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Question:

Consider a spherical gaseous cloud of mass density ρ(r) in a 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 their common centre 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?

Kπr²m²G

K²πr²m²G

K⁶πr²m²G

3Kπr²m²G

Solution:

Correct option is D. K²πr²m²G
GMmr²=mv²/r=2r(1/2mv²)
⇒GMmr²=2Kr
⇒M=2Kr/Gm
⇒dM=2KGmdr
⇒4πr²drρ=2KGmdr
∴ρ=K/(2πGmr²)
⇒ρ/m=K/(2πGm²r²)
Alternative:
GM(r)r²=V²/r
Where M(r)=total mass upto radius(r)
⇒K=GMm/(2r)
⇒M(r)=2Kr/Gm
⇒dM(r)=2KGmdr=ρdV=ρ4πr²dr
⇒ρ=KG/(2πr²m)
⇒ρ/m=K/(2πGm²r²)
Correct option 4.