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ARTES
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1837
SCIENTIA
VERITAS
OF THE
UNIVERSITY OF MICHIGAN
CALURIBUS
THEBOR
SIQUERIS FENINSULAM AMŒNAM
CIRCUMSPICE
PROF.
THE GIFT OF
ALEXANDER ZIWET
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Alexandu Ziwek

A TREATISE
ON
THE MOTION OF A RIGID BODY.
BY
Nathaniel
WILLIAM N. GRIFFIN, B.D.,
W
FELLOW OF ST JOHN'S COLLEGE.
CAMBRIDGE:
PRINTED AT THE UNIVERSITY PRESS
FOR DEIGHTONS;
SOLD BY
SIMPKIN, MARSHALL AND CO., AND GEORGE BELL, LONDON.
M.DCCC.XLVII.
Prof. Alex. Ziwet
1-9-192-3
7-33
IT has been the wish of the compiler of the following
work to exhibit in a brief and compact form the leading
principles of the Motion of a Rigid Body, under an im-
pression that a syllabus of this kind is best adapted for
the combination of oral teaching and reference to books
which is the present practice of the University. While
in this view it appears convenient to present the propo-
sitions of the subject apart from their applications, it will
not be supposed that extensive practice in the solution
of problems can be dispensed with by any who would
understand the theory of this or any other portion of
Mechanics. The author has not however thought it
requisite to include in the present book any specimens
of the use of its principles, from an expectation that
Mr Walton's collection will be in the hands of every
reader and supply him with guidance and exercise.
A few pages of examples have been added as supple-
mentary to those already accessible. Most of them are
aimed at particular difficulties and misconceptions to
which readers have in experience been found liable, with
a belief that questions of this kind are most valuable as
a means of starting reflection, whereby the student is
enabled to detect and rectify his misapprehensions.
ST JOHN'S COLLEGE,
October, 1847.
414595
ERRATUM.
Page 61, line 7 from bottom for rotation read translation.
SECTION I.
GEOMETRICAL PROPERTIES OF A RIGID BODY.
1.
DEF. A body of finite dimensions in which all points
retain an invariable position with respect to one another is
called a rigid body.
A rigid body is thus distinguished from bodies flexible,
compressible, extensible and fluid; because in these, assigned
points may have different relative positions while the con-
tinuity of the substance is preserved, whereas the particles of
a rigid body may occupy different positions in space but still
preserve the same relative positions among one another.
Without asserting that any substance in nature strictly
fulfils this definition of a rigid body, mathematical calculations
proceed upon this conception of it with a view to distinctness
and simplicity. When the results of these calculations are
employed in practice, they require small corrections, by reason
of the substances to which they are applied not possessing
completely the properties of the imaginary body on which we
have reasoned. If however the forces in action are not
adequate to produce an appreciable change of form, the body
subject to them may be regarded during their action as per-
fectly rigid and incapable of any change in shape or con-
sistence. Iron, for instance, will sensibly depart from its
primitive form when a force sufficiently great is applied to it,
and is found to be flexible, extensible, and compressible, but
so long as the forces acting upon it are so small as to be inca-
pable of altering its shape to a measurable extent, it may be
regarded as rigid according to the definition.
2. The definition thus given of a rigid body is only one
out of many instances where, in the application of Analysis to
Physical Science, hypotheses are needful to simplify the pro-
1
2
GEOMETRICAL PROPERTIES
!
1?
blems which practice presents. Strings, for instance, are con-
sidered weightless and perfectly flexible, the surfaces of bodies
smooth, planes accordant with their geometrical definition,
none of which circumstances are ever perfectly realized. The
results of calculation therefore on these or similar hypotheses
are first approximations to the solution of the questions which
really arise, requiring corrections to bring them to accuracy
whenever the conditions on which they rest are appreciably
violated. This manner of proceeding, which in any case would
perhaps have the recommendation of simplicity, is unavoidable
while Analysis is imperfect. Problems thus reduced and
simplified may be within the reach of our instruments of calcu-
lation, when they would be beyond it if they retain all the
complexity which their natural state presents.
3. A body of finite size may be conceived to consist of an
infinite number of elementary portions each of inappreciable
volume but finite mass, and exhibited in position by reference
to co-ordinate axes, so that the place of any one of these
elements may be represented by the co-ordinates of any point
of it. Such elements are sometimes termed the particles of
the body.
OBS.
The axes of co-ordinates are to be considered rect-
angular when the contrary is not expressed.
4 Moment of Inertia.
DEF. The moment of inertia of a rigid body about a
given axis is the sum of the products of the mass of each
particle of the body and the square of the distance of that
particle from the axis.
Thus if m be the mass of a particle of a rigid body, r its
distance from a fixed straight line, Σ(mr) is the moment of
inertia of the body about that line, the summation extending
throughout the whole body. If p be the density at the point
x, y, z of the body, fffpr is the moment of inertia of the
body with respect to the axis from which is measured, the
integration being performed throughout the body. The defi-
OF A RIGID BODY.
3
nition given above is to be regarded as a statement in words of
this analytical expression, which has required nomenclature by
the frequency of its occurrence in dynamical investigations.
The computation of the moment of inertia of a given
body about a given axis thus amounts to the evaluation of a
definite integral which is equivalent to the summation of
elements, but it is often facilitated by some of the following
properties.
5. To connect the moment of inertia of a rigid body
about any axis with its moment of inertia about a parallel axis
through the centre of gravity.
Let P be a particle whose mass is m (fig. 1), A, G the
traces of two straight lines on a plane through P perpendi-
cular to each, the latter of these straight lines being drawn
through the centre of gravity of the rigid body of which P is
a particle. Join PA, PG, and draw PN perpendicular to
AG or AG produced.
Then PA PG² + AG2+2 AG. GN,
= AG²
and
(m. GN) = 0,
Σ (m . PA²) = Σ (m . PG²) + AG³. Σ (m),
or the moment of inertia of the body about the axis A is
greater than that about the axis G by the product of the mass
of the body and the square of the distance between the axes.
6. The moment of inertia of a body of mass M about an
axis through its centre of gravity is usually denoted by Mk,
k being dependent on the direction of the axis. The moment
of inertia of the body about a parallel axis at distance h will
then be M (k² + h²). The length h+k is sometimes
called the radius of gyration of the body about the axis con-
sidered, and k similarly the radius of gyration about a parallel
axis through the centre of gravity.
2
7. If a circular cylinder be described so that the centre
of gravity of the body is in its axis, the moment of inertia of
4
GEOMETRICAL PROPERTIES
the body about all generating lines of this cylinder is the
same.
8. If there be any two parallel axes A and B at distances
a and b from the centre of gravity of the body whose mass
is M,
moment of inertia about A
Ma² = moment about B – Mb”.
Hence when the moment of inertia of a body about any
axis is given, that about any other parallel axis can be deduced
if the position of the centre of gravity be known.
9. DEF. A solid of finite size bounded by two parallel
planes, the distance between which is infinitely less than the
other dimensions of the body, is called a lamina.
The moment of inertia of a lamina about an axis perpen-
dicular to its plane is the sum of its moments of inertia about
any two perpendicular axes in its plane drawn from the point
where the former axis meets the plane.
For if P (fig. 2) be a particle of the lamina of mass m,
PN, PM perpendiculars on two straight lines AB, AC at right
angles to one another and in the plane of the lamina, and if
AP be joined
AP² = PM² + PN;
Σ (m. AP²) Σ (m.PM²) + Σ (m.PN³),
i.e. the moments of inertia about AB and AC are together
equal to the moment of inertia about an axis through A perpen-
dicular to the plane of the lamina.
10. To find the moment of inertia of a rigid body about
a given straight line through the origin of co-ordinates to
which the body is referred.
Let x, y, ≈ be the co-ordinates of a particle of mass m,
X y z the equations to the given straight line,
a
β
Y
where a² + ß² + y² = 1.
OF A RIGID BODY.
5
LO
The distance of the point x, y, z from this line
= {x² + y² + x² - (ax + By + yx)²}}.
= {a²(y²+s²)+ẞ²(x² +≈²) + y² (x² + y²) −2ßyyz-2ayxz-2aßxy}³.
Let A = Σm (y² + s²)
be the moments of inertia of the
B = Σm (x² + *²) | body about the co-ordinate axes,
C = Σm (x² + y²)
Ση(
Q the moment of inertia of the body about the given axis.
Q=Aa²+Bß* +Cy² −2ßy Σ(myz) – 2ay Σ(mxx) – 2aßΣ(mxy).
11. Principal Axes.
DEF. If x, y, z be co-ordinates of a particle of mass m
belonging to a rigid body, and if Σ (myx) = 0, Σ(mxx)=0,
Σ (mxy) = 0, the summations extending throughout the whole
body, the axes of co-ordinates are then called principal axes at
the point which is adopted as origin.
OBS. Principal axes are not defined separately, but in a
system. Wherefore if one of the preceding conditions be ful-
filled it may not thence be concluded that one of the axes of
co-ordinates is a principal axis.
12. Through any point in space there may be drawn at
least one system of principal axes of a given body.
If Q be the moment of inertia of the body about the axis
whose direction cosines are a, ß, y,
Q=Aa²+BB²+Cy² - 2 ẞy Σ(myx) −2 ay Σ(mxx) −2 aßΣ(mxy).
Now this is the equation to a surface of the second order,
where Q-represents the radius vector in the direction ex-
pressed by a, ß, y. But one system at least of co-ordinate
axes, the principal axes of the surface, can be adopted for
which the equation to the surface will not involve the pro-
ducts By, ay, aß; therefore with these co-ordinate axes
Σ(mxy) = 0, Σ(mxx)=0, Σ (mxy) = 0, or the axes are a
system of principal axes.
6
GEOMETRICAL PROPERTIES
Since moments of inertia are necessarily positive quan-
tities, the surface is an ellipsoid.
A demonstration of this proposition without reference to
the properties of the ellipsoid will be found in the Cambridge
Mathematical Journal, Vol. I.
13. If the ellipsoid become a figure of revolution, which
will be the case if the expressions
A +
Σ(mxy).
Σ(myx)
(mxx)
B+
Σ(mxy).Σ(myx)
Σ (mxx)
C +
Σ (mxx). Σ (myx)
Σ(mxy)
be equal and finite, (Hymers' Geometry of three dimensions,
149) then while one principal axis is fixed in direction, any
pair of rectangular axes perpendicular to it form with it a
system of principal axes.
14. If the ellipsoid become a sphere, which will be the
case if
A = B = C, (myx) = 0, Σ (mxx) = 0, Σ (mxx) = 0,
then any system of rectangular axes is a system of principal
axes at the origin.
Hence when any point is adopted for origin, there is always
at least one determinate system of principal axes belonging to
it, and in certain cases an infinite number of such systems.
OBS. The origin of co-ordinates in this proposition is not
necessarily a point in the body or in any way restricted.
Z
15. If the axis of x be a principal axis, (Σ mxx) = 0,
Σ (myx) = 0, for any system of rectangular co-ordinate axes.
For let x'y' designate the co-ordinates of m referred to the
principal axes;
.. Σ(mx'x) = 0, Σ (my'z) = 0.
OF A RIGID BODY.
7
Let x and x make an angle 0,
...
x = x′ cos 0 - y' sin 0,
y
= x′ sin 0 + y'cos ◊ ;
X
(Σ mxx) = Σ (mx'x) . cos 0 − Σ (my'x) . sin 0 = 0,
Σ (myx) = Σ (m x'x) . sin 0 + Σ (myx). cos 0 0.
It does not follow, however, that Σ (mxy) = 0 for every
The truth of this additional equa-
system of principal axes from any
system of axes of x and y.
tion then distinguishes the
other.
16. The determination of the principal axes of a given
body at a proposed point is identical with finding the position
of the axes of figure of an ellipsoid whose general equation
referred to its centre is presented. Since, however, when
these axes are employed with practical advantage, one of them
can in most of such cases be assigned by inspection from the
form of the body, the following proposition is then useful
to complete the determination of the system.
17. Given one principal axis of a body at a proposed
point, to find the other two.
Let the body be referred to a system of rectangular axes
at the proposed point so that the given principal axis is the
axis of %. The required principal axes lie in the plane xy.
Let them make an angle with the axes of x and y respect-
ively, and let ', y' be the co-ordinates of a particle m referred
to them.
Then
x' = x cos 0 + y sin 0,
y' = x sin ✪ + y cos 0;
.. 0 = Σ(mx'y') = Σ (mxy). cos 20 + ¦ Σm (y² − x²). sin 20.
.. tan 20
=
2Σ (mxy)
Σm (x² - y²)
1
00
GEOMETRICAL PROPERTIES
Also Σ(mx'x') = 0, Σ(my'x') = 0; otherwise no system
of principal axes would exist whereof the axis of ≈ is one. (15).
The values of which the preceding equation assigns give
the same pair of principal axes presented in a different order,
unless the expression is indeterminate, when (mxy) = 0,
Σ (mx²) = Σ (my), and then 0 = Σ (mx'y') is satisfied inde-
pendently of 0, so that any pair of axes in the plane ay forms
a system of principal axes with z.
OBS. The angle is measured from the positive direction
of the axis of x towards the corresponding portion of the axis
of y.
18.
Σ
Hence whenever the form of the body allows an axis
of ≈ to be taken for which (mxx) = 0, Σ(myx) = 0, what-
ever be the axes of x and y, then the other two principal axes
at the origin can at once be assigned.
19. A straight line perpendicular to the plane of a lamina
will be a principal axis at the point where it meets the lamina.
20. The axis of a solid of revolution bounded by planes
perpendicular to this axis is a principal axis at any point in
its length.
21. DEF. The moments of inertia of a body about its
principal axes at any point are called its principal moments at
that point.
If the principal axes be co-ordinate axes, A, B, C the
principal moments, and Q the moment of inertia about any
axis through the origin whose direction cosines are a, ß, y,
Q = A c° + B3* +C
2
Hence when the moments of inertia of a body about its
principal axes at any point are known, its moment of inertia
about any axis through that point results.
22. If an ellipsoid be constructed about the origin as
centre whose principal semi-axes are A-1, B-1, and C-¹, a
radius vector of this ellipsoid is the inverse square root of the
OF A RIGID BODY.
moment of inertia of the body about the direction of that
radius vector. This is called the central ellipsoid of the body
at the origin adopted.
23. Let the principal moments be unequal.
Since Q = Aa + BB + C,
and
.ˆ.
1 = a² + ß² + y²;
0 = (A − Q) a² + (B − Q) ẞ² + (C – Q) y³.
A
Hence of the quantities 4- Q, BQ, C- Q, one must
always have a contrary sign to the other two, or Q is between
the greatest and least of the magnitudes A, B, C.
Hence of the moments of inertia of a body about axes
through a given point, the moment of inertia about one of the
principal axes is greatest and about another is least.
Conversely, axes about which the moment of inertia is
greatest or least are principal axes.
The axes, about which the moment of inertia has a given
magnitude, lie on a cone of the second order with the origin as
vertex. (Gregory's Solid Geometry, 80.)
24.
If two of the principal moments are equal, if A = B,
for instance,
Q = A (1 − y³) + Cy² ;
therefore Q depends on y only, or the moment of inertia is the
same about all axes which lie on a right circular cone about
that principal axis with respect to which the moment of inertia
is not equal to those about the other two.
25. If all the principal moments of inertia are equal, the
moment of inertia of the body about all axes through the
origin is the same.
2
SECTION II.
D'ALEMBERT'S PRINCIPLE.
26. In examining the motion of a body of finite size we
have first to consider the geometrical character of its motion,
and thus to fix upon certain elements of that motion, from the
determination of which the position, direction, and velocity of
any proposed point of the body will at any instant be known.
These elements of the motion will be selected according to the
particular circumstances of the body, and the problem of
determining the body's motion will consist in so connecting
these with the given forces which are in action upon it, that by
a knowledge of the latter and the state of the body at some
epoch, the former can at once be assigned.
27. Now the causes which influence the motion of each
particle of a rigid body are, 1º. forces extraneous to the body,
and 2º. forces resulting from its connection with the other
particles of the body, and called in consequence molecular.
Of these latter we know neither the law nor the intensity, and
therefore the computation of the motion of the body by calcu-
lating the motions of its individual particles is impossible. This
difficulty is eluded by the principle which follows, called
D'Alembert's principle.
28. The dynamical circumstances of the motion of a
particle shall first be reviewed.
I.
Let the forces acting on the particle be finite, or such
as can produce no change in its state except in an appreciable
time.
1.
If x, y, z be the rectangular co-ordinates of the
D'ALEMBERT'S PRINCIPLE.
11
particle's position at time t from a fixed epoch, the accelerating
forces on it in directions of these co-ordinates are
d²x
dt2
dy
d² &
d t²
dt2
and if m be the mass of the particle, the moving forces upon
it in the same directions are
m
d² x
dť
,
m
day
dt
d t²
d²
z
m
dt2
The directions of the forces are those in which the co-ordi-
nates are measured, independently of the direction of the
particle's motion.
2. If v be the velocity of the particle at time t when it
has described along its path a lengths from a fixed point
thereof, and p the radius of absolute curvature of its path at
its present position, the accelerating forces upon it are
v2
p
d² s
d t²
in direction of its motion,
in the normal and towards the centre of curvature.
These multiplied by m give the moving forces in the same
directions.
3.
If r, be the polar co-ordinates of a particle moving
in one plane, the accelerating forces upon it resolved in and
perpendicular to the radius vector are
dr
dt2
2
ᏧᎾ
d Ꮎ
dr do
dt
and r
dt
+ 2
d t d t'
the former estimated from the pole, the latter in the direction
in which is measured. These, as before, may be converted
into the moving forces in the same directions.
Adopting then any one of these three modes of resolution,
we have analytical expressions for the accelerating and the
12
D'ALEMBERT'S PRINCIPLE.
moving forces which must act on a particle in motion, suppos-
ing them finite.
II. If the forces acting on the particle be impulsive, or
such as can produce a finite alteration in its state in an inap-
preciable time, then if the particle has velocities u, v, w, in
directions of the three rectangular co-ordinate axes, and if these
suddenly become u', v', w', the force upon it consists of
impulsive forces in directions of the same axes whose numerical
representations are
u, v' v, w' – w,
or m (u' – u), m (v′ – v), m (w' – w),
according as they are estimated with respect to the velocity
only, or to the momentum which they can produce, i.e. as
accelerating or moving impulsive forces.
29. DEF.
Extraneous forces acting on a rigid body are
called impressed forces.
DEF. The force under which a particle of a body if free
might move as it really moves is called the effective force on
that particle.
Of this force the analytical representations have just been
given (28).
30.
D'Alembert's Principle.
If the effective moving forces belonging to the several
particles of a body be applied to them respectively in directions
contrary to the directions of the same forces, the conditions of
statical equilibrium of the body are satisfied if it be under the
action of these supposed forces and the impressed forces which
are really applied to it.
31. Ex. A wheel is revolving uniformly about its centre.
Let v be the velocity of a particle of it at distance from the
centre. Then since the particle is describing a circle with uniform
velocity v, the force which if it were free must act upon it that
D'ALEMBERT'S PRINCIPLE.
13
v2
it might have this motion is towards the centre of the wheel.
This then is the effective accelerating force on the particle in
question regarded as a molecule of the rigid body. In this
instance the effective force on each particle is wholly perpendi-
cular to the direction of its motion.
Ex. 2.
If a body is moving in such a manner that the
direction of each particle remains unchanged, the effective
force on each particle is in direction of its motion. If the
motion of each particle remains constant as to velocity as well
as direction, the effective force on each particle is zero.
Ex. 3.
If a wheel be sliding uniformly along its axis and
also revolving uniformly about it, the effective force on a
འདུ°
molecule is acting perpendicularly to the axis, r being the
r
particle's distance from the axis, and v the part of its velocity
perpendicular to the axis.
Ex. 4. If a body have its motion so altered by impulses
that the velocity v of a particle m is suddenly changed to v',
its direction being unaltered, the effective impulsive force on
the particle is v' – v or m (v' – v) the force being estimated as
an accelerating or moving force respectively.
32. Thus the principle of D'Alembert is the means of
evading the unknown molecular forces which unite the parts of
a rigid body. It may thus be illustrated. If P (fig. 3) be
a molecule of a rigid body, I the impressed force upon it, and
M the molecular force arising from the connection of P with
contiguous parts of the body, then E the resultant of these
will be the effective force on P. If E' be equal and opposite
to E, the assertion of D'Alembert's principle is that the system
of forces I and the system of forces E' taken throughout the
rigid body satisfy the conditions of statical equilibrium. The
forces I and E' will not generally balance one another on any
assigned particle, but when the whole body is contemplated,
the aggregate of the forces M may be removed.
14
D'ALEMBERT'S PRINCIPLE.
33. The proof of the principle of D'Alembert will rest,
like the laws of motion of a molecule, on the verification by ob-
servation of nature, of results obtained from calculations based
upon it. The continued accordance of complex results as ob-
served and as predicted by calculation is our security that the
principle on which our calculations have proceeded cannot be
untrue.
