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Unformatted text preview: 11 Angular Momentum CHAPTER OUTLINE 11.1 The Vector Product and Torque 11.2 Angular Momentum 11.3 Angular Momentum of a Rotating Rigid Object 11.4 Conservation of Angular Momentum 11.5 The Motion of Gyroscopes and Tops ANSWERS TO QUESTIONS Q11.1 No to both questions. An axis of rotation must be defi ned to calculate the torque acting on an object. The moment arm of each force is measured from the axis, so the value of the torque depends on the location of the axis. *Q11.2 (i) Downcrossleft is away from you: = j i k ( ) answer (f ), as in the fi rst picture. (ii) Leftcrossdown is toward you: ( ) = i j k answer (e), as in the second picture. *Q11.3 (3 m down) (2 N toward you) = 6 N m left. The answers are (i) a (ii) a (iii) f *Q11.4 The unit vectors have magnitude 1, so the magnitude of each cross product is |1 1 sin | where is the angle between the factors. Thus for (a) the magnitude of the cross product is sin 0 = 0. For (b), |sin 135| = 0.707 (c) sin 90 = 1 (d) sin 45 = 0.707 (e) sin 90 = 1. The assem- bled answer is c = e > b = d > a = 0. Q11.5 Its angular momentum about that axis is constant in time. You cannot conclude anything about the magnitude of the angular momentum. Q11.6 No. The angular momentum about any axis that does not lie along the instantaneous line of motion of the ball is nonzero. *Q11.7 (a) Yes. Rotational kinetic energy is one contribution to a systems total energy. (b) No. Pulling down on one side of a steering wheel and pushing up equally hard on the other side causes a total torque on the wheel with zero total force. (c) No. A top spinning with its center of mass on a fi xed axis has angular momentum with no momentum. A car driving straight toward a light pole has momentum but no angular momentum about the axis of the pole. 283 FIG. Q11.2 13794_11_ch11_p283-310.indd 283 13794_11_ch11_p283-310.indd 283 1/8/07 8:52:33 PM 1/8/07 8:52:33 PM 284 Chapter 11 Q11.8 The long pole has a large moment of inertia about an axis along the rope. An unbalanced torque will then produce only a small angular acceleration of the performer-pole system, to extend the time available for getting back in balance. To keep the center of mass above the rope, the performer can shift the pole left or right, instead of having to bend his body around. The pole sags down at the ends to lower the system center of gravity. *Q11.9 Her angular momentum stays constant as I is cut in half and doubles. Then (1 / 2) I 2 doubles. Answer (b). Q11.10 Since the source reel stops almost instantly when the tape stops playing, the friction on the source reel axle must be fairly large. Since the source reel appears to us to rotate at almost constant angular velocity, the angular acceleration must be very small. Therefore, the torque on the source reel due to the tension in the tape must almost exactly balance the frictional torque. In turn, the frictional torque is nearly constant because kinetic friction forces dont depend on velocity, and...
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