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Natural Period of Vibration for Mass with Torque

  1. Jun 2, 2010 #1
    1. The problem statement, all variables and given/known data

    A body of arbitrary shape has a mass m, mass center at G with a distance of D and a radius of gyration about G of K. It is hanging, and pinned at the top. If it is displaced a slight amount of angle P from it's equilibrium position and released, determine the natural period of vibration.

    2. Relevant equations



    3. The attempt at a solution

    Sum of Torques = Ia. Hence, I = K2m

    The torque applied by gravity: -mgDSinP = K2ma

    Hence, mgDSinP + K2ma = 0
    By the small angle approximation..

    mgDP + K2mP'' = 0

    Is this correct so far? I am lost at what to do next,

    Cheers,
    Adrian
     
    Last edited: Jun 2, 2010
  2. jcsd
  3. Jun 2, 2010 #2
    I just have to ask, you have not defined K so I'm not sure your derivation is correct.

    Assuming it is correct, you just got your answer. HINT: Compare your final differential equation with the harmonic oscillator differential equation.

    i.e.Compare
    [itex]
    \ddot{x} + \omega^{2}x = 0
    [/itex]

    with yours:

    [itex]
    \ddot{P} + \frac{K^{2}}{g D} P = 0
    [/itex]
    the rest is just identification.

    Tell me if this was helpful, good luck.
     
    Last edited: Jun 2, 2010
  4. Jun 3, 2010 #3
    Hello,

    Thanks for that! I am a little confused about how the K2 coefficient moved from the angular acceleration to the angle, though? ^^
     
  5. Jun 3, 2010 #4
    Oh my bad, just a typo. Here's the good one:

    [itex]
    P + \frac{K^{2}}{g D} \ddot{P} = 0
    [/itex]

    Just got the P miced up with the P'' for a second.
     
  6. Jun 3, 2010 #5
    Cheers, thanks for that, I got it! :)

    Thanks very much for that, I appreciate your help
    Adrian
     
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