Chapter XVII: Part II: Kinetics (2)

In astronomical and other investigations relating to central forces it is often convenient to use polar co-ordinates with the centre of force as pole. Let P, Q be the positions of a moving point at times t, t + [delta]t, and write OP = r, OQ = r + [delta]r, [angle]POQ = [delta][theta], O being any fixed origin. If u, v be the component velocities at P along and perpendicular to OP (in the direction of [theta] increasing), we have

[delta]r dr r[delta][theta] d[theta]
u = lim.-------- = --, v = lim. --------------- = r --------. (13)
[delta]t dt [delta]t dt

Again, the velocities parallel and perpendicular to OP change in the time [delta]t from u, v to u - v[delta][theta], v + u[delta][theta], ultimately. The component accelerations at P in these directions are therefore

du d[theta] d²r /d[theta]\² \
-- - v -------- = --- - r ( -------- ), |
dt dt dt² \ dt / |
> (14)
dv d[theta] 1 d / d[theta]\ |
-- + u -------- = --- --- ( r² -------- ), |
dt dt r dt \ dt / /

respectively.

In the case of a central force, with O as pole, the transverse acceleration vanishes, so that

r²d[theta]/dt = h, (15)

where h is constant; this shows (again) that the radius vector sweeps over equal areas in equal times. The radial resolution gives

d²r /d[theta]\²
--- - r ( -------- ) = -P, (16)
dt² \ dt /

where P, as before, denotes the acceleration towards O. If in this we put r = 1/u, and eliminate t by means of (15), we obtain the general differential equation of central orbits, viz.

d²u P
--------- + u = ----. (17)
d[theta]² h²u²

If, for example, the law be that of the inverse square, we have P =
[mu]u², and the solution is of the form

[mu]
u = ------ {1 + e cos ([theta] - [alpha])}, (18)
h²

where e, [alpha] are arbitrary constants. This is recognized as the
polar equation of a conic referred to the focus, the half latus-rectum
being h²/[mu].

The law of the inverse cube P = [mu]u³ is interesting by way of
contrast. The orbits may be divided into two classes according as h²
<> [mu], i.e. according as the transverse velocity (hu) is greater or
less than the velocity [root]([mu]·u) appropriate to a circular orbit
at the same distance. In the former case the equation (17) takes the
form

d²u
-------- + m²u = 0, (19)
d[theta]²

the solution of which is

au = sin m ([theta] - [alpha]). (20)

The orbit has therefore two asymptotes, inclined at an angle [pi]/m.
In the latter case the differential equation is of the form

d²u
--------- = m²u, (21)
d[theta]²

so that

u = A e^(m[theta]) + B e^(-m[theta]) (22)

If A, B have the same sign, this is equivalent to

au = cosh m[theta], (23)

if the origin of [theta] be suitably adjusted; hence r has a maximum
value [alpha], and the particle ultimately approaches the pole
asymptotically by an infinite number of convolutions. If A, B have
opposite signs the form is

au = sinh m[theta], (24)

this has an asymptote parallel to [theta] = 0, but the path near the
origin has the same general form as in the case of (23). If A or B
vanish we have an equiangular spiral, and the velocity at infinity is
zero. In the critical case of h² = [mu], we have d²u/d[theta]² = 0,
and

u = A[theta] + B; (25)

the orbit is therefore a "reciprocal spiral," except in the special
case of A = 0, when it is a circle. It will be seen that unless the
conditions be exactly adjusted for a circular orbit the particle will
either recede to infinity or approach the pole asymptotically. This
problem was investigated by R. Cotes (1682-1716), and the various
curves obtained arc known as _Coles's spirals_.

A point on a central orbit where the radial velocity (dr/dt) vanishes is called an _apse_, and the corresponding radius is called an _apse-line_. If the force is always the same at the same distance any apse-line will divide the orbit symmetrically, as is seen by imagining the velocity at the apse to be reversed. It follows that the angle between successive apse-lines is constant; it is called the _apsidal angle_ of the orbit.

If in a central orbit the velocity is equal to the velocity from
infinity, we have, from (5),
_
h² / [oo]
-- = 2 | P dr; (26)
p² _/ r

this determines the form of the critical orbit, as it is called. If P = [mu]/r^[n], its polar equation is

r^m cos m[theta] = a^m, (27)

where m = ½(3 - n), except in the case n = 3, when the orbit is an equiangular spiral. The case n = 2 gives the parabola as before.

If we eliminate d[theta]/dt between (15) and (16) we obtain

d²r h²
--- - -- = -P = -f(r),
dt² r³

say. We may apply this to the investigation of the stability of a
circular orbit. Assuming that r = a + x, where x is small, we have,
approximately,

d²x h² / 3x\
--- - -- ( 1 - -- ) = -f(a) - xf´(a).
dt² r³ \ a /

Hence if h and a be connected by the relation h² = a³f(a) proper to a
circular orbit, we have
_ _
d²x | 3 |
--- + | f´(a) + --- f(a)| x = 0. (28)
dt² |_ a _|

If the coefficient of x be positive the variations of x are
simple-harmonic, and x can remain permanently small; the circular
orbit is then said to be stable. The condition for this may be written
_ _
d | |
-- | a³f(a) | > 0, (29)
da |_ _|

i.e. the intensity of the force in the region for which r = a, nearly,
must diminish with increasing distance less rapidly than according to
the law of the inverse cube. Again, the half-period of x is
[pi]/sqrt[f´(a) + 3^{-1}f(a)], and since the angular velocity in the
orbit is h/a², approximately, the apsidal angle is, ultimately,
_ _
/ | f(a) |
[pi] / | --------------- |, (30)
\/ |_ af´(a) + 3f(a) _|

or, in the case of f(a) = [mu]/r^n, [pi]/[root](3 - n). This is in
agreement with the known results for n = 2, n = -1.

