The mass moment of inertia of a nonhomogeneous and/or complex-shaped body of revolution about the axis of rotation can be determined by first finding its natural frequency of torsional vibration about its axis of rotation. In the torsional system shown in Fig. 2.121, the body of revolution (or rotor), of rotary inertia \(J\), is supported on two frictionless bearings and connected to a torsional spring of stiffness \(k_{t}\). By giving an initial twist (angular displacement) of \(\theta_{0}\) and releasing the rotor, the period of the resulting vibration is measured as \(\tau\).

a. Find an expression for the mass moment of inertia of the rotor \((J)\) in terms of \(\tau\) and \(k_{t}\).

b. Determine the value of \(J\) if \(\tau=0.5 \mathrm{~s}\) and \(k_{t}=5000 \mathrm{~N}-\mathrm{m} / \mathrm{rad}\).

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Bearing CB Body of revolution (S) KA Bearing FIGURE 2.121 Body of revolution. Torsional spring (k,)