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

Given: A homogeneous disk of mass $m$ and with an outer radius of $R$ is pinned to ground at its center O$.$ Block A $($of mass $m)$ is connected to grounded springs of stiffnesses $k$ and $2k,$ as shown, as it slides on a smooth horizontal surface. The disk and block A are in contact as the system moves, with the disk not slipping on the surface of block A $.$ A torque $M(t)$ acts on the disk. Let $x$ measure the position of block A$,$ with $x$ measured positively to the right and with $x=0$ corresponding to the springs being unstretched.

Find: For this problem:

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

$($b$)$ Determine the natural frequency of the system; and,

$($c$)$ Determine the particular solution of the EOM corresponding to an input torque of $M(t)=M_{0}sint.$