The pulley of mass (M) shown in Figure 5.126 has a radius of (r). The mass moment

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The pulley of mass \(M\) shown in Figure 5.126 has a radius of \(r\). The mass moment of inertia of the pulley about the point \(\mathrm{O}\) is \(I_{\mathrm{O}}\). A translational spring of stiffness \(k\) and a block of mass \(m\) are connected to the pulley, as shown. Assume that the pulley rolls without slipping. Derive the equation of motion by using

(a) the force/ moment approach and

(b) the energy approach.

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