34. When the body degenerates into a particle, it will
be seen that the three laws of a particle's motion are included
in this principle. It is an extension of those laws, and like
them depends for its certainty on the accordance of calcula-
tions in which it is assumed with observations of nature.
35. D'Alembert's principle has been stated without re-
spect to the manner in the which particles of the body under
consideration are connected or act on one another. Therefore
although at present the principle is only employed in ex-
amining the motion of a rigid body, it must be recollected to
be no less true of an extensible, elastic or fluid substance.
SECTION III.
THE MOTION OF A RIGID BODY ABOUT A FIXED
AXIS.
36.
GEOMETRICAL nature of the motion of a rigid body
about a fixed axis.
When a body revolves about a fixed axis, every point of
it describes a circle about that axis in a plane perpendicular to
the axis. The position of the body at any instant is designated
by the angle which a plane fixed in the body and parallel to
the axis of rotation makes with a plane fixed in space and also
parallel to the same axis.
37. The angular velocity of the body is the rate at
which this angle changes. Its amount at any instant is
measured, 1º. when it is uniform, by the angle described in
any unit of time, 2º. when it is variable, by the angle which
would be described in an unit of time if the body retained
through that unit the angular motion which it has at the
instant considered.
38.
Let o be the angular velocity.
(1) If the angular velocity be constant we mean that the
body revolves through an angle w in the unit of time, and the
angle is described equably; therefore w is the ratio of the
angle described in any time to that time.
(2) If the angular velocity be variable, let ◊ be the
angle described by the body at time t, de the angle described
thence in time St at the end of which the angular velocity
becomes w + dw. Then if St be sufficiently small
Se lies between
ω.δι,
(w + Sw)st;
16
MOTION OF A RIGID BODY
so
W
lies between
δε
w + dw
+ow
.. in the limit when Sw vanishes
se
de
Ꮎ
W limit
St
dt
OBS. The direction in which
is measured is fixed in
space and independent of the direction in which the body is
revolving. The body's motion will be in or contrary to the
do
dt
direction in which is measured according as is positive
or negative.
W
39. If w be the angular velocity of the body, the velocity
of a point at distancer from the axis is wr in a direction
perpendicular both to that distance and the axis. Hence if w
is determined, the velocity of every particle is known. Also if
the position of the assumed plane in the body be assigned,
then by the geometrical form of the body the position in
space of every point of it is also assigned. The position there-
fore of this plane and the angular velocity are the elements
by which the body's dynamical state is determined.
40. If the body has an angular velocity w about the
axis of ≈ in a direction from the positive part of the axis of x
towards that of y, the velocity of a particle whose co-ordinates
are x, y is wy parallel to x, and wa parallel to y.
GRAND
Motion of a body about an axis under finite forces.
41. To determine the motion of a rigid body revolving
about a fixed axis under given forces.
Let the axis about which the body revolves be made the
axis of %, and let x, y, z be the co-ordinates at time t of a
particle of the body of mass m, r its distance from the axis of
≈, and ✪ the inclination of r to the plane xx, X, Y, Z the im-
pressed accelerating forces on this particle parallel to the
ABOUT A FIXED AXIS.
17
Then the effective moving forces on this
co-ordinate axes.
do 2
d Ꮎ
particle are mr in direction of r, and mr perpen-
dt
dt2
dicular tor and in the direction in which is measured (28).
If such effective forces belonging to each particle be supposed
to act on it in directions contrary to their actual directions,
the conditions of statical equilibrium are satisfied between
these and the impressed forces on the body, among which
latter the reaction of the axis is included.
Hence if moments be taken about the axis of x,
-
d Ꮎ
d
0 = Σ {m(Yv − Xy)} − 2 (mr². 10),
-
the summation extending through all particles of the body.
Now let a fixed plane in the body parallel to ≈ be inclined
to the plane of ≈≈ at an angle measured in the same
direction with 0, and let +a, where a depends on the
position in the body of the molecule considered and so by the
=
condition of rigidity is independent of t;
d20 аф
dť
dt2
And since is independent of any particular particle,
d² o Σm (Yx - Xy)
dt2
Σ(mr²)
Hence if the forces acting on each particle of the body in
any position, excluding the reaction of the axis, be given, and
so be known functions of and the position of the molecules
in the body, since the reaction of the axis has no moment
about the axis and does not appear in the latter member of
this equation, integration will give
аф
,
dt
and then in terms of
t, and so determine the velocity and position of the body at
any proposed instant.
42. To find the pressure on the axis of revolution.
Since any system of forces acting on a rigid body is at
most reducible to two single forces, let the pressure on the
3
18
MOTION OF A RIGID BODY
axis be reduced to a force at the origin and another at a
known distance a in the positive direction of the axis of ≈.
Let F, G, H be components of the former parallel to the co-
ordinate axes, F, G, H' similar components of the latter.
Forces equal and opposite to these are the reactions of the
axis upon the body. The equations of equilibrium give
-
2
Σ(mX) — F — F' = - mr cos 0
- 2
{mr cose (de)" } -
\dt
Ꮎ
(mr sine. de),
Σ(mY) – G – G′ = mr cos 0 θ
t2
de) -
Σ
{mr sine (de)"},
dt
Σ(mZ) - H-H' = 0.
Ꮎ
Σm (Zy - Y₂) + G'a=-2{mxr cose e} + {marsine. (de)"},
d Ꮎ
dt²
Σ
0.
dt
đ² Ꮎ
Σm(Xx-Zx)−F'a=-Σmxr cos
dt
--> {marcos e (de) } -> {marsine de
dt
The sixth equation of equilibrium has been already em-
ployed in (41).
If x, y be the co-ordinates of the centre of gravity of the
body, M its mass, these equations become
αφ
2
ďo
Σ(mX) – F – F' = − Mã
Mx
ミ
​J
My
dt
dt2
Σ(mY) - G- G' = Mx·
d² p
dt
My (dp)",
2
dt
Σ(mZ) - H - H' = 0.
Σm (Zy - Yx) + G'a = − Σ(mxx)
dt2
аф
Σ (mxx) (10)
Σm (Xx - Zx) – F'a = − Σ (mxx)
dt
2
G
From the third equation the sum of H and H' is known,
their separate values under this condition being indeterminate.
ф
+ Σ(myx) (dp)
2
dt
ď o
Σ(myx).
dt2
ABOUT A FIXED AXIS.
19
The other four equations give F, G, F', Gʻ, since
thence
αφ
dt
2
d²
φ
and
dt
are already known and the expressions Σ(mxx),
Σ(my) can be evaluated by integration according to the known
form of the body.
43. If no force be impressed on the body besides the
reaction of the axis,
ďo
dt2
F + F'
аф
0, = w is constant, and
>
dt
Mxw²,
G+ G' = Myw²,
H + H =
0,
G'a = Σ(myx) w²,
Σ(myz)w³,
F'a=
Σ(mxx) w³.
Two particular cases deserve notice:
(1) If the fixed axis pass through the centre of gravity of
the body the former three equations shew that the pressure on
the axis forms a couple.
(2) If the fixed axis be a principal axis, since the assumed
axes of x and y may be so taken as to coincide with other
principal axes of the body at the instant considered, so that
Σ(mxx)=0, Σ(myx) = 0, the latter two equations shew that
the pressure on the axis is reducible to a single force.
If the conditions subsist together, there is no pressure on
the axis.
44. Forces parallel to the axis of rotation do not in-
fluence the motion of the body, but affect the pressure on the
axis.
45. If the symmetry of the body or any other cause
indicates that the pressure on the axis is reducible to a single
20
MOTION OF A RIGID BODY
force, the preceding investigation has a corresponding simpli-
fication.
46. If the whole pressure on the axis be reduced to the
two R and S, the former R in a plane through the centre of
gravity and from the axis, and the latter S perpendicular to
this plane, and in the direction in which the inclination of
this plane to the horizon is measured, and if P, Q be the
impressed forces in the same directions, the distance of the
centre of gravity from the axis.
P+MT
аф
(dt
S = Q - M7 d² $
r
dt
2
>
47. The indeterminateness which arises in the pressure
on the axis in direction of its length even when the points
are assigned where we suppose it applied, like a similar inde-
terminateness in the equilibrium of a rigid body capable of
revolving about an axis, is contrary to experience; for we
know that if a solid body be suspended at two points so as to
be capable of turning about the straight line which joins them
as an axis, the pressure on each of these points when the body
is at rest or in motion is under similar circumstances fixed and
determinate. The contradiction which practice gives to the
theoretical result arises from the imperfect rigidity of the
bodies which are under our observation. The law according
to which pressure is distributed through bodies of this kind
which do not completely fulfil the conception of a rigid body
on which our calculations have proceeded, would supply if it
were within the reach of mathematical calculation, another
equation of condition, and make the two pressures determinate
of which in the case of a perfectly rigid body the sum only is
assigned. (Poisson. III. 1.)
48.
When a body rotates about a horizontal axis under
the action of gravity alone, the case deserves a separate inves-
ABOUT A FIXED AXIS.
21
tigation on account of the importance of its practical applica-
tions.
Leth be the distance of the centre of gravity from the
axis, the angle at which this distance is inclined to the
The effective moving forces on a particle
horizon at time t.
of mass m being
mr
(dp)
dt
2
ďo
and mr in and perpendi-
d t²
cular to its distance r from the axis, the equation of moments
about the axis of rotation gives
0 = Mgh cos & - Σ (m r²)
орф
dt²
If Mk2 be the moment of inertia of the body about an
axis through its centre of gravity parallel to the axis of
rotation,
Σ(mr²) = M (k² + h³),
ď²gh cos
&
and
dť k² + h²
This is the equation of motion of a particle hung by an
inextensible thread of length
k² + h²
h
and swinging under the
action of gravity. For this reason, if in the line h produced
a point be taken at a distance
k² + h²
h
from the axis, this point
is called the centre of oscillation of the body with respect to
this axis. The following is its definition:
DEF. When a rigid body moves about a fixed hori-
zontal axis under the action of gravity, in the straight line
which is drawn through the centre of gravity perpendicular to
the axis a point can be found such that if the mass of the
body were collected there and hung by a thread from the axis,
the angular motion of the point would, under the same initial
circumstances, be the same with that of the body, and this
point is called the centre of oscillation of the body with respect
to the axis.
22
MOTION OF A RIGID BODY
49. Hence when a body makes small oscillations about a
fixed horizontal axis, it is only necessary to calculate the
k² + h²
value of the expression
I suppose, and the time of a
h
small oscillation is. The double of this is the interval
g
in which the body returns to the same place in the same state
of motion.
In adopting the approximate expression π
g
as the
time of oscillation the proportionate error is of the order
(sin)* where 2a is the whole angle through which the
pendulum vibrates. (Poisson. II. 5). By this test the ex-
tent of vibration for which the approximate time of vibra-
tion may be taken can in any proposed case be assigned. If
the vibration, for instance, on each side of the vertical is
within 5º, the error will not exceed 2000th part of the time of
oscillation.
1
50. If h' be the distance of the centre of oscillation from
the centre of gravity when h is the distance of the axis,
h'
k² + h²
h
k2
h
or hh' k2.
=
h
Hence the body will oscillate about a parallel axis through
the centre of oscillation in the same time as it oscillates about
the original axis, the extent of vibration being the same.
=
51. If in any straight line through the centre of gravity
perpendicular to the direction to which k has reference points
be taken at distances h, h' from the centre of gravity, and on
opposite sides of it, such that hh' k², then the length of the
simple equivalent pendulum when an axis in the said direction.
passes through either of these points is h+h'; so that the time
of vibration about each of these two axes is the same, and the
length of the equivalent simple pendulum is in each case the
distance between the axes. Each of these points is the centre
ABOUT A FIXED AXIS.
23
•
of oscillation in reference to the other as a centre of suspen-
sion.
The centres of oscillation and suspension are therefore
convertible; and if two parallel axes of a body can be found
about which the time of a small oscillation is the same, the dis-
tance between them, if it pass through the centre of gravity of
the body, is the length of the simple isochronous pendulum.
52. The convertibility of the centres of oscillation and
suspension was employed by Captain Kater in finding the
length of a simple pendulum vibrating seconds, and conse-
quently the force of gravity at the place of observation. It is
impossible to find the centre of oscillation of a body by calcu-
lation with sufficient accuracy for this purpose by reason of
inequalities of form and structure which no workmanship can
obviate. The position of that centre was therefore obtained
by observation in the following manner.
A bar of brass one inch and a half wide and one eighth of
an inch thick was pierced by two holes through which tri-
angular wedges of steel called knife edges were inserted, so
that the pendulum could vibrate on the edge of either of these
as an axis.
On either of these the pendulum could be hung
between two fixed horizontal agate plates upon which it then
vibrated. The axes were about 39 inches apart. Three weights.
were attached to the bar. One, much the largest, was im-
moveable, fixed near one knife edge and beyond it. The
other two were between the knife edges, and capable of small
motions along the bar by screws. These latter were so ad-
justed by trial that the time of a small vibration through an
angle of about 1° was the same when either knife edge was the
axis, so that each gave the centre of oscillation belonging to
the other. The distance of the knife edges was obtained by
placing the pendulum so that the edges were viewed by two
fixed microscopes, each furnished with a micrometer of ascer-
tained value, and afterwards placing a scale of known accuracy
in a similar position under the same microscopes. If l be this
length, t the time of vibration about either axis, then
t = π
يال
1 80
π?
or g =
g = 1.
g
t²
24
MOTION OF A RIGID BODY
and the length of the seconds pendulum is
length in which g is expressed.
201
2
in the unit of
The time of oscillation of the pendulum was thus found.
The pendulum was suspended in front of a clock pendulum,
oscillating in very nearly the same time, so that both when
at the lowest points of their oscillation were in the field
of view of a fixed telescope. The times by the clock were
noted when the two pendulums coincided so that one covered
the other in their passage over the field. The rates of the two
were so nearly equal that between two such successive times
one pendulum had gained a complete oscillation upon the
other; and then the number of oscillations of the clock pen-
dulum in this interval being known by the clock face, the
number of the oscillations of the other was also known, and
consequently its time of vibration. Phil. Trans. 1818.
53. A determination by experiment of the length of a
simple pendulum vibrating seconds has two important uses.
(1) It gives a value from experiment of the force of
gravity at a given place, and thus supplies a test of theories
respecting the constitution of the earth whereby a value of the
force of gravity is computed.
(2) It has been adopted as a standard of length on ac-
count of being invariable and capable at any time of recovery.
An act of parliament, 5 Geo. IV. defines the yard to con-
tain 36 such parts, of which parts there are 39-1393 in the
length of a pendulum vibrating seconds of mean time in the
latitude of London in vacuo at the level of the sea at tempe-
rature 62 F. The commissioners, however, appointed to con-
sider the mode of restoring the standards of weight and
measure which were lost by fire in 1834, report that several
elements of reduction of pendulum experiments are yet doubt-
ful or erroneous, so that the results of a convertible pendulum
are not so trustworthy as to serve for supplying a standard of
length; and they recommend a material standard, the distance
ABOUT A FIXED AXIS.
25
namely between two marks on a certain bar of metal under
given circumstances, in preference to any standard derived
from measuring phenomena in nature. (Report, 1841.)
54. Since metals expand as their temperature increases,
the position of the centre of oscillation and consequently the
time of vibration through a given arc of a metallic pendulum
will change with the temperature of the atmosphere. In order
to employ metallic pendulums in clocks where uniformity of
the time of their vibration is essential, contrivances are em-
ployed whereby the expansion of different parts so counteract
the effect of one another as to leave the centre of oscillation at
a nearly constant distance from the axis of suspension and the
time of the pendulum's vibration in consequence permanent.
55. The compensation pendulum of Harrison called from
its form the gridiron pendulum (fig. 4) consists of five ver-
tical rods of steel and four of brass placed alternately, and
united by horizontal rods of brass. The central rod sup-
ports a disc or lens of metal called the bob of the pendulum.
The pendulum, as it is figured, vibrates in the plane of the
paper.
The principle on which the compensation is produced may
thus be explained. In a system of parallel metallic rods in
figure 5 let A be the fixed point about which the pendulum
vibrates, G the bob. If CD alone expand in consequence of
an increase of temperature, the points D, E and consequently
G are raised. If AB and EF were alone to expand, G would
sink by the increase of length of each of these rods. Hence if
DC be of brass, and AB, EF of steel whose expansibility is
little more than half that of brass, it is possible that the up-
ward expansion of the brass rod may so correct the downward
expansions of the steel rods as to leave the centre of oscillation
nearly unaffected. Rods of brass and steel united on this
principle form the gridiron pendulum of Harrison.
56. A rod of brass is increased by 00018782 of its length,
and one of steel by 00010791 of its length when its tempera-
4
26
MOTION OF A RIGID BODY
ture is increased by 10 centigrade. (Annuaire par le bureau
des longitudes, 1846). These at least are mean values, from
which, however, the expansibilities of different specimens of
these metals may differ so far as to demand an independent
determination in individual cases whenever accuracy is re-
quired.
57. The mercurial pendulum of Graham is a vertical
cylinder of glass or cast iron containing mercury, and sus-
pended by a steel rod. By the expansion of the steel rod on
an increase of temperature the centre of oscillation is lowered
while by the expansion of the mercury in the cylinder it is
elevated. Since mercury expands much faster than steel, the
quantity of mercury may be so adjusted that the centre of
oscillation is nearly permanent under ordinary changes of
temperature.
The mercurial pendulum has this advantage over that of
Harrison, that it admits of more easy adjustment by a screw
affecting the attachment of the cylinder, by which the distance
of the cylinder from the point of suspension may be altered.
Motion of a body about an axis under impulsive forces.
58. To determine the change in the motion of a rigid
body revolving about a fixed axis after given impulses have
been applied to it.
Let the axis about which the body is revolving with an
angular velocity w be made the axis of ≈; let x, y, ≈ be the
rectangular co-ordinates of a particle of the body whose mass
is m, r its distance from the axis, m X, m Y, m Z the impressed
impulses upon it parallel to the co-ordinate axes. If ' be the
initial angular velocity of the body after the impulse, the
effective impulsive force on that particle is mr(w' - w) in
direction of its motion; and if such effective forces on the
several particles of the body be supposed to act on them in
directions contrary to their own, between such forces and the
impressed forces the conditions of equilibrium of the body are
satisfied.
ABOUT A FIXED AXIS.
27
... 0 = Σm (Yx - Xy)
Σm(Yx Xy) – Σmr² (w' – w) ;
or w' - w
Σm (Yx - Xy)
Σ(mr²)
Since the impulsive reactions of the axis have no moment
about it and so do not appear on the latter side of the equa-
tion, w' is known if the impressed impulsive forces be given.
59. To find the impulse on the axis.
Since the impulses on the axis can be reduced to two
forces at most, let these act at the origin and at an assumed
distance a in the positive direction of the axis of ≈, and let
F, G, H be components of the former, F, G', H' of the latter.
Then the equations of equilibrium give
0 = Σ (mX) - F - F' + Zmy (w' — w),
0 = Σ (m Y) - G – G' – Σm x (w' – w),
0 =
Σ (mZ) - H - H',
0 = Σm (Zy - Yx) + G'a + Σm x x (w' – w),
0 = Em (Xx - Zx) - F'a + Emyz (w' – w),
the sixth equation being that employed in (58).
If x, y be co-ordinates of the centre of gravity of the
body, M its mass, these equations become
0 = Σ (mX) - F - F' + Mỹ (w' — w),
0 = Σ (m Y) – G – G' – Mx (w' – w),
.
0 = Σ (mZ) - H - H',
0 = Σm (Zy - Y%) + G'a + (w' — w) Σ (mxz),
0 = Σm (Xx − Z x) − F'a + (w' – w) Σ (myz).
Zx) –
From the third equation the sum of H and H' is found,
their separate values under this condition remaining unknown.
28
MOTION OF A RIGID BODY
The other four equations give F, F, G, G'. The explana-
tion given by Poisson (47) applies to this anomaly.
60. If a body at rest, but capable of turning about a
fixed axis, can be so struck that there may be no impulse on
the axis, a point where the impulse is applied to the body
in this case is called a centre of percussion.
Let the axis about which the body can revolve be the axis
of x, and let x, y, z be the rectangular co-ordinates of a particle
whose mass is m, X, Y, Z the components of the impulse acting
on the body at the point x'y'x'. Then this impulse must be
at equilibrium with the effective impulsive force on each
particle applied in a reversed direction, which if w be the
body's initial velocity will be - mwr, where r is the particle's
distance from the axis of x.
Hence 0 = X + Σ (mwy),
0 = Y-Σ (mw x),
0 = Z,
0 = Zý – Yx' + Σ (mwxz),
0 = Xx′ − Z∞′ + E (mwyz),
0 = Yx′ – Xy' – Σ (mwr²).
