$3-26.$ Consider a train consisting of an engine and a car, as shown in Fig.

$3$P$\_76$

Figure $3$P$-26$

A controller is applied to the train so that it has a smooth start and stop,

along with a constant$-$speed ride. The mass of the engine and the car are $M$

and $m,$ respectively. The two are held together by a spring with the stiffness

coefficient of $K.F$ represents the force applied by the engine, and $$

represents the coefficient of rolling friction. If the train only travels in one

direction:

$($a$)$ Draw the free$-$body diagram.

$($b$)$ Find the state variables and output equations.

$($c$)$ Find the transfer function.

$($d$)$ Write the state space of the system.