Solving Mass Collapse Problem: Total Time & Homogeneity

[rsd_sosu]

Consider a ball of dust mass M, Radius R stats from rest and collapses due to gravity.

Q. Find total time of collapse.
This I am currently working on I believe I can show this using conservation of momentum.


Q. Prove that it remains homogeneous
Dose anyone have any ideas on this one, I am sure I have done this before. I have tried proving it using gauss's law by setting up gauss sphere's inside and out but I am missing something.

Thanks for any help
Robert

 

[Tide]

If the ball starts off with uniform density you should find that the gravitational force at any point within the sphere is directly proportional to the distance from the center. You can set up the equation of motion for test particles at any distance from the center and the equation will essentially look like a harmonic oscillator!

You should be able to see from the solution that the position of anyone of those test particles remains directly proportional to its starting position throughout its fall. E.g. a particle starting near the surface always remains twice as far from the surface as a particle that started halfway between the center and the surface. Apply the argument to all test particles and you have - UNIFORMITY! :-)