2(ot^i) ^(wJig) l.{mAs)^
X fx 1/ _
and therefore, if /r = ^,^ + ^2^ + /^3^ we have
k, h, /«3
That is, the position of the plane is constant during the motion ; it
is called the plane of maximum areas, or the " Invariable plane."
We may remark that the position in space of the plane has
not been proved constant, but merely the direction of its normal ;
all planes parallel to this would have the same property.
If at any instant we know the positions, magnitudes, velocities,
and directions of motion of the several particles of such a system
as we have been considering, we can find the position of this invari-
able plane, with reference to co-ordinate axes arbitrarily chosen;
and if at any future time we take other co-ordinate axes arbitrarily
chosen, we can also determine with reference to them the position of
the invariable plane, and we know that these two positions must
be absolutely the same, as far as regards direction, whatever the
co-ordinates by which they are determined. Thus we see that the
system itself furnishes a plane of reference which is invariable in
direction, and can always be determined from the state of the system
at any instant.
The preceding are the principal general properties of a system
of particles moving under their mutual actions ; the actual determi-
nation of the motion of each particle in anything like the general
case far surpasses our powers of analysis. The solution of a number
of simple cases is the only means of acquiring a familiarity with the
treatment of problems of this sort.
87- We have hitherto supposed the system to be acted on by
no forces but those which arise from the mutual attractions of its
several particles ; in addition to these, all or any of the particles of
the system may be acted on by forces arising from causes extraneous
to the system, as, for instance, the attraction of some body not
128 A TREATISE ON DYNAMICS. [cHAP.
forming a part of the system ; or by forces arising from geometrical
constraints, such as, that some of them move in tubes or on surfaces.
In these cases if the forces are given, they will enter our equa-
tions as known quantities, and we shall still have as many equations
as we have co-ordinates to determine.
If the forces are not given explicitly, but are known to arise from
the particles being compelled to satisfy certain geometrical conditions,
they will then enter the equations as unknown quantities; and,
corresponding to each unknown force, there will be an equation
expressing the geometrical condition from which it arises ; so that
the whole number of equations will in every case be the same as the
whole number of unknown quantities, and will thus be sufficient
to express every such unknown quantity as a function of t.
We shall not, however, enter upon any of these cases here, as our
object is not to obtain solutions of particular problems, but to explain
the principles by which those solutions are to be obtained, and to
investigate certain properties which belong to extensive classes of
such problems.
VI.] A TREATISE ON DYNAMICS. 129
CHAPTER VI.
88. In the preceding pages the motion of a single material
particle, and also of a system of several particles has been consi-
dered, when the particles were acted on by various forces and subject
to different conditions; it remains to investigate the methods of
determining the motion of bodies of finite magnitude.
Now, when a body of finite magnitude is in motion, unknown
forces act on every particle of it arising from the action of every
other particle. The determination of these forces from the equations
expressing the conditions to which the system is subject presents
insuperable difficulties. The consideration of these forces is avoided
by making use of a principle first stated by D'Alembert, which we
will now proceed to explain.
When a material system is in motion under the action of forces
arising from causes external to the system, these external forces are
called "impressed forces;" and those of them which act on any one
particle of the system are called the impressed forces on that particle.
Every particle besides being acted on by the impressed forces, is
also acted on by forces arising from its connexion with the other
parts of the system ; these are called " internal forces."
The whole force acting on any particle is the resultant of the
impressed and internal forces on that particle, and is called the
"effective" force on that particle.
Conceive an equal isolated particle to be moving in the same
direction and with the same velocity as the proposed particle, and to
be acted on by a single force equal to this resultant and in the same
direction; the two particles Avill be in the same state, one being
acted on by a number of forces, the other by a single force which is
their resultant; the motions of the two particles will therefore be
the same. Hence then we sometimes have the following definition
of effective force.
That force which acting alone on any particle of a system sup-
posed isolated would cause it to move as it does when it forms part
of the system; that is, which would generate in the small time 8/
the same change in velocity and direction of motion that takes place
when the particle forms part of the system, is called the " effective"
force on that particle.
9
130 A TREATISE ON DYNAMICS. [cHAP.
Now D'Alembert's principle asserts that if at any instant we
apply to every particle of a system forces equal and opposite to the
eftective forces on that particle, the system, if it were at rest in the
position which it occupies at that instant, would remain in equili-
brium under the action of these together with the impressed forces.
We have enunciated this principle in the form in which it will be
most convenient for obtaining equations of motion for a system. We
will now explain the nature of the assertion contained in it.
Let mhe a particle of the system ; / the
resultant impressed force acting on that
particle ; M the resultant of the internal
forces on it; JE the resultant of M and /,
and therefore the effective force on the par-
ticle ; R the reversed effective force, that is, II
a force equal to E acting in the opposite direction.
Then if the particle m were at rest, and the forces /, M, and R
acted on it, they would keep it in equilibrium : and the same would
be true of every particle.
If therefore the whole system were at rest, and each particle
acted on by the impressed forces, the effective forces reversed, and
the same internal forces that acted on it when the system was in
motion, every particle would be separately in equilibrium, and
therefore the whole system would be in equilibrium.
Instead of this suppose the system to be at rest, and each particle
to be acted on by the impressed forces and reversed effective forces
that correspond to it, and by such internal forces as naturally arise
from this condition ; then D'Alembert's Principle asserts that the
system will in this case also be in equilibrium.
The Principle may also be considered in the following light, in
which it possibly appears less arbitrary.
A system is in motion under the action of impressed forces and
internal forces arising from the connexion of its parts. Let the
impressed forces be replaced by the effective forces: in this case,
since each particle will move under the action of its effective force
alone consistently with the connexion of the parts of the system, the
internal forces arising from this connexion will not be called into
action, and the change of motion in the small time ht will be the
same as under the action of the impressed forces.
Now, forces equal and opposite to the effective forces, acting
together with the effective forces on the system at rest, keep it in
equilibrium ; will they keep it in equilibrium when they act together
with the impressed forces ? D'Alembert's Principle asserts that they
VI.] A TREATISE ON DYNAMICS. 131
will. It states that a set of forces which balances one of two sets that
produce the same change of motion in a system will also balance the
other.
D'Alembert's Principle, like the laws of motion and the theory
of gravitation, is proved by the agreement between the results of
calculation and observation. We are not, however, able in this case
to avail ourselves of Astronomical observation. The comparison
has been principally made by means of a machine invented by
Atwood, by which the motion of a system of bodies can be very
accurately observed in a great variety of cases.
89. To form the equations of motion by the aid of this principle,
we must express the effective force on each particle in terms of the
co-ordinates of that particle : if m be the mass of a particle, x, y, z
its co-ordinates, the resolved parts of this force at the time t will be
d-x d-y , d'z „,
m-T-j, m -— , and m -^ . We must now suppose the system to be at
rest in the position it occupies at the time t, and apply to every par-
ticle a force equal and opposite to the effective force on it The
system will then be in equilibrium under the action of these and the
impressed forces : and the equations expressing the conditions of
equilibrium of the system will be the equations of motion.
These equations will be different for different systems of bodies.
They will be different according as there is one body or more, as the
bodies are elastic or inelastic, solid or fluid, rigid or flexible.