From the third equation Z = 0, or the impulse on the
body must be perpendicular to the axis about which it can
rotate. There then remain five equations to determine x', y', z'
and w, which will only in certain cases be consistent, or in
certain cases only and not in all will a centre of percussion
exist.
From these equations
wΣ (my z)
X
Σ (myx)
Σ (my)
ABOUT A FIXED AXIS.
29
+
22.
wΣ (mxx)
Y
Σ (mxx)
Σ (mx)
$
If the body be so constituted that these are equal to one
another, either of them gives and then the last equation
0 = x′ Σ (mx) + y' Σ (my) − Σ (mr²)
gives a straight line parallel to the plane of xy, any point of
which rigidly connected with the body is a centre of per-
cussion.
SECTION IV.
MOTION OF A RIGID BODY ABOUT A FIXED POINT.
61.
GEOMETRICAL nature of the motion of a body about
a fixed point.
A conception of the motion of a rigid body about a fixed
axis having been already acquired, the motion of a body about
a fixed point will be reduced to this and explained by it.
When a rigid body moves so that one point of it is fixed in
space, every particle of the body retains an invariable distance
from that point. This is the geometrical condition of the
motion. It will be shewn that at every instant a certain
straight line can be assigned, by rotation about which as if it
were fixed the motion of the body at that instant is properly
represented. This line is called an instantaneous axis.
62. In problems of this class there will generally be
occasion to introduce two systems of rectangular co-ordinate
axes, both originating in the fixed point, one system fixed in
space, another fixed in the body and moveable with it.
Let a particle m have co-ordinates x, y, z referred to the
former system, x', y', ' referred to the latter. Then x', y', '
time, and only
are geometrical quantities independent of the
altered in passing from one point of the rigid body to another;
but x, y, ≈ are dynamical elements and functions of the time,
designating the position of m in space, and so varying with
different positions of the body, although the same particle of it
be kept under contemplation.
If the position of the axes of x', y', ', or of any two assumed
lines in the body, be assigned with respect to the axes of x, y, ≈,
the position of the body is determined.
MOTION OF A RIGID BODY ABOUT A FIXED POINT.
fixed point.
31
63. To prove that the motion of a body about a fixed
point may be exhibited at any instant by rotation about
an axis.
(1) The fixed point being the origin, let x, y, ≈ be the
rectangular co-ordinates of a particle of the body referred to
axes fixed in space, x', y', x' co-ordinates of the same particle
referred to rectangular axes fixed in the body and moveable
with it;
x′ = α₁x + b₁y + C₁%,
y
Z'
=
ɑ2x + b₂y + cqz,
ɑ3x + bzy + C3& ;
where the coefficients are functions of the angles which the
moveable co-ordinate axes make with the fixed axes and are
thus functions of the time, but they respect only the position.
of the body in space, and not any particular particle of it,
and are connected by the conditions
2
2
2
2
a₁² + a²² + a²² = 1 = b₁² + b²² + b²² = c₁² + C₂² + C3²,
2
2
2
2
2
C
dx
dy
dz
Now, 0 = α1
+ b,
+ C₁
dt
dt
dt
a₁b₁+а₂b₂+ab₂=0, а₁С₁+α₂с₂+аz Cz=0, b₁ C₁+b₂ C₂+bz Cz=0. (A).
da₁ db
1
dc₁
+ 20 +y· + z
(1)
dt
dt
dt
dx
dy
dz
da2
db2 dc2
α = α 2
0
+ b²·
ხა
+ C₂
+ x
dt
dt
dt
dt
+ y
dt
+x
(2)
dt
dx
dy
dz
0 =
Az
+ b²·
+
dt
dt
C3
dt
dx
db₁
0 = +ya₁ + α2
dt
dt
db₂
dbz
dc
dc.
dc3
+ az
+ zα1
+ a2
dt
dt
dt
dt
dt
db3
Y-
+ Y dt
a¸ ⋅ (1) + a¸ . (2) + a. (3) gives by virtue of (A)
da3
dc3
+ z
(3)
+ x
dt
dt
32
MOTION OF A RIGID BODY
dc₁
dcz
dc3
Let b₁
+ b₂
+b3
= Wx
C₁
db₁
+ C₂
dt
dt
dt
dt
da
da₂
da3
C₁
+ C2
+ C 3
=
Wy
dt
dt
dt
(
dc₁
dc2
αι
+ α₂
dt
dt
db2
dt
+ C3
db3
dt
dc3
dt
+ az
db₁
db₂
db3
da
das
2
b₁
+ b₂
2
+b3
das).
Ꮹ .
+ α 2
+ Az
ωχ
dt
dt
dt
dt
dt
dx
Wyz wxY,
dt
dy
So
W
= wzX - Wrz,
dt
dz
wry — wyx
dt
Hence to determine points which if they be supposed rigidly
united with the body are at rest at the instant under considera-
tion, we have
0 = wyz
wy X — WxY,
0 = w xx
wxz,
0 = wxY
Wy X.
These equations not being independent but reducible to the
X y
two,
Wx
Wy
Z
ωχ
give a straight line through the origin as
the locus of points which have no velocity.
(2)
The direction of the motion of a point x, y, z is
given by the equations
X - x
Y-y
2-z
wy z
<
w zY
Wzx
(1) Z
x
wxY — wyx
—
Since w₁(wy - w₂Y) + wy(w₂x − w₂x) + w₂ (w₂Y − wyx) = 0,
this direction is perpendicular to the straight line which has
been already determined as the locus of particles at rest,
ABOUT A FIXED POINT.
33.
be the distance of a particle x, y, z from the
(3) If
P
X Y
Ꮓ
line
Wx
Wy W%
p² = x² + y² + ș²
(xwx + Ywy + zwz)²
ωπ Wy
2
2
wx² + w₂² + w₂²
and the velocity of the same particle
2
do
dt
2
2
+
2
(dy)
+
dt
dt
(2x)
dz
2
2

= √(wx² + w₂® + wx) (x² + y² + x²) − (xw, + Ywy + %wz)²
2
2
2
=
= p√ ww² + wy² + w%² = pw.
From these three results it appears that at the instant
considered the motion of the body is the same as if it were
rotating about an axis through the fixed point with an angular
velocity w. (39).
Since the quantities wa, wy, we are in general variable, the
position of the instantaneous axis and the angular velocity
about it change in general in successive instants.
64. Equations (4) can be transformed into the following
system:
2
2
2
2
2
2
2
2
2
a₁² + b²² + c₁₂² = a₂² + b²² + c₂² = α3² + b²² + C3² = 1,
ɑ2ɑ3 + b₂b3 + C2C3 = 0,
α1α3 + b₁b3 + C1 C3 = 0,
а¸ª½ + b₁b₂ + C₁ C₂ = 0.
(Vide Appendix.)
Now
α1
dai
+ a2
dt
daz
da3
(1)
+ az
0,
dt
dt
da
da2
da3
(2)
b₁
+ b₂
+ bz
b3
dt
dt
dt
da
da₂
da3
(3)
C1
+ C₂
+ C3
Wy
dt
dt
dt
5
34
MOTION OF A RIGID BODY
.•. α₂(1) + b₁(2) +c₁(3) gives
da,
dt
= c₂wy - b₁ w₂
Wy
Similar values result for the differential coefficients of the
remaining eight cosines.
65. We will now moreover shew that the motion of the
body which we have proved to be one of rotation about an axis,
may also be represented by three coexistent rotations about
the co-ordinate axes.
Suppose the body to have angular velocities
x
a about the axis of y
Wx
wy from
y
≈ towards
E
ωχ
Wy
X
8.
These angular velocities if they exist separately produce
linear velocities in the particle x, y, z,
-
Y ∞ ગે
x parallel to
x
z wyl
జలు
- x wy
x wzl
- ywz
parallel to
*
;
a
;
parallel to {3. (40).
Hence if the three angular velocities be supposed to co-
exist without interference, the velocities of x, y, z are
-
Z Wy y wz
xww parallel to
wz
ZWx
y w x - xw
y,
Z.
Hence it appears that when a body rotates about a fixed
point, its motion may be truly represented by supposing three
angular velocities about perpendicular axes to coexist. These
angular velocities are known at each instant if the velocities
at that instant of any particle of the body be given; and, con-
versely, if they be known, the motion of each particle of the
body is determined.
ABOUT A FIXED POINT.
35
66. Since at any instant the motion of the body is
correctly exhibited by rotations properly chosen about three
perpendicular axes, and since the position of these axes is
arbitrarily adopted, the motion at different instants can also
be represented by rotations about different systems of axes
originating in the fixed point. Hence the body's motion may
furthermore be exhibited by rotations about axes which them-
selves move after any assignable law, a statement which will
include the case of axes so moving in space as to retain fixed
positions in the body. If these moving axes were at any
instant to become fixed, the angular velocities about them
would by their coexistence exhibit the motion of the body at
that instant.
67. Composition and resolution of angular velocities.
Since the motion of the body is exhibited either by a single
angular velocity w about an axis whose direction cosines are
ωχ (63), or by three coexistent rotations w, wy, wz
Wx
W
Wy
W
W
about the co-ordinate axes (65), one of these modes of re-
presenting the same motion may be replaced by the other.
Hence if a rigid body have an angular velocity w about an
axis through a fixed point whose direction cosines are a, ß, y,
this velocity may be resolved into the coexistent angular velo-
cities aw, Bw, yw about the axes of x, y, z respectively; or
these coexistent velocities may be compounded into the single
angular velocity w.
γω
68. To determine the motion of a body fixed at one
point when two given angular velocities about given axes
coexist in it.
Let w, w' be the given angular velocities about axes
through the fixed point whose direction cosines are a, ß, y,
and a', B', y'.
These angular velocities are equivalent to
a'w'
X,
Bw and B'w' about the axis of y,
a w
βω
y wl
ω
j'w')
z;
36
MOTION OF A RIGID BODY
and since they coexist, they are therefore equivalent to the
single angular velocities
aw + a w
βω β'
a,
w+ B'w' about the axis of y,
γωγω
z.
These are equivalent to the angular velocity,
√ (aw + a' w')² + (ßw + ẞ'w')² + (y w + y 'w')³
α
2
√ w² + w²² + 2 ww' (aa' + ßß' + ɣy'),
about the axis whose equations are
X
Y
'w'
aw + a w Bw + B' w
Z
γω + γω
(B)
69. The angular velocity of a rigid body about an axis
may be properly represented by a straight line in direction of
that axis. For the length of this line will represent the magni-
tude of the angular velocity according to some assumed
standard, and the direction of the line will represent the direc-
tion of the angular velocity, if the convention be made that
when the eye looks in the direction which is accounted positive
in the measurement of the line, then the particles of a body
having a positive angular velocity appear to rotate from right
to left, or oppositely to the hands of a watch.
70. By help of this convention since aa' + ßß' + yy' is
the cosine of the angle between the two given axes in (68),
equation (B) indicates the following property:
If two straight lines drawn from a point represent in mag-
nitude and direction two angular velocities coexistent in a rigid
body of which that point is fixed, and if the parallelogram of
which these lines are adjacent sides be completed, the resultant
angular velocity of the body will be represented in quantity
and direction by that diagonal of the parallelogram which is
drawn from the intersection of the same two straight lines.
ABOUT A FIXED POINT.
37
71. If there be more than two angular velocities co-
existent in the body whereof one w is about the axis whose
direction cosines are a, ß, y, the resultant angular velocity
2
2
= √ {Σ (aw)}² + {Σ(Bw)}² + {Z(yw) } ²
about the axis
X
Y
Z
Σ (αω)
Σ (βω) Σ (γω)
72. Let ABC be a triangle, in which C is a right angle.
Let AB represent the angular velocity of the body in
magnitude and direction. If the parallelogram with sides AC,
CB be completed, it is seen that the rotation AB is equivalent
to rotations represented by AC, CB coexistent in the body.
Now BC is perpendicular to AC; hence AC is the component of
the body's angular velocity about an axis in this direction.
73. If a body has rotations wr, wy, w₂ about rectangular
co-ordinate axes, its angular velocity about any axis whose
direction cosines are a, ß, y is the sum of the components
of those rotations about this axis, or awr + ẞwy + ywx•
The analogy of these propositions with the resolution of
forces and of linear velocities will readily be traced.
74. The motion of a body turning about a fixed point
may be expressed by coexistent angular motions about rectan-
gular axes originating in that point, fixed relatively to the
body and moving with it. (66). When such angular velo-
cities are known the velocity of every particle is determined,
and its direction is also known with reference to the moveable
axes. It remains to be shewn how from the same angular
velocities the position of the body in space may be ascertained.
The following method is taken from Prof. O'Brien's tract on
Precession and Nutation.
75. Let a sphere be described with radius unity about
the fixed point which is the origin of co-ordinates. Let the
38
MOTION OF A RIGID BODY
fixed co-ordinate axes meet this sphere in the points X, Y, Z,
and let three rectangular axes fixed in the body and moveable
with it meet the surface in A, B, C (fig. 6). Let w₁, w2, W3
be the given angular velocities of the body about the latter,
by the supposed coexistence of which its motion is exhibited,
(66). Let the great circles ZC, BA, produced if necessary,
meet in E. Let XZC = √, ZC = 0, ECA = p. If these
angles be known the position of the body in space is com-
pletely assigned. (62).
Now if C be a point of the body, or be rigidly united
ᏧᎾ
along ZC,
dt
with it, the velocity of C =
بله
sin perpendicular to ZC.
dt
But the velocity of C is likewise exhibited by w₁ about A
in direction BC, and w
about B in direction CA.
ᏧᎾ
WI
sin 18+
$ + w₂ cos &,
dt
df
sin
w₁ cos & + w₂ sin p.
dt
(1)
(2)
Again, the velocity of the point corresponding to E in
do
direction AB is
dy
+ cos 0;
dt
dt
dp dy
+
dt dt
cos
= W3.
ვ.
(3)
W3
If then w₁, w2, we be given these equations after elimination
and integration give 0, 0, y, and determine the position of the
body. The constants which integration introduces are to be
assigned from the given position of the body at some given
epoch.
76. Hence the angular velocities of the body about a
system of rectangular axes fixed in itself may be considered
the elements of its motion, from which the position, velocity
and direction of any point of it at a given instant can be
known.
ABOUT A FIXED POINT.
39
Equations of motion.
77. The preceding propositions are designed to give a
conception of the geometrical nature of a body's motion about
a fixed point. We have now to compute its motion from a
knowledge of the forces which cause it. The properties of
principal axes make it most convenient to investigate the
angular velocities about such a system, and to exhibit the
body's motion by these angular velocities in the sense ex-
plained in (66).
I. Finite forces.
78. When a rigid body is in motion about a fixed point
under given finite forces, to find its angular velocities about a
system of principal axes through that point.
Let the fixed point be made origin, and let x, y, ≈ be the
co-ordinates at time t of a particle of the body of mass m,
referred to fixed rectangular axes, X, Y, Z the given im-
pressed accelerating forces on this particle parallel to these
axes.
Then if the effective moving forces, represented generally
d2%
d² y
m be applied to the particles to which
d² x
by m m
d t2 dt2 d t²
>
they belong in directions contrary to their own, the conditions
of equilibrium will be satisfied among these and the impressed
forces (30);
.. Emly
Ση
d² z
=
Σm (Xx - Zx) = M,
૪
d²y
*
Σm(Zy – Yx) = L,
dt2
dt2
Σm z
(
d² x
d² z
d t²
Σmx
(
d² y
d²x
Y
=
dt2
dt
i
d t2
Σm(Yx − Xy) = N,
L, M, N being known from the given forces applied to
the body, the unknown reaction of the fixed point having no
moment about the co-ordinate axes.
€
40
MOTION OF A RIGID BODY
Let wx, wy, wz be the coexistent rotations about the co-
ordinate axes of x, y, z which can represent the body's
motion (65);
dx
dt
= Z Wy — YWz,
d² z
d t
y
•
dy
X
= xWx
Z Wxr
dt
dz
= yw₂ x wy
—
dt
d w x
dt
-
x
dwy
dt
doz
-
+ w₂ (x w% − zwx) − wy (zw, − ywz),
+ wz (z wy − y w z) — wx (Ywx -
-
x wy)
d² y
dt2
d w z
= x
dt
dt
dws
.. Σm (y² + x²).
+ Σm (y² − x²). wy Wz
dt
dwy
dwz
-
Σ(mxx).
dt
dt
-Σ(mxy).
Σ(mxx).wxWy
+ Σ(mxy).wxwx − Σ(mx x). wzwy
2
-Σ(myz).w₂² + Σ(myx). w₂²
2
2
- Σ(myx) w,² +Σ(myx)w,² = L.
-
The other two equations of moments may be similarly
transformed.
Let w₁, w, w be the angular velocities about a system of
principal axes through the origin which may likewise represent
the motion of the body at time t. (66).
If a₁, b₁, c₁, a₂, by, cy, az, b3, C3, be the direction cosines of
these axes with reference to the fixed axes,
W x = A₂ W₁ + Az Wz + Az Wz⋅
ABOUT A FIXED POINT.
41
d w x
dwr
d wz
dws
+ α.2
+ α3
аз
dt
dt
dt
dt
da
+ (α₂wx + b₂wy + C₂ wx)
Wx
+ (α₂wx + b₂wy + C₂ Wx)
+ (α zwz + bz wy + Cz Wz)
dt
da2
dt
da3
dt
da₁
da2
da3
0,
Here the coefficient of wx α1
+ Az
+ Az-
dt
dt
dt
ad₁
daz
da3
wy = b₁
+ ba
+b3
dt
dt
dt
dai
daz
+ C3
+ C2
Wx =
C1
dt
da3
dt
=
Wy
dt
dw.
dwi
d we
dw3
+ az
ai
+ a2
dt
dt
d t
dt
dwy
dwi
d wz
dw3
So
= b₁
+ b₂
+
b3
2
dt
dt
dt
dt
dwz
dwi
d w2
dw3
+ C2
+ C3
= C₁
dt
dt
dt
dt
Now let the axes of x, y, ≈ have been so assumed that at
the time t the principal axes coincide with them.
= 0, Σ (mxx) = 0, Σ (mxy) = 0,
Then Σ(myx) = 0,
Wx = WI,
Σ (mxx) = 0,
Wy
= W2,
and since a₁=b₂=c3=1, a‚=0, ɑ3=0, b₁=0,
Wz = Wzr
b=0, c₁=0, C2=0,
dw
dwi
do
day - day
dwx
dwz
dt
dt
dt dt
dt
dt
6
42
MOTION OF A RIGID BODY
Hence if A, B, C be the moments of inertia of the body
about the principal axes to which w₁, w2, w3 respectively refer,
A dł w₁ + (C − B) w₂ w3 = L,
B dł w₂ + (A − C) W1 W3
=
M,
C dł wz + (B − A) w₁ w₂ = N.
These equations do not recognize the axes of x, y, z, and
therefore are not restricted by the particular position in which
those axes were for convenience assumed. They are therefore
the three equations for determining the body's angular velo-
cities about the given principal axes, about which L, M, N
are the moments of the impressed forces.
2
2
2
79. When w₁, w₂, we are thus determined, the body's motion
is represented by a single angular velocity w²+ w₂² + wś
about a straight line whose equations referred to the principal
The position of the body in space
axes are
X У
W1 W2
2
W3
will be known from the equations of (75).
80.
To find the pressure on the fixed point.
If Σ (mX), Σ (mY), Σ (mZ) be the sum of the com-
ponents of the given forces resolved parallel to the co-ordinate
axes, F, G, H the pressures in the same directions on the
fixed point,
W
Σ (mX) - F = Σ
md2x
dt2
Σ (mY) - G = Σ
m d² y
d t²
-
Σ (m Z) – H = Σ
m d² z
"
dť
d²x
d²y
d² z
whereby, if the values of
in terms of the
df
dt dt
angular velocities be substituted (78), the pressures on the
point are known.
ABOUT A FIXED POINT.
43
If a body turns about its centre of gravity, the pressures
on that point in direction of the co-ordinate axes are
Σ (mX), Σ(mY), Σ(mZ).
81.
In the case where no force acts on the rigid body
except at the fixed point, the equations of motion lead to some
remarkable results which shall now be exhibited.
If no forces act on the body which would tend to produce
motion in it if it were at rest, L = 0, M = 0, N = 0;
N=0;
dw
A
-
+ (C − B) w₂ w3 =
0,
(1)
dt
d wz
B
+ (A − C) w₁ w3 = 0,
(2)
dt
dws
C- + (B − A) w1 w₂ = 0 ;
(3)
dt
Hence w₁ (1) + w₂ (2) + ws (3),
and Aw₁. (1) + Bw₂ . (2) + Cw.. (3), give
dt
dwi
dwz
dw3
Aw + Bw₂
+ Cw3
=
0,
dt
dt
d wz
dw3
A²wi + B²w₂
+ C² w3
0;
dt
dt
dwr
dt
whence by integration,
2
2
A w₁² + Bw₂² + Cw3² = h²,
2
2
A² w₁² + B² w₂² + C²wz² = k² ;
the latter members of these equations being positive quantities
depending on the primitive circumstances of the motion.
