Consider the system with transfer function

$G(s)=\frac{1}{s(s+5)}$

shown below. Assume $G_c(s)$ is a proportional controller $($i$.$e$.,G_c(s)=K).$

$($a$)$ For what range of values of K is the closed$-$loop system stable?

$($b$)$ For what value of K does the closed$-$loop system have a natural frequency of vibration ${}_n=10ra\frac{d}{s}?$

$($c$)$ In the case of $($b$),$ what is the damping factor, $,$ and what does this say about the damping of the system?

$($d$)$ Now assume $G_c(s)$ is a proportional$-$derivative controller $($i$.$e$.,G_c(s)=K_1+K_{2}s).$ Chose values of $K_1$ and $K_2$ to achieve ${}_n=4ra\frac{d}{s}$ and $=0\mathrm{.}6.$