A 2.0-kg block travels around a 0.50 m radius circle with an

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1 1 A 3-kg particle moves at a constant speed of 4 m/s around a circle of radius 5 m. (a) What is its angular momentum about the center of the circle? (b) What is its moment of inertia about an axis through the center of the circle and perpendicular to the plane of the motion? (c) What is the angular velocity of the particle? 5 A 2.0 kg block travels around a 0.50 m radius circle with an angular speed of 12 rad/s. The circle is parallel to the xy plane and is centered on the z axis, 0.75 m from the origin. What is the magnitude of the angular momentum around the origin? 10.8 kg.cm 2 /s (a) 60 kg.m 2 /s (b) 75 kg.m 2 /s (c).8 rad/s 6 A 2.0-kg block travels around a 0.50 m radius circle with an angular speed of 12 rad/s. The circle is parallel to the xy plane and is centered on the z axis, a distance of 0.75 m from the origin. What is the z component of the angular momentum around the origin? 2 Find the angular momentum due to the daily rotation of the earth about its axis. Data: M e = 6 x kg, R e = 6.4 x 10 6 m,? = 1/86,400 rev/s. Assume the earth to be a uniform sphere. 6.0 kg.cm 2 /s x10 33 kg.m 2 /s A 2.0-kg block travels around a 0.50-m radius circle with an angular velocity of 12 rad/s. What is its angular momentum about the center of the circle? 7 A 2.0-kg block travels around a 0.50 m radius circle with an angular speed of 12 rad/s. The circle is parallel to the xy plane and is centered on the z axis, 0.75 m from the origin. What is the magnitude of the component in the xy plane of the angular momentum around the origin? 9.0 kg.m 2 /s 6.0 kg.m 2 /s 4 A 15 g paper clip is attached to the rim of a phonograph record with a radius of 30 cm, spinning at 3.5 rad/s. What is its angular momentum in kg.cm 2 /s? 8 The angular momentum of a flywheel about its axis is 925 kg. m 2 /s. Its moment of inertia about the same axis is 2.50 kg.m 2. What is its angular velocity? 370 rad/s 4.7 x 10-3

2 9 In part a of the figure, particles 1 and 2 move around point 0 in opposite directions, in circles with radii 2 m and 4 m. In part b, particles 3 and 4 travel in the same direction, along straight lines at perpendicular distances of 4 m and 2 m from point 0. Particle 5 moves directly away from 0. All five particles have the same mass and the same constant speed. (a) Rank the particles according to the magnitudes of their angular momentum about point 0, greatest first. (b) Which particles have negative angular momentum about point 0? 11 In the figure, a disk, a hoop, and a solid sphere are made to spin about fixed central axes (like a top) by means of strings wrapped around them, with the strings producing the same constant tangential force F on all three objects. The three objects have the same mass and radius, and they are initially stationary. (a) Rank the objects according to their angular momentum about their central axes, when the strings have been pulled for a certain time t. (b) Rank the objects according to their angular speed, greatest first, when the strings have been pulled for a certain time t. 10 (a) 1 and 3 tie, then 2 and 4 tie, then 5 (zero); (b) 2 and 3 The figure shows an overhead view of two particles moving at constant momentum along horizontal paths. Particle 1, with momentum magnitude p 1 = 5.0 kg.m/s, has position vector r 1 and will pass 2.0 m from point 0. Particle 2, with momentum magnitude p 2 = 2.0 kg.m/s, has position vector r 2 and will pass 4.0 m from point 0. What are the magnitude and direction of the net angular momentum L about point 0 of the two-particle system? 12 (a) all tie; (b) sphere, disk, hoop The figure shows three particles of the same mass and the same constant speed moving as indicated by the velocity vectors. Points a, b, c, and d form a square, with point e at the center. Rank the points according to the magnitude of the net angular momentum of the three-particle system when measured about the points, greatest first. Note Look at total angular momentum at each point a, then b and c tie, E then d (zero) (a) 2 kg.m 2 /s (b) out of the page

3 13 The figure shows an overhead view of a rectangular slab that can spin like a merry-go-round about its center at 0. Also shown are seven paths along which wads of bubble gum can be thrown (all with the same speed and mass) to stick onto the stationary slab. (a) Rank the paths according to the angular speed that the slab (and gum) will have after the gum sticks, greatest first. (b) For which paths will the angular momentum of the slab (and gum) about 0 be negative from the view of the figue? 15 The figure shows a rigid structure consisting of a circular hoop of radius R and mass m, and a square made of four thin bars, each of length R and mass m. The rigid structure rotates at a constant speed about a vertical axis, with a period of rotation of 2.5 s. (a) Assuming R = 0.50 m and m = 2.0 kg, calculate the structure's rotational inertia about the axis of rotation and (b) Assuming R = 0.50 m and m = 2.0 kg, calculate its angular momentum about that axis. (a) 4, 6, 7, 1, then 2, 3, and 5 tied (zero) (b) 4, 1, 7 14 In the instant of the figure, two particles move in an xy plane. Particle P 1 has mass 6.5 kg and speed v 1 = 2.2 m/s, and it is at distance d 1 = 1.5 m from point 0. Particle P 2 has mass 3.1 kg and speed v 2 = 3.6 m/s, and it is at distance d 2 = 2.8 m from point 0. (a) What is the magnitude of the net angular momentum of the two particles about 0? (b) What is the direction of the net angular momentum of the two particles about 0? (a) 1.6 kg.m 2 /s (b) 4.0 kg.m 2 /s 16 A uniform rod rotates in a horizontal plane about a vertical axis through one end. The rod is 6.00 m long, weighs 10.0 N, and rotates at 240 rev/min. (a) Calculate its rotational inertia about the axis of rotation. (b) Calculate the magnitude of its angular momentum about that axis. (a) 12.2 kg.m 2 (b) 308 kg.m 2 /s (a) 9.8 kg.m 2 /s; (b) positive z direction 17 Look up information about planet rotation in your text. (a) Use the data to compute the total of the magnitudes of the angular momenta of all the planets due to their revolution about the Sun. (b) What fraction of this total is associated with the planet Jupiter? (a) 3.14 x kg.m 2 /s (b) 0.614

4 18 For an 84 kg person standing at the equator, what is the magnitude of the angular momentum about Earth's center due to Earth's rotation? 21 Why do quarterbacks spiral a football when they throw a pass? 19 In a common physics lecture demonstration, the lecturer sits on a stool that can rotate freely about a vertical axis on lowfriction bearings. The lecturer holds with extended arms two dumbbells, each of mass m, and kicks the floor so as to achieve an initial angular speed? 1. The lecturer then pulls in the dumbbells, so that their distances from the rotation axis decrease from the initial value R 1 to the final value R 2. Determine the final angular speed? 2, assuming that the moment of inertia about the rotation axis of the lecturer's body plus the stool does not change in the process. Then evaluate K 1 and K 2, the initial and final kinetic energies in the system. What is the source of the additional kinetic energy? The spiraling football conserves angular momentum. Thus, the direction of the axis of rotation stays fixed, preventing the ball from tumbling and being more subjected to air resistance and harder to control. Compare this to an arrow, which does not spin. The shaft of the arrow always remains tangent to the trajectory during the flight, so that the point implants in the ground upon landing. The football does not do this. The direction of the rotation axis stays fixed and does not remain tangent to the trajectory. 22 A student volunteer is sitting stationary on a piano stool with her feet off the floor. The stool can turn freely on its axle. The volunteer is handed a non-rotating bicycle wheel, which has handles on the axle. Holding the axle vertically with one hand, she grasps the rim of the wheel with the other and spins the wheel clockwise (as seen from above). She now grasps the ends of the vertical axle and turns the wheel until the axle is horizontal. What happens? See Schaums 20 A student volunteer is sitting stationary on a piano stool with her feet off the floor. The stool can turn freely on its axle. The volunteer is handed a non-rotating bicycle wheel, which has handles on the axle. Holding the axle vertically with one hand, she grasps the rim of the wheel with the other and spins the wheel clockwise (as seen from above). What happens to the volunteer as she does this? When the student volunteer was first handed the wheel she is spinning because the total angular momentum of the system is zero The angular momentum of the wheel is now horizontal. The volunteer's vertical component of angular momentum must now be zero, so she stops spinning. 23 Why do automobile engines have flywheels? Since the axle of the piano stool is frictionless, there are no vertical torque's extended on the stool-volunteer system, so the vertical component of angular momentum is conserved. (There is horizontal torques; these result from forces that the floor exerts on the base of the stool.) The initial angular momentum is zero, so the final angular momentum must also be zero. Therefor, the volunteer spins counterclockwise. The flywheel in an engine uses the principle of inertia to smooth out the discontinuous impulses from the engine s individual cylinders. Without the flywheel, the crankshaft would jerk every time a spark plug fired. The same principle is used in a potter s wheel.

