A solid uniform disk with radius $\backslash ($ R $\backslash ),$ mass $\backslash ($ M $\backslash ),$ and centroidal moment of inertia $\backslash ($ M R$^\{2\}/2\backslash )($see Appendix C$)$ rolls along the ground without slipping $($Figure P$4\mathrm{.}8).$ Attached to the disk is a point of mass $\backslash ($ m $\backslash )$ distance $\backslash ($ r $\backslash )$ from the center. Because of the no$-$slip assumption, the angle $\backslash (\backslash$theta $\backslash )$ can function as the single generalized coordinate needed to describe the motion. The gravity force on $\backslash ($ m $\backslash )$ will cause the disk to roll with an unsteady angular velocity or to oscillate back and forth.

$($a$)$ Find an expression for the kinetic energy of the system. The diagram on the left shows how the vector velocity $\backslash (\backslash$mathbf$\{$v$\}\_\{$m$\}\backslash )$ for $\backslash ($ m $\backslash )$ is composed from the velocity of the disk center and the velocity with respect to the center due to $\backslash (\backslash$theta $\backslash ).$

Figure P$4\mathrm{.}8$

$($b$)$ Find the potential energy for the system.

$($c$)$ Find the equation of motion using Lagrange's equation.