82. Also when the body is referred to axes fixed in
space,
Σmy
d² z
dt2
Z
d²y
= 0,
dt2
44
MOTION OF A RIGID BODY
dz
dy
.. Ση
Y
I
***********
h₁.
dt
dt
· So Emz
(
dx
dz
at
=
h₂,
dt
dt
Σm (x
dy
dx
y
-
dt
dt
ha
h1, h2, h3 being constants dependent on the primitive circum-
stances of motion.
Let another system of fixed rectangular axes from the
same origin be adopted, and let a, b₁, c1, a2, b2, C2, ɑ3, b3, C3,
be their direction cosines in reference to the former system,
x', y', 'the co-ordinates of m referred to them. Since
(y at
dz dy
dt
dt
૪
St
(3
dz
dy'
dt
St,
dt
are ultimately doubles of the sectorial areas described in time
St by the projection of m upon the planes of yx, y'z', ...
dx
dy'
·· h₁ = Σmy'
dt
dt
a₁ + Σm (x²
da'
dx'
Em
a2
dt
dt
+ Σm x
(v
dy'
dx'
Y
az.
dt
dt
Now Em (y-
dy'
2
dt
dt
-
Σm(y² + x²²)w, − Σ(mx'y')wy – Σ(m x'x') wz,
and the other two functions may be similarly transformed.
If then the second system of co-ordinate axes be so chosen
as to coincide at the instant considered with principal axes of
the body,
83.
So
h₁ = Aw₁ α₁ + Bw₂αz + Cwzaz.
h₂ = Aw₁b₁ + Bw₂b₂ + Сwzbз,
2
h₂ = Aw₁c₁ + Bw₂C₂ + CwzCz.
2
2
2
Hence h² + h² + h3² = k².
ABOUT A FIXED POINT.
45
84. Also ha₁ + h₂b₁ + h₂C₁
h3C2
=
A w₁
B
h₁ α₂ + h₂ b₂ + h₂C₂ = В w₂,
h₁ αз + h₂ bз + hz Cz
=
Cw3.
85. If H be the double of the velocity with which
sectorial areas are described by the molecule m on any plane.
whose direction cosines are a, ß, Y,
Σ(mH) = h₁a + h₂ß + h₂y.
α
B
Y
Σ(m H) will be greatest when
1
h₂ h₂ h3
1
h₁² + h₂²² + h₂³
2
86. This plane, whose equation is 0=h,x+hy + h3≈,
is denominated the invariable plane of the body, on account.
of properties which it will hereafter be seen to possess. If
we denote its normal by I, whose direction cosines are
hh,
and call a the principal axis to which A belongs,
k' k k
hg
we have
h₁a₁ + h₂b₁ + h3C1
cos (I, a)
k
Awi
k
(84).
Also with the notation and figure of (75), by supposing
the invariable plane to coincide with that of a y, and the point
A to belong to the principal axis a,
cos (1, a) = cos ZA
sin e cos ;
:. A wi
Αωι
k sin
cos .
do
Hence
dt
So Bw₂ = k sin 0 sin o,
and Cw₂ = k cos 0.
- k sin 0. sin cos (1)
o o
B
df
d sin 0 = k sin 0
cos²
2
sin²
dt
✪ = k (
dø dy
+
A
B
k cos 0
+
cos
se
(75.)
dt
dt
C
46
MOTION OF A RIGID BODY
87. When the principal axes are axes of co-ordinates, the
equation to the central ellipsoid is
| 8
Ax² + By² + Cx² = 1. (22).
The instantaneous axis referred to the same axes having
y
Z for its equations, meets this ellipsoid in a point
W1
W2
W3
whose co-ordinates are
Οι W2
W3
h
h
h
The tangent plane to the ellipsoid at this point is
A w₂ x + Bw₂y + Cwz≈ h,
=
and when axes fixed in space are adopted as co-ordinate axes
the equation to this tangent plane becomes
Aw₁(α₁x+b₁y+c₁*)+Bw₂(α₂x+b₂y+C₂≈)+Cw3(α3x+b3y+c3x)=h,
or h₁x + h₂y + hz≈ = h. (82).
The tangent plane to the central ellipsoid at the point
where the instantaneous axis meets it is therefore fixed in
space during the motion.
88. The motion of the central ellipsoid therefore is under
these two conditions: (1) its centre is fixed, (2) it always
touches a fixed plane. Since also the point of contact lies in
the instantaneous axis, that point has no velocity. Hence the
motion of the central ellipsoid is such that while its centre is
fixed it rolls upon a fixed plane whose distance from its centre
is
h
k
3
The plane h+h₂y + h₂ = 0 is from this property
termed the invariable plane of the rigid body. (86).
89. When the body is a solid of revolution turning about
a point in its axis, the central ellipsoid becomes a spheroid
whose axis is that of the body, and the instantaneous axis
ABOUT A FIXED POINT.
47
describes a circular right cone whose base is parallel to the
invariable plane.
90. If the central ellipsoid becomes a sphere in conse-
quence of the principal moments of inertia of the body at the
point about which it turns being equal, then the instantaneous
axis remains fixed in space as well as in the body.
91. The equation to the invariable plane referred to axes
fixed in space and so taken that the principal axes of the body
coincided with them at a given epoch is
0 = AQ₁x + BQ2Y + CQ3≈,
where N1, N2, N, are the values of w₁, w₂, w, at that instant.
92. If the instantaneous axis
Ω
20 y Z
lie in the
Ως
2
2
2
invariable plane, AQ² + BQ² + C'Q,² = 0, which is impossible
2
since A, B, C are positive quantities.
The instantaneous axis will be perpendicular to the in-
variable plane if the straight lines
y I
X
y
22
and
2
Ωι
Ωρ
Ως
ΑΩ,
BQ2
CQ3
coincide.
This will happen (1) if A = B = C ;
(2) if two of the quantities N1, N2, N3
vanish together.
93. The radius vector of the central ellipsoid drawn in
direction of the instantaneous axis,
1
2
{√ (w₂² + w₂ ² + w₂²),
and is therefore proportional to the angular velocity of the
body (63, 66).
48
MOTION OF A RIGID BODY
94. The following is an additional illustration of the
nature of the body's motion.
Since
A
A
k?
2
2
Aw₁² + Bw₂² + Cw;² = h²
2
2
2
A w₁² + B² w² + C² w₂² = k²
2
1
2
C
2
=
1 (4-1) w' + B (B-) m² + C (-1) w² - 0.
h²
Wi
k? h2
h²
Now the equations to the instantaneous axis referred to
the principal axes of the body are
Xxx
y Z
૨ |
A
.. A
WI W2 W3
4(4-1)+B(B-) + C(-)-0, (a)
k² h2
k² h²
y²
k² h
=
Hence the instantaneous axis describes in the body a
conical surface of the second order. It likewise describes in
space a fixed conical surface, and these two cones have always
a common generatrix, viz. the instantaneous axis, and every
point of this instantaneous axis if it be rigidly united to the
body has no velocity. Hence if we suppose the conical
surface whose equation is (a) to roll on the fixed conical
surface which is the locus of the axis in space, the motion of
the body will be exhibited by supposing it rigidly attached to
and carried along with the former rolling cone.
This geometrical illustration of rotatory motion about a
point is due to Poinsot, and exhibits the double motion of the
instantaneous axis in space and in the body. In space that
axis describes a fixed conical surface; in the body it describes
a cone of the second order which may in particular cases
degenerate into a plane or a line when one or two of the co-
efficients in (a) vanish.
95. The motion of the Earth as a spheroid in consequence
of Precession, which is one of the most important applications
of the equations of this chapter, may be thus conceived.
ABOUT A FIXED POINT.
49
Describe a circular cone whose vertex is at the centre of
the spheroid, and its axis perpendicular to the ecliptic, and
also another circular cone about the axis of the spheroid with
the same vertex. Let the difference of their semi-vertical
angles be the obliquity of the ecliptic, and the ratio of the semi-
vertical angle of the latter to that of the former the ratio of a
day to the period of the Precession of the Equinoxes. Then
if the latter cone roll within the former fixed, with the Earth's
mean angular velocity, it may complete its circuit in the period
of the Equinoxes and revolve in space in one day, and there-
fore if the Earth be rigidly attached to it and carried with it,
the motion of the Earth about its own centre in consequence
of Precession is correctly exhibited.
96. The body, as has been seen (88), will permanently
rotate about any of its principal axes at the fixed point about
which it turns, but the three principal axes which form a
system have different characters with respect to the stability of
the rotation about them.
The path of the pole of the instantaneous axis on the
surface of the central ellipsoid is given by the equations
Ax² + By² + СC²²
=
1,
k²
h2
A²x² + B²y² + C² ≈²
Its projections therefore on the principal planes are
−
B (4 − B)y² + C (4 − C) &² = (
(A
k2
A
h²
A (B − A) x² + C (B − C)≈² = (B
− − - (B-),
▲ (C − ▲) x² + B (C − B) y² = (C
A A)
h²
k2
C
h²
Let A, B, C be in descending order of magnitude.
7
50
MOTION OF A RIGID BODY
(1) The projection of the pole's path on the plane yx is an
ellipse, whose eccentricity
√
B-C B+C - A
A - C
C
If this
eccentricity is small, the position of the instantaneous axis will
never deviate far from that of the principal axis of maximum
moment if it is once near to it.
(2) The projection of the path on the plane xy is an
ellipse whose eccentricity
ང
A − B A + B − C
A - C A
If this
eccentricity be small, a similar conclusion may be drawn respect-
ing the axis of minimum moment.
Hence, under the conditions prescribed imposed on the
eccentricities of these ellipses, rotation about the axis of maxi-
mum or minimum moment is said to be stable; for if the body
be rotating about either and be slightly disturbed so that the
position of the instantaneous axis is thrown slightly out of
coincidence with either, it will still never deviate far from that
with which it previously coincided.
The degree of stability about the two axes will be com-
pared by means of the ratio of the eccentricities of the two
ellipses which the pole's projections describe, or by

A B C B + C - A
C'A-B'A+B-C'
(3) The projection of the path on the plane az is an
hyperbola, and therefore the projection of the pole may depart
widely from the centre, if it was originally near to it, so that
rotation about the axis of mean moment will not generally be
stable.
97. In one particular case however rotation about the
mean axis may be stable.
Let k² = Bh²,
2
A(A – B)w¸² = C (B − C) wz²,
.. ·
2
AB(A – C) w¸² = (B − C) {k² – B³ w₂²};
-
2
B³w½³}, (1)
2
BC (A – C) wz² = (A − B) {k² – B² w³}.
ABOUT A FIXED POINT.
51
dw₂
B
dt
= F
B (A - C) w¸w.s

(A
√√√(4 – B) (B − C')
AC
the upper or lower sign being used as
(k² – B³w,²),
w, and we have like or
2
> B'w; .. w₂ con-
W2
unlike signs. Since by virtue of (1) k²
tinuously decreases or increases with the time in these cases.
respectively.
B
d wa
Now
F
k2
B³ w₂
2
dt
B
(A-B)(B-C)
AC
= n suppose.
k + B wz
k
-
= C&F 2k nt,
Bwz
C' being an arbitrary constant.
Hence as t is indefinitely increased w approaches to
as its limit, and w₁, wę to zero.
=
k
B
Hence if k² Bh², the mean axis is an axis of stable
rotation, in this sense that if a body revolving about it be
disturbed into revolving about another axis so that the con-
dition aforesaid is preserved, the instantaneous axis will con-
tinually tend to restore itself into coincidence either with its
original position or that position reversed by being produced
backwards, and the cases are distinguished by the circumstance
of w₁ and w, having at the beginning of the disturbance unlike
or like signs respectively.
In this case the instantaneous axis constantly lies in one
of the planes
X
√ C(B - C)
Z
√A(AB)
Each of these
planes cuts the central ellipsoid in a curve such that the per-
pendicular from the centre on the tangent plane at any point
of the curve has the same length B-
52
MOTION OF A RIGID BODY
98. In reviewing then the results of this section it ap-
pears that
(1) When a rigid body revolves under any circumstances
about a fixed point, there is at every instant a line of particles
at rest, and the motion of the body is at the instant identical
with that of a body revolving about this line as a fixed axis.
(63). The instantaneous axis is in general variable in position
and traces out a conical surface of some species with the fixed
point as vertex.
(2) When the forces acting on the rigid body have no
moment about any straight line through the fixed point, then
the motion is such that the central ellipsoid rolls upon a fixed
tangent plane (88), the instantaneous axis being the diameter
at the point of contact (88) and the angular velocity being
proportional to the length of this diameter (93).
When the motion of the instantaneous axis in the body is
considered, i.e. its motion as it would appear to an observer
unconscious of the body's motion and employing for reference
lines fixed in it as if they were fixed in space, then in this view
the instantaneous axis describes a conical surface of some
species about the fixed point as vertex, and in the case of the
impressed forces having no moment about the fixed point, this
cone is one of the second order (94).
II. Impulsive forces.
99. To determine the change of motion under given im-
pulses of a body of which one point is fixed.
Wy
The fixed point being origin let x, y, ≈ be the rectangular
co-ordinates of a particle whose mass is m at the moment when
the impulses act. Let wa, wy, w, be angular velocities.
about the axes of x, y, z which represent the body's motion
before the impulses, and let them be suddenly changed into
ω, ω, ως
ABOUT A FIXED POINT.
53
Now the effective impulsive moving forces on m are
m {(wý – wy)x − (w% – w₁)y} in direction of ≈,
m {(w½' — w₂) x − (w,' — w₂) %}
m {(w,' — wz) y − (wý – wy)x}
y,
Z.
(28. 63.)
And between these in reversed directions and the impressed
impulses the conditions of statical equilibrium are satisfied.
If then L, M, N be the moments of the given impulses
about the axes, since the unknown reaction of the fixed point
has no moment about these axes,
L = Σ[my {(w,' — w₂) y − (w,' — wy)x}
− mx {(wz − w₂ ) x − (w,' — w₂) ≈}],
-
=Σm (y²+x²). ( wr'−wz) − Σ (myx). (w,' — w₁) −Σ (mx x). (wz' — w%),
or if A, B, C be the moments of inertia of the body about the
axes of x, y, z;
then
A' = Σ(myx), B' = Σ(mxx), C= Σ(m xy),
L = A (w' — w₂) − C′ (w, – wy) – B' (w%' – wz),
M = B(w₁ − w₁) − C' (w.' – w₂) – A' (w%' – wz),
– −
N = C(ww₂) – B' (w.' - ws) - A' (wy – wy).
When we', w, w are obtained from these equations, the
initial motion of the body after the impulses is expressed by
an angular velocity √//w," + w," + w;
Wx
12
y
12
wx
about the axis
wx
Wy
W%
Hence the velocity and direction of every point of the
body are known.
54
MOTION of a rigID BODY
100. If the axes of co-ordinates be so selected as to
coincide with principal axes of the body at the origin,
A (w,' — w₂) = L, B(wy – wy) = M, C(w₂' — w₂) = N.
101. To find the impulses on the fixed point.
Let F, G, H be the impulses on the fixed point, and these
in reversed directions the reactions on the body. Then if
Σ(X), Σ(Y), Σ(Z) are the algebraic sums of the given im-
pulses parallel to the co-ordinate axes, x, y, z co-ordinates of
the centre of gravity of the body,
Σ(X) - F = Σ{mx (wý – w,) – my (w%' – wz) }
=
= {(wý – wy)z − (wz' — w₂)ÿ} . Σ(m),
Σ(Y) – (w₂'
- G = {(w% − wz) x − (w;' — w₂) z} . Σ(m),
– G
-
Σ(Z) – H = {(w,' – wz)ÿ − (wy – wy)x} .Σ(m);
whereby the impulses on the fixed point are known, w,', wy', wź
being already determined. (99).
If the axes be principal axes,
M%
F = 2(X)
Ny z (m),
B
G = E(Y)
Lz
(NT - LE) 2 (m),
C A
Mx
H = Σ(Z)
2(Z) - {Ly - M
A B
Σ(m).
102. If the axes be principal axes, the equations to the
initial instantaneous axis of a body previously at rest are
Ax
By
Cz
L
M
Ν
N'
ABOUT A FIXED POINT.
55
The forces acting on the body consist of a force at the
origin and a couple whose plane is
Lx + My + N≈ = 0.
Hence if the central ellipsoid be described whose equation is
Ax² + By + C≈² = 1,
the plane of the couple is diametral in this or any similar,
concentric, and similarly situated ellipsoid to the instantaneous
axis.
The instantaneous axis is thus never perpendicular to the
plane of the couple except when that plane is perpendicular to
a principal axis of the body at the fixed point.
SECTION V.
MOTION OF A FREE RIGID BODY.
103.
Geometrical conception of the motion.
The most general motion of an unconstrained rigid body
may be represented by supposing two motions coexisting;
(1)
A motion of translation, whereby each particle has
the velocity in magnitude and direction of an assumed point
which either belongs to the body or is in imaginary rigid
union with it, and
(2)
A motion of rotation about an axis through the said
assumed point, whereby each particle has the velocity in mag-
nitude and direction arising from rotation about this axis.
The possibility of correctly exhibiting a body's motion
by this means appears from the following proof.
Let x, y, be rectangular co-ordinates at time t of a
particle of the body referred to axes fixed in space, a, B, y
co-ordinates of another point which is made the origin of a
system of rectangular co-ordinates moveable with the body.
Let x', y', ' be co-ordinates referred to these latter.
ß)
Then x = a₁(x − a) + b₁ (y - ẞ) + c₁ (≈ − y),
a₂ (x − a) + b₂ (y − ẞ) + c₂ (≈ − y),
y = a₁₂ (x
≈′ =
A 2
2
αz (x − a) + b¸ (y − ẞ) + c3 (≈ − y);
where a₁, b₁, ... depend on the position of the whole body and
not on any particular particle of it, and are connected by the
conditions
2
2
2
2
2
a₁² + a₂² + a‚² = b₁² + b₂² + b²² = c₁² + c₂² + C3² = 1,
C2²
a₁b₁ + a₂ b₂ + a3b3 = 0,
а₁С1 + α₂С2 + A3 C3 = 0,
(A)
b₁c₁ + b₂c₂ + b3c3 = 0.
MOTION OF A FREE RIGID BODY.
57
d (x − a)
d(y
B)
... 0 = α₁
+ b₂
dt
dt
d(x − y)
dt
+ C1
da
db,
dc₁
dt
d (x
a)
d(y-B)
0 = a₂
+b.
dt
dt
+ (x − a) +1 + (y − (ß)
d (x − y)
-
dt
+ (x − y)
(1)
dt
dt
+ C₂
da
dba
+ (x − a)
+ (y - B)
(x) +
dt
dt
dc2
(2)
dt
d (x - a)
d (y - ẞ)
d(x − y)
0 = α3
+ b3
+ C3
dt
dt
dt
d a3
d b₂
dc3
+ (x − a)
-
+ (y − B)
-
+(87)
(3).
dt
dt
dt
a₁ (1) + α₂(2) + a(3) gives by virtue of (4),
db₁ db2 db3
+ az + az
dt
0 =
d (x − a)
dt
+(y-B) a₂
(a
а1
dt
dt
(an
dc
dc2 des
dt
+ Az + αz
dt
dt
+ (≈ − y) (
+ (≈ − y) w₁ - (y - ẞ) wx, suppose,
(x −
У
+(x − α) wz − (≈ − y) wx,
a)
dx
da
or
dt
dt
dy
dß
SO
dt
dt
dz
dy
+ (y − ẞ) wx − (x − a) wy;
dt
dt
wherein wa, wy, we are dependent on the position of the body
in space.
Now in each of these expressions the first terms are the
component velocities of the assumed point a, ẞ, y; on sup-
8
58
MOTION OF A FREE RIGID BODY.
position of its rigid union with the rest of the body; the latter
two terms are the velocities which would result from the body
Wx
2
2
2
having an angular velocity w² + w + w about an axis
X Y Ꮓ
through a, ß, y parallel to the straight line
Wx
ω Wy W%
Therefore the motion of the body is such as would result
from the co-existence of the motion of all its particles in
parallel directions and the motion of rotation aforesaid. The
former motion is called the motion of translation, and then the
proposition may be stated that the motion of a rigid body may
be always supposed to result from the coexistence of a motion
of translation and a motion of rotation. These will be the
elements of the body's motion from a determination of which
the place, velocity, and direction of every point of it will be
known.
104. For reasons which will appear hereafter it will be
usually expedient to allow the assumed point, mentioned in
the preceding article as determining by its motion the motion
of translation of the body, to be the centre of gravity of the
body.
105. If a rigid body has two coexistent angular velocities
about parallel axes, to find the nature of its motion.
P.
