A rack and pinion is a pair of gears that converts rotational motion into translation. As shown in Figure 5.130, a torque \(\tau\) is applied to the shaft. The pinion rotates and causes the rack to translate. The mass moment of inertia of the pinion is \(I\) and the mass of the rack is \(m\). Draw the free-body diagram and derive the differential equation of motion.

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FIGURE 5.130 Problem 12. m