Lagrangian problems, constrained point masses

Problem:

A bead of mass m slides without friction on a circular loop of radius r.  The loop lies in a vertical plane and rotates about a vertical diameter with constant angular velocity ω.

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(a)  For angular velocity ω larger than some critical value ωc, the bead can undergo small oscillations about an equilibrium point θ0 ≠ 0.  Find ωc and θ0(ω).
(b)  Obtain the equations of motion for the small oscillations about θ0 as a function of ω and find the period of the oscillations.

Solution: