If it's worth saying, but not worth its own post, then it goes here.
Notes for future OT posters:
1. Please add the 'open_thread' tag.
2. Check if there is an active Open Thread before posting a new one. (Immediately before; refresh the list-of-threads page before posting.)
3. Open Threads should start on Monday, and end on Sunday.
4. Unflag the two options "Notify me of new top level comments on this article" and "
Cool insight. We'll just pretend constant density of 3M/4r^3.
This kind of integral shows up all the time in E and M, so I'll give it a shot to keep in practice.
You simplify it by using the law of cosines, to turn the vector subtraction 1/|r-r'|^2 into 1/(|r|^2+|r'|^2+2|r||r'|cos(θ)). And this looks like you still have to worry about integrating two things, but actually you can just call r' due north during the integral over r without loss of generality.
So now we need to integrate 1/(r^2+|r'|^2+2r|r'|cos(θ)) r^2 sin(θ) dr dφ dθ. First take your free 2π from φ. Cosine is the derivative of sine, so substitution makes it obvious that the θ integral gives you a log of cosine. So now we integrate 2πr (ln(r^2+|r'|^2+2r|r'|) - ln(r^2+|r'|^2-2r|r'|)) / 2|r'| dr from 0 to R. Which mathematica says is some nasty inverse-tangent-containing thing.
Okay, maybe I don't actually want to do this integral that much :P
EDIT: On second thoughts most of the following is bullshit. In particular, the answer clearly can't depend logarithmically on R.
I had a long train journey today so I did the integral! And it's more interesting than I expected because it diverges! I got the answer (GM^2/R^2)(9/4)(log(2)-43/12-log(0)). Of course I might have made a numerical mistake somewhere, in particular the number 43/12 looks a bit strange. But the interesting bit is the log(0). The divergence arises because we've modelled matter as a continuum, with parts of it getting arbitrarily close... (read more)