Homework H$\mathrm{.}6.$C

Given: Two homogeneous cylinders, each of mass $m$ and having outer radii of $3R$ and $R,$ are able to roll without slipping on a rough inclined surface. Bar B $,$ having a mass of $m,$ is connected on one end to the center of one disk and is supported by the top surface of disk A $,$ as shown in the figure. Bar B does not slip on disk A as the system moves. A spring of stiffness $k$ is attached between the center O of the upper disk and ground. Let $x$ represent the motion of the disk center O along the incline, where $x=0$ when the spring is unstretched.

Find: For this problem:

$($a$)$ Derive the dynamical equation of motion $($EOM$)$ of the system in terms of the coordinate $x$;

$($b$)$ Write down the dynamical EOM of the system in terms of the coordinate $z=x-x_{st},$ where $x_{st}$ is the $x-$position of O when the system is in static equilibrium.

$($c$)$ Determine undamped natural frequency ${}_n$ for the system.