Problem $4.$ EOM Derivation

Derive the equation of motion for the following system: We want to measure the spin rate

$$ of the rotating shaft of a turbine. To do it we push a small mass $m$ against the end of

the shaft. The shaft makes friction contact with the drag cup mounted on the mass. The

frictional torque is $b(-\frac{d}{dt}).$ Assume that the rotational spring torque is $k$ and that the

moment of inertia of mass $m$ is $J.$