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Rotation condition?

  1. Dec 22, 2008 #1

    KFC

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    If I know the equation of motion of the following form

    [tex]\ddot{\theta} + k^2\sin\theta = 0[/tex]

    (for pendulum for example). What's the condition (minimum angular velocity) to keep it rotate instead of just oscillation?
     
  2. jcsd
  3. Dec 23, 2008 #2
    Writing [tex]\omega = \dot \theta[/tex], that equation becomes (using the chain rule)

    [tex]\omega \frac{d\omega}{d\theta} + k^2 \sin \theta = 0[/tex]

    and solving that differential equation gives

    [tex]\frac12 \omega^2 = k^2 \cos \theta + C.[/tex]

    Once you have that, what condition on C do you need for [tex]\theta = \pi[/tex] to be possible? (Note that each side of this equation corresponds nicely to energy.)
     
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