Let the rigid body have angular velocites w, w' in the
same direction about axes whose traces on a plane perpendicu-
lar to each are A, B (fig. 7.) In AB produced take any point
Then from these angular velocities separately the veloci-
ties of P regarded as a point of the body or rigidly united
with it, would be w. AP and w'. BP in the same direction in
the plane on which A, B are traces. Hence if the two angular
velocities coexist the velocity of P is
w. AP + w' .BP.
The point P has no velocity if
w.AP+w'. BP = 0,
w'.AB
or AP
w + w
MOTION OF A FREE RIGID BODY.
59
and the same will be true of every point in the straight line
through P perpendicular to the assumed plane; therefore the
body will rotate about this line.
If w₁ be the angular velocity about the axis through P,
then from the two expressions for the velocity of A,
W1
AP = w'. AB,
106. If w= - a,
-
.. ω: = w + w'.
, these results are nugatory. In this
case the velocity of any point P in AB or AB produced
= w . AP + w'. BP – w. AB.
All points therefore in the plane of the two axes A, B
have the same velocity in a direction perpendicular to that
plane. The motion of the body is therefore a motion of
translation.
DEF. Equal angular velocities existing in opposite direc-
tions in a rigid body form a couple of rotatory motion.
Hence a velocity of translation may be replaced by a pair
of equal and opposite angular velocities about axes which are
parallel to one another and perpendicular to the direction of
that velocity, the distance of the axes being the velocity of
translation divided by the angular velocity.
107. Hence the motion of a body can always be reduced
to two rotations at most about two axes. For after its motion
has been reduced to a translation and a rotation, the former
may be replaced by rotations about two axes, of which one
meets the axis of rotation, and the three rotations may then be
compounded into two at most. (103, 106, 68.)
108. Couples of rotatory motion are equivalent if the
planes through the axes are parallel and the product of the
distance of the axes and the angular velocity about each be
the same. A couple of rotatory motion may be resolved in any
60
MOTION OF A FREE RIGID BODY.
plane different from its own by multiplying its magnitude by
the cosine of the angle between that plane and its own.
109. When the motion of a rigid body referred to rect-
angular co-ordinates is expressed by the velocities u, v, w of
the origin of co-ordinates, which is either actually or hypo-
thetically in rigid union with the body, together with angular
velocities wr, wy, w about the same axes, these angular veloci-
ties are the same whatever origin be adopted.
Wx
%
For let u', v', w' be the linear velocities of a point a, ß, y,
wx,
,wy, w the angular velocities about it when it is made
origin. Then expressing the velocities of a particle x, y, ≈ on
each supposition we have
u + zw₁ − yw₂ = u' + (x − y) w,' - (y – B) wś
- Y
a)w%' (8
v + x w% − zw₂ = v′ + (x − a) w₂ − (≈ − y) we'
w + ywx − x wy= w' + (y − ß) w,' − (x − a) wy
-—
These equations hold for all values of x, y, and z.
Wx
Wx
Wy
Wy
ωχ ω wx
20
y
Since the angular velocities are independent of x, y, z and
the latter are unconnected,
..
ω.
·· wx = wx', wy = wy', w₂ = wx.
Thus whatever origin be adopted the motion of rotation
whereby the body's motion is exhibited is the same, while the
motion of translation in general varies.
u = u + γ = βοη
110. Since
γων
v = v + a wz
-
γως
w' = w + ßwx − awy
the linear velocity is the same for all origins along any straight
line in direction of the axis of rotation.
MOTION OF A FREE RIGID BODY.
61
111.
If the origin a, ß, y be such that the motion of
translation of that point is in direction of the axis of rotation
about it,
11
w'
W%
v
(၂) ရှာ
Wy
u + yw y Bw x
v + awx yw x
w + Bwx - awy
W%
or
Wy
wx
so that the locus of origins with this property is a straight
line in direction of the axis of rotation.
12
112. The same result is obtained by making u²²+v²²+w
a minimum by the alteration of a, ß, y; so that origins which
give the least velocity of translation give that translation in
direction of the axis of rotation.
DEF. The axis of rotation thus determined may be called
the central axis of the motion by analogy to the line which
receives the same name in Statics.
113. If V and be the velocities of translation and
rotation which express the motion of the body when the axis
of the latter is in direction of the former, the velocity of any
particle at distance from this central axis is √/V² + Q²r².
Particles therefore which have the same velocity lie in cylin-
drical surfaces about the central axis.
114. The motion of a free body is thus exhibited by the
velocity of any assumed point really or hypothetically in rigid
union with the body, and a velocity of rotation about an axis
through that point. For certain positions of the point the
Hence, in
axis is also the direction of the velocity oftram the point the
the latter view of it, the motion of the body is reduced to a
screw like motion, a motion of rotation about an axis co-exist-
ing with a motion in direction of that axis.
115. A body possesses given motions of translation and
rotation to find the condition that its motion may be ex-
hibited by rotation about a single axis.
62
MOTION OF A FREE RIGID BODY.
Let the body have velocities of translation u, v, w, in
directions of the rectangular axes of co-ordinates, and rotations
w, wy, w about axes parallel to these originating in the point
a, B, y, whose velocity, supposing it in rigid union with the
body, is the velocity of translation.
If possible let the body's motion be exhibited by a single
angular velocity compounded of three rotations N, Ny, Qx
about axes parallel to the co-ordinate axes and meeting in a
point a', B', y'.
The motion of the body will not be affected by pairs of
opposite angular velocities equal to these being impressed
Its motion then consists of
about axes meeting in α, B, Y.
(1) rotations, Q, Q, about the point a, ß, y,
(2) translations ₂ (B − ß′) − 2, (v −y′) in direction of x,
Ω (γ-γ) - Ω, (α - α')
y,
2₁(α-a') - 2 (B-B')
Ꮖ
Z.
If these then are equivalent to the given motions of the
body,
x
= Wx) Qay = Wy'
= Wxs
Q₂ (ẞ − ẞ') - Q₁ (Y — Y') = U,
2
Qx (y − y) − 2₂ (a − a′) = v,
Q₁ (aa) - 2x (ß – B') = w ;
Y
-
•. 0 = U . wc + v. wy + w.
This condition of the motion being reducible to a single
rotation is necessary but not sufficient: for it may be satisfied
by the evanescence of the velocities of rotation while the trans-
lations remain.
MOTION OF A FREE RIGID BODY.
63
116. The analogy of several of the preceding results to
propositions in the theory of the equilibrium of a rigid body
will hardly have escaped notice. The reader will probably
have been reminded of the composition of two parallel forces
(105), of the illusory case whereby a couple is presented
(106), of the laws of composition of statical couples (108), of
the reduction of any system of forces to a single force and a
single couple (103), or to two forces (107) in all cases, and
to a single force in certain cases (115), the single force being
independent of the origin adopted (109) while the couple
varies in general as the origin is changed and has its least
value when the origin lies in the line which Poinsot designates
the central axis (112).
Equations of Motion.
117. With these geometrical conceptions of the motion
of a free body, we may proceed to the calculation of that
motion under given forces, (1) finite, (2) impulsive.
i
118. The equations which determine the translation of
a rigid body and its rotation about its centre of gravity are
each the same as if the other did not exist.
I. Finite forces.
Let X, Y, Z be the finite accelerating forces parallel to the
rectangular co-ordinate axes which are impressed forces on a
particle of the body whose mass is m, and co-ordinates x, y, z.
The effective moving forces on this particle in the same direc-
ď² z
m ; and since the conditions of
dt2
tions are m
ď²x
dt'
dť
m
d² y
dt2
statical equilibrium are satisfied between those forces reversed
and the impressed moving forces taken throughout the whole
body,
64
MOTION OF A FREE RIGID BODY.
d2x
.::
.. 0 = Σm (X
dt2
y
0 = 2m (Y-d)
=
dt2
0 - 2m (z - da).
Ꮓ
Y
>
p) },
dt2
d²
0
o = 2m {(Z - A) y - (1 -
Σm {(x − da) x − (z
- das) *}
0 = Ση
0 = Ση
Y
Ꮓ
dť
· {(x - dy ) - ( x − da) y}.
X
dť
dt2
Let x, y, z be the co-ordinates of the centre of gravity of
the body, a', y', ' those of m referred to axes originating at
the centre of gravity and parallel to the former co-ordinate
axes;
:. x = x + x, y =ÿ + y',
..
Σ(mx') = 0, Σ(my') = 0,
x=x+ *',
Σ(mx') = 0,
Σ (mda) = 0, 2 (mdv) = 0, x (max)
Ση
=
=
Σ
Hence the former three equations give
X
Σ
= 0.
0 = Σm ( x − da ) = Σ(m X)
dx
Σ(m),
dt2
d t²
d²y
0 = Σ(m Y) ·
Σ(m),
(4).
dt2
d2z
0 = Σ(m2) -
Σ(m).
dt
MOTION OF A FREE RIGID BODY.
65
}
These are the equations of motion of a free molecule in
the position of the centre of gravity acted on by the whole of
the forces impressed on the body. Hence the equations for
determining the motion of the centre of gravity are the same
as if the body had no rotation.
Again, my
Σ
(my
d² z
d2%
d2
= y
dť
d t²
Σ(m) + Σ (my
dt2
day
d³y
Σmx
d t²
d t
Σ (m) + Σ (mx up)
d² y'
df
0 = Ση
*) y' -
dx) v -
(Y
-
- 1) +}
Y
) 5 - ( x
Y
−
1
) = }
Hence the latter three equations give
o - Σm {( Z -
Ꮓ
{(z.
+ 2 m ( z −
dt2
d²
1
z
dt²
d²
- Σm {(Z) - (Y -
Ση Ꮓ - y'
dt2
So 0 = Zm
0
= Σm {(x − da ')
dt2
d²y'
d²
d
d t²
d² y
1)} by (4).
dt2
d²
x'
-
(Z
-
A) ~},
(B).
X X
; = 2 m {(Y - 1 x ) x' - ( x − 1 ) y}.
d t²
d t2
Now these are the equations of motion which, if the centre
of gravity were a fixed point, would arise for determining the
motion of the body about it (78): therefore the equations
for determining the body's rotation about its centre of gravity
are the same as if there were no translation.
119. OBS. When the motions of translation and rotation
about the centre of gravity are said to be independent, the
meaning is only that the equations belonging to these two
parts of the motion are independent at any particular instant,
9
66
FREE RIGID BODY.
MOTION OF A FRE
and not that each motion continues for a finite time as if the
other had no existence. The forces which produce the rotation
for instance, depend generally on the position in space of the
particle on which they act, and thus they will vary with the
position of the centre of gravity, so that the rotation is thus
indirectly influenced by the translation. At any instant how-
ever the equations of a free particle's motion apply to the
centre of gravity, and the equations of motion about a fixed
point are true for the motion about the centre of gravity.
(Poisson, Iv. 5.)
II. Impulsive forces.
120. Let X, Y, Z be impulsive accelerating forces
parallel to the rectangular co-ordinate axes which are im-
pressed on a particle of the body x, y, z whose mass is m.
Let u, v, w the component velocities of this particle in the
same directions be suddenly changed to u', v', w' so that
m(u′ – u), m(v' — v), m(w' - w) are the effective impulsive
forces on m. Since the conditions of statical equilibrium are
satisfied by the impressed forces and the reversed effective
forces, both taken throughout the body,
–
0 = Σm (X − u' + u),
0 = Σm(Y - v' + v),
0 = Σm (Z - w' + w),
0 = Σm {(Z − w' + w) y − (Y − v' + v)≈},
-
0 = Σm {(X − u' + u) ≈ − (Z − w' + w)x},
-
0 = Σm { (Y − v' + v) x − ( X − u' + u)y}.
Let u, v, wo be the component velocities of the centre of
gravity which by the impressed impulses are suddenly changed
to u', v', w'.
.. Σm(u'
Σm (u' — u) = Σ(m). (u' — u).
-
Σm (v' — v) = Σ(m). (v′ – v),
Σm (w' — w) = Σ(m). (w' – w).
MOTION OF A FREE RIGID BODY.
67
Hence the first three equations give
u)
0 = Σ (mX) - Σ(m). (ù' – ù)
-
0 = ·Σ(mY) (m). (v' – v)
(4).
·
0 = Σ(mZ) - Σ(m). (w′ – w)
acted on by the whole of
Therefore the equations
motion of the centre of
These are the equations of motion of a free molecule in
the position of the centre of gravity
the impulses impressed on the body.
for determining the change in the
gravity are the same as if the body had no rotation.
Also let x, y, z be co-ordinates of the centre of gravity,
x = x + x', y = y + y', x=% + x'.
..the fourth equation gives
=
0 = Σ {m (Z − w' + w)y' – (Y − v' + v)x'}
+ÿΣm(Z − w' + w) − zΣm(Y − v' + v).
Σm {(Z – w' + w)y' − (Y − v' + v) x'} by (4)
Also 0 = Σm {(X — u' + u) ≈' — (Z — w' + w)y'}
0 = Σm { (Y − v' + v) x' − ( X − u' + u) y'}
v′ (X
-
(B).
These are the equations of motion which if the centre of
gravity were a fixed point would arise for determining the
change of motion of the body about it (99); therefore the
equations for determining the change in the body's rotatory
motion about its centre of gravity are the same as if there
were no translation.
121. The preceding theorems shew the advantage of
adopting the centre of gravity for the point whose motion is
the motion of translation (104).
122. If a rigid body move from rest by the action of a
couple it will turn about its centre of gravity.
68
MOTION OF A FREE RIGID BODY.
123. The proposition given in (118) supplies the equa-
tions which, when integrated and corrected by given cir-
cumstances of the motion at a given epoch, determine the
translation and rotation of a rigid body under given finite
forces. The equations of (120) lead to a determination of
the change in the body's state under given impulses in the
following manner.
124. A free body is struck by given impulses; to de-
termine the change in its motion.
Let the motion of the body before the impulses be ex-
hibited by translations u, v, w in direction of three co-ordinate
axes through the centre of gravity with rotations wa, wy, wx
about these same axes, and let accents denote the values into
which these elements of the motion are instantaneously con-
verted. Let Σ(X), Σ(Y), Σ(Z) be the algebraic sums of
the impulses resolved in directions of the axes, L, M, N their
moments about the axes.
(1) The centre of gravity has its motion altered as if the
impulses were applied to the whole mass collected at that
point: (120)
.. Σ (m). (u' — u) = E(X),
Σ (m). (v′ – v) = Σ (Y),
Σ (m). (w' — w) = Σ (Z).
(2) The rotation about the centre of gravity is affected as
if the centre of gravity were fixed. Hence with the notation
of (99),
L = A (w,' — w₂) - C′ (wy –
· C'
-
-
wy) – B' (w,' — w₂),
–
M = B (wý - w₁) - C' (wr' — w₂) - A' (w,' — w₂),
N = C (w,' - w₂) - B' (ws' - w₂) - A' (w,' – wy).
-
The new elements of the motion being thus obtained, the
MOTION OF A FREE RIGID BODY.
69
velocities of any particle x, y, z in directions of the co-
ordinate axes are known, being
u' + zw, - yw,' in direction of x,
v² + x W x
Z W
w' + yw; − xw,
y,
Z.
125. If two linear relations in x, y, ≈ make these three
velocities vanish, such relations are the equations to a line of
particles which are at rest immediately upon the application of
the impulses. Such a line is called an axis of spontaneous.
rotation and the initial motion is exhibited by a single rotation
about this. If such a line exist the preceding expression shew
that
0 = u w; + v' wy + w w %'
agreeably with (115).
126. If the axes of co-ordinates be the principal axes at
the centre of gravity, the equations of rotatory motion of a
body previously at rest become
Aw = L, Bw, M, Cw = N,
and the condition of the motion being one of rotation about a
single axis determinate in position, gives
L.E(X). M.Σ(Y)
0 =
A
+
+
B
N.Σ(Z)
C
127. Since the direction cosines of the instantaneous axis
L M N
are proportional to w', wy, w, and therefore to A'B'C
while the direction cosines of the resultant impressed impulse
are proportional to Σ(X), Σ(Y), Σ(Z), the preceding equation
states that the spontaneous axis, if it exists, is perpendicular
to the resultant impulse when the impulses admit of reduction
to a single blow. (Cambridge Math. Journal, Vol. IV.)
128. If the axis of spontaneous rotation were a fixed
axis, the point where the impulse is applied is a centre of
percussion relatively to it (60).
SECTION VI.
MOTION OF A SYSTEM OF RIGID BODIES.
129. WHEN a system of rigid bodies in motion influence
one another by certain given connections, by attraction, or
contact, or the action of strings or rods for instance, they will
exert two and two opposite forces on one another. Each body
when the forces thus in action upon it are taken into account,
may be treated as a free body, and the unknown mutual forces
are to be eliminated between the systems of mechanical equa-
tions which the several bodies supply and the geometrical
equations which express the relations of the parts of the
system to one another or to other bodies.
Similarly if constraints extraneous to the system act upon
parts of it, if the forces which these constraints exert be repre-
sented by certain assumed quantities and thus taken into
account, the system may then be treated as dynamically free,
though subject still to the geometrical conditions of position
which the said restraints impose.
Thus the equations of motion of each body of the system
may be constructed according to the principles of the preceding
sections. The equations which arise from mechanical prin-
ciples are limited in number, six namely at most for each body
of the system, because six equations are the necessary and
sufficient conditions of equilibrium of a rigid body. The
equations which geometry supplies are not limited in number,
and, if the problem be determinate, will with the dynamical
equations already mentioned make up a number of equations
equal to the number of quantities whose determination is
requisite for the complete solution of the problem.
MOTION OF A SYSTEM OF RIGID BODIES.
71
130. Since each body of the system would be separately
at equilibrium if the effective force on each molecule were
applied in a reversed direction (30), the whole system would in
this case be at equilibrium, and its state of equilibrium would
not be disturbed by supposing any or all of its parts rigidly
united. Hence the equations of equilibrium of the whole.
system may be constructed as if it were a single rigid body,
mutual impressed forces being hereby left out of consider-
ation.
131. If two bodies press against one another, whereof
one at least is smooth, and if a finite area of the surface of
each is in contact, the resultant mutual action will be at some
unknown point of this common area, the determination of
which is a part of the problem. If their surfaces have only a
point in common, the resultant action is in the normal to either
of the surfaces at this point, supposing that there is a deter-
minate normal to one of the surfaces at this point or to both
of them in common. A cube pressing against a sphere first at
a corner, secondly at a point of one of its faces, is an instance
of these kinds of mutual action respectively. The geome-
trical condition of this case is that the velocity of the two
points of contact in direction of the mutual action is the same.
If the normal to neither surface at the point of contact is
determinate, as when two cubes press against one another at
an angular point of each, the mutual action is unknown in
direction as well as magnitude, and part of the complete solu-
tion of the problem will be the determination of its direction
as well as its magnitude.
132.
When the surfaces of the bodies which press against
one another are both rough, in addition to the action already
described there will be mutual force perpendicular to it caused
by friction. When the points in contact have different velo-
cities the bodies are said to slide, and in this case the friction
= μ × normal pressure, where μ is a constant dependent on
the nature of the two substances only, and independent of
their velocities, and of the shape or extent of the surfaces in
72
MOTION OF A SYSTEM of RIGID BODIES.
contact*. When the surfaces touch at a point and when the
point of contact has the same velocity in each, the bodies are
said to roll upon one another. In this case the amount of
friction is unknown, and will be one of the circumstances which
the solution of the problem is to determine.
The preceding statement includes the case of a body press-
ing on a fixed surface. The geometrical condition of sliding
will be that the point or points of contact in the body have no
velocity perpendicular to the fixed surface at the point or
respective points of contact, and the condition of rolling that
the point or points of contact of the rolling body have no
velocity.
133. The forces arising from mutual action which have
been hitherto considered are called finite forces, having this
property, that they can produce no finite effect unless they
have acted during a finite time. Each particle of a system
under the action of forces of this kind in passing from one
velocity to another has had in succession all degrees of velocity
between these, and its path has been a continuous curve. Now
cases of mutual action occur to which this description will not
apply. If two bodies strike one another, if they be connected
by a string which after being loose becomes at last tight, if an
explosion occur in the interior of a body, then at the instant
of these events, the motion of any particle generally undergoes
a sudden change both as to velocity and direction, a change
not indeed really instantaneous, but occupying a time which is
inappreciable. The forces which act in this case are impulsive
forces, their definition being that they can produce a finite
effect in an inappreciable time.
If the laws of the internal constitution of bodies and of the
action of their molecules upon one another were known, such
circumstances as have been mentioned would introduce no new
feature. At any time during a collision or explosion the force
acting on any molecule would be known and its effect might
* These laws have been verified with great accuracy by M. Morin. An abstract
of his results is given by Mr Moseley. Illustrations of Mechanics. APPENDIX.
MOTION OF A SYSTEM OF RIGID BODIES.
