We have a random 2D distribution of many particles (a cloud) within distance "R" from the origin (forming a disk shape of points with the origin in the middle). Total mass of the system is “M”.

QUESTION: Considering that the total mass of the system is distributed evenly throughout this volume, what is the linear orbital speed of a particle "P", with mass “m”, at distance “r” from the origin?

Givens:

Assume r < R

Assume M>>m

Requests:

Newtonian gravity only please (F=GmM/rr), though using calculus fine.

If possible, please show how you derived your answer.

Would you want to just do a numerical solution?

I agree that the configuration seems unstable.

I'd expect that interactions would tend to circularize the orbits,

but that you'd also get spreading beyond the R boundary, while other particles

migrate toward the center.