$4\mathrm{.}53$ In Figure P$4\mathrm{.}53$ a motor supplies a torque $T$ to turn a drum of radius $R$ and

inertia I about its axis of rotation. The rotating drum lifts a mass $m$ by means

of a cable that wraps around the drum. The drum's speed is $.$ Viscous

torsional damping $c_T$ exists in the drum shaft. Neglect the mass of the cable.

$($a$)$ Obtain the equation of motion with the torque $T$ as the input and the vertical

speed $v$ of the mass as the output. $($b$)$ Suppose that $m=50kg,R=0\mathrm{.}3m,$

$I=1\mathrm{.}1kg*m^2,$ and $c_T=0\mathrm{.}2N*m*s.$ Find the speed $v(t)$ if the system is initially

at rest and the torque $T$ is a step function of magnitude $500N*m.$

Figure P$4\mathrm{.}53$

While possible to use equivalent mass and equivalent torque, an easier way may be to

draw two FBDs even though there is $1$ DOF. If you do use the equivalent mass and

equivalent force, be careful with the directions $($i$.$e$.,$ signs$)$ and don't forget the power

done by the weight $($ mg $)$ due to gravity.

Use a Laplace transform approach to solve part b$.$

Use a transfer function approach with the Matlab step command to plot the result.

Compare the steady state response of the system using the final value theorem and also

with the numerical Matlab step command.