73
be computed. But in want of such knowledge we are obliged
to reason upon the entire impulsive action from the beginning
to the end of its existence. This duration we neglect as inap-
preciable, and measure an impulsive force by the momentum
which it is capable of generating instantaneously on a free
particle on which it acts.
134. In problems where mutual impulsive action arises
the elasticity of the bodies is an important element. The
collision of two bodies produces a compression of each to an
extent depending on their hardness. Inelastic bodies are those
which after being compressed or extended have no power of
regaining their shapes: with such bodies therefore the condi-
tion of the end of impact is that the points in contact have the
same velocity in the direction in which the impulsive action
has taken place. Elastic bodies again exert a further action
on one another additional to this, and bearing a ratio to it
which depends only on their materials. Thus if R be the
impulse between two bodies, R₁ the impulse which would
have existed if they had been inelastic, R = (1 + e) R₁, where
e depends only on the substances of the two bodies.
1
135. In determining the instantaneous change of the
motion of bodies produced by impulse, finite forces may be
left out of consideration, for the effect of these latter is not
developed except in a finite time. The examination of the
motion of bodies in the course of which mutual impulsive
action occurs will usually consist of three stages, a calculation
(1) of the motion before the impulses under the finite forces
which may be in action, whereby the state of motion of the
system immediately before the impulses is known, (2) of the
state of motion into which the former is suddenly altered by
the impulses, and (3) of the subsequent motion under the finite
forces which may still continue in action.
136. Cases where the mutual force approaches to the
nature of impulse as here defined, but cannot be safely re-
garded as instantaneous, as for instance when a ram drives a
10
74
MOTION OF A SYSTEM OF RIGID BODIES.
pile, are discussed in Professor Whewell's Mechanics of Engi-
neering. Chap. XII.
137. In examining the action of impulsive forces we are
led to consider bodies which do not exactly fulfil the condition
of rigidity, but are capable of sustaining small compressions
and again of assuming their former forms by their elasticity.
The principle of D'Alembert, it will be remembered, includes
cases like these. (35).
138. The general method of determining the motion of a
system of bodies under the action of one another, or of given
fixed restraints, such as surfaces along which they roll or slide
has been already described (129). The differential equations
will be of the second order, and the unknown forces of con-
straint or mutual action must be removed by elimination be-
fore the integration can be attempted. This elimination may
often be avoided by aid of the general principles contained in
the following articles. These principles, which can be enun-
ciated as practical rules, enable us to arrive at once at the
result of one integration of the differential equations of motion,
and generally reduce the problem to the solution of a diffe-
rential equation of the first order.
It must be observed that these principles give no result
which cannot be drawn from the differential equations of
motion. They do no more than expedite the analytical opera-
tion of solution. This remark shews how any uncertainty in
the application of these principles can always be resolved, and
any results obtained from the use of them be checked.
I.
Conservation of the motion of the centre of gravity.
139. The centre of gravity of a system of moving bodies
connected in any manner has the same motion as if the mass of
the whole were collected at that point and acted upon by the
impressed moving forces which are extraneous to the system.
If the effective force on every particle act upon it in a
direction contrary to its real direction, the conditions of equi-
MOTION OF A SYSTEM OF RIGID BODIES.
75
librium of each body are satisfied between these and the whole
of the impressed forces (30). The conditions of equilibrium
are therefore also satisfied if the bodies become united into one
rigid body, in which case the mutual forces become molecular
and so cease to appear in those equations. In this case if
P, Q, R be the sums of the extraneous impressed forces re-
solved parallel to the rectangular co-ordinate axes of x, y, z,
m the mass of a particle x, y, z, M the mass of the system,
x, y, ≈ its centre of gravity,
0 =
O - P - Z (m)
d
x
ď x
P - M.
dt2
0 = Q - Σ (1
day
d²y
Ση
Q - M.
dt2
dt2
ď z
0 = R - Σ
Σ m
de) = Q - M.
d2%
•
dt
These are the equations of motion of a particle whose mass
is M, at x, y, z, under the forces P, Q, R. Hence the motion
of the centre of gravity of the system is that of a particle in
its position under the action of the whole of the impressed
forces which are extraneous to the system.
140. Since this investigation is independent of the nature
and magnitude of internal forces, the result is true if any col-
lisions among the parts or explosions arise. The motion of
the centre of gravity will not be influenced by action of this
kind.
141. If no forces extraneous to the system are acting,
the centre of gravity is either at rest or moves uniformly in a
straight line. This invariability of the motion of the centre
of gravity has given rise to the name of the principle now
under consideration.
76
MOTION OF A SYSTEM OF RIGID BODIES.
II. Conservation of areas.
142. If a system of bodies be in motion under forces
which, if the bodies were rigidly connected, would not tend to
produce rotation about a certain fixed straight line, then the
sum of the products of the mass of each particle and the area
which its projection on any plane perpendicular to that line
describes about the line is proportional to the time during
which the motion is considered.
Let the fixed straight line be the axis of x, m the mass of
a particle whose co-ordinates at time t are x, y, z; X, Y, Z
the accelerating forces impressed upon it in directions of the
co-ordinates. Since the forces would not tend to produce
rotation in the system about the axis of x, if the parts of it
were rigidly connected,
Σm(Zy – Yx) = 0.
.. Em y
Ση9
(3
d² &
dy
= 0, (130).
d t2
dť²
Σ
Em (y
dz
dy
Z
= c, a constant.
dt
dt
Let
plane y
A be the area which the projection of m on the
describes about a from the fixed time t₁ to the time t,
d
Σ (m +4)
= C,
dt
Σ(mA) = c(t − t₁) ;
or if T be the duration between the instants represented by
t and t₁,
143.
Σ(mA) = c T ∞ T.
α
The result of this proposition continues to apply
so long as the condition Em (Zy - Yx) = 0 is preserved, and
therefore is not affected by any changes which the system may
MOTION OF A SYSTEM OF RIGID BODIES.
77
undergo from any mutual internal forces, whatever be their
intensity or duration, such as those which arise in collision
between parts of the system or explosion.
144. If the three conditions
Σm(Zy - Yx)=0, Σm (Xx-Zx) = 0, Σm (Yx - Xy)=0, (1)
are simultaneously satisfied, so that if the system were rigidly
united the forces would not tend to produce motion about the
origin, then the sum of the masses of every particle each
multiplied into the area which its projection on any plane
through the origin describes about the origin, is proportional
to the time during which the motion is considered.
For if A, A, A, designate the doubles of the areas which
the particle m describes in the interval T on the co-ordinate.
planes, to each of which the reasoning of (142) now applies,
Σ(m Ax) Σ(m Ay)___ Σ(m Az)
C1
C2
C3
T;
and if A be the area similarly described in projection on a
plane whose direction cosines are l, m, n,
Σ(mA) = Σm (A¸ . l + Ay . m + Az. n)
(c₁l + c₂m + c¸n) T∞ T.
α
145. The expression c₁l + c₂m + c¸n is a maximum while
1² + m² + n² = 1,
1, if
m
N
1
2
C1
C2 C3
C₁² + C₂² + C3
2
2
Since these values of the direction cosines are invariable,
the plane on which the sum of the masses of the molecules
each multiplied into the projected area which it describes
thereon in a given time, is the greatest possible, remains per-
manent in position so long as the original conditions (1) are
fulfilled. The plane is hence called the invariable plane of the
system with reference to the assumed origin of co-ordinates.
78
MOTION OF A SYSTEM OF RIGID BODIES.
In examining the system of the universe where all the
actions are mutual and the conditions (1) therefore obtain,
this plane supplies a plane of reference which is unaffected by
any alterations which take place in the constitution of any
parts of the system. Its utility in the particular case of a single
body in place of a system of bodies has been seen in (86, 87).
146. (mA) = 0 if c₁l + c₂m + cn= 0, that is, if the
projections are made on a plane perpendicular to the invariable
plane.
147. Since the functions C1, C2, C3 are generally different
when the system is referred to different origins, with respect
to which the moments of the forces vanish, hence the in-
variable plane generally differs in position when it is determined
in reference to these different origins.
If c, c, c' be the values of the parameters when the
origin is moved to the point a, b, c; M the mass of the whole
system and u, v, the velocities of its centre of gravity
parallel to the co-ordinate axes, which are supposed to retain
permanent directions,
W
dy
dz
=
cí = Em
2 C
-y
b
dt
dt
c₁ – M (cò – bπ),
c = c - Mao cũ),
c3′
α
C3 C3M(bu - аõ).
=
Hence the position of the invariable plane is the same for
all origins along a line in direction of the motion of the centre
of gravity. If the centre of gravity has no velocity, the
position of the invariable plane is independent of the origin
selected.
148. The invariable plane at the origin a, b, c, is
X{c₁ - M(co - bw)}
+ Y{c₂ − M(aw cũ)}
+ Z {c. - M(bu – а v)}
M (bù − a )} = a + be + CCgo
CC3.
MOTION OF A System of RIGID BODIES.
79
149. The sum of the masses of each particle of the
system multiplied each into the area which its projection de-
scribes in a given time on a plane through the origin a, b, c,
is greatest, as has been seen, when that plane is the invariable
plane, and in that case = √c," + c₂+c". If now the origin
a, b, c be such that this expression, the maximum with respect
to different planes at that origin, may be the minimum with
respect to different origins,
C1
12
12
0 = c₁'dc'' + c₂ dc2' + c3'dcg',
0 = c(vdc - wdb),
+ c₂ (wda - ūdc),
+cs (udb - vda),
whence since da, db, de are independent,
cí
C3
И
v
พ
or the origin required has this property, that the invariable
plane belonging to it is perpendicular to the direction of the
motion of the centre of gravity. Such an origin will have for.
its locus a line in direction of the motion of the centre of
gravity.
150. If x, y,
be co-ordinates of the centre of gravity of
the system, a', y, z′ those of m referred to parallel axes origi-
nating at the centre of gravity,
Σm (
ar
dy
dt
dx
dy
dx
Y
X
y
Σ (m)
dt
dt
dt
x'
dt
dy
da
Hence the function Em x
dt
dt
+ Σm (a'dy - y' da)
dt
y is evaluated by
adding together its values obtained on the separate suppositions,
(1) that the whole system was condensed at its centre of gravity
80
MOTION OF A SYSTEM OF RIGID BODIES.
into a point, (2) that the system is in motion about its centre
of gravity fixed, the two portions of the function in short due
to the motions of translation and rotation.
III. Principle of Vis Viva.
151. DEF. The vis viva of a particle in motion is the
product of its mass and the square of its velocity.
The vis viva of a system in motion is the sum of the
vires vivæ of its particles.
152. If X, Y, Z be the accelerating forces on a free
particle in directions parallel to the co-ordinate axes, m the
mass of the particle, the change of its vis viva in passing from
one position to another in the course of its motion
= 2m f(Xdx + Ydy + Zdx),
the limits of integration being those two positions.
153. If a rigid body be revolving about an axis with an
angular velocity w, and if r be the distance from that axis of
a molecule whose mass is m, the vis viva of the particle is
mw²r² (39), and the vis viva of the body is Σ(mr³). w², or
the body's moment of inertia about the axis multiplied by the
square of the angular velocity.
154.
The vis viva of a body is the sum of the vires
vivæ which it would separately have if it possessed the motion
of translation of the centre of gravity or the motion of rotation
about that centre separately.
Let x, y, ≈ be the co-ordinates at time t of a particle of
the body of mass m, x, y, ≈ those of the centre of gravity,
x', y', 'the co-ordinates of m referred to axes through the
centre of gravity parallel to the former;
Σ(mx') = 0,
. . x = x + x',
Σ(my') = 0,
y
y = ÿ + y',
z
Σ(mx') = 0.
≈ = ≈ + x',
MOTION OF A SYSTEM OF RIGID BODIES.
81
Σ (mar) = 0,
2
2 (mdv) = 0, 2 (mdx) = 0.
Σ
..
dx'
dt
dy'
2
dt
dz
dt
Σ {m (de) } - (de) " . 2 (m) + Σ {m (da")"},
dt
…. the vis viva of the body
Ση
2
dx
dy)?
- Σm { (da) " + (1/4)² + (dt)"}
=
dx
2
dt
dy
2
dt
{(147)* + (17)² + (da)} = (m)
dx
2
dt
dy'
2
t
2
dt
+ Σm{ (da)² + (dx)" + (dz)}.
dt
dt
dt
=
Of the two terms of this expression the former is the vis
viva which the body would have if every particle possessed
the velocity of the centre of gravity, the latter is the vis viva
which the body would have if it were moving about the centre
of gravity fixed. The sum of these is the vis viva of the
body.
155. Statement and proof of the principle of vis viva.
If a system of rigid bodies be in motion under the action
of finite forces, the geometrical connexion of the different
parts being expressed by equations which do not contain the
time explicitly and remain invariable in form, the change of
vis viva of the system in passing from any one position to
another is the same as if the particles had been free and had
been acted upon by the same impressed forces through the
same spaces.
Let x, y, z be rectangular co-ordinates of a particle of the
system whose mass is m, X, Y, Z the accelerating impressed
forces on the same in directions of the co-ordinate axes.
11
82
MOTION OF A SYSTEM OF RIGID BODIES.
d2x
Then m
d² y
m
m
,
,
d t
డిజ
dt2 d t²
are the effective moving forces
on this particle in the same directions, and if such effective
forces be applied in reversed directions to the particles to
which they belong, the conditions of equilibrium are satisfied
among the reversed forces and the impressed forces (30, 130).
Let the system receive a change of position in consistence
with the connexions of its parts and let the displacement of m
parallel to the co-ordinate axes be ultimately expressed by
Sx, Sy, Sz as the alteration of position is reduced indefinitely.
Then by the principle of virtual velocities
бх
Sy
0 - 2 m {(x - 1) dx + (Y - 1) by + ( Z - dx) dx}
d
dť
Let L = 0 be one of the equations expressing the geo-
metrical connexions of the system by exhibiting a relation
among the co-ordinates of certain of its particles. Since the dis-
placement already imagined is consistent with such connexions,
L
0 =
d L
dx
d L
Sx+
Sy +
dy
(1).
Now in the actual motion of the system such equations as
O continue true by the limitation of the statement of the
motion considered. Hence L must be invariable under a
change of time and consequent changes of position;
or 0 =
d L
dt
St +
dL dx
dx dt
St+
dL dy
dy dt
St +
(2).
d L
Hence if
dt
O, and if the same be the case in the other
equations of its class, we can in consistence with equations of
the kind (1) and (2) assume
dy
dx
бас
=
Sta
бу
St....
dt
dt
or by reason of the expressions of connexion between the
MOTION OF A SYSTEM OF RIGID BODIES.
83
parts not involving the time explicitly, we may ultimately
identify the hypothetical displacements which the principle of
virtual velocities allows with the real changes of position which
the actual motion produces. Denoting these latter by da,
we then have
dy,
Ση
(dx dx dy dy dz dz
de +
dť² dt
Ση
d t² dt
dx
dt
2
+
+
dt dt
(dy)
dt
2
+
St=Σm(Xdx+Ydy+Zdx);
(4)
~
2Σm f(Xdx + Ydy + Zdz), (A),
the integral being taken between the two positions between
which the system is considered to pass, so that the correction
involved is the vis viva in the former of those positions. The
former member of the equation is the vis viva of the system
in the latter of the two positions.
Now 2mЛ(Xdx + Ydy + Zdx) is the change which would
have arisen in the vis viva of the particle m if it had been free
and had passed from one to the other of its positions under
the accelerating forces X, Y, Z, (152). Hence the change of
the vis viva of the system between the two positions is the
same as if the particles had been free and had been acted on
by the same impressed forces through the same spaces.
156. A reference to the principle of virtual velocities on
which the preceding proposition has been demonstrated, will
shew that forces of the following kinds will not appear in the
latter member of equation (4).
(1) Mutual forces arising from the pressure of parts in
contact or the tensions of inextensible strings.
(2) Reactions of fixed rough surfaces on which parts of
the system roll, or of fixed axes, because the points where these
forces act have no velocity (132).
84
MOTION OF A SYSTEM OF RIGID BODIES.
(3) Reactions of fixed smooth surfaces along which parts
of the system slide, because the points where these forces act
have no velocity except in directions perpendicular to the
directions of the forces.
By evading forces of these kinds the principle of vis viva
shews its usefulness, and expedites the solution of problems.
It must be remembered that the application of the principle is
limited to cases where the actual motion is of a kind which
may be employed as a hypothetical change of position ad-
missible in using the principle of virtual velocities.
157. The expression Xdx + Ydy + Zdx can be inte-
grated as a perfect differential whenever the forces tend to
fixed centres at finite distances, and are functions of the dis-
tances from the centres, under which characters all forces
existing in nature except molecular force will be comprehended.
For if X, Y, Z be components of a force tending to a
centre whose co-ordinates are a, b, c and acting at x, y, x, and
in intensity a function of its distance r from this centre,
X
Y
X
α y − b
Z
Z C
= − P(r)
while r² = (x − a)² + (y − b)² + (≈ − c)³.
·. (x − a) dx + (y − b) dy + (x − c) dz = r.dr,
.. Xdæ + Ydy+ Zd
-
= − q(r). dr.
If the forces X, Y, Z are the components of several forces
of this character, Xdx + Ydy + Zdx is the sum of a cor-
responding number of perfect derivatives.
158. Also Σm (Xdx + Ydy + Zds) will be a perfect
differential as far as it results from the attractions of one
particle of the system on another at a finite distance.
MOTION OF A SYSTEM OF RIGID BODIES.
85
For if x, y, z be co-ordinates of m, x', y', x' of m', then
the terms of the function which depend on their mutual
attraction are
x'
-
y
m (m' * = * da + m'¹º = " dy + m' * = * dz) ¢(1)
+ m² (m
since
X
r
dx
2°
-
2
m³ = "'dy' + m² = * dz') ((r)
dx' + m
ጥ
-mm'p(r). dr,
159.
r
о
p² = (x′ − x)² + (y′ − y)² + (≈′ − 2)³.
Hence under the forces which exist in nature,
change in vis viva
=
2Σm fp (r) dr
= Σmf(x, y, ≈) – Σmf(x。, Yo, %),
where ≈。, y。, ≈。 are the co-ordinates in the initial position of
the molecule m whose present co-ordinates are x, y, z, the
symbol expressing a geometrical integration performed
through the whole system in either position.
Since the change of vis viva depends only on the initial
and final positions of the system and on the law of the forces
to which it is subject, it is therefore independent of the paths
which the molecules have described in passing from the one
position to the other.
160. When a force acts upon a body in motion the effi-
ciency or 'work' of the force between two positions of the
body is measured by the product of the force and the space
through which its point of application moves in its direction,
when the force and its direction are constant: when either of
these varies, the efficiency is obtained by integrating the ele-
mentary value of it obtained on the preceding definition.
86
MOTION OF A SYSTEM of RIGID BODIES.
161. Let mX, mY, m Z be the pressures acting at time
t parallel to the co-ordinate axes on a particle m of the system.
Then as the system passes into a consecutive position
m(Xd + Ydy+Zds)
is the efficiency expended by the forces applied to m, and
Em(Xd + Ydy+Zds)
is the efficiency expended on the whole system.
Hence the efficiency expended between any two positions
of the system = Σm f(Xdx + Ydy + Zdx), the integral being
taken from one of those positions to the other, and if the
motion of the system obeys the principle of vis viva, this is
half the change of vis viva in passing between the same states.
162. The principle of vis viva in the form of the last
article has an important use in calculating the motion of
machines which consist of rigid parts united together and com-
municating motion one to another, provided the nature of the
connexions fulfils the conditions of motion by which the truth
of the principle of vis viva is limited. In cases like these the
function Em(Xdx + Ydy + Zdz) consists of three parts
arising (1) from the forces applied to the machine as the
primary cause of its motion, (2) the internal forces from the
weight and friction of the parts which are not included in the
exceptions of (156), and (3) the reactions of the bodies on
which the machine is designed to act and in which it produces.
motion. Forces of the first kind tend to augment the vis viva
of the system, while those of the latter two generally tend to
diminish it, and therefore the efficiency of these latter is in
general negative. Hence the change of vis viva of the machine
between two states is double the excess of the efficiency applied
to it by the original moving agent over the sum of the effici-
ency consumed within itself and that exhibited on the objects
on which it is employed.
When a machine is working uniformly the efficiency
applied to it in any interval equals the sum of the efficiency
motion of a system of rigid bodies,
87
consumed internally and that transmitted effectively. When
the action of the machine is periodic so that its vis viva re-
turns to the same state after given intervals, the same state-
ment applies to each of such intervals. If the machine was at
first at rest, efficiency must have been expended in generating
the vis viva which is thence maintained uniform or periodic in
the manner above mentioned.
Since the internal resistances of a machine usually consume
part of the efficiency applied to it, the efficiency of an agent is
generally impaired when it is transmitted through machinery,
and in the most favourable case can only equal the efficiency
exhibited. A machine therefore is advantageous not in in-
creasing the efficiency of the agent which works it, but in
applying the power of the agent more conveniently.
