A rotational motor$-$load system is shown in Figure Q$2($a$)$ where $J$ is the moment of inertia, $B$ is the viscous friction, $T_e$ is the electromagnetic torque, $T_L$ is the load torque and ${}_m$ is thespeed of the motor. The speed$-$time profile of the rotational system is as shown in Figure Q$2($b$).$

Figure Q$2($a$)$: A rotational motor$-$load system

Figure Q$2($b$)$: Motor$-$load speed$-$time profile

i$)$ Write the mathematical expression that describes the rotational motion.

$(3$ marks$)$

ii$)$ With the following values; $J=0\mathrm{.}01kg-m^2,B=0\mathrm{.}015k\frac{g}{m.s},T_L=8Nm,$ analyse the torquetime profile characteristic of the rotational system. Plot the complete torque$-$time profile of the rotational system.

$(15$ marks$)$