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

Given: Two blocks, having masses of $m$ and $2m,$ are connected by an inextensible cable, with the cable being pulled over a frictionless pulley having negligible mass. The upper block is able to slide along the smooth upper surface of the lower block, with the lower block being able to slide along a smooth horizontal surface. A spring of stiffness $k$ is attached between the upper block and ground, and a dashpot with a damping coefficient $c$ is attached between the lower block and ground. Forces $F$ and $2F$ act to the right on the upper and lower blocks, respectively. Let $x$ represent the motion of the upper block, where $x=0$ when the spring is unstretched.

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