A thin uniform disk of mass $\backslash ($ m $\backslash )$ and radius $\backslash ($ r $\backslash )$ rolls without slipping inside the circular surface of a moving cart of mass $\backslash ($ M $\backslash ).$ The semi$-$circle in the cart has a radius of $\backslash ($ R $\backslash ).$ The disk is restrained by a spring of stiffness $\backslash ($ k $\backslash )$ that is stretched from point $\backslash ($ A $\backslash )$ to the disk center at $\backslash ($ C $\backslash ).$ The spring is unstretched when the disk is at its highest spot on the cart, $\backslash (\backslash$theta$=\backslash )0.$ A second spring of stiffness $\backslash ($ k $\backslash )$ connects the cart to a stationary wall; the spring is unstretched when $\backslash ($ x$=0\backslash ).$ The cart is excited by a horizontal force $\backslash ($ F $\backslash )$ as shown. Gravity acts in the vertical direction and cannot be ignored. We wish to model this system using generalized coordinates $\backslash ($ x $\backslash )$ and $\backslash (\backslash$theta $\backslash ).$

$($a$)$ Find the kinetic energy of the system

$($b$)$ Find the potential energy of the system