Chapter 10 | Rotation Around A Fixed Axis | Angular Momentum

Transcript

Chapter 10: Conservation of Angular Momentum Section: 10-1 Topic: The Vector Nature of Rotation Type: Conceptual 1. Ans: r A force F x in the negative x direction is applied to a particle in the xy plane. The arrow r that best represents the torque produced by F x on the particle with respect to the origin is r r r r r A) 1 B) 2 C) 3 D) 4 E) 5 A Section: 10-1 Topic: The Vector Nature of Rotation Type: Conceptual 2. Ans: A phonograph turntable in the xz plane is rotating clockwise as viewed from above. The vector that represents the torque with which the motor turns the table is r r r r r A) 1 B) 2 C) 3 D) 4 E) 5 B Section: 10-1 Topic: The Vector Nature of Rotation 3. Page 1 Type: Conceptual Chapter 10: Conservation of Angular Momentum Ans: A torque is applied to a bolt by hanging a weight w from the end of the wrench, as shown. The coordinate axis along which the torque vector is directed is A) y B) x C) –y D) –x E) z D Section: 10-1 Topic: The Vector Nature of Rotation Type: Conceptual 4. Ans: r As a particle with a velocity v in the negative x direction passes through the point (0, 0, 1), it has an angular velocity relative to the origin that is best represented by vector r r r r A) 1 B) 2 C) 3 D) 4 E) zero A Section: 10-1 Topic: The Vector Nature of Rotation 5. Page 2 Type: Conceptual Chapter 10: Conservation of Angular Momentum A) B) C) Ans: r The vector C represents r r A ´ B r r B · A r r A ´ B D Section: 10-1 r r D) B ´ A E) None of these is correct. Topic: The Vector Nature of Rotation Type: Conceptual 6. Ans: r r r r r Vectors I and II lie in the xy plane. The vector product II ´ ( I ´ II ) could be represented by vector r r r r r C A B A) B) C) D) D E) E A Section: 10-1 Topic: The Vector Nature of Rotation 7. Page 3 Type: Conceptual Chapter 10: Conservation of Angular Momentum Ans: A 7-kg mass and a 4-kg mass are mounted on a spindle that is free to turn about the x axis as shown. Assume the mass of the arms and the spindle to be negligible. If the system is free to rotate and is released from rest, there will initially be a resultant torque in which of the following directions? A) z B) –z C) y D) –x E) x E Section: 10-1 Topic: The Vector Nature of Rotation Type: Conceptual 8. Ans: A wheel is rotating clockwise on a fixed axis perpendicular to the page (x). A torque that causes the wheel to slow down is best represented by the vector r r r r r 3 5 1 2 4 A) B) C) D) E) A Section: 10-2 Topic: Angular Momentum 9. Page 4 Type: Conceptual Chapter 10: Conservation of Angular Momentum Ans: r A particle of mass m is moving with a velocity v , in the yz plane as shown. The vector that most nearly represents the angular momentum about the x axis is r r r r r A) 1 B) 2 C) 3 D) 4 E) 5 D Section: 10-2 Topic: Angular Momentum Type: Conceptual 10. Ans: A wheel is set spinning and is then hung by a rope placed at one end of the axle. If the wheel is spinning as shown, the angular momentum of the wheel could be represented by vector r r r r r 3 1 2 A) B) C) D) 4 E) 5 E Section: 10-2 Topic: Angular Momentum Type: Conceptual 11. A disk rotates clockwise in the plane of the page. What is the direction of the angular momentum vector? A) clockwise D) out of the page B) counterclockwise E) Angular momentum has no direction. C) into the page Ans: C Section: 10-2 Topic: Angular Momentum Page 5 Type: Conceptual Chapter 10: Conservation of Angular Momentum 12. A) B) C) Ans: Particles 1, 2, and 3 have equal masses and equal speeds. The angular momentum with respect to the origin for these three masses is the same for each particle. D) greatest for particle 3. greatest for particle 1. E) least for particle 2. greatest for particle 2. C Section: 10-2 Topic: Angular Momentum Type: Numerical 13. The angular momentum of a flywheel about its axis is 925 kg · m2/s. If its moment of inertia about the same