Serway & Jewett — Angular Momentum: Problem 2

A disk with moment of inertia \(I_1\) rotates about a frictionless, vertical axle with angular speed \(\omega_i\). A second disk, this one having moment of inertia \(I_2\) and initially not rotating, drops onto the first disk (Fig. TP11.2). Because of friction between the surfaces, the two eventually reach the same angular speed \(\omega_f\). Discuss in your group the following. (a) Calculate \(\omega_f\). (b) What fraction of the initial kinetic energy of the two-disk system remains after the disks rotate with the same angular speed? (c) Find the value of the answer in part (b) for the following limits: (i) \(I_2 \to 0\), (ii) \(I_1 = I_2\), (iii) \(I_2 \to \infty\), and (iv) \(I_1 \to \infty\). (d) Explain how each of the results in part (c) makes sense. (e) In the general case in which the kinetic energy of the system decreases in the process, where does that energy go? (f) What If? In Figure TP11.2, what is \(\omega_f\) if the second disk is also rotating, but in the clockwise direction, opposite that of disk 1, with an angular speed of \(\omega'\) before the collision?