Consider a vertical circular hoop of radius \(R\) rotating about its vertical symmetry axis with constant angular velocity \(\Omega\). A bead of mass \(m\) is threaded onto the hoop, so is free to move along the hoop. Let the angle \(\theta\) of the bead be measured up from the bottom of the hoop.

(a) Write the Lagrangian in terms of the generalized coordinate \(\theta\). Are there any firstintegrals of motion?

(b) Show that there are two equilibrium angles of the bead for sufficiently small angular velocities \(\Omega\), but that there are four equilibrium angles if \(\Omega\) is sufficiently large.

(c) For each of these equilibrium angles, find out whether that position of the bead is stable or unstable. That is, if the bead is displaced slightly from equilibrium, does it tend to move back toward the equilibrium angle, or does it depart farther and farther from it?