A double star system consists of two stars having masses M and 2M. The distance between their centers is r. They revolve under their mutual gravitational attraction. Then the incorrect options are____

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Heavier star revolve in an orbit of radius $\frac{2 r}{3}$

Both stars revolve with the same period which is equal to $2 π {(\frac{3 r^{3}}{2 G M})}^{1 / 2}$

Kinetic energy of heavy star is twice that of light star.

Angular momentum of both stars about common center of mass are not equal

answer is A, B, C, D.

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Detailed Solution

(i) they revolve about their C.O.M with same ‘ $ω$ ’

$r_{1} = \frac{2 M}{M + 2 M} r = \frac{2 r}{3} = r a d i u s o f s m a l l s t a r r_{2} = \frac{M}{M + 2 M} r = \frac{r}{3} = r a d i u s o f b i g s t a r (i v) L_{1} = M {(\frac{2 r}{3})}^{2} ω L_{2} = 2 M {(\frac{r}{3})}^{2} ω ∴ L_{1} \neq L_{2} (i i) M . (\frac{2 r}{3}) . ω^{2} = \frac{G . M .2 M}{r^{2}} \Rightarrow ω^{2} = \frac{2 G M}{r^{2} . \frac{2 r}{3}} \Rightarrow ω^{2} = \frac{3 G M}{r^{3}} ∴ T = 2 π \sqrt{\frac{r^{3}}{3 G M}} (i i i) K . E_{s m a l l} = \frac{1}{2} (M {(\frac{2 r}{3})}^{2}) ω^{2} = \frac{2 M r^{2} ω^{2}}{9} K . E_{b i g} = \frac{1}{2} (2 M {(\frac{r}{3})}^{2}) ω^{2} = \frac{M r^{2} ω^{2}}{9}$