Two massive particles of masses M & m (M > m) are separated by a distance $\ell$. They rotate with equal angular velocity under their gravitational attraction. The linear speed of the particle of mass m is

$\sqrt{\frac{G M^2}{(M+m) \ell}}$

The system rotates about the centre of mass. The gravitational force acting on the particle m accelerates it towards the centre of the circular path, which has the radius

$R=\frac{M \ell}{M+m}$

$\Rightarrow F=\frac{mv^2}{r} \Rightarrow \frac{GMm}{\ell^2}=\frac{mv^2}{\frac{M \ell}{M+m}}$

$\Rightarrow v=\sqrt{\frac{GM^2}{(M+m) \ell}}$