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**given by right-hand rule**

Dot product scalar (commutative) Cross product vector (anti-commutative) Direction: given by right-hand rule

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**Linear momentum Angular momentum Note that need to define a center**

with respect to which L is calculated For rigid objects with moment of inertia I, rotating around a symmetry axis:

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**Linear momentum Angular momentum Force Torque**

Note that need to define a center with respect to which L is calculated For rigid objects with moment of inertia I, rotating around a symmetry axis: Force Torque

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**Translational angular momentum**

An object moving at constant velocity (straight line) does have angular momentum with respect to point A. Point A. Sometimes convenient to use rperp Angular momentum wrt point B is zero! Point A. Point A. Point B.

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C13T.2 If you are standing 30 m due east of a car traveling at 25 m/s southwest, what is the direction of the car's angular momentum wrt you? A. Southwest B. Northwest C. East D. West E. Up F. Down

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C13T.2 If you are standing 30 m due east of a car traveling at 25 m/s southwest, what is the direction of the car's angular momentum wrt you? A. Southwest B. Northwest C. East D. West E. Up F. Down

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C13T.3 You are standing 30 m due east of a 50-kg person who is running at a speed of 2 m/s due west. What is the magnitude of that person's angular momentum about you? A kg m2/s B kg m2/s C. 300 kg m2/s D. 100 kg m2/s E. 0 kg m2/s F. other

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C13T.3 You are standing 30 m due east of a 50-kg person who is running at a speed of 2 m/s due west. What is the magnitude of that person's angular momentum about you? A kg m2/s B kg m2/s C. 300 kg m2/s D. 100 kg m2/s E. 0 kg m2/s F. other

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C13.4 The lengths of the hour and minute hands of a clock are 4cm and 6cm, respectively. If the vector vec(u) and vec(w) represent the hour and minute hands, respectively, then vec(u) cross vec(w) at 5 o'clock is: A. 24 cm^2 up. B. 24 cm^2 down. C. 21 cm^2 up. D. 21 cm^2 down. E. 12 cm^2 up. F. 12 cm^2 down.

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C13.4 The lengths of the hour and minute hands of a clock are 4cm and 6cm, respectively. If the vector vec(u) and vec(w) represent the hour and minute hands, respectively, then vec(u) cross vec(w) at 5 o'clock is: A. 24 cm^2 up. B. 24 cm^2 down. C. 21 cm^2 up. D. 21 cm^2 down. E. 12 cm^2 up. F. 12 cm^2 down.

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C13.4 (follow up) The lengths of the hour and minute hands of a clock are 4cm and 6cm, respectively. If the vector vec(u) and vec(w) represent the hour and minute hands, respectively, then vec(u) cross vec(w) at 5:30 is: A. 24 cm^2 up. B. 24 cm^2 down. C. 21 cm^2 up. D. 21 cm^2 down. E. 12 cm^2 up. F. 12 cm^2 down.

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C13.4 (follow up) The lengths of the hour and minute hands of a clock are 4cm and 6cm, respectively. If the vector vec(u) and vec(w) represent the hour and minute hands, respectively, then vec(u) cross vec(w) at 5:30 is: A. 24 cm^2 up. B. 24 cm^2 down. C. 21 cm^2 up. D. 21 cm^2 down. E. 12 cm^2 up. F. 12 cm^2 down.

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**C13T. 6 A disk with a mass of 10 kg and a radius of 0**

C13T.6 A disk with a mass of 10 kg and a radius of 0.1 m rotates at a rate of 10 turns per second. The magnitude of the disk's total angular momentum is A. 63 kg m^2/s. B. 31 kg m^2/s. C. 10 kg m^2/s. D. 6.3 kg m^2/s. E. 3.1 kg m^2/s. F. 1.0 kg m^2/s.

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**C13T. 6 A disk with a mass of 10 kg and a radius of 0**

C13T.6 A disk with a mass of 10 kg and a radius of 0.1 m rotates at a rate of 10 turns per second. The magnitude of the disk's total angular momentum is A. 63 kg m^2/s. B. 31 kg m^2/s. C. 10 kg m^2/s. D. 6.3 kg m^2/s. E. 3.1 kg m^2/s. F. 1.0 kg m^2/s.