The equations expressing the conditions of equilibrium assume
different forms in these different cases, and consequently the equa-
tions of motion obtained from them by the aid of D'Alembert's
principle are of different forms.
90. We will now apply this principle in the form in which we
have enunciated it to the solution of a problem ; viz. to calculate the
motion of a uniform cylinder with its axis horizontal, rolling down
an inclined plane so rough as to prevent all sliding.
Let the figure represent a section of the cylinder and inclined
plane; and let a be the radius of the cylinder, h its length, a the
inclination of the plane to the horizon.
Let any point B in the plane be the origin, BP the axis of x, a
normal to the plane atB the axis of ^ ; Q an element of the cylinder,
^7)1 its mass, x, i/, z its co-ordinates ; then BN = x, QN=y.
Let BP^x, QC = r, QCA^tp.
ACP = 6 the angle through which the cylinder has rolled.
9—2
132 ■ A TREATISE ON DYNAMICS. [cHAP,
Since z is constant -7-^ — , and the resolved parts of the effective
at' ^
force on Q are Im -y— and Im ~~, parallel to the axes of a; and y ;
and the impressed force is g^tti vertically downwards. Also on the
line of particles in contact with the plane there will be other im-
pressed forces arising from the reaction of the plane : let the resolved
parts of one of these parallel to PC and PB be R' and F\
Now apply to every particle the reversed effective forces on that
particle, and write down the equations of equilibrium : they ai*e
1.R' - "EZjn-— — ^^me^cosa — 0,
(It
1.F' + 1.^7)1 -j-^ ~ I-hm g sin a = 0.
1.F'a + IhynQc- x)—^^-^lm{y -")ji2 +'^^mg cos a (x - x)
+ 28 mg sin « (^y — a) = 0.
To these equations we must add the geometrical relations
X ^x — r sin (6 + (p), y = a- r cos {Q + (p),
from which we obtain
•i , -.fdey . .d'0
r^-^'-^\dl)^^-'^-''^dF
de di-
d'y . .fdOy . rl^d
dF- ~(^-"Kdi)-^'-''^de'
Let 2/2' = 72, ^F'^F, 1.1m =M, then substituting these values
in the equations we have obtained, and bearing in mind that ^, y,
VI.] A TUEATISE ON DYNAMICS. 133
and d refer to the whole cyhnder, and that the center of gravity of
the cylinder is in the axis, and therefore thSt
1.l7n(x-w) = 0, 1.d7n(i/-a)-0,
we have R - Mg cos a = (1 )
M~ = Mgdna-F (2)
'^^hnr"-=Fa (3)
also X — aO + c (4).
Equation (l) determines the whole pressure on the plane, which
is constant.
Also 'Zcmr^^Ma%
therefore equation (3) becomes
Ma d^_
2 de~
Eliminating F between this equation and (2) and reducing by
means of (4), we have
d^O _ 9.g sin a ^
dQ _2g sxna.t
" dt ~ Sa '
— is called the angular velocity of the cylinder.
C and C may be determined from the values of 6 and -j- corre-
sponding to any value of t ; and the motion is then completely de-
termined.
We see from the preceding example, that even in a simple case
the equations given by the immediate application of the principle
are complicated, and require considerable reduction before they can
be integrated. In the case of rigid bodies, however, this reduction
can be effected generally, as follows:
91. Let there be one rigid body, and let 7/?, be the mass of a
particle, x,, i/i, ri its co-ordinates at time /, X, Yi Zj the resolved
parts of the impressed forces that act on it. The resolved parts of
the effective force on it are
d'x, d'y, d'z,
""^Tt^' "''rfF' "'^di'-
134 A TREATISE ON DYNAMICS. [CHAP.
If we apply these in the opposite directions, the resolved parts
of the whole force on the particle will be
,„,--^andZ.-^,-^
and the body will be in equilibrium under the action of these and
similar forces applied to every particle.
Hence then the six equations of equilibrium give
(I)
^(i
mA--
dt-
)}
(II)
These are the six equations of motion for one rigid body, and
when these equations are solved the motion of the body is com-
pletely determined.
92. The preceding equations contain the co-ordinates of every
particle of the body, the use of them is greatly facilitated by the
following reduction.
Let X, y, J be the co-ordinates of the center of gravity of the
body at the time t ; x' , y', z' the co-ordinates of a particle of the
body referred to axes whose origin is the center of gravity and which
are parallel to the original ones : then
and therefore
d'x d^x (fx'
dt' dt'
df
d^J^ d^
de dt^ de
dt'
di'' "^ dt'
VI.] A TREATISE ON DYNAMICS. 135
Substituting in equations (I), we have
dl^ dt^
but since the origin of x' y' z' is the center of gravity
and therefore
„ d'x' ^ _ d'y' ^ ^ d'z' ^
di' dl' dr
also X, y, z are the same for every "particle of the body, therefore
writing M for 2?h, we have
^^§=-^4 (Ill)
dt'
These three equations are in the form in which they are most
available for determining the motion of the body. They are the
same as those for the motion of a particle whose mass is M acted on
by the forces 1.X, 1.Y, and 2Z,
Hence then, if a body of finite magnitude be acted on by any
forces, its center of gravity will move as a particle of equal mass
Avould if acted on by all the forces impressed on the body.
We also see that the motion of the center of gravity of the body
will not be affected by changing the points of application of any of
the impressed forces, provided their intensities and directions are
not altered.
If all the forces X, Y and Z be given explicitly as functions of
the variables, these equations will give the motion of the center
of gravity at once by integration.
When this is not the case, that is, when unknown forces enter,
arising from geometrical conditions to which the motion is subject,
it will generally be necessary to combine them with the remain-
ing three equations before the integration can be effected.
136
93. Substituting a; + a;', 7/+ y, 2' + s for
(II) we have
X, y, an
2{(i + /)(x-
d'x
-(i + xo(z-
d'l
2|(3 + x0(f-
d'y
-""dP-
d'y'\
-^de)-
.G^+y)(x-
^)}-
d'y'-
dt
Now in these equations x, y, i and their differential coefficients
are common to all the particles of the system, they may therefore
be placed before the symbol, 2.
Also, 2 m x' = 0, 2 my' = 0, 1.mz' = ;
and therefore,
_ d'x' ^ _ n't/ ^ ^ d'2' ^
By means of these relations, and also the equations (III), the
preceding equations may be reduced to
,(IV)
Now we observe that the position of the center of gravity does not
enter into these equations either explicitly or implicitly.
If then the center of gravity of the body were fixed, and the body
were acted on at the same points by the same impressed forces that
act on it when it is free, the equations of motion would be the same,
since the forces introduced by the condition of the center of gravity
being at rest would all act at the center of gravity, the co-ordinates
of their points of application would be zero, and they would therefore
not enter into the equations.
Hence then we conclude that the motion of the body about its
center of gravity is the same as if the center of gravity were fixed,
and the body acted on by the same impressed forces.
These two results are expressed by saying that motions of
translation and rotation are independent. They are of the utmost
importance in the solution of every problem about the motion of one
rigid body.
VI.] A TREATISE ON DYNAMICS. 137
We will for distinctness enunciate them again.
