﻿ Three particles, each of mass m, are situates at the vertices of an equilateral triangle of side length a. The only force acting on the particles are their mutual gravitational forces. It is desired that each particle move in a circle while maintaining the original mutual separation a. Find the initial velocity that should be given to each particle and also the time-period of the circular motion. : Kaysons Education

# Three Particles, Each Of Mass m, Are Situates At The Vertices Of An Equilateral Triangle Of Side Length a. The Only Force Acting On The Particles Are Their Mutual Gravitational Forces. It Is Desired That Each Particle Move In A Circle While Maintaining The Original Mutual Separation a. Find The Initial Velocity That Should Be Given To Each Particle And Also The Time-period Of The Circular Motion.

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Let the particles be situated at the corners AB and C of equilateral triangle, where AB = BC = CA = a. The circular path of the particles, such that their mutual separation a is maintained, will be the circle circumscribing the triangle. Let O be the centre and r the radius of this circle. Then

Let v be the speed of each particle. Let us consider the particle (mass m) at A. The gravitational force on it due to particles at B and C (each of massm) are

This, for preserving the circular motion, must be equal to the centripetal force mv2/r. Thus

The time period of the circular motion is

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