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[QU/CM-08002]

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Consider a hoop of radius $a$ in a vertical plane  with angular velocity $\omega$ about the vertical diameter. Consider a bead of mass $m$ which slides without friction on the hoop as indicated in figure.
Under what condition will the equilibrium of the bead at  $\theta=0$ be stable?

  1. Find another value of $\theta$ for which, in certain circumstances, the bead will be in stable equlibrium. Indicate the values of $\omega$ for which this stable equilibrium takes place.
  2. Explain your answer with the aid of appropriate graphs of the potential energy versus $\theta$ as measured in the rotating frame.                                                                      [Lim 1097]

Problem-CM-08002

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