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Chapter 10 Problems 1, 2, 3 = straightforward, intermediate, challenging = full solution available in Student Solutions Manual/Study Guide = coached solution with hints available at www.pop4e.com = computer useful in solving problem = paired numerical and symbolic problems = biomedical application Section 10.1 Angular Position, Speed, and Acceleration 1. During a certain period of time, the angular position of a swinging door is described by θ = 5.00 + 10.0t + 2.00t2, where θ is in radians and t is in seconds. Determine the angular position, angular speed, and angular acceleration of the door (a) at t = 0 and (b) at t = 3.00 s. Section 10.2 Rotational Kinematics: The Rigid Object Under Constant Angular Acceleration 2. A dentist’s drill starts from rest. After 3.20 s of constant angular acceleration, it turns at a rate of 2.51 × 104 rev/min. (a) Find the drill’s angular acceleration. (b) Determine the angle (in radians) through which the drill rotates during this period. 3. An electric motor rotating a grinding wheel at 100 rev/min is switched off. The wheel then moves with a constant negative angular acceleration of magnitude 2.00 rad/s2. (a) During what time interval does the wheel come to rest? (b) Through how many radians does it turn while it is slowing down? 4. A centrifuge in a medical laboratory rotates at an angular speed of 3 600 rev/min. When switched off, it rotates through 50.0 revolutions before coming to rest. Find the constant angular acceleration of the centrifuge. 5. The tub of a washer goes into its spin cycle, starting from rest and gaining angular speed steadily for 8.00 s, at which time it is turning at 5.00 rev/s. At this point, the person doing the laundry opens the lid and a safety switch turns off the washer. The tub smoothly slows to rest in 12.0 s. Through how many revolutions does the tub turn while it is in motion? 6. A rotating wheel requires 3.00 s to rotate through 37.0 rev. Its angular speed at the end of the 3.00-s interval is 98.0 rad/s. What is the constant angular acceleration of the wheel? 7. (a) Find the angular speed of the Earth’s rotation on its axis. As the Earth turns toward the east, we see the sky turning toward the west at this same rate. (b) The rainy Pleiads wester And seek beyond the sea The head that I shall dream of That shall not dream of me. —A. E. Housman (© Robert E. Symons) Cambridge, England, is at longitude 0° and Saskatoon, Saskatchewan, is at longitude 107° west. How much time elapses after the Pleiades set in Cambridge until these stars fall below the western horizon in Saskatoon? Section 10.3 Relations Between Rotational and Translational Quantities The cyclist pedals at a steady cadence of 76.0 rev/min. The chain engages with a front sprocket 15.2 cm in diameter and a rear sprocket 7.00 cm in diameter. (a) Calculate the speed of a link of the chain relative to the bicycle frame. (b) Calculate the angular speed of the bicycle wheels. (c) Calculate the speed of the bicycle relative to the road. (d) What pieces of data, if any, are not necessary for the calculations? 8. Make an order-of-magnitude estimate of the number of revolutions through which a typical automobile tire turns in 1 yr. State the quantities you measure or estimate and their values. 9. A disk 8.00 cm in radius rotates at a constant rate of 1 200 rev/min about its central axis. Determine (a) its angular speed, (b) the tangential speed at a point 3.00 cm from its center, (c) the radial acceleration of a point on the rim, and (d) the total distance a point on the rim moves in 2.00 s. 10. A wheel 2.00 m in diameter lies in a vertical plane and rotates with a constant angular acceleration of 4.00 rad/s2. The wheel starts at rest at t = 0, and the radius vector of a certain point P on the rim makes an angle of 57.3° with the horizontal at this time. At t = 2.00 s, find (a) the angular speed of the wheel, (b) the tangential speed and the total acceleration of the point P, and (c) the angular position of the point P. 11. Figure P10.11 shows the drivetrain of a bicycle that has wheels 67.3 cm in diameter and pedal cranks 17.5 cm long. Figure P10.11 12. A digital audio compact disc carries data, each bit of which occupies 0.6 μm along a continuous spiral track from the inner circumference of the disc to the outside edge. A CD player turns the disc to carry the track counterclockwise above a lens at a constant speed of 1.30 m/s. Find the required angular speed (a) at the beginning of the recording, where the spiral has a radius of 2.30 cm, and (b) at the end of the recording, where the spiral has a radius of 5.80 cm. (c) A full-length recording lasts for 74 min 33 s. Find the average angular acceleration of the disc. (d) Assuming that the acceleration is constant, find the total angular displacement of the disc as it plays. (e) Find the total length of the track. 