SM_chapter11

SM_chapter11 - 11 Angular Momentum CHAPTER OUTLINE 11.1...

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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 deF 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) Down–cross–left is away from you: −×− =− ˆˆ ˆ jik ( ) answer (f ), as in the F rst picture. (ii) Left–cross–down is toward you: −×− () = ˆˆ ˆ ij 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 θ | where is the angle between the factors. Thus for (a) the magnitude of the cross product is sin 0° = 0. ±or (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 The angular momentum of the mouse-turntable system is initially zero, with both at rest. The fric- tionless axle isolates the mouse-turntable system from outside torques, so its angular momentum must stay constant with the value zero. (i) The mouse makes some progress north, or counterclockwise. Answer (a). (ii) The turntable will rotate clockwise. The turntable rotates in the direction opposite to the motion of the mouse, for the angular momentum of the system to remain zero. Answer (b). 251 FIG. Q11.2 continued on next page
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252 Chapter 11 (iii) No, mechanical energy changes as the mouse converts some chemical into mechanical energy, positive for the motions of both the mouse and the turntable. (iv) No, momentum is not conserved. The turntable has zero momentum while the mouse has a bit of northward momentum. The sheave around the turntable axis exerts a force northward to feed in this momentum. (v) Yes, angular momentum is constant with the value zero. *Q11.8 (i) The angular momentum is constant. The moment of inertia decreases, so the angular speed must increase. Answer (a). (ii) No, mechanical energy increases. The ponies must do work to push themselves inward. (iii) Yes, momentum stays constant with the value zero. (iv) Yes, angular momentum is constant with a nonzero value. *Q11.9 Angular momentum is conserved according to the equation I 1 ω 0 + 0 = ( I 1 + I 2 ) f . Solving for f gives answer (c). Q11.10 The suitcase might contain a spinning gyroscope. If the gyroscope is spinning about an axis that is oriented horizontally passing through the bellhop, the force he applies to turn the corner results in a torque that could make the suitcase swing away. If the bellhop turns quickly enough, anything at all could be in the suitcase and need not be rotating. Since the suitcase is massive, it will want to follow
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This note was uploaded on 01/28/2011 for the course PHYS 011 taught by Professor Nianlin during the Fall '08 term at HKUST.

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SM_chapter11 - 11 Angular Momentum CHAPTER OUTLINE 11.1...

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