the
body will ascend to an equal height on the other branch of the
•cycloid in a time equal to that of the descent, or Tt\ — \ ; after
which it will return to the point of starting, and so on continu-
ously. These recurring movements are called oscillations, and
since they are performed in equal times, 2;rl— 1 , the cycloid is
•called a tautochronous curve.
The time down any inclined right line / is, Eq (377),
^ = (7^^)*' (389)
therefore down any radius of curvature of the cycloid it is
(390)
dx
whence, substituting the value of p, Eq. (385), and of — , Eq.
CONSTRAINED MOTION. 1 59
{384), and reducing, we get
=(?)' <-)
g
which is the time of fall down the maximum radius of curvature,
or twice the diameter of the generating circle. Therefore the
times of descent down all radii of curvature of the cycloid are
•equal.
141. On the Concave Side of a Circular Arc in a Vertical Plane,
— Let a be the radius, and take the origin at the highest point
with the axis of 2: vertical and positive downward. The equation
of the circle is
X* = 2az—z*, (392)
from which we have
dx dz ds
a ^ z X
(393)
Let the weight be the only force acting, and denote the ve-
locity at the origin by V^. Then we have
MV'-MV:=M{2gz\ (394)
or
v' = 2gz+v: (395)
For the normal reaction we have
^ = _.. _ jr^_ = ^ J.^. _ Mg^-. . (396)
The limit of N being zero, we have for the corresponding
value of the velocity at the origin
V, = V^. (397)
i6o
MECHANICS OF SOLIDS.
That is, the body will not follow the curve continuously in one
direction unless the velocity at the highest point be at least
equal to ^ag. The corresponding value of the velocity at the
lowest point is ^^ag.
Constrained Motion about a Fixed Axis.
The Compound Pendulum. — When a rigid solid oscillates
freely about a horizontal axis under the action of
its weight, it is called a compound pendulum.
Let G, Fig. 55, be the centre of gravity, h its dis-
tance from the axis, and ^ the angle which h
makes with the vertical plane through the axis.
Then we have, Eq. (257),
Fig. 55.
M.
Mgh sin
■ (398)
or, taking ^ so small that it may be substituted for its sine,
d^^ _ gh^
(399)
Multiplying by '2dif) and integrating, supposing the body to
start from rest when ip — a, ^q have
#'^ S^^ (a'~0')- (4oo>
and
dt
= i/^
-\- h^ -dtp
-1 ^/a" - f
(401)
CONSTRAINED MOTION ABOUT A FIXED AXIS. l6l
Hence
d^
^ gh J |/a» _ ^» ^ gh J / _^»
gh a ^ '
The time of one oscillation is therefore
^, = ^y ^^ > (403)
which is the integral between the limits ip =. a and ^ = — or.
The oscillations of a compound pendulum may therefore be
considered isochronal when the arcs of vibration are very small.
143. The Equivalent Simple Pendulum. — If in Eq. (403) we
make k^ = o, we shall have the time of oscillation of a material
point about a horizontal axis with which it is connected by a
line without weight. Such a pendulum is called a simple pendu-
lum. Denoting its length by /, we have for its time of oscillation
/' = 'rV^ (404)
ii
For a simple and a compound pendulum which are isochronal
with each other we have
or
/=^^; (405)
and / is called the equivalent simple pendulum of the given com-
pound pendulum.
1 62 MECHANICS OF SOLIDS,
A point of the compound pendulum on tlie line h at the dis-
tance / from the axis is called the centre of oscillation, and a line
through this centre and parallel to the axis of suspension is
called the axis of oscillation.
144. From Eq. (405) we have
{l-h)/i = k; (406)
Hence, as regards their distances from the centre of gravity,
the axes of suspension and oscillation are connected by the
same law as the line of impact and spontaneous axis. Therefore
the axes of suspension and oscillation are reciprocal and convertible, and
the times of oscillation about them are the same.
145. Adding 2k^ to both members of Eq. (406), we have,
after reduction,
/=.., + (A^* (407)
The minimum value of / is 2/^^, and this occurs when h — k^
r=i I — h. Therefore, since / is a minimum when / is a mini-
mum, the time of oscillation is a minimum wheri the axis of suspension
passes through the centre of gyration 7vith respect to an axis through the
centre of gravity and parallel to the axis of suspension. It is evidently
a minimum minimum for that centre of gyration which corre-
sponds to the least central principal moment of inertia of the body.
146. The Simple Seconds Pendulum. — The simple seco?ids pendu-
lujn is a simple pendulum whose time of oscillation is one mean
cr
solar second. Its length, L — — 2, obtained by making / = i in
Eq. (404), varies directly with the acceleration due to gravity.
Let / be the equivalent simple pendulum of a given com-
pound pendulum, and we have
CONSTRAINED MOTION ABOUT A FIXED AXIS.
163
and substituting in the value of Z, we get
^ - ;r' - /-
{409)
Hence to find L it is necessary simply to find the time of
oscillation of a compound pendulum and the length of its equiva-
lent simple pendulum. To do this Kater
used the pendulum represented in Fig 56.
Its centre of gravity is removed from its
middle point by the heavy bob BB'\ S
and O are two knife-edged prismatic axes
of hardened steel, permanently attached
to the rod and having their edges turned
towards each other ; w is a small ring-
shaped mass which can be moved up and
down so as to change slightly the position
of G and the value of ^^, and is arranged
with clamps and a screw to fix it in any de-
sired position. For any assumed position
of ni let GC be the gyratory circumference
which, from the construction of the pen-
dulum, always lies between S and G^ and
beyond O from G, as in the figure.
It is evident that with this arrange-
ment every position of m gives a different
compound pendulum for each axis, and
that there is but one among these whose
equivalent simple pendulum is the length
h-\-h' between the fixed axes, and for which
the times of oscillation about 6" and O would be the same. It
was this particular compound pendulum that Kater desired to
find experimentally by moving the sliding mass m until it was
placed in the required position. This he was enabled to do by
the application of the principles stated in Arts. 144 and 145^
By the first principle, the times of oscillation about S and O
Fig. 56.
164 MECHANICS OF SOLIDS.
are equal only when these lines accurately coincide with the
axes of suspension and oscillation. On trial the times about
S and O would in general be found to differ slightly, and by the
second principle a displacement of m towards either axis would
cause both times to lengthen or both to shorten, but unequally.
And since the gyratory circumference is nearer O than S, a
greater change for the same displacement takes place with re-
spect to the latter than to the former, from the principle of max-
ima and minima. The pendulum was mounted on the axis S
and permitted to oscillate through small arcs, the number of
oscillations, being counted; then it was suspended from the axis
and oscillated, the corresponding number in the same time
being determined. Repeated trials enabled Kater to find the
position of m for which the distance between S and (9, or
// -j- //' z= /, was accurately the length of the equivalent simple
pendulum, and its time of oscillation was known from observa-
tion. This method does not require the determination of the
exact position of G, but has the disadvantage of exacting accu-
rate adjustment of the mass m, an operation requiring very care-
ful and repeated manipulation.
In order to count the number of oscillations the method of
coincidences was used. Thus the pendulum was mounted in
front of a clock whose pendulum, beating seconds, could be seen
by means of a telescope behind the position of the Kater pendu-
lum, when it passed the lowest point of its arc of oscillation. At
a certain second, indicated by the clock, the two pendulums
would coincide, and after an exact number of oscillations of the
clock pendulum they would again coincide. The number of
oscillations of the clock pendulum in this period, if the duration
of the clock pendulum's oscillation was less than that of the
Kater pendulum, would be two viore, and if greater two less, than
the Kater pendulum. The clock gives its own indication, and
hence the other is at once determined. In Kater's experiment
the entire duration of each trial lasted about thirty-five minutes,
corresponding to five coincidences, or four intervals of 530 sec-
onds each.
CONSTRAINED MOTION ABOUT A FIXED AXIS. 165
147. Another and less tedious method is to use a reversible
pendulum of such a form that its centre of gravity may be de-
termined with considerable accuracy, and whose axes are not
reciprocal. Then we have, Eq. (403),
k^ -f /^' = Lhf', \
and eliminating k^^
h^ -h
L
= ht-' _ ^Y«;
L~ 2 h^h' '^ 2 h-'h''
As the error in this method is due to the approximate values
of h and h\ t and f should be as nearly equal as practicable, and
h and h* should differ as much as practicable, thus making the
term which depends on ^ — h* very small,
148. The Value of G. — Having found L by experiment, the
acceleration due to gravity is found from
^=^'^ (412)
When g has been determined for one locality we may find its
value for any other place by means of any compound pendulum.
For the two places we have
-y^-^ and f = n\/-±^.. . (413)
1 66 MECHANICS OF SOLIDS.
Hence
t'U*'v.g^:g (414)
or
^ =^5-6" (415)
If N and JSP be the numbers of oscillations per hour at the
two places, these become
N'^:N^v.g^ '.g', (416)
N
^' = -w^^ (417)
The formula Eq. (59), taken from Everett's Units and Physi-
cal Constants, gives, for all latitudes,
g = 32.173 — 0.0821 cos 2/1 — .000003-^, . . (418)
pr
and from L = —^ we have the corresponding value for the simple
seconds pendulum,
Z = 3.2597 — .0083 cos 2A, — .0000003/^. . . (419)'
Substituting the value of the latitude of West Point, 4i°23'3i",
we derive
^ = 32.163/.^. and Z= 3.2587//. . . (420)
for the values of ^ and Z at West Point.
149. Length of the Equivalent Simple Pendulum. — The length of
the equivalent simple pendulum of any compound pendulum is,
Eq. (409),
l=Lt\ (42i>
CONSTRAINED MOTION ABOUT A FIXED AXIS. 1 6/
in terms of its time of oscillation and the length of the simple
seconds pendulum at the place of observation.
150. British Standard of Length. — In 1824 an act of Parlia-
ment defined the Imperial Standard Yard " to be the straight line
or distance between two points in the gold studs in the straight
brass rod" known as the "Standard Yard, 1760," at 62° F., and
designated the ratio of its length to that of a simple pendulum
vibrating mean seconds, in vacuo, at sea-level at the latitude of
London, to be as Tf^ '•Z9-^Z9Z- This standard was destroyed in
the burning of the Houses of Parliament in 1834. Upon the
recommendation of a commission of scientific men appointed to
restore the standards of weights and measures, the act of 1855
defined the Imperial Standard Yard oi Great Britain to be "the
straight line or distance between the centres of the two gold
plugs or pins in the bronze bar deposited in the office of the
Exchequer," at 62° F. Its restoration in case of loss or destruc-
tion is provided for, by reference to its numerous copies. The
present standard is therefore not referred to tlie length of the
simple seconds pendulum.
151. To determine the Mo mefit of Inertia of a body by the princi-
ples of the Compound Pendulum. — It is often necessary to find the
moment of inertia of a body which is not homogeneous nor of a
regular form, and to which therefore the methods of Art. 101
will not apply. Whenever the body can be mounted so as to
oscillate about a horizontal axis under the action of its own
weight, we may apply the principles of the compound pendulum
and find its moment of inertia no matter what its form or sub-
stance may be.
Thus, multiplying Eq. (405) by M, the mass of the body, and
clearing of fractions we have
Mlh = M{k; + h') = 2mr''] (422)
that is, the product of the mass of the body, the length of its equivalent
simple pendulum, and tiie distance of the axis from the centre of
gravity, is the moment of inertia with respect to the axis about
1 68 MECHANICS OF SOLIDS.
which the body is oscillated. The first two quantities are ob-
tained frcm the equations
W
M = — and I - Lt\ (423)
in which W is the weight in pounds, ^ the acceleration due to
gravity at the place of observation, derived from Eq. (418), Z the
length of the simple seconds pendulum, and t the time of oscilla-
tion of the body about the axis in question. The value of h can
be found by measurement, provided the exact position of the
centre of gravity is known. When this is not known h may be
found as follows: Attach a dynamometer to the
extremity farthest from the axis of suspension,
as in Fig. 57, and take its reading when the
I^sbody has been lifted by the dynamometer to a
^^ position such that the axis and centre of gravity
Fig. 57. are in a horizontal plane; this position is reached
when the reading of the dynamometer is a maximum. Then if
R be the reading in pounds and a the horizontal distance TS in
feet, we have, by the equality of moments,
Ra = Wh,
whence
k=^.. ...... . (424)
Substituting these values of M, I and hy we have
Mlh = 2n.>' = ^.^.Lt' = p. . . . (425)
To find the moment of inertia with respect to an axis through
the centre of mass, the body must be mounted on a parallel axis,
with reference to which its moment of inertia may be found by
CONSTRAINED MOTION ABOUT A FIXED AXIS. 1 69
the above method. Its moment of inertia with respect to the
axis through the centre of mass is then
Mk^ = Mlh - Mh^ (426)
For bodies of small mass it is sometimes more convenient to
attach them to a pendulum w^hose moment of inertia with respect
to its axis is known. The moment of inertia of the combination
may then be determined, and the difference between this result
and the moment of inertia of the known pendulum is the mo-
ment of inertia of the body with respect to the axis of the pen-
dulum, from which the required moment may be found.
152. The Conical Pendulum. — Let a simple pendulum have a
component vibration about two horizontal axes at right angles
to each other. The path of the material point will in general be
a curve of double curvature on the surface of a sphere whose
radius is the length of the pendulum, this length describing a
cone with its vertex at the point of suspension. Such a pendu-
lum is called a conical pendulum.
The equation of the projection on a horizontal plane of the
curve of double curvature may be determined as follows:
Let ^ and be the arcs of oscillation about the two axes re-
spectively; then sin ^ and sin will be the co-ordinates of the
projected path of the material point of the pendulum. If the
arcs be taken so small that the oscillations may be considered
isochronal, then ^ and may be taken as co-ordinates of the
projected path, instead of sin ^ and sin 0.
Let P and a be the maximum values of 0and ^ respectively.
When = yS we will have ^ = o, and = o when ^ = or. Tak-
which / is measured, we have, Eq. (402),
ing ip = a when / = o, and taking y — seconds as the unit in
o
/=cos~^-; (427)
170 MECHANICS OF SOLIDS.
and taking = /? when ^ = o, or / = \7t, we have
/=cos-^| + i;r (428)
Eliminating /, we get
cos-i| = cos-i-^ + i;r; (429)
and taking the cosines of both members,
(430)
or
rfi'i-cP'a'--a^^\ (431)
the equation of an ellipse referred to its centre and axes.
153. While the point moves in azimuth about the vertical
through the point of suspension, the change in azimuth from
7t
the time when tp ^= a until = /? is -. But this deduction is
made under the supposition that the vibrations of a simple pen-
dulum are isochronal. However, as the time of vibration in-
creases with the length of the arc, the shorter component vibra-
tion in the conical pendulum will be completed first, counting the
time from any assumed epoch. Therefore the change in azi-
muth from the time when tp = a until tp = — a \s greater than
7t, and that from 7p = a until cp = /3 is greater than — . Hence
the axis of the ellipse has a motion in azimuth in the same di-
rection as that of the pendulum.
A closer approximation shows that the change in azimuth
EQ UILIBRIUM. 1 7 1
7t ( ■? \ ^
from ^ = a until = /? is -( i + f^/^j instead of -. The el-
lipse therefore makes one complete revolution in azimuth in
8 . . . J. •
— - times Its periodic time.
Za6
Equilibrium.
154. When a body is in a state of equilibrium, Art. 68, the
acceleration factors in the general equation of energy become
separately equal to zero, and there can then be no change of po-
tential into kinetic energy, or the reverse. Therefore during
equilibrium the general equation of energy reduces to
2Idp = o. , : (S)
In general a free body is never in a state of equilibrium, since
it is subjected to the action of forces the resultant of which is not
in general zero. Hence a body in equilibrium must be under
constraint. But since any case of constrained motion may be
discussed as one of free motion by introducing the normal reac-
tion of the constraining curve, the general equation of energy
may thus be made to apply to all cases of equilibrium.
155. There are three cases in which the acceleration may be-
come zero, viz.:
(i) When the resultant is zero, the body being at rest or hav-
ing uniform motion.
(2) When the resultant of the system of forces reverses its
direction as it passes through zero, thus changing the sign of the
resultant acceleration.
(3) When the resultant becomes zero but does not pass
through it; in this case there is no change of sign of the result-
ant acceleration.
First. When the body is at rest the forces are called siresseSy
and they produce changes of form and volume, these effects
1/2 MECHANICS OF SOLIDS.
usually being known as strains; their investigation properly be-
longs to Applied Mechanics.
When the accelerations become zero because of the uniform
motion of the body, Eq. (S) simply asserts that the quantity of
work done positively by some of the forces is equal to that done
negatively by the others, the whole quantity of energy added to
the system being zero. If a body be supposed at rest at any
point of that portion of its path over which it has uniform mo-
tion, it will evidently remain there if subjected only to the sys-
tem of forces which caused it to follow that path. In sucli a
•case the body is said to be in neutral or indifferent equilibrium.