5 24 A student volunteer is sitting stationary on a piano stool with her feet off the floor. The stool can turn freely on its axle. The volunteer is handed a non-rotating bicycle wheel, which has handles on the axle. Holding the axle vertically with one hand, she grasps the rim of the wheel with the other and spins the wheel clockwise (as seen from above). She now grasps the ends of the vertical axle and turns the wheel until the axle is horizontal. Next she gives the rotating wheel to the instructor, who turns the axle until it is vertical with the wheel rotating clockwise, as seen from above. The instructor now hands the wheel back to the volunteer. What happens? 26 A student volunteer is sitting stationary on a piano stool with her feet off the floor. The stool can turn freely on its axle. The volunteer is handed a non-rotating bicycle wheel, which has handles on the axle. Holding the axle vertically with one hand, she grasps the rim of the wheel with the other and spins the wheel clockwise (as seen from above). She now grasps the ends of the vertical axle and turns the wheel until the axle is horizontal. Next she gives the rotating wheel to the instructor, who turns the axle until it is vertical with the wheel rotating clockwise, as seen from above. The instructor now hands the wheel back to the volunteer? The volunteer grasps the ends of the axle and turns the axle until it is horizontal. What happens now? When the wheel is handed back to the volunteer, the system of the wheel and volunteer has a downward vertical angular momentum, all contributed by the wheel. The volunteer remains stationary. Since the vertical component of the total angular momentum must not change, the volunteer must rotate clockwise. 25 Why are divers able to do more rotations in the tuck position than in the layout position? 27 The Voyager 2 spacecraft had to have thruster jets turned on when its high-speed tape recorder was activated. Why? As a diver leaves the platform or diving board, he or she has angular momentum from the torque exerted on the body by pushing off. By going into the tuck position, where the knees are brought close to the chest, the moment of inertia of the diver is significantly reduced. In order to conserve angular momentum, the angular velocity of the diver must increase. Thus during the time determined by the height of the platform and the translational kinematics of falling to the water, the diver can do more rotations than in the layout position, where the moment of inertia is large and the angular velocity correspondingly small. When the tape recorder was turned on, part of the spacecraft (the tape recording mechanism) was set into a rotational motion. By Newton s third Law, fi the spacecraft exerts a torque to start the tape mechanism rotating, the mechanism will exert an equal and opposite torque on the spacecraft. Thus, the spacecraft will start rotating in the opposite direction

6 28 A student volunteer is sitting stationary on a piano stool with her feet off the floor. The stool can turn freely on its axle. The volunteer is handed a non-rotating bicycle wheel, which has handles on the axle. Holding the axle vertically with one hand, she grasps the rim of the wheel with the other and spins the wheel clockwise (as seen from above). She now grasps the ends of the vertical axle and turns the wheel until the axle is horizontal. Next she gives the rotating wheel to the instructor, who turns the axle until it is vertical with the wheel rotating clockwise, as seen from above. The instructor now hands the wheel back to the volunteer? The volunteer grasps the ends of the axle and turns the axle until it is horizontal. She continues turning the axle until it is vertical but with the wheel rotating counterclockwise as viewed from above. What is the result? 31 A man stands on a frictionless platform that is rotating with an angular speed of 2.0 rev/s. His arms are outstretched, and he holds a heavy weight in each hand. The moment of inertia of the man, the extended weights, and the platform is 5 kg.m 2. When the man pulls the weights inward toward his body the moment of inertia decreases to 2 kg.m. (a) What is the resulting angular speed of the platform? (b) What is the change in kinetic energy of the system? (c) Where did this increase in energy come from? (a) 10π rad/sec 31.4 rad/s (b) 394 kg.m 2 /s 2 (c) 907 kg.m 2 /s 2a 29 The wheel's angular momentum is now upward. The volunteer must therefore have a downward vertical angular momentum to keep the total angular momentum pointing down. She must therefore spin clockwise (her spin rate is twice as fast as in part (d).) A large wooden wheel of radius R and moment of inertia I is mounted on an axle so as to rotate freely. a bullet of mass m and speed v is shot tangential to the wheel and strikes its edge, lodging in the rim. If the wheel were originally at rest, what would be its rotational rate just after collision. 32 A man of mass m stands at the edge of a rotating circular turntable. The turntable has a radius R and a moment of inertia I, and it rotates without friction. The angular velocity about the vertical axis through the center of the turntable is 8 rad/s. The man walks radially inward. What is the final angular velocity of the system? Assume that mr 2 = 3I and that the man has a moment of inertia of I/10 when he is standing at the center Rad/s 30 w = mvr/(mr 2 + I) A 20 kg boy stands on and near the edge of a small merry-goround with the system at rest. The system's total moment of inertia about the center is 120 kg.m 2. The boy, at a radius of 2.0 m, jumps off the merry-go-round in a tangential direction with a speed of 1.5 m/s. How fast will the merry-go round be rotating after the boy leaves it? 33 A professor stands at the center of a turntable that can rotate without friction. She begins to rotate a heavy ball on the end of a 0.8 m chain about her head. The ball has a mass of 2 kg, and it makes one revolution every 3 s. The professor and platform have a moment of inertia of 0.5 kg.m 2. (a) What is the angular speed of the professor? (b) What is the total kinetic energy of the ball, professor, and platform? -1.5 rad/s (a) rad/s (b) 9.91 J

7 34 The sun's radius is 6.96 x 10 8 m and it rotates with a period of 25.3 d. Estimate its new period of rotation if it collapses with no loss of mass to become a neutron star of radius 5 km x 10-4 s 38 A merry-go-round in a park consists of an essentially uniform 200-kg solid disk rotating about a vertical axis. The radius of the disk is 6.0 m and a 100 kg man is standing on its outer edge when it is rotating at a speed of 0.20 rev/s. (a) How fast will the disk be rotating if the man walks 3.0 m in toward the center along a radius? (b) What will happen if the man drops off the edge? (c) Is it allowable to assume that the man acts like a point particle? 35 The polar ice caps contain about 2.3 x kg of ice. This mass contributes essentially nothing to the moment of inertia of the earth because it is located at the poles, close to the axis of rotation. Estimate the change in the length of the day that would be expected if the polar ice caps were to melt and the water were distributed uniformly over the surface of the earth rev/s If the man falls off, rather than pushes off the disk rotates as before and the angular velocity does not change. So long as the man's moment of inertia about a vertical axis passing through his center of mass is small compared with his mass times the distance of his center of mass to the vertical axis of the merry-go-round, squared, we can treat him as a point mass. This follows from the parallel-axis theorem. 36 A man sits on a stool on a frictionless turntable holding a pair of dumbbells at a distance of 2.5 m from the axis of rotation. He is given an angular velocity of 2 rad/s, after which he pulls the dumbbells in until they are.5 m from the axis. The moment of inertia of the man plus stool plus turntable is 6 kg. m 2 and may be considered constant. The dumbbells may be considered point masses of 3.5 kg each. (a) What is the initial angular momentum of the system? (b) What is the final angular velocity of the system? 39 A woman stands over the center of a horizontal platform that is rotating freely with speed 2.0 rev/s about a vertical axis through the center of the platform and straight up through the woman. She hold s two 5 kg masses in her hands close to her body. The combined moment of inertia of platform, woman, and masses is 1.2 kg.m 2. The woman now extends her hands so as to hold the masses far from her body. In so doing, she increases the moment of inertia of the system by 2.0 kg.m 2. (a) What is the final rotational speed of the platform? (b) Was the kinetic energy of the system changed during the process? Explain (a) (b) 37 An Ice skater spins with arms outstretched at 1.9 rev/s Her moment of inertia at this time is 1.33 kg.m 2. She pulls in her arms to increase her rate of spin. If her moment of inertia is 0.48 kg.m 2 after she pulls in her arms, what is her new rate of rotation? (a) 0.l75 rev/s (b) Because the Kinetic Energy is changed the system did work in extending the 5-kg masses, thereby decreasing the kinetic energy 5.26 rev/s rad/s

8 40 A man stands on a frictionless rotating platform which is rotating with an angular speed of 1.0 rev/s; his arms are outstretched and he holds a weight in each hand. With his hands in this position the total rotational inertia of the man, the weights and the platform is 6.0 kg.m 2. If by moving the weights the man decreases the rotational inertia to 2.0 kg.m 2. What is the resulting angular speed of the platform? 43 A wheel, mounted on a vertical shaft of negligible rotational inertia, is rotating at 500 rpm. Another identical (but not rotating) wheel is suddenly dropped onto the same shaft as shown. What will be the speed at which the resultant combination of the two wheels and shaft rotate? (a) 3 rev/s 41 Two skaters, each of mass 50 kg, approach each other along parallel paths separated by 3.0 m. They have equal and opposite velocities of 10 m/s The first skater carries a long light pole, 3.0 m long and the second skater grabs the end of it as he passes (Assume frictionless ice). (a) Describe quantitatively the motion of the skaters after they are connected by the pole. (b) By pulling on the pole, the skaters reduce their separation to 1.0 m. What is their motion then? rpms A playground merry-go-round has a radius of 3.0 m and a rotational inertia of 600 kg.m 2. It is initially spinning at 0.80 rad/s when a 20-kg child crawls from the center to the rim. When the child reaches the rim, what will be the angular velocity of the merry-go-round? (a) rad/s (b) 60 rad/s 1.06 rev/s 0.61 rad/s 42 Two disks are mounted on low-friction bearings on a common shaft. The first disc has rotational inertia I and is spinning with angular velocity?. The second disc has rotational inertia 2I and is spinning in the same direction as the first disc with angular velocity 2? as shown. The two disks are slowly forced toward each other along the shaft until they couple. What is their common angular velocity? 45 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, what will be their final angular velocity? 2.0 rad/s. counterclockwise 5/3?

