Consider the system shown in Figure 5.21. Let \(x_{1}\) and \(\theta\) be the displacement and angle of rotation of the pulley measured from their original locations when the spring is undeformed. Let \(x_{2}\) be the displacement of the cylinder, also measured from its original location when the spring is undeformed. The pulley has mass \(M\), radius \(R\), and its mass moment of inertia about its centroidal axis is \(I_{G}\). The cylinder has mass \(m\). Find

(a) the static equilibrium position of the system using the principle of virtual work, and

(b) the equation of motion of the system for generalized coordinate \(x_{2}\) using d'Alembert's principle.

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ellel x m k M. R. IG. x2 Figure 5.21: Pulley-cylinder coupled system.