$4.[30$ points$]$ Consider a wheel $($not necessarily a disk$)$ of mass $\backslash ($ M $\backslash ),$ radius $\backslash ($ R $\backslash ),$ and moment of inertia $\backslash ($ I $\backslash )$ about its center, that rolls without slipping. One end of a spring is attached to the center of the wheel, and the other to a fixed wall.

$($a$)$ Using Lagrange's method, determine the equation of motion.

$($b$)$ Determine the frequency of oscillations about the equilibrium position.