156. To investigate the second csiSQ let us resume Eq. (132).
The forces of gravitation, electricity, etc., or what are known
generally -as forces of nature, are taken to be constant or to vary
as some function of the distance; and therefore Eq. (132) is ap-
plicable to a body subjected to their action, the normal reaction
of the curve on which the body moves being considered as one
of the extraneous forces.
Assuming consecutive values of the kinetic energy of the
body, we have, after developing the difference of the correspond-
ing, states of the function by Taylor's theorem,
WV: - WV: = n^, + dx^, y^ 4- dy^, z^ + ^5,)- F{x^, y^, z^}
= Xdx,+ Ydy^ + Zdz^±{Adx^^-\- Bdy^-^ Cdz^)-\- etc. (432)
If \MV^ be a maximum or a minimum, we shall have as a
condition
Xdx, -f Ydy^ + Z^^, = o; (433)
that is, the body will be in equilibrium when it reaches a position
where it has a maximum or minimum kinetic energy.
Let this condition be fulfilled, and we have
\MV^ - \MV^ = ± {Adx^ + Bdy^' + Cdz^) -f etc. (434)
EQUILIBRIUM. 1 75
If \MV^ be a maximum the second member of this equation
will have the negative sign, and whatever be the value of V^, it
must be greater than V ^ ; and if the body be slightly displaced
from its position of equilibrium and then move from rest under
the action of the given system of forces, the direction of the re-
sultant must be such as to bring it back again. In this case the
body would oscillate to and fro through its position of equilib-
rium, and could never depart far from it; the equilibrium of a
body is therefore said to be stable when it occupies a position,
corresponding to a maximum value of its kinetic energy.
If \MV^ be a minimum the second member of Eq. (434) will
be positive, and hence V^ must be greater than F, ; and if the
body move from rest and from a point very near that corre-
sponding to the minimum value of F, the resultant of the system
will act in such a direction as to move it away from its position
of equilibrium, to which it would never return. The equilibrium
is therefore said to be unstable when it occupies a position corre-
sponding to a minimum value of its kinetic energy.
157. In the third case, since a function is not necessarily a
maximum or a minimum when its differential coefficient is equal
to zero, it is evident that cases may arise in which a body will be
in equilibrium when its kinetic energy is neither a maximum nor
a minimum. For example, a body is in equilibrium as it passes
a point of inflection at which the resultant is normal to the path.
In this case the equilibrium is stable in one direction and un-
stable in the other.
158. When 2ifree body passes a point in its path correspond-
ing to a maximum or a minimum value of its kinetic energy, this
point would be a position of stable or unstable equilibrium, if we
suppose the body to be moving on a rigid curve coincident with
its path.
Let us take the general case of a body acted upon by its own
weight, and subjected to any condition of constraint whatever.
Since this condition can have no influence on the velocity, the
change of kinetic energy between any two points of the path
174 MECHANICS OF SOLIDS.
will be that due to the weight acting over the vertical distance
through which the body moves; hence we may write
WV; - mV^ = Mg{z, - ^.), .... (435)
z being taken vertical and positive downward. From this we see
that if the body be in stable equilibrium its centre of mass must
occupy a point in its path which is lower than the consecutive
points on either side; and similarly if it be in unstable equilib-
rium it must be at a point which is higher than the consecutive
points.
Thus, a pendulum is in stable equilibrium when its centre of
gravity occupies the lowest point of its possible path, and in un-
stable equilibrium when it is at the highest point it can reach
under the given conditions of constraint. A homogeneous el-
lipsoid of three unequal axes, resting on a horizontal plane, has
two positions of unstable and one of stable equilibrium; an
oblate spheroid has one of stable and many of unstable equilib-
rium; a prolate spheroid has one of unstable and many of stable
equilibrium, and a sphere is an example of indifferent equilib-
rium. If the centre of gravity of the sphere be not coincident
with the centre of figure, it will be in stable equilibrium when
the centre of gravity is at the lowest point it can reach, and un-
stable when at the highest.
Examples, i. A particle on the concave surface of a sphere
is acted on by its weight and by a repulsion from the lowest
point of the surface, the latter varying inversely as the square of
the distance. Find the position of rest.
Take the origin at the lowest point and the axis of z vertical
and positive upward; let r be the distance of the point of rest
from the origin, a the radius of the surface, and ^ the intensity
of the repulsive force at the distance unity; then we have the
equation of the surface,
^^ +y + -2;' — 2az = o;
EQUILIBRIUM.
175
the intensity of the repulsion at the distance r,
and Eqs. (T^,),
2az
x = J^^
2az ' r
N'
o;
N^- =0;
2az r a
(436)
from which we have, substituting in the last of Eqs. (436) the
value of N obtained from one of the others and reducing,
^ = ^ and
w
2 aw
(437)
that is, the particle will remain at rest at any point in the cir-
cumference of-fi horizontal circle whose plane is at the distance
given by Eq. (437) above the lowest point of the surface.
If another repellent force whose intensity at a unit's distance
is /i' be supposed to act on the particle, the value for /-' would
be r'' =
7a '
hence the ratio of the intensities of these forces at
or their intensities are
the distance unitv is given by — =
directly as the cubes of the distances at which a heavy particle
would remain at rest on the surface of a sphere due to their
action. The quadrant electroscope, consisting of a light pith
ball joined to a point by a thread, measures the relative inten-
sities of strong electrical charges by the principles of this prob-
lem.
176
MECHANICS OF SOLIDS.
2. To find the position of rest of a heavy particle m on a.
given rigid curve AB^ Fig. 58, when acted on
by its weight w and a constant attraction t
toward the origin. Let O be the origin, Ox and
Oz the co-ordinate axes, the latter being vertical
and positive downwards, and /;/ the supposed
position of rest of the particle; let OA = a,
Then we have
Fig. 58.
Om = r, and mOn = 6.
X= -/sin 6 -\-N
ds
Z=ze/ — /cos d — N
dx
Ts
o:
(438)
from which we get
{w — t cos 6)dz — / sin 6dx = wdz — / = o. (439)
But
xdx 4" zdz = rdr,
and Eq. (439) reduces to
wdz — tdr = o, .
(440)
(441)
which is the condition of equilibrium. In order that there may
be a position of rest on the curve this condition must be satisfied
by the co-ordinates of one of its points.
Let the curve be a hyperbola, and we have
^V - «V = a'd^; (442)
^'' = ^-^+^«=:.V-^'; (443)
rdr = e^zdz; (444)'
EQ U I LIBRIUM. I *J*J
and Eq. (441) becomes
rwdz — te^zdz = o (44S)
Solving Eq. (443) with respect to z and substituting in the result-
ing equation the value of r^ obtained from Eq. (445), we have
The equilibrium in this case evidently requires w '>et.
If it be required to find the equation of the curve on all
points of which m will be at rest, we have, from Eq. (441),
wz — tr = 3. constant, (447)
which may be written
wz — tr — {w — t)a (448)
By substituting for z its value /- cos 6, this equation becomes
(■-?)
^ = ' (449)
I cos 6
the polar equation of an ellipse, parabola, or hyperbola, accord-
ing as w is less than, equal to, or greater than /, the pole being
at the focus.
12
178 MECHANICS OF SOLIDS.
^ The Potential. •
159, The general theory of attraction embraces the consider-
ation of those forces by which matter attracts or repels other
matter, and whose intensities are functions of the masses, and of
the distances which separate them. The general term attraction
is used for both repellent and attractive forces; the former being
affected by the positive and the latter by the negative sign, to
distinguish them.
In the following discussion the theory of attraction is limited
to those forces whose intensities are expressed by the law, Eq. (2),
^ mm'
Electrical and magnetic forces are sometimes attractive and
sometimes repellent; and since their intensities vary according
to the law of the inverse square of the distance, Eq. (2) may
also be applied to them by considering m and m' to be quantities
of free electricity or magnetism instead of masses; the unit quantity
being that which will attract an opposite or repel a similar unit
quantity with a unit intensity, at a unit's distance apart. In the
following discussion the term mass will then for convenience be
taken to apply to the quantities m and m\ As in gravitation,
these mutual attractions or repulsions are equal in intensity,
opposite in direction, and exert their own influence whether
other forces act on the masses or not.
160. Component Attractions. — Let m be the attracted mass, m'
one of the attracting masses, r the distance between them, and
X, J, z the co-ordinates of m' referred to m. We have then
^« = ^«+y + ^«; (450)
dr _x dr _y dr _ z . .
d^~P ~^~7' ~d^~~r' ' ' ' ' ^^^'^
■&)=-..
^^ (452)
THE POTENTIAL.
179
The component attraction of m* for m in the direction of the
co-ordinate axis x will then be
mm' X
X = — 7[-U — = — mm -.
r r dr ax
(453)
and similarly, for the other axes,
.,/(^). ,- ..,'(^)
K= — mm*ii-^\ Z
dy
mm }i
dz
(454)
The sums of the component attractions of all the molecules
m\ m'\ m"'y etc., for tn are therefore
-y =
( 4) -^W ^
_,„^\^„'._ + ^"__ + etc.y
/- ^
m' fx
+ 0,
(459)
or, between the limits r and co ,
, /''■ dr
m* II
. (460)
THE POTENTIAL. l8l
If the unit mass be subjected to the attraction of all the
masses in\ m'\ etc., then we have
^n ^ ^-2-- = ixV, (461)
r
which becomes, when the absolute intensity }a is taken to be
unity^
27r=:2—= F. (462)
That is, the potential is equal to the change in the potential energy of
a unit mass when the latter is moved to infinite distance from any dis-
m*
tance r against the decreasing attraction — -2— j, or when the unit mas^
r
is brought from infinite distance to any distance r in opposition to the in-
creasing repulsion^ 2-^ \ the absolute intensity being taken as unity in
r
both cases.
For any definite system of masses and forces governed by
the assumed law the potential at a point has a definite value and
can be expressed in units of work. When the potential is known
the component attraction in any direction can be readily ob-
tained by means of Eqs. (457).
162. Equi'potential Surfaces. — Let R be the intensity of the
resultant attraction of the system of masses for m at the dis-
tance r\ then, after multiplying Eqs. (457) by dx^ dy, dz, respec-
tively, and adding, we have
fdV dV dV \
Xdx + Ydy -\- Zdz = — mpiy—dx -\- -j-dy + —j-^A^ (4^3)
or
Rdr = — jnfidV^ (464)
1 82 MECHANICS OF SOLIDS.
whence
P r^ — ^m IJ.
dr
dV
^ = - "^i^iz (465)
Let s be the path of m as it changes its position in any direc-
tion; d the angle which s makes with r\ and /the component of
R in the direction of s. Then we have
I ■=^ R cos cy = — m)j.—— cos U =. — in}x—-. . . (466)
Hence the component attraction in any direction varies directly with
the first differential coefficient of the potential regarded as a function
of the path in that direction.
An equi-potential surface is one for which the potential is con-
stant for each of its points. If the path j be a line of such a sur-
face, we have
Ids Rdr .
dV — = = o, (467)
m^ mjx \-r 1/
and hence
F= - — fids = -— fRdr =C. . . (468)
A surface which fulfils for each of its points this condition
is an equi-potential surface for the system of attractions. As
any value may be attributed to C between its greatest and least
values, there will be an indefinitely great number of equi-poten-
tial surfaces, corresponding to any given system of attractions^
each of which will be a closed surface.
From Eq. (466) it is evident that /becomes zero when ^ is
90°, and is the resultant attraction, or R^ when Q is zero; hence
THE POTENTIAL. 1 83
the equi-potential surface cuts at right angles the direction of R
at every point of its surface. These action lines of R are called
lines of force, and any collection of them passing through an
elementary portion of the surface is called a tube of force.
If the surface be supposed perfectly smooth, vi would remain
at rest on every point of it if subjected only to the system of at-
tractions, and the surface would resist pressure only in the di-
rection of the normal. For this reason it is called a level surf ace
or a surface of equilibrium. No two surfaces belonging to the
same system can intersect or have a common point, for Eq. (468)
cannot be satisfied for the same values of x, y, z, and give C
dissimilar values. Of any two surfaces, the interior one corre-
sponds to the greater resultant attraction and the less value of
the potential when the attraction is negative, and to the greater
repulsion and greater value of the potential when the attraction
IS positive.
163. The determination of the values of the potential and the
attractions for any given system of masses or quantities 2m^
depends on the solution of the equations
V= 2--, R=- nifx— and / = - w/^-^. (469)
When the quantities w' are elements of a quantity whose
density and boundary vary by continuity, we may write
m' = dM' = ddv; (47°)
in which dv represents the elementary volume of M\ In such
cases we have, when rectangular co-ordinates are employed,
^^jSJ^^jSdxdydz ^^^^^
1 84
MECHANICS OF SOLIDS.
If it be desirable to use polar co-ordinates, we have for the
z value of the elementary volume,
Fig. 59,
= r'dr sin OdcpdO; ] ^"^^^^
for the edge of the infinitesimal
cube in the direction of r is dr,
the horizontal edge perpendicular
to r is r sin 6dcp, and that per-
pendicular to the plane of these two is rdS; and hence, in polar
co-ordinates,
= / drdr sin 6d(pd6 (473)
164. Examples. — In the following examples the mass m will
be taken to be the unit mass, and the absolute intensity fx to be unity
also. By multiplying the results obtained by ?nfJL we readily get
the attractions for any mass and any intensity.
I. The Potential and Attractions of straight Rod at an External
Point. — Let go be the area of the rod's cross-section, y the dis-
tance of the external point m, Fig. 60, from
the rod, and r its distance from any element
of the rod; and let the axis of the rod be a
taken as the axis x, and the axis y pass °
through the external point. The element ^"^- ^°
volume of the rod is then oodx^ and the distance r = Vy^ -\- x'^;
then we have
V=Sgd T-
= =(y&?iog ^. ^ ,, i . (474)
4// + X
THE POTENTIAL. 1 85
which, taken between the limits x** and — x\ corresponding to
the extremities of the rod, becomes
= ,Ja,log-^-^^== (475)
The component attraction in the direction of the rod is
To find that perpendicular to the rod we have, from Eq. (474),
dy dy
Therefore between the limits ^" and — x' we have
i86
MECHANICS OF SOLIDS.
When the point m is on the perpendicular bisecting the rod,
we have
X = o\ Y= .——=== = sine';. . (479)
in which 6 is the angle included between the bisecting perpen-
dicular and the line drawn from m to the extremity of the rod.
2. The Potential and Attractions of a Circular Arc at its Centre.
— Let d be tlie angle subtended by any
portion of the arc estimated from its mid-
dle point, Fig. 61. The element volume is.
then oordd, and we have
V=dGD r dd = 26006',
(480)
which is independent of the radius of the
arc. The resultant attraction is evidently in the direction CO^
or along the radius drawn to its middle point. Its value is
doo
L
^' rdO
2600
r
(481)
From this we see that the attraction of the arc at the centre is.
the same as the attraction of the straight rod AOA' . Since also
the masses of the elements//' and PP' have the ratio
CP
Cp :CPstca = Cj> : CP^ = C/ : CP\ . . (482).
their attractions on m at Care equal; whence, any portion of the
right line tangent to the arc at O, as PP\ attracts m at C with
the same intensity as the corresponding arc//'.
THE POTENTIAL. \%J
3. The Potential and Attraction of a uniform Circular Ring at
a Point on the Perpendicular to its Plane through its Centre. — Let a
be the radius of the ring, and c the dis-
tance of the point (9, Fig. 62. Then
•^'^' 27taSG0 , „ ,
V= 2— = ^ . (483)
r ^a' + c' ^ ^
When c = o, this becomes a maxi-
mum, or 27rSco, which, being independent Fig. 62.
of the radius, shows that Fis a constant for all concentric rings
at the centre.
As the sum of the component attractions of the elements of
the ring in the plane of the ring is zero, the resultant attraction
is in the direction of the perpendicular to the plane of the ring..
Its value is evidently
R=-2^co%QOC = -,-2m' = j^^-j^^ . . (484)
4. The Potential and Attraction of a Circular Plate at a Point
on the Perpendicular to its Plane through its Centre. — Let / be the
thickness of the plate, Fig. 62, and suppose the plate made up of
separate rings whose width is da. Then we have 00 = tda, and
the element volume is 27ttada\ whence
V=2n6tr-^^^ = 2nSt(\fT-:^-c\,. (485)
t/o ya -\- c
which for the centre of the plate becomes
F= 2ndta (486)
The resultant attraction is
2 7ttada y^^r^ A P^ ada
R
n^ 2ntada /^ aa
= 01 5 — cos QOC = 27tdtc / —_ —
e/o r J^ V^-f c^
= 27ttc6[ , ^ X =27t6t{l ^ - V . (487)
1 88 MECHANICS OF SOLIDS.
