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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 center. 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

K2πr2m2G


Step 1: Given and assume

A spherical gaseous cloud

ρ(r) is the mass density in free space.

r is the radial distance from its center.

K is the kinetic energy.

m is the mass of the particles.

Particle number density, n(r)=ρ(r)m

Step 2: Calculation

We know that,

Gravitational force, GMmr2

Centripetal force, mv2r

Now,

Gravitationalforce=CentripetalforceGMmr2=mv2rGMmr2=2×12×mv2rGMmr2=2krM=2KrGMdM=2KGmdr...(1)

Now,

ρ=mvm=ρ×vdm=ρ.4πr2.drρ.4πr2.dr=2KGmdrρ=2KGmdr4πr2.dr=K2πGmr2.drρ(r)m=K2πGmr2.dr

Therefore, option B is the correct answer.


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