A binary star system consists of two stars having masses M and 2M. The distance between their
centers is equal to R. They revolve about their common center of mass under their mutual
gravitational force of interaction. Which of the following is/are correct ?

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Both the stars will revolve with the same time period which is equal to $\frac{2 π}{\sqrt{3 GM}} R^{3 / 2}$

Kinetic energy of lighter star is twice that of the other.

Heavier star revolves in orbit of radius $\frac{R}{3}$

Kinetic energy of lighter star is 4 times that of the other.

answer is A, B, C.

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Detailed Solution

$\frac{G 2 MM}{R^{2}} = 2 M ω^{2} \frac{R}{3} \Rightarrow ω = \sqrt{\frac{3 GM}{R^{3}}}$

$K_{M} = \frac{1}{2} M {(ω \times \frac{2 R}{3})}^{2}$

And $K_{2 M} = \frac{1}{2} \times 2 M {(ω \times \frac{R}{3})}^{2}$

So, $\frac{K_{2 M}}{K_{M}} = \frac{1}{2}$

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