But since
^a'' + c
= = cos QOC,
it follows that the attraction of all circular plates of the same
thickness and density, for a given molecule on a perpendicular
to its plane through its centre, is the same for equal angles, sub-
tended by the plate at the molecule. Therefore, if a molecule
be at the vertex of a right cone with a circular base, the attrac-
tions of the normal sections of equal thickness of the cone for
the molecule are equal. If the radius of the plate be infinite the
attraction becomes 27cdi, which is independent of the position of
m with respect to the plate.
5. The Potential and Attractions of a Spherical Shell at any
Point. — Taking the centre of the shell as the origin of a system
of polar co-ordinates, let a be the radius and p the distance of
the point from the centre; then the volume element is, Eq. (472),
aV sin ddddcf),
and
r=z{a'' - 2ap cos 6 + py-, .... (488)
whence
cv o /»*'' n"" sin Odd del)
Integrating first with respect to 0, we have
- , />Tr sin Odd . .
V=27t6ta-^ — ; . . (490)
e/o (a^ — 2ap cos 6 -\- pry
and then with respect to 6,
THE POTENTIAL. 1 89
There are two cases to consider: (i) when m is within the
surface, or p < «; and (2) when m is without the surface, or p > «.
In the first case we have
r=^— -^[« + p- (^-p)] = 4^(^/«; . . (492)
and in the second.
[« + p-(p-«)] =——— = --; (494)
9 "- P 9
tdta' _ A
dp p^ ~ p^
■7;: = -^ = ^-7^ = --^'^ • (495)
M being the mass of the shell. Hence the potential of the
interior space is constant and the resultant attraction zero, while
all external space is made up of concentric spherical equi-poten-
tial surfaces, and the attraction at any point is the same in
intensity and direction as though the whole quantity M were
concentrated at the centre of the shell.
6. The Potential and Attraction of a Thick Homogeneous Spheri-
cal Shell at any Point. — Let the radii of the exterior and interior
surfaces of the shell be a' and a" respectively. The potential
will, in general, consist of two parts, one corresponding to the
shell within the spherical surface containing the point and the
other to the shell without it.
For the first part we have (Ex. 5)
dV='-I^da: (496)
and for the second,
dV = 47rdada; (497)
IQO MECHANICS OF SOLIDS.
Jience
A.7td f*^ />*'
V = / a^da 4- attS I ada
P Ja" J?
If the point be wholly without the shell, then, M being the
mass of the shell, we have
V
= 4^^ r'a^^a = i^^(«- - an = ^; (499)
P J a" 3P ^ P ^ ^'
^ = ^; (500)
and if wholly within,
V = 27td{a" - a"')', (501)
^ = o (502)
If the attracting quantity be a homogeneous sphere a" = o,
>and at an interior point
V=,„Sa"-'-^^; (S03)
Ii = ±nSp = ^p- (S04)
o
a
and at an exterior point
^=f (507)
p ^>
THE POTENTIAL.
191
For p =. a' vje, have, from both sets,
r=4^; . . (508) i?=^; . . . (S09)
hence both Fand R are continuous functions.
From this we see that, considering the earth as homogeneous,
we may take its potential and attraction at any external point
as though the whole mass of the earth were concentrated at its
centre; while the attraction at an interior point is directly pro-
portional to its distance from the centre.
165. The Theorem of Laplace. — Let S be any closed surface,
and M any attracting quantity wholly external to S\ let V be
the potential of J/, and p the normal to the surface reckoned
outward. Then it is to be proved that
ndV
dS - o.
(510)
From any molecule w', Fig. 63, of M draw a right line pierc-
ing the surface S. It will pierce the surface in an even number
Fig. 63.
of points. Let/' and/" be a pair of these points, 6' and ^" the
angles made by the normals at /' and /" with the intersecting
192 MECHANICS OF SOLIDS.
line, and r' and r" the distances of/' and/" from m\ Then we
have
With m' as a vertex, conceive an indefinitely small cone whose
solid angle is cd to be described about w'/'. The areas inter-
cepted by the cone on S at the points/' and/" are
dS' = -^, and ^y = -^-; . . . (512)
cos d' cos 6" ^'^ '
and hence
^J^,dS' = m'm; ^-dS" = - m'm; .... (513)
g:.5'+^/5'-o; (5x4)
and this result is true for every pair of points.
Now suppose the cone whose vertex is w' envelops the whole
surface S; then its solid angle is made up of an indefinitely
great number of elementary cones whose solid angles are c», to
,each of which the above reasoning will apply. Therefore we
have
/
^■5=»- fe">
166. Poisson's Extension of Laplace's Theorem. — If any portion
of M, as M\ be contained within the closed surface 6", then it is
to be proved that
/
'l^dS=-A7tM' (516)
THE POTENTIAL. 1 93
Let m' be the mass of one of the molecules of M\ and let a
right line be drawn from it, piercing S. It will pierce S in an
odd number of points, which may be arranged in pairs as above,
with one point remaining.
For the pairs Ecj. (515) will be true, and for the odd point
we have
;^' = -"^- • • (517)
Integrating for the whole space about w', we have
—dS=—Pt'l doD = — ^7tm'\ . . . (518)
and for all the molecules of J/',
/
'^-f^dS=-4nM' (S19)
Hence we conclude, Eqs. (515) and (519), that the sum of the
attractions of a mass J/, estimated along the normals at all
points of a closed surface, is zero when the attracting matter M
is wholly external to Sy and is — ^7tM' when the closed surface
S contains any portion, J/', of M.
167. This theorem is also expressed by the equations
d'V , d'V . d'V ^ / X
-d^^W^~dF = '' ^^ =-A7t6., . (520)
To show this, let x, y, z be the co-ordinates of the attracting
molecule, of density (J, without or within the surface, which may
be taken as the surface of a rectangular parallelopipedon dxdydz.
Then
/
dp
13
194 MECHANICS OF SOLIDS.
for the face dydz^ is
and for the opposite face is •
Sd'^v. ,dv),
-and for this pair of faces is
-j^dxdydz',
and similarly for the other faces we have
d'^V
-—^dxdydz
and
d'^V
a dxdydz.
Hence the integral
Placing the second member equal to o and — ^nM' in succes-
sion, we have, when the attracting matter is external to a closed
surface,
d^V , d'V d'V . .
MOTION OF A SYSTEM OF BODIES.
195
and when it contains it, wholly or in part,
(522)
These theorems find their most frequent application in elec-
tricity and magnetism.
Motion of a System of Bodies.
168. The conclusions of Arts. 78 and 82 with respect to the
motion of a single body under the action of extraneous forces
are similarly true of a group or system of bodies when the mo-
tion of its centre of mass determines the translation of the
system in space, and the rotation is estimated about that centre.
To show this, let x^,y^,z^be the co-ordinates of the centre of
mass of the system referred to any fixed origin; x,y, z, the co-
ordinates of the centre of mass of each body referred to the
fixed origin, and x\y\ z\ when referred to the centre of mass
of the system; and let M be the type-symbol of the masses of
the bodies. Applying Eqs. (T^,) to each mass and summing the
results, we have
^X = 2M'
df
:EY=:2Mg,
2z = :2M
dt""
(523)
and since
x = x,-\-x\ y=y^ +/, zz=^z,-\-z\ . (524)
d^x = d'x^-\-dW, dy = dy^-{-dy, d'z = d\+d*z', (525)
196
MECHANICS OF SOLIDS.
these become, by applying the principle of the centre of mass,
2V
2Z =
df '
d'
dt'
'-2M.
(526)
Similarly applying Eqs. (Tm') when impulsions alone act, we
have
at
dz
-^^M = 2MF.;
(527)
hence the conclusions of Art. 78 follow from Eqs. (526) and (527)
with respect to the centre of mass of the system.
169. If each of Eqs. (526) be multiplied in succession by the
two co-ordinates which it does not contain, and the difference of
the products be taken, then the summation of these differences
for all the bodies of the system gives
:s(y,,-jr,,) = 2Jf(..5'/-,,5<)
2(Xz, - Zx,)
2(Zy, - Yz,)
d^x^ d"^ z
■ (528)
from which the motion of the centre of mass about the fixed
origin may be found.
MOTION OF A SYSTEM OF BODIES.
197
In the same way we have, from Eqs. (Tm),
(529)
Substituting in these Eqs. (529) the values of the co-ordinates
and accelerations from Eqs. (524) and (525) and reducing by
the principle of the centre of mass, we obtain
2{ Yx' - xy) = 2M (^'^ -y^);
(530)
Since these equations are independent of the co-ordinates of
the centre of mass of the system, we conclude that the motion
of rotation of the centres of mass of the constituent bodies about
the centre of mass of the system is the same as if this point were
at rest, and their motion is therefore entirely independent of
the motion of translation of the latter point, a conclusion pre-
cisely similar to that of Art. 82.
170. Conservation of t/ie Motion of the Centre of Mass, — If we
suppose that a material system has been put in motion and
then subjected only to the mutual attractions of its own bodies,
we shall have
2x = o, :2Y^o, :ez =
(531)
198 MECHANICS OF SOLIDS.
then there can be no accelerations of the centre of mass of the
system, and Eqs. (526) become
d^'x, d^y, d'^z,
df ' dt' ' dt'
o; . . . . {532)
from which we have
sc,- at-Y a\ y^=z bt-Y b\ z^ = a + c';. . (534)
- (535)
x^-a' _ y^ - b' _ z^
a b c
That is, if a system of masses be subjected only to its mutual attrac-
tions^ its centre of 7nass will either be at rest or more uniformly in a
right line. This is called the principle of the conservation of the
motion of the centre of mass.
171. If the masses of the solar system be subjected only to
their mutual attractions of gravitation, the conditions of Eqs.
(531) are satisfied for this system, and therefore its centre of
mass must have uniform and rectilinear motion, or be at rest.
Since the mass of the sun is very much greater than the sum
of all the other masses of the constituents of the system, the
error of assuming the centre of mass of the solar system to
be coincident with that of the sun is slight. Calculations found-
ed on the observations of astronomers show that this latter point
is moving through space with a velocity of very nearly five miles
per second, but sufficient data is not yet available to determine
whether its path is a right line or an arc of small curvature; the
latter being the more probable, owing to the extraneous forces
of attraction of other systems.
By the same principle, the motion of the centre of mass of
MOTION OF A SYSTEM OF BODIES.
199
the earth is uninfluenced by earthquakes or volcanic explosions
occurring upon it, and that of the centre of mass of a projectile
is not affected by its explosion, since the impulsive forces in each
of these cases are mutually counterbalanced.
172. Conservation of Moments, Invariable Axis, and Invariable
Plane. — If the forces acting on the system be the mutual attrac-
tions of its masses, we have the conditions
:2{Yx* - Xy') = o:
2{Xz' - Zx') = o;
2{Z/ - Vz') = o;
(536)
which reduce, Eqs. (530), to
2M\
2M\
and which, by integration, become
> ,
(S37)
at
:^M - - c ,
at
(S38)
That is, when the forces acting on the system are the mutual
attractions of its masses, the algebraic sums of the moments of the
momenta of the masses of the systetn with respect to any set of rectangu-
lar co-ordinate axes at the centre of mass of the system are constant;
this is called the principle of the conservation of moments.
200 MECHANICS OF SOLIDS.
This principle may also be stated as follows: If the bodies of
the system be supposed at rest in any one of its configurations,
a definite system of impulsions would give each body its actual
velocity. Eqs. (538) show that the sum of the component mo-
ments of these impulsions with respect to each co-ordinate axis
is constant. The resultant moment of the system is also con-
stant, and is given by
C = VC + C + C' (539)
As the co-ordinate planes may be assumed at pleasure, it is
evident that the constants C, C", C" will in general change
with each set of co-ordinate axes. The resultant axis of the sys-
tem on which C would be measured is normal to that plane with
reference to which the sum of the products of the projected
areas by the masses is the maximum constant, and is the com-
mon intersection of those planes on which these sums are zero.
This axis, and the normal plane through the centre of mass of
the system, are called the invariable axis and invariable plane of
the system of masses; the equation of the latter,
Cz' + cy + c"v = o, . . . . . (540)
is found by multiplying each of Eqs. (538) by the co-ordinate
which it does not contain and adding the results together.
173. Conservation of Areas. — Eqs. (538) express another prin-
ciple, which is known as the conservation of areas. Let radii-
vectores r be drawn from the centre of mass of each body to that
of the system, supposed at rest ; then changing x'dy'—y'dx' into
its equivalent expression in polar co-ordinates with the pole at
the centre of mass of the system, we have
x' =■ r cos 6, dx' = dr cos — r sin ddd\ \ ^ ^
y = r sin e, d/ = dr sin e + r cos edd; j * ^^"^^^
x'dy' - fdx' = r\iB (542)
MOTION OF A SYSTEM OF BODIES. 201
But r^dd is twice the projection on the plane 3(^y' of the sectoral
area described by the radius vector of M* in the time dt\ and the
corresponding factors in the other equations are similarly the
projections of double the differential areas on the planes x*z* and
y'z* respectively. Let the type symbols of twice these projec-
tions on the planes x'y\ x'z\ y'z' be denoted respectively by
dAz, dAy^ and dA^ ; Eqs. (538) then become
at
2M^^ = C";
at
if A
at
(543)
Inregrating between the limits corresponding to the interval /,
we have
'2MA, =6-7;^
:2MAy = C't', I (544)
:2MA^ = C"V.
That is, if a system of masses be subjected only to its mutual attrac-
tions^ the sum of the products of each mass by the projection of its secto-
ral area about the centre of the system on any plane varies directly
with the time. This statement of the principle is called the con-
servation of areas.
If the resultant of a system of extraneous forces act through
the centre of the system, Eqs. (536) will be satisfied and the con-
clusions of Arts. 170, 172 will apply to this case also.
174. Relative Acceleration. — If one of two bodies be supposed
fixed and all the motion be attributed to the other, the accelera-
tion which the latter would have under this supposition is called
its relative acceleration. To find the relative acceleration of one body
of the system with reference to the centre of mass of any other, let
202 MECHANICS OF SOLIDS.
Mbe the mass to which the motion is referred, and call this body
the central; M\ the mass of the moving body, called l\\^ primary \
M'\ the type-symbol of the masses of the remaining bodies of
the system, called Xh^ perturbating bodies. Let the symbol (AfM^)
represent the intensity of the reciprocal attraction of the masses
M and J/' along the line joining their centres, and the same
symbol with a subscript letter, as (MM')^, the component inten-
sity in the direction of the corresponding axis; and similarly for
the other masses and directions. Let x, y, z be the co-ordinates
of J/' referred to M. The component relative acceleration of
J/' with respect to Mis the sum of their actual component
accelerations due to their mutual attraction, plus the difference
between the components of their actual accelerations due to the
attractions of the perturbating bodies. The actual accelerations
of M and J/' due to their mutual attraction are ^ — r^r-— and
M
(MM') , , , , , ,. . ^ ..
-^^ — T77 » and those due to one of the perturbatmg bodies are
(MM'') , (M'M") ^, , . , .
^^ — TF— and -^^ — , \ The component relative accelerations are
therefore
d'xr (MM% (MM^- l r 2{MM")^ 2(M'M")^
df~V M '^ M' J + L M M'
dy_ r{MM% {MM ')y-\ r :E(MM")y _ :2{M'M")y
df'V M '^ M' J + L M M'
d'z_\- (MM '), ^ {MM')r \ , y ^{MM'% 2{M'M")
dt
+
V {MM '), iMM%-\ r 2(MM'% _ :2{M'M"), -]
\_ M ^ M' J + L M M' J
^(545)
175. The path of the centre of mass of the primary with
respect to the central is called the relative orbit of the primary ;
its relative path influenced by the action of the perturbating
bodies is called the disturbed or actual orbit, and if the action of
these latter bodies be neglected the resulting relative path is
CENTRAL FORCES.
203
called the undisturbed orbit. The differential equations in this
case become
df +
d\
-df=^
(MM') {MM')-lx ^
M
+
M
P]?,
\MM') {MM' y
. J/ '^~ M' "
) (^^j/'r
V{ MM'
(546)
in which the upper sign corresponds to attraction and the lowti
to repulsion, and r is the radius vector of M' . When the law ot
the reciprocal attraction or repulsion is known the value of its
intensity may be substituted for its symbol {MM'), and the
resulting equations being integrated twice, there will result the
component relative velocities and co-ordinates of the centre of
mass of the primary referred to the centre of mass of the central
body.
Central Forces.
176. A central force is one whose action-line is directed to or
from a fixed point called the ce7itre of force, and whose intensity
is a function of the distance of the body acted on from that
point. The force is attractive or repulsive according as its
action-line is directed toward or from the centre.
