Solve the preceding problem if instead of being forced to oscillate in the horizontal direction, the upper end of the rod is forced to oscillate in the vertical direction with \(y=A \cos \omega t\).

Data from preceding problem

A plane pendulum consists of a light rod of length R supporting a plumb bob of mass \(m\) in a uniform gravitational field \(g\). The point of support of the top end of the rod is forced to oscillate back and forth in the horizontal direction with \(x=A \cos \omega t\). Using the angle \(\theta\) of the bob from the vertical as the generalized coordinate,

(a) find the Lagrangian of the plumb bob.

(b) Are there any conserved dynamical quantities?

(c) Find the simplest differential equation of motion of the bob.