Two stars of masses m and 2 m at a distance $d$ rotate about their common

(1)

$\Rightarrow \frac{\mathrm{G}(\mathrm{m})(2 \mathrm{~m})}{\mathrm{d}^{2}}=\mathrm{m} \omega^{2} \times \frac{2 \mathrm{~d}}{3}$

$\Rightarrow \frac{2 \mathrm{Gm}}{\mathrm{d}^{2}}=\omega^{2} \times \frac{2 \mathrm{~d}}{3}$

$\Rightarrow \omega^{2}=\frac{3 \mathrm{Gm}}{\mathrm{d}^{3}}$

$\Rightarrow \omega=\sqrt{\frac{3 \mathrm{Gm}}{\mathrm{d}^{3}}}$

we know that, $\omega=\frac{2 \pi}{T}$ so $T=\frac{2 \pi}{\omega}$

$\Rightarrow T=\frac{2 \pi}{\sqrt{\frac{3 G m}{d^{3}}}} \Rightarrow T=2 \pi \sqrt{\frac{d^{3}}{3 G m}}$