177. Laws of Central Forces. — (i) Let the two masses M and
M' be subjected to the action of their mutual attraction or re-
pulsion. Then the motion of one, relative to the centre of mass
of the other, may be considered as resulting from the action of
a central force whose centre is the centre of mass of that body
which is considered as fixed; Eqs. (546) are then applicable.
Multiply the first by j and the second by x, and take the differ-
ence of the products; then multiply the first by z and the third
by Xy taking the difference of the products; and lastly, multiply
204
MECHANICS OF SOLIDS.
the second by z and the third by_>', taking the difference of the
products, and we shall obtain
//>
d-'x
''ae
^-d¥
—
o;
d'x
d'z
''dt^-
=
o;
d'z
^dt^
^dt'
=
o.
* • •
(S47)
Integrating, we have
dy dx ,,
dt -^dt
dx dz ,,,
'dt-''-di = '' '
^dt dt
(548)
Multiplying these equations by z, y and x^ respectively, we have,
by addition,
h'z-\-h''y-\-h'"x = o, (549)
the equation of an invariable plane. Hence the orbit of a body
acted on by a central force is contained in a fixed plane through the
centre of force.
(2) Take xy to be the plane of the orbit; then Eqs. (548)
reduce to the single equation
dy dx . , V
or, in polar co-ordinates.
,dQ
dt
= >^;
(551)
CENTRAL FORCES. 20$
in which r is the radius vector of M\ 6 is the variable angle
made by r with a fixed line of reference, and h is double the
sectoral area described by the radius vector in its own plane in
the unit time.
Integrating Eq. (551) between the limits corresponding to
values of the radius vector r' and r" and the angles 6' and 6'\
we have
r'^'r^dd^hif ^f') = ht', .... (552)
dr"9"
or, the sectoral area described by the radius vector in the plane of the
orbit varies directly with the time.
Reciprocally, when this law is fulfilled we have, by differen-
tiating Eq. (550) and multiplying by M\
M'^,x-M'^y=Yx-Xy = o;. '. . (533)
and the orbit is described under the action of a central force.
(3) From Eq. (551) we have
du h / X
or, the angular velocity of the body in its orbital motion about the centre
of force varies inversely as the square of the radius vector,
(4) Since the velocity of the body in its orbit is
ds ds dd , V
we have, from Eq. (554),
h ds I r\
'' = PTff (SS6)
206 MECHANICS OF SOLIDS.
Let/ be the length of the perpendicular from the centre "of
force to the tangent to the orbit at the body's place; then
(557)
. n rdd
^ = r sm = r-^,
and hence
h
"" P'
(558)
or, the velocity of the body varies inversely as the perpendicular distance
fro7ti the centre of force to the tangent to the orbit at the place of the
body.
(5) Let A be the relative acceleration at any point of the
orbit, and p the corresponding radius of curvature; then the
component relative acceleration in the direction of p is
^^ = -^'; (559)
from which we have
4-M • (560)
But 2p— is the length of the chord of curvature drawn through
the centre of force to the place of the body. Comparing Eq.
(560) with V"^ = 2ghf we see that the actual velocity of the body at
any point of its orbit is that due to a height equal to one fourth of the
chord of curvature drawn through the centre of force ; the body starting
from rest and the intensity of the central force remaining constant over
this distance. If the orbit be circular, R its radius, and its centre
coincide with the centre of force, the velocity becomes constant
and is
V'^AR (561)
CENTRAL FORCES. 20/
The acceleration in the direction of the tangent to the orbit
is
_ = _ ^_ . .... . . . (s6.)
Multiplying by M' and ds and integrating between limits, we
have
W{V.'-V,') = -M'£'Adr, . . . (563)
the equation of energy. Hence the orbital velocity is inde-
pendent of the path described and varies with the distance of
the body from the centre of force. In any closed orbit, therefore,
when the body returns successively to the same position in its
orbit, the velocity will always be the same as before.
These are the general laws of central forces, and are seen to
be independent of the character and law of variation of the cen-
tral force.
178. The Differential Equation of the Orbit. — Assuming the co-
ordinate plane xy as the plane of the orbit and employing polar
co-ordinates, we have, from the law of areas, Eq. (551),
which becomes, when r is replaced by — ,
^^f'"' (S64)
Differentiating
n COS 6 / , X
x-rcose= ; (565)
208 MECHANICS OF SOLIDS.
and dividing by dt^ we obtain
dt u" 'dt
= — hiu sin 6 + cos 6-\ .... (566)
and hence
= -/^v(«cos^ + cos^^). . . . (567)
Placing this value equal to that of the relative acceleration
d^x
—-- in the first of Eqs. (546), we have, after dividing by cos d
X
and remembering that — = cos ^,
{MM') , {MM') . -^ u, J ^ ^M , ^ox
the differential polar equation of the orbit of a body under the
action of a central force.
179. To solve the direct and inverse problems in the case of
a central force we proceed as follows:
(i) To find the equation of the orbity substitute for ^ in Eq.
(568) its value in terms of «, and integrate twice; the resulting
equation expressing the relation between u and B is the equation
of the orbit. The two arbitrary constants which appear in the
integration are determined by the initial conditions, viz., the
initial values of the radius vector and velocity, and the initial
direction of motion.
(2) To find the law of the force necessary to cause a body to
CENTRAL FORCES. 209
describe a given orbit, differentiate the polar equation of the
orbit twice and substitute the resulting value of — ^ in Eq. (568);
then eliminate 6^, and the result will give ^ in terms of r.
180. Particular Cases of the Direct Problem, — (i) To find the
orbit due to an attractive central force whose intensity varies directly
with the distance of the body from the centre. Let the centre of
force be the origin; }x\ the measure of the intensity of the central
attraction for a unit mass at a unit's distance. From the law of
the force and the differential equation of the orbit we have
or
^=/'''-=^'-M"+^) (569)
d^u a'
Multiplying by idu and integrating, we get
5^ + " =-^' + ^- (57')
Let jR be the initial value of the radius vector, and take the
initial direction of motion perpendicular to ^. Then at the
time / = o we have ^ = o and « = - ; hence ^ = ipi + ^ ,
and
^ + « -^»+-;^-^i. . . . . (572)
Let Fbe the initial value of the velocity, and from Eq. (558)
we have h := RV, Substituting this value of h in Eq. (572),
14
2IO MECHANICS OF SOLIDS.
multiplying through by u^^ and changing the form of the result-
ing equation, we have
Solving with reference to dQ^ and multiplying by 2, we have
ludu
2de-
^ \ 2V'R' I \ 2V'R' I
2 V'R'
,2udu
. (574)
, '^ ^ V y - m'R' I
Integrating between limits corresponding to / = o and /, re-
membering that the initial radius vector coincides with 7?, we
have
„ . 2V'R''u'' -{V^ ^'R-") . . ,
2^ = sm-^ ___Jy_^ ^-sin-^i; (575)
whence we have
sin (90° + 26/) = cos 2^ = --^—^Jl L, (576)
Clearing of fractions and changing to rectangular co-ordi-
nates, we have, recalling that
cos"" 6 = -^-^- — 3 and sin" 6 = -—-- ,
V
2 V'S' -{V' + lx'IP){x'' +/) = (V'- m'H^Xx' -/); (577)
CENTRAL FORCES, 211
whence we have
;^^ + ^ = i, (578)
the equation of an ellipse referred to its centre and axes; hence
the orbit of a body acted on by a central attraction varying directly with
the dista?ice, is an ellipse whose centre is at the centre of force.
To find the velocity at any point of the orbit. Let v be the gen-
eral value of the velocity of the body in its orbit; then from
Eq. (558) we have
h^
^'=J.' (579)
and from Eq. (556) we get
I _ ds" _ r'de''-^d r' _ du^
Hence
But from Eq. (572) we have
and therefore
V' = V + ix'{R- - r'), (583)
which gives v when r is known.
To find the Periodic Time. The semi-axes of the orbit are,
y
E<^- (578), J^ and -—=: The periodic time, or the time required
212 MECHANICS OF SOLIDS.
for the body to complete its orbit, is, by the law of areas, therefore
V
"[^-^^ _2nRV__jn_
Hence the periodic time is independent of the dimensions of the
orbit.
Examples of orbits under this law are found in molecular vi-
brations; in the small vibrations of elastic bodies, such as tun-
ing-forks, stretched strings, etc.; and in the oscillations of a
pendulum through small arcs.
181. (2) If the central force be repellent, the discussion above
may be made applicable by changing the sign of /i' in Eqs. (568)
and (578); the equation of the orbit then becomes
^ = .; (585)
hence the orbit is an hyperbola whose centre is at the centre of force,
182. (3) To find the orbit when the central force is attractive and
varies inversely as the square of the distance. Assume the same no-
tation as in the previous problem, and let the direction of the
initial velocity make any angle with the prime radius vector H.
Then we have
^ = ^' = ,,V = /5v(J + «), . . . (586)
or
dd" ' '^~ h^
%^ = ^ (587)
CENTRAL FORCES, 21 3
Multiplying by 2du and integrating, we have
From the initial conditions and Eq. (558) we have
^=¥-1^' (589)
and therefore
#
5gi + «=^ + -^-^ (S90)
This equation may be written
by assuming
^^ = c'-{u-by (591)
¥ = ^ and _-^_ + ^ = ... . . (59.)
From Eq. (591) we have, taking the negative sign of the radical,
■vrT§T^^ = ''-' (593)
and by integration,
cos-^^y- = e + y, (594)
in which y, the constant of integration, is the initial angle which
214 MECHANICS OF SOLIDS.
the prime radius vector makes with the fixed line of reference.
From Eq. {594) we get
... . u=^b-]-c cos {O-i-y), (595)
or
'' = l, + ccoHd + y) ' (596)
and substituting the values of b and c,
which may be written
"^ ^ 'rJjT^ + I . cos (B-^y)
Comparing this with the polar equation of a conic section
referred to the focus as a pole,
'•-T+T^o70' (599)
we see that (598) is the equation of a conic section referred to
the focus as a pole, in which
'^y r/ +^> ('°^>
= (9 + X (602)
CENTRAL FORCES. 21 5
Therefore the orbit will be an ellipsCy parabola or hyperbola accord-
ing as
V'R — 2)U' < o, =0 or > o;
that is, as
r<|/^, =|/? or >|/^. . .
183. To determine the meaning of y -~ , we have
(603)
R
1? ~ ~~?'
(604)
Multiplying by 2dr and integrating between the limits r = 00
and r = jRy we have
or
.=/?
(606)
Thus y ^ is shown to be the velocity which the body would
have if it should move from rest at infinity to the distance J^
under the action of the central force; it is called l/ie velocity from
infifiity at the distance R.
Hence we conclude that the orbit of a body under the action of a
central attraction varying inversely as the square of the distance will
be an ellipse, parabola or hyperbola according as the i?iitial velocity is
less than, equal to or greater than the velocity from infinity at the ini-
tial point.
2l6 MECHANICS OF SOLIDS.
184. To find the velocity at any point of the orbit. For the ve-
locity at any point we have, Eq. (581),
^du
and hence from Eq. (590) we have
z/' = F'' + 2^'u - ^' (608)
or
z-'=F' + 2A<'(i-^] (609)
from which the velocity corresponding to any radius vector r can
be found.
We also see from Eq. (609) that the velocity at any point of
the orbit will always conform to that which characterizes the
particular orbit in question; that is, if the orbit be a parabola,
for example, the velocity at any point whose radius vector is r
2U'
will always be equal to — at that point, and, similarly, less than
-^ for the ellipse and greater than -^ for the hyperbola.
185. To find the time of description of any portion of the orbit.
(i) The Elliptical Orbit. — We have for the equation of the orbit
I I +
and hence for the time from the extremity of the conjugate to
the farther extremity of the transverse axis we have
C-+')*^
,= ^ + AV-, (
CENTRAL FORCES. 219
From these values we see that the Velocity decreases from the
nearer to the farther extremity of the transverse axis, and then
increases to the nearer extremity.
The trajectory of a projectile in vacuo, under the supposi-
tion that its weight acts with constant intensity in parallel
directions, was shown in Art. 93 to be a parabola. The pre-
ceding discussion shows that this trajectory will be an arc of an
ellipse having the earth's centre at the farther focus, when grav-
ity is considered as a central force varying inversely as the
square of the distance from the centre of the earth. The trans-
verse axis of the ellipse is the vertical through the highest point
of the trajectory.
(2) The Parabolic Orbit. — We have for the equation of the
orbit
r = -^—, (6^4)
I -j- cos u
in which 2a is the semi-parameter. We therefore have
^ = ^.^^ +.]=-- = --,; , . .^ (625)
hence
and
/*' = — , or h" = 2a^', (626)
2a
P
We have from Eq. (551)
^ =j»=n^ +^j=-7 (^^7)
dt = -de\
n
hence
220 MECHANICS OF SOLIDS.
~ ^ '^'M, (i + cos ey
= y —j-i tan -» — tan -^ + - tan' -' tan' -' . (629)
n n
Place tan -" = /, and tan -^ = t^^ then the equation above be-
comes
=t/g«. ->,)(■ +'••+'•;■+'•) (630)
Let/ = I 4- ^'^ ^ , and we have
4
Let ^ be the length of the chord joining the extremities of the
radii vectores r, and r^\ then we have
^• = r.'-ir.r,cos( = fr^" ^'" ^* "^ ^T^''* * ' ' ^^^^^
Therefore
= 4a\t,-i,Y/; (636)
hence
c=2a(f,-ljy (637)
and
= ^''\y+'-^'Yi (638)
and similarly
r, + r,-c=2a^y-' i^^ ^ (639)
Substituting in Eq. (631), we have finally
' = ^j *'(v+^T^-'»^(v+^^^'}; (640)
from which / can be found in terms of the radii vectores and the
chord of the parabolic arc.
VM'^(e'' - i)
rdr
222 MECHANICS OF SOLIDS.
(3) The Hyperbolic Orbit. — We have for the equation of the
orbit
r = — ^^ —
I 4- ^ cos 6^'
or
I e
U = — r^ r + --7-5 T cos ^; . . . . (641)
a(e^ — I) a(f — i) v -^ /
and therefore
_ />« r'^dd
/^V^(.-i)4/{r + ay -^V
= 4/^ { 4/ (.+.r-.v -.iog -+-+^(;+-)' i:£:f! [ ; (64.)
from which the time corresponding to the description of that
part of the orbit from the vertex to the point corresponding to
any radius vector r can be found.
186. The Anomalies. — When the central attraction varies in-
versely as the square of the distance, the position of the body in
its orbit is generally referred to the right line coinciding with
the least radius vector. This line is called the line of apsides ^ and
its intersections with the orbit are called apses; the one nearer
to the focus being the lower^ and the other the higher^ apsis. The
angle included between the line of apsides and the radius vector
is called the anomaly^ and is measured from the lower apsis as an
origin.
Let us place
cos-i— ^j- = «^, (643)
an auxiliary angle; then we have
r = a{\ — e COS, n) (644)
CENTRAL FORCES.
223
Substituting the value of u in Eq. (620), we have, after placing
nt = u — e s\n u (^45)
Equating the values of r from Eqs. (644) and (610), we have
. . . (646)
1 —e'
=z I — e cos u,
whence
and therefore
or
I -{- e cos d
_ (i — e)(i -fcos u)^
I + cos 6
I — cos 6
1 — e cos u
(i + ^) (i — cos u)
I — e cos
— cos 6 I 4" ^ I — cos u
I -\- cos 6 1—^*1 + cos «*'
tan-^
2
./i + e I
y — ! — tan -u.
(647)
— e
(648)
(649)
The angle 6 is called the true anomaly, u the eccentric anomaly^
and ;«/ the mean ano??ialy.
From Eqs. (644), (645) and (649) the values of the true
anomaly ^and the radius vector r can be found in terms of the
eccentricity and the mean anomaly. (See Price's Calculus, vol.
iii. pp. 561-567.) These are
d=nt-\-2e sin nt-\- —e* sin 2nt-\ (13 sin snt—^ sin nt)-\-etc.;
•=«( I — ^cos nt-\ — (i —cos 2nt) — - (cos 3«/— cos «/)-|-etc. J.
(650)
224
MECHANICS OF SOLIDS.
Hence, knowing the mean motion, the time since the epoch,
and the eccentricity, the true anomaly or the angular distance of
the body from the line of apsides, and the distance of the body
from the focus at once results. The difference between the true
and mean anomaly d — 7it, called the Equation of the Centre, is
evidently a function of the eccentricity, and its value is obtained
at any time / from the first of Eqs. (650).