9 46 A man stands on the center of a platform that is rotating on a frictionless bearings at a speed of 1.00 rad/s. Originally his arms are outstreched 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.m 2. The original distance of the weights from the axis is m and their final distance is m. What is the final angular velocity of the system? 50 The lower disk in the figure to the left is coasting with an angular speed? 1. The combined moment of inertia of it and its axle is I 1. A second disk of moment of inertia I 2 is dropped onto the first and ends up rotating with it. (a) Find the angular velocity of the combination if the original angular velocity of the upper disk waszero (b) Find the angular velocity of the combination if the original angular velocity of the upper disk was? 2 in the same direction as? 1, (c) Find the angular velocity of the combination if the original angular velocity of the upper disk was? 2 in the opposite direction as? rad/s 47 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. What is the angular speed of the coupled wheels? 10 rad/sec 48 A merry-go-round with a moment of inertia of 6.78 x 10 3 kg.m 2 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. What is the radius of the merry-go-round? (a)? = (I 1 ω 1 )/I 1 + I 2 ) (b)? = (I 1 ω 1 I 2 ω 2 )/(I 1 + I 2 ) (c)? = (I 1 ω 1 - I 2 ω 2 )/(I 1 + I 2 ) m It has been surmised that the sun was formed in the gravitational collapse of a dust cloud, which filled the space now occupied by the solar system and beyond. If we assume that the original cloud was a uniform sphere of radius R 0 with an average velocity of wo, how fast should the cloud be rotating now? For the present purposes, ignore the small mass of the planets and assume the sun to be a uniform sphere of radius R s. 51 As shown in the figure to the left, sand drops onto a disk rotating freely about an axis. The moment of inertia of the disk about this axis is I, and its original rotational rate was? 0, What is its rate of rotation after a mass M of sand has accumulated on the disk at radius b?? = (R o /R s ) 2 ω o 2? =? o /(1 + Mb 2 /l)? = I? b /(I + Mb 2 ) or

10 52 A rhinoceros beetle rides the rim of a small disk that rotates like a merry-go-round. The beetle crawls toward the center of the disk. (a) Does the rotational inertia (relative to the central axis) increase, decrease, or remain the same for the beetle-disk system? (b) Does the angular momentum (relative to the central axis) increase, decrease, or remain the same for the beetle-disk system? (c) Does the angular speed (relative to the central axis) increase, decrease, or remain the same for the beetle-disk system? 54 The figure shows a particle moving at constant velocity, v and five points with their xy coordinates. Rank the points according to the magnitude of the angular momentum of the particle measured about them, greatest first. (a) decreases; (b) same; (c) increases b, then c and d tie, then a and e tie (zero) 53 Figure a shows a student, again sitting on a stool that can rotate freely about a vertical axis. The student, initially at rest, is holding a bicycle wheel whose rim is loaded with lead and whose rotational inertia I wh about its central axis is 1.2 kg.m 2. The wheel is rotating at an angular speed? wh of 3.9 rev/s; as seen from overhead, the rotation is counterclockwise. The axis of the wheel is vertical, and the angular momentum L wh of the wheel points vertically upward. The student now inverts the wheel (figure b) so that, as seen from overhead, it is rotating clockwise. Its angular momentum is now -L wh. The inversion results in the student, the stool, and the wheel's center rotating together as a composite rigid body about the stool's rotation axis, with rotational inertia I b = 6.8 kg.m 2. (The fact that the wheel is also rotating about its center does not affect the mass distribution of this composite body; thus, I b has the same value whether or not the wheel rotates.) With what angular speed? b and in what direction does the composite body rotate after the inversion of the wheel? 55 A rhinoceros beetle rides the rim of a horizontal disk rotating counterclockwise like a merry-go-round. The beetle then walks along the rim in the direction of the rotation. (a) Will the magnitude the angular momentum of the beetledisk system, (measured about the rotation axis) increase, decrease, or remain the same (the disk is still rotating in the counterclockwise direction)? (b) Will the magnitude of the angular momentum and angular velocity of the beetle (measured about the rotation axis) increase, decrease, or remain the same (the disk is still rotating in the counterclockwise direction)? (c) Will the magnitude of the angular momentum and angular velocity of the disk (measured about the rotation axis) increase, decrease, or remain the same (the disk is still rotating in the counterclockwise direction)? (d) What are your answers if the beetle walks in the direction opposite the rotation? (a) same (b) increase (c) decrease (d) same, decrease, increase 1/4 rev/s

11 56 The rotor of an electric motor has rotational inertia I m = 2.0 x 10-3 kg.m 2 about its central axis. The motor is used to change the orientation of the space probe in which it is mounted. The motor axis is mounted along the central axis of the probe; the probe has rotational inertia I p = 12 kg.m 2 about this axis. Calculate the number of revolutions of the rotor required to turn the probe through 30 0 about its central axis. 59 A track is mounted on a large wheel that is free to turn with negligible friction about a vertical axis (see the figure). A toy train of mass m is placed on the track and, with the system initially at rest, the train's electrical power is turned on. The train reaches speed 0.15 m/s with respect to the track. What is the angular speed of the wheel if its mass is 1.1m and its radius is 0.43 m? (Treat the wheel as a hoop, and neglect the mass of the spokes and hub.) 5.0 x 10 2 rev 57 Two disks are mounted (like a merry-go-round) 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 rotational inertia 3.30 kg.m 2 about its central axis, is set spinning counterclockwise at 450 rev/min. The second disk, with rotational inertia 6.60 kg.m 2 about its central axis, is set spinning counterclockwise at 900 rev/min. They then couple together. (a) What is their angular speed after coupling? If instead the second disk is set spinning clockwise at 900 rev/min, what is their (b) If instead the second disk is set spinning clockwise at 900 rev/min, what is their angular speed? (c) If instead the second disk is set spinning clockwise at 900 rev/min, what is their direction of rotation after they couple together? (a) 750 rev/min; (b) 450 rev/min; (c) clock-wise rad/s The rotational inertia of a collapsing spinning star drops to 1/3 its initial value. What is the ratio of the new rotational kinetic energy to the initial rotational kinetic energy? 3 A horizontal vinyl record of mass 0.10 kg and radius 0.10 m rotates freely about a vertical axis through its center with an angular speed of 4.7 rad/s. The rotational inertia of the record about its axis of rotatior is 5.0 x 10-4 kg.m 2. A wad of wet putty of mass kg drops vertically onto the record from above and sticks to the edge of the record. What is, the angular speed of the record immediately after the putty sticks to it? 58 A wheel is rotating freely at angular speed 800 rev/min on a shaft whose rotational inertia is negligible. A second wheel, initially at rest and with twice the rotational inertia of the first, is suddenly coupled to the same shaft. (a) What is the angular speed of the resultant combination of the shaft and two wheels? (b) What fraction of the original rotational kinetic energy is lost? (a) 267 rev/min (b) rad/s A horizontal platform in the shape of a circular disk rotates on a frictionless bearing about a vertical axle through the center of the disk. The platform has a mass of 150 kg, a radius of 2.0 m, and a rotational inertia of 300 kg.m 2 about the axis of rotation. A 60 kg student walks slowly from the rim of the platform toward the center. If the angular speed of the system is 1.5 rad/s when the student starts at the rim, what is the angular speed when she is 0.50 m from the center? 2.6 rad/s

12 63 If we are given r, p, and?, we can calculate the angular momentum of a particle from the equation r = mv sin?. Sometimes, however, we are given the components (x, y, z) of r and (v x, v y, v z ) of v instead. (a) Show that the components of L along the x, y, and z axes are then given by L x = m(yv z - zv y ), L y = m(zv x - xv z ), and L z = m(xv y - yv x ). (b) Show that if the particle moves only in the xy plane, the angular momentum vector has only a z component. 66 The velocity of a particle of mass m is v = 5i + 4j + 6k when at r = -2i + 4j + 6k. Find the angular momentum of the particle about the origin M(42j - 28k) 67 A 1000 kg airplane is flying in a straight line at 80 m/s, 1.3 km above the ground. What is the magnitude of its angular momentum with respect to an observer on the ground directly under the path of the plane. 64 A particle P with mass 2.0 kg has position r and velocity v as shown in the figure. It is acted on by the force F. All three vectors lie in a common plane. Presume that r = 3.0 m, v = 4.0 m/s and F = 2.0 N. (a) Compute the angular momentum of the particle. (b) Compute the torque acting on the particle. (c) What are the directions of these two vectors? x 10 8 kg.m 2 /s A 3.0 kg particle at x = 3.0 m, y = 8.0 m with a velocity v = 5i - 6 j m/s. It is acted on by a 7.0 N force in the negative x direction (a) What is the angular momentum of the particle? (b) What torque acts on the particle? (c) At what rat is the angular momentum of the particle changing with time? (a) -174 k Nm (b) -56 k Nm (c) -56 k Nm (a) 12 kg.m 2 /s out of the page (b) 3 N. m out of the page (c) Right hand rule Out of the Page 69 A 6-kg particle moves to the right at 4 m/s as shown. What is its angular momentum in kg.m 2 /s about the point O? 65 A rocket, of mass 106 kg, has a speed of 500 m/s in the horizontal direction. If its altitude (y) is 10 km and its horizontal distance (x) from the chosen origin is 10 km, what is its angular momentum with respect to this origin? -5.3 x 10 8 kg.m 2 /s. 144 kg.m 2 /s

