MECHANICS 719 .stem Let, at time t, ?, y n , z n be the coordinates of the particle free whose mass is m n , and let </> (D) be the law of attraction. Let p r q trticles. express the distance between the particles m p and m q ; then we have, for the motion of ?,, , .... (1), .... (2), .... (3), with similar equations for each of the others, the summations being taken throughout the system. Before we can make any attempt at a solution of these equations, we must know their num ber, and the laws of attraction between the several pairs of particles. But some general theorems, independent of these data, may easily be obtained. First, we have Conservation of Momentum. In the expression - ; , 1 or m p *- , we have a term m p m q (f> ( p r q ) onser-
- ition of
LJL IHp iomen- dt- and in m<, -3 we have dt* p q X,, Xq m q m p <p ( q r p ) q> p Hence, if we add all the equations of the form (1) together, the result will be Similarly = 0, and where ( 109) x, y, z is the centre of inertia of the system. These equations show that the speed of the centre of inertia parallel to each of the coordinate axes remains invariable during the motion ; that is, that the centre of inertia of the system remains at rest, or moves with constant speed in a straight line. Next we have Conservation ofMoment of Momentum. For if we moment multiply in succession equation (1) by y^ and equation (2) by x lt f mo- and subtract, and take the sum of all such remainders through the iieiitum, system of equations of the forms (1) and (2), we have S[m(a$ - yx)} . Integrating once, we have where the left-hand member is the moment of momentum of the system about the axis of z. This equation shows (since xy is any plane) that gene rally in the motion of a free system of particles, subject only to their mutual attractions, the moment of momentum about every axis remains constant. f energy. Finally, we have Conservation of Energy. Multiply (1) by 1 , (2) by 1 , (3) by Clt iit
- and, treating similarly all the other
equations, add them all together. Let us consider the result as regards the term on the right-hand side involving the product m p m q . Written at length it is dx p - at dx q rr at + similar terms in y and z| ; and the portion in brackets is equal to ] (x q - x p ^ C -(x q - x p ) + similar terms in y, z > ; tic d , , or - pr q ~( p r q ) ; _ fdx d 2 x dy d^ii dz d-z ) hence 2 m -
at dt - at at 2 dt df ) )
+ 2 m f m^<f> ( p r q )( p r q } j =0; therefore, on integration, !2(rar 2 ) + 2{m ; >m < ,0(pr,,)} = 11 . We see therefore that the change in the kinetic energy of the system in any time depends only on the relative distances of the particles at the beginning and end of that time. Another general expression for the kinetic energy of a system of Virial. particles, in terms of a function of the mutual forces, and the con straining forces if there be such, is readily found as follows. If x, y, z be the coordinates, at time t, of the particle m, we have / d 2 T 2?n(x 2 + y- + z 2 ) = 22w(x 2 + ff~ + s 2 ) + 22wi(ccx + yy + zz) .
ai j
But if X, Y, Z be the components of the forces (of whatever kind) acting on m, we have ( 119) ?nx = X, mi} = y, mj = Z. Thus This expression was originally devised by Clausius for application to the kinetic theory of gases. The quantity 2,m(x^ + y- + z 2 ) is obviously half the sum of the three principal moments of inertia of the group of particles about the origin ( 234). In all cases of motion of a group, in which this sum is either con stant or oscillates in an extremely short period about a constant value, the left-hand side may be regarded as (on the average at least) a vanishing quantity. Thus an equivalent of the kinetic energy is expressible as This expression is called the "virial." In so far as it arises from the mutual action between two par ticles m p and m q , its value is (in the notation above) -i(a p + m q with corresponding terms in y and z, altogether 5 ~ Xq x q } , p r i J Hence if we write, generally, r for the distance between two of the particles, and It for the stress between them as depending on their mutual action, the corresponding part of the virial is 42(Rr). This is positive when the stresses are of the nature of tension. When the mutual action is due to gravity only, 0Vf) = -^r > p 1 i and the part of the virial corresponding to this is m p m q / p r {1 , expressing half the exhaustion of the potential energy of the system. When the particles are in very great numbers, and enclosed in a Virial of vessel from the sides of which they rebound as is supposed in the a gas kinetic gas theory the pressure p, per unit of surface, on the walls contained of the vessel must be taken into account. If Z, m, n be the direc- in a tion cosines of the normal to the element rfS of the wall of the vessel. vessel whose coordinates arc x, y, z, the corresponding part of the virial is extended over the whole internal surface. We here assume that^> is constant. But Ix + my + nz is the perpendicular from the origin on the plane of rfS, so that the integral expresses three times the volume V of the vessel. Hence this part of the virial is |j>V. Thus, in the case of a gas not acted on by external forces, the kinetic energy is Impact of Smooth Spheres. 180. There remains to be treated, so far as particle Impact. dynamics is concerned, the self-contained subject of Impact. In connexion with it we must once more refer to the second and third of Newton s laws. We are now dealing with forces which produce, in finite masses, finite changes of momentum in excessively short periods of time. It is clear from this statement that their effects may be treated altogether independently of finite forces, which may be acting along with them, but which produce during the very short periods in question only infinitesimal results. And, as in general we have no knowledge of the actual force exerted at any instant during the impact, nor of the time during which the action lasts, we confine ourselves to the quantity, called the "impulse," which measures the Impulse. amount of momentum lost by one of the impinging bodies
and acquired by the other.