Finding the total gravitational potential energy of a gas cloud

[TheTourist]

An interstellar gas cloud can be roughly described as spherical with a uniform density. Its radius is R and its total mass M.
By considering the gravitational potential energy of a thin spherical shell, show that the total potential energy of the cloud is given by:

E_grav=-$$\frac{3}{5}$$*$$\frac{GM^2}{R}$$​

Ok, so I believe that I need to find the gravitational force acting on this shell, which I have found to be

F=4$$\pi$$GM(r)$$\rho$$(r)$$\delta$$r​

and I must integrate this to find energy of the shell, and then integrate over the mass to find the total energy, but I am failing to get the desired result.

 

[nickjer]

Think of it more as if you had a shell with mass dM. And you brought it in from infinity to a solid sphere of mass M.

So write out the differential change in potential energy to bring a shell of mass dM from infinity to 'r'.

This is what you will want to integrate.