Consider a spherical gaseous cloud of mass density `rho(r)` in a free space where r is the radial distance from its centre. The gaseous cloud is made of particle 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 `rho(r)` is constant with time. the particle number density n(r)=`rho(r)` /m is : (g =universal gravitational constant)

Options

A. `(3K)/(pir^2m^2G)`
B. `K/(2pir^2m^2G)`
C. `K/(6pir^2m^2G)`
D. `K/(pir^2m^2G)`

Answer

B

Explanation

A particle at distance r will experience gravitational force by the sphere of radius r (or mass say M)
`therefore (GM)/r^2 m =(mv^2)/r rArr M=v^2/(Gm) r =("2K"/m)r/"Gm"`
`rArr dM=(2K)/(Gm^2) dr rArr rho4pir^2dr =(2K)/(Gm^2) dr therefore rho =K/(2pir^2m^2G)`