13 70 Two objects are moving in the xy plane as shown. What is the magnitude of their total angular momentum (about the origin O) is (in kg.m 2 /s)? 74 A 2.0 kg stone is tied to a 0.50 m string and swung around a circle at a constant angular velocity of 12 rad/s. The circle is parallel to the xy plane and is centered on the z axis, 0.75 m from the origin. What is the magnitude of the torque about the origin? 0 75 A 2.0 kg block starts from rest on the positive x axis 3.0 m from the origin and thereafter has an acceleration given by a = 4.0î - 3.0, in m/s 2. What is the torque, relative to the origin, acting on it at the end of 2.0 s? 6 kg.m 2 /s A 2.0-kg block starts from rest on the positive x axis 3.0 m from the origin and thereafter has an acceleration given by a = 4.0 i j, in m/s 2. What is its angular momentum about the origin at the end of 2.0 s? (-36 kg.m 2 /s) k A playground merry-go-round has a radius of 3.0 m and a rotational inertia of 600 kg.m 2. When the merry-go-round is at rest, a 20-kg child runs at 5.0 m/s along a line tangent to the rim and jumps on. What is the angular velocity of the merrygo-round? N.m A 0.3 kg billiard ball of radius 3 cm is given a sharp blow by a cue stick. The applied impulse is horizontal and passes through the center of the ball. The initial velocity of the ball is 4 m/s. The coefficient of kinetic friction is 0.6. (a) For how many seconds does the ball slide before it begins to roll without slipping? (b) How far does it slide? (c) What is its velocity once it begins rolling without slipping? 0.38 rad/s 77 A single force acts on a particle P. Rank each of the orientations of the force shown to the left according to the magnitude of the time rate of change of the particle's angular momentum about the point O, least to greatest. 73 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? 1 and 2 tie, then 4, rad/s

14 78 A particle of mass 3 kg moves with velocity v = 2 m/s i along the line z = 0, Y = 4.3 m. (a) Find the angular momentum L relative to the origin when the particle is at x = 12 m, y = 4.3 m. (b) A force F = -3 N i is applied to the particle. Find the torque relative to the origin due to this force. 82 A particle travels in a circular path. (a) If its linear momentum p is doubled, how is its angular momentum affected? (b) If the radius of the circle is doubled but the speed is unchanged, how is the angular momentum of the particle affected? 79 A 2.0 kg stone is tied to a 0.50 m string and swung around a circle at a constant angular velocity of 12 rad/s. What is the torque on the stone about the center of the circle? 83 A particle P with mass 3.0 kg has position r and velocity v as shown in the figure. It is acted on by the force F. All three vectors lie in a common plane. Presume that r = 3.0 m, v = 6.0 m/s and F = 3.0 N. (a) Compute the angular momentum of the particle. (b) Compute the torque acting on the particle. (c) What are the directions of these two vectors? 0 80 If we are given P 1 and?, we can calculate the angular momentum of a particle from the following equations L = rp sin? Sometimes, however, we are given the components of (x,y,z) for r and (P x, P y, P z ) of p instead. (a) Show that the components of L along the x, y and z axis are given they by lx = yp z - zp y ly = zp x - xp z lz = xp y - yp x (b) Show that if the particle moves only in the x-y plane, the resultant angular moment has only a z component. (a) out of the page (b) out of the page (c) Right hand rule Out of the Page 81 A 3-kg particle moves at constant speed of 4 m/s along a straight line. (a) What is its angular momentum about a point 5 m from the line? (b) Describe qualitatively how its angular velocity about that point varies with time.

15 84 If we are given P 1 and q, we can calculate the angular momentum of a particle from the following equations L = rp sin? Sometimes, however, we are given the components of (x,y,z) for r and (P x, P y, P z ) of p instead. (a) Show that the components of l along the x, y and z axis are given they by lx = yp z - zp y 87 In the overhead view of the figure, four thin, uniform rods, each of mass M and length d = 0.50 m, are rigidly connected to a vertical axle to form a turnstile. The turnstile rotates clockwise about the axle, which is attached to a floor, with initial angular velocity? i = -2.0 rad/s. A mud ball of mass m = 1/3 M and initial speed v i = 12 m/s is thrown along the path shown and sticks to the end of one rod. What is the final angular velocity? f of the ball-turnstile system? ly = zp x - xp z lz = xp y - yp x (b) Show that if the particle moves only in the x-y plane, the resultant angular moment has only a z component. 85 A large wooden wheel of radius R and moment of inertia I is mounted on an axle so as to rotate freely. a bullet of mass m and speed v is shot tangential to the wheel and strikes its edge, lodging in the rim. If the wheel were originally at rest, what would be its rotational rate just after collision. w = mvr/(mr 2 + I) 88 A 2.0 kg particle-like object moves in a plane with velocity components v x = 30 m/s and v x = 60 m/s as it passes through the point with (x, y) coordinates of (3.0, -4.0) m. Just then, in unit-vector notation, what is its angular momentum relative to (a) Just then, in unit-vector notation, what is its angular momentum relative to the origin? (b) Just then, in unit-vector notation, what is its angular momentum relative to the point (-2.0, -2.0) m? 86 A 20 kg boy stands on and near the edge of a small merry-goround with the system at rest. The system's total moment of inertia about the center is 120 kg.m 2. The boy, at a radius of 2.0 m, jumps off the merry-go-round in a tangential direction with a speed of 1.5 m/s. How fast will the merry-go round be rotating after the boy leaves it? (a) (6.0 x 10 2 kg.m 2 /s) k (b) (7.2 x 10 2 kg.m 2 /s) k -1.5 rad/s 89 At one instant, force 4.0j N acts on a 0.25 kg object that has position vector r = (2.0i - 2.0k) m and velocity vector v = (-5.0i + 5.0k) m/s. About the origin and in unit vector notation, what is (a) About the origin and in unit vector notation, what is the object's angular momentum? (b) About the origin and in unit vector notation, what is the torque acting on the object? (a) 0 (b) (8.0 N.m) i + (8.0 N.m) k

16 90 A uniform thin rod of length 0.50 m and mass 4.0 kg can rotate in a horizontal plane about a vertical axis through its center. The rod is at rest when a 3.0 g bullet traveling in the rotation plane is fired into one end of the rod. As viewed from above, the bullet's path makes angle? = 60 0 with the rod (see the figure). If the bullet lodges in the rod and the angular velocity of the rod is 10 rad/s immediately after the collision, what is the bullet's speed just before impact? 92 In the figure, a 30 kg child stands on the edge of a stationary merry-go-round of mass 100 kg and radius 2.0 m. The rotational inertia of the merry-go-round about its rotation axis is 150 kg.m 2. The child catches a ball of mass 1.0 kg thrown by a friend. Just before the ball is caught, it has a horizontal velocity v of magnitude 12 m/s, at angle? = 37 0 with a line tangent to the outer edge of the merry-go-round, as shown. What is the angular speed of the merry-go-round just after the ball is caught? 1.3 x 10 3 m/s 91 The figure is an overhead view of a thin uniform rod of length m and mass M rotating horizontally at angular speed 20.0 rad/s about an axis through its center. A particle of mass M/3.00 initially attached to one end is ejected from the rod and travels along a path that is perpendicular to the rod at the instant of ejection. If the particle's speed v p is 6.00 m/s greater than the speed of the rod end just after ejection, what is the value of v p? rad/s A 2.50 kg particle that is moving horizontally over a floor with velocity (-3.00 m/s)j undergoes a completely inelastic collision with a 4.00 kg particle that is moving horizontally over the floor with velocity (4,50 m/s)i. The collision occurs at xy coordinates ( m, m). After the collision and in unit-vector notation, what is the angular momentum of the stuck-together particles with respect to the origin? (5.55 kg.m 2 /s) k 11.0 m/s

