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Question

# Consider a spherical gaseous cloud of mass density ρ(r) in free space, where r is the radial distance from its centre. The gaseous cloud is made of particles of equal mass m moving in circular orbits about the common center with the same kinetic energy K. The force acting on the particles is their mutual gravitational force. If ρ(r) is constant in time, the particle number density n(r)=ρ(r)/m is [G is universal gravitational constant]

A
3Kπr2m2G
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B
K2πr2m2G
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C
K6πr2m2G
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D
Kπr2m2G
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Solution

## The correct option is B K2πr2m2GFor a particle rotating in the circular orbit of radius r due to the gravitational attraction of inner cloud of mass M, GMmr2=mv2r M=v2rG=2mv2r2Gm As K=12mv2= constant, then M=2KrGm or dM=2KdrGm Correspondingly dM=ρ(r)×4πr2dr ρ(r)⋅4πr2dr=2KdrGm ρ(r)m=K2πGm2r2

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