A kid, modelled as a point mass with $m=10kg,$ is sitting at end B of a rigid rod AB of negligible mass. The rod is supported by a spring at end A and a pin at point $O.$ The system is in static equilibrium when the rod is horizontal. Someone pushes the kid vertically downwards by a small distance y and lets go$.$ Given that $AB=3m$;$AC=0\mathrm{.}5m$;$k=1k\frac{N}{m}$; find

a$)$ the unstretched $($relaxed$)$ length of the spring,

b$)$ the equation of motion $($a differential equation relating the position of the mass to its acceleration$)$ of the system, and

c$)$ the natural frequency of the system.

d$)$ If the rod is pinned at the midpoint instead of at $O,$ what is the natural frequency of the system?