$1[20$ points$]$ Inverted Pendulum

Consider the system in Figure $1$ that depicts an inverted pendulum coupled with a mass spring mechanism. The rod has a length of $\backslash ($ l $\backslash ),$ with spring $\backslash ($ k$\_\{3\}\backslash )$ connected at the midway point and spring $\backslash ($ k$\_\{2\}\backslash )$ at the end of the rod. Assume that all displacements of the rod are small enough that the spring forces remain horizontal. Assume negligible friction for each surface.

$1$

a$.$ Construct the Free Body Diagrams $[4$ points$]$

b$.$ Find, but do not linearize, the equations of motion for $\backslash ($ m$\_\{1\}\backslash )$ and $\backslash ($ m$\_\{2\}\backslash )[7$ points$]$

c$.$ Linearize the equations of motion $[5$ points$]$

d$.$ Put the equations of motion into the linear matrix format. $[3$ points$]$