Question

Two particles of equal mass go around in a circle of radius $r$ under the action of their mutual gravitational attraction. If the mass of each particle is $m$, the speed of each particle is

• A. $\sqrt{\frac{Gm}{r}}$
• B. $\sqrt{\frac{Gm}{2r}}$
• C. $\sqrt{\frac{Gm}{4r}}$
• D. $\sqrt{\frac{2Gm}{r}}$

Answer 1

The correct option is C $\sqrt{\frac{Gm}{4r}}$

Centripetal force acting on the body is balanced by the gravitational force acting between them i.e. $F_c=F_g$

Centripetal force $F_c=\frac{m{v}^2}{r}$

where $v$ is the speed of the each particle.

Gravitational force $F_g=\frac{Gmm}{(2r{)}^2}$

$\therefore$ $\frac{m{v}^2}{r}=\frac{Gmm}{(2r{)}^2}$

$⟹$ $v=\sqrt{\frac{Gm}{4r}}$