JEE Physics Practice Question

Consider a spherical gaseous cloud of mass density $\rho(r)$ 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 $\rho(r)$ is constant in time, the particle number density $n(r)=\rho(r) / m$ is [ $G$ is universal gravitational constant]

1. A.$\frac{K}{2 \pi r^{2} m^{2} G}$
2. B.$\frac{K}{\pi r^{2} m^{2} G}$
3. C.$\frac{3 K}{\pi r^{2} m^{2} G}$
4. D.$\frac{K}{6 \pi r^{2} m^{2} G}$