$6\mathrm{.}3.$ A ball$-$screw linear actuator is shown in Fig. $6\mathrm{.}26.$ The actuator uses a dc electric motor, with an electromotive coupling constant $K_a,$ armature resistance $R,$ and inductance $L,$ and driven by a voltage source $V_s(t).$ The motor drives a threaded shaft with $N$ threads $\frac{?}{cm}$ on which the ball$-$nut moves linearly. The mechanical load is a mass $m$ which slides on a surface with viscous damping coefficient $B.$

$($a$)$ Construct a model for the ball$-$screw that relates the shaft torque and angular velocity to the nut linear velocity and force. Write the equations for the transduction process and show that they are energy conserving. Is this a transforming or gyrating relationship?

$($b$)$ Generate a set of state equations that describe the dynamics of the system.