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Particle oscillating around equilibrium radius

  1. Jul 9, 2012 #1
    1. The problem statement, all variables and given/known data
    A particle of mass m moving in three dimensions is attracted to the origin by the gravitational force of a much heavier object. It can be shown that the radial motion is governed by the following equation
    [itex]m\ddot{r}=-\frac{k}{r^{2}}+\frac{l^{2}}{mr^{3}}[/itex]

    where k is a constant and l is the angular momentum. Determine an equilibrium radius [itex]r_{0}[/itex] in terms of k, l, and m. If the particle is put near that equilibrium radius, [itex]r=r_{0}+\epsilon[/itex](where [itex]\epsilon << r_{0}[/itex]), it will have an oscillatory radial motion about [itex]r_{0}[/itex]. What will be the frequency of that oscillation?

    3. The attempt at a solution
    Attached to thread as I'm horribly slow at typing latex.
     

    Attached Files:

  2. jcsd
  3. Jul 9, 2012 #2

    gabbagabbahey

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    Your final answer looks correct, but I'm not quite sure what it is you've done to get it. Specifically, why do you assert that [itex]\omega=\sqrt{\frac{1}{m}\frac{d^2U_{eff}}{dr^2}}[/itex]? Is the RHS of this equation even a constant?

    The method I would suggest is to just plug [itex]r=r_0+\epsilon[/itex] into your equation of motion and Taylor expand the RHS of it in powers of [itex]\frac{\epsilon}{r_0}[/itex] (since you know that it is much smaller than one).
     
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