17 94 Two particles, each of mass 2.90 x 10-4 kg and speed 5.46 m/s, travel in opposite directions along parallel lines separated by 4.20 cm. (a) What is the magnitude L of the angular momentum of the two-particle system around a point midway between the two lines? (b) Does the value of L change if the point about which it is calculated is not midway between the lines? (c) If the direction of travel for one of the particles is reversed, what would be the answer to part (a)? (d) If the direction of travel for one of the particles is reversed, what would be the answer to part the answer to part (b)? 97 At one instant, a 0.80 kg particle is located at the position r = (2.0 m)i + (3.0 m)j. The linear momentum of the particle lies in the xy plane and has a magnitude of 2.4 kg.m/s and a direction of measured counterclockwise from the positive direction of x. What is the angular momentum of the particle about the origin, in unit-vector notation? (7.4 kg.m 2 /s) k (a) 6.65 x 10-5 kg.m 2 /s (b) no (c) 0 (d) yes 98 A 1200 kg airplane is flying in a straight line at 80 m/s, 1.3 km above the ground. What is the magnitude of its angular momentum with respect to a point on the ground directly under the path of the plane? 1.2 x 10 8 kg.m 2 /s 95 A particle of mass M = 0.25 kg is dropped from a point that is at height h = 1.80 m above the ground and horizontal distances = 0.45 m from an observation point 0, as shown in the figure. What is the magnitude of the angular momenum of the particle with respect to point 0 when the particle has fallen half the distance to the ground? 99 In a playground, there is a small merry-go-round of radius 1.20 m and mass 180 kg. Its radius of gyration is 91.0 cm. A child of mass 44.0 kg runs at a speed of 3.00 m/s along a path that is tangent to the rim of the initially stationary merry-go-round and then jumps on. Neglect friction between the bearings and the shaft of the merrygo-round. (a) Calculate the rotational inertia of the merry-goround about its axis of rotation. (b) Calculate the magnitude of the angular momentum of the running child about the axis of rotation of the merry-go-round. (c) Calculate the angular speed of the merry-go-round and child after the child has jumped onto the merry-go-round kg.m 2 /s (a) 149 kg.m 2 (b) 158 kg.m 2 /s (c) rad/s 96 A girl of mass M stands on the rim of a frictionless merry-goround of radius R and rotational inertia I that is not moving. She throws a rock of mass m horizontally in a direction that is tangent to the outer edge of the merry-go-round. The speed of the rock, relative to the ground, is v. Afterward, what is (a) Afterward, what is the angular speed of the merry-goround? (b) Afterward, what is the linear speed of the girl? 100 The angular momentum of the propeller of a small airplane points forward. (a) As the plane takes off, the nose lifts up and the airplane tends to veer to one side. To which side does it veer and why? (b) If the plane is flying horizontally and suddenly turns to the right, does the nose of the plane tend to move up or down? Why? (a) mvr/(i + MR 2 ); (b) mvr 2 /(I + MR 2 )

18 101 A wheel is set spinning and is then hung by a rope placed at one end of the axle. The wheel is spinning as shown, What is the direction of the angular momentum vector? 104 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, what happens to the front end of the suitcase? 102 A disk rotates clockwise in the plane of the page. What is the direction of the angular momentum vector? 105 If the sum of the torques on a body about a fixed axis is not zero, what happens to the body? into the page experiences angular acceleration 103 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? 106 A wheel of moment of inertia kg.m 2 is spinning with an angular speed of 5000 rad/s. A torque is applied torque about an axis perpendicular to the spin axis. The applied torque has a magnitude of 67.8 N.m. What the angular velocity of precession? up rad/s 107 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 what happens to the front end of the plane? Explain. it tends to dive.

19 108 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 what direction? Explain. 111 A gyroscopic wheel spins clockwise as shown. Describe the set of vectors that correctly describes the directions of the torque?, angular momentum L, and angular velocity of precession? p. -z t(+y);l(-x);w(-z) 109 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 act on the front end of the boat? Explain what happens. 112 A gyroscopic toy is spinning as shown. The torque?, angular momentum of the wheel L, and angular precession velocity? p are in which directions? cause the front of the boat to rise. 110 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 in what direction? Explain. 5 to the left. 113 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 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. What is the angular velocity of precession? rad/s

20 114 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? 116 A bicycle wheel of radius 30 cm is mounted at the middle of an axle 60 cm long. The tire and rim weigh 36 N. The wheel is spun at 10 rev/s, and the axle is then placed in a horizontal position with one end resting on a pivot. (a) What is the angular momentum due to the spinning of the wheel? (Treat the wheel as a hoop.) (b) What is the angular velocity of precession? (c) How long does it take for the axle to swing through 360o around the pivot? (d) What is the angular momentum associated with the motion of the center of mass, that is, due to the precession? (e) In what direction is this angular momentum? 115 y A car is powered by the energy stored in a single flywheel with an angular momentum L. Discuss the problems that would arise for various orientations of L and various maneuvers of the car. For example, what would happen if L points vertically upward and the car travels over a hilltop or through a valley, or what would happen if L points forward or to one side and the car attempts to turn to the left or right. In each case, consider the direction of the torque exerted on the car by the road. L points vertically upward and the car is moving directly away from you rounding the top of the hill it rolls to the left 117 (a) 20.8 kg.m 2 /s b/ /52- Rads c/ 12.1 sec d..172 kg m 2 /s upward (downward when the angular momentum de to the spin of the wheel is away (toward) the pivot A uniform disk of mass 2 kg and radius 6 cm is nted in the center of a 10 cm axle and spun at 900 rev/min. The axle is then placed in a horizontal position with one end resting on a pivot. The other end is given an initial horizontal velocity such that the precession is smooth with no nutation. (a) Find the angular velocity of precession. (b) What is the speed of the center of mass during the precession? (c) What are the magnitude and direction of the acceleration of the center of mass? (d) What are the vertical and horizontal components of the force exerted by the pivot? (a) 2.89 rad/s (b).145 m/s (c).418 m/s 2 directed toward the pivot d. F vert = 19.6 N F Hor =.836 N

21 118 A top spins at 30 rev/s about an axis that makes an angle of 30 0 with the vertical. The mass of the top is 0.50 kg, its rotational inertia about its central axis is 5.0 x 10-4 kg.m 2 and its center of mass is 4.0 cm from the pivot point. (a) If the spin is clockwise from an overhead view, what is the precession rate? (b) If the spin is clockwise from an overhead view, what is the direction of the precession as viewed from overhead? 122 The angular momentum of a body about a particular axis as a function of time is shown in the graph. What is the external torque acting on the body along this axis at t = 2 s? (a) 0.33 rev/s (b) clockwise A certain gyroscope consists of a uniform disk with a 50 cm radius mounted at the center of an axle that is 11 cm long and of negligible mass. The axle is horizontal and supported at one end. If the disk is spinning around the axle at 1000 rev/min, what is the precession rate? 0.43 rev/min A solid cylinder is spinning counterclockwise about a longitudinal axis when a net torque? is applied, as shown. What happens to the cylinder? N/m A uniform disk of radius in and mass 5 kg is pivoted such that it rotates freely about its axis. A string wrapped around the disk is pulled with a force of 20 N.. (a) What is the torque exerted on the disk? (b) What is the angular acceleration of the disk? (c) If the disk starts from rest, what is its angular velocity after 3 s? (e) What is its angular momentum after 3 s? (f) Find the total angle? the disk turns through in 3 s. 121 speeds up. A constant torque of 15 Nm acts for 3.0 s on a system of mass 2.0 kg. What is the change in angular momentum of the system during this period of time? (a) 2.4 N.m (b) 66.7 rad/s 2 (c) 200 rad/s (d) 720 J (e) 7.2 kg.m 2 /s (f) 300 rad 45 kg.m 2 /s

22 124 A particle is traveling with a constant velocity v along a line that is a distance b from the origin 0. Let da be the area swept out by the position vector from 0 to the particle in time dt. Show that da/dt is constant in time and equal to 1/2 L/m, where L is the angular momentum of the particle about the origin. 127 A uniform slender rod 1.00 m long is initially standing vertically on a smooth horizontal surface. It is struck a sharp horizontal blow at the top end, with the blow directed at right angles to the rod axis. As a result, the rod acquires an angular velocity of 3.00 rad/s. What is the translational velocity of the center of mass of the rod after the blow?.500 m/s 128 A uniform slender rod 1.00 m long is initially standing vertically on a smooth horizontal surface. It is struck a sharp horizontal blow at the top end, with the blow directed at right angles to the rod axis. As a result, the rod acquires an angular velocity of 3.00 rad/s. Which point on the rod is stationary just after the blow? 125 Given that r = ix + jy + kz and F = if x + jf y + kf z. (a) Find the torque t = r x F. (b) Show that if r and F lie in a given plane then t has no component in that plane. 2/3 m from the struck end. 129 A uniform bar of length l and mass m is suspended from a very thin axle that passes through a hole near the top end A of the bar. How far from A should a blow be applied at right angles to the bar in order to start the bar rotating about A without breaking the axle? 126 How far above its center should a billiard ball be struck in order to make it roll without an initial slippage? Denote the ball's radius by R, and assume that the impulse delivered by the cure is purely horizontal. d = 2l/e h = 2R/5

