Question

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

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

Answer 1

Answer: (B)

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