Question

A double star is a composite system of two stars rotating about their centre of mass under their mutual gravitational attraction. Let us consider such a double star which has two stars of masses $m$ and $2m$ at a separation $l$. If $T$ is the time period of rotation about their entre of mass then.

• A. $T=2\pi \sqrt{\frac{l^3}{mG}}$
• B. $T=2\pi \sqrt{\frac{l^3}{2mG}}$
• C. $T=2\pi \sqrt{\frac{l^3}{3mG}}$
• D. $T=2\pi \sqrt{\frac{l^3}{4mG}}$

Answer 1

Answer: (C)

$T=2\pi \sqrt{\frac{l^3}{3mG}}$

The masses will be rotating about their Center of Mass which is at distance of 2l/3 from the mass m. The force on mass m is the gravitational force which is also providing the force for circular motion.

$m\left(2l/3\right){\omega }^2=G\frac{m\times 2m}{l^2}$
$\omega =\sqrt{G\frac{3m}{l^3}}$
$T=2\pi /\omega$