We have seen that under the law of the inverse square all finite
orbits are elliptical. The question presents itself whether there
then is any other law of force, giving a finite velocity from
infinity, under which all finite orbits are necessarily closed curves.
If this is the case, the apsidal angle must evidently be commensurable
with [pi], and since it cannot vary discontinuously the apsidal angle
in a nearly circular orbit must be constant. Equating the expression
(30) to [pi]/m, we find that f(a) = C/a^n, where n = 3 - m². The
force must therefore vary as a power of the distance, and n must be
less than 3. Moreover, the case n = 2 is the only one in which the
critical orbit (27) can be regarded as the limiting form of a closed
curve. Hence the only law of force which satisfies the conditions is
that of the inverse square.

At the beginning of § 13 the velocity of a moving point P was represented by a vector [->OV] drawn from a fixed origin O. The locus of the point V is called the _hodograph_ (q.v.); and it appears that the velocity of the point V along the hodograph represents in magnitude and in direction the acceleration in the original orbit. Thus in the case of a plane orbit, if v be the velocity of P, [psi] the inclination of the direction of motion to some fixed direction, the polar co-ordinates of V may be taken to be v, [psi]; hence the velocities of V along and perpendicular to OV will be dv/dt and vd[psi]/dt. These expressions therefore give the tangential and normal accelerations of P; cf. § 13 (12).

In the motion of a projectile under gravity the hodograph is a
vertical line described with constant velocity. In elliptic harmonic
motion the velocity of P is parallel and proportional to the
semi-diameter CD which is conjugate to the radius CP; the hodograph is
therefore an ellipse similar to the actual orbit. In the case of a
central orbit described under the law of the inverse square we have v
= h/SY = h. SZ/b², where S is the centre of force, SY is the
perpendicular to the tangent at P, and Z is the point where YS meets
the auxiliary circle again. Hence the hodograph is similar and
similarly situated to the locus of Z (the auxiliary circle) turned
about S through a right angle. This applies to an elliptic or
hyperbolic orbit; the case of the parabolic orbit may be examined
separately or treated as a limiting case. The annexed fig. 70 exhibits
the various cases, with the hodograph in its proper orientation. The
pole O of the hodograph is inside on or outside the circle, according
as the orbit is an ellipse, parabola or hyperbola. In any case of a
central orbit the hodograph (when turned through a right angle) is
similar and similarly situated to the "reciprocal polar" of the orbit
with respect to the centre of force. Thus for a circular orbit with
the centre of force at an excentric point, the hodograph is a conic
with the pole as focus. In the case of a particle oscillating under
gravity on a smooth cycloid from rest at the cusp the hodograph is a
circle through the pole, described with constant velocity.

§ 15. _Kinetics of a System of Discrete Particles._--The momenta of the several particles constitute a system of localized vectors which, for purposes of resolving and taking moments, may be reduced like a system of forces in statics (§ 8). Thus taking any point O as base, we have first a _linear momentum_ whose components referred to rectangular axes through O are

[Sigma](m[.x]), [Sigma](m[.y]), [Sigma](m[.z]); (1)

its representative vector is the same whatever point O be chosen. Secondly, we have an _angular momentum_ whose components are

[Sigma]{m(y[.z] - z[.y])}, [Sigma]{m(z[.x] - xz[.z])}, [Sigma]{m(x[.y] - y[.x])}, (2)

these being the sums of the moments of the momenta of the several particles about the respective axes. This is subject to the same relations as a couple in statics; it may be represented by a vector which will, however, in general vary with the position of O.

The linear momentum is the same as if the whole mass were concentrated at the centre of mass G, and endowed with the velocity of this point. This follows at once from equation (8) of § 11, if we imagine the two configurations of the system there referred to to be those corresponding to the instants t, t + [delta]t. Thus

__ / [->PP] \ __ [->GG´]
\ ( m·-------- ) = \ (m)·--------. (3)
/__ \ [delta]t / /__ [delta]t

Analytically we have

d d[|x]
[Sigma](m[.x]) = --- [Sigma](mx) = [Sigma](m)·-----. (4)
dt dt

with two similar formulae.

Again, if the instantaneous position of G be taken as base, the angular momentum of the absolute motion is the same as the angular momentum of the motion relative to G. For the velocity of a particle m at P may be replaced by two components one of which (v) is identical in magnitude and direction with the velocity of G, whilst the other (v) is the velocity relative to G. The aggregate of the components mv of momentum is equivalent to a single localized vector [Sigma](m)·v in a line through G, and has therefore zero moment about any axis through G; hence in taking moments about such an axis we need only regard the velocities relative to G. In symbols, we have

/ d[|z] d[|y]\
[Sigma]{m(y[.z] - z[.y])} = [Sigma](m)·( y ----- - z ----- ) + [Sigma]{m([eta][zeta] - [.zeta][eta])}. (5)
\ dt dt /

since [Sigma](m[xi]) = 0, [Sigma](m[xi]) = 0, and so on, the notation being as in § 11. This expresses that the moment of momentum about any fixed axis (e.g. Ox) is equal to the moment of momentum of the motion relative to G about a parallel axis through G, together with the moment of momentum of the whole mass supposed concentrated at G and moving with this point. If in (5) we make O coincide with the instantaneous position of G, we have [|x], [|y], [|z] = 0, and the theorem follows.

Finally, the rates of change of the components of the angular momentum of the motion relative to G referred to G as a moving base, are equal to the rates of change of the corresponding components of angular momentum relative to a fixed base coincident with the instantaneous position of G. For let G´ be a consecutive position of G. At the instant t + [delta]t the momenta of the system are equivalent to a linear momentum represented by a localized vector [Sigma](m)·(v + [delta]v) in a line through G´ tangential to the path of G´, together with a certain angular momentum. Now the moment of this localized vector with respect to any axis through G is zero, to the first order of [delta]t, since the perpendicular distance of G from the tangent line at G´ is of the order ([delta]t)². Analytically we have from (5),

d / d[|z]² d²[|y] \ d
--- [Sigma] {m (y[.z] - z[.y])} = [Sigma](m)·( y ------ - z ------- ) + --- [Sigma] {m([eta][zeta - [zeta][.eta])} (6)
dt \ dt² dt² / dt

If we put x, y, z = 0, the theorem is proved as regards axes parallel to Ox.

