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AbigailM
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Homework Statement
A large, spherically symmetric collection of point particles of mass m move in circular orbits about a common center each with the same kinetic energy. If the only force acting is the mutual gravitational attraction of the particles, find the particle density (in the continuum limit) as a function of r from the center in order that the density remain constant in time.
Homework Equations
[itex]F=\frac{GMm}{R^{2}}[/itex] ,where M is the total mass.
[itex]\frac{v^{2}}{R}=\frac{GM}{R^{2}}[/itex]
The Attempt at a Solution
[itex]\frac{F}{m}=\frac{GM}{R^{2}}[/itex]
[itex]\frac{4\pi F}{m}=\frac{4\pi GM}{R^{2}}[/itex]
[itex]dM=\rho(R)R^{2}dR4\pi[/itex]
[itex]dM=\frac{v^{2}}{G}dR[/itex]
[itex]\rho(R)=\frac{v^{2}}{4\pi GR^{2}}[/itex]
[itex]\frac{1}{2}mv^{2}=\frac{GMm}{r}[/itex] solve for v and substitute into prev. equation.
[itex]\rho(R)=\frac{GM}{2\pi R^{3}}[/itex]
Is this looking ok? Thanks for the help