Three particles, each of the mass `m` are situated at the vertices of an equilateral triangle of side `a`. The only forces acting on the particles are their mutual gravitational forces. It is desired that each particle moves 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. `(F=(Gm_(1)m_(2))/(r^(2)))`

Options

A. `sqrt((GM)/(a))`
B. `sqrt((3GM)/(a))`
C. `3sqrt((GM)/(a))`
D. `sqrt((GM)/(3a))`

Answer

a

Explanation

`F_(R)=sqrt(F^(2)+F^(2)+2FF cos 60)`
`=sqrt3F`
`F_(R)= sqrt3(GM^(2))/(a^(2))`
`"Since "F_(R)=(MV^(2))/(r)`
`cos30=(a//2)/(OA)`
`therefore" "OA=(a//2)/(sqrt3//2)=(a)/(sqrt3)`
`r=(a)/(sqrt3).`