$8.$ Determine the E$.$O$.$M$.$ for the center of the disk, $\backslash ($ x $\backslash ),$ of the system shown below. This system consists of a disk of mass M and radius R that rolls without slip on a rail. It also carries a mass $\backslash ($ m $\backslash )$ by a way of massless arm of length $\backslash ($ l $\backslash ),$ which is rigidly attached $($welded$)$ to the center of the disk.

$-$ First derive the nonlinear EOM

$-$ Then linearize that assuming small angles of rotation.