Next consider the kinetic energy of the system. If from a fixed point O we draw vectors [->OV1], [->OV2] to represent the velocities of the several particles m1, m2 ..., and if we construct the vector

[Sigma](m·[->OV])
[->OK] = ----------------- (7)
[Sigma](m)

this will represent the velocity of the mass-centre, by (3). We find, exactly as in the proof of Lagrange's First Theorem (§ 11), that

½[Sigma](m·OV²) = ½[Sigma](m)·OK² + ½[Sigma](m·KV²); (8)

i.e. the total kinetic energy is equal to the kinetic energy of the whole mass supposed concentrated at G and moving with this point, together with the kinetic energy of the motion relative to G. The latter may be called the _internal kinetic energy_ of the system. Analytically we have _ _ | /d[|x]\² /d[|y]\² /d[|z]\ | ½[Sigma]{m([.x]² + [.y]² + [.z]²)} = ½[Sigma](m)·| ( ----- ) + ( ----- ) + ( ----- ) | |_ \ dt / \ dt / \ dt / _|

+ ½[Sigma] {m([zeta]² + [.eta]² + [zeta]²)}. (9)

There is also an analogue to Lagrange's Second Theorem, viz.

[Sigma][Sigma] (m_p m_q·V_p V_q²)
½[Sigma](m·KV²) = ½ --------------------------------- (10)
[Sigma]m

which expresses the internal kinetic energy in terms of the relative velocities of the several pairs of particles. This formula is due to Möbius.

The preceding theorems are purely kinematical. We have now to consider the effect of the forces acting on the particles. These may be divided into two categories; we have first, the _extraneous forces_ exerted on the various particles from without, and, secondly, the mutual or _internal forces_ between the various pairs of particles. It is assumed that these latter are subject to the law of equality of action and reaction. If the equations of motion of each particle be formed separately, each such internal force will appear twice over, with opposite signs for its components, viz. as affecting the motion of each of the two particles between which it acts. The full working out is in general difficult, the comparatively simple problem of "three bodies," for instance, in gravitational astronomy being still unsolved, but some general theorems can be formulated.

The first of these may be called the _Principle of Linear Momentum_. If there are no extraneous forces, the resultant linear momentum is constant in every respect. For consider any two particles at P and Q, acting on one another with equal and opposite forces in the line PQ. In the time [delta]t a certain impulse is given to the first particle in the direction (say) from P to Q, whilst an equal and opposite impulse is given to the second in the direction from Q to P. Since these impulses produce equal and opposite momenta in the two particles, the resultant linear momentum of the system is unaltered. If extraneous forces act, it is seen in like manner that the resultant linear momentum of the system is in any given time modified by the geometric addition of the total impulse of the extraneous forces. It follows, by the preceding kinematic theory, that the mass-centre G of the system will move exactly as if the whole mass were concentrated there and were acted on by the extraneous forces applied parallel to their original directions. For example, the mass-centre of a system free from extraneous force will describe a straight line with constant velocity. Again, the mass-centre of a chain of particles connected by strings, projected anyhow under gravity, will describe a parabola.

The second general result is the _Principle of Angular Momentum_. If there are no extraneous forces, the moment of momentum about any fixed axis is constant. For in time [delta]t the mutual action between two particles at P and Q produces equal and opposite momenta in the line PQ, and these will have equal and opposite moments about the fixed axis. If extraneous forces act, the total angular momentum about any fixed axis is in time [delta]t increased by the total extraneous impulse about that axis. The kinematical relations above explained now lead to the conclusion that in calculating the effect of extraneous forces in an infinitely short time [delta]t we may take moments about an axis passing through the instantaneous position of G exactly as if G were fixed; moreover, the result will be the same whether in this process we employ the true velocities of the particles or merely their velocities relative to G. If there are no extraneous forces, or if the extraneous forces have zero moment about any axis through G, the vector which represents the resultant angular momentum relative to G is constant in every respect. A plane through G perpendicular to this vector has a fixed direction in space, and is called the _invariable plane_; it may sometimes be conveniently used as a plane of reference.

For example, if we have two particles connected by a string, the
invariable plane passes through the string, and if [omega] be the
angular velocity in this plane, the angular momentum relative to G is

m1[omega]1r1·r1 + m2[omega]r2·r2 = (m1r1² + m2r2²)[omega],

where r1, r2 are the distances of m1, m2 from their mass-centre G.
Hence if the extraneous forces (e.g. gravity) have zero moment about
G, [omega] will be constant. Again, the tension R of the string is
given by

m1m2
R = m1[omega]²r1 = ------- [omega]²a,
m1 + m2

where a = r1 + r2. Also by (10) the internal kinetic energy is

m1m2
½ ------- [omega]²a².
m1 + m2

The increase of the kinetic energy of the system in any interval of time will of course be equal to the total work done by all the forces acting on the particles. In many questions relating to systems of discrete particles the internal force R_pq (which we will reckon positive when attractive) between any two particles m_p, m_q is a function only of the distance r_pq between them. In this case the work done by the internal forces will be represented by _ / -[Sigma] | R_(pg) dr_(pq), _/

when the summation includes every pair of particles, and each integral
is to be taken between the proper limits. If we write
_
/
V = [Sigma] | R_(pq) dr_(pq), (11)
_/

when r_pq ranges from its value in some standard configuration A of the system to its value in any other configuration P, it is plain that V represents the work which would have to be done in order to bring the system from rest in the configuration A to rest in the configuration P. Hence V is a definite function of the configuration P; it is called the _internal potential energy_. If T denote the kinetic energy, we may say then that the sum T + V is in any interval of time increased by an amount equal to the work done by the extraneous forces. In particular, if there are no extraneous forces T + V is constant. Again, if some of the extraneous forces are due to a conservative field of force, the work which they do may be reckoned as a diminution of the potential energy relative to the field as in § 13.