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**C13T. 7 Imagine that you are the pitcher in a baseball game**

C13T.7 Imagine that you are the pitcher in a baseball game. The batter hits a foul ball vertically in the air. If the ball has a weight of 2 N and an initial upward velocity of about 30 m/s, and you are 40 m from where the ball is hit, what is the gravitational torque (magnitude and direction) on the ball about you just after it is hit? A N.m upward. B N.m to your left. C N.m to your right. D N.m upward. E N.m to your left. F N.m to your right.

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**C13T. 7 Imagine that you are the pitcher in a baseball game**

C13T.7 Imagine that you are the pitcher in a baseball game. The batter hits a foul ball vertically in the air. If the ball has a weight of 2 N and an initial upward velocity of about 30 m/s, and you are 40 m from where the ball is hit, what is the gravitational torque (magnitude and direction) on the ball about you just after it is hit? A N.m upward. B N.m to your left. C N.m to your right. D N.m upward. E N.m to your left. F N.m to your right.

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As the ball described above continues to rise, the magnitude of the torque on the ball about you due to the ball’s weight A. Increases. B. Essentially remains the same. C. Decreases.

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As the ball described above continues to rise, the magnitude of the torque on the ball about you due to the ball’s weight A. Increases. B. Essentially remains the same. C. Decreases.

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C13T.9 A cylinder rolls without slipping down an incline directly toward you. The contact interaction between the cylinder and the incline exerts a friction torque on the cylinder about the cylinder’s center of mass. What is the direction of this torque? Toward you B. Away from you. C. To your right. D. To your left. E. Upward. F. Downward.

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C13T.9 A cylinder rolls without slipping down an incline directly toward you. The contact interaction between the cylinder and the incline exerts a friction torque on the cylinder about the cylinder’s center of mass. What is the direction of this torque? FN Toward you B. Away from you. C. To your right. D. To your left. E. Upward. F. Downward. Ffr Fg

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C13T.10 Imagine that you are looking down on a turntable that is spinning counterclockwise. If an upward torque is applied to the turntable, its angular speed A. Increases. B. Essentially remains the same. C. Decreases.

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C13T.10 Imagine that you are looking down on a turntable that is spinning counterclockwise. If an upward torque is applied to the turntable, its angular speed A. Increases. B. Essentially remains the same. C. Decreases.

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**A wheel of radius 50 cm rotates freely on an axle of radius 0. 5cm**

A wheel of radius 50 cm rotates freely on an axle of radius 0.5cm. If you want to slow the wheel to rest with your hand, you can either exert a friction force with your hand on the wheel’s rim (call the magnitude of this force Frim) or exert a force on the wheel’s axle (call the magnitude of this force Faxle). If you had to bring the wheel to rest in 2.0 s either way, how would the forces compare? A. Frim =100Faxle. B. Frim =10Faxle. C. Frim =Faxle. D. Frim =0.1Faxle. E. Frim =0.01Faxle.

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**A wheel of radius 50 cm rotates freely on an axle of radius 0. 5cm**

A wheel of radius 50 cm rotates freely on an axle of radius 0.5cm. If you want to slow the wheel to rest with your hand, you can either exert a friction force with your hand on the wheel’s rim (call the magnitude of this force Frim) or exert a force on the wheel’s axle (call the magnitude of this force Faxle). If you had to bring the wheel to rest in 2.0 s either way, how would the forces compare? A. Frim =100Faxle. B. Frim =10Faxle. C. Frim =Faxle. D. Frim =0.1Faxle. E. Frim =0.01Faxle.

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CT14.1 Imagine that you are carrying a suitcase containing a gyroscope whose angular velocity is horizontal and points directly away from your legs as you carry it with your right hand. Imagine that you make a 90 degree turn to the left as you walk. What does the suitcase do? A. It turns to the left along with you. B. It turns to the right as you try to turn left. C. Its bottom edge flips up away from your legs as you make the turn. D. Its bottom edge flips in to tangle with your legs. E. Its front end dips toward the ground. F. Its back end dips toward the ground.

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CT14.1 Imagine that you are carrying a suitcase containing a gyroscope whose angular velocity is horizontal and points directly away from your legs as you carry it with your right hand. Imagine that you make a 90 degree turn to the left as you walk. What does the suitcase do? A. It turns to the left along with you. B. It turns to the right as you try to turn left. C. Its bottom edge flips up away from your legs as you make the turn. D. Its bottom edge flips in to tangle with your legs. E. Its front end dips toward the ground. F. Its back end dips toward the ground.