23 130 Two cylinders having radii R 1 and R 2 and rotational inertia I 1 and I 2 respectively, are supported by axes perpendicular to the plane of the figure shown to the left. The large cylinder is initially rotating with angular velocity wo. The small cylinder is moved to the right until it touches the large cylinder and is caused to rotate by the frictional force between the two. Eventually, slipping ceases, and the two cylinders rotate at constant rates in opposite directions. Assume that the total angular momentum is conserved. (a) Find the final angular velocity? 2 of the smaller cylinder in terms of I 1, I 2, R 1, R 2 and? 0. (b) Is the total angular momentum conserved in this case? 132 The figure shows the position vector r of a particle at a certain instant, and four choices for the direction of a force that is to accelerate the particle. All four choices lie in the xy plane. (a) Rank the choices according to the magnitude of the time rate of change (dl/dt) they produce in the angular momentum of the particle about point 0, greatest first. (b) Which choice results in a negative rate of change about 0? (a) 3, 1, then 2 and 4 tie (zero) 131 How far above its center should a billiard ball be struck in order to make it roll without an initial slippage? Denote the ball's radius by R, and assume that the impulse delivered by the cure is purely horizontal. 133 In the figure, a penguin of mass m falls from rest at point A, a horizontal distance D from the origin 0 of an xyz coordinate system. (The positive direction of the z axis is directly outward from the plane of the figure.) (a) What is the angular momentum L of the falling penguin about 0? (b) About the origin 0, what is the torque? on the penguin due to the gravitational force F g? h = 2R/5

24 134 George Washington Gale Ferris, Jr., a civil engineering graduate from Rensselaer Polytechnic Institute, built the original Ferris wheel (see the figure) for the 1893 World's Columbian Exposition in Chicago. The wheel, an astounding engineering construction at the time, carried 36 wooden cars, each holding as many as 60 passengers, around a circle of radius R = 38 m. The mass of each car was about 1.1 x 10 4 kg. The mass of the wheel's structure was about 6.0 x 10 5 kg, which was mostly in the circular grid from which the cars were suspended. The wheel made a complete rotation at an angular speed? F in about 2 min. (a) Estimate the magnitude L of the angular momentum of the wheel and its passengers while the wheel rotated at? 0. (b) Assume that the fully loaded wheel is rotated from rest to? F in a time period t 1 = 5.0 s. What is the magnitude? avg of the average net external torque acting on it during t 1? 136 The figure gives the angular momentum magnitude L of a wheel versus time t. Rank the four lettered time intervals according to the magnitude of the torque acting on the wheel, greatest first. D, B, then A and C tie 137 In the figure, a kg ball is shot directly upward at initial speed 40.0 m/s. (a) What is its angular momentum about P, 2.00 m horizontally from the launch point, when the ball is at maximum heigh? (b) What is its angular momentum about P, 2.00 m horizontally from the launch point, when the ball is halfway back to the ground? (c) What is the torque on the ball about P due to the gravitational force when the ball is at maximum height? (d) What is the torque on the ball about P due to the gravitational force when the ball is halfway back to the ground? 135 The angular momenta L(t) of a particle in four situations are (1) L = 3t + 4; (2) L = -6t 2 ; (3) L = 2; (4) f = 4/t. (a) In which situation is the net torque on the particle zero? (b) In which situation is the net torque on the particle positive and constant? (c) In which situation is the net torque on the particle negative and increasing in magnitude (t > 0)? (d) In which situation is the net torque on the particle negative and decreasing in magnitude (t > 0)? (a) 0 (b) -22 kg.m 2 /s (c) N.m (d) -784 N.m (a) 3 (b) 1 (c) 2 (d) 4

25 At the instant the displacement of a 2.00 kg object relative to the origin is (2.00 m)i + (4.00 m)j - (3.00 m)k, its velocity is v = -(6.00 m/s)i + (3.00 m/s)j + (3.00 m/s)k and it is subject to a force (6.00 N)i - (8.00 N)j + (4.00 N)k. (a) Find the acceleration of the object. (b) Find the angular momentum of the object about the origin. (c) Findthe torque about the origin acting on the object. (d) Find the angle between the velocity of the object and the force acting on the object. (a) (3.00 m/s 2 ) i - (4.00 m/s 2 ) j + (2.00 m/s 2 ) k; (b) (42.0 kg.m 2 /s) i + (24.0 kg.m 2 /s) j + (60.0 kg.m 2 /s) k; (c) ( N.m) i - (26.0 N.m) j - (40.0 N.m) k; (d) A particle is acted on by two torques about the origin:? 1 has a magnitude of 2.0 N.m and is directed in the positive direction of the x axis, and? 2 has a magnitude of 4.0 N.m and is directed in the negative direction of the y axis. In unit vector notation, find dl/dt, where L is the angular momentum of the particle about the origin. 141 A particle is to move in an xy plane, clockwise around the origin as seen from the positive side of the z axis. (a) In unit vector notation, what torque acts on the particle if the magnitude of its angular momentum about the origin is 4.0 kg.m 2 /s? (b) In unit vector notation, what torque acts on the particle if the magnitude of its angular momentum about the origin is 4.0t 2 kg.m 2 /s? (c) In unit vector notation, what torque acts on the particle if the magnitude of its angular momentum about the origin is 4.0 t kg.m 2 /s? (d) In unit vector notation, what torque acts on the particle if the magnitude of its angular momentum about the origin is 4.0/t 2 kg.m 2 /s? (a) 0; (b) (-8.0 N.m/s) t k; (c) -2.0t k in newton meters for t in seconds; (d) 8.0t - 3 k in newton meters for t in seconds (2.0 N.m) i - (4.0 N.m) j 142 At time t, r = 4.0t 2 i - (2.0t + 6.0t 2 )j gives the position of a 3.0 kg particle relative to the origin of an xy coordinate system (r is in meters and t is in seconds). (a) Find an expression for the torque acting on the particle relative to the origin, (b) Is the magnitude of the particle's angular momentum relative to the origin increasing, decreasing, or unchanging? 140 A 3.0 kg particle with velocity v = (5.0 m/s)i - (6.0 m/s)j is at x = 3.0 m, y = 8.0 m. It is pulled by a 7.0 N force in the negative x direction. (a) About the origin, what is the particle's angular momentum? (b) About the origin, what is the torque acting on the particle? (c) About the origin, what is the rate at which the angular momentum is changing? (a) (-1.7 x 10 2 kg.m 2 /s) k (b) (+56 N.m) k (c) (+56 kg.m 2 /s) k 143 (a) (48 N.m/s)t k; (b) increasing A sanding disk with rotational inertia 1.2 x 10-3 kg.m 2 is attached to an electric drill whose motor delivers a torque of magnitude 16 N.m about the central axis of the disk. (a) About that axis and with the torque applied for 33 ms, what is the magnitude of the angular momentum? (b) About that axis and with the torque applied for 33 ms, what is the magnitude of the angular velocity of the disk? (a) 0.53 kg.m 2 /s; (b) 4.2 x 10 3 rev/min

26 144 The figure gives the torque? that acts on an initially stationary disk that can rotate about its center like a merry go-round. (a) What is the angular momentum of the disk about the rotation axis at time t = 7.0 s? (b) What is the angular momentum of the disk about the rotation axis at time t = 20 s? 147 A 3.0 kg toy car moves along an x axis with a velocity given by v = -2.0t 3 i m/s, with t in seconds. (a) For t > 0, what is the angular momentum L of the car? (b) For t > 0, what is the torque? on the car, both calculated about the origin? (c) What is L about the point (2.0 m, 5.0 m, 0)? (d) What is? about the point (2.0 m, 5.0 m, 0)? (e) What is L about the point (2.0 m, -5.0 m, 0)? (f) What is? about the point (2.0 m, -5.0 m, 0)? (a) 0 (b) 0 (c) (-30t 3 kg.m 2 /s) k (d) (90t 2 N.m) k; (e) (30t 3 kg.m 2 /s) k; (f) 90t 2 N.m) k (a) 24 kg.m 2 /s (b) 1.5 kg.m 2 /s 145 At time t = 0, a 2.0 kg particle has position vector r = (4.0 m)i - (2.0 m)j relative to the origin. Its velocity is given by v = (-6.0t 2 m/s)i for t > = 0 in seconds. About the origin, what is (a) About the origin, what is the particle's angular momentum L? (b) About the origin, what is the torque? acting on the particle, both in unit-vector notation and for t > 0? (c) About the point (-2.0 m, -3.0 m, 0), what is L? (d) About the point (-2.0 m, -3.0 m, 0), what is? for t > 0? 148 A 4.0 kg particle moves in an xy plane. At the instant when the particle's position and velocity are r = (2.0i + 4.0j)j m and v = -4.0j m/s, the force on the particle is F = -3.0i N. (a) At this instant, determine the particle's angular momentum about the origin. (b) At this instant, determine the particle's angular momentum about the point x = 0, y = 4.0 m. (c) At this instant, determine the torque acting on the particle about the origin. (d) At this instant, determine the torque acting on the particle about the point x = 0, y = 4.0 m. (a) (-24t 2 kg.m 2 /s) k; (b) (-48t N.m) k; (c) (12t 2 kg.m 2 /s) k (d) (24t N.m) k (a) (-32 kg.m 2 /s) k (b) (-32 kg.m 2 /s) k (c) (12 N.m) k; (d) A wheel rotates clockwise about its central axis with an angular momentum of 600 kg.m 2 /s. At time t = 0, a torque of magnitude 50 N.m is applied to the wheel to reverse the rotation. At what time t is the angular speed zero? 149 A wheel of radius m, which is moving initially at 43.0 m/s, rolls to a stop in 225 m. (a) Calculate the magnitudes of its linear acceleration. (b) Calculate the magnitudes of its angular acceleration. (c) The wheel's rotational inertia is kg.m 2 about its central axis. Calculate the magnitude of the torque about the central axis due to friction on the wheel. 12 s (a) 4.11 m/s 2 (b) 16.4 rad/s 2 (c) 2.55 N.m

