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