Gravitational Energy Released from Sphere

[CedarPark]

Homework Statement

"How much energy is released when a sphere of constant density (p) with mass (M) and radius (R) is put together gravitationally? What you should do is to think of the energy released when a shell is brought in from infinite distance (where potential energy of zero) to the current surface of radius r of the sphere. What is the gravitational force on a sphere as it moves inwards, what is the differential mass of the infinitesimally thin shell, what distance is it brought to, and hence what is the differential energy for bringing in the shell? Then, integrate the differential energy over radius to get total energy,

Homework Equations

The Attempt at a Solution

I am pretty lost on the solution.

I have: F_g=GMm/r^2

Integrated from infinity to r (GMm/r^2)
This, however, gives potential, energy, which is 0.Any help would be appreciated, I'm very lost on where to even begin on the problem.

Thanks!

 

[PeroK]

Do you understand the question?

 

[haruspex]

This, however, gives potential, energy, which is 0.

Gravitational potential energy, by convention, is zero at infinity. On moving from infinity to radius r, it must have lost some and now be negative. Please post you working.

 

[CedarPark]

Do you understand the question?

Not really;
I understand there is a relationship between work and the volume of an object. I'm struggling to visualize the problem though.

 

[haruspex]

Not really;
I understand there is a relationship between work and the volume of an object. I'm struggling to visualize the problem though.

The scenario is that a sphere starts from nothing and gradually grows by accretion of matter that is falling in from very far away. To make life simpler, we consider one thin uniform shell falling in at a time. So suppose at some stage it has reached radius r. Assume some density for the accreted material, and that this does not change. Find the mass of the sphere so far, and hence the gravitational potential at its surface.
Now let a shell of radius r and thickness dr accrete. What is the mass of this shell? What GPE did it lose in falling in from infinity?