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! :-)

 

[GSXR750]

To find the total time of collapse, you can use the equation for gravitational potential energy: U = -G(Mm)/r, where G is the gravitational constant, M is the mass of the ball of dust, m is the mass of the collapsing particle, and r is the distance between the two masses. As the particle collapses, its distance from the center of the ball decreases, causing the potential energy to increase. This increase in potential energy is converted into kinetic energy, causing the particle to gain speed. You can use conservation of energy to set the initial potential energy (when the particle is at rest) equal to the final kinetic energy (when the particle reaches the center of the ball) to solve for the total time of collapse.

To prove that the ball of dust remains homogeneous during the collapse, you can use the fact that the gravitational force is directly proportional to the mass of the object. As the particle collapses, its mass increases, causing the force of gravity to increase as well. However, this increase in force will be evenly distributed throughout the entire ball of dust, ensuring that the density remains constant. Therefore, the ball of dust remains homogeneous throughout the collapse.

Alternatively, you can use Gauss's law to prove homogeneity. As the particle collapses, the gravitational field inside the ball of dust will also increase. However, this increase in field strength will be evenly distributed throughout the ball, maintaining homogeneity. You can set up a Gaussian surface inside the ball and use Gauss's law to show that the gravitational field is constant at any point inside the ball, thus proving homogeneity.

I hope this helps. Good luck with your calculations!