163. Hence no machine can by itself maintain its own
motion for an indefinite time and also do work, or what is
understood by perpetual motion is physically impossible under
the forces presented in nature. See a paper by Mr Airy on
this subject in the third volume of the Transactions of the
Cambridge Philosophical Society.
164. A fourth general principle of the motion of a system
called the principle of least action, which has generally been
added to those already given, is here omitted, because as an
auxiliary in the solution of problems it is nearly useless, and
as a theory, it is only a particular case of a more general
principle developed by Sir W. R. Hamilton, in the Philo-
sophical Transactions of 1834 and 1835, and to these students
for whom such reading is expedient are referred.
APPENDIX.
THE following proposition, which is frequently required
when two systems of rectangular co-ordinate axes are employed,
is introduced in this place to avoid interrupting the subject of
the section in which it arose.
The convertibility of the two groups of equations between
the direction cosines in (63) and (64) is manifest from each
being the expression of rectangularity of a system of axes
referred to another rectangular system, and one or the other
group of equations arises as one or the other system of axes is
made the system of reference. The following however is an
algebraic proof.
;
2
2
2
2
2
2
(a₁² + b₁² + c₁² − 1)² + (a½² + b²² + c₂² − 1)² + (a²² + b²² + c3² −1)²
2
+ 2(a¸α₂+b3b₂+C3C2)²+2 (α₁α3+b¸ b²+С₁C3)²+2 (a¸α₂+b¸ b₂+c¸С₂)²
=
2
=
2
1
Σ (a¹) + 2(a¸² b¸²) − 2 Σ (a²) + 3
2
+ 2 Σ (a,₂² α²²) + 4 Σ (ɑ2α¸ b₂b3)
2
= Σ (a¹) + 2 Σ (a,³ a¸³) − 2 Σ (a²) + 3
3
1
+ 2 Σ (a¸² b₁²) + 4 Σ (ɑ₂b₂ ɑ¸b¸)
2
2
2
(a₁² + a₂² + a3² − 1)² + (b,² + b² + b²² − 1)² + (c²² + c₂² + c3² − 1 )²
3
+2(b₁C1+b₂C₂+b3C3)²+2(a¿C1 + A2 C2 + A3C3)² +2 (ɑ₁b₁+ɑ½b₂+ɑ¸b₂)².
2
Hence if one member of this equality vanishes by virtue
of the one of the two groups of equations which is given, the
six squares which form the other member being in consequence
separately zero, give the six equations which form the other
group whose truth is to be established.
EXAMPLES.
REFERENCES are generally attached to the following ex-
amples in order to suggest the principles which they are seve-
rally intended to illustrate or apply. The order of arrangement
adopted in the preceding sections of this book has been fol-
lowed as far as possible.
1.
I. Geometrical properties of a rigid body.
Evaluations of the functions (mys), (mxx),
Σ(mxy) in particular instances, the axes being rectangular
and the body homogeneous; M the mass of the body.
If p be the density of the body at the point x, y, ≈, these
functions are equivalent to the definite integrals z√y SzpYx,
Sz Sy Szρ xx, Safy Szρxy respectively, the limits of integration
including the whole body.
(a) A circular lamina in the plane xy, a, b co-ordi-
nates of its centre.
Σ (myx) = 0, Σ (mxx) = 0, Σ (mxy) = Mab.
(B) A lamina in the plane ry whose boundary is sym-
metrical with regard to the axes of x and y.
Σ(myx) = 0, Σ (mxx) = 0, Σ(mxy)
= 0.
(y) A right-angled triangle whose sides a, b are in
the axes of x and y,
Σ(myx) = 0, Σ (mxx) = 0, Σ (mxy) = 1½ Mab.
2
12
90
EXAMPLES.
(8) A straight line where a, b, c and a', b', c' are the
co-ordinates of its extremities,
Σ (myx) = † M {2 (bc + b'c') + bc′ + b′c}
Σ (mxx) = } M {2(ac + a'c') + ac' + a'c
Σ(mxy) = }M {2(ab+a'b') + ab' + a'b}.
(e) A sphere whose centre is a, b, c,
Σ(mys) __ Σ(mx x) Σ (mxy) M.
bc
ac
ab
=
(8) In a plane lamina referred to such rectangular
axes in its plane that lines parallel to them through the centre
of gravity (xy) are lines of symmetry,
Σ (mxy) = Mxy.
(1) A circular right cone is referred to its vertex as
origin and such axes that the plane of xy passes through its
axis and a generating line is the axis of x.
3
Σ (mxy) = M sin a. cos a a² -
15
GO
3
20
where a is the semi-vertical angle of the cone, a its altitude
and r the radius of its base.
2.
ཉ.
In any body Σ(mx²). Σ (my²) > {Σ(mxy)}².
Σ (mx²). Σ (mx²) > {E (mxx)}².
Σ (mx²). Σ (my³) > {Σ (mxy)}².
The moment of inertia of a circular arc about an axis
through one extremity of it perpendicular to its plane
= 2 Mr² (1 - sin 0},
Ꮎ
r being the radius and ✪ the angle subtended. (8).
GEOMETRICAL PROPERTIES OF A RIGID BODY.
91
4. The moment of inertia of the whole arc of a circle
about a diameter = 1 Mr². (9).
5. An oblate spheroid consists of similar strata each in-
definitely thin and uniform in density. Its moment of inertia
about its axis of figure
8
π
√1 −
1 - e³ √ pa²,
α
where p is the density of the stratum of which 2a is the equa-
toreal diameter, e the eccentricity of the strata.
sin na
Let P = C
α
for instance.
6. The sum of the moments of inertia of a rigid body
about any three rectangular axes at the same origin is the
(10).
same.
7. If a point is taken in one of a system of three principal
axes, axes through this point parallel to the other two original
principal axes form a new system of principal axes when the
first axis passes through the centre of gravity of the body, but
not generally in other cases.
8. If a body satisfy the conditions Σ (mx²) = Σ (my³)
Σ (m²) when referred to a system of principal axes, any
other rectangular system of axes with the same origin are also
principal axes.
9. The principal axes of a right-angled triangle at the
right angle are one perpendicular to its plane, and two others
in its plane and inclined to its sides at the angle
tan - 1
ab
2 (a² – b³) *
(17, 18).
10. If a point in the circumference of an elliptic board
92
EXAMPLES.
be made origin, the position of the principal axes in its plane
is given by
tan 20 2
α
a²
2aß
· B² − 1 (b² — a³)
where a, ẞ are co-ordinates from the centre of the assumed
origin.
11.
(17).
The moment of inertia of a cube about a diagonal
= }Ma², (25), and about a diagonal of one of its faces
5
= 12 Ma²,
Ma³, a being an edge of the cube.
12.
(25, 5).
The moment of inertia of a circular right cone about
6a² + c²
a generating line
3
20
28 Mc².
a being the altitude and
a² + c²
(24).
e the radius of the base.
13. The moment of inertia of an elliptic area about any
diameter is proportional to the square of that diameter in-
versely. (21).
14. The moment of inertia of an ellipsoid about a
diameter = (a + b² + c² - p²) M, where p is the perpendicular
from the centre on the tangent plane perpendicular to the
diameter. (21).
15. The moment of inertia of a spheroid about all axes
through one of its poles is the same. Its axes are in the
ratio of 1: 6. (25).
16. A circular right cone whose altitude is half the
radius of its base has the same moment of inertia about every
axis through its vertex.
II.
(25).
Motion of a body about a fixed axis.
1. A body revolving about an axis has a velocity a sec² A.
It will make each revolution in half the time in which it would
revolve with the uniform angular velocity a. (38).
MOTION OF A BODY ABOUT A FIXED AXIS.
93
2. If a rigid body has an angular velocity w about the
x y Z
straight line
the velocities of a point x, y, z in
1 m
n
direction of the co-ordinate axes are
mz ny
nx โช
ω,
W₂
√ 1² + m² + n²
√ F² + m² + n²
ly – mx
√ 1² + m² + n²
ω.
3. A hemispherical surface whose radius is r makes small
oscillations about a diameter of its base in time π
42°
3g
. (49).
4. A wire is bent into the form of an arc of a circle, and
makes small oscillations about a horizontal axis through its
middle point perpendicular to its plane in 1.5708 seconds.
The radius of the arc is four feet nearly. (49).
5. A sphere revolves about a given chord as a vertical
axis. Find the pressure on the axis and explain the result
when the chord becomes a tangent. (42).
6. A square revolves about a vertical edge. Shew
whether under any assumption of angular velocity the result-
ant pressure on the axis can ever act (1) through the highest
point, (2) at the middle point. (42).
7. A rod revolves in a vertical plane about its upper end.
Shew in what positions of the rod the pressure on the point
of support is (1) in direction of the rod, (2) perpendicular
to the rod. (46).
8. A cylinder whose axis √x radius of base, is
capable of revolving about a horizontal diameter of its base
and falls from the position wherein the axis is horizontal. If
the axis and the direction of the pressure upon the fixed
diameter make with the horizon the angles and respec-
tively in any position of the body, 4 tan (-4) = cot p.
(46).
94
EXAMPLES.
9. When the centre of gravity of a revolving body is
vertically under the horizontal axis the pressure on the axis
resolved horizontally does not vanish unless the origin can be
so taken that (mxx) = 0, x being the axis of revolution and
a also horizontal.
(42).
A hemisphere revolves about an axis which coincides with
a diameter of its base and is inclined at an angle a to the
vertical. If it swings through 180° the whole pressure on the
axis in the lowest position
64
{(109 sin a)² + (64 cos a)²} × weight of the body. (42).
10. An oblate spheroid revolves about a horizontal axis.
Find the position of the axis when the time of small oscillation
is the least possible. (49).
11. A cube revolves about a horizontal edge so as to
make complete revolutions. The axis must at least be able to
bear four times the weight of the body. (41, 46).
12. A sphere swings about a horizontal tangent. Its
pressure on the axis in its lowest position is less than the pres-
sure which it would exert if its whole mass were collected at its
centre of oscillation and oscillated by a weightless string by
Mro, r being the radius of the sphere and w its angular
velocity in the position contemplated. (46).
& w
13. Two equal rods can turn freely in their own plane
about a hinge at the lower extremity, and have their upper
extremities connected by a horizontal string. If the rods
revolve about a vertical axis through the hinge and in their
plane with an uniform angular velocity w, the tension of the
string
= 1 Mg tan a + M w² a sin a,
where M is the mass and a the length of each rod, 2a the
angle between the two.
MOTION OF A BODY ABOUT A FIXED AXIS.
95
14. A pendulum consists of a cylinder of one metal hung
by a rod of another metal which passes through an orifice in
direction of its axis and is attached to the centre of its base.
Find the relation between the dimensions of the parts and
expansibilities of the metals that the system may form a com-
pensation pendulum.
15. Two spheres are connected by a thin rod whose ends
penetrate each of them and are attached to them at their
centres. Find the relation between the dimensions and expan-
sibilities of the parts that this system may form a compensation
pendulum when it oscillates about a given point of the rod.
=
16. The two parts m, m' of a compound vibrating body
have their centres of gravity H, H', centres of oscillation O, O'
and point of connexion P all in a straight line through S the
point of suspension: SH=h, SH' h', SO = l, SO′ = l,
SP e, rr' the expansions of linear units of the masses for a
given rise of temperature. The length L of the compound
pendulum is unaltered by temperature if
=
L
2
mrhl + m'r' h' l' + m'h'e (r− r)
mrh + m'r'h' + m'e (r − r')
17. A rod is supported at its two extremities and rests in
a horizontal position. If one of the supports is removed so
that the rod begins to turn about its other extremity, the pres-
sure on the other support is suddenly diminished by one half.
(46).
18. A rod is swinging about one end and when it is
horizontal, the other end is suddenly fixed. The impulses on
the points of attachment of the two ends are as 2: 1. (59).
19. A square is revolving about one of its sides as a
fixed axis with a given angular velocity. Find the blow which
applied at one of its angular points perpendicularly to its
plane will at once destroy its motion.
the accompanying impulse on the axis.
(58).
(59).
Determine also
96
EXAMPLES.
20. A circle is revolving about a tangent line in its plane.
If it be struck at any point except in the diameter perpen-
dicular to that tangent, there must be a fracture at the point
of attachment. (59).
21. An elastic rod swinging about its upper end impinges
on a fixed vertical plane perpendicular to the plane of its
motion. Its angular velocity is reversed and altered in the
ratio of 1 e by the impulse. (58).
22. A rod of mass M and length 2a hangs vertically
downwards from one end. It will be thrown into its position
of unstable equilibrium by the blow M√3ag, the axis
sustaining no impulse. (41, 58, 59).
=
23. A circle hanging at rest is capable of turning about
a horizontal tangent line. Find how it is to be struck that
there may be no impulse on the axis and that it may rise
through 90º. (41, 58, 59).
24. A rectangle capable of turning about one of its
diagonals is struck perpendicularly at one of its angular points.
Find the impulse on the axis and shew whether any form of
the rectangle will allow it to vanish. (59).
25. When a lamina in the plane of xy capable of turning
about the axis of y has a centre of percussion, the blow must
be perpendicular to its plane and at a point whose co-ordinates
are
a'
Σ(m x²)
Σ (mx)
y
Σ(mxy)
Σ(mm)
(60).
26. A sector of a circle whose radius is a and angle a, is
capable of turning about the axis of which is in its plane
and perpendicular to one of its bounding radii. If this radius
be axis of x, and the centre the origin, the co-ordinates of the
centre of percussion are
За α
30 (a + cos a),
8 sin
મ
3 a
sin a.
(60).
8
MOTION OF A BODY ABOUT A FIXED AXIS.
97
27. A circular right cone is capable of turning about an
axis through its vertex. If it be struck perpendicularly to its
curved surface, it does not admit a centre of percussion. (60).
28. A body capable of turning about a fixed axis is
struck by a couple in the plane Lx + My + Nx=0, some point
of the axis being origin and the principal axes at the same
point the axes of co-ordinates. Prove that the greatest angular
velocity will be produced if the axis be so taken that its equa-
Ax By Cz
A, B, C being the principal moments
tions are
of inertia.
L M N
>
[Compare (102)].
III. Motion of a rigid body about a fixed point.
. 1. When a body is turning about the origin fixed the
velocities of points of it which are in the axes of x, y, z, and
each at a distance a from the origin are at a given instant
v1, v2, v3 respectively. Find the equations to the instantaneous
axis at that instant. (63).
2. If wx, wy, w, be the angular velocities about the co-
ordinate axes by which the motion of a body about the
origin may be exhibited, find the locus of the particles whose
velocity is a w (63).
3. The locus of points in a body revolving about a fixed
point which have at a proposed instant the same given velocity
is a circular cylinder.
4. If the motion of a body about a fixed point be ex-
hibited by angular velocities wa, wy, w, about the fixed co-
ordinate axes, or by angular velocities w₁, w2, w3 about rect-
angular axes fixed in itself,
2
2
2
2
2
wx² + wy² + wz² = w¸² + w₂² + wz².
5. Find the nature of the motion in space of a body
turning about a fixed point when its angular velocities about
its principal axes are w₁ = a sin nt, w₂ = a cos nt, w;= n a con-
13
98
EXAMPLES.
stant. (75).
(75). Shew that the instantaneous axis describes a
circular cone in the body with uniform motion, and that the
angular velocity about it is invariable.
6. If the body in (75) moves so that (1) the inclination
of the planes AB and XY increases uniformly, and (2) the
plane AC constantly passes through Z,
W1
@₁ =
d X
dt
dy
•
sin (at + B), w₂ = α,
W3
Wz =
cos (at + B).
dt
7. A rod can turn freely about its upper end. Find the
nature of its motion when it revolves so as to retain a constant
inclination to the vertical. Find also the pressure on the
upper end.
8.
If in the circumstances of (81),
1
1
(-1) tan" --
B
A C
at any instant when is finite, then will be invariable and
will increase uniformly. Find the values of w₁, w2, wg in this
case.
and
9. If in the circumstances of (75) and (78), Ø is invari-
able while
increase uniformly, L ∞ sin o, M ∞ cos
If the figure be one of revolution about the
axis C, the forces acting on the body must be reducible to a
force at the origin and a couple in the plane ZC.
and N∞ sin 2p.
If and increase uniformly while is invariable,
If the body be one
L ∞ cos o, M ∞ sin o, and N ∞ sin 20.
of revolution about C, the forces must be reducible to a force
at the origin and a couple in the plane through C perpen-
dicular to ZC.
M
10. If in the notation of (78), A = B, L = a sin nt,
= a cos nt, N = 0, and if when t= = 0 the instantaneous
axis was coincident with the principal axis to which C belongs,
α
A wr
(cos nt - cos mt),
m
n
MOTION OF A RIGID BODY ABOUT A FIXED POINT.
99
α
A w2
(sin mt - sin nt),
m
n
where Am (A — C') w3.
Examine the forms which w₁, we assume when m = n.
11. A cube fixed at its centre of gravity and acted on
by no force besides the reaction of the fixed point and gravity,
will continue to revolve about any axis about which it was
originally put in motion. (90). The pressure on the fixed
point is equal to the weight of the body.
12. If a circular right cone whose altitude a is double
the radius of its base turn about its centre of gravity fixed,
and be originally put in motion about an axis inclined at an
angle a to its axis of figure, the vertex of the cone will de-
scribe a circle whose radius is a sin a. (89, 91).
3
13.
If a right circular cone have a motion impressed
upon it about a given axis through the centre of gravity
whose equations when the centre of gravity is origin and the
cone's axis the axis of x are
% the invariable plane
is
X y
m n
0 = 1} (a² + 4b³) (lx + my) + b²nz
a being the altitude of the cone, b the radius of its base. (91).
14. With the notation and circumstances of (81)
k²
is
h2
intermediate to the greatest and least of the three quantities
A, B, C.
15. A circular plate revolves about its centre of gravity
fixed. If an angular velocity w were originally impressed
upon it about an axis making an angle a with its plane, a
normal to the plane of the disc will make a revolution in space
in time
(86).
2π
W √1 + 3 sin² a
100
EXAMPLES.
16.
A solid of revolution moving about its centre of
gravity has an angular velocity impressed upon it about a line
between its axis of figure and that of ≈. If its axis of figure
originally makes an angle a with the fixed axis of ≈ and an
angle B with the initial axis of rotation, and if
A tan ẞ = C tan a,
the axis will uniformly describe a circular right cone about
the fixed axis of x, C, A being the body's moments of inertia
about its own axis of figure and about a perpendicular line.
(89, 91).
17. If a solid of revolution moving about its centre of
gravity be originally put in motion about an axis with respect
to which the moment of inertia is Q, and if C be the moment
of inertia about the axis of figure, A that about a perpendicular
axis through the centre of gravity, then
(1) The instantaneous axis will describe a circular cone
in space, and the vertical angle of this cone is greatest when Q
is a harmonic mean between A and C. (89, 91).
(2) The axis of figure will describe a circular cone in
2
space whose semivertical angle is tan-14 (Q − C) } }
C¹² (A — Q)
(3)
The instantaneous axis will describe a circular cone
relatively to the axis of figure whose semivertical angle is
tan-1
=
Q - C₁ &
A
(94).
18. When a body moves about a fixed point under no
force besides the reaction of that point, shew that no point of
it will in general describe areas uniformly on the co-ordinate
planes.
19. A cube fixed at its centre of gravity is struck in
direction of an edge: find the axis about which it begins to
rotate. (99).
MOTION OF A RIGID BODY ABOUT A FIXED POINT. 101
20. A cube fixed at its centre of gravity is struck by
three equal blows in directions of three of its edges which
neither meet nor are parallel. Determine the axis about
which it begins to revolve. (99).
21. A rod capable of turning about its centre of gravity
is so set in motion as to describe a given right cone. Find
the nature of the impulsive forces which originally caused its
motion. (91, 99).
22. A cone is moving about its vertex. Shew that it
cannot be struck perpendicularly to its surface so that its
motion may be destroyed without producing an impulse at the
vertex.
23. If a rigid body capable of turning about a fixed point
be struck by impulses, the sums of whose components in
directions of the co-ordinate axes are X, Y, Z, and if F, G, H
be the consequent impulses in similar directions on the fixed
point, x, y, ≈ co-ordinates of the centre of gravity, the axes
being principal axes,
0 = −
(X - F)π + (Y − G) ÿ + (Z – H)≈. (101).
ვ
If w₁, wa, w be the initial angular velocities about the co-
ordinate axes,
0 = (X − F)w₁ + (Y - G) w₂ + (Z − H) w3.
w3•
24. A triangle ABC fixed at its centre of gravity is
struck perpendicularly to its plane at its right angle C. The
initial instantaneous axis is parallel to the hypothen use.
(99).
25. The motion of a body revolving about the origin
fixed is exhibited by the angular velocities w, wy, w about
the rectangular cordinate axes. If the line
X y
1 m N
sud-
denly becomes a fixed axis, obtain equations for determining
the impulses on this axis and the new angular velocity about it.
