$[40]$ Find the closed$-$loop transfer function, $T(s)=\frac{C(s)}{R(s)},$ for the system shown in the Fig. $1,$ using block diagram reduction.

$[40]$ Find the transfer function, $G(s)=\frac{x_1(s)}{F(s)},$ for the translational mechanical system shown in the Fig. $2.$

Fig. $2$ Translational mechanical system

$[20]$ A DC motor's system and its torque$-$speed curve are given in Fig. $3($a$)$ and Fig. $3($b$),$ respectively. Determine the system's transfer function, $\frac{{}_L(s)}{E_a(s)}$

Hints for problem No$.3$: