(Solved): 3) [30 pts] Given the pulley system on the left. A spring is inserted between point A (the center o ...

3) [30 pts] Given the pulley system on the left. A spring is inserted between point A (the center of the lower pulley) and the end $B$ of the cord. The cord is inextensible. The parameters are $m=4\mathrm{kg},k=4000\mathrm{N}/\mathrm{m}$. a) Derive the equation of motion of the mass $m$ in free vibration using any method you prefer. b) We now wish to connect a damper in parallel with the spring to make the system underdamped so that the magnitude of the first peak in free vibration is reduced by $60\%$ in the second peak. What should the damping constant of the damper be?