Q$3.$ Below figure shows an inverted pendulum supported by a linear spring of stiffness

$k.$ If $$ is the angle between the vertical through the hinge connection at $O$ and the

pendulum, determine

a$)$ The equilibrium position in terms of ${}_0$

b$)$ The differential equation of motion for small angular motion ${}_1$ about ${}_0$

c$)$ The stability condition such that the motion ${}_1$ is harmonic.

d$)$ The natural frequency of the oscillation ${}_1$