Question

Compare the angular momentum of the two stars about their common centre of mass by calculating the ratio $L_m/L_M$.

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

$\frac{Gm\left(2m\right)}{d^2}=2m{\omega }^{2}R=m{\omega }^{2}r\to \left(1\right)$

$\to r=2R$

Angular momentum , $L_m=m\omega r^2=4m\omega R^2$

similarly, $L_M=2m\omega R^2$

Taking ratio, $\frac{L_m}{L_M}=\frac{4m\omega R^2}{2m\omega R^2}=2$