Question

A binary star is a system of two stars rotating about their centre of mass only under mutual gravitational attraction. Let the stars have mass m and $2m$ and let their separation be $l$. Their time period of rotation about their centre of mass will be proportional to

• A. $l^2$
• B. $l^{3/2}$
• C. $l^{1/2}$
• D. $l$

Answer 1

The correct option is B $l^{3/2}$


$r_1=\frac{l}{3},r_2=\frac{2l}{3}$
$\frac{G\cdot 2m\cdot m}{l^2}=2m\cdot {\omega }^2\cdot \frac{l}{3}=m{\omega }^2\cdot \frac{2l}{3}$
$\frac{3GM}{l^3}={\omega }^2\Rightarrow \omega \propto \frac{1}{l^{3/2}}$
$T=\frac{2\pi }{\omega }\Rightarrow T\propto l^{3/2}$