Question

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. $2\pi \sqrt{\frac{d^3}{3Gm}}$
• B. $\frac{1}{2\pi }\sqrt{\frac{3Gm}{d^3}}$
• C. $\frac{1}{2\pi }\sqrt{\frac{d^3}{3Gm}}$
• D. $2\pi \sqrt{\frac{3Gm}{d^3}}$

Answer 1

The correct option is A $2\pi \sqrt{\frac{d^3}{3Gm}}$


This is a double star system in which both the stars revolve around the centre of mass of the system with the same angular frequency $\omega$ $($say$)$.

On planet of mass $m$ :

$F_{\text{gravitation}}$ $=$ $F_{\text{centripetal}}$

$\Rightarrow \frac{G\left(m\right)\left(2m\right)}{d^2}=m{\omega }^2\times \frac{2d}{3}$

$[$ Distance of COM of the system from the star of mass $m$ $=$$\frac{2d}{3}]$

$\Rightarrow \frac{2Gm}{d^2}={\omega }^2\times \frac{2d}{3}$

$\Rightarrow {\omega }^2=\frac{3Gm}{d^3}$

$\Rightarrow \omega =\sqrt{\frac{3Gm}{d^3}}$

We know that,
$\omega =\frac{2\pi }{T}$

So, $T=\frac{2\pi }{\omega }$

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

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