axis is 2.50 kg · m2, its angular velocity is A) 370 rev/min B) 62 rev/min C) 36 rev/min D) 2210 rad/s E) 370 rad/s Ans: E Section: 10-2 Status: New to 5th edition Topic: Angular Momentum Type: Numerical 14. We would like to compare the angular momentum of Mars about its axis of rotation with that of the Earth's. The mass of Mars is 11% that of the Earth, with a radius 53% that of the Earth, and a rotational period 103% that of the Earth. Assuming both planets to be uniform spheres calculate the ratio of the angular momentum of Mars to that of the Earth. A) 6.0 ´ 10-2 B) 5.7 ´ 10-2 C) 3.2 ´ 10-2 D) 3.0 ´ 10-2 E) 3.1 ´ 10-2 Ans: D Section: 10-2 Status: New to 5th edition Topic: Angular Momentum Type: Numerical 15. Large meteors can impact the earth with speeds of 80,000 km/h. If such a meteor were to impact at a point on the plane of the Earth's equator resulting in a change of the Earth's angular momentum by 80 × 1012 kg•m2/s (an insignificant change, thankfully), calculate the mass of the meteor. (The radius of the Earth = 6.38 ´ 106 m.) A) 3.4 ´ 104 kg D) 4.5 ´ 104 kg B) 5.6 ´ 102 kg E) 56 kg C) 3.6 ´ 103 kg Ans: B Section: 10-2 Status: New to 5th edition Page 6 Topic: Angular Momentum Chapter 10: Conservation of Angular Momentum Type: Numerical 16. Let us compare the angular momentum of Mars (LM) in its orbit around the Sun to that of the Earth (LE). The mean orbital speed of Mars is 24 km/s, whereas that of the Earth is 30 km/s. The mean orbital radius of Mars is 228 ´ 106 km, whereas that of the Earth is 150 ´ 106 km. If the mass of Mars is 11% that of the Earth, calculate the ratio LM / LE. A) 0.21 B) 0.090 C) 7.7 D) 11 E) 0.13 Ans: E Section: 10-3 Topic: Torque and Angular Momentum Type: Conceptual 17. If the angular momentum of a system is constant, which of the following statements must be true? A) No torque acts on any part of the system. B) A constant torque acts on each part of the system. C) Zero net torque acts on each part of the system. D) A constant external torque acts on the system. E) Zero net torque acts on the system. Ans: E Section: 10-3 Topic: Torque and Angular Momentum Type: Conceptual 18. The angular momentum of a system is conserved only if A) the angular velocity is a function of time. B) the sum of the external torques equals the sum of the internal torques. C) the moment of inertia of the system is constant. D) the sum of the external torques is zero. E) the sum of the internal torques is zero. Ans: D Section: 10-3 Topic: Torque and Angular Momentum Type: Numerical 19. A constant torque of 15 N · m acts for 3.0 s on a system of mass 2.0 kg. The change in angular momentum of the system during this period of time is A) 5.0 kg · m2/s D) 23 kg · m2/s 2 B) 7.5 kg · m /s E) 45 kg · m2/s C) 10 kg · m2/s Ans: E Section: 10-3 Topic: Torque and Angular Momentum 20. Page 7 Type: Numerical Chapter 10: Conservation of Angular Momentum Ans: The angular momentum of a body about a particular axis as a function of time is shown in the graph. The external torque acting on the body along this axis at t = 2 s is A) 0 B) 5 N · m C) 10 N · m D) 20 N · m E) 40 N · m B Section: 10-3 Topic: Torque and Angular Momentum Type: Conceptual 21. If the sum of the external torques acting on an isolated system of particles is zero, it must be true that A) the system can have no kinetic energy. B) the angular momentum of the system does not change. C) the system can have no angular velocity. D) the system can have no linear velocity. E) the angular momentum of the system must be continually decreasing. Ans: B Section: 10-3 Topic: Torque and Angular Momentum Type: Numerical 22. A disk-shaped grindstone of mass 3.0 kg and radius 8.0 cm is spinning at 600 rev/min. After the power is shut off, a man continues to sharpen his axe by holding it against the grindstone until it stops 10 s later. What was the stone's initial kinetic energy when the power was turned