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(D) The two stars revolve around their common center of mass $(COM)$ with the same angular velocity $\omega$.Let $r_1$ and $r_2$ be the distances of m

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$\sqrt{\frac{3 G M}{d^3}}$

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(D) The two stars revolve around their common center of mass $(COM)$ with the same angular velocity $\omega$.Let $r_1$ and $r_2$ be the distances of masses $M$ and $2M$ from the $COM$,respectively.We know that $r_1 + r_2 = d$ and $M r_1 = (2M) r_2$.From these,$r_1 = \frac{2M}{3M} d = \frac{2}{3} d$.The gravitational force between the stars provides the necessary centripetal force for the star of mass $M$:$F_g = F_c$$\frac{G(M)(2M)}{d^2} = M \omega^2 r_1$Substituting $r_1 = \frac{2}{3} d$:$\frac{2 G M^2}{d^2} = M \omega^2 \left(\frac{2}{3} d\right)$$\frac{2 G M}{d^2} = \omega^2 \left(\frac{2}{3} d\right)$$\omega^2 = \frac{2 G M}{d^2} \cdot \frac{3}{2 d} = \frac{3 G M}{d^3}$$\omega = \sqrt{\frac{3 G M}{d^3}}$

Two stars of masses $M$ and $2M$ are at a distance $d$ apart and are revolving around their common center of mass. The angular velocity of the system of the two stars is ($G$ is the universal gravitational constant).

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