Question

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

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

Answer 1

Answer: (A), (D)

The correct options are
A $l^{\frac{3}{2}}$
D $m^{-\frac{1}{2}}$

$\because F_{\text{gravitational}}=F_{\text{centripetal}}$
$\frac{Gm\left(2m\right)}{l^2}=m\left(\frac{2l}{3}\right){\omega }^2$ $\frac{m}{l^3}\propto {\omega }^2\propto \frac{1}{(\text{Time period}{)}^2}$
$T\propto l^{\frac{3}{2}}$ and $T\propto \frac{1}{\sqrt{m}}$