PROBLEM $2.(20$ points$)$ To impress the family you decide to upgrade the oven, adding an electric motor and a rigid shaft, for making rotisserie turkey. A schematic, including the turkey, is shown in Figure $2.$ The motor has internal resistance $R_M$ and inductance $L_M.$ Motor constants are $K_t$ and $K_b.$ DC supply voltage is $E_M.$ Viscous friction coefficient for the shaft is $B.$ Assume rotational inertia of motor and rigid shaft is negligible so $J_M=0.$ Rotational inertia of the turkey around the motor shaft is $J_T.$

Figure $2.$ Oven with rotisserie turkey on a rotating shaft.

a$)(5$ points$)$ Write the equation$($s$)$ describing the dynamics of motor current $i(t)$ and shaft rotational speed $(t).$