Question

Two stars of masses $m$ and $2m$ at a distance $d$ rotate about their common centre of mass in free space. The period of revolution is :

• A. $\frac{1}{2\pi} \sqrt{\frac{d^3}{2Gm}}$
• B. $2\pi \sqrt{\frac{d^3}{Gm}}$
• C. $2\pi \sqrt{\frac{d^3}{3Gm}}$
• D. $\frac{1}{2\pi} \sqrt{\frac{2 d^3}{Gm}}$

Hint:

In a binary star system, both stars have the same angular velocity $\omega$ and the same period $T$, but they revolve in orbits of different radii.

Answer 1

The Correct Option is C

Step 1: The centripetal force for mass $m$ is provided by gravitational attraction: $\frac{G(m)(2m)}{d^2} = m\omega^2 r_1$.
Step 2: Distance of mass $m$ from center of mass $r_1 = \frac{2m(d)}{m+2m} = \frac{2d}{3}$.
Step 3: $\frac{2Gm^2}{d^2} = m\omega^2 (\frac{2d}{3}) \Rightarrow \omega^2 = \frac{3Gm}{d^3}$.
Step 4: $T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{d^3}{3Gm}}$.