Consider continuous rolling mill depicted in above figure. Suppose that the motion of the adjustable roller has a

damping coefficient $b,$ and that the force exerted by the rolled material on the adjustable roller is proportional

to the material's change in thickness Fs $=c(T-x).$ Suppose further that the DC motor has torque constant Kt and

a back emf constant Ke and that the rack$-$and$-$pinion has an effective radius of R $.$

a$)$ What are the inputs to this system? The output?

b$)$ Without neglecting the effect gravity on the adjustable roller, draw a block diagram of the system

that explicitly shows the following quantities $Vs(s),Ia(s),Fm(s)($the force the motor exerts on the

adjustable roller $)$ and $x(s).$

c$)$ Simplify your block diagram as much as possible while still identifying the output and each input.