$3\mathrm{.}42$ Figure P$3\mathrm{.}42$ illustrates a pendulum with a base that moves horizontally. This is a simple model of an overhead crane carrying a suspended load with cables. The load mass is $m,$ the cable length is $L,$ and the base acceleration is $a(t).$ Assuming that the cable acts like a rigid rod, derive the equation of motion in terms of $$ with $a(t)$ as the input.

Figure P$3\mathrm{.}42$