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Lagrangian Problem. Two masses on a massless circle

  1. Nov 16, 2011 #1
    1. The problem statement, all variables and given/known data
    Two equal masses are glued to a massless hoop of radius R that is free to rotate about its center in a vertical plane. The angle between the masses is 2*theta. Find the frequency of small oscillations.


    2. Relevant equations
    [tex]\frac{d}{dt} \frac{∂L}{∂\dot{q}}=\frac{∂L}{∂q}[/tex]

    3. The attempt at a solution

    So since the masses are glued to the hoop, the radius is constant. So I am assuming my professor is referring to the oscillations in the angle? Meaning the hoop will just turn back and fourth?

    So going by that (r=constant) My lagrangian is..
    [tex]L=\frac{1}{2}mr^2 \dot{\theta}^2+2mr^2 \dot{\theta}^2+mgr(cos\theta +cos2\theta)[/tex]
    Which can be simplified obviously. I left it this way so it is easier to see my logic in the formulation of the Lagrangian? Is this correct? For the second mass, I used 2θ.

    Now when I take the respective derivatives I get..
    [tex]\ddot{\theta}-\frac{5g}{9r^2}\theta=0[/tex]

    The problem is, this is not an oscillator! If that was a plus sign I would be good to go. Could anyone help me out in figuring out this problem? Where was my logic flawed? :\??

    Thanks!!
     
  2. jcsd
  3. Nov 16, 2011 #2
    In the way you are choosing to define your coordinate system, is g negative? If so, I think you're good to go. If not, consider your Lagrangian; there may be a plus sign where there ought to be a minus, especially before the potential energy term. It all comes down to how you chose to define your system, but as long as you were consistent with it, you should obtain oscillatory motion.
     
  4. Nov 16, 2011 #3
    *Sigh* I got it, I made an algebra error. How embarrassing! :p!
    Thanks! :D
     
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