§ 16. _Kinetics of a Rigid Body. Fundamental Principles._--When we pass from the consideration of discrete particles to that of continuous distributions of matter, we require some physical postulate over and above what is contained in the Laws of Motion, in their original formulation. This additional postulate may be introduced under various forms. One plan is to assume that any body whatever may be treated as if it were composed of material particles, i.e. mathematical points endowed with inertia coefficients, separated by finite intervals, and acting on one another with forces in the lines joining them subject to the law of equality of action and reaction. In the case of a rigid body we must suppose that those forces adjust themselves so as to preserve the mutual distances of the various particles unaltered. On this basis we can predicate the principles of linear and angular momentum, as in § 15.

An alternative procedure is to adopt the principle first formally enunciated by J. Le R. d'Alembert and since known by his name. If x, y, z be the rectangular co-ordinates of a mass-element m, the expressions m[:x], m[:y], m[:z] must be equal to the components of the total force on m, these forces being partly extraneous and partly forces exerted on m by other mass-elements of the system. Hence (m[:x], m[:y], m[:z]) is called the actual or _effective_ force on m. According to d'Alembert's formulation, the extraneous forces together with the _effective forces reversed_ fulfil the statical conditions of equilibrium. In other words, the whole assemblage of effective forces is statically equivalent to the extraneous forces. This leads, by the principles of § 8, to the equations

[Sigma](m[:x]) = X, [Sigma](m[:y]) = Y, [Sigma](m[:z]) = Z, \
> (1)
[Sigma]{m(y[:z] - z[:y]) = L, [Sigma]{m(z[:x] - x[:z]) = M, [Sigma]{m(x[:y] - y[:x]) = N, /

where (X, Y, Z) and (L, M, N) are the force--and couple--constituents of the system of extraneous forces, referred to O as base, and the summations extend over all the mass-elements of the system. These equations may be written

d d d
--- [Sigma](m[.x]) = X, --- [Sigma](m[.y]) = Y, --- [Sigma](m[.z]) = Z, \
dt dt dt | } (2)
> (2)
d d d |
--- [Sigma]{m(y[.z] - z[.y]) = L, --- [Sigma]{m(z[.x]-x[.z]) = M, --- [Sigma]{m(x[.y] - y[.x]) = N, /
dt dt dt

and so express that the rate of change of the linear momentum in any fixed direction (e.g. that of Ox) is equal to the total extraneous force in that direction, and that the rate of change of the angular momentum about any fixed axis is equal to the moment of the extraneous forces about that axis. If we integrate with respect to t between fixed limits, we obtain the principles of linear and angular momentum in the form previously given. Hence, whichever form of postulate we adopt, we are led to the principles of linear and angular momentum, which form in fact the basis of all our subsequent work. It is to be noticed that the preceding statements are not intended to be restricted to rigid bodies; they are assumed to hold for all material systems whatever. The peculiar status of rigid bodies is that the principles in question are in most cases sufficient for the complete determination of the motion, the dynamical equations (1 or 2) being equal in number to the degrees of freedom (six) of a rigid solid, whereas in cases where the freedom is greater we have to invoke the aid of other supplementary physical hypotheses (cf. ELASTICITY; HYDROMECHANICS).

The increase of the kinetic energy of a rigid body in any interval of time is equal to the work done by the extraneous forces acting on the body. This is an immediate consequence of the fundamental postulate, in either of the forms above stated, since the internal forces do on the whole no work. The statement may be extended to a system of rigid bodies, provided the mutual reactions consist of the stresses in inextensible links, or the pressures between smooth surfaces, or the reactions at rolling contacts (§ 9).

§ 17. _Two-dimensional Problems._--In the case of rotation about a fixed axis, the principles take a very simple form. The position of the body is specified by a single co-ordinate, viz. the angle [theta] through which some plane passing through the axis and fixed in the body has turned from a standard position in space. Then d[theta]/dt, = [omega] say, is the _angular velocity_ of the body. The angular momentum of a particle m at a distance r from the axis is m[omega]r·r, and the total angular momentum is [Sigma](mr²)·[omega], or I[omega], if I denote the moment of inertia (§ 11) about the axis. Hence if N be the moment of the extraneous forces about the axis, we have

d
--- (I[omega]) = N. (1)
dt

This may be compared with the equation of rectilinear motion of a particle, viz. d/dt·(Mu) = X; it shows that I measures the inertia of the body as regards rotation, just as M measures its inertia as regards translation. If N = 0, [omega] is constant.

As a first example, suppose we have a flywheel free to rotate about a
horizontal axis, and that a weight m hangs by a vertical string from
the circumferences of an axle of radius b (fig. 72). Neglecting
frictional resistance we have, if R be the tension of the string,

I[.omega] = Rb, m[.u] = mg - R,

whence
mb²
b[.omega] = ------- (2)
1 + mb²

This gives the acceleration of m as modified by the inertia of the
wheel.

