Consider a spherical gaseous cloud of mass density ρ(r) in free...

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 $‘M’$
$⇒ \dfrac{GM ⋅ m}{{r}^{2}} = \dfrac{m{v}^{2}}{r}$ ....(1)
But, kinetic energy, $K = \dfrac{1}{2}m{v}^{2}$
$⇒ m{v}^{2} = 2K$
Put it in $( 1 )$
$⇒ \dfrac{GMm}{{r}^{2}} = \dfrac{2K}{r}$
$⇒ M = \dfrac{2Kr}{Gm}$
Differentiating it we get,
$dM = \dfrac{2K}{Gm} ⋅ dr$ .....(2)
Now, if density of cloud is ′{ρ}^{′},
Then elementary mass,
$dM = \left(4\pi{r}^{2}dr\right) ⋅ ρ = \dfrac{2K}{Gm} ⋅ dr$
$⇒ ρ = \dfrac{K}{2\piGm{r}^{2}}$