187. To illustrate the geometrical meaning of these quanti-
ties let APB'A^ Fig. 64, be the elliptical orbit, .V the centre of
force afthe focus, P the position of the body, SP ~ r, PSA =
6 and AC = a. On AA^ describe a semicircle APA\ From
the properties of the ellipse we have
SP = a- eCM,
and therefore
r •=^a — ae cos QCM
= « — ae cos u, .
(651)
(652)
From Eq. (616) we see that the periodic time is independent
of the eccentricity of the ellipse; it is therefore the same as that
in the circle whose radius is a; but in this case e = o, r = a, 6 =1
u = nt. Hence nt represents the arc of
the circle which would be described
uniformly by a body in the same time
as that in which the elliptic arc is de-
scribed, both bodies starting from A^
and both reaching^' at the same time;
71 is therefore called the mean motion
of the body. Since sin u is positive in
the first two quadrants, we see, Eq.
(645), that u is greater than nt while
the body is describing that part of its
orbit from A to A\ and less than nt from A' to A; therefore the
true place of the body is in advance of its mean place in the first
and second quadrants, and behind in the third and fourth; nt is
Fig. 64.
THE SOLAR SYSTEM. 22$
therefore called the mean anomaly, and since u depends on the
value of ^, «* is called the eccentric anomaly. Both the velocity
and the angular velocity are greatest at A and least at A\ as is
seen from the equations giving these values in the laws of cen-
tral forces.
The Solar System.
l88. The Solar System consists of the sun and other bodies
whose relative positions and motions are mutually dependent,
and which taken together may be considered as a single system
of bodies in space. It derives its name from the sun, the great
central body about which all the other members of the system,
called /r/Vz/^ry or secondary bodies, revolve.
Th.Q primary bodies are:
(i) The four inner or lesser planets. Mercury, Venus, the Earth
and Mars, named in order of their distance from the sun.
(2) A group of minor planets called Asteroids, of which over
two hundred and sixty have so far been noted and catalogued.
(3) The four outer or greater planets, Jupiter, Saturn, Uranus,
and Neptune.
(4) A number of Comets and Meteors, or bodies having masses
much smaller, and generally orbits of much greater eccentricity,
than those of the planets and asteroids.
The secondary bodies are the Satellites or Moons of the planets,
which describe orbits about the latter and are carried with them
in orbital motion about the sun; of these now known three be-
long to the inner and seventeen to the outer planets.
All the bodies of the solar system are spheroidal in form, and
their diameters are very small compared with the distances
which separate them from each other. In addition to their or-
bital motions they have a motion of rotation about their axes.
The mass of the sun is more. than seven hundred and forty
times as great as the sum of the masses of all the other bodies
of the system. Owing to this fact, and to the relative positions.
15
226 MECHANICS OF SOLIDS.
of the planets, the centre of mass of the entire system lies within
the sun's volume and not far from its own centre.
189. Kepler s Laws. — John Kepler, of Wurtemburg, was the
first to announce the laws governing the motion of the planets
about the sun. This announcement was the result of more than
twenty years' faithful and laborious study of the observations
collected by his predecessor Tycho Brahe. Kepler's investiga-
tions were principally directed to the explanation of the appar-
ent irregularities of the motion of the planet Mars, whose orbit
was at that time supposed to be an epicycloid. These laws of
Kepler not only completely accounted for the motion of Mars, but
also satisfactorily explained the motions of all the other planets
about the sun, and those of the satellites about their respective
primaries. These laws are:
(i) The orbit of each planet about the sun is an ellipse^ having one
of its foci in the suits centre.
(2) The areas described by the radius vector of each planet in its
orbital motion vary directly as the times of describing them.
(3) The squares of the periodic times of the planets are directly
proportional to the cubes of their fnean distances from the suns centre.
190. If we assume, for tlie present, that these laws are accu-
rately true, we readily deduce the following consequences, viz.:
(i) The orbit of each planet being an ellipse having the sun's
centre in one focus, it follows that the value of the relative ac-
celeration becomes, Eq. (613),
^ _ hW _ h' I
that is, the relative acceleration varies inversely as the square of the
distance of the planet from the centre of the sun.
(2) From the second law, or that of equal areas described in
equal times, we have
r^dd = hdt = xdy — ydx (654)
THE SOLAR SYSTEM. 22/
Whence, after differentiating and dividing by
hence, since the masses of the planets are known to be unequal,
we must conclude that Kepler's third law is not rigidly true.
If the masses of the sun, 1,000,000,000, of Jupiter, 954,305, the
greatest, and of Mercury, 200, the least of the planets, be substi-
tuted in (661), we find
^ = 1.000954, (662)
and this ratio will be more nearly equal to unity for any other
pair of planets. The discrepancy in assuming unity for the ratia
^ for the planets of the solar system is therefore in general
negligible, and the consequence of Kepler's third law may be
taken as true within sensible limits.
191. -Law of Gravitation. — Later and more accurate observa-
tions than those which Kepler employed show that his laws are
not exactly true, but are only very close approximations to the
truth. The single law which governs planetary motions and
definitely fixes their actual departures from the positions as-
signed by Kepler's laws is that of universal gravitation, which
is thus enunciated by Isaac Newton:
That every particle of matter in the universe attracts every other
particle^ with an intensity which varies directly as the product of their
masses^ and ifiversely as the square of the distance which separates them.
Newton deduced this law from his investigations of the rela-
tive acceleration of the moon, in a direction normal to its orbit
THE SOLAR SYSTEM. 229
about the earth. He proved tliat the earth's relative attraction
on the moon caused it to fall towards the earth, with an acceler-
ation due to gravity, modified only by the increased distance and
greater mass of the moon, precisely as a body near the earth's
surface falls with its particular acceleration; and therefore con-
cluded that the law of attraction between the earth and moon
was essentially the same as that between the earth and body.
From this deduction the generalization to the enunciated law of
gravitation followed.
The intensity of the reciprocal attraction between the sun
and a planet, whose masses are M and M' respectively, under
the law of gravitation is therefore
G = -—t-m; (663)
/f in this expression being the intensity of the reciprocal attrac-
tion of a unit mass for another unit mass at a unit's distance.
Therefore the partial differential equations of the undisturbed
orbit of a planet about the sun, Eqs. (546), become, under the law
of gravitation,
d'x MM'
-}XX\
dt"" r
d\ MM'
d^'z _ MM'
dt' - -7^^^'
(664)
Kepler's first and second laws can be deduced directly from
these equations, and hence are simply the consequences derived
from the undisturbed orbit of a primary about the sun under the
supposition that the law of gravitation is the governing law of
their inutual attraction.
The differential equations of the actual or disturbed orbit can
be obtained immediately from Eqs. (545), by substituting the
values which the symbols (MM'), {MM") and (M'M") take
230 MECHANICS OF SOLIDS.
under the law of gravitation, and the resulting equations will
differ from (664) only in the third and fourth terms. The latter
being computed and applied properly to the undisturbed orbit
will give the actual orbit of the planet. These terms, called the
perturb ating functions of the orbit, depend upon the relative attrac-
tions of the other planets for tlie sun and for the planet whose
orbit is to be determined. Owing to the relatively great mass
of the sun, and to the immense distances which separate the
planets from each other, \\i^ perturbations or actual displacements
of a planet from its undisturbed orbit are sensibly infinitesimal
quantities, compared with the actual distance of the planet from
the sun. Hence the component rectangular displacements due
to the perturbating action of each planet may be computed for
each planet separately as if it alone acted; then the algebraic sum
of the separate perturbations in any direction may be taken as
the resultant perturbating effect in that direction due to the
simultaneous action of all the planets, without the least appreci-
able error. Because of this fact, tlie problem is called the prob-
lem of three bodies^ viz., the sun, the planet and the perturbating
body.
The theoretical deductions which flow from the assump-
tion of the law of universal gravitation as the governing law of
planetary motion have been amply confirmed by the accurate
astronomical predictions of the positions and motions of the
planetary bodies, made years in advance, and markedly so, by
the circumstances attending the discovery of the planet Neptune;
so that the law itself is at present accepted as the fundamental
law of physical astronomy.
192. Planetary Orbits. — The undisturbed orbit of each of the
bodies of the solar system has been shown to be a plane curve,
whose plane passes through the sun's centre; but it is ascer«
tained, by observation, that no two of these planes are coincident.
In order to find the relative positions and motions of the bodies
of the solar system at any time, it is necessary to refer them to
the surface of the celestial sphere by some system of spherical
co-ordinates.
THE SOLAR SYSTEM. 23 1
The celestial sphere is usually taken to be that sphere which
is enclosed by the surface of the visible heavens; but it may be
taken to be any sphere whose centre is tlie position of the ob-
server, and whose radius is entirely arbitrary. The Ecliptic is the
great circle of intersection of the celestial sphere, by the plane
of the earth's orbit. The Celestial Equator, or Equinoctial, is the
great circle of intersection of the celestial sphere, by the plane of
tiie earth's equator. The poles of the heavens are the poles of
the Equinoctial, or are the points in which the earth's axis pro-
duced pierces the celestial spiiere. T ho. Equinoxes are the points
in which the equinoctial and ecliptic intersect; the Vernal Equi-
nox being that point in which the sun appears at the beginning
of spring, and the Autu?nnal Equinox that in which it appears in
the beginning of autumn. Celestial longitude and latitude are
spherical co-ordinates by which any point is referred to the plane
of the Ecliptic, and to that of a great circle of the celestial sphere
perpendicular to the ecliptic, passing through the vernal equi-
nox. Celestial longitude is the angular distance from the vernal
equinox, measured on the ecliptic eastward ly in direction, to the
circle of latitude which passes through the position in question;
-eiud celestial latitude \s the angular distance to the given point,
from the ecliptic, measured on that circle of latitude which
passes through the given point. The line of intersection of the
plane of a planet's orbit with the plane of the ecliptic is called
\.\\Qline of nodes; the ascending node being the point of the planet's
orbit at which the planet passes from south to north of the
ecliptic, the other being the descending node. The nearest point
of a planet's orbit to the sun is C3.\\&6. perihelion, and the farthest
is called aphelion.
The elements of a planefs orbit are seven in number, viz.:
(i) The inclination of its plane to the plane of the ecliptic.
(2) The longitude of the ascejiding node.
(3) The orbit longitude oi perihelion.
(4) The mean distance of the planet from the sun, or the semi-
transverse axis of the planet's orbit.
(5) The eccentricity of the orbit.
232 MECHANICS OF SOLIDS.
(6) T\\Q position of the planet at any given time, as at \.\\q epoch.
(7) The ?}ica?i orbital fnotion.
The first two of these elements fix the plane of the orbit with
reference to the plane of the ecliptic; the third fixes the position
of the line of apsides in this plane and from which the anomalies
are reckoned; from the fourth and fifth the form and dimensions
of the ellipse are determined; and the last two, called the ele-
ments of position, locate the planet in its orbit at any time.
The elements of any planetary orbit are deduced from three
consecutive observations of its position in right ascension and
declination and the times of observation, by methods which will
be explained in the course in Astronomy.
Since the mean motion depends on the periodic time, and the
latter, by Eq. (616), depends on the mass of the planet, it is nec-
essary to explan how the mass of a planet is ascertained. The
masses of the planets are so small, in comparison with the mass
of the sun, that their values cannot be ascertained from Eq.
(657) after substituting the observed periodic times and mean
distances. But if we consider the planet Jupiter and one of its
satellites, we will have, for the periodic time T' of the latter
about Jupiter,
^' = ^T-.^ (^^3)
in which a' is its mean distance from Jupiter, and m its mass; /(,
the attraction of a unit mass at a unit's distance, being here
taken as unity. Similarly, for the periodic time of Jupiter about
the sun, we have
whence we have
THE SOLAR SYSTEM, 233
if ;;/ be supposed small compared with M\ and M' small com-
pared with M. Substituting the known values of T^ T', a and
^', we have the ratio of the mass of Jupiter to that of the sun.
In the same way the masses of all the planets having satellites
may be compared with that of the sun; and if the mass of one of
these be found, that of the remaining planets will at once result.
To find the mass of the planets Mercury and Venus^ which have
no satellites, recourse must be had to their perturbating effects
on the other planets.
The mass of the earth has been ascertained by direct meas-
urement of its figure, magnitude and density. From the direct
geodesic measurement of the arcs of the meridian in England,
France, Russia, India and Africa, the form and dimensions of
the earth have been determined. The density has been directly
investigated by means of Dr. Maskelyne's observations with the
pendulum near Schehallien Mountain in Scotland; and also by
the experiments of Cavendish and Bailey from the attraction of
leaden balls on small masses. From these, the mass of the earth
having been found, the masses of the sun and the other planets
are readily obtained. The further discussion of planetary mo-
tions is reserved for the course in Astronomy.
THEORY OF MACHINES,
193. Thus far we have regarded bodies as rigid solids, eitlier
wholly or partially free to move under the action of extraneous
forces. But the various devices or machines designed for the
transfer of energy from one system of masses to another are
made up of parts which are neither free nor rigid; and in their
use certain resistances are developed by the active extraneous
forces, which malce the actual results differ from the theoretical
more or less widely. It is the office of Experiment to find the
values of these resistances, to tabulate the results, and to deduce
therefrom the experimental Imvs covering all cases that may arise
in practice. We will consider briefly the resistances of Friction
and Stiffness of Cordage.
Friction.
194. Friction is the resistance which the surface of one body
offers to the sliding or rolling upon it of any other body. It is
due to the roughness of the surfaces of bodies; for as no degree
of polish can make any surface perfectly smooth, there will
always be minute projections on one surface which interlock
with those of the other. These projections must be broken
down, abraded or lifted over each other before motion can take
place. When the roughness of any two surfaces is diminished
by polish or lubricants the friction between them decreases.
The friction which opposes a change of the body from rest
to motion is called static friction, and that which accompanies
motion is called kinetic friction. The latter may be either sliding
or rolling; thiis a heavy body dragged over a surface, an axle
FRICTION. 235
turning in a journal-box, and a vertical shaft turning on a hori-
zontal plane, give examples of sliding friction; while a wheel
rolling over the surface of the ground is resisted by rolling fric-
tion. Sliding friction, being the more common, will be alone
discussed.
The action-line of friction coincides with the tangent to the
surfaces at the point of contact, and its direction is always op-
posite to that of the motion. The intensity of friction must al-
ways be determined by experiment, or may be assumed trom
previous experiments on similar bodies and surfaces. Such
results iiave been generalized into what are known as the laws
of friction. The accepted laws have been deduced from the ex-
periments made by Coulomb in 1781, and from those made by
Morin under the direction of the French Government in 1830-4.
They are :
(i) The intensity of friction^ for the same material surfaces, varies
directly with the normal pressure.
(2) The intensity of friction is independent of the area of contact of
the surfaces.
(3) The intensity of friction is independent of the velocity of motion
of the rubbing surfaces.
Recent experiments indicate that the last law is only ap-
proximately true for velocities less and greater than those em-
ployed in the experiments of Morin; and since the laws are
wholly experimental, we are warranted in accepting them only
within the limits covered by the experiments from which they
were deduced. In static friction there are also variations de-
pending on the length of time during which the surfaces have
been in contact, and therefore greater discrepancies occur in this
kind than in kinetic friction.
195. Coefficient of Friction. — If iV represent the normal pres-
sure, F the intensity of friction for any two surfaces due to Ny
and /"the intensity of friction for a unit of normal pressuie, we
have, from the first law,
F=fN, or /=^. (668)
236 THEORY OF MACHINES.
f is called the coefficient of friction, and when known for any two
surfaces the total friction for those surfaces can be found for any
normal pressure. Its value depends upon the nature of the rub-
bing surfaces, upon their smoothness, and upon the degree of
lubrication given lo them.
To find /experimentally in any particular case, let the body
be placed upon a plane surface, Fig. 65, and let the latter be
gradually inclined to the horizon until the body is in a state
bordering on motion ; then if motion
be given the body, it will descend the
^ws$7m plane uniformly. The forces are the
weight of the body acting vertically
downward, and the friction resisting up
the plane. The normal component of
the weight, if a be the angle made by the
^^^- ^5- plane with the horizon, is ^cos a, and
hence the friction \s fW cos a. The component of the weight
parallel to the plane urges the body down the plane, and is
W sin a\ and since the motion is uniform the intensity of the
parallel component must be equal to that of the resistance due
to friction. Hence we have
f IV cos a = JV sin a, (669)
and therefore
/= tan a (670)
That is, t/ie coefficient of friction for any two materials is equal to
ihe natural tangent of the inclination to the horizon which a plane of
one of the substances must make in order that a body of the other sub-
stance may descend uniformly due to its weight when resisted only by
friction.
The inclination of the plane is usually called the attgle of fric-
iion^ and sometimes the limiting angle of resistance. To explain
the meaning of this latter term, let a body rest on a plane in-*
clined at the angle of friction; it is then said to be in a state
FRICTION. 237
bordering on motion downward. The angle at the centre of
gravity of tlie body, made by the direction of the weight and the
normal to the surface, is then equal to or, the angle of friction.
