Question:
Two stars of masses $\mathrm{m}$ and $2 \mathrm{~m}$ at a distance $d$ rotate about their common centre of mass in free space. The period of revolution is -
Correct Option: 1
Solution:
(1)
$\Rightarrow \frac{\mathrm{G}(\mathrm{m})(2 \mathrm{~m})}{\mathrm{d}^{2}}=\mathrm{m} \omega^{2} \times \frac{2 \mathrm{~d}}{3}$
$\Rightarrow \frac{2 \mathrm{Gm}}{\mathrm{d}^{2}}=\omega^{2} \times \frac{2 \mathrm{~d}}{3}$
$\Rightarrow \omega^{2}=\frac{3 \mathrm{Gm}}{\mathrm{d}^{3}}$
$\Rightarrow \omega=\sqrt{\frac{3 \mathrm{Gm}}{\mathrm{d}^{3}}}$
we know that, $\omega=\frac{2 \pi}{T}$ so $T=\frac{2 \pi}{\omega}$
$\Rightarrow T=\frac{2 \pi}{\sqrt{\frac{3 G m}{d^{3}}}} \Rightarrow T=2 \pi \sqrt{\frac{d^{3}}{3 G m}}$