102
EXAMPLES.
26. An elliptic lamina fixed at its centre is struck and
begins to rotate about the diameter ay be 0. The blow
must have been applied at some point of the diameter
ay + bx
= 0,
the axes of the figure being axes of co-ordinates. (99, 102).
Will the body continue to rotate about this diameter if
no force except the reaction of the fixed point henceforth acts
upon it? (88). Will the angular velocity about its instan-
taneous axis continue invariable?
27. An ellipsoid whose centre is fixed receives a normal
blow so that the initial instantaneous axis lies in the conical
surface
x²
y2
+
+
0.
J
b² c²
c² - a²
a² - b²
The blow must have been given at some point where the
ellipsoid meets the conical surface
b² - c²
a¹
c²
a²
b4
a²
b?
C4
+
+
2
y²
(a² + b²)*
²
0. (99).
(b² + c²)²' x² (c² + a²)²
28. A solid ellipsoid whose centre is fixed receives a
normal blow at a point a, ß, y of its surface: the equations
to the initial instantaneous axis when the axes of the body are
co-ordinate axes are
χα
b² + c²
a² b² - c²
b² c²
yß c² + a²
a²
zy a² + b²
c² a² - b²
(99).
If the ellipsoid thus put in motion moves henceforth under
no force but the reaction of its fixed centre, the equation to
its invariable plane when the original directions of its axes are
co-ordinate axes is
20
Y
0 = aº (b² − cº) ~ + b² (c² – a³) 1½ + c² (a² − bº) ≈. (91).
α
—
−
Y
MOTION OF A FREE RIGID BODY.
103
1.
IV. Motion of a free rigid body.
If the velocities of two points of a rigid body be
given in magnitude and direction, shew that the translation and
rotation which exhibits its motion can in general be determined.
Shew in what cases these data are insufficient.
A body has a motion of rotation w about the axis
2.
a
y - B
m
ج
Y
where l² + m² + n² = 1. The motion
n
is equivalent to rotations lw, mw, nw about the co-ordinate.
axes, and translations (my – nß)w, (na – ly) w, (Iẞ – ma) w
in direction of them. (106).
3. A circular disc revolves with an uniform angular
velocity w about an axis through its centre perpendicular to
its plane, while its centre describes a circle of radius a with
another uniform angular velocity Q about a point in the plane
of the disc. The motion is at any instant exhibited by a
single rotation, and the locus of the instantaneous axis is a
cylinder of radius
αω
ω + Ω
Ω
Explain the result when w, are
equal and contrary in direction.
4. The motion of the earth will not generally be ex-
hibited by rotation about a single axis, neglecting the nutation
of its axis and supposing its centre to describe an ellipse.
(95, 115).
5. The motion of a rigid body is exhibited by the co-
existence of given rotations about a series of given parallel
axes; determine the single rotation to which the motion is in
general reducible.
6. A body has equal angular velocities about two axes
which neither meet nor are parallel. Prove that the central axis
of the motion is equally inclined to each of the axes. (112).
104
EXAMPLES.
7. A given weight hangs by a string which is wrapped
round a wheel whose plane is vertical. Compare the motions
produced in the wheel when its centre is fixed and when it is
allowed to fall freely, the system being originally at rest.
(118).
8. A wheel revolves about an axis through its centre
perpendicular to its plane with a velocity increasing uniformly
from zero, while its centre moves with an uniform velocity
along a straight line in the plane of the wheel. The instan-
taneous axis has a hyperbolic cylinder for its locus, and the
forces producing such a motion must form a couple of constant
moment.
9. If a body be under no forces in motion about a
principal axis through the centre of gravity, that axis will
continue fixed in space and will always be the axis of in-
stantaneous rotation about which the body turns with uniform
angular velocity. (88, 118).
10. A hemisphere is acted upon by a force constant in
magnitude and direction applied at the centre of its base. If
the radius in which its centre lies is originally inclined at a
small angle to the direction of the force, the body will make
small oscillations about this direction in the time π
83 α
120 f
while the centre of gravity moves in the same direction, a
being the radius of the hemisphere and ƒ the given accelerating
force. (118). In what unit of time will this result be ex-
pressed?
11. How must a free cube be struck that it may begin to
rotate (1) about a diagonal, or (2) about an edge? (125).
12. A rectangular lamina whose sides are a, b is struck
perpendicularly at a point whose co-ordinates from the fixed
centre parallel to the sides of the rectangle are h, k. The
rectangle will begin to revolve about the line o
b² x
a²y
+
k
h
MOTION OF A FREE RIGID BODY.
105
If the body were free, it would begin to revolve about a line in
itself parallel to the former and at the distance 1
from it. (100, 125).
h2 k² - }
1/2 +
a4 b²
13. A free spheroid is put in motion so that when it is
thenceforth acted on by no force, its centre moves along a
given straight line while its axis describes a circular cone
about the same line. Find the nature of the impulsive forces
which produced its motion. (89, 124).
14. A circular lamina of radius a, revolving about a
diameter with a given angular velocity w, is struck perpendicu-
larly to its plane at the extremity of the diameter at right
angles to the former, and afterwards has an angular velocity w'.
The distance between the original and latter axis of rotation
a w
W (124).
4
ω
•
15. Find at what point a rod AB may be struck per-
pendicularly that one end of it A may be initially at rest.
(124). Shew that the point of impact is the centre of per-
cussion when the rod can turn about A fixed. If the rod move
henceforth freely, the path of A is given by
x = α COS
cos
-1
Y
a² - y²,
a
where the axis of x is measured from the original position of
the centre of gravity in direction of the blow, and 2a is the
length of the rod. (123).
16. A cube is struck by three equal blows in directions of
three of its edges which neither meet nor are parallel. De-
termine its initial instantaneous axis. (124, 126).
14
106
MISCELLANEOUS EXAMPLES.
V.
Miscellaneous examples.
1. A circular disc of radius a rolls with uniform velocity
V along a horizontal plane: the effective force on any particle
V2
a²
of it at distance r from the centre is r acting in the line
joining the particle with the centre of the disc. (132).
2. A sphere rolls on a rough board which itself can slide
horizontally on a fixed smooth horizontal plane on which it
lies, the motions of the sphere and board being in one vertical
plane. Determine the motion. (132).
3. A sphere can turn about its centre which is fixed.
Another sphere is placed upon it at a given point and rolls
down it. Determine the motion. (132).
4. A sphere rolls on a horizontal plane which is itself made
to move horizontally with an uniformly increasing velocity.
Determine the friction between the sphere and plane. (132).
If the plane is suddenly brought to rest, find the change in the
velocity of the sphere.
5. A cylinder rolls down a plane inclined at 30° to the
horizon. If a be the radius, I the space through which it has
descended, x, y the co-ordinates of a point of the cylinder
referred to axes parallel and perpendicular to the plane origi-
nating in the line of contact, then the effective forces on that
particle in these directions are
gy
2 glx
3 a
3a²
2gl
and + (a − y).
3a²
gx
3 a
6. A rough sphere rolls on a horizontal plane under the
action of a centre of force in the plane. The equation to the
projection on the plane of the path of the sphere's centre is
d2 u
+ u =
d 02
P
h² u²
parallel to the plane.
where P is the resultant force resolved
MISCELLANEOUS EXAMPLES.
107
7. If a wheel rolls on a plane the velocity of translation
of its centre is equal to the product of its angular velocity
about its centre by its radius. (132).
8. Given the motions of each of a system of unconnected
particles, find the impulse to be applied to each that the centre
of gravity of the system may afterwards be at rest.
9.
A bomb shell projected in a given manner explodes
in its flight. If a fragment of given mass impinges on the
ground with a given velocity and is suddenly brought to rest
find the change in the direction of the parabolic path which
the centre of gravity of the whole has described. (139, 140).
10. Two equal particles are describing an ellipse in the
same direction about the same centre of force at their centre of
gravity. If they suddenly become rigidly united they will
then revolve with the uniform angular velocity
ab √μ
22
where
μ is the absolute force, a, b the semi-axes of the ellipse, and
2r the distance of the particles when the change takes place.
(142, 143).
11. If a small planet were to explode and if the fragments
then describe ellipses about the sun's centre, a, b being the
semi-axes of the ellipse described by a fragment of mass m in a
plane inclined to the ecliptic at an angle i, Σ( (m
m
b?
√
a
COS
i)
is independent of the place where the explosion takes place or
the manner in which the body is separated by it, the sum-
mation indicated being made through all the portions of the
original body. (142, 143).
12. A cylinder, capable of turning about its horizontal
axis, has a cylindrical shell enclosing it and fitting closely to it
so that there is friction between them. Given the different
angular velocities with which each is at a given time revolving
about the axis, find the common angular velocity which they
ultimately attain. (142).
108
MISCELLANEOUS EXAMPLES.
13. A series of concentric spherical shells fit closely one
within another and influence one another's motion by friction.
If the shells have given rotations impressed upon them about
given diameters, no extraneous force acting on the system,
they will ultimately revolve about a common axis which is
perpendicular to the plane
0 = xΣ(Alw) + yΣ(Amw) + xΣ(Anw);
where A is the moment of inertia about a diameter of a shell
which originally has an angular velocity w about the diameter
X Y ૪ the centre of the shells being origin. (90, 144).
1
m
N
14. A sphere whose radius is a and mass M has an
angular velocity w about its diameter which is perpendicular
to the plane of xy while its centre is describing a circle about
the origin in the plane of xy with an angular velocity . For
this body the value of the function 2m(
&Ma³w+ Mb²Q, (150).
dy
dx
is
20
Y
dt
dt
[This example suggests an oversight in Laplace's deter-
mination of the invariable plane of the solar system. Mech.
Celeste. Liv. VI. ch. 17.*]
15. A wheel rolls on a horizontal plane. The sum of its
vires vivæ from the translation of its centre of gravity and its
rotation about its instantaneous axis exceeds the vis viva of
the body by Mv², M being the mass of the body and v the
velocity of its centre. (154).
16. A sphere revolving about a diameter and acted on
by no extraneous force expands symmetrically. Its vis viva
is proportional to its moment of inertia about a diameter in-
versely. (142, 153).
17. A wheel rolls on a horizontal plane. Prove that its
velocity is uniform.
* Poinsot. Elemens de Statique. Appendix. 1842.
MISCELLANEOUS EXAMPLES.
109
18.
Find the time in which a sphere will roll down a
given inclined plane, and the amount of friction at any instant
of the motion.
19. A solid cylinder fits exactly within a hollow cylinder
so that their axes coincide and no friction can take place between
them. The whole rolls down a rough inclined plane. Compare
the time in which the system descends through a given space
with that in which it would descend through the same space
if the two cylinders were rigidly united. (154, 156).
20. A rod of length 2a descends in a vertical plane, its
lower end sliding on a smooth horizontal plane. If the rod is
originally inclined at 30° to the horizon, its angular velocity
3g
when it becomes horizontal is
4 a
21. When a sphere rolls on a fixed plane under the
action of any forces through its centre, the instantaneous axis.
is always perpendicular to the direction of motion of the centre
of gravity, and the friction is opposite to the direction of the
resultant force parallel to the plane.
22. A sphere rolls on an inclined plane. The path of
its centre of gravity is in general a parabola.
23. Determine the motion of a sphere rolling down a
smooth inverted cycloid whose axis is vertical. (156).
24. Shew that the principle of vis viva is not true in the
following cases.
A sphere slides along a horizontal plane which rises with
a velocity proportional to the fourth power of the time.
Two particles are connected by a string which increases
uniformly in length. One of them hangs over the edge of a
table and moves the other which lies on the table.
A sphere expands while it falls freely under the action of
gravity.
110
MISCELLANEOUS EXAMPLES.
A sphere rolling down an inclined plane contracts in
size symmetrically, so that a(t) is its radius at time t
from the beginning of the motion.
OBS. In each of these cases it will be instructive to
ascertain the equation of the kind L = 0, involved in the proof
of the principle of vis viva, which does not fulfil the conditions
on which the proof rests.
25. A carriage with four wheels descends an inclined
plane. Find the velocity at the bottom. Compare this with
the velocity which would be acquired (1) if two of the wheels
be prevented from turning, (2) if the plane were smooth. (156).
26. Two wheels work together by means of a band, so
that a weight hanging from one raises another weight hanging
from the other. Determine the motion (156).
27. A machine wherein the efficiency exhibited is to that
applied as a : 1, raises a volume V of water through a height
h, the efficiency applied to it
☛ being the specific
gravity of water (162).
o Vh
σ
α
An efficiency 33000, a foot and a pound being the units,
exerted in a minute of time is termed a horse power. Hence
if the water in the preceding question be raised in t minutes,
o V h
the agent in operation exerts
33000 at
horse powers, σ V and
h being expressed with reference to a pound and a foot.
28. A globe of given exterior radius rolls down an in-
clined plane d feet long in n seconds. Shew how to ascertain
from these data whether the globe is hollow or solid; and if
the former be the case, shew how the interior radius may be
known, the density being uniform.
1
29. Two equal rods united at their ends at a given angle
rest over a fixed horizontal cylinder at stable equilibrium in a
vertical plane. Find the time of a small oscillation when the
system is slightly displaced in the vertical plane through the
rods.
MISCELLANEOUS EXAMPLES.
111
30. A rod of length a is placed in a smooth hemisphere
whose radius is a, so that a plane through the rod and the
centre of the hemisphere is always vertical. The rod will
oscillate in the same time as a pendulum whose length is
√3.a.
§
31. A horizontal rod of length 2a hangs by two parallel
strings each of length b attached to its ends. If it be slightly
displaced in direction of its length it will make small oscillations
in the time π
If it be twisted horizontally through a
g
small angle, it will make small oscillations in the time
b
T
(156).
32.
3g
A smooth ball is placed within a spherical surface
and lies at rest in it. If the spherical surface be made to
move horizontally so as to have at any time the velocity which
gravity would generate in a falling body, the centre of the ball
will rise into the same horizontal plane with the centre of the
spherical surface.
33. If a figure of revolution resting at stable equilibrium
on its vertex on a horizontal plane be slightly disturbed, it
will make small oscillations in the time
if the plane
k?
C
is smooth, or \
√
k² + c²
C
if it is rough, where c is the dis-
tance of the centre of gravity from the centre of curvature of
the generating curve at the vertex. Are the results altered if
the plane has a given uniform horizontal motion?
34.
The half of a circular cylinder cut by a plane along
its axis rocks on a smooth horizontal table. The instantaneous
axis lies always on the surface of a circular cylinder whose
radius is to that of the body as 3: 8.
112
MISCELLANEOUS EXAMPLES.
35. A wire bent into the form of a circular arc makes
small oscillations in a vertical plane between two smooth in-
clined planes. The length of the simple equivalent pendulum
radius × arc
is
chord
36. Two equal weights connected by a weightless rod of
length a√3 are placed in a smooth hemisphere whose radius
is a, so that a plane through the rod and the centre of the
hemisphere is vertical; the rod if disturbed from the position
of equilibrium in the same plane will make oscillations in the
same time as a simple pendulum of length 2 a.
37. A rough hemisphere rests at stable equilibrium on a
fixed rough sphere. If the former be slightly displaced de-
termine the subsequent motion (1) when its flat base (2) when
its curved surface is in contact with the fixed sphere.
38. A rough plank capable of turning about a horizontal
axis through its centre of gravity is in a horizontal position.
Determine the motion of a rough sphere placed upon it at a
given point.
39. Determine the motion of a rod which passes through
a small fixed ring while its lower end slides on a smooth
horizontal plane.
40. A wheel is set in motion on a rough horizontal plane
with given velocities of translation and rotation. Find the
locus of the instantaneous axis until friction reduces the motion
to that of rolling.
41. A heavy right angled cone resting on a horizontal
plane with its vertex downwards is bisected by a vertical plane
through its axis. The relative instantaneous decrease of
and the initial mutual pressure
pressure on the plane
at the vertex
42.
1
2π
4
3 π² 2
weight of cone.
Three equal spheres lie on a table in contact, their
centres forming an equilateral triangle. A fourth equal sphere
MISCELLANEOUS EXAMPLES.
113
drops and impinges on the three similarly and simultaneously.
The impinging ball will entirely lose its velocity if e =
If the three quiescent balls had a string passing round
them in the plane of their centres, find the impulsive tension
which the string sustains when the impact takes place.
43. Two equal smooth balls moving in parallel straight
lines with a common velocity V and impinge at once on a third
equal ball lying at rest, so that the centres of the three at the
moment of impact form an equilateral triangle in the plane of
the motion of the two balls, to one of the two sides of which
the direction of V is perpendicular. The latter ball will begin
to move with the velocity (1 + e) V.
44. Two equal spheres are in contact: find how one of
them should be struck that after impact it may move in a given
direction.
45. A sphere rolling on a plane impinges on a perfectly
rough point. Find the condition that it may be brought to
rest.
46. Two rods are moving in the
velocities of translation and rotation.
strikes the centre of gravity of the other
sequent motion.
same plane with given
The end of one rod
determine the sub-
47. A cylinder sliding in a direction perpendicular to its
length along a smooth plane suddenly comes to a rough part
of the plane whereby it is compelled to roll. Prove that it
loses at once one third of its velocity.
48.
A hoop is rolling on a horizontal plane. If it be
struck at a given point of its circumference, find the limi-
tations to the direction of the blow that the motion may con-
tinue that of rolling.
15
114
MISCELLANEOUS EXAMPLES.
49. A rod slides between a smooth horizontal and vertical
plane until a string connecting its lower end with the intersec-
tion of those planes becomes tight, the string and the motion
of the rod being in one vertical plane. The rod may be
brought to rest by the impulse if tan² a < 2, a being the rod's
inclination to the horizon when the impulse takes place.
50.
A rod descends in a vertical plane, its lower end
sliding on a smooth horizontal plane, and impinges on a given
fixed peg. Determine its motion immediately after the impulse.
51. An inelastic rod rests in a horizontal position on two
pegs with its centre of gravity equidistant from them. If it
be turned about one of them and then allowed to fall, sliding
being prevented by friction, its motion will cease or not after
the first impact as the distance between the pegs is greater or
1
less than × the length of the rod.
52.
V3
A circular disc revolving in its own plane drops on
a rough inelastic inclined plane whose intersection with the
horizon is perpendicular to the plane of the disc. The disc
will run up or down the plane according as aw> or < V cosi,
V and w being the velocities of translation and rotation of the
disc before impact, a its radius and i the inclination of the
plane to the vertical.
53. A chain coiled loosely on a horizontal table is drawn
up by a very thin string at one end of it, passing over a pulley
and having a given weight at the other end. The pulley is
vertically above the coil of chain and the size of the coil
may be neglected so that each link begins to rise vertically.
Determine the velocity of the chain when it reaches the pulley.
Determine the motion also if the chain originally lies in a
horizontal line of which one end is vertically under the pulley.
54. A rod is constrained to move vertically by two fixed
rings through which it slides, while its lower end presses upon
MISCELLANEOUS EXAMPLES.
115
an inclined plane sliding on a horizontal table. Find the
velocity generated in the inclined plane when the rod leaves it
and the pressure on each ring at any instant. If at a given
instant the plane be suddenly brought to rest by a fixed
obstacle, determine the impulse on each ring.
55. Two equal rods connected by a hinge are placed on
two pegs in the same horizontal line, the rods having the same
inclination to the vertical. Determine the angle through which
they open.
If at a given point of the motion a string connecting the
lower ends of the rods becomes tight, find the impulse on each
peg.
56. Three equal rods are placed at given equal in-
clinations to the horizon and to one another, united by a
common joint at their lowest point. A sphere is placed
between them. Determine its motion. If the upper ends of
the rods be united two and two by equal strings which are
originally slack, determine the impulsive tension which each
sustains when they become tight.
57. An inelastic and smooth ellipsoid whose semi-axes are
a, b, c having a velocity of rotation w about one axis e impiuges
with a velocity v on a sphere of equal mass. At the instant of
impact the sphere touches the ellipsoid at the extremity of the
latus rectum of its principal section containing the axes a and
b, and the axis a is in direction of the motion of the centre of
gravity. If the eccentricity of that principal section is ä,
and if the ellipsoid has no rotation after impact, 2 aw = v.
58. A circular hoop revolves about a vertical diameter
with uniform angular velocity, and a smooth heavy ring slides
upon it. Find the condition that the ring may perform oscil-
lations about the lowest point.
59. Two equal rods are similarly placed in one vertical
plane so that their lower ends are on a smooth horizontal
table and their upper ends press against one another. Find
116
MISCELLANEOUS EXAMPLES.
whether the rods will cease to be in contact before they fall to
the table.
60. Two particles connected by an elastic string move
with equal velocities in the same straight line, their distance
being the length of the string when it sustains no tension.
If the hinder of the two be suddenly stopped, find how far
the other body will move.
THE END.

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