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CT14.2 Experienced players can throw a Frisbee using forehand motion that causes the Frisbee to rotate counterclockwise (when thrown by a right-handed person.) If a Frisbee thrown in this way is to skip off the ground, which edge has to hit the ground first? A. The back edge. B. The front edge. C. The left edge. D. The right edge. E. One can't make the Frisbee thrown in this way skip.

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CT14.2 Experienced players can throw a Frisbee using forehand motion that causes the Frisbee to rotate counterclockwise (when thrown by a right-handed person.) If a Frisbee thrown in this way is to skip off the ground, which edge has to hit the ground first? A. The back edge. B. The front edge. C. The left edge. D. The right edge. E. One can't make the Frisbee thrown in this way skip.

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CT14.3 Which of the following changes in a top's design will cause the largest decrease in the top's precesion rate? A. Increasing its mass by 10%. B. Decreasing its height by 10%. C. Increasing its radius by 10%. D. Increasing its angular speed by 10%. E. Changes under B, C, D have the same effect. F. None of these design changes modify the precesion rate.

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CT14.3 Which of the following changes in a top's design will cause the largest decrease in the top's precesion rate? A. Increasing its mass by 10%. B. Decreasing its height by 10%. C. Increasing its radius by 10%. D. Increasing its angular speed by 10%. E. Changes under B, C, D have the same effect. F. None of these design changes modify the precesion rate.

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CT14.4 A person is sitting at rest on a stool that is free to rotate about a vertical axis while holding in one hand a bicycle wheel that is rapidly spinning counterclockwise when viewed from above. The person then stops the wheel with the other hand. What happens to the person as a result? A. The person must rotate counterclockwise. B. The person must rotate clockwise. C. The person will rotate in a direction that depends on which hand does the stopping. D. Nothing; the wheel's angular momentum is carried away by external interactions. E. Nothing; the wheel's angular momentum is simply dissipated by the friction interaction.

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CT14.4 A person is sitting at rest on a stool that is free to rotate about a vertical axis while holding in one hand a bicycle wheel that is rapidly spinning counterclockwise when viewed from above. The person then stops the wheel with the other hand. What happens to the person as a result? A. The person must rotate counterclockwise. B. The person must rotate clockwise. C. The person will rotate in a direction that depends on which hand does the stopping. D. Nothing; the wheel's angular momentum is carried away by external interactions. E. Nothing; the wheel's angular momentum is simply dissipated by the friction interaction.

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C14.T5 An astronaut floating at rest in space throws a ball to another astronaut, using a side-arm motion that ends up releasing the ball well to the astronaut's right as he or she faces the direction of the ball's forward motion. After throwing the ball, the astronaut (when viewed from above). A. Remains completely at rest. B. Rotates clockwise, but his or her center of mass remains at rest. C. Rotates counterclockwise, but her or his center of mass remains at rest. D. Rotates clockwise and her or his center of mass moves opposite to the ball's motion. E. Rotates counterclockwise and her or his center of mass moves opposite to the ball's motion.

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C14.T5 An astronaut floating at rest in space throws a ball to another astronaut, using a side-arm motion that ends up releasing the ball well to the astronaut's right as he or she faces the direction of the ball's forward motion. After throwing the ball, the astronaut (when viewed from above). A. Remains completely at rest. B. Rotates clockwise, but his or her center of mass remains at rest. C. Rotates counterclockwise, but her or his center of mass remains at rest. D. Rotates clockwise and her or his center of mass moves opposite to the ball's motion. E. Rotates counterclockwise and her or his center of mass moves opposite to the ball's motion.

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C14.T6 If global warming proceeds during the next century as anticipated, it is possible that the polar ice caps will melt, substantially raising sea levels around the world. This would: A. Lengthen the day slightly. B. Shorten the day slightly. C. Have strictly no effect on the length of the day.

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C14.T6 If global warming proceeds during the next century as anticipated, it is possible that the polar ice caps will melt, substantially raising sea levels around the world. This would: A. Lengthen the day slightly. B. Shorten the day slightly. C. Have strictly no effect on the length of the day.

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FROM LAST CLASS: Consider a specific point P on a bicycle wheel as the wheel turns about a fixed axis, neither speeding up nor slowing down. Compare.

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