13. A car traveling on a flat (unbanked) circular track accelerates uniformly from rest with a tangential acceleration of 1.70 m/s2. The car makes it one fourth of the way around the circle before it skids off the track. Determine the coefficient of static friction between the car and track from these data. Section 10.4 Rotational Kinetic Energy 15. This problem describes one experimental method for determining the moment of inertia of an irregularly shaped object such as the payload for a satellite. Figure P10.15 shows a counterweight of mass m suspended by a cord wound around a spool of radius r, forming part of a turntable supporting the object. The turntable can rotate without friction. When the counterweight is released from rest, it descends through a distance h, acquiring a speed v. Show that the moment of inertia I of the rotating apparatus (including the turntable) is mr2(2gh/v2 – 1). 14. Rigid rods of negligible mass lying along the y axis connect three particles (Fig. P10.14). The system rotates about the x axis with an angular speed of 2.00 rad/s. Find (a) the moment of inertia about the x axis and the total rotational kinetic energy evaluated from ½ Iω2 and (b) the tangential speed of each particle and the total kinetic energy evaluated from 1 2 mi vi 2 . Figure P10.15 Figure P10.14 16. Big Ben, the Parliament tower clock in London, has an hour hand 2.70 m long with a mass of 60.0 kg and a minute hand 4.50 m long with a mass of 100 kg (Fig. P10.16). Calculate the total rotational kinetic energy of the two hands about the axis of rotation. (You may model the hands as uniform long, thin rods.) (John Lawrence/Stone/Getty Images) Figure P10.16 Problems 10.16, 10.42, and 10.64. 17. Consider two objects with m1 > m2 connected by a light string that passes over a pulley having a moment of inertia of I about its axis of rotation as shown in Figure P10.17. The string does not slip on the pulley or stretch. The pulley turns without friction. The two objects are released from rest separated by a vertical distance 2h. (a) Use the principle of conservation of energy to find the translational speeds of the objects as they pass each other. (b) Find the angular speed of the pulley at this time. Figure P10.17 18. As a gasoline engine operates, a flywheel turning with the crankshaft stores energy after each fuel explosion, providing the energy required to compress the next charge of fuel and air. For the engine of a certain lawn tractor, suppose a flywheel must be no more than 18.0 cm in diameter. Its thickness, measured along its axis of rotation, must be no larger than 8.00 cm. The flywheel must release energy 60.0 J when its angular speed drops from 800 rev/min to 600 rev/min. Design a sturdy, steel flywheel to meet these requirements with the smallest mass that you can reasonably attain. Assume that the material has the density listed for iron in Table 15.1. Specify the shape and mass of the flywheel. 19. A war-wolf or trebuchet is a device used during the Middle Ages to throw rocks at castles and sometimes now used to fling pianos as a sport. A simple trebuchet is shown in Figure P10.19. Model it as a stiff rod of negligible mass, 3.00 m long, joining particles of mass 60.0 kg and 0.120 kg at its ends. It can turn on a frictionless horizontal axle perpendicular to the rod and 14.0 cm from the large-mass particle. The rod is released from rest in a horizontal orientation. Find the maximum speed that the small-mass object attains. Figure P10.20 21. Find the net torque on the wheel in Figure P10.21 about the axle through O, taking a = 10.0 cm and b = 25.0 cm. Figure P10.19 Section 10.5 Torque and the Vector Product 20. The fishing pole in Figure P10.20 makes an angle of 20.0° with the horizontal. What is the torque exerted by the fish about an axis perpendicular to the page and passing through the angler’s hand? Figure P10.21 22. Given M 6iˆ 2ˆj kˆ and N 2iˆ ˆj 3kˆ , calculate the vector product M N. 23. A force of F ( 2.00iˆ 3.00ˆj) N is applied to an object that is pivoted about a fixed axle aligned along the z coordinate axis. The force is applied at the point r ( 4.00iˆ 5.00ˆj) m. Find (a) the magnitude of the net torque about the z axis and (b) the direction of the torque vector . 