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Small oscillations of constrained particle

  1. Dec 10, 2008 #1
    1. The problem statement, all variables and given/known data
    Consider a particle of mass m constrained to move on the surface of a paraboloid whose equation (in cylindrical coordinates) is [tex]r^2=4az[/tex]. If the particle is subject to a gravitational force, show that the frequency of small oscillations about a cirrcular orbit with radius [tex]\rho=\sqrt{4az_0}[/tex] is

    [tex]\omega=\sqrt{\frac{2g}{a+z_0}}[/tex]


    2. Relevant equations



    3. The attempt at a solution

    I'm aware that the [tex]\omega[/tex] must come from the equation:

    [tex]\ddot{x}+\omega x=0[/tex]

    This DiffEq comes from [tex]F=m\ddot{x}[/tex], so we need [tex]F=-\frac{2gm}{a+z_0}[/tex].

    The only definition of g goes to [tex]\frac{MG}{R^2}[/tex], so we must be looking for a force [tex]F=-\frac{mMG}{R^2}*\frac{2}{a+z_0}[/tex]

    My attempt is based on looking at the radial component of a gravitational force pulling out of the plane along [tex]\hat{s}[/tex], however, every time, I get:

    [tex]\vec{F}=-\frac{mMG}{S^2}\hat{s}[/tex], where the radial component would be [tex]\vec{F}\bullet\hat{r}[/tex]


    In a nutshell...my answer keeps turning up wrong.
     
  2. jcsd
  3. Dec 10, 2008 #2

    mukundpa

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    Homework Helper

    The particle is rotating about z axis and in equilibrium at z = z0. First consider equilibrium of particle in the circular orbit and then think of its oscillation on the surface.
     
  4. Dec 11, 2008 #3
    That's exactly what I'm doing, but I can't get the correct answer.
     
  5. Jan 11, 2009 #4
    I'm gonna make a late bump on this thread because I still haven't solved this problem.
     
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