VIEWS: 93 PAGES: 12 POSTED ON: 9/15/2012
Angular Momentum of a rigid object rotating about a fixed axis L I dL d I dt dt But for any rigid object the rotational inertia is a constant dL d I dt dt dL I dt Newton’s dL dp Second Law Analogous to Fnet dt dt What if the system is isolated and closed? Isolated – no external torques Closed – no change in the mass dL dt dL 0 dt L constant Law of Conservation of Angular Momentum In any closed, isolated system, the angular momentum is constant Conservation of Angular Momentum Examples 1. The spinning volunteer. Li L f Iii I f f Ii i I f f Conservation of Angular Momentum Examples 2. Stabilizing a Frisbee®. The classic Frisbee® has a heavy outer ridge which increases its rotational inertia and is then spun resulting in a large angular momentum which resists changes to its motion. Conservation of Angular Momentum Examples 3. Two disks are mounted on low friction bearings on the same axle and can be brought together so that they couple and rotate as one unit. The first disk, with mass 2.0 kg and radius 0.50 m, is set spinning at 450 rev . The second disk, min with mass 4.0 kg and radius 0.50 m, is set spinning at 900. rev in the same min direction as the first. They then couple together. a. What is their angular speed after coupling? Li L f l1i l2i L f I11i I 22i I1 I 2 f 1 mR 2 1 2m R 2 1 mR 2 1 2m R 2 1i 2i f 2 2 2 2 1i 22i 3 f Conservation of Angular Momentum Examples 3. Two disks are mounted on low friction bearings on the same axle and can be brought together so that they couple and rotate as one unit. The first disk, with mass 2.0 kg and radius 0.50 m, is set spinning at 450 rev . The second disk, min with mass 4.0 kg and radius 0.50 m, is set spinning at 900. rev in the same min direction as the first. They then couple together. a. What is their angular speed after coupling? 1i 22i 3 f 1i 22i f 3 f 450 rev min 2 900. rev min 3 f 750 rev min Conservation of Angular Momentum Examples 3. Two disks are mounted on low friction bearings on the same axle and can be brought together so that they couple and rotate as one unit. The first disk, with mass 2.0 kg and radius 0.50 m, is set spinning at 450 rev . The second disk, min with mass 4.0 kg and radius 0.50 m, is set spinning at 900. rev in the same min direction as the first. They then couple together. b. If instead the second disk is set spinning at 900.rev in the opposite direction min of the first disk' s rotation, what is their angular speed after coupling? f 450 rev min 2 900. rev min 3 f 450 rev min Conservation of Angular Momentum Examples 2. Two children, each with mass M, sit on opposite ends of a narrow board with length L and mass M. The board is pivoted at its center and is free to rotate in a horizontal circle without friction. (Treat the board as a thin rod.) a. What is the rotational inertia of the board plus the children about a vertical axis through the center of the board? l 2 I total I board 2 I child M M 2 I total 1 l Ml 2M 2 l 12 2 1 1 2 I total Ml Ml 2 12 2 7 I total Ml 2 12 Conservation of Angular Momentum Examples 2. Two children, each with mass M, sit on opposite ends of a narrow board with length L and mass M. The board is pivoted at its center and is free to rotate in a horizontal circle without friction. (Treat the board as a thin rod.) b. What is the magnitude and direction of the angular momentum of the system if it is rotating with angular speed ωo in a clockwise direction as seen from above? L I M M 7 l L Ml 2o Downward 12 Conservation of Angular Momentum Examples While the system is rotating, the children pull themselves toward the center of the board until they are half as far from the center as before. c. What is the ratio of the new rotational inertia to the initial rotational inertia? l 2 4 7 I i Ml 2 1 l I f Ml 2M 2 12 12 4 M M 5 I f Ml 2 l 24 5 If Ml 2 24 Ii 7 Ml 2 12 If 5 Ii 14 Conservation of Angular Momentum Examples While the system is rotating, the children pull themselves toward the center of the board until they are half as far from the center as before. d. What is the resulting angular speed in terms of ωo? l Li L f 4 Iio I f f M M l Ii f o If 14 f o 5 Conservation of Angular Momentum Examples While the system is rotating, the children pull themselves toward the center of the board until they are half as far from the center as before. e. What is the change in kinetic energy of the system as a result of the children changing their position? (From where does the added kinetic energy come?) l 4 Ek Ekf Eki 1 1 M M Ek I f f I ii2 2 2 2 l 2 1 5 2 14 1 7 Ml 2 2 Ek Ml o o 2 24 5 2 12 *Note: L = constant The added energy comes 21 2 2 Ek Ml o from the work done by the Ek = increases children when pulling 40 themselves forward.