$3\mathrm{.}41$ Work this problem as specified in the problem statement in the textbook, and as specified below.

For problem $3\mathrm{.}41,$ in addition to the requirements for this problem stated in the textbook, also do following:

i$)$ Find the equilibrium positions of the pendulum.

ii$)$ Linearize the equation of motion about $=0,$ using the Taylor Series. You must show all steps in this linearization process, including the Taylor Series.

iii$)$ If $a(t)$ is constant, and $a>0,$ use the linearized equation of motion to determine the stability of the unforced system, about $=0,$

You must determine stability from the characteristic roots, and you must show the characteristic equation, and the steps used to find the characteristic roots.

$($Note: Express the moment of inertia of the pendulum in terms of the pendulum bob mass and the pendulum rod length.$).$Figure P$3\mathrm{.}41$ illustrates a pendulum with a base that moves. The base

acceleration is $a(t).$ Derive the equation of motion in terms of $$ with $a(t)$ as the

input. Neglect the mass of the rod.$|$