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 gra

Step-by-step solution

Given that all the particles of gaseous cloud has mass ‘ m ’ and are moving due to mutual attraction with same kinetic energy ‘ K ’ . Now in that cloud, let us consider a sphere of radius ‘ r ’ containing a group of particles adding to a total mass ‘ M ’ . Now, another particle of mass ‘ m ’ is moving with speed ‘ V ’ due to gravitational attraction of M , in a circle of radius ‘ r ’ . ∴ Centripetal force is here given by ‘ M ’ ⇒ G M ⋅ m r 2 = m v 2 r ....(1) But, kinetic energy, K = 1 2 m v 2 ⇒ m v 2 = 2 K Put it