When a body is in motion under the action of any forces, its
center of gravity moves as if the whole mass were collected there and
acted on by iill the forces which act on the body. And the motion
of the body about its center of gravity is the same as if the center of
gravity were fixed, and the body acted on by the same forces that do
act on it
94. The equations (IV) involve the co-ordinates of every par-
ticle in the body. To determine from them the motion of the body,
they must be reduced to others containing only co-ordinates common
to the whole body.
We shall not enter upon this reduction in the general case at
present, as the process is complicated, but shall confine our attention
to the cases where all the particles of the body move in parallel planes.
Let the plane of xi/ be that to which the motion is parallel ; then
the ordinate parallel to z of every particle is constant. Equations
(IV) are therefore reduced to
or, 2,«(^x'— -y^j^2(x'I-yZ).
Let x' = r cos d, y' = r sin 0,
then for each particle r is constant ; so that we have
d-x' . .,rQ aM^\'
-^ = -rs.ne^-rcose[-),
cVy' .(f-d . ./ddV
dt^ dl- \dtj
Making these substitutions, we have
d^O
Now since it is a rigid body — , and therefore j-r is the same
for every particle of the body, we may therefore place it before 2.
So that we have
'^^j ^mr^ = ^{x'Y-y'X).
Now tlie right-hand side of this equation is the resultant moment
of the impressed forces round an axis through the center of gravity
138 A TREATISE ON DYNAMICS. [CHAP.
perpendicular to the plane of motion; calling this L and putting
Mk^ for 2r«r^ we have
at
Combining with this the first two of equations (III), we have
(V)
These three equations are sufficient to determine the motion of a
rigid body which takes place in one plane, whenever the forces
which act on it are given as functions of the variables.
This is seldom the case. Unknown forces generally enter into
these equations, arising from conditions to which the motion of the
body is subject. There will, however, in every case be a geometrical
equation corresponding to each unknown force so introduced, so
that the number of equations will always be equal to the number of
unknown quantities.
In effecting this last transformation all the geometrical conditions
of the rigidity of the body were introduced when we considered r
to be constant for each particle, and -j- to be the same for all the
particles.
95. In the preceding reduction the expression j-^'Lmr^ occurs,
where r is the distance of the particle m from a certain fixed line
in the body, vir^ is called the "Moment of Inertia" of the particle
m round this line, and 'Zmr^ is the moment of inertia of the whole
body round the same line ; it is the sum of the moments of inertia
of every particle. This quantity will frequently occur; it is different
for different bodies and for different lines in the same body. It is
generally written MF, in which case k is called the "Radius of
Gyration" of the body round the particular line. The Moment of
Inertia of any body may be found by the ordinary methods of inte-
j a
gration. Again, — is called the angular velocity of the body ; if we
conceive a plane fixed in the body perpendicular to the plane of
. motion to be inclined at an angle to a fixed plane also perpen-
VI.]
A TREATISE ON DYNAMICS.
139
dicular to the plane of motion, -7- is the rate at which the angle 6 is
increasing, and the propriety of the term angular velocity is manifest.
Sometimes -jt, is called the angular accelerating force.
96. Let us now take an example of the application of these
equations in the case of motion in one plane.
A uniform heavy rod slides down between a smooth vertical and
a smooth horizontal plane, to determine the motion.
Let the figure represent the
rod at the time t.
AB = 2a,
DN=x, NC = y,
ACN=d,
M the mass of the rod,
R, K' the reactions of the
planes, then from equations (V)
we have
>4-^'..
Mk
de
Ra smd-R'a co&Q
(1)
•(2)
.(3)
and corresponding to the two unknown forces R, R' we have the
two geometrical relations
x = a5\nd, y = acosB, (4)
which express that the ends of the rod are in contact with the
planes.
Eliminating R and R' from the equations, we get
^j^^fd^^dx d^ydj^ d^de\_ dy
^^^ W dt'' dt' dt^^deTtJ-'^^^Tf
Integrating this, we have
and using equations (4),
<--')e
C — 2ag COS0.
140 A TREATISE ON DYiSAMICS. [cHAP.
If we suppose the sliding to commence when the rod is vertical,
we have -r-=0 when 6 = 0:
at
0=C~2ag;
dey 2ag . ..
dtj a^ + k'
dd
dl
V a' + k'
(5)
Or, since k^ — -^
dd_ /Si
dt~\ a
H
which gives the angular velocity in any position ; the equations
dx ^dd dt/ . „dd
dJ=''''''^irt^ 57=-^^^"^d7'
give the velocity of the center of gravity.
The integral of equation (5) is
log tan ? = . /5 t + C.
On attempting to determine C by the condition used beforej
we get C = — CO . This might have been expected since the position
from which the motion was assumed to commence is one of equi-
librium, so that the motion would not take place at all. We may,
however, suppose the motion to commence in a position differing
very slightly from that of equilibrium, in which case the velocity
we have obtained will only be affected by a small quantity of the
same order of magnitude, and C will be rendered finite : or we may
suppose a velocity equal to that indicated in the solution to have
been communicated to the rod in the corresponding position.
To find the pressures on the two planes in any position, we have
\-n) •= x^ (1 -cosy);
\dtj 2a ^ ^'
d'0 3g . .
dv 4«
also from (4) a' = a sin (), y — a cos Q ;
d^x So- . ^,
.•.-^=-^sm0(3cos0-2),
i| = -?l(l+2cosa-3 cos^ &),
at 4 '
VI.] A TREATISE ON DYNAMICS. 141
substituting these values in equations (1) and (2), we have
R'=.^-^ sm 6(3 cos 0-2),
4
R=—^(l-6cosd + 9 cos' e),
Mo-
= — ^ {1 + 3 cos e (3 cos 6 - 2)},
which give the pressures on the planes corresponding to any value
of 0.
When S cos 6 — 2 = 0, R' = 0, and as 6 increases R' becomes
negative : this result shews that a force would be required to keep
the point B in contact with the vertical plane : as this force is not
supposed to exist the rod will at that point leave the plane, and the
motion will be different afterwards.
From the expression for R we see that it continues positive up to
this point.
To determine the motion after the rod leaves the plane BD we
have the equations
M2 = R-Mg,
Mk'~=Rasm0,
y — a cos 6.
dx
From the first equation it appears that -j- remains constant Avith
the value it had when the rod left the plane.
Now at that time
dx ,dd
Tt=''''''^dt'
= cos0y^(l_cos0),
and putting for cos 6 its value | we get
This will continue to be the velocity of the center of gravity in a
horizontal direction.
142 A TREATISE ON DYNAMICS. [CHAP.
From the other equations we have
.-. d" (sin^ + 1) (^Y= C-^ag cos Q.
Now when the rod left the vertical plane
... (3sin^0.l)Q^|f(8-9cos0),
which gives the value of -j- in any position, and we may find the
value of R as in the previous case.
97- When the motion of every particle is parallel to one plane,
we have only the three equations (V) instead of the six equations
(III) and (IV).
These six equations however must always be satisfied, let us
therefore consider what they become in this case.
The last of equations (III) becomes
2Z = 0,
which shews that the sum of the resolved parts parallel to the axis of
z of all the forces which act on the body must be zero.
If the body is free this is one of the conditions that must hold
among the forces acting on it that the motion may be of the kind we
have supposed.
If the body is constrained to move in the manner supposed this
will be one of the equations which determine the forces called into
action by this constraint.
The first two of equations (IV) become
2:,«.'g- = 2;(/7-yz),
^mz' -y-^ = ^{z'X-x'Z).