A "compound pendulum" is a body of any form which is free to rotate
about a fixed horizontal axis, the only extraneous force (other than
the pressures of the axis) being that of gravity. If M be the total
mass, k the radius of gyration (§ 11) about the axis, we have

d / d[theta]\
--- ( Mk² -------- ) = -Mgh sin [theta], (3)
dt \ dt /

where [theta] is the angle which the plane containing the axis and the
centre of gravity G makes with the vertical, and h is the distance of
G from the axis. This coincides with the equation of motion of a
simple pendulum [§ 13 (15)] of length l, provided l = k²/h. The plane
of the diagram (fig. 73) is supposed to be a plane through G
perpendicular to the axis, which it meets in O. If we produce OG to P,
making OP = l, the point P is called the _centre of oscillation_; the
bob of a simple pendulum of length OP suspended from O will keep step
with the motion of P, if properly started. If [kappa] be the radius of
gyration about a parallel axis through G, we have k² = [kappa]² + h²
by § 11 (16), and therefore l = h + [kappa]²/h, whence

GO·GP = [kappa]². (4)

This shows that if the body were swung from a parallel axis through P
the new centre of oscillation would be at O. For different parallel
axes, the period of a small oscillation varies as [root]l, or
[root](GO + OP); this is least, subject to the condition (4), when GO
= GP = [kappa]. The reciprocal relation between the centres of
suspension and oscillation is the basis of Kater's method of
determining g experimentally. A pendulum is constructed with two
parallel knife-edges as nearly as possible in the same plane with G,
the position of one of them being adjustable. If it could be arranged
that the period of a small oscillation should be exactly the same
about either edge, the two knife-edges would in general occupy the
positions of conjugate centres of suspension and oscillation; and the
distances between them would be the length l of the equivalent simple
pendulum. For if h1 + [kappa]²/h1 = h2 + [kappa]²/h2, then unless h1 =
h2, we must have [kappa]² = h1h2, l = h1 + h2. Exact equality of the
two observed periods ([tau]1, [tau]2, say) cannot of course be secured
in practice, and a modification is necessary. If we write l1 = h1 +
[kappa]²/h1, l2 = h2 + [kappa]²/h2, we find, on elimination of
[kappa],

l1 + l2 l1 - l2
½ ------- + ½ ------- = 1,
h1 + h2 h1 - h2

whence

4[pi]² ½ ([tau]1² + [tau]2²) ½ ([tau]1² - [tau]2²)
------ = --------------------- + --------------------- (5)
g h1 + h2 h1 - h2

The distance h1 + h2, which occurs in the first term on the right hand
can be measured directly. For the second term we require the values of
h1, h2 separately, but if [tau]1, [tau]2 are nearly equal whilst h1,
h2 are distinctly unequal this term will be relatively small, so that
an approximate knowledge of h1, h2 is sufficient.

As a final example we may note the arrangement, often employed in
physical measurements, where a body performs small oscillations about
a vertical axis through its mass-centre G, under the influence of a
couple whose moment varies as the angle of rotation from the
equilibrium position. The equation of motion is of the type

I[:theta] = -K[theta], (6)

and the period is therefore [tau] = 2[pi][root](I/K). If by the
attachment of another body of known moment of inertia I´, the period
is altered from [tau] to [tau]´, we have [tau]´ = 2[pi][root][(I +
I´)/K]. We are thus enabled to determine both I and K, viz.

I/I´ = [tau]²/([tau]´² - [tau]²), K = 4[pi]²[tau]²I/([tau]´² - [tau]²). (7)

The couple may be due to the earth's magnetism, or to the torsion of
a suspending wire, or to a "bifilar" suspension. In the latter case,
the body hangs by two vertical threads of equal length l in a plane
through G. The motion being assumed to be small, the tensions of the
two strings may be taken to have their statical values Mgb/(a + b),
Mga/(a + b), where a, b are the distances of G from the two threads.
When the body is twisted through an angle [theta] the threads make
angles a[theta]/l, b[theta]/l with the vertical, and the moment of the
tensions about the vertical through G is accordingly -K[theta], where
K = M gab/l.

For the determination of the motion it has only been necessary to use one of the dynamical equations. The remaining equations serve to determine the reactions of the rotating body on its bearings. Suppose, for example, that there are no extraneous forces. Take rectangular axes, of which Oz coincides with the axis of rotation. The angular velocity being constant, the effective force on a particle m at a distance r from Oz is m[omega]²r towards this axis, and its components are accordingly -[omega]²mx, -[omega]²my, O. Since the reactions on the bearings must be statically equivalent to the whole system of effective forces, they will reduce to a force (X Y Z) at O and a couple (L M N) given by

X = -[omega]²[Sigma](mx) = -[omega]²[Sigma](m)[|x], Y = -[omega]²[Sigma](my) = -[omega]²[Sigma](m)[|y], Z = 0,

L = [omega]²[Sigma](myz), M = -[omega]²[Sigma](mzx), N = 0, (8)

where [|x], [|y] refer to the mass-centre G. The reactions do not therefore reduce to a single force at O unless [Sigma](myz) = 0, [Sigma](msx) = 0, i.e. unless the axis of rotation be a principal axis of inertia (§ 11) at O. In order that the force may vanish we must also have x, y = 0, i.e. the mass-centre must lie in the axis of rotation. These considerations are important in the "balancing" of machinery. We note further that if a body be free to turn about a fixed point O, there are three mutually perpendicular lines through this point about which it can rotate steadily, without further constraint. The theory of principal or "permanent" axes was first investigated from this point of view by J. A. Segner (1755). The origin of the name "deviation moment" sometimes applied to a product of inertia is also now apparent.

Proceeding to the general motion of a rigid body in two dimensions we may take as the three co-ordinates of the body the rectangular Cartesian co-ordinates x, y of the mass-centre G and the angle [theta] through which the body has turned from some standard position. The components of linear momentum are then M[.x], M[.y], and the angular momentum relative to G as base is I[.theta], where M is the mass and I the moment of inertia about G. If the extraneous forces be reduced to a force (X, Y) at G and a couple N, we have

M[:x] = X, M[:y] = Y, I[:theta] = N. (9)

If the extraneous forces have zero moment about G the angular velocity [.theta] is constant. Thus a circular disk projected under gravity in a vertical plane spins with constant angular velocity, whilst its centre describes a parabola.

