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(D) Let the distances of the stars of masses $M$ and $2M$ from their center of mass be $r_1$ and $r_2$ respectively.By the definition of center of mas

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$\sqrt{\frac{3GM}{d^3}}$

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(D) Let the distances of the stars of masses $M$ and $2M$ from their center of mass be $r_1$ and $r_2$ respectively.By the definition of center of mass,$M r_1 = (2M) r_2$ and $r_1 + r_2 = d$.Solving these,we get $r_1 = \frac{2d}{3}$ and $r_2 = \frac{d}{3}$.The gravitational force between the stars provides the necessary centripetal force for their circular motion.For the star of mass $M$: $F_g = M \omega^2 r_1$.Substituting the values: $\frac{G(M)(2M)}{d^2} = M \omega^2 \left(\frac{2d}{3}\right)$.$\frac{2GM^2}{d^2} = M \omega^2 \left(\frac{2d}{3}\right)$.$\omega^2 = \frac{3GM}{d^3}$.Therefore,$\omega = \sqrt{\frac{3GM}{d^3}}$.

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 two stars is ($G$ is the universal gravitational constant):

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