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The mass density of a spherical galaxy varies as Kr 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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a

T∝R

b

T2∝1R3

c

T2∝R3

d

T2∝R

answer is D.

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Detailed Solution

Gravitational force on star FG=GmMr2        , where  M  is  mass  of  galaxy within radius r​So, M= 0r4πr2drKr=4π2kr22⇒FG=mω2r​⇒FG=m.r4π2T2​⇒T2=4π2mrFG=4π2mrGmM/r2=4π2r3G4π2kr22⇒T2  ∝  r
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The mass density of a spherical galaxy varies as Kr 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