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Q.

In a double star system, two stars of masses $m_{1} {and m}_{2}$ separated by a distance d rotate about their centre of mass. Then their common angular velocity would be

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a

\sqrt{\frac{G (m_{1} + m_{2})}{d^{3}}}

b

\sqrt{\frac{d^{3}}{G (m_{1} + m_{2})}}

c

\sqrt{\frac{{Gm}_{1}}{d^{3}}}

d

\sqrt{\frac{{Gm}_{2}}{d^{3}}}

answer is C.

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Detailed Solution

The C.M of the star

d_{1} = \frac{d}{(M + m)} m

$d_{2} = \frac{d}{(M + m)} \times M$

$\frac{d_{1}}{d_{2}} = \frac{m}{M}$

$F_{C} = F_{G}$

$\frac{GMm}{d^{2}} = {Mw}_{1}^{2} d_{1}$

$w_{1}^{2} = \frac{Gm}{d^{2} d_{1}} = \frac{Gm}{d^{2}} \times \frac{M + m}{d \times m}$

$w_{1}^{2} = \frac{G (M + m)}{d^{3}}$

$w_{1} = \sqrt{\frac{G (M + m)}{d^{3}}}$

$w_{1} = \sqrt{\frac{G (m_{1} + m_{2})}{d^{3}}}$

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In a double star system, two stars of massesm1 and m2 separated by a distance d rotate about their centre of mass. Then their common angular velocity would be