$(20$ pts$)$ Consider the system shown in Fig. $2$ below. The disk of mass m has radius r and

rolls on a $($frictionless$)$ inclined plane. Coordinate xx$=0$ g is acting in the downward vertical direction. Note: the moment of inertia of the

disk J$=(1)/(2)$mr$^(2).$

For simplicity, assume that the "ground reference" for the gravitational potential energy of

the mass is at the height of the dashed line, i$.$e$.,$ the gravitational potential energy of the

mass is zero at this vertical location.

$($a$)(12$ pts$)$ Determine an expression for the Lagrangian L$=$KE$-$PE$.$

$($b$)(5$ pts$)$ Use the Euler$-$Lagrange equation

$($d$)/($dt$)(($delL$)/($delq$\_($i$)^()))-($delL$)/($delq$\_($i$))+($delD$)/($delq$\_($i$)^())=0$

where q$\_($i$)$ represents the i$^()$ th generalized coordinate to determine the differential equations

describing the system dynamics.

$($c$)(3$ pts$)$ Express the equations obtained in $($b$)$ in state$-$space form.