Now from the relations
x' = r cos d, v' = r sin d.
VI.] A TREATISE ON DYNAMICS. 143
the left-hand sides of these equations can be expressed in terms
of -J- and T— , which are known by (V) ; and the equations then
give two more conditions to be satisfied by the impressed forces.
Thus whenever the motion is of this sort, the six equations (III)
and (IV) are reduced by relations between the impressed forces to no
more than three independent equations.
98. It frequently happens that the conditions of equilibrium of
the system under consideration have been found in a form simpler
than the six general equations of equilibrium. Now, as in consi-
dering any particular case of equilibrium we may employ either the
six general equations, or the more simple ones that have been
deduced for the class to which the particular case under considera-
tion belongs, so we may employ either set of equations of equilibrium
to form the equations of motion.
The case of a body moving about a fixed axis affords an illus-
tration of this remark, and deserves consideration from its own
importance.
In the case of a body moveable about a fixed axis, we may
determine the conditions of equilibrium from the six general equa-
tions, in which case we introduce unknown forces for the pressures
exerted by the axis on the body ; or we may take the condition
which has been deduced from these, that the body will be in equi-
librium when the resultant moment round the fixed axis is zero.
We will consider the motion of a body about a fixed axis in both
these points of view.
Let r be the distance from the axis of a particle whose mass is m,
since its motion is at every instant perpendicular to the axis and the
radius vector, the resolved part of the effective force in this direc-
tion is
(Ps , . - cVe
m-, which = ^r^^^,.
If therefore we apply a force equal to this in the opposite direction,
72/1
its moment round the axis will be — mr^ —-^ .
Let L be the whole moment of the impressed forces, then the
condition that the whole moment round the axis is zero gives the
equation
144 A TREATISE ON DYNAMICS. [cHAP.
or, since -y— /and therefore— -^ is common to every particle of the
system
or, putting Mk^ for 2m r'
iV«-^=L.
From this equation the motion of the body may be completely
determined.
We will now see how the same result may be obtained from the
six general equations.
Let the fixed axis be taken as the axis of z, and a plane perpen-
dicular to it through the center of gravity for the plane of xt/; let
X, y be the co-ordinates of the center of gravity, Q the angle which
a line joining the center of gravity and the origin makes with the
axis of .r.
We may suppose the body to be fixed to the axis at two points,
let these be at distances c, and c' from the origin, and let the re-
solved parts parallel to the axes of the forces which these points
exercise on the body be -R^, Ky, jR,, J?/, Hy, -R/.
Then we have from equations (III), since -r-j — 0,
M^ = 2X4-i?. + i?,' (1)
3f^ = 2:F+2?, + i?; (2)
= 2Z + 7i', + i?/ (3)
df
d^z
also from equations (II), since ,72 =0 foi* every particle, we have
-cE,-c'i?/+2(^Z-.y) + 27«.|^ = (4)
cR, + c'RJ + ^{zX-xZ)-l.mz^^ = (5)
^{xY-yX)-^(^nx^,-my^:;) = (6)
equation (6) may be reduced to
VI.] A TREATISE ON DYNAMICS. 145
This is the same as the equation
which was obtained by the other method : it is sufficient to deter-
mine the motion.
The remaining five equations enable us to determine the un-
known forces /?,, Ry, RJ , RJ , and R^ + R/ ; just as in the statical
problem the general method gives the pressures on the axis, while
the particular method tacitly eliminates them.
We will return to these equations, and put them into a more
convenient form for determining the unknown pressures in particular
cases, but for the present will consider the motion of the body.
99. A particular case of motion round a fixed axis worthy
of consideration is where the only impressed force is gravity, and the
axis is horizontal.
Let h be the distance of the center of gravity from the fixed
axis, 9 the angle between a vertical plane through the axis, and a
plane through the axis and the center of gravity.
Then L =-Mgh sin 6, so that the equation is
•••(;
:^:)'=^-f^-^ (^)-
C must be determined from the angular velocity corresponding to
some known value of 6.
Now the equation of motion of a point in a circular arc of radius
I is
comparing this with equation (l) we see that they are identical, if
'4-
Hence then the time of oscillation of the rigid body will be the
same as that of a particle attached to a string whose length is -j ,
for the same amplitude. Tliis is called the isochronous simple pen-
dulum, the body itself being called a compound pendulum.
10
146 A TREATISE ON DYNAMICS. [cHAP.
The equation . -rji+-jj sin =
can only be solved so as to give the i-elation between & and t \n a.
series.
When however Q is always small, so that powers higher than the
first may be neglected, we can find an approximate solution.
Writing for sin d, which amounts to neglecting 0\ we have
l-^4^^- (^)^
Let the body start from rest when 6= a, then
a = A cos B ; . -
.-. jB = and A = a:
a COS . / ~ t ;
■when 6 = we must have
A''=(^->>i=
let ^1, ts, &c. be the successive corresponding values of t, then we
have _
'^-" SI Rh'
2 V gh
_S7r 11^
~ 2" V gh
_57r 111^
~ % SI gh
1^
This shews that the center of gravity of the body will attain the
loAvest position after successive intervals of tt sj — j '
VI.] A TREATISE ON DYNAMICS. ^ 147
We also see that, to the order of approximation, to which we have
proceeded, the time is independent of the value of o, or the ampli-
tude of the oscillation.
We shall return to this subject afterwards when we treat of pen-
dula, and shall then shew how a more approximate value of the
time of oscillation may be obtained; before leaving it, however, there
are a few general properties which deserve notice.
Let the figure represent a section
of the body made by a plane through
its center of gravity perpendicular to the
axis round which it oscillates ; and let C
be the point in which the axis cuts this
plane, G the center of gravity.
The two equations
will be identical if Z = -7- .
h
If therefore we take in CG produced a point such that CO = I,
the position and motion at any time will be the same under the same
initial circuiftstances as if the whole mass were collected at 0.
O is called the center of oscillation.
If k^ be the radius of gyration round an axis through G parallel
to the fixed one through C,
and therefore I = -^ + h.
h
Now if the body were suspended by an axis through parallel
to the same axis, the length of the corresponding isochronous simple
pendulum would be
which is equal to I.
Hence then oscillations about this axis take place in the same
time as those of the same amplitude about the first axis through C.
C is called the center of suspension, and this property is ex-
pressed by saying that the centers of oscillation and suspension ai'e
reciprocal.
10-5
148 A TREATISE ON DYNAMICS. [cHAP.
This property might also have been deduced in the following
manner. Suppose it required to find the distance from G of an
axis parallel to a given direction, about which the oscillations will
take place in the same time as those of a simple pendulum whose
length is /.
Let k be this distance; then h will be determined from the
equation
or h^-lh + ki'' = 0.
Now this being a quadratic equation gives two values of h,
hi and li^ such that hi + hi = /.
So that an oscillation about an axis at distance h^ takes place in
the same time as one about an axis at distance l-h^.
Also, since in this investigation nothing has been introduced to
indicate in which direction from G the required axis lies, we con-
clude that any direction will satisfy the condition. If therefore with
center G we describe circles in the plane of the paper at distances
GO and GC, small oscillations will take place in the same time about
any parallel axis through any point in either of these circles.
