Problem $1$ Consider the pendulum shown in Fig $1$ that consists of a massless $($i$.$e$.$ with negligible moment of

inertia$)$ straight rigid rod AOB with a point mass $m$ attached at the bottom point $A.$ The pendulum rotates without

friction about the point $O$ and it is initially at vertical equilibrium. A horizontal spring $k$ is attached at the top

point $B$ and sets the pendulum to motion with the application of a horizontal displacement input $z(t).$ The spring

$k$ is unstretched at $t=0$ and the pendulum is vertical. Determine the linearized equation of motion of the

pendulum about the vertical position.