Assuming small swing angles $($i$.$e$.,sin()$~~$,cos()$~~$1)$ for the pendulum with a mass of $m,$ and ignoring friction and drag,

a$.$ Derive the equations of motion $($EOM$)$ for the $2-$DOF double pendulum system shown below $($hint: linearize the EOM by ignoring non$-$linear terms, e$.$g$.,*{}^{}{?}^2,2$ pts$.).$

b$.$ Based on your EOM of the previous part, indicate whether the system is dynamically and$/$or statically coupled $(1$ pts$.).$

c$.$ Determine the natural frequencies and associated mode shapes if $k=m\frac{g}{L}(2$ pts$.).$