Question

A double star is a system of two stars of masses m and $2$m, rotating about their centre of mass only under their mutual gravitational attraction. If r is the separation between these two stars then their time period of rotation about their centre of mass will be proportional to.

• A. $r^{\frac{3}{2}}$
• B. r
• C. $m^{\frac{1}{2}}$
• D. $m^{-\frac{1}{2}}$

Answer 1

Answer: (A), (D)

$r^{\frac{3}{2}}$
$m^{-\frac{1}{2}}$

Consider a double star system

This system rotates about its com which is $\frac{2m}{3m}\times r=\frac{2r}{3}m$ from object of mass $m$

Now for rotation

$\frac{2r}{3}m{\omega }^2=\frac{G.m.2m}{r^2}\Rightarrow {\omega }^2=\frac{3Gm}{r^3}\Rightarrow \omega \alpha \frac{1}{r^{\frac{3}{2}}}$

and $\omega \alpha m^{\frac{1}{2}}$

$\Rightarrow T\alpha r^{\frac{3}{2}}andT{\alpha }^{\frac{-1}{2}}$