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 δt 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 δt 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 δt 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 ω be the angular velocity in this plane, the angular momentum relative to G is
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, ω will be constant. Again, the tension R of the string is given by
| R = m1ω2r1 = | m1m2 | ω2a, |
| m1 + m2 |
where a = r1 + r2. Also by (10) the internal kinetic energy is
| 12 | m1m2 | ω2a2 |
| 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 Rpq (which we will reckon positive when attractive) between any two particles mp, mq is a function only of the distance rpq between them. In this case the work done by the internal forces will be represented by
when the summation includes every pair of particles, and each integral is to be taken between the proper limits. If we write
when rpq 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 mẍ, mÿ, mz̈ 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 (mẍ, mÿ, mz̈) 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
| Σ(mẍ) = X, Σ(mÿ) = Y, Σ(mz̈) = Z, | |
| Σ {m (yz̈ − zÿ) } = L, Σ {m (zẍ − xz̈) } = M, Σ{m (xÿ − yẍ) } = 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
| ddt Σ(mẋ) = X, ddtΣ(mẏ) = Y, ddtΣ(mż) = Z, | |
| ddtΣ {m (yż − zẏ) } = L, ddtΣ {m (zẋ − xż) } = M, ddtΣ {m (xẏ − yẋ) } = N, ddt |
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 θ through which some plane passing through the axis and fixed in the body has turned from a standard position in space. Then dθ/dt, = ω say, is the angular velocity of the body. The angular momentum of a particle m at a distance r from the axis is mωr·r, and the total angular momentum is Σ(mr 2)·ω, or Iω, 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
| ddt(Iω) = N. | (1) |
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, ω is constant.
