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The mass density of a spherical galaxy varies as $\frac{K}{r}$ over a large distance ‘r’ from its centre. In that region, a small star is in a circular orbit of radius R. Then the period of revolution, T depends on R as

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T∝R

T2∝1R3

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T2∝R

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detailed solution

Correct option is D

Gravitational force on star FG=GmMr2 , where M is mass of galaxy within radius rSo, M= 0r4πr2drKr=4π2kr22⇒FG=mω2r⇒FG=m.r4π2T2⇒T2=4π2mrFG=4π2mrGmM/r2=4π2r3G4π2kr22⇒T2 ∝ r

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