We may apply the equations (9) to the case of a solid of revolution
rolling with its axis horizontal on a plane of inclination [alpha]. If
the axis of x be taken parallel to the slope of the plane, with x
increasing downwards, we have

M[:x] = Mg sin [alpha] - F, 0 = Mg cos [alpha] - R, M[kappa]²[:theta] = Fa (10)

where [kappa] is the radius of gyration about the axis of symmetry, a
is the constant distance of G from the plane, and R, F are the normal
and tangential components of the reaction of the plane, as shown in
fig. 74. We have also the kinematical relation [.x] = a[.theta]. Hence

a² [kappa]²
[:x] = ------------- g sin [alpha], R = Mg cos [alpha], F = ------------- Mg sin [alpha]. (11)
[kappa]² + a² [kappa]² + a²

The acceleration of G is therefore less than in the case of
frictionless sliding in the ratio a²/([kappa]² + a²). For a
homogeneous sphere this ratio is 5/7, for a uniform circular cylinder
or disk 2/3, for a circular hoop or a thin cylindrical shell ½.

The equation of energy for a rigid body has already been stated (in effect) as a corollary from fundamental assumptions. It may also be deduced from the principles of linear and angular momentum as embodied in the equations (9). We have

M([.x][:x] + [.y][:]y) + l[.theta][:theta] + X[.x] + Y[.y] + N[.theta], (12)

whence, integrating with respect to t,

½ M([.x]² + [.y]²) + ½I[.theta]² = [int](X dx + Y dy + Nd[theta]) + const. (13)

The left-hand side is the kinetic energy of the whole mass, supposed concentrated at G and moving with this point, together with the kinetic energy of the motion relative to G (§ 15); and the right-hand member represents the integral work done by the extraneous forces in the successive infinitesimal displacements into which the motion may be resolved.

The formula (13) may be easily verified in the case of the compound
pendulum, or of the solid rolling down an incline. As another example,
suppose we have a circular cylinder whose mass-centre is at an
excentric point, rolling on a horizontal plane. This includes the case
of a compound pendulum in which the knife-edge is replaced by a
cylindrical pin. If [alpha] be the radius of the cylinder, h the
distance of G from its axis (O), [kappa] the radius of gyration about
a longitudinal axis through G, and [theta] the inclination of OG to
the vertical, the kinetic energy is 1/2M[kappa]²[.theta]² +
½M·CG²·[.theta]², by § 3, since the body is turning about the line of
contact (C) as instantaneous axis, and the potential energy is--Mgh
cos [theta]. The equation of energy is therefore

½ M([kappa]² + [alpha]² + h² - 2 ah cos [theta]) [.theta]² - Mgh cos [theta] - const. (14)

Whenever, as in the preceding examples, a body or a system of bodies, is subject to constraints which leave it virtually only one degree of freedom, the equation of energy is sufficient for the complete determination of the motion. If q be any variable co-ordinate defining the position or (in the case of a system of bodies) the configuration, the velocity of each particle at any instant will be proportional to [.q], and the total kinetic energy may be expressed in the form ½A[.q]², where A is in general a function of q [cf. equation (14)]. This coefficient A is called the coefficient of inertia, or the reduced inertia of the system, referred to the co-ordinate q.

Thus in the case of a railway truck travelling with velocity u the
kinetic energy is ½(M + m[kappa]²/[alpha]²)u², where M is the total
mass, [alpha] the radius and [kappa] the radius of gyration of each
wheel, and m is the sum of the masses of the wheels; the reduced
inertia is therefore M + m[kappa]²/[alpha]². Again, take the system
composed of the flywheel, connecting rod, and piston of a
steam-engine. We have here a limiting case of three-bar motion (§ 3),
and the instantaneous centre J of the connecting-rod PQ will have the
position shown in the figure. The velocities of P and Q will be in the
ratio of JP to JQ, or OR to OQ; the velocity of the piston is
therefore y[.theta], where y = OR. Hence if, for simplicity, we
neglect the inertia of the connecting-rod, the kinetic energy will be
½(I + My²)[.theta]², where I is the moment of inertia of the flywheel,
and M is the mass of the piston. The effect of the mass of the piston
is therefore to increase the apparent moment of inertia of the
flywheel by the variable amount My². If, on the other hand, we take OP
(= x) as our variable, the kinetic energy is 1/2(M + I/y²)[.x]². We
may also say, therefore, that the effect of the flywheel is to
increase the apparent mass of the piston by the amount I/y²; this
becomes infinite at the "dead-points" where the crank is in line with
the connecting-rod.

If the system be "conservative," we have

½ Aq² + V = const., (15)

where V is the potential energy. If we differentiate this with respect to t, and divide out by [.q], we obtain

dA dV
A[:q] + ½ -- q² + -- = 0 (16)
dq dq

as the equation of motion of the system with the unknown reactions (if any) eliminated. For equilibrium this must be satisfied by [.q] = O; this requires that dV/dq = 0, i.e. the potential energy must be "stationary." To examine the effect of a small disturbance from equilibrium we put V = f(q), and write q = q0 + [eta], where q0 is a root of f´(q0) = 0 and [eta] is small. Neglecting terms of the second order in [eta] we have dV/dq = f´(q) = f´´(q0)·[eta], and the equation (16) reduces to

A[:eta] + f´´(q0)[eta] = 0, (17)

where A may be supposed to be constant and to have the value corresponding to q = q0. Hence if f´´(q0) > 0, i.e. if V is a minimum in the configuration of equilibrium, the variation of [eta] is simple-harmonic, and the period is 2[pi][root][A/f´´(q0)]. This depends only on the constitution of the system, whereas the amplitude and epoch will vary with the initial circumstances. If f´´(q0) < 0, the solution of (17) will involve real exponentials, and [eta] will in general increase until the neglect of the terms of the second order is no longer justified. The configuration q = q0, is then unstable.

As an example of the method, we may take the problem to which equation
(14) relates. If we differentiate, and divide by [theta], and retain
only the terms of the first order in [theta], we obtain

{x² + (h - [alpha])²} [:theta] + gh[theta] = 0, (18)

as the equation of small oscillations about the position [theta] = 0.
The length of the equivalent simple pendulum is {[kappa]² + (h -
[alpha])²}/h.

