The slender bar as shown in below figure $4($a$)$ has a mass of $m$ and a length of $l.$ A force of $F$ is

statically applied to the bar at $P$ then removed. The following oscillations of $P$ are monitored,

and the acceleration data is measured as shown in the figure $4(b).$

$($a$)$ Derive the equation of motion for the system and the undamped nature frequency.

$($b$)$ Use the data in figure $4($b$)$ to determine the damping ratio $$ and the damped natural

frequency.

$($c$)$ When $m=31kg,l=2\mathrm{.}6m,$ calculate the spring stiffness $k$ and the damping coefficient $c.$

$($d$)$ If $F=50N,$ obtain the value of $a$ in the figure $4($b$).$