off? A) 19 J B) 3.8 ´ 10–3 J C) 4.8 ´ 10–5 J D) 1.9 ´ 10–3 J E) 2.4 ´ 10–2 J Ans: A Section: 10-3 Topic: Torque and Angular Momentum 23. Page 8 Type: Conceptual Chapter 10: Conservation of Angular Momentum Ans: A spinning bicycle wheel is supported as shown by a line fastened to one end of its axle. The resultant torque acting on the wheel lies along which of the following axes? A) x B) y C) –y D) z E) –z E Section: 10-3 Topic: Torque and Angular Momentum Type: Conceptual 24. The angular momentum vector for a spinning wheel lies along its axle and is pointed east. To make this vector point south, it is necessary to exert a force on the east end of the axle in which direction? A) up B) down C) north D) south E) east Ans: A Section: 10-3 Topic: Torque and Angular Momentum Type: Conceptual 25. If the sum of the external torques on a system is zero, there is A) a change in the system's moment of inertia. B) no change in the system's moment of inertia. C) a change in the system's angular momentum. D) no change in the system's angular momentum. E) a precessional angular velocity. Ans: D Section: 10-3 Topic: Torque and Angular Momentum Type: Conceptual 26. If the sum of the torques on a body about a fixed axis is not zero, the body most certainly A) experiences translational acceleration. D) experioences rotational inertia. B) experiences angular acceleration. E) remains in equilibrium. C) experiences precession. Ans: B Section: 10-4 Topic: Conservation of Angular Momentum Type: Conceptual 27. A woman sits on a spinning piano stool with her arms folded. When she extends her arms, which of the following occurs? A) She increases her moment of inertia, thereby increasing her angular speed. Page 9 Chapter 10: Conservation of Angular Momentum B) C) D) E) Ans: She increases her moment of inertia, thereby decreasing her angular speed. She decreases her moment of inertia, thereby increasing her angular speed. She decreases her moment of inertia, thereby decreasing her angular speed. Both her moment of inertia and her angular speed remain constant. B Section: 10-4 Topic: Conservation of Angular Momentum Type: Conceptual 28. A man turns with an angular velocity on a rotating table, holding two equal masses at arms' length. If he drops the two masses without moving his arms, his angular velocity A) decreases. B) remains the same. C) increases. D) increases as the angular velocity of the masses decreases. E) decreases as the angular velocity of the masses increases. Ans: B Section: 10-4 Topic: Conservation of Angular Momentum Type: Numerical 29. A man stands on the center of a platform that is rotating on frictionless bearings at a speed of 1.00 rad/s. Originally his arms are outstretched and he holds a 4.54-kg mass in each hand. He then pulls the weights in toward his body. Assume the moment of inertia of the man, including his arms, to remain constant at 5.42 kg · m2. If the original distance of the weights from the axis is 0.914 m and their final distance is 0.305 m, the final angular velocity is A) 1.14 rad/s B) 1.27 rad/s C) 1.58 rad/s D) 2.08 rad/s E) 7.70 rad/s Ans: D Section: 10-4 Topic: Conservation of Angular Momentum Type: Numerical 30. Ans: A woman sits on a stool that can turn friction-free about its vertical axis. She is handed r a spinning bicycle wheel that has angular momentum L 0 and she turns it over (that is, through 180º). She thereby acquires an angular momentum of magnitude r r r r A) 0 B) ½ L 0 C) L 0 D) 2 L 0 E) 4 L 0 D Section: 10-4 Topic: Conservation of Angular Momentum Type: Conceptual 31. Two identical cylindrical disks have a common axis. Initially one of the disks is Page 10 Chapter 10: Conservation of Angular Momentum A) B) C) D) E) Ans: spinning. When the two disks are brought into contact, they stick together. Which of the following is true? The total kinetic energy and the total angular momentum are unchanged from their initial values. Both