$(10$ pts$)$ The figure below depicts a rack and pinion connection where a viscous friction coefficient B

and a spring with stiffness $K$ are resisting the motion of the rack. The inertias of the disk and pinion

are $J_m$ and $J_p$; respectively, and any twisting of the shaft connecting the disk and pinion is ignored

$($i$.$e$.,$ this link is assumed rigid$).$ Assuming perfect rolling contact between the rack and pinion, derive

the equations of motion for this system with an applied torque $T(t)$ as the input variable and the

rack's linear displacement $x(t)$ as the output variable.