Question 20 of 223

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

A$2\pi \sqrt{\dfrac{{d}^{3}}{3Gm}}$B$2\pi \sqrt{\dfrac{3Gm}{{d}^{3}}}$C$\dfrac{1}{2\pi }\sqrt{\dfrac{3Gm}{{d}^{3}}}$D$\dfrac{1}{2\pi }\sqrt{\dfrac{{d}^{3}}{3Gm}}$

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Answer: (A) $2\pi \sqrt{\dfrac{{d}^{3}}{3Gm}}$

$F=\dfrac{G\left(2m\right)m}{{d}^{2}}=\left(2m\right){\omega }^{2}\left(d/3\right)$
$\dfrac{Gm}{{d}^{2}}={\omega }^{2}\dfrac{d}{3}$
$\Rightarrow {\omega }^{2}=\dfrac{3Gm}{{d}^{3}}$
$\Rightarrow \omega =\sqrt{\dfrac{3Gm}{{d}^{3}}}$
$\Rightarrow T=\dfrac{2\pi }{\omega }=2\pi \sqrt{\dfrac{{d}^{3}}{3Gm}}$

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