the total kinetic energy and the total angular momentum are reduced to half of their original values. The total angular momentum is unchanged, but the total kinetic energy is reduced to half its original value. The total angular momentum is reduced to half its original value, but the total kinetic energy is unchanged. The total angular momentum is unchanged, and the total kinetic energy is reduced to one-quarter of its original value. E Section: 10-4 Topic: Conservation of Angular Momentum Type: Numerical 32. A wheel is rotating freely with an angular speed of 20 rad/s on a shaft whose moment of inertia is negligible. A second identical wheel, initially at rest, is suddenly coupled to the same shaft. The angular speed of the coupled wheels is A) 10 rad/s B) 14 rad/s C) 20 rad/s D) 28 rad/s E) 40 rad/s Ans: A Section: 10-4 Topic: Conservation of Angular Momentum Type: Numerical 33. A merry-go-round with a moment of inertia of 6.78 ´ 103 kg · m2 is coasting at 2.20 rad/s. When a 72.6-kg man steps onto the rim, the angular velocity decreases to 2.0 rad/s. The radius of the merry-go-round is A) 3.06 m B) 3.66 m C) 4.27 m D) 4.88 m E) 5.49 m Ans: A Section: 10-4 Topic: Conservation of Angular Momentum Type: Numerical 34. In a playground there is a small merry-go-round of radius 1.25 m and mass 175 kg. Assume the merry-go-round to be a uniform disk. A child of mass 45 kg runs at a speed of 3.0 m/s tangent to the rim of the merry-go-round (initially at rest) and jumps on. If we neglect friction, what is the angular speed of the merry-go-round after the child has jumped on and is standing at its outer rim? A) 0.82 rad/s B) 2.4 rad/s C) 0.49 rad/s D) 1.2 rad/s E) 0.41 rad/s Ans: A Section: 10-4 Topic: Conservation of Angular Momentum Type: Numerical 35. A hoop rotates about an axis through its center with an angular velocity of 40.0 rad/s. If the rotational kinetic energy of the hoop is 400 J, its angular momentum is A) 800 kg · m2/s D) 20 kg · m2/s B) 400 kg · m2/s E) 5 kg · m2/s 2 C) 200 kg · m /s Ans: D Section: 10-4 Topic: Conservation of Angular Momentum Page 11 Type: Conceptual Chapter 10: Conservation of Angular Momentum 36. Which of the following are required for the total momentum (both angular and linear) of a system to be conserved? A) B) C) Ans: 1. The sum of the external torques acting on the system must be zero. 2. The sum of the external forces acting on the system must be zero. 3. The total kinetic energy must remain constant. 4. There can be no external torques or forces acting on the system. 5. There can be no internal torques or forces acting on the system. 1 and 2 D) 1, 2, 3, and 4 1, 2, and 3 E) 1, 2, 3, 4, and 5 1, 2, and 4 A Section: 10-4 Topic: Conservation of Angular Momentum Type: Conceptual 37. A) B) C) D) E) Ans: A man is walking north carrying a suitcase that contains a spinning gyroscope mounted on an axle attached to the front and back of the case. The angular velocity of the gyroscope points north. The man now begins to turn to walk east. As a result, the front end of the suitcase resists his attempt to turn and tries to remain pointed north. fights his attempt to turn and pulls to the west. rises upward. dips downward. does nothing whatever unusual. D Section: 10-4 Topic: Conservation of Angular Momentum Type: Conceptual 38. Two wheels with identical moments of inertia are rotating about the same axle. The first is rotating clockwise at 2.0 rad/s, and the second is rotating counterclockwise at 6.0 rad/s. If the two wheels are brought into contact so that they rotate together, their final angular velocity will be A) 2.0 rad/s, counterclockwise. D) 5.0 rad/s, clockwise. B) 3.0 rad/s, clockwise. E) 6.0 rad/s, clockwise. C) 4.0 rad/s, counterclockwise. Ans: C Page 12 Chapter 10: Conservation of Angular Momentum Section: 10-4 Topic: Conservation of Angular Momentum Type: Numerical 39. A wheel of moment of inertia 0.136 kg · m2 is spinning