If any force P whose action-line lies ivithin the angle a be supposed
introduced at the centre of gravity, it is evident that the addi-
tional developed friction arising from the normal component of
P is greater than the component of P parallel to the plane;
therefore the body will be no longer in a state bordering on
motion. If the action-line of a force fall outside of the same
angle on the side of the weight, the additional friction developed
by its normal component will be less than the intensity of its
parallel component ; therefore the body will move down the
plane with accelerated motion. If the action-line of any force
coincide with that of the weight, the body will still be in a state
bordering on motion downward, and it put in motion it will
descend the plane uniformly.
196. Problems involving Frictioti. — (i) Motion on a Plane Surface^
Let a body resting on a plane. Fig. dd, be
acted on by its weight, W^ and let any ex-
traneous force whose action-line lies in a
vertical plane, and whose intensity is /, be
applied to it; let / be the inclination of the
plane to the horizon, and d the angle made
by the force with the plane. If the body ^'^- ^^•
be bordering on motion downward the friction will act up the
plane, and we shall have, from the principles of equilibrium,
/( J^cos /— /sin 6^) = fFsin / — /cos ^, . . (671)
whence
/= ^^ sin/-/ cosj;
If the body be bordering on motion up the plane, the action-
line of friction will change direction by 180°, and the value of /
can be obtained from the preceding by changing the sign of/; or
sin/+/cos/
238
THEORY OF MACHINES.
Therefore the value of / may vary between these limits without
causing motion in the body.
Considering / a function of 6^, we have, after differentiating
the last equation,
dl . . . . J. .V sin — f cos d tr \
-:Ja = ^(sin ^ +/C0S t)-, a , r . n-.^ • • (674)
dd ^ ' -^ ^(cos ^+/sm^)^ ^ ' ''
Applying the condition for maxima and minima, by placing
the second member equal to zero, we find
/ = tan 6, or 6 = tan - i/,
(675)
which corresponds to a minimum value of /; therefore a force is
applied to the best advantage in moving a body up a plane when
its action-line makes an angle with the plane equal to the angle
of friction. As this result is independent of /, it is true what-
ever be the inclination of the plane to the horizon, and is there-
fore true if the plane be horizontal.
(2) To find the Friction on a Trunnion. — The cylindrical pro-
jections at the extremities of an axle are called trunnions; the
cylindrical box upon which a trunnion is supported is called a
trunnion-bed or pillow-block. When the axle is supported on its
end, the latter is called 2, pivot.
Let A^ Fig. 67, be the trunnion, B the trunnion-bed, and C
any element of contact during rotation; let R
be the resultant of all the extraneous forces
acting on the trunnion, excluding friction; N
and T the normal and tangential components
of R respectively. If the trunnion rotate, it
will rise in its box until the developed friction
F is equal to 7", after which sliding will occur
at the element of contact. Then the resultant
of R and F will be normal to the surface of the trunnion at the
\ i
>
A
\,
iM
1;
^
s->
^ B
'\1
FRICTION. 239
element of contact, and will be equal to N. Let a be the angle
between R and N\ then we have
N=RcQsa and T = R sia a =1 F\ . . (676)
whence
_=/=tanaf (677)
Therefore the element of contact during rotation is that at which
the normal to the trunnion makes an angle with the resultant
equal to the angle of friction.
To find the friction we have
(678)
I + tan'' a 1+/"' *
multiplying by/'^' and extracting the square root, we have
Fr=fN=fR cos a =/R '._. = R-j^-, (679)
y I + tan" a Vi +/"
That is, the friction on trunnions is equal to the resultant of the
extraneous forces multiplied bv , in which / is the co-
efficient of friction for the materials which compose the trunnion
and box.
The moment of friction on trunnions, if r be the radius of
the trunnion, is
Rr / , (680)
and the work consumed by friction in n complete turns is
R27trn—-£=^. (681)
240 THEORY OF MACHINES.
From the second law of friction we see that the intensity of
friction and the work are independent of the length of the trun-
nion.
(3) Friction on a Circular Pivot. — Let the shaft be vertical and
its pivot end a circle of radius a^ Fig. 68, resting on a
horizontal surface; and let the centre of the circle of
contact be the origin of the polar co-ordinates r and 6^,
which fix the position of any elementary area of contact.
Fig 68 Let/ be the intensity of the normal pressure on each
unit of area, which is equal to the whole normal pres-
sure divided by the area of tiie pivot surface.
The expression for any elementary area is
rdrdd',
the normal pressure upon it is
prdrdQ\
the developed friction is
fprdrdB',
and the moment of this friction with respect to the axis of the
shaft is
fpr'^drdd.
Integrating this last expression between proper limits, we have
for the resultant moment of the friction on a pivot
M= I I fpr'drdd = %fp7ta\ . . . (682)
If iV represent the whole normal pressure, we have
N=p7ta\ {eZz}
whence
M^fN^a (684)
FRICTION. 241
Therefore the moment of friction on a pivot is equal to the pro-
duct of the coefficient of friction for the materials of which the
shaft and support are made, the total normal pressure, and two
thirds of the radius of the pivot surface.
From this we see that the moment of friction may be dimin-
ished by decreasing the circular area of the pivot, provided the
diminished area be sufficient to withstand the normal pressure
without penetrating the surface on which it rests. The distance
\a is called the viean lever of friction on pivots, and we see that
if the whole friction be supposed concentrated on an arc of this
radius its moment will be the same as the resultant moment of
all the elementary frictions.
The work consumed by friction in 11 revolutions is
27tnfNla. (685)
(4) Friction on a Ring Pivot. — Let the inner and outer radii of
the ring be a and a' respectively; then (Eqs. (682) and (683))
M=\fpn{a" -a') (686)
N=pn{a"-ay, (687)
whence
^=t/^S^ (^^*)
If b be the breadth of the ring and rbe its mean radius, we have
^' = r + \b,
a = r — \b\
which substituted in Eq. (688) give
^=/A-(r+^g. ..... (689)
r6
242
THEORY OF MACHINES.
The factor
Ki^f)
is called the mean lever of friction for
a ring pivot.
The quantity of work consumed by friction on pivots in n
revolutions of the shaft, / being the mean lever of friction, is
27tnfNl. (690)
(5) Friction of a Cord on a Cylinder. — Let R be the radius of
the cylinder, s the length of the cord in contact, and 7\ and T^
the tensions at first and last points of contact, T^ being greater
than T^. If there were no friction T^ would be equal to T^\
hence the excess T^ — T^, when the cord is in a state bordering
on motion, is due to friction. Let T be the tensions at the ex-
tremities of the elementary arc ds^ and let 6 be the angle included
between their action-lines. Their resultant N is then
JV= VT''-\-2TTcose-\-T'' = TV2{i + COS 6) = 2 T cos id. {691)
If 0, Fig. 69, be the angle at the centre of
the cylinder subtended by ds, we have
cos id = sin i(p.
Whence, since = T^. (692)
The friction due to iVis then
/N=fT
and this being the increment of the tension, we have
,ds
dT=fT
R
{^93)
(694)
FRICTION. 243
Whence
Integrating, we have
or
-^ =/^- (695)
logT'^^' + logC, (696)
T — Ce^ (697)
When J = o we have T— T^, and when 5 = 5 we have T =. T^\
therefore
T,= T,eR=.T,ef-n-, (698)
n being the number of times the cord is wrapped around the
cylinder. This relation may be written
^^e^-"- (699)
From which it is seen that as the number of turns increases in
arithmetical progression 7", increases in geometrical progression.
We see, also, how it is possible for a man exerting a tension T^
on the free end of a rope wound several times around a pile to
hold in equilibrium the very much greater tension 7", of the
other end caused by the stoppage of a boat at a wharf.
244 THEORY OF MACHINES.
Stiffness of Cordage.
197. In theoretical mechanics a cord is defined to be a collec-
tion of molecules so united as to form a perfectly flexible mate-
rial line. Considering it also to be without weight and to be
inextensible, the effect of a force is supposed to be transmitted
along its length without loss.
The tension of a cord is the intensity of the force which tends
to separate any two of its adjacent sections.
Cordage is a term applied to all varieties of lines, cord, and
rope, formed by twisting together the textile fibres of hemp, flax,
cotton, etc. Since these fibres are neither perfectly flexible nor
inextensible themselves, cordage must be much less so, and hence
will offer a resistance to being bent from the direction which it
naturally assumes. By stiffness of cordage is meant the resistance
which it offers when it is forced to take a curved form in adapt-
ing itself to the surfaces of wheels and pulleys.
The law of this resistance has been deduced by Coulomb from
numerous experiments made on different kinds of cordage. He
found that the stiffness of cordage is composed of two parts, viz.,
one, a constant depending on the natural torsion of its fibres; the
other, a variable depending on the intensity of the stretching
force applied to the cord. He also found that for the same cord
it varies inversely with the diameter of the wheel around which
the cord is bent. If ^S be the stiffness, K the constant part due
to the natural torsion of the fibres, / that due to a unit of ten-
sion, W the total tension, and D the diameter of the wheel,
Coulomb's experimental law for a particular cord is expressed
by the formula
*5=— ^ (700)
The quantities K and /, according to Morin, ought properly
to be expressed in terms of the number n of yarns of whicli the
STIFFNESS OF CORDAGE. 245
rope is composed. Making use of Coulomb's results, Morin found
that if / be assumed to vary with «, and if K be taken to consist
of two terms, one proportional to n and the other to «", the values
of 6" derived from
11
»$■= -^(0.002148 + O.OOI772W -|- 0.0026256 ^) . (701)
would conform to all of Coulomb's results for ordinary new
white rope, and
6* = -^(0.01054 + o.oo25« + 0.003024 fF) . . . (702)
to those for tarred rope. These formulas of Morin are identically
those of Coulomb, when for white rope we place
K = «(o.oo2i48 -|- o.ooi772«) and / =: «(o.oo3024),
and for tarred rope
J^ = ^(0.01054 -|- 0.0025//) ^"d -^ — «(o.oo3024).
The values of S in pounds, for both kinds of rope, bent over a
wheel or axle one foot in diameter, under a tension of one pound,
are given in Table VI. For other axles and tensions these values
substituted in the above formulas will give the desired results.
The stiffness of partly worn or oily rope is less than that of
new rope; it may be only one half as great. The *' natural stiff-
ness" ^of wet rope is twice that of dry, while the value of / is
the same for both.
The stiffness of Cordage consumes work when the cord is
wound on the wheel so as to adapt itself to the circumference.
This work is equal to the product of the intensity of the resist-
ance by the path described by its point of application, estimated
246 THEORY OF MACHINES.
in the direction of the resistance. The path is evidently equal ta
the length of that portion of the cord wound on the wheel, which
is the actual distance passed over by the resistance. Then if it
be the number of turns of the wheel, the cord wound is in length
equal to tmtR^ and the quantity of work consumed by the stiff-
ness of the cordage will be, for new white rope,
= {K^ IW)n7t (7o3>
From this we see that the work consumed in n revolutions
is independent of the radius of the wheel; as it should be, since
the increased stiffness for a wheel of smaller radius is compen-
sated by the less path over which the resistance works in making
one complete turn.
Machines.
198. A machine is any instrument or device designed to
receive energy from some source, and to overcome certain resist-
ances in transferring this energy to other bodies. Every
machine consists of three essential parts, viz., the driving point,
the working point, and the train. The first is the point at which
the energy is received, the second that at which the transmitted
energy is applied, and the third is the series of parts connecting
the first and second.
Tlie operating forces in a machine are classified as Powers
and Resistances. A power is a force which increases or tends to
increase the momentum of the parts of a machine; a resistarice
is a force which diminishes or tends to diminish their momentum.
Those resistances which the machine is primarily designed to
overcome are Ccilled useful, and the energy expended in overcom-
ing them is called useful work : all other resistances are called
prejudicial or wasteful, and the energy expended in overcoming
MACHINES. 247
them is called wasteful or lost work. From these definitions we
see that the action-lines of the powers must either coincide, or
make acute angles, with their corresponding virtual velocities,,
and consequently (Art. 68) their elementary quantities of work
will be positive; and that the action-lines of the resistances must
either be opposite to, or make obtuse angles with, their respec-
tive virtual velocities, and hence their elementary quantities of
work will be negative. Recalling the fundamental principle that
enersry can neither be created nor destroyed, we see that in the
discussion of a machine it is necessary to ascertain what amount
of energy it has received from the source, and how much of this
has been exchanged for useful and lost work, and how much still
continues in tlie machine as potential or kinetic energy. The
energy received by the machine is generally designated as the
work of the powers; that which has been distributed by the ma-
chine to masses forming no part of the machine is the work of the
resistances,
199. Theory of Machines. — Resuming the Equation of Energy,
we may apply it to any machine by letting P and dp represent
the type-symbols of the intensities and projected virtual veloci-
ties of the powers, Q and dq those of the resistances, and m the
mass of any particle of the machine; then we have
:2Pdp - :2Qdq =^ ^m-^Js (704)
Integrating, we have
:2jpdp-:2jQdq^\^mv'^C . . . (705)
for the general equation of energy applied to machines. If z\ be
the type-symbol of the velocities of the masses j?i at the instant
248 THEORY OF MACHINES.
tlie machine begins to receive energy from the source, we have,
since the work of the powers and resistances is then zero,
C — — \^mv^ (706)
Substituting this value of C in the general equation, we have
^fPdp - ^jQdq = i2mv' - i:^mv;. . . (707)
If the machine start from rest, thenz;„= o, and Eq. (707) becomes
2rFdp-2fQdq = i:2mv' (708)
If the integration be taken between any limits corresponding to
the states i and 2, Eq. (705) will become
^ rPdp - 2 rqdq = \^mv^ - i2mv,\ . (709)
The theory of machines is embodied in the general equation
(707), and may be derived from either of its special forms, Eq.
(708) or (709). Taking (709), which applies to any machine in
operation, we see that if its kinetic energy increase between any
two successive states, the increment is exactly equal to the ex-
cess of the work of the powers over that of the resistances in the
intervening interval of time. If its kinetic energy diminish, then
the loss is equal to the excess of the work of the resistances over
that of the powers, and should this condition continue, the ma-
chine will come to rest when the total kinetic energy is wholly
absorbed in making good the deficiency. If there be no change
in the kinetic energy the total work of the powers is exactly
equal to that of the resistances during the interval in which the
kinetic energy is invariable.
MACHINES. 249
Hence, it appears that when the work of the powers, in any
interval whatever, exceeds that of the resistances, the excess is
stored up as kinetic energy in the masses which constitute the
machine; and when more work is required by the resistances
than is supplied by tlie powers in a given interval of time, the
deficiency is made good by the withdrawal of kinetic energy
from the parts of the maciiine. The total quantity of work done
by tlie powers from the instant the machine starts from rest un-
til it comes to rest again, or from any particular state of motion
to the same state again, is precisely equal to the work of the re-
sistances in this interval. Therefore, whatever energy is received
by the machine is employed in making resistances perform work,
and the object of any machine is to get as much useful work done
by the expenditure of a definite quantity of energy as is possil)!e.
Whatever kinetic energy remains in the machine at any instant
is simply the work of the powers which has not heretofore been
used in overcoming recurring resistances, and hence is continu-
ally accumulating, to be afterwards utilized as necessity requires.
200. Use of Fly-7vheel. — Due to the construction and applica-
tion of many machines, it often happens that the energy received
from the source, and that consumed by the resistances, vary
with the time; and, in addition, these variations of supply and
demand may neither be equal nor simultaneous. In such cases,
if the machine be of relatively small mass the acceleration of ve-
locity of its parts will be correspondingly great, and the whole
machine will be subject to rapid changes of motion, wliich are
often detrimental. To obviate such a defect, the mass of the ma-
chine may be increased by the addition of 2i fly-wheel. This con-
sists of a mass of matter distributed in the form of a ring and
suitably connected with the rotating shaft on which it is mounted.
We have seen that the kinetic energy of rotation is measured by
^Qo^'^mr'^, in which co is the angular velocity and "^Jtir^ the mo-
ment of inertia of the rotating mass with respect to the axis of
rotation. Hence the changes in g?, due to any change in kinetic
energy, may be made as small as we please by suitably increas-
ing ^Av/-"; this may be done by increasing either the mass or the
250 THEORY OF MACHINES,
radius of gyration with respect to the axis. By the introduction
of a suitable fly-wheel the changes of velocity may thus be di-
minished to any desired degree. The greater the moment of in-
ertia of the fly-wheel, the greater will be the quantity of work
which it will store up for a given increase in its angular velocity,,
and, similarly, the more it will yield for a given decrease. As
the change of level in a reservoir, due to the addition or dis-
charge of a given quantity of water, will be less noticeable as
the surface area is greater, so likewise will be the changes of
velocity in the moving parts of a machine, due to a difference
between the work of the powers and that of the resistances, ac-
cording as the moment of inertia of its fly-wheel is greater.
