A spherical gaseous cloud has mass density ρ(r). It consists of particles of equal mass m moving in circular orbits about the common centre, all with the same kinetic energy K. If ρ(r) is time-independent, find the particle number density n(r) = ρ(r)/m. G is the gravitational constant.
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Question

A spherical gaseous cloud has mass density ρ(r). It consists of particles of equal mass m moving in circular orbits about the common centre, all with the same kinetic energy K. If ρ(r) is time-independent, find the particle number density n(r) = ρ(r)/m. G is the gravitational constant.

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G1 Physics · Accepted Answer

All particles have the same kinetic energy, so v² = 2K/m is constant. For circular motion at radius r, GM(r)/r² = v²/r, hence M(r) = v²r/G = 2Kr/(mG). Differentiate: dM/dr = 2K/(mG). But dM/dr = 4πr²ρ = 4πr²mn(r). Therefore n(r) = K/(2πr²m²G). Correct answer: D.