Q$5.$ A small $40-kg$ block $A$ is attached to the rod $BC$ of negligible mass which is supported at $B$ by a pin and bracket and at $C$ by a spring of constant $k=4k\frac{N}{m}.$ The system can move in a vertical plane and is in equilibrium when the rod is horizontal. The rod is acted upon at $C$ by a periodic force of magnitude $P=P_{m}sin{}_{f}t,$ where $P_m=10N$ and ${}_f=16ra\frac{d}{s}.$

a$)$ What are the period values $($in seconds$)$ of the given system when when $b=200mm,400mm$ and $600$ mm respectively for free vibration $($i$.$e$.$ in the absence of P $)?$

b$)$ In the presence of $P($i$.$e$.$ forced vibration$)$ and $b=400mm,$ obtain the differential equation of motion.

c$)$ Solve the differential equation of motion in part b$.$

d$)$ Plot $y_c($i$.$e$.$ vertical deflection at C $)$ versus time graph by using the results of part C and find the maximum vertical tip deflection.