From this analogy, the fly-wheel may be regarded as a reservoir
of work in a machine.
201. Efficiency. — If W be the whole amount of energy sup-
plied to a machine in a given time, during which its kinetic
energy remains constant, and Wu and Wi be that employed in
overcoming the useful and wasteful resistances respectively, in
the same time, then
W-Wu^ Wi. (710).
The ratio of the total work to that of the useful resistances,.
or-^j^, is called the modulus or efficiency oi the machine. This
ratio is evidently always less than unity, which is its maximum,
limit, and which it can never reach, since Wi can never in actual
machines become zero.
Wi can be diminished in value:
(i) By avoiding all unnecessary friction.
(2) By diminishing the intensity of the necessary iv\Q.x!\ovi\ this.
may be accomplished by selecting material for the contact sur-
faces whose friction-coefficients are small, or reducirig these co-
efficients by the application of lubricants.
(3) By decreasing the moments of friction in rotating parts,.
SIMPLE MACHINES. 2$!
either by decreasing the coefficients as above, or by shortening^
the lever-arm of friction, or both.
(4) By such an assemblage of parts, arrangement of supports
and solidity of foundation, as to avoid sudden and unnecessary
vibrations and shocks. These consume work which is dissipated
in the form of heat-energy.
The principal sources of energy, whether from fuel, air in mo-
tion, animal or water power, etc., have received their supply
either directly or indirectly from the sun, which is constantly
parting with a portion of its energy in the form of heat. In es-
timating energy the foot-pound has been assumed as the unit.
This unit does not take into consideration the time in which the
quantity of work is expended, and since the element of time is
important in the use of machines, a different unit from the foot-
pound is required in measuring their efficiency. Such a unit is.
the //^rj-f'/^Tf^'^r, which corresponds to the expenditure of 550 foot-
pounds of work in a second of time. An engine of ten horse-
power is one which is capable of doing 5500 foot-pounds of work
in a second.
Simple Machines.
202. The Sifnple Mac/itneSy or, as they are sometimes called, the
Mechanical Powers, are the Cord, Lever, Pulley, Wheel and Axky
Inclined Plane, Wedge, and Screw. All other machines are formed
of combinations of these, and when the relations existing between
the powers and resistances are known in the simple machines, the
corresponding relations in compound machines may be derived.
There is said to be Si gain of power in a machine when the inten-
sity of the power is less than that of the corresponding useful
resistance with which it is compared; and a loss of pouter when the
intensity of the power is greater than that of the resistance.
These terms are technical, and are used merely to compare the
intensities of the powers and resistances, and not to compare the
work done by these forces. If we take the quantity of work done
by the power and by the useful resistance to be equal, as in the
252 THEORY OF MACHINES.
limiting case, it is evident that when there is a gain of power the
path described by its point of application, estimated along its
action-line, must be greater than that of the resistance; that is,
its velocity will be greater than that of the point of application
of the resistance, since these unequal distances are described in
the same time; this is technically called a loss of speed. When
there is a loss of power there will be a corresponding gain of
speed.
The object of discussing Simple Machines is to find the rela-
tion existing between the intensity of the power and that of the use-
ful resistance. To do this, we first place the work of the powers
equal to the sum of the work done by all the resistances; the
equation so formed we know to be true, whilst the parts of the
machine have uniform motion, for during this time the energy
received by the machine is wholly absorbed by the work of the
resistances.
This condition will always be presupposed, and Eq. (709) will
then take the form
2£Fdp = 2£Qdg (711)
We are then often able to eliminate the path factor, and get an
equation from which the desired relation between the intensities
may be obtained. Passing then to the theoretically perfect ma-
chine by supposing all the wasteful resistances to be neglected,
we find the theoretical ratio of the intensities of the power and
useful resistance.
In the following discussion dp and dq dive taken to be the pro-
jected elementary paths of the points of application of P and Q
on the action-lines of these forces respectively, in the time dty
during which the forces are supposed to remain constant in in-
tensity; and ds is the path described by a point at a unit's dis-
tance from the axis of rotation in the same time.
THE LEVER.
253
The Lever.
203. The lever is any solid bar, straight or curved, capable of
rotating about a fixed point or line under
the action of a power. The point or axis
of rotation is called the fulcrum. Let AB^
Fig. 70, be the axis of a lever, O the axis of
the trunnions supporting it, P and Q the
power and resistance, / and q their lever-
arms with respect to (9, and r the radius of
the trunnion. The resistances, omitting
that of the air, are the useful resistance
Q^ and the wasteful resistance friction on
trunnions, whose intensity will be designated by F. To find F,
let 6 be the angle included between the action-lines of P and Q\
then if N be the intensity of the resultant pressure on the trun-
nions, we have
Fig. 70.
N1/P'-irQ^-\-2PQcoi d (712)
and
F=N-
f
Vi+f
(713)
The action-line of N passes through C, the intersection of the
action-lines of P and Q, and through the point of contact of the
trunnion with the fixed support. Hence CN is the action-line
of the resultant pressure. The lever-arm of friction on trun-
nions is r\ therefore the elementary work consumed by friction is
Frds',
(7i4>
that absorbed by Q is
Q = Qq^Fr, , (717)
Whence
M+f ? <'-«
p
Therefore the ratio -^ of the lever can be found when the quan-
tities P^ q,p, r, /and 6 are known.
J?
In practice both factors of the last term of Eq. (718), — and
— , are much less than unity, and in ordinary cases their product
/
is negligible. The limit of the ratio of the power to the resist-
ance is the reciprocal of the ratio of their lever-arms; or
f =? ■ • <'■'>
Either P or Q may be taken as the power or resistance, but,
to agree with the convention established heretofore, the power is
taken to be that force whose virtual moment is positive. When
p> q there is a gain of power, and when/ < q there is a loss of
power.
If the power or resistance, or both, be applied in a plane ob-
lique to the axis of the trunnion, the forces must be resolved
THE LEVER.
255
into components parallel and perpendicular to the axis. The
perpendicular components will replace P and Q in the above
discussion, and those which are parallel to the axis will either
cause motion in the direction of the axis or produce pressure
on the side supports, giving rise to sliding friction, which can
readily be computed. The work consumed by this friction will
•appear among those of the resistances in the equation of equi-
librium.
Levers are commonly divided into three classes or orders.
In those of the first class the fulcrum is between the power and
resistance; in the second the resistance acts between the fulcrum
and the power; and in the third the power is applied between
the fulcrum and the resistance. As there is no difference in
principle in these orders this classification is unimportant.
204. The principles of the lever are involved in the construc-
tion of the common balance. To find
the conditions of equilibrium, let O, Fig.
71, be the point of suspension, G the
•centre of gravity of the balance un-
loaded, AC—CB—a, OC = c, OG=h.
Let the balance be loaded with unequal
-weights Qy- F, and suppose that it has q"
taken its position of equilibrium as in
the figure. Then the moments of the forces about O must be
•equal; whence we have
Fig. 71.
Q{a cos ^ — ^: sin ^) = F{a cos ^ + ^ sin 6) -{-wh sin ^,
and therefore
tan 6 =
c(Q -\- F)-h 7v/i'
(720)
The conditions required in a balance are (i) horizontality of
the beam when the arms are equal in length and the weights In
256 THEORY OF MACHINES.
the scale-pans are equal; (2) sensibility, which is estimated by the
value of S for a given difference in the weights; (3) stability ^ or
the tendency to return to horizontality when the weights are
removed. The balance is so constructed that the first condition
is satisfied when Q=^ P. The second depends on the greater or
less value of 6 for a small value oi Q — P. Assuming a constant
difference Q — F, we see that tan 6 will increase, and hence
also,
(i) when a is large; that is, when the arms are long;
(2) when c is small, that is, by moving the point of suspension
nearer the beam;
(3) when P -{- Q\'i small, or the sum of the weights small;
(4) when 7v, the weight of the balance, is small;
(5) when h is small; that is, when G is not far below the beam.
The sensibility of the balance may be very great and the
balance have no stability; that is, no tendency to return to its
primitive position after the removal of the weights, and the
former will have to be modified to satisfy the latter condition,
which is of course essential.
The stability increases with OG, and the sensibility decreases
as OG increases. In any particular case the conditions of stabil-
ity and sensibility must be determined by the uses for which
the balance is designed. Thus for rapid weighing of large
masses, where great accuracy is not important, the stability
must be great; while for the weighing of the precious metals,
drugs in small quantities, etc., great sensibility is of primary
importance. There is generally some device attached to the;
balance to check oscillations when the stability is slight.
THE WHEEL AND AXLE. 257
The Wheel and Axle.
205. This machine consists of a wheel W^ Fig. 72, firmly
attached to a cylinder C, whose trunnion-ends t and f rest on
trunnion-b^s. The power /^ maybe applied
through the intervention of a rope passing
over a groove in the wheel, or by a crank, or "^TTTTTTTr
by capstan-bars; its action-line is generally
tangent to the circumference of the wheel.
in*
4-10
w
m
The useful resistance Q is applied tangen- n*
tially to the cylinder C by means of a rope £]q L
wound upon the cylinder. The principal re-
sistances are Q^ and the two wasteful resist- ^'^ 72.
ances, the stiffness of cordage caused by Q^ and the friction on trun-
nions. Let R, r and p be the radii of the wheel, the cylinder
and the trunnions, respectively; then the elementary work of the
power is
Pdp^PRds\ (721)
that of Q is
Qdq = Qrds\ (722)
that of stiffness of cordage is
^£±I^rds^\l,K^IQ)ds, . . . . (723).
the proper values of jFT and / being assumed for the kind of rope
used; that of friction on trunnions is
f{Ar^jv^)pds, (724)
in which N" and iV' are the pressures at / and t* due to w, the
17
258 THEORY OF MACHINES.
weight of the machine, and to the forces P and Q\ and/' is a
symbol for - Placing the work of the power equal to
the sum of the works of the resistances, and omitting the com-
mon factor ds, we have
PR^Qr^\{K-^IQ)^f{N^N')fi. . . (725)
The limiting ratio of the power to the useful resistance, ob-
tained by neglecting friction on trunnions and stiffness of cord-
-age, is
P r . ..
Q=R'- ■ ■ (7^')
the same as in the lever; as it should be, since the principle of
the two machines is essentially the same.
To find the resultant pressures iV^and iV' at / and /', let /be
the length of the axis between the middle points of the trunnions,
at which iVand iV' may be supposed applied; let a, b and c be
the distances, estimated parallel to the axis, from the point of
•application of iV to the action-lines of Q^ w and P, respectively;
the corresponding distances of N' will be I — a, I— b and l — c.
Let the action-lines of w and Q be vertical, and let that of P
make an angle with the vertical. The components of P will
then be P cos 0, vertical, and P sin 0, horizontal. By the prin-
ciples of parallel forces, the components of the vertical forces g,
m and P cos 0, at /, will be
Q — - — , w — —- and P — j— cos 0,
and the component of P sin / sin 6^' and cos6'>/sin^,
2/0 THEORY OF MACHINES.
or
cot 6' > / and cot6/>/. . . , . (767)
Substituting for/ its value tan a, we have
cot 6^ > tan a and cot 6 > tan or,
or
^' < 90° - a and 6 < go° - a. . . . (768)
Hence
CO < 180° — 2a (769)
That is, z'n order that the wedge may be driven 171, the angle of the
wedge must be less than 180° diminished by twice the a?igle of frictiofi.
If the wedge be in a state bordering on motion outward, the
friction terms in Eq. (766) will change their signs, and we have
_ iV[(i — f) sin GO — 2f cos gd\
~ cos ^'+7 sin d'
__ iV^'[(i — f^) sin GO — 2/ cos gd\ . .
~ cos6>+/sin 6 • • • ^^^°^
If some pressure be required to prevent the wedge from flying
out, we must have ,
(i —f^) sin 00 > 2/ cos GJ,
or
tan a . .
tanc»>2 r— ^— ; (771)
I — tan a \i > 1
which reduces to
tan 00 > tan 2af,
or
03 > 2a (772)
THE SCREW. 271
Hence we see that in order that the wedge may be held in its
place when the external pressure is removed^ the angle of the wedge
7mcst be less than twice the angle of friction.
When the wedge is used as a power iVand N' are generally
nearly equal, and either may be considered as the resistance to
be overcome. The particular form of the wedge in any case de-
pends on the special use for which it is intended. Thus in split-
ting wood it is usually made isosceles, and the power P is ap-
plied to the back of the wedge as an impulsion. The axe, chisel,
engraver, knife, tool of a plane, and the raised projections of a
file, are examples of wedges, whose forms are modified in accord
with the above principles, for the particular purposes for which
they are designed. Taking the wedge to be isosceles and N —
N\ we have for the ratio of P Xo N
-P _ (i -/') sin GO-\r 2/coscj ^
N cosica-Zsin^G? » • • • \11^)
and omitting friction,
= 2 sin i(i? (774)
N cos \0D
Hence the gain of power increases very rapidly as the angle
of the wedge diminishes.
The Screw.
214. The screw combines the principles of the lever and
inclined plane. It consists usually of a solid circular cylinder,
called the newel, on the surface of which is a thread or fillet,
whose section by a plane through the axis of the cylinder is
usually either a rectangle or a triangle. The thread of the screw
is a volume which may be generated by a rectangle or triangle
272
THEORY OF MACHINES.
Fig. 8o.
Iiavipig its base on the cylindrical surface and always parallel to
the axis of the newel, moving uniformly
around and along the axis. Every point
of the generating area will therefore
describe a helix, and the upper and under
sides will describe helicoidal surfaces^
The distance between the successive posi-
tions of the same point of the generating
area, measured in the direction of the
axis of the newel, after one complete rev-
olution, is called the pitch of the screw,
or the helical interval. In screws with rect-
angular threads the pitch must be at least
equal to twice the base of the generating
rectangle; in triangular threads it is usu-
ally equal to the base of the generating
triangle.
The screw is engaged in a nut whose interior cylindrical sur-
face is screw-cut in such a manner as to fit the fillet accurately.
The useful resistance to be overcome, if the nut be fixed in posi-
tion, is applied to the foot of the screw so that its action-line
may be in the direction of the axis of the newel ; if the nut
have freedom of motion and the screw is fixed, the useful resist-
ance is applied to the nut.
Take the axis of the newel as the axis of z^ and let abc^ Fig.
80, be the generating area. Let
P, be the constant angle made by ab in all of its positions
with z.
r, the distance of any helix from the axis, constant for the
same helix, but variable for different helices.
y^ the constant angle made by any assumed helix with the
horizontal plane.
0, the angle through which the screw or nut is rotated.
/, the lever arm of the power.
For the elementary work of the power we have
FdJ> = Fld(P, (775)
THE SCREW. 273
and for the work of the useful resistance, Q,
Qdz = Qrd^ tan y = Qr^d
' + cot-^ \
/ \ sin/-/sinV|/i + tanV + cotVV
If the thread be rectangular, /? = 90°, and we have
^^g^a^Y^ /^FWT^V . . (,86)
^ \ sin ;/' — /sin^;/ r I + tan'' ;/'/
hence, all other things being equal, the screw with a rectangular
thread is more advantageous than one with a triangular thread.
If we suppose the friction to be neglected, then/= o, and the
limiting ratio of the power to the resistance is
P _r tan y
(7S7)
Q I
Multiplying and dividing the second member by 2K, we have
P_^^anr (^88)
Q 27tl
THE CORD.
275
or when friction is neglected the power is to the useful resistance as the
helical interval is to the circumference described by the extremity of the
lever arm of the power; hence there is usually a great gain of
power in the application of this machine.
The Cord.
216. Let a perfectly flexible and inextensible cord assume a
position of equilibrium under the action of any forces whatever.
The resultant of the forces acting at either extremity must be in
the direction of the cord at that extremity; for, if it have a com-
ponent perpendicular to the cord, the latter, being perfectly
flexible, must move in the direction indicated by the perpendic-
FlG. 81.
ular component. Let i 2 3, Fig. 81, be the cord, the resultant
R^ being in the direction 2 i. Since the point of application of
a force may be taken to be any point of its action-line within the
limits of the body on which it acts, the resultant R^ maybe con-
sidered as applied at the point 2. Then the resultant of all the
forces acting at 2, including ^,, must be in the direction 3 2; and
this resultant, i?„ may be considered as applied at the point 3.
2'J^ THEORY OF MACHINES.
