$(2)$ A very typical problem of electromechanical position control is an electric motor driving a load that has one dominant vibration mode. The problem arises in computer$-$disk$-$head control, reel$-$to$-$reel tape drives, and many other applications. A schematic diagram is sketched below. The motor has an electrical constant $K_e,$ a torque constant $K_t,$ an armature inductance $L_a$ and a resistance $R_a.$ The rotor has an inertia $J_1,$ and a viscous friction $B.$ The load has an inertial $J_2.$ The two inertias are connected by a shaft with a spring constant $k$ and an equivalent viscous damping $b.$ Write the equations of motion.