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Coupled Oscillator

  1. Feb 17, 2009 #1
    1. The problem statement, all variables and given/known data
    A thin hoop of radius R and mass M oscillates in its own plane with one point of the hoop fixed. Attached to the hoop is a small mass M constrained to move (in a frictionless manner) along the hoop. Consider only small oscillations, and show that the eigenfrequencies are blah blah blah (two eigenfrequencies).



    2. Relevant equations



    3. The attempt at a solution

    My difficulties are in setting up this problem. I believe that I am picturing the system correctly, but I can't quite figure out how to do it. I need to find the Lagrangian of the system first, but I am having a hard time with the kinetic energy part.


    The oscillation of the hoop if I am not mistaken will be like that of a pendulum.

    Hoop: T = 1/2*Iw^2 = 1/2 m * R^2 * [tex]\omega ^2[/tex]
    Small Mass: T = 1/2 mR^2 [tex]\theta '[/tex]

    I have set up theta as the angle between the center of the hoop and the position of the small mass. The problem is that I don't quite know how to get the second generalized coordinate -- I am assuming there are two generalized coordinates because this is a coupled oscillator problem and I am given two eigenfrequencies.

    I am tempted to say that [tex]\omega = R \theta ' [/tex] but that doesn't yield a correct answer and I don't think it's right to begin with...

    Any help?
     
  2. jcsd
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