with an angular speed of 5000 rad/s. A torque is applied about an axis perpendicular to the spin axis. If the applied torque has a magnitude of 67.8 N · m, the angular velocity of precession will be A) 1.00 rad/s B) 0.100 rad/s C) 10.0 rad/s D) 100 rad/s E) 1000 rad/s Ans: B Section: 10-4 Topic: Conservation of Angular Momentum Type: Conceptual 40. A certain airplane engine rotates counterclockwise when viewed from aft (that is, from the back of the airplane). When the plane turns to the left, A) the engine makes it turn faster than when it turns to the right. B) the engine makes it turn slower than when it turns to the right. C) it tends to dive. D) it tends to climb. E) the engine has no effect on the turn. Ans: C Section: 10-4 Topic: Conservation of Angular Momentum Type: Conceptual 41. A) B) C) Ans: A solid cylinder is spinning counterclockwise about a longitudinal axis when a net torque t is applied, as shown. The cylinder speeds up. D) precesses about a horizontal axis. slows down. E) does none of these. precesses about a vertical axis. A Section: 10-4 Topic: Conservation of Angular Momentum 42. Page 13 Type: Conceptual Chapter 10: Conservation of Angular Momentum Ans: A wheel is set spinning and then is hung by a rope placed at one end of the axle. The precession vector of the spinning wheel points in the direction of A) z B) –y C) –z D) –x E) y C Section: 10-4 Topic: Conservation of Angular Momentum Type: Conceptual 43. The propeller of a motorboat turns clockwise relative to a water skier being towed by the boat. As the boat makes a sharp turn to the left, gyroscopic action tends to A) cause the front of the boat to rise. B) cause the front of the boat to dip. C) cause the boat to tip to the left. D) cause the boat to tip to the right. E) keep the boat headed in its original direction. Ans: A Section: 10-4 Topic: Conservation of Angular Momentum Type: Conceptual 44. Ans: A wheel is rotating in the direction indicated. If you pull down on the end of the axle nearest you, that end of the axle tends to move A) up. B) to the right. C) down. D) to the left. E) directly inward. D Section: 10-4 Topic: Conservation of Angular Momentum Page 14 Type: Conceptual Chapter 10: Conservation of Angular Momentum 45. A) B) C) Ans: A gyroscopic wheel spins clockwise as shown. The set of vectors that correctly r r describes ther directions of the torque t , angular momentum L , and angular velocity of precession w p, is r r r r r r D) t (–y); L (–z); w (–x) t (+z); L (–x); w (+y) r r r r r r E) t (+y); L (–x); w (–z) t (–z); L (+x); w (–y) r r r t (+y); L (–x); w (+z) E Section: 10-4 Topic: Conservation of Angular Momentum Type: Conceptual 46. A gyroscopic toy is spinning as shown. The torque t, angular momentum of the wheel r L , and angular precession velocity wp are in which directions? 1 2 3 t –z –x –x r L wp y –y –y Page 15 x –z z Chapter 10: Conservation of Angular Momentum 4 5 A) 1 Ans: B) 2 C) 3 x x D) 4 y y E) 5 z –z E Section: 10-4 Topic: Conservation of Angular Momentum Type: Numerical 47. A spinning bicycle wheel with a loaded rim (essentially a hoop) is supported by a line at one end of its axle. The radius of the wheel is 0.305 m, and the wheel has a mass of 3.63 kg. It is spinning at 80.0 rad/s, and the center of mass is 15.2 cm from the point of support. The angular velocity of precession is A) 0.0125 rad/s D) 0.100 rad/s B) 0.0318 rad/s E) 0.625 rad/s C) 0.200 rad/s Ans: C Section: 10-4 Topic: Conservation of Angular Momentum Type: Conceptual 48. Ans: The figure shows vectors representing the angular velocity of precession wp and the spin velocity ws. The associated torque vector points along which of the axes? A) –x B) y C) z D) –z E) None of these is correct. B Section: 10-4 Status: New to 5th edition Topic: Conservation of Angular Momentum Type: Numerical 49. Consider the Earth as a uniform sphere of diameter 12.7 ´ 106 m, with a mass of the 