A system of physical chemistry - Index of

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A system of physical chemistry - Index of

EQUIPARTITION OF KINETIC ENERGY 27

attraction can only hold when the particle is not far removed from the

centre, i.e. when r is small. If A is this attractive force per unit dis-

tance from the centre, then on the assumption just made, the force

acting inwards at a distance r is A/*. We have now to calculate this

force in terms of the motion of the particle. Consider the particle

traversing the circular path shown in the figure (Fig. 4).

Suppose that the particle is at the point a, travelling with a velocity

u in the direction ax. After a short interval of time 8/ it is at b travel-

ling with a velocity u in the direction ib. The arc at = u8f, and so the

radian measure of the angle aob or cTd is 86= u^ijr. If Tc = Td=u, then

the velocity has changed from one represented in magnitude and direc-

tion by Tc to one represented by Td. By the triangle of velocities, the

change in velocity is represented in magnitude and direction by the line

cd. The direction of cl is the same as that of to ; and its magnitude is

2u sin 86/2. In the limit when 8/ is infinitesimally small sin Sdj-, == 8^/2,

or the change of velocity is u80, i.e., u^f/r.

Fig. 4.

Hence the acceleration inward (viz. the velocity inward divided by

8/ is equal to u'^Jr. But force = mass x acceleration. Hence the

force acting inwards and preventing the particle from flying oiff is mu^/n

This must be identical with Ar. That is—

mu^/r = Ar.

Further, the potential energy of the particle at a (namely, the work

which must be done upon the particle to bring it from the position of

rest, the centre of the circle, to the point a on the circumference) is the

product of the force acting into the distance traversed. The lorce varies

at every stage of the radius, so that it is necessary to integrate the work

expression lor each increment dr in order to obtain an expression for the

potential energy of the particle at the point a. That is the potential

energy of the particle when it is on the circumference is Ardr which

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