A double star consists of two stars having masses m and 2m. The distance between their centres is r. They revolve in circular orbits keeping a constant radius. Then choose the correct statement(s):

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The heavier star revolves in a circle of radius $\frac{r}{3}$

Both the stars revolve with the same period

The period of revolution of each star is $\frac{2 π r^{3 / 2}}{\sqrt{\frac{2 G m}{3}}}$

The period of revolution of each star is $\frac{2 π r^{3 / 2}}{\sqrt{3 G m}}$

answer is A, B, D.

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Detailed Solution

$\begin{array}{l} r_{1} + r_{2} = r a n d \frac{r_{1}}{r_{2}} = \frac{m_{2}}{m_{1}} = \frac{2}{1} \\ \Rightarrow r_{1} = \frac{2 r}{3} (s m a l l s t a r) a n d r_{2} = \frac{r}{3} (b i g s t a r) \end{array}$

$\begin{array}{l} G r a v i t a t i o n a l f o r c e p r o v i d e s c e n t r i p e t a l f o r c e . \\ \Rightarrow \frac{G (2 m) m}{r^{2}} = m (\frac{2 r}{3}) ω^{2} \\ \Rightarrow ω = \sqrt{\frac{3 G m}{r^{3}}} \\ \Rightarrow T = \frac{2 π}{ω} = 2 π \sqrt{\frac{r^{3}}{3 G m}} \end{array}$