$(1)$ For the system shown in Fig. $1,$ find the following

$($a$)$ EOM with respect to $$

$($b$)$ The natural frequency of the system

$($c$)$ The response of the system if $=5$rad and ${}^{}=0$

Let $J_0=0\mathrm{.}5kg-m^2,K=1200\frac{N}{m},m=10kg,r=15cm$

Figure $1$: pulley and mass system