A bead of mass \(m\) is placed on a vertically-oriented circular hoop of radius \(R\) that is forced to rotate with constant angular velocity \(\omega\) about a vertical axis through its center.

(a) Using the polar angle \(\theta\) measured up from the bottom as the single generalized coordinate, find the kinetic and potential energies of the bead. (Remember that the bead has motion due to the forced rotation of the hoop as well as motion due to changing \(\theta\).)

(b) Find the bead's equation of motion using Lagrange's equation.

(c) Is its energy conserved? Why or why not?

(d) Find its Hamiltonian. Is \(H\) conserved? Why or why not? Is \(E=H\) ? Why or why not?

(e) Find the equilibrium angle \(\theta_{0}\) for the bead as a function of the hoop's angular velocity \(\omega\). Sketch a graph of \(\theta_{0}\) versus \(\omega\). Notice that there is a "phase transition" at a certain critical velocity \(\omega_{\text {crit }}\).

(f) Find the frequency of small oscillations of the bead about the equilibrium angle \(\theta_{0}\), as a function of \(\omega\).