(Take home) A rotor, consisting of a solid disc having a mass, M, and radius, R,

Question

(Take home) A rotor, consisting of a solid disc having a mass, M, and radius, R, is attached to one end of an axle as shown in Figure 1. The opposing end of the axle rests on a pedestal. The axle is, and remains in, the xy plane. The axle has a length of/ and can be considered massless. A massless cord is wound around the outside of the rotor. The free end of the cord is attached to a mass m0. The mass m0 is allowed to drop through a height of h, thus imparting an angular speed to the rotor. The axle does not rotate with the rotor but is free to rotate about the vertical axis. What is the rate at which the rotor and axle rotate about the vertical axis? The system is completely free of friction or drag forces. Note: as the mass m0 is falling, the axle is held in place. When m0 has fallen through the height h, the cord and m0 fall away from the disc and the axle is released. Here: R = 0.7 m, M = 100 kg, l = 1 m, h = 45 m, m0 = 404.08 kg.

Explanation / Answer

Assuming that there is no slipping between the cord andthe rotor disc, v = R v = final vertical speed of the mass mo, R = radius of rotor disc. = final angular speed of rotor disc aboutaxle. Since mechanical energy of the mass-rotor system is tobe conserved, mogh = mov2/2 + I2/2 I = MR2/2 for a cylindrical disc. So, 2R2[mo/2 + M/4] = mogh =[{4mogh/(2mo+M)}]/R about theaxle. Since the initial angular momentum of the system alongthe vertical axis is zero, there is no applied moment aboutit, the final angular moment and angular speed should alsobe zero.

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