The value of h which will make /, and therefore the time of
oscillation a minimum may be easily determined : since
k ^
l^^ + h,
that I may be a minimum, we have
k *
-^ + 1=0,
or h = ki,
and therefore l = 2h or S^i-
In this case therefore the two circles we have mentioned
coincide.
The axes determined by these conditions are not axes about
which the time of oscillation is the least possible for the body, but
only for axes in the body parallel to a given direction. To find those
about which it will be absolutely the least, we must select that
direction for which k is the least. This we know will be one of the
principal axes through the center of gravity of the body.
A TREATISE ON DYNAMICS. 149
CHAPTER VII.
100. To find the conditions of equilibrium of a system of rigid
bodies we have for each body of the system six equations of equi-
librium, as if that particular body were the only one of the system.
And besides these we have equations expressing the geometrical
connexions of the bodies.
The same method must be adopted to determine the motion of a
system of rigid bodies ; we must write down for each body of the
system the six equations (III) and (IV), or the three equations (V),
and then the equations expressing the geometrical conditions which
the bodies satisfy.
When the equations are once written down the difficulties of
solving them are merely analytical, and such artifices must be used
as the forms of the equations suggest.
The constants introduced in integration must be determined in
the same way as in the case of a single body.
When any system is in equilibrium we may either consider each
body separately, and write down for it the six equations of equi-
librium ; in which case the actions of the other bodies of the system
on it will enter as external forces ; or we may consider the whole
system as one body and write down for it the six equations of equi-
librium; in which case the mutual actions of the different parts of
the system will not appear in the equations; or we may consider
any two or more of the bodies as one, and write down for them the
equations of equilibrium.
These different methods may also be adopted in Dynamics. The
dynamical equations, as first written down, are strictly equations of
equilibrium, and it is only from the analytical character of the
quantities that enter them that they enable us to determine the
motion of the system.
Though the methods to be employed in solving the equations
of motion must in each case be suggested by the forms of those
equations, there are certain artifices by which the first integrals may
often be obtained, the applicability of which can be determined at
once from an inspection of the system. The difficulty generally
arises from there being unknown forces which have to be eliminated
150 A TREATISE ON DYNAMICS. [CHAP.
by means of geometrical relations. These artifices enable us in
certain cases to effect that elimination at once.
101. When a body or system of bodies is in motion under the
action of any forces, if x, y, i are the co-ordinates of the center of
gravity at the time t, and M the mass of the system, we have the
equations
(III)
in which no forces appear but such as arise from causes external to
the system ; that is, no pressure of one part of the system against
another enters, no mutual attraction, no tension of a string con-
necting two parts of the system. But all forces do enter which
arise from causes external to the system, as pressures of fixed sur-
faces, or fixed points, tensions of strings passing over fixed pulleys,
&c., together with the given external forces whatever they may be.
Now equations (III) shew that when
2X=0, 2r=0, 2Z = 0,
the center of gravity will move uniformly in a straight line. Thus,
whatever the internal forces of the system, so long as there are no
external forces the center of gravity will move in a straight line
with uniform velocity. This is called the Principle of the Con-
servation of the motion of the center of gravity.
dx
We see from equations (III) that if SX= 0, j- is constant, what-
ever 2 Y and 2Z are. Hence if the resolved parts of all the external
forces actino- on a system parallel to any line have a resultant zero,
the motion of the center of gravity parallel to that line will be
uniform. Similarly, if two of them vanish the resolved part of the
motion parallel to that plane will be uniform and rectilinear. Thus,
for instance, if any number of balls be piled up on a perfectly smooth
horizontal plane and then left to themselves, they will be acted on
by no external horizontal force, they will therefore fall down in
such a manner that their center of gravity will describe a vertical
line.
In the cases that will most frequently occur the center of gravity
of the system will be at rest at the commencement of the motion,
and will therefore continue so throughout.
VII.] A TREATISE ON DYNAMICS. 151
102. The next principle that we shall mention depends on
equations (II), which may be written
Whenever the right-hand sides of these equations are zero the
equations themselves become
_ / d'x d'z\ ^
^ / d'y d'x\ ^
the integrals of which are
^ ( dz dy\ ,
.(1)
Sot
(2)
Now if A^ be the area described in any time by the projection
on the plane of yz of the line joining the origin and the pai'ticle
whose co-ordinates are x, y, z,
^ dt dt dt '
Employing a similar notation for the other co-ordinate planes,
and substituting in equations (2), we have
from which 2 2rn^, = /«,< + C,,
(3)
152 A TREATISE ON DYNAMICS. [CHAP.
or, if we suppose J^, &c., to be the areas described since the instant
from which t is measured,
2'2mA^ = hJ,[ (4)
2l.mA, = kst.)
In these equations or in equations (3), consists the principle
which is called the Principle of the Conservation of Areas.
We may state it in words as follows :
The area described by the projection on the plane o? yz of the
radius vector of any particle, is called the area described by the
particle round the axis of x. Since any fixed line may be taken as
the axis of x, this definition is applicable to any line. In certain
cases the sum of the products of the mass of each particle of the
system, and the area described by that particle round some fixed
line is proportional to the time; whenever this is the case, the
principle of the conservation of areas is said to hold for that system,
round that line.
This may be the case for only one line, or for any line whatever,
or for lines satisfying certain conditions ; let us consider generally
how to determine in what cases it does hold.
We see by the equations from which it is deduced, that it holds
for the axis of x whenever
^{yZ-zY) = 0.
Now in this expression no mutual actions or other internal forces
enter. It is the expression for the resultant moment of the impressed
forces round the axis of x. The principle will therefore hold for any
line round which the moment of the impressed forces is zero.
First, let there be no external forces, or such as would keep the
system in equilibrium in any position, if placed at rest in that
position. In this case whatever line be chosen, the moment of the
forces round that line is zero, and consequently the principle holds
for any line whatever.
Again, let the resultant of the impressed forces always pass
through a fixed point. In this case the moment round any line
through that point is zero, and therefore the principle holds for any
line through that point.
If all the impressed forces which act on the system are parallel,
the principle will hold round any line parallel to their common
direction.
VII.] A TREATISE ON DYNAMICS. 153
If the resultant of the impressed forces always passes through
a fixed line, the principle will hold round that line. It is needless
to enumerate any more particular cases.
103. When the resultant of the impressed forces acting on the
system is either zero, or passes always through a fixed point, we
have
2 2m J, = ^3^
for any directions whatever of the axes ; and in the former case for
any origin, in the latter only when the fixed point is origin.
hi, h.2, hs, are constants introduced by integration, which may
be determined from the circumstances of the motion at some known
instant ; and will be different for different axes of co-ordinates.
If A, fx, V be the direction-cosines of any line passing through the
origin, the sum of the products of the masses and areas described
by them round this line
= \.2'2.mA^ + fx.2l.mAy-i-v.Q'2mA,
= (\hi + fxh2 + vh3)t
where ¥ = hi + /?/ + h^.
Now -T- J -r J -r i^^y he considered as the direction-cosines of
some line, and if be the angle between this line and that round
which the areas are reckoned, the sum in question is h I cos Q.
The greatest value of this is when cos0= 1, which is the case
when
hi K h,
^ = J' "-^ir " = !'
Hence then the line whose direction-cosines are -j^ , ^, -^ is
h h h
such that the sum of the products of the areas round it and the
masses is a maximum.