Thus in any case each successive resultant may have its point
of application transferred until all the action-lines have a com-
mon point, and the conditions of equilibrium will be the same
as before. Therefore the conditions of equilibrium for a perfectly
flexible and inextensible cord, under the action of any forces what-
ever, are the same as if all the forces were applied at a single point,
their intensities and directions remaining unchanged.
2I7. Let T^ be the tension of the cord at the origin, assumed
at any point; /, the type-symbol of the intensities of the extrane-
ous forces; T, the tension at any point; and let d^, By, dz\ oc, /?,
y\ 0j;, 0jj/» 0z> be the angles which T^, I and T make with the
co-ordinate axes, respectively. Then from the equilibrium of the
svstem we have
T cos 0^ = 7; cos 6^ + ^7 cos or; \
Tcos (f)y = 7; cos 6y -{- 2/cos fi;l . . . (789)
Tcos d)^ = T cos e^ 4- ^7 cos y; )
the last terms comprising all the extraneous forces between the
origin and the point where the tension is T.
If forces act at all points of the cord in such a manner as to
make the tension vary by continuity, then the cord will assume
the form of a curve, and Eqs. (789) become
dx
T— = 7; cos 6^-^21 cos a;
T^ = 7; cos dy + :5'7cos /?;
ds~ -^^ — -y
dz_
ds
dz
T— = r„ cos e, 4- :2'7cos y.
(790)
Such a curve is called a funicular curve^ and Eqs. (790) are its
differential equations.
If the extraneous forces be parallel and coplanar, we may
THE CORD.
277
assume the curve in the plane xz^ with the forces parallel to z^
and we have
hence
cos a = cos a:" = etc. = o;
dx ^
T—z- = T^ cos ^x = a constant;
(791)
that is, the componefit of the tension perpendicular to the direction of
the forces and in their plane is constant.
218. Let the cord abed, Fig. 82, be in equilibrium, the length
of each branch representing its own tension, and the tensions being as-
sumed constant throughout. Let R^ and R^ be equal and opposite
Fig. 82.
to the resultants of the forces acting at the points b and c, and
let the symbols (/, /,), (/, R^, etc., represent the angles made by
the corresponding lines in the figure. When three forces acting
at a single point are in equilibrio, their intensities are inversely
as the sines of the opposite angles, and we have
sin(/,^,) sin(/,i?,) sin (tf^ '
. (792)
R.
sin (/',i?J sin (/,/?J sin (/,/,)' '
. (793)
278 THEORY OF MACHINES.
and since the tensions are equal,
sin (t,R,) = sin (/.^,); ) , .
sin (t,J{,) = sin (/,Ji,). f ^'^'^^
That is, w^en the tensions are equal throughout, the resultant of the
forces at any point bisects the angle made by the adjacent branches of the
cord.
219. Let circumferences be passed through each vertex and
the two adjacent ones, and denote their radii by r^ and r,, and
let s be the length of one branch of the cord. Then
^ cos i(//J = \s\ \ , X
^. cos KVa) = 4^. ^ • • • • • ^^^^^^
We have also, from the figure,
/, cos i(/.0 = i^.: ) , ..
h cos 4(^3) = \R,. i. ^'^ ^
From the latter equations we have
cos i(//.) " cos \{t,t,y (^^^^
which by Eqs. (795) reduce to
r^R^ = r^R, (798)
That is, the intensities of the resultants are inversely as the radii of the
circumferences passing through their points of application and the two
adjacent vertices.
220. From Arts. 218 and 219 we conclude that when the fu-
nicular curve has a constant tension throughout, the resultants of
the forces acting at the different points are normal to the curve, and
their intensities vary inversely as the radii of curvature at their points of
application.
THE CATENARY CURVE. 279
The Catenary Curve.
221. The curve assumed by a heavy flexible and inextensible
cord under the action of its own weight is called a catenary curve.
Assume the curve in the plane xz^ and we have for its differential
equations
dx
T-^^ =T^QOse^=c; (799)
T^ = CgdcDds + 7; cos 6^^ ; . . . . (800]
in which (J, go and ds are the density, cross-section and length of
an elementary portion of the curve.
From Eq. (799) we see that the horizontal component 0/ the ten-
sion is constant : and since at the lowest point -^- = i, the horizon-
as
tal component of the tension at any point is equal to the tension at the
lowest point.
Taking the origin at the lowest point, we have T^ cos 6^ = o,
and from Eq. (800) we have
ds ^e/
Soods (801)
That is, the vertical co??tponent of the tension at any point is equal
to the weight of that portion of the cord between this point and the lowest
point.
Having the vertical and horizontal components, the tension
is readily constructed.
222. The Common Catenary (Fig. 83). — When d and oo are con-
stant the curve is called the common catenary^ and we have
dz
r-^=gS%— Continued.
Substances.
Density.
Water at 62° F.
Weight of
Cubic Foot
in Pounds.
Miscellaneous — Continued.
Gneiss
Graniie
Gunpowder, press-cake
ordinary grained
Ice
I ndia-rubber
Ivory
Limestone, marble
common building.
Nitre, crystallized
Quartz
Sand, dry to wet
Sandstone, building
Slate
Sulphur
Wax, various kinds
2.6 -2.8
162-175
2.5 -2.9
156-1S1
1.70 -1.85
106-115
.875- .900
55- 56
.92
57
.95
59
1.8
112
2.65 -2.85
165-178
2.4 -2.9
150-181
1.9
118
2.65
165
1.5 -2.
94-125
2.1 -2.7
131-168
2.7 -2.9
168-181
2.
125
.9 -I.
56- 62
LIQUIDS.
Acid, hydrochloric, muriatic, sat. sol
" nitric, concentrated
" sulphuric, concentrated
Alcohol, absolute
* ' proof spirit
Ether, sulphuric, common
Glycerine
Mercury
Nitro-glycerine
Oil, illuminating
" linseed
" olive
" (" spirits") of turpentine
Water, distilled
" sea
1.21
1-5
1.84
•795
.92
.72
1.27
3.6
1.6
.8
•94
.92
.87
1.027
75
93
114
49
57
44
79
848
99.8
49.9
58.6
57^3
54-2
62.4
64.0
286
APPENDIX.
TABLE \,— Continued.
GASES.
Air, dry, at 60° F. and 30 in. Bar., density
" " " 32° F. - 30 " "
813.
770.89
= .001229
.001279
Times
maximum
density
of water.
Air, dry
Carbonic acid, CO2.
oxide, CO
Coal gas
Hydrogen ,
Marsh gas, CH4. . .
Nitrogen
defiant gas, CqH4.
Oxygen
Steam (ideal)
" at 212° F...,
Density of
Hydrogen
14.422
22
14
4.76-5.77
I
8
14
14
16
9
Density of
Air
33-
.525
.970
.40
.069
•555
.970
.970
.109
.624
Weight of
Cubic Foot
60° F., 30".
Ozs. avoir.
1.226
1.870
1. 190
.40-. 49
.085
.680
1. 190
1. 190
1.360
.765
.592
Weight of
Cubic Meter
0° C, 76 cm.
Grams.
1293
1973
1255
427-517
89.
717
1255
1255
^435
807
624
APPENDIX.
287
TABLE 11.
THE METRIC SYSTEM.
The metric system of weights and measures is founded on the meter as a
unit of length. The units of the system are as follows:
Length : The Meter = length of standard bar preserved at Paris.
Area: The Are = lOO square meters.
Volume : The Stere = i cubic meter.
Capacity: The Liter = i cubic decimeter.
Mass and Weight: The Gram = the mass or weight of i cubic centimeter of
distilled water at the temperature of maximum density.
It is a decimal system. The prefixes denoting multiples are derived from
the Greek, and are: deka, ten; hecto, hundred; kilo, thousand; and myria, ten
thousand. Those denoting sub-multiples are taken from the Latin, and are:
deci, tenth; centi, hundredth; and 7nilli, thousandth.
The following table includes all the measures of the system in use:
No. of
the
Unit.
Length.
Area.
Volume.
Capacity.
Mass and
Weight.
lOOOO
Myriameter.
Myriagram.
1000
Kilometer.
Kilogram, kg.
100
Hectometer,
Hectare, ha.
Hectoliter, hi.
Hectogram.
10
Dekameter.
Dekastere.
Dekaliter, dal.
Dekagram.
I
Meter, m.
Are, a.
Stere, s.
Liter, 1.
Gram, g.
.1
Decimeter, dm.
Declare.
Decistere.
Deciliter, dl.
Decigram, dg.
.01
Centimeter, cm.
Centiare.
Centiliter, cl.
Centigram, eg.
.001
Millimeter, mm.
Milligram, mg.
The are and its derivatives are used only for land measure. In other cases
area is expressed in terms of the square whose side is a measure of length —
e.g., square meter, m'; square centimeter, cm'-^, etc.
The stere is rarely used except in measuring firewood. In other cases a
cube whose edge is a unit of length is used — e.g., cubic meter, m^; cubic
dekameter, dm^ etc. Cubic dekameter, cubic hectometer, etc., are not used.
A Mikron, //, = .001 mm.
A Tonne, t, or millier, = 1000 kg.
A Metric Quintal, q, = 100 kg.
288
APPENDIX.
to
It
I
•= o
3 3
E B «= g
u S
S E
3 =3 3 "
cr cr cr ^
c/5 C/) c/3 3-
U U U t/5
ro \0
M N
? 2- §^
d d M
lO On ro O
"* q 00 ■*
yo 6 6 6
VO CO •>«■•«*■ VO' 00 M
o t^ >o
d d ro
M M ro
s ^
0^ \0
o PI rv fi
00 n o> ■<^
'<^ o ro N
•<1- M CO o
M »o 00 lO
o T»- 00 in
00 00
O 00
•«^ On ro
O VO ON O-
5: ^
^ H O*
a 3
C i5 .
r<. IT) 10 •* 6
•* O ro 6 ^
'g 13; ^ 8 o
d « 6 0- M
d
d
d
d
M
fo
d
■♦
d
d
vd
W \0 ^0 VO
r^
>o
t^
to
•*
v8
PI
\
N
g
iT «* S vo vo"
*o
00
N
IT
00 ■♦ VO NO 5
**
■♦
**
VO
N
IN w IM (■♦
1 M
'"
1 H
IN
IN
M
f)
10
2*
fO
in
m
M
VO
VO
•s
r<1 m rn M 00
8^
s
VO
•* -^ M fl
00
'§^
«s 3; ^ s; §;
fO
m
M IN 1 f«1 IM
'"
'"
"
IH
IN
w
IN
« in ■*• m -"I-
•^*-
^
s
!>
0!
^
s
%
VO
5.
? ? 8 vS 'S^
w
«
00
•s.
V>
xS-
S*
s
%
%
IH
^
«o « *
t^
M
"
fn
tn
-*■
%
gram,
kilogram,
tonne (millier).
tonne (raillier).
5
a
a
2
s
u
2
2
?
a
'c'c
a
•2h
!3
4>
i
1
i
u
eS
IS
R
M
is
il
y
i
■sS
1?
. c c = =
'^
^
V
3 3
^?
^
c
"^ItT
'• "^^
^
OJ3
0.5
a X x • - • -
2 U ^^ U !«■
II
c_y
•2^;
n
^i
3 V
3 V
u
r?5
u i^
<2
i-S
t) ~
d"S
(U-=
iX!
t« 5 £
c 111 1
2
•0
•2f
7S
JJ
£6
•S i
l^
-Si
•56
Q.
§
g&
2li
2S
?s
2,b
r^
V,?
-I
^2
C he
s) su
spun
spun
suie
2
UT3
u.:4
-§2
II
ri
•S2
•S2
c be
2i
K
A
.c
3
3
c
-3
u 3
.-Q
£
CU 0, H
(X,
b.
b
fc
(X
cu
H
cu
ou
H
0.
290
APPENDIX.
TABLE IV.
GRAVITY,
g = Acceleration due to gravity, in feet per second.
L = Length of simple seconds pendulum, in feet.
A = Latitude.
k = Height above sea-level, in feet.
g — 32.173 — 0.0821 cos 2X — 0.000003/^.*
L = 3.2597 — 0.0083 cos 2/1 — 0.0000003>^.*
Values of ^ = 32.173 — 0.0821 cos 2A and L = 3.2597 — 0.0083 cos 2A.
Latitude.
£r.
L.
0'
32.091 f. s.
3-2514 f.
5
32.092
3-2515
ID
32.096
3-2519
15
32.102
3-252S
20
32.110
3-2533
25
32.120
3-2544
30
32.132
3-2556
35
32.145
3.2569
40
32.159
3.2583
45
32.173
3.2597
50
32.187
3.2611
55
32.201
3.2625
60
32.214
3.2638
65
32.226
3.2650
70
32.236
3.2661
75
32.244
3 . 2669
80
32.250
3.2675
85
32.254
3.2679
90
32.255
3.2680
The value of g is affected to some extent by the character and arrangement
of the local geological strata. The variation from the tabular value may be as
great as 10 units of the last place of figures, but rarely exceeds 5 of these units.
* Encyclopaedia Britannica, art. Gravitation.
APPENDIX.
291
TABLE V.
FRICTION *
Substances.
Angle of
Repose.
Coefficient of
Friction.
-f-
1
Vi +/«
14 -26.5
II-5
II. 5-31.
26.5-31.
11-5
II. 5-14.
26.5
16.5
8.5-11.5
16.5
4.-4.5
3.
1.7
.25-.5
.2
.2 -.6
.5 -.6
.2
.2 -.25
•5
.3
.I5-.2
.3
.07-. 08
.05
.03
.24-. 45
.20 1
.20-. 51
.45--5I
.20
.20-. 24
.45
.29
.15-. 20
.29
.07-. 08 j
.05
.03
i
" " " soaoed
Wood on metals, dry
Metals on oak, dry
' * * ' * ' soaped
Hempen cord on oak dry
" " " wet
Metals on metals, dry
" " " wet
Smooth surfaces, occasion'y greased
'* ** continually "
best results
These values are for low velocities and pressures at ordinary temperatures.
The coefficient for smooth metal bearings, well oiled, varies somewhat with the
pressure and velocity, being generally less than the above. It also varies con-
siderably with the temperature, which affects the lubricant. In favorable cases
it has been as low as .002 [Thurston].
Mostly from Rankine's " Rules and Tables," and Trautwine's " Engineer's Pocket-Book.'
592
APPENDIX,
TABLE VI.
STIFFNESS OF CORDAGE FOR WHITE AND TARRED
ROPE.
Morin's Formulas.*
6" = — — = ^(0.002148 4- 0.001772W -[" 0.0026256 W) for white rope;
K = «(o.oo2i48 -f- o.ooi772«), and / = «(o.oo26256).
S = — ^ = —(0.01054 4- o.oo25« -\- 0.003024 ^) for tarred rope;
K = «(o.oio54 -f- o.oo25»), and / = «(o.oo3024).
Values
OF ^AND /IN Lbs.,
FOR ROPK WOUND ON A
XLE I Foot
IN Diameter.
j
i
No. of
1 Yarns.
Ordinary White Rope.
Tarred Rope.
Circum-
ference
in inches
Natural
Stiffness.
Stiffness due
to Tension of
lib.
/.
Circum-
ference
in inches
Natural
Stiffness,
Stiffness due
to Tension of
lib.
/.
i
=
/if in lbs.
=
A' in lbs.
.4524 v^
.5378 Vn.
6
1. 10
0.07668
0.015754
1.32
15324
O.O18146
i 9
1.36
0.16286
0.023630
1. 61
0.29736
0.027219
12
1.57
0.28094
0.031507
1.86
0.48648
0.036292
15
1.74
0.43092
0.039384
2.08
0.72060
0.045365
18
1.92
0.61279
0.047261
2.28
0.99972
0.054438
21
2.08
0.82656
0.055138
2.46
1.32384
O.063511
24
2.21
1.07222
0.063014
2.64
I .69296
0.072584
27
2.35
1.34978
0.070891
2.80
2. 10708
0.081657
30
2.47
1.65924
0.078680
2 95
2 . 56620
0.090730
33
2.60
2.00059
0.086645
3.09
3.07032
0.099803
36
2.72
2.37204
0.094522
3.23
3.61944
0.108876
39
2.84
2.77888
0.102398
3.36
4.21356
0.117949
42
2.94
3.21602
O.IIO275
3.48
4.85268
0.127022
45
3.05
3 • 68496
O.I18152
3.61
5-53680
0.136095
48
3-17
4.18579
0.126929
3.73
6.26592
O.145168
51
3-26
4.71852
0.133906
3.84
7.04004
o.T542+r
54
3-35
5.28314
O.141782
3-95
7.85916
0.163314
57
3 45
5.87966
0.149659
4.07
8.72328
0.172387
60
3.54
6 . 50808
0.157536
4.17
9.63240
0.181460
* Adapted from Morin's formulas, " Cours de M^canique," vol. ii., Dulos, pp. 193, 194.
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