27 150 A projectile of mass kg is fired from the ground with an initial speed v 0 = 12.6 m/s and an initial angle? 0 = above a horizontal x axis (the y axis extends upward). (a) Find an expression for the magnitude of the projectile's angular momentum about the firing point as a function of time. (b) Find the rate at which the angular momentum changes with time. (c) Evaluate the magnitude of r X mg' directly and compare the result with (b). (d) Why should the results of (b) and (c) be identical? 152 How far above its center should a billiard ball be struck in order to make it roll without an initial slippage? Denote the ball's radius by R, and assume that the impulse delivered by the cure is purely horizontal. (a) (-17.1t 2 kg.m 2 /s) k (b) (-34.2t kg.m 2 /s) k (c) (-34.2t N.m) k h = 2R/5 151 With axle and spokes of negligible mass and a thin rim, a certain bicycle wheel has a radius of m and weighs 37.0 N; it can turn on its axle with negligible friction. A man holds the wheel above his head with the axle vertical while he stands on a turntable that is free to rotate without friction; the wheel rotates clockwise, as seen from above, with an angular speed of 57.7 rad/s, and the turntable is initially at rest. The rotational inertia of wheel + man + turntable about the common axis of rotation is 2.10 kg.m 2. The man's free hand suddenly stops the rotation of the wheel (relative to the turntable). (a) Determine the resulting angular speed of rotation of the system. (b) Determine the resulting direction of rotation of the system. 153 A uniform slender rod 1.00 m long is initially standing vertically on a smooth horizontal surface. It is struck a sharp horizontal blow at the top end, with the blow directed at right angles to the rod axis. As a result, the rod acquires an angular velocity of 3.00 rad/s. What is the translational velocity of the center of mass of the rod after the blow?.500 m/s (a) 12.7 rad/s; (b) clockwise 154 A uniform slender rod 1.00 m long is initially standing vertically on a smooth horizontal surface. It is struck a sharp horizontal blow at the top end, with the blow directed at right angles to the rod axis. As a result, the rod acquires an angular velocity of 3.00 rad/s. Which point on the rod is stationary just after the blow? 2/3 m from the struck end.

28 155 A uniform bar of length l and mass m is suspended from a very thin axle that passes through a hole near the top end A of the bar. How far from A should a blow be applied at right angles to the bar in order to start the bar rotating about A without breaking the axle? 157 The two uniform disks shown in the figure to the left rotate separately on parallel axles. The upper disk has angular speed wo, and the lower disk is at rest. Now the two disks are moved together so that their rims touch. After a short time the two disks, now in contact, are rotating without slipping. Find the final rate of rotation of the upper disk. d = 2l/e w 1 = (I 1 w o )/(I 1 + (a 2 I 2 )/b 2 ) 156 Two cylinders having radii R 1 and R 2 and rotational inertia I 1 and I 2 respectively, are supported by axes perpendicular to the plane of the figure shown to the left. The large cylinder is initially rotating with angular velocity? 0. The small cylinder is moved to the right until it touches the large cylinder and is caused to rotate by the frictional force between the two. Eventually, slipping ceases, and the two cylinders rotate at constant rates in opposite directions. Assume that the total angular momentum is conserved. (a) Find the final angular velocity? 2 of the smaller cylinder in terms of I 1, I 2, R 1, R 2 and? 0. (b) Is the total angular momentum conserved in this case? 158 Two wheels A and B are connected by a belt as in the figure to the left. The radius of B is three times the radius of A. (a) What would be the ratio of the rotational inertia I A /I B if both wheels have the same angular momenta? (a) 1/3 159 A pulley with radius R and rotational inertia I is free to rotate on a horizontal fixed axis through its center. A string passes over the pulley. Mass m1 is attached to one end and mass m2 is attached to the other. At one time m1 is moving downward with speed v. If the string does not slip on the pulley, the magnitude of the total angular momentum, about the pulley center, of the masses and pulley, considered as a system, is given by: (m 1 + m 2 )?R + I?/R

29 kg metal hoop with a radius of 0.5 m has a translational velocity of 2.0 m/s as it rolls without slipping. What is the angular momentum of this hoop about its center of mass? 162 Answer the following questions: (a) Assuming that the incline in the figure above is frictionless and that the string passes through the center of mass Of m2, find the net torque acting on the system (the two masses and the pulley) about the center of the pulley. (b) Write an expression for the total angular momentum of the system about the center of the pulley when the masses are moving with a speed v. Assume the pulley has a moment of inertia I and a radius r. (c) Find the acceleration of the masses from your results for parts a and b by setting the net torque equal to the rate of change of the angular momentum of the system. 1.0 kg.m 2 /s 161 A 15 g coin of diameter 1.5 cm is spinning at 10 rev/s about a vertical diameter at a fixed point on a tabletop. (a) What is the angular momentum of the coin about its center of mass? (b) What is its angular momentum about a point on the table 10 cm from the coin? If the coin spins about a vertical diameter at 10 rev/s, but it also travels in a straight line across the tabletop at 5 cm/s? (c) What is the angular momentum of the coin about a point on the line of motion? (d) What is the angular momentum of the coin about a point 10 cm from the line of motion? (There are two answers to this question. Explain why and give both.) a x 10-5 kg.m 2 /s b x 10-5 kg.m 2 /s c x 10-5 kg.m 2 /s d. L c, 8.83 z 10-5 kg.m 2 s 163 T org (m 2sci A 2 kg mass attached to a string of length 1.5 m moves in a horizontal circle as a conical pendulum shown to the left. The string makes an angle 0 = 30 0 with the vertical. (a) Show that the angular momentum of the mass about the point of support P has a horizontal component toward the center of the circle as well as a vertical component, and find these components. (b) Find the magnitude of dl/dt. and show that it equals the magnitude of the torque exerted by gravity about the point of support. pr =6.18 x 1-5 kg.m 2 /s depending on the direction of the velocity of the center of mass.

30 164 A cue ball of radius r is initially at rest on a horizontal pool table. It is struck by a horizontal cue stick that delivers an impulse of magnitude P 0 = F av Dt. (We use P 0 for the impulse rather than I 0 to avoid confusion with the moment of inertia I) The stick strikes the ball at a point h above the ball's point of contact with the table). Show that the ball's initial angular velocity wo is related to the initial linear velocity of its center of mass? 0 = 5v 0 (h r)/2r In the figure, three particles of mass m = 23 g are fastened to three rods of length d = 12 cm and negligible mass. The rigid assembly rotates around point 0 at angular speed? = 0.85 rad/s. (a) About 0, what is the rotational inertia of the assembly? (b) About 0, what is the magnitude of the angular momentum of the middle particle? (c) About 0, what is the magnitude of the angular momentum of the asssembly? (a) 4.6 x 10 3 kg.m 2 (b) 1.1 x 10 3 kg.m 2 /s; (c) 3.9 x 10 3 kg.m 2 /s 165 The figure shows three rotating, uniform disks that are coupled by belts. One belt runs around the rims of disks A and C. Another belt runs around a central hub on disk A and the rim of disk B. The belts move smoothly without slippage on the rims and hub. Disk A has radius R; its hub has radius R; disk B has radius R; and disk C has radius 2.000R. Disks B and C have the same density (mass per unit volume) and thickness. What is the ratio of the magnitude of the angular momentum of disk C to that of disk B? 167 A Texas cockroach first rides at the center-of a circular disk that rotates freely like a merry-go-round without external torques. The cockroach then walks out to the edge of the disk, at radius R. The figure gives the angular speed w of the cockroach-disk system during the walk. When the cockroach is on the edge at radius R, what is the ratio of the bug's rotational inertia to that of the disk, both calculated about the rotation axis?