Since its direction-cosines are independent of t its position is
constant, and from its nature its direction in space is independent of
the axes to which it is referred. We see therefore that in every
such system there exists a line whose direction in space is invariable
154 A TREATISE ON DYNAMICS. [cHAP.
and can be determined at any instant;, from the circumstances of the
system at that instant.
A plane perpendicular to this line is called the " invariable plane/'
and the line itself is called the " invariable axis."
These results agree with those which were determined previously
in a particular case.
The sum of these products round any line inclined at an angle 6
to this is ?it cos ; and therefore becomes zero for any of the lines at
right angles to this one.
Many other curious properties of the motion connected with this
line might be deduced from these equations, these, however, are the
most important.
104. When a system is in equilibrium under the action of forces
X, F, Z, &c. "We have by the principle of virtual velocities
^{Xlx+Yly + Zlz)=0.
We have therefore by D'Alembert's principle for a system in motion
We will make a few remarks on this equation.
No internal forces appear in it which arise from invariable
geometrical connexions of the parts, that is, which arise from con-
nexions of the parts which are geometrically the same before and
after the iiidefinitely small displacement which is supposed to take
place. Such, for instance, as the tensions of flexible and inextensible
strings ; the pressures of smooth surfaces which continue in contact,
or of rough surfaces where the displacement may have been sup-
posed to take place by rolling ; the pressures of fixed points, &c.
Also we may omit any external forces, the displacements of the
points of application of which are such that their virtual velocities
ai*e zero.
Now since in this equation the displacement and therefore the
values of Zx, oi/, &c, are arbitrary, being subject only to the condi-
tion of not violating the geometrical relations referred to above, we
may suppose the displacement to be that which actually takes place
in the indefinitely small time ht, that is, for ex, by, Iz
.^ dx^, dy ^, dz ^
we mav write -7- d ^ , -~-ot, -y.ct,
dt at dt
VII.] A TREATISE ON DYNAMICS. 155
provided the geometrical relations of the system at the beginning
and end of the time ^t are the same. The equation thus becomes
_ fcPx dx d'y dy d^z dz\^^ -f^dx ^dy rydz\^^
the integral of which is
Hit) * (Mj - (s)} - ^ ^ ^ ^^ ('^''^ - ^''^ ^ ""^-
or 27W v»= C + 2 /2 (Xdx + Ydy + Zdz).
Now in this equation no forces appear which are either internal
forces such as we have already mentioned, or external forces such
that their virtual velocities arising from the actual displacement are
zero. Such forces are the tensions of inextensible strings attached to
fixed points ; pressures of smooth fixed surfaces ; pressures of fixed
points against smooth surfaces, &c., where the actual displacement
of the point of application of the force is perpendicular to the direc-
tion of the force ; and also forces where there is no displacement of
the point of application of the force, as when a body moves about a
fixed point or axis ; or rolls without sliding on a rough surface, &c.
We may remark that the internal forces do not enter into the
equation, because if they did for each force there would be an equal
force whose virtual velocity would have a contrary sign, so that they
would enter in pairs which would destroy each other ; and that the
external forces which we have mentioned do not enter because the
virtual velocity of each is zero.
The product of the mass of a particle and the square of the
absolute velocity with which it is moving is called the vis viva of
the particle.
Hence, then, ^mv" is the vis viva of the whole system, and
27«y= = C + 2/2 {Xdx + Ydtj + Zdz)
is an equation which gives the vis viva of the whole system in any
position, whenever the integration on the right-hand side can be
performed. This is easily done in most of the cases that will occur
in practice.
In order that the integration may be possible, the expression
2 {Xdx + Ydy + Zdz)
must be a perfect differential of some function of the co-ordinates of
the different particles.
This will be the case whenever the forces X, F, Z, &c. are
constant, or arise from the attractions of fixed centers, or from the
156 A TREATISE ON DYNAMICS. [cHAP.
mutual attractions of the different particles of the system, with forces
varying as some powers of the distance. It will not be the case
when the forces arise from the attractions of moveable centers not
forming part of the system, or when the forces are functions of the
velocities, as when the motion takes place in a resisting medium, &c.
Let it be a perfect differential, and let its integral be
Zi, a^s, t/s, Sg. . .),
then ^mv" = C + 2(p (or,, t/^, 0, . . .)
Let Oj, bi, Cj. . . be the values of .Tj, ?/„ ^,. . . when the velocities
of the different particles are F^, F^.. . then
Stwd^ - Stw F' = 20 (x„ y,...)-2
' (5)
Again, "'{^^-y^^.| + ^W^=0.
From which, integrating and changing to polar co-ordinates,
we have
(mr' + Mk')j^ = C,
at the commencement of the motion
(7na^ + Mk')o>=C;
de ma' + Mk"
•■• "".57.
dt mr" + Mk
■2 0)
(6)
11—2
164 A TREATISE ON DYNAMICS. [cHAP.
J a
Eliminating -y- from equations (5) and (6), we have
fdr
\dt
Hence the velocity along the tube and the angular velocity cf
the tube are known for any position of the particle, and depend
only on its distance from A.
We cannot integrate these equations in finite terms so as to
obtain the position in terms of the time.
108. Hitherto the effective force on any particle forming part
of a system has been resolved in the directions of three rectangular
axes fixed in space. When the motion of the particle is in one
plane it is often convenient to employ polar co-ordinates, and the
resolved parts of the effective force in the direction of the radius
vector, and perpendicular to that direction.
The equations of motion of a single particle acted on by a force
P along the radius vector from the pole, and a force T perpendicularly
to that direction are
dr de d-6 „,
^'''Ttdt^''''dt^-^'
Hence then from the definition of effective force if r and 6 are
the polar co-ordinates of a particle m forming part of a system,
the effective force along the radius vector is
and that in a direction perpendicular to it is
^ dr de d'0
"-'"dtdt'-'"'d?-
Another mode of resolving the effective force is also useful,
namely, in the directions of the tangent and normal of the path of
the particle. These resolved parts are readily obtained as follows.
The parts parallel to the axes of x and y are »j -r^ and m -~„ .
^ ^ -^ dt- df
Hence the part in direction of the tangent
^ d^x dx d^y dy
dt^ ds dt- ds
~^d^\dt^'dt'^d? dt
d's
Til.] A TREATISE ON DYNAMICS. 165
And the part along the uoimal
d'x dy J-y dx
dl ds dl ds
dj^ ((£21 dx d'x dy\
~"' ds W dt df dtj'
Let p be the radius of curvature of the path of the particle, v its
velocity
" d^ dx d'x drf^
dJ 'dl~'dJ~ dt
and the expression for the force along the normal becomes
vxv^ m (ds'
T "'■ p U
When the particle moves in a circle 7: = ^ and the two pairs of
expressions coincide Avith each other.
These expressions for the effective force can in any case be used
instead of those referred to rectangular co-ordinates.
109. These modes of resolving the effective force, afford an
explanation of a term, which, originally founded in error, has a ten-
dency to convey an erroneous impression, viz.. Centrifugal Force.
If a smooth straight tube is moved with uniform angular velocity
in a horizontal plane about one extremity, and a particle is placed in
it, the particle will commence moving from the fixed end of the tube,
and will move with accelerated motion till it ultimately flies out of
the tube.
