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

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

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

Options

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 $\dfrac{1}{2\pi}\sqrt{\dfrac{{d}^{3}}{3Gm}}$

Correct answer

A. $2\pi\sqrt{\dfrac{{d}^{3}}{3Gm}}$

Step-by-step solution

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

$F = \dfrac{G\left(2m\right)m}{{d}^{2}} = \left(2m\right){\omega}^{2}\left(d / 3\right)$

$\dfrac{Gm}{{d}^{2}} = {\omega}^{2}\dfrac{d}{3}$

$⇒ {\omega}^{2} = \dfrac{3Gm}{{d}^{3}}$

$⇒ \omega = \sqrt{\dfrac{3Gm}{{d}^{3}}}$

$⇒ T = \dfrac{2\pi}{\omega} = 2\pi\sqrt{\dfrac{{d}^{3}}{3Gm}}$

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