Question

Two stars of masses $M$ and $3M$ respectively are going around each other, in near circular orbits, with period $T$. The separation between them is given $D=\sqrt[3]{\frac{kGM{T}^2}{{\pi }^2}}.$ The value of $k$ is

• A. 0.5
• B. 1
• C. 1.5
• D. 3

Answer 1

Answer: (B)

The centre of mass of the system will be located at distance $\frac{D}{4}$ from the heavier mass.
Centripetal force is provided by the mutual gravitation.
$\frac{GM(3M)}{D^2}=(3M)\frac{D}{4}{\omega }^2$
$\Rightarrow \frac{GM}{D^2}=\frac{D4{\pi }^2}{4T^2}$
$D^3=\frac{GM{T}^2}{{\pi }^2}$
$\Rightarrow D=\sqrt[3]{\frac{GM{T}^2}{{\pi }^2}}$