The initial angular velocity of the system is i = 0.20 rev/s = 1.26 rad/s, and we find the final angular velocity via conservation of angular momentum, as f = I i I f i = 370 130 (1.26 rad/s) = 3.6 rad/s. (b) The change in kinetic energy is KE f - KE i = 1 2 I f f 2 - 1 2 I i i 2 , or = 1 2 (130 kg m 2 )(3.58 rad/s) 2 - 1 2 (370 kg m 2 )(1.26 rad/s) 2 = 540 J. This difference results from work done by the man on the system as he walks inward.
• A cylinder with moment of inertia I 1 rotates about a vertical, frictionless axle with angular velocity v i . A second cylinder, this one having moment of inertia I 2 and initially not rotating, drops onto the first cylinder. Because of friction between the surfaces, the two eventually reach the same angular velocity v f . • (a) Calculate v f . • (b) Show that the kinetic energy of the system decreases in this interaction and calculate the ratio of the final to the initial rotational energy.
(a) From conservation of angular momentum for the system of two cylinders: 1 2 1 f i I I I or 1 1 2 f i I I I (b) 2 1 2 1 2 f f K I I and 2 1 1 2 i i K I so 2 1 1 2 2 1 1 2 1 1 2 1 2 1 2 which is less than 1 f i i i K I I I I K I I I I I .
• A puck of mass m is attached to a cord passing through a small hole in a frictionless, horizontal surface. The puck is initially orbiting with speed v i in a circle of radius r i . The cord is then slowly pulled from below, decreasing the radius of the circle to r . • (a) What is the speed of the puck when the radius is r ? • (b) Find the tension in the cord as a function of r .
(a) sin180 0 r F rF Angular momentum is conserved. f i i i i i L L mrv mrv rv v r (b) 2 2 3 i i m rv mv T r r FIG. P11.49
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