Consider the system shown in Figure 5.73, in which a uniform sphere of mass \(m\) and radius \(r\) rolls along an inclined plane of \(30^{\circ}\). A translational spring of stiffness \(k\) is attached to the sphere. Assuming that there is no slipping between the sphere and the surface, draw the necessary free-body diagram and derive the differential equation of motion.

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FIGURE 5.73 Problem 15. 30