If a particle be fastened by a string to a fixed point on a smooth
horizontal table, and then be projected so as to move round this
point, it will cause a tension of the string, depending on the mass of
the particle and the rapidity with which it moves.
From these and similar cases, it was concluded that bodies moving
round a center have a tendency to fly off from that center, and in
consequence exercise a force to which the name centrifugal force was
given: and its magnitude Avas found to be mro)', where 7* is the
distance from the fixed center, and w the angular velocity with which
the body is moving round that center.
166 A TREATISE ON DYNAMICS. [cHAP.
The true explanation is found in the following method of view-
ing it.
In the former of the preceding examples, let R be the pressure
which the side of the tube exerts on the particle. This is the only
impressed force on it.
Then reversing the effective forces on it, we obtain the equations
of motion
„ ^ dr dO d'0 ^
^-^''TtTt-''''df=^'
From equation (1)
mj^ = mr.\
which shews that the motion along the tube will be the same as if
the tube were at rest, and the particle were acted on by a force mrta-
along it.
Again, in the other case, let T be the tension of the string.
Applying the effective forces reversed we have
0,
: and from the latter
Hence the first equation becomes
T^mru>\
That is, the tension of the string is the same as it would be if the
particle were at rest and acted on by a force mrio^.
A similar result will follow in any other case in which the
equations are applied to the ntiotion of a particle, either alone or
forming part of a system. So that though no such force as centri-
fugal force really exists, yet when a system in motion is supposed at
rest, and the effective forces are applied in the opposite directions to
balance the impressed forces, a portion of one of these effective forces
will correspond in direction and magnitude to the fabulous centri-
fugal force.
-T-
d'r fddV ,
-'''7it^-''"'Kdt) ='
2
dr dd d'd ^
Now
ris
constant, therefore -j- =
equation
dO
is constant.
VII.] A TREATISE ON DYNAMICS. 107
Thus along the radius vectoi*, the term '"''(y:) in the expression
(Pr fdey J . ,, , .1, 1. , . mv'
-7M-7-5 +wirl — j, and m the normal the whole expression
correspond to the centrifugal forces in those directions respectively.
The name " Centrifugal force/' is very objectionable, from the
plausibihty of the reasoning by which the idea is supported. But it
is found convenient in several classes of problems to have a name for
the portion of the reversed effective force represented by inru^, and
with this meaning the term centrifugal force is retained.
168 A TREATISE ON DYNAMICS. [cHAP.
CHAPTEE VIII.
110. In discussing the elementary principles, it was remarked
that there are cases in which the whole effect produced by a force is
known J when the magnitude of the force and the time during which
it acts are not known ; that is, when we know the definite integral
/ ' Pdt without knowing either P or /o— /j. The name "impulse"
was given to this definite integral, and the force P was called an
impulsive force.
Let us now consider under what circumstances forces of this sort
are called into action.
We have used the term " rigid body/' and have investigated the
motions of rigid bodies in several cases, and have given the equations
for determining that motion in all cases. Now the idea which we
have attached to the term rigid is this : a body is said to be rigid
when all the points preserve the same invariable mutual distances,
whatever the forces which act on the different points of the body.
This idea of rigidity is suggested by the first touch of external
objects ; though a slight examination is sufficient to convince us that
it is not strictly applicable to anj' of them.
Different substances require different degrees of force to produce
any sensible change in their form, some require very great force to
do so. This approximate rigidity enables us to form the idea of the
perfect quality, even though there is no substance in existence
endued with that quality ; and a great many substances are so nearly
rigid as to convince us that calculations effected on the hypothesis
of their perfect rigidity cannot be affected with sensible errors on
account of the deviations from it.
In the branch of the subject however, which is now under con-
sideration, this deviation from rigidity forms a principal feature of
the problem. When pressure is exerted on a body by any means,
the body slightly changes its form, or is compressed ; also when the
pressure ceases to act, the body wholly or partially recovers its
original form. When a ball in motion strikes another ball at rest, a
pressure immediately takes place between the two balls, each ball
pressing on the other, and so flattening it at the point of contact.
This pressure is gradually communicated from one particle to
Vm.] A TREATISE ON DYNAMIC'S. 1G9
another of each of the balls^, so that velocity is communicated to the
one which was at rest, and the velocity of that which was in motion
is lessened ; and this action continues as long as the hinder ball
moves faster than the other. When the balls have acquired the
same velocity, the flattening of the two is at its greatest, and if the
balls had no tendency to recover their original forms the pressure
between them would cease, and the two would move on together
with the same velocity. Most substances however have a tendency
to recover their original forms, and in consequence of this the action
between the two continues till they cease to touch each other ; the
one which was originally at rest having the greater velocity of the
two.
Now all this that we have been describing takes place in a very
short time indeed : so short as to be considered instantaneous with-
out any sensible error.
If our knowledge of the structure of matter were perfect, and if
also our mathematical analysis were perfect, we should be able to
investigate, from the principles already laid down, the exact nature
and magnitude of the action which takes place between the two
bodies, and also the degree and manner in which a change of form
takes place. As it is, however, we are obliged to rest on experiment
for the facts which we have stated, and for the laws which we are
going to state.
Let /i be the time when the first contact takes place, /„ the time
when the two bodies are moving with the same velocity, and ^3 the
time when they cease to touch at all. Then the whole action which
takes place during compression, as it is termed, will be represented
by I Pelt and the whole action during restitution (that is, during
the recovery of the original form) by / Pdl.
The experimental law which forms the basis of our calculations
is that
rh (t-2
\ Pdl = e Pdl:
where e is a multiplier less than unity, which is constant for all
values of P, so long as the substances between which the impact
takes place remain unchanged.
The less or the greater the change of form that takes place for a
given value of I Pdt, the harder or softer the body is said to be.
170
A TREATISE ON DYNAMICS.
[chap.
The greater or less e is, the greater or less is said to be the
" elasticity" of the bodies.
If e = 1, the elasticity is said to be perfect.
e is called the coefficient of elasticity, or sometimes the elasticity.
The definite integrals are generally written R and B', so that
i2'= eR.
R and jR' are called the "impulses" during compression and
restitution, and R + R', or {\+e)R is called the whole impulse.
The law has been enunciated and the action between the bodies
explained in a particular case only ; it is however applicable, with a
very slight extension, to all cases.
111. Since an impulsive force is only a force of great intensity,
D'Alembert's principle is applicable to it, as to all other forces.
If we take the case of one rigid body, that is, rigid in the modified
sense in which we have explained the term, the equations of motion
will be the same as before, and may, as in that case, be reduced to
M
dt'
2X,
And 2
M^=2F,
de ^^'
{,(.-.S)-.(r-4|)}=o.|
{.(x-.$)-.(.-.g)}.0.
{.(r-.g)-,(x-,
'5f)}-.|
(III)
(IV)
Now in this case, 2X, 2F and 2Z include all the forces which
act on the body, impulsive or otherwise ; and if we take the definite
integrals of the first three equations between the times /j and tz, the
beginning and end of impact, we have
m(m,-«o = 2/ xd/,
M(v,-v,)=l. j^Ydi,
M{w„-Wi)-
Zdt,
YIII.] A TREATISE ON DYNAMICS. 171
or using A'', Y', Z' to denote the " Impulses,"
M{y,-v,^ = ^Y'\ (VI)
The limits t^ and ^ must be taken so as to include the whole of
every impulse, and consequently may include a longer time than is
occupied by some of them. This however will produce no error;
for suppose one of the impulses as X' to commence at t( and end at
^2' both of Avhich times are between t^ and t^.