The equations which express the change of motion (in two dimensions) due to an instantaneous impulse are of the forms

M(u´- u) = [xi], M([nu]´ - [nu]) = [eta], I([omega]´ - [omega]) = [nu]. (19)

Here u´, [nu]´ are the values of the component velocities of G just before, and u, [nu] their values just after, the impulse, whilst [omega]´, [omega] denote the corresponding angular velocities. Further, [xi], [eta] are the time-integrals of the forces parallel to the co-ordinate axes, and [nu] is the time-integral of their moment about G. Suppose, for example, that a rigid lamina at rest, but free to move, is struck by an instantaneous impulse F in a given line. Evidently G will begin to move parallel to the line of F; let its initial velocity be u´, and let [omega]´ be the initial angular velocity. Then Mu´ = F, I[omega]´ = F·GP, where GP is the perpendicular from G to the line of F. If PG be produced to any point C, the initial velocity of the point C of the lamina will be

u´ - [omega]´·GC = (F/M)·(I - GC·CP/[kappa]²),

where [kappa]² is the radius of gyration about G. The initial centre of rotation will therefore be at C, provided GC·GP = [kappa]². If this condition be satisfied there would be no impulsive reaction at C even if this point were fixed. The point P is therefore called the _centre of percussion_ for the axis at C. It will be noted that the relation between C and P is the same as that which connects the centres of suspension and oscillation in the compound pendulum.

§ 18. _Equations of Motion in Three Dimensions._--It was proved in § 7 that a body moving about a fixed point O can be brought from its position at time t to its position at time t + [delta]t by an infinitesimal rotation [epsilon] about some axis through O; and the limiting position of this axis, when [delta]t is infinitely small, was called the "instantaneous axis." The limiting value of the ratio [epsilon]/[delta]t is called the _angular velocity_ of the body; we denote it by [omega]. If [xi], [eta], [zeta] are the components of [epsilon] about rectangular co-ordinate axes through O, the limiting values of [xi]/[delta]t, [eta]/[delta]t, [zeta]/[delta]t are called the _component angular velocities_; we denote them by p, q, r. If l, m, n be the direction-cosines of the instantaneous axis we have

p = l[omega], q = m[omega], r = n[omega], (1)
p² + q² + r² = [omega]². (2)

If we draw a vector OJ to represent the angular velocity, then J traces out a certain curve in the body, called the _polhode_, and a certain curve in space, called the _herpolhode_. The cones generated by the instantaneous axis in the body and in space are called the polhode and herpolhode cones, respectively; in the actual motion the former cone rolls on the latter (§ 7).

The special case where both cones are right circular and [omega] is
constant is important in astronomy and also in mechanism (theory of
bevel wheels). The "precession of the equinoxes" is due to the fact
that the earth performs a motion of this kind about its centre, and
the whole class of such motions has therefore been termed
_precessional_. In fig. 78, which shows the various cases, OZ is the
axis of the fixed and OC that of the rolling cone, and J is the point
of contact of the polhode and herpolhode, which are of course both
circles. If [alpha]be the semi-angle of the rolling cone, [beta] the
constant inclination of OC to OZ, and [.psi] the angular velocity with
which the plane ZOC revolves about OZ, then, considering the velocity
of a point in OC at unit distance from O, we have

[omega] sin [alpha] = ±[.psi] sin [beta], (3)

where the lower sign belongs to the third case. The earth's
precessional motion is of this latter type, the angles being [alpha] =
.0087´´, [beta] = 23° 28´.

If m be the mass of a particle at P, and PN the perpendicular to the instantaneous axis, the kinetic energy T is given by

2T = [Sigma] {m([omega]·PN)²} = [omega]²·[Sigma](m·PN²) = I[omega]², (4)

where I is the moment of inertia about the instantaneous axis. With the same notation for moments and products of inertia as in § 11 (38), we have

I = Al² + Bm² + Cn² - 2Fmn - 2Gnl - 2Hlm,

and therefore by (1),

2T = Ap² + Bq² + Cr² - 2Fqr - 2Grp - 2Hpq. (5)

Again, if x, y, z be the co-ordinates of P, the component velocities of m are

qz - ry, rx - pz, py - qx, (6)

by § 7 (5); hence, if [lambda], [mu], [nu] be now used to denote the component angular momenta about the co-ordinate axes, we have [lambda] = [Sigma][m(py - qx)y - m(rx - pz)z], with two similar formulae, or

[dP]T \
[lambda] = Ap - Hq - Gr= -----, |
[dP]p |
|
[dP]T |
[mu] = -Hp + Bq - Fr = -----, > (7)
[dP]q |
|
[dP]T |
[nu] = -Gp - Fq + Cr = -----. |
[dP]r /

If the co-ordinate axes be taken to coincide with the principal axes of inertia at O, at the instant under consideration, we have the simpler formulae

2T = Ap² + Bq² + Cr², (8)

[lambda] = Ap, [mu] = Bq, [nu] = Cr. (9)

It is to be carefully noticed that the axis of resultant angular momentum about O does not in general coincide with the instantaneous axis of rotation. The relation between these axes may be expressed by means of the momental ellipsoid at O. The equation of the latter, referred to its principal axes, being as in § 11 (41), the co-ordinates of the point J where it is met by the instantaneous axis are proportional to p, q, r, and the direction-cosines of the normal at J are therefore proportional to Ap, Bq, Cr, or [lambda], [mu], [nu]. The axis of resultant angular momentum is therefore normal to the tangent plane at J, and does not coincide with OJ unless the latter be a principal axis. Again, if [Gamma] be the resultant angular momentum, so that

[lambda]² + [mu]² + [nu]² = [Gamma]², (10)

the length of the perpendicular OH on the tangent plane at J is

Ap p Bq q Cr r 2T [rho]
OH = ------- · -------[rho] + ------- · -------[rho] + ------- · -------[rho] = ------- · -------, (11)
[Gamma] [omega] [Gamma] [omega] [Gamma] [omega] [Gamma] [omega]

where [rho] = OJ. This relation will be of use to us presently (§ 19).

