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Angular Oscillation of a rod in a circle

  1. Apr 19, 2013 #1
    1. The problem statement, all variables and given/known data

    A uniform rod moves in a vertical circle .Its ends are constrained to move on the track without friction.Find the angular frequency of small oscillation .


    2. Relevant equations



    3. The attempt at a solution

    Suppose the rod of length L moves in a circle of radius R .
    Let the equilibrium position of the rod be AB .X be the mid point .CD is the position of the rod when it displaced by an angle θ .Y is the mid point.

    The mechanical energy of the rod in position CD is denoted by E .

    The moment of inertia of the rod about its CM (the middle point) is Icm
    The moment of inertia of the rod about O is I .

    [itex]I_{cm} = ML^2/12 [/itex]

    [itex]I=I_{cm} + Md^2[/itex]

    [itex]I=M(R^2-\frac{L^2}{6})[/itex]

    [itex]E= mgd(1-cos\theta)+(1/2)I\dot\theta^2[/itex]

    Differentiating E w.r.t time ,we get

    [itex]dE/dt = mgdsin\theta\dot\theta+(1/2)I(2\dot\theta\ddot\theta) [/itex]

    Since Mechanical energy remains conserved ,

    Putting dE/dt=0 ,we get

    [itex]\ddot\theta = -\frac{mgdsin\theta}{I}[/itex]

    Using small angle approximation , sinθ≈θ

    [itex]\ddot\theta = -\frac{mgd\theta}{I}[/itex]

    Is my approach correct ?
     

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    Last edited: Apr 19, 2013
  2. jcsd
  3. Apr 19, 2013 #2

    haruspex

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    Looks right to me. (It's clearly the same as making a pendulum out of the rod by attaching a light bar length d rigidly, at right angles, to its centre.)
     
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