Then since
{h ffi' r^i ['2
Xdt = Xdt + Xdt + Xdt,
•'/, hi J ti Jt.j
and since X is zero between /j and //, and also between // and to,
we
have Xdt= Xdt = X'
,•(■>
Also since 1 Xdt has a sensible magnitude in consequence of X
being very large ; this integral will not have a sensible magnitude
for those forces which are not impulsive ; in other words, these forces
will not produce a sensible effect on the velocity during the short
time of impact, and therefore do not appear in equations (VI).
In the same way we may integrate equations (IV) between /, and
t^; and since the interval to — t^ is extremely short x, y, z, &c., may
be considered constant during that interval, so that we have
'^{y\_Z'-m{io'-wY\-z [F'- m (?;'- v)]} = 0, "j
l.{z\X' -in{u' -xi)-\- x{Z' -m{w' -w)-^ = Q, 1 . . . (VII)
2 {x \Y' - m iy'- vY\ ~y \_X' - m (?/ - «)]} = 0. i
In these equations x, y, z are referred to the center of gravity as
dx
origin, ii is the value of-j- before impact, and u' its value after
impact, and similarly for the other letters.
(VI) and (VII) give the state of motion of the body immediately
after impact, without any integration beyond that expressed by 2.
When the motion of every particle before and after impact is
parallel to one plane, equations (VII) are reduced to the one equation
l.m{x{v'-v)-y{u'-u)\ = ^{xY' -yX'),
172 A TREATISE ON DYNAMICS. [cHAP.
Let 0), and w. be the angular velocities of the body before and
after impact, then
11 =- t/w^, u' = — y w.^, v — xm^^ v' = xw.2.
Let L' be the resultant moment of the impulses, and let r^=x°+i/^,
then this equation becomes
1.1117'- (uy^— w,) -L'.
And the three equations in this case are
M(v,-v,) ^1Y', [ (VIII)
112. If the system consists of more than one body, the equations
must be written down for each body separately, and the unknown
forces which enter the equations, must be determined by the relations
existing between the velocities of the points at which impact takes
place. These will be different according as the bodies are elastic or
inelastic, and as their surfaces are rough or smooth.
When the bodies are inelastic and smooth, the whole impulse will
be in the direction of the normal to the surface at the point of con-
tact, and must be determined from the condition that the resolved
parts in the direction of this normal of the velocities of the points in
contact must be equal. If the surfaces are rough, the whole velocities
of the two points will be equal and in the same direction.
When the bodies are elastic, and the coefficient of elasticity is
given, the normal impulse may be found from the consideration that
the normal velocities of the points in contact are the same at the end
of compression, and that the impulse during restitution is in a con-
stant ratio to the impulse during compression : when the surfaces
are rough, the velocities of the points in contact resolved along the
tangent plane Avill be equal at the conclusion of the impact, while
the resolved parts along the normal will be equal only at the end of
compression.
If the surfaces are partially rough, as is always the case in nature,
let /A be the coefficient of friction and R the normal portion of the
impulsive force ; then the tangential portion of the impulsive force
cannot exceed fxR, and therefore the whole tangential impulse cannot
exceed/ ixRdt.
Little is known experimentally of the value of ^x for forces so
great as to be impulsive ; so far as the experiments go, however,
Viri.] A TREATISE ON DYNAMICS. 173
they indicate a diminution of m for large values of the normal pres-
sure. Considering /,t to be approximately constant, we have
rt-2 rh
Rdl = nR',
where R' is the normal impulse. This is the greatest amount of
tangential impulse that can take place for a given amount of normal
impulse ; all will not necessarily be exerted in any particular instance.
The preceding remarks will be better understood by a consideration
of the following examples.
113. Let a ball whose mass is m^ moving with a velocity t\ over-
take a ball Avhose mass is vk moving with a velocity v^, and let e be
the coefficient of elasticity.
Let R be the impulse during compression, u the common velocity
of the two balls after compression, ?/, and j/g their velocities after
impact : then from equations (VIII),
7W, {u -?;,)=— R,
vi^ (m - t'g) = R,
from which Ave have u = =-^ ,
nil + W'a
Vli + 7»2
the whole impulse is (I + e) R, and the velocities after impact are
given by the equations
7«, (?/i -vi) = -(l + e) -^ '^ ,
^ ^ ^ ^ nii + mg ^
W2 (^2 - v'2) = (l+e) -^-^^' ^-'- .
If the balls are equal and the elasticity perfect, these equations
become Uy-Vi, and xi2=Vx.
114. Again, let a sphere of radius a revolving with the angular
velocity w about a horizontal axis fall vertically on a horizontal
plane ; and let /a and e be the coefficients of friction and elasticity
between the sphere and the plane.
Suppose, to fix the ideas, that the axis of x is horizontal, and the
plane of xt/ perpendicular to the axis of rotation.
Let M be the mass of the sphere, V the velocity of its center of
gravity before impact ; v^. , x\ the resolved parts of the velocity, and
174 A TREATISE ON DYNAMICS. [CHAP.
to' the angular velocity after compression: %i^, iiy and w the same
quantities when the impact is concluded ; B. and F the normal and
tangential impulses during compression ; then
Mv, = F (1)
M(vy+F) = R (2)
Mk^{co'-w) = Fa (3)
and the condition that the point of contact is at rest gives
v^ + a)'a = 0, Vy=.0 (4)
we also have the condition that F cannot exceed n R.
From equations (2) and (4), we have
R = MV,
also from (1), (3) and (4), we have
F Fa' ^
Ma Ir a,
~ a^ + k- '
provided this is not greater than MF. Let this be the case, then if
F' is the tangential impulse during restitution
Mu^=F + F' (5)
M(Uy+V) = (l + e)R (6)
iWF(w-(o) = (F + F')a (7)
with the condition Uj. + -aa^O (8)
Equation (6) gives Uy = eV,
also from (5), (7), and (8)
and therefore F' = 0, and from (5) and (7)
and the initial motion after impact is completely determined.
We may remark here that the whole change of angular velocity
took place during compression ; if — — ^ had been greater than V
but less 'than {l+e)V, part of this change would have taken place
during restitution, and the final result would have been the same.
VIII.] A TREATISE ON DYNAMICS. 175
Let J — p be greater than (l+e) F; in this case instead of the
equation u^ + wa = 0, we have
F+F' = -fx{l + e)MF;
.'. n^ = - fx(\ +c) V and w = w - -^ — r— ,
and Uy = eV as before.
115. As a conchiding example we will take the system described
in Art. 106, and preserve the same notation.
Let the cylinder fall from rest so as to strike the plane along a
horizontal line.
Let F be the velocity of the cylinder before impact; v^, Vy the
resolved parts of the velocity of its center of gravity after compres-
sion, 0) its angular velocity, v the velocity of the plane at the same
time; it^, %iy, tn and u the same quantities after restitution.
Then we have
Mv^ = F cosa — R sin a,
M(Vy+ F) = Fsma + R cos a,
Mk''