5.98 ´ 1030 kg, and a rotational period of 1 day. If the Earth suddenly had a diameter of half this value without any loss of mass, calculate the new period of rotation. A) 0.5 days B) 2.0 days C) 0.25 days D) 4.0 days E) 1.4 days Ans: C Section: 10-4 Status: New to 5th edition Topic: Conservation of Angular Momentum Type: Numerical 50. If an object were to suddenly shrink and decrease its moment of inertia by a factor of 3, Page 16 Chapter 10: Conservation of Angular Momentum A) B) C) Ans: what is the difference in energy between the final and initial rotational kinetic energies? 3 times initial Krot D) 2 times initial Krot 9 times initial Krot E) zero, Krot remains the same 8 times initial Krot D Section: 10-5 Topic: Quantization of Angular Momentum Type: Factual 51. Spin-½ particles A) are called bosons. B) have spin angular momenta that can be changed by applying a net torque to them. C) can have angular momenta that change continuously from one value to another. D) can very accurately be thought of as spinning spheres. E) are described by none of the above. Ans: E Section: 10-5 Topic: Quantization of Angular Momentum Type: Conceptual 52. Which of the following statements is true? A) The angular momentum of a particle due to its motion is its orbital angular momentum. B) Spin-½ particles are called fermions. C) The fundamental unit of angular momentum is U. D) The units of U are J·s. E) All of these are correct. Ans: E Section: 10-5 Topic: Quantization of Angular Momentum Type: Conceptual 53. Which of the following statements is true? A) Stable matter consists of electrons, protons, and neutrons. B) Electrons, protons, and neutrons have an intrinsic angular momentum that is called spin. C) Bosons have zero spin or integral spin. D) The spin angular momentum of a particle is a fundamental property of the particle and as such cannot be changed. E) All of these are correct. Ans: E Section: 10-5 Topic: Quantization of Angular Momentum Type: Conceptual 54. Which of the following statements is not true? A) Stable matter consists of electrons, protons, and neutrons. B) Electrons, protons, and neutrons have an intrinsic angular momentum that is called spin. C) Bosons have zero spin or integral spin. D) An electron is well known to have a finite size. E) The spin angular momentum of a particle is a fundamental property of the particle and as such cannot be changed. Ans: D Page 17 Chapter 10: Conservation of Angular Momentum Section: 10-5 Topic: Quantization of Angular Momentum Type: Factual 55. Ans: The energy-level diagram that most closely represents a rotating molecule with constant moment of inertia is A) 1 B) 2 C) 3 D) 4 E) 5 A Section: 10-5 Status: New to 5th edition Topic: Quantization of Angular Momentum Type: Numerical 56. In a rotating atomic nucleus shaped like a football the quantized rotational energy levels have angular momentum values l = 0, 2, 4, 6, 8, etc. Given that the energy of each level is equal to a constant times l ´ (l + 1), calculate the ratio of the l = 4 to l = 2 energy levels. A) 4.0 B) 0.3 C) 2.0 D) 0.5 E) 3.3 Ans: E Section: 10-5 Status: New to 5th edition Topic: Quantization of Angular Momentum Type: Numerical 57. In a rotating atomic nucleus shaped like a football the quantized energy levels have rotational quantum number values of 0, 2, 4, etc. If the energies of the first three states are 0 keV, 138 keV, and 460 keV, calculate the moment of inertia of the nucleus. A) D) 1.43 ´ 10-17 eV•s2 9.41 ´ 10-36 keV•s2 B) E) 9.41 ´ 10-33 keV•s2 9.41 ´ 10-36 eV•s2 C) 1.43 ´ 10-17 keV•s2 Ans: B Section: 10-5 Status: New to 5th edition Topic: Quantization of Angular Momentum Type: Numerical 58. For a rotating molecule with a constant moment of inertia what is the energy difference between the l = 5 and l = 7 energy levels? A) 2 E0r B) 24 E0r Page 18 Chapter 10: Conservation of Angular Momentum C) D) Ans: 86 E0r 26 E0r F) 35 E0r D Page 19