Simple derivation: sphere contracting under gravity

1. Feb 25, 2016

HowardTheDuck

Hi Guys, I'm reading Roger Penrose's book "The Road to Reality" at the moment and I wonder if you could help me out with a pretty simple derivation which he doesn't describe in complete detail.
On page 399 he considers a sphere of mass contracting under gravity, and says "The rate of volume reduction is 4πGM". Can you help me derive this, please? It shouldn't be too difficult, it's just beyond me!

OK, what do we know. The volume of a sphere is 4/3πr3. So dV/dr is 4πr2.
And the acceleration (d2r/dt2) due to Newton's law of gravity would be GM/r2. Can we combine these via the chain rule or similar to get the rate (acceleration?) of volume reduction equal to 4πGM.

Thanks a lot.

Last edited: Feb 25, 2016
2. Feb 25, 2016

HowardTheDuck

I can see that if you simply multiply the 4πr2 by the GM/r2 then you get the right answer. But I don't see how that's a valid thing to do.
I don't see how it's valid to multiply dV/dr by d2r/dt2 (by the chain rule?) to get the right answer.
Any help appreciated.