31 168 Large machine components that undergo prolonged, high speed rotation are first examined for the possibility of failure in a spin test system. In this system, a component is spun up (brought up to high speed) while inside a cylindrical arrangement of lead bricks and containment liner, all within a steel shell that is closed by a lid clamped into place. If the rotation causes the component to shatter, the soft lead bricks are supposed to catch the pieces for later analysis. In early 1985, Test Devices, Inc. ( was spin testing a sample of a solid steel rotor (a disk) of mass M = 272 kg and radius R = 38.0 cm. When the sample reached an angular speed? of rev/min, the test engineers heard a dull thump from the test system, which was located one floor down and one room over from them. Investigating, they found that lead bricks had been thrown out in the hallway leading to the test room, a door to the room had been hurled into the adjacent parking lot, one lead brick had shot from the test site through the wall of a neighbor's kitchen, the structural beams of the test building had been damaged, the concrete floor beneath the spin chamber had been shoved downward by about 0.5 cm, and the 900 kg lid had been blown upward through the ceiling and had then crashed back onto the test equipment (see the figure). The exploding pieces had not penetrated the room of the test engineers only by luck. (a) When the rotor exploded, how much angular momentum, Given that r = xi + yj + zk and F = F x l + F y J + F z k, show that the torque? = r X F is given by? = (yf z - zf y )i + (zf x - xf z )j + (xf y - yf x )k. An impulsive force F(t) acts for a short time t on a rotating rigid body of rotational inertia L Show that (for the given equation) where? is the torque due to the force, R is the moment arm of the force, F avg is the average value of the force during the time it acts on the body, and? i and? f are the angular velocities of the body just before and just after the force acts. (The quantity? dt = RF avg t is called the angular impulse, analogous to F avg t, the linear impulse.) 172 The angular momentum of a flywheel having a rotational inertia of kg.m 2 about its central axis decreases from 3.00 to kg.m 2 /s in 1.50 s. (a) What is the magnitude of the average torque acting on the flywheel about its central axis during this period? (b) Assuming a constant angular acceleration, through what angle does the flywheel turn? (c) How much work is done on the wheel? (d) What is the average power of the flywheel? (a) 2.9 x 10 4 kg.m 2 /s (b) 1.2 x 10 6 N.m (a) 1.47 N.m; (b) 20.4 rad; (c) J; (d) 19.9 W 169 If Earth's polar ice caps fully melted and the water returned to the oceans, the oceans would be deeper by about 30 m. What effect would this have on Earth's rotation? Make an estimate of the resulting change in the length of the day. (Concern has been expressed that warming of the atmosphere resulting from industrial pollution could cause the ice caps to melt.) rotational speed would decrease; day would be about 0.8 s longer

32 173 A man stands on a platform that is rotating (without friction) with an angular speed of 1.2 rev/s; his arms are outstretched and he holds a brick in each hand. The rotational inertia of the system consisting of the man, bricks, and platform about the central vertical axis of the platform is 6.0 kg.m 2. By moving the bricks the man decreases the rotational inertia of the system to 2.0 kg.m 2. (a) What is the resulting angular speed of the platform? (b) What is the ratio of the new kinetic energy of the system to the original kinetic energy? (c) What source provided the added kinetic energy? 175 A Texas cockroach of mass 0.17 kg runs counterclockwise around the rim of a lazy Susan (a circular disk mounted on a vertical axle) that has radius 15 cm, rotational inertia 5.0 x 10-3 kg.m 2, and frictionless bearings. The cockroach's speed (relative to the ground) is 2.0 m/s, and the lazy Susan turns clockwise with angular velocity? 0 = 2.8 rad/s. The cockroach finds a bread crumb on the rim and, of course, stops. (a) What is the angular speed of the lazy Susan after the cockroach stops? (b) Is mechanical energy conserved as it stops? 174 (a) 3.6 rev/s; (b) 3.0; (c) forces on the bricks from the man transferred energy from the man s internal energy to kinetic energy In the figure, two skaters, each of mass 50 kg, approach each other along parallel paths separated by 3.0 m. They have opposite velocities of 1.4 m/s each. One skater carries one end of a long pole of negligible mass, and the other skater grabs the other end as she passes. The skaters then rotate around the center of the pole. Assume that the friction between skates and ice is negligible. (a) What is the radius of the circle? (b) What is the angular speed of the skaters? (c) What is the kinetic energy of the two-skater system? Next, the skaters pull along the pole until they are separated by 1.0 m. (d) What then is their angular speed? (e) What then is the kinetic energy of the system? (f) What provided the energy for the increased kinetic energy? 176 (a) 4.2 rad/s; (b) no, because energy is transferred to the cockroach s internal energy The figure is an overhead view of a thin uniform rod of length m and mass M rotating horizontally at 80.0 rad/s counterclockwise about an axis through its center. A particle of mass M/3.00 and traveling horizontally at speed 40.0 m/s hits the rod and sticks. The particle's path is perpendicular to the rod at the instant of the hit, at a distance d from the rod's center. (a) At what value of d are rod and particle stationary after the hit? (b) In which direction do rod and particle rotate if d is greater than this value? (a) m; (b) clockwise (a) 1.5 m; (b) 0.93 rad/s; (c) 98 J; (d) 8.4 rad/s; (e) 8.8 x 10 2 J; (f) internal energy of the skaters

33 177 A cockroach of mass m lies on the rim of a uniform disk of mass 4.00m that can rotate freely about its center like a merrygo-round. Initially the cockroach and disk rotate together with an angular velocity of rad/s. Then the cockroach walks halfway to the center of the disk. (a) What then is the angular velocity of the cockroach-disk system? (b) What is the ratio K/K 0 of the new kinetic energy of the system to its initial kinetic energy? (c) What accounts for the change in the kinetic energy? 179 The uniform rod (length 0.60 m, mass 1.0 kg) in the figure rotates in the plane of the figure about an axis through one end, with a rotational inertia of 0.12 kg.m 2. As the rod swings through its lowest position, it collides with a 0.20 kg putty wad that sticks to the end of the rod. If the rod's angular speed just before collision is 2.4 rad/s, what is the angular speed of the rod-putty system immediately after collision? (a) rad/s; (b) 1.33; (c) energy is transferred from the internal energy of the cockroach to kinetic energy 1.5 rad/s 178 In the figure a l.0 g bullet is fired into a 0.50 kg block attached to the end of a 0.60 m nonuniform rod of mass 0.50 kg. The block-rod-bullet system then rotates in the plane of the figure, about a fixed axis at A. The rotational inertia of the rod alone about that axis at A is kg.m 2. Treat the block as a particle. (a) What then is the rotational inertia of the block-rod-bullet system about point A? (b) If the angular speed of the system about A just after impact is 4.5 rad/s, what is the bullet's speed just before impact? 180 A uniform disk of mass 10 m and radius 3.0r can rotate freely about its fixed center like a merry-go-round. A smaller uniform disk of mass m and radius r lies on top of the larger disk, concentric with it. Initially the two disks rotate together with an angular velocity of 20 rad/s. Then a slight disturbance causes the smaller disk to slide outward across the larger disk, until the outer edge of the smaller disk catches on the outer edge of the larger disk. Afterward, the two disks again rotate together (without further sliding). (a) What then is their angular velocity about the center of the larger disk? (b) What is the ratio K/K 0 of the new kinetic energy of the twodisk system to the system's initial kinetic energy? (a) 18 rad/s; (b) 0.92 (a) 0.24 kg.m 2 (b) 1.8 x 10 3 m/s

34 181 Two 2,00 kg balls are attached to the ends of a thin rod of length 50.0 cm and negligible mass. The rod is free to rotate in a vertical plane without friction about a horizontal axis through its center. With the rod initially horizontal (see figure), a 50.0 g wad of wet putty drops onto one of the balls, hitting it with a speed of 3.00 m/s and then sticking to it. (a) What is the angular speed of the system just after the putty wad hits? (b) What is the ratio of the kinetic energy of the system after the collision to that of the putty wad just before? (c) Through what angle will the system rotate before it momentarily stops? 184 The figure shows an overhead view of a ring that can rotate about its center like a merry-go-round. Its outer radius R 2 is m, its inner radius R 1 is R 2 /2.00, its mass M is 8.00 kg, and the mass of the crossbars at its center is negligible. It initially rotates at an angular speed of 8.00 rad/s with a cat of mass m = M/4.00 on its outer edge, at radius R 2. By how much does the cat increase the kinetic energy of the cat-ring system if the cat crawls to the inner edge, at radius R 1? (a) rad/s (b) (c) J 182 In the figure, a small 50 g block slides down a frictionless surface through height h = 20 cm and then sticks to a uniform rod of mass 100 g and length 40 cm. The rod pivots about point 0 through angle? before momentarily stopping. Find?. 185 Wheels A and B in the figure are connected by a belt that does not slip. The radius of B is 3.00 times the radius of A. What would be the ratio of the rotational inertias I A /I B if the two wheels had (a)what would be the ratio of the rotational inertias I A /I B if the two wheels had the same angular momentum about their central axes? (b) What would be the ratio of the rotational inertias I A /I B if the two wheels had the same rotational kinetic energy? 32 0 (a) 0.333; (b) A uniform block of granite in the shape of a book has face dimensions of 20 cm and 15 cm and a thickness of 1.2 cm. The density (mass per unit volume) of granite is 2.64 g/cm 3. The block rotates around an axis that is perpendicular to its face and halfway between its center and a corner. Its angular momentum about that axis is kg.m 2 /s. What is its rotational kinetic energy about that axis? 0.62 J

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