Two stars of masses m and 2m at a distance d | vedclass

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(B) The gravitational force between the two stars provides the necessary centripetal force for their circular motion about the common centre of mass (

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$2 \pi \sqrt{\frac{d^{3}}{3Gm}}$

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(B) The gravitational force between the two stars provides the necessary centripetal force for their circular motion about the common centre of mass (c.o.m.).The distance of mass $m$ from the c.o.m. is $r_1 = \frac{2m}{m+2m} d = \frac{2d}{3}$.The gravitational force is $F = \frac{G(m)(2m)}{d^2} = \frac{2Gm^2}{d^2}$.For the star of mass $m$,the centripetal force is $F = m \omega^2 r_1 = m \omega^2 (\frac{2d}{3})$.Equating the forces: $\frac{2Gm^2}{d^2} = m \omega^2 (\frac{2d}{3})$.Simplifying: $\frac{Gm}{d^2} = \omega^2 \frac{d}{3} \implies \omega^2 = \frac{3Gm}{d^3}$.Thus,$\omega = \sqrt{\frac{3Gm}{d^3}}$.The period of revolution $T$ is given by $T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{d^3}{3Gm}}$.

Two stars of masses $m$ and $2m$ at a distance $d$ rotate about their common centre of mass in free space. The period of revolution is

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