Mass within a gravitating shell

Mass within a gravitating shell - Mass within a gravitating...

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Mass within a gravitating shell Now let us consider the potential for a particle inside such a shell. Figure %: Particle m inside thin shell. The only change in the mathematics is now that l extends from R - r to R + r and hence: U = dl = (2 r ) = Thus the potential inside the sphere is independent of position--that is it is constant in r . Since F Equations of motion We can write expressions for both the angular momentum and the total energy. If p θ is the magnitude of the momentum in the tangential direction, then since this perpendicular to , L = rp θ . But p θ = mv θ = m = mr = mr . Hence L = r ( mr ) = mr 2 . Hence: L = mr 2 We can also write an expression for the total energy as a sum of the radial kinetic energy term, the angular kinetic energy term and the potential term: E = 1/2 m + - Rearranging and dividing through the left side by m 2 r 4 and the right by L 2 , and canceling factors of dt 2 we find: = - - To find the equations of motion we want to find r in terms of θ . In principle we could take the square root of both sides of the above equation, separate the variables, integrate to find θ ( r ) , and

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