Potential Energy to bring in sphere from infinity

[Sentin3l]

Homework Statement

Given a uniform sphere of mass M and radius R. Use integral calculus and start with a mass dm in the sphere. Calculate the work done to bring in the remainder of the mass from infinity. By this technique show that the self-potential energy of the mass is:

$P = -\frac{3}{5} \frac{GM^{2}}{R}$​

Homework Equations

$W = \int\vec{F} \bullet d\vec{r}$
$F = \frac{GMm}{r^{2}}$​

The Attempt at a Solution

First let me say that this is a cosmology question. I began by considering a differential mass near or at the center of the sphere. Using the above equations for force and work, I derived:

$W = - \int \frac{GM(dm)}{r^{2}} \hat{r}$​

Since the sphere is uniform, it has a constant mass to radius ratio $λ = \frac{M}{R} = \frac{dm}{dr}$. So using this I found:

$W = -λ\int \frac{GM}{r^{2}}dr = -3λ \frac{GM}{r^{3}}$​

If we substitute $λ = \frac {M}{R}$ and $r=R$, we get the result:

$W = -3 \frac{GM^{2}}{R^{3}}$​

Here is where i think I went wrong, I don't know if I need to deal with $\hat{r}$ and if so, I'm not sure how to approach that.

I think that once I get the work, you use the work-energy theorem, and intial/final KE is 0 so the potential energy equals the work, please correct me if I'm wrong in that.

 

[TSny]

Since the sphere is uniform, it has a constant mass to radius ratio $λ = \frac{M}{R} = \frac{dm}{dr}$.

Uniform mass distribution means that the mass to volume ratio is constant.

 

[vela]

The problem is worded kind of strangely. Consider when the mass that's already been brought in occupies a sphere of radius $r$. How much work is required to bring in the mass to increase the radius by dr?

 

[Sentin3l]

Uniform mass distribution means that the mass to volume ratio is constant.

Should I use my previous integral, taking note that $dV = 4 \pi r^{2} dr$ ?

The problem is worded kind of strangely. Consider when the mass that's already been brought in occupies a sphere of radius $r$. How much work is required to bring in the mass to increase the radius by dr?

I agree that the problem is worded strangely, it's a recurring problem with this textbook. Also I'm not sure what your mean in your suggestion, did you mean "How much work is required to bring in the mass to decrease the radius by dr"?

 

[vela]

No. Why would you want to decrease the amount of mass there when you're building it up?

 

[Sentin3l]

I have solved the problem, thanks to you both for the assistance.