$6\mathrm{.}1.$ The dynamic performance of actuator systems is affected by the equivalent inertia driven by the motor. Consider the two systems illustrated in Figure $6\mathrm{.}24.$ In $($a$)$ a servo motor, which may be considered as a torque source, drives a load inertia $J$ through a $10$:$1$ frictionless gearbox. In $($b$)$ a similar servo motor drives a translating mass $m$ through an ideal rack and pinion of radius $r_p.$

$($a$)$

$($b$)$

Figure $6\mathrm{.}24$: Two inertial loads on a servo motor.

$($a$)$ Draw the linear graph for each system.

$($b$)$ Determine the equivalent rotational inertia $J_{eq}$ reflected through the transduction element to the drive motor.