Two identical particles each of mass m go round a circle of radius...

The gravitational force $\left({F}_{G}\right)$ between the particles is given by

$\begin{cases}{F}_{G} & = & \dfrac{G{m}^{2}}{{\left(2a\right)}^{2}} \\ & = & \dfrac{G{m}^{2}}{4{a}^{2}} . . . \left(1\right)\end{cases}$

The centripetal force \left({F}_{C}\right)$of each particle can be written as</p><p>${F}_{C} = m{\omega}^{2}a . . . \left(2\right)$</p><p>Under balanced condition, equate equation (1) and equation (2) and simplify to obtain the angular speed of each particle.</p><p>$\begin{cases}m{\omega}^{2}a & = & \dfrac{G{m}^{2}}{4{a}^{2}} \\ & ⇒ & {\omega}^{2} = \dfrac{Gm}{4{a}^{3}} \\ & ⇒ & \omega = \sqrt{\dfrac{Gm}{4{a}^{3}}}\end{cases}