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]

Engineering

Physics

Gravitational Force and Field

Question

$\frac{K}{2 {πr}^{2} m^{2} G}$

$\frac{K}{{πr}^{2} m^{2} G}$

$\frac{3 K}{{πr}^{2} m^{2} G}$

$\frac{K}{6 {πr}^{2} m^{2} G}$

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Let total mass included in a sphere of radius r be M.

For a particle of mass m,

$\frac{GMm}{r^{2}} = \frac{{mv}^{2}}{r}$

$\Rightarrow \frac{GMm}{r} = 2 K \Rightarrow M = \frac{2 Kr}{Gm}$

$∴ dM = \frac{2 Kdr}{Gm}$

$\Rightarrow (4 {πr}^{2} dr) ρ = \frac{2 Kdr}{Gm}$

$\Rightarrow ρ = \frac{K}{2 {πr}^{2} Gm}$

$∴ n = \frac{ρ}{m} = \frac{K}{2 {πr}^{2} m^{2} G}$