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A smooth uniform circular hoop of mass M and radius $R$ swings in a vertical plane about a point $O$ at which it freely hangs to a fixed  support.  A bead B of mass $M$ slides without friction on the hoop. Let $\theta$ and $\phi$ be the angles as shown in the Figure below.

 

  1. Set up the Lagrangian and find the equations of motion.
  2. Find the characteristic frequencies and the normal modes of smalloscillations about the position of stable equilibrium.

 

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4727:Diamond Point

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