Question

The period of rotation of the stars about their common centre of mass in terms of $d,m,G$ is given as $T=\frac{2\pi d^{3/2}}{\sqrt{xGm}}$. Find $x$

Answer 1

Both the stars have same angular velocity $\omega$
Let the radii of stars are $r$ and $R$ for masses $m$ and $2m$ respectively
So$,r+R=d$. Force on each star $F=\frac{Gm\left(2m\right)}{d^2}$
Now, this $F$ acts as centripetal force for either stars and
$\frac{Gm\left(2m\right)}{d^2}=2m{\omega }^{2}R=m{\omega }^{2}r....................\left(1\right)$

$\to r=2R$ and $2R+R=d\to R=\frac{d}{3},r=\frac{2d}{3}$

From $\left(1\right),\frac{Gm\left(2m\right)}{d^2}=\frac{2m{\omega }^{2}d}{3}$

Solving this , $\omega =\sqrt{\frac{3Gm}{d^3}}$

So, $T=\frac{2\pi }{\omega }=\frac{2\pi d^{1.5}}{(3Gm{)}^{0.5}}$ and $x=3$.