The motion of a rigid body in the most general case may be specified by means of the component velocities u, v, w of any point O of it which is taken as base, and the component angular velocities p, q, r. The component velocities of any point whose co-ordinates relative to O are x, y, z are then

u + qz - ry, v + rx - pz, w + py - qx (12)

by § 7 (6). It is usually convenient to take as our base-point the mass-centre of the body. In this case the kinetic energy is given by

2T = M0(u² + v² + w²) + Ap² + Bq² + Cr² - 2Fqr - 2Grp - 2Hpg, (13)

where M0 is the mass, and A, B, C, F, G, H are the moments and products of inertia with respect to the mass-centre; cf. § 15 (9).

The components [xi], [eta], [zeta] of linear momentum are

[dP]T [dP]T [dP]T
[xi] = M0u = -----, [eta] = M0v = -----, [zeta] = M0w = -----, (14)
[dP]u [dP]v [dP]w

whilst those of the relative angular momentum are given by (7). The preceding formulae are sufficient for the treatment of instantaneous impulses. Thus if an impulse ([xi], [eta], [zeta], [lambda], [mu], [nu]) change the motion from (u, v, w, p, q, r) to (u´, v´, w´, p´, q´, r´) we have

M0(u´- u) = [xi], M0(v´- v) = [eta], M0(w´- w) = [zeta], \
> (15)
A(p´ - p) = [lambda], B(q´- q) = [mu], C(r´- r) = [nu], /

where, for simplicity, the co-ordinate axes are supposed to coincide with the principal axes at the mass-centre. Hence the change of kinetic energy is

T´- T = [xi] · ½(u + u´) + [eta] · ½(v + v´) + [zeta] · ½(w + w´),
+ [lambda] · ½(p + p´) + [mu] · ½(q + q´) + [nu] · ½(r + r´). (16)

The factors of [xi], [eta], [zeta], [lambda], [mu], [nu] on the right-hand side are proportional to the constituents of a possible infinitesimal displacement of the solid, and the whole expression is proportional (on the same scale) to the work done by the given system of impulsive forces in such a displacement. As in § 9 this must be equal to the total work done in such a displacement by the several forces, whatever they are, which make up the impulse. We are thus led to the following statement: the change of kinetic energy due to any system of impulsive forces is equal to the sum of the products of the several forces into the semi-sum of the initial and final velocities of their respective points of application, resolved in the directions of the forces. Thus in the problem of fig. 77 the kinetic energy generated is ½M([kappa]² + Cq²)[omega]´², if C be the instantaneous centre; this is seen to be equal to ½F·[omega]´·CP, where [omega]´·CP represents the initial velocity of P.

The equations of continuous motion of a solid are obtained by substituting the values of [xi], [eta], [zeta], [lambda], [mu], [nu] from (14) and (7) in the general equations

d[xi] d[eta] d[zeta] \
----- = X, ------ = Y, ------- = Z, |
dt dt dt |
> (17)
d[lambda] d[mu] d[nu] |
--------- = L, ----- = M, ----- = N, |
dt dt dt /

where (X, Y, Z, L, M, N) denotes the system of extraneous forces referred (like the momenta) to the mass-centre as base, the co-ordinate axes being of course fixed in direction. The resulting equations are not as a rule easy of application, owing to the fact that the moments and products of inertia A, B, C, F, G, H are not constants but vary in consequence of the changing orientation of the body with respect to the co-ordinate axes.

An exception occurs, however, in the case of a solid which is
kinetically symmetrical (§ 11) about the mass-centre, e.g. a uniform
sphere. The equations then take the forms

M0[.u] = X, M0[.v] = Y, M0[.w] = Z,
C[.p] = L, C[.q] = M, C[.r] = N, (18)

where C is the constant moment of inertia about any axis through the
mass-centre. Take, for example, the case of a sphere rolling on a
plane; and let the axes Ox, Oy be drawn through the centre parallel to
the plane, so that the equation of the latter is z = -a. We will
suppose that the extraneous forces consist of a known force (X, Y, Z)
at the centre, and of the reactions (F1, F2, R) at the point of
contact. Hence

M0[.u] = X + F1, M0[.v] = Y + F2, 0 = Z + R, \
C[.p] = F2a, C[.q] = -F1a, C[.r] = 0. / (19)

The last equation shows that the angular velocity about the normal to
the plane is constant. Again, since the point of the sphere which is
in contact with the plane is instantaneously at rest, we have the
geometrical relations

u + qa = 0, v + pa = 0, w = 0, (20)

by (12). Eliminating p, q, we get

(M0 + Ca^-2)[.u] = X, (M0 + Ca^-2)[.v] = Y. (21)

The acceleration of the centre is therefore the same as if the plane
were smooth and the mass of the sphere were increased by C/[alpha]².
Thus the centre of a sphere rolling under gravity on a plane of
inclination a describes a parabola with an acceleration

g sin [alpha]/(1 + C/Ma²)

parallel to the lines of greatest slope.

Take next the case of a sphere rolling on a fixed spherical surface.
Let a be the radius of the rolling sphere, c that of the spherical
surface which is the locus of its centre, and let x, y, z be the
co-ordinates of this centre relative to axes through O, the centre of
the fixed sphere. If the only extraneous forces are the reactions (P,
Q, R) at the point of contact, we have

M0[:x] = P, M0[.y] = Q, M0[:z] = R, \
|
a a a > (22)
Cp = ---(yR - zQ), C[.q] = ---(zP - xR), C[.r] = ---(xQ - yP), |
c c c /