Objective: $(1)$ Calculating the maximum power of a DC motor as a function of its parameters.

$(2)$ Designing and analyzing a closed$-$loop controller

Problem $1.$ A DC motor's angular speed decreases linearly with the load torque as shown in the diagram. The output power of the motor is calculated from $P_{\text{o}\text{u}\text{t}}=T_l.$ There is a particular load torque at which the output power is the maximum.

Calculate the maximum power of a DC motor as a function of applied voltage, $V,$ torque constant, $K_t,$ back$-$emf constant, $K_b,$ coil resistance, $R,$ damping constant $C,$ and friction torque $T_{\text{f}\text{r}\text{i}\text{c}\text{i}\text{c}}.$

Hint: In the figure below, find the value of $x$ which would maximize the area