One end of a wire is tied to a point A on the ceiling and the other end is tied to a point on a ring of radius \(R\) and negligible mass. The ring therefore hangs from the wire in a vertical plane and in a gravitational field \(g\). A bead of mass \(m\) is threaded onto the ring so it can slide around the ring without friction. The lowest point on the ring is then tied to a second wire whose opposite end is attached to point B on the floor, where point \(B\) is directly beneath point \(A\). The wires are then drawn taut. If the ring and attached wires are made to twist sideways through an angle \(\varphi\) away from equilibrium, a potential energy \((1 / 2) \kappa \varphi^{2}\) is set up in the wire.

(a) Using angles \(\theta\) and \(\varphi\) as generalized coordinates, where \(\theta\) is the angle of the bead down from the top of the ring, find the kinetic and potential energies of the bead.

(b) Find the equations of motion using Lagrange's equations. Assume that during the motion of the bead it remains entirely on one side of the ring, so it does not meet the wires at \(\theta=0\) and \(\theta=\pi\).