$\text{The distance of COM from mass m is }\frac{2d}{3}.\text{ Thus, m is moving in uniform circular motion of radius }\frac{2d}{3}$

$\text{Gravitational force on m is providing centripetal force.}$

$\begin{array}{l}\Rightarrow \frac{G\left(m\right)\left(2m\right)}{d^2}=m\times {\omega }^2\times \frac{2d}{3}\\ \Rightarrow \frac{2Gm}{d^2}={\omega }^2\times \frac{2d}{3}\\ \Rightarrow {\omega }^2=\frac{3Gm}{d^3}\\ \Rightarrow \omega =\sqrt{\frac{3Gm}{d^3}}\\ \text{ Hence, }T=\frac{2\pi }{\omega }=2\pi \sqrt{\frac{d^3}{3Gm}}\end{array}$