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^(2)m^(2)G)`
B. `K/(2pir^(2)m^(2)G)`
C. `K/(pi^(2)r^(2)G)`
D. `K/(6pir^(2)m^(2)G)`

Answer

B

Explanation

`(GMm)/(r^(2))=(mv^(2))/r`
`=2/r (1/2 mv^(2))`
`implies (GMm)/(r^(2))=(2K)/r`
`implies M=(2Kr)/(Gm)`
`implies dM=(2K)/(Gm)dr`
`implies 4pir^(2)d r rho =(2K)/(Gm) dr`
`:. rho=K/(2piGmr^(2))`
Alternative. `(GM(r))/(r^(2))=(V^(2))/r` where M=total mass upto radius (r)
`implies K=(GMm)/(2r)`
`M(r)=(2Kr)/(Gm)`
`implies dm(r) (2K)/(Gm) dr =rho dVp=rho 4pi^(2) d r`
`implies rho =K/(G 2pir^(2)m)`
`implies (rho)/m=K/(2pi Gm^(2)r^(2))`