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

Given: Collar A $($having a mass of $m)$ is able to slide along a smooth, fixed horizontal rod, with springs of stiffnesses $2k$ and $k$ attached between the left and right ends of the collar, respectively, and ground, as shown in the figure below. A homogeneous disk $($of mass $m$ and radius $R)$ is pinned to collar A at its center O $.$ The disk is able to roll without slipping on a rough, horizontal surface. Let $$ measure the rotation of the disk, with $=0$ when the two springs are unstretched.

Find: For this problem, derive the dynamical equation of motion for the system in terms of the coordinate $.$