24. Two vectors are given by A 3iˆ 7 ˆj 4kˆ and B 6iˆ 10ˆj 9kˆ . Evaluate the following quantities. (a) cos 1 A B / AB and (b) sin 1 A B / AB . (c) Which give(s) the angle between the vectors? 25. Use the definition of the vector product and the definitions of the unit vectors î , ĵ , and k̂ to prove Equations 10.23. You may assume that the x axis points to the right, the y axis up, and the z axis toward you (not away from you). This choice is said to make the coordinate system right-handed. Figure P10.26 27. A uniform beam of mass mb and length ℓ supports blocks with masses m1 and m2 at two positions as shown in Figure P10.27. The beam rests on two knife edges. For what value of x will the beam be balanced at P such that the normal force at O is zero? Section 10.6 The Rigid Object in Equilibrium 26. In exercise physiology studies, it is sometimes important to determine the location of a person’s center of mass, which can be done with the arrangement shown in Figure P10.26. A light plank rests on two scales, which read Fg1 = 380 N and Fg2 = 320 N. A distance of 2.00 m separates the scales. How far from the woman’s feet is her center of mass? Figure P10.27 28. A uniform plank of length 6.00 m and mass 30.0 kg rests horizontally across two horizontal bars of a scaffold. The bars are 4.50 m apart, and 1.50 m of the plank hangs over one side of the scaffold. Draw a free-body diagram of the plank. How far can a painter of mass 70.0 kg walk on the overhanging part of the plank before it tips? 29. Figure P10.29 shows a claw hammer as it is being used to pull a nail out of a horizontal board. A force of 150 N is exerted horizontally as shown. Find (a) the force exerted by the hammer claws on the nail and (b) the force exerted by the surface on the point of contact with the hammer head. Assume that the force the hammer exerts on the nail is parallel to the nail. 31. A uniform sign of weight Fg and width 2L hangs from a light, horizontal beam hinged at the wall and supported by a cable (Fig. P10.31). Determine (a) the tension in the cable and (b) the components of the reaction force exerted by the wall on the beam, in terms of Fg, d, L, and θ. Figure P10.31 Figure P10.29 30. A uniform ladder of length L and mass m1 rests against a frictionless wall. The ladder makes an angle θ with the horizontal. (a) Find the horizontal and vertical forces the ground exerts on the base of the ladder when a firefighter of mass m2 is a distance x from the bottom. (b) If the ladder is just on the verge of slipping when the fire-fighter is a distance d from the bottom, what is the coefficient of static friction between ladder and ground? 32. A crane of mass 3 000 kg supports a load of 10 000 kg as shown in Figure P10.32. The crane is pivoted with a frictionless pin at A and rests against a smooth support at B. Find the reaction forces at A and B. Figure P10.32 35. An electric motor turns a flywheel through a drive belt that joins a pulley on the motor and a pulley that is rigidly attached to the flywheel, as shown in Figure P10.35. The flywheel is a solid disk with a mass of 80.0 kg and a diameter of 1.25 m. It turns on a frictionless axle. Its pulley has much smaller mass and a radius of 0.230 m. The tension in the upper (taut) segment of the belt is 135 N, and the flywheel has a clockwise angular acceleration of 1.67 rad/s2. Find the tension in the lower (slack) segment of the belt. Section 10.7 The Rigid Object Under a Net Torque 33. The combination of an applied force and a friction force produces a constant total torque of 36.0 N · m on a wheel rotating about a fixed axis. The applied force acts for 6.00 s. During this time the angular speed of the wheel increases from 0 to 10.0 rad/s. The applied force is then removed, and the wheel comes to rest in 60.0 s. Find (a) the moment of inertia of the wheel, (b) the magnitude of the frictional torque, and (c) the total number of revolutions of the wheel. 34. A potter’s wheel—a thick stone disk of radius 0.500 m and mass 100 kg—is freely rotating at 50.0 rev/min. The potter can stop the wheel in 6.00 s by pressing a wet rag against the rim and exerting a radially inward force of 70.0 N. Find the effective coefficient of kinetic friction between wheel and rag. Figure P10.35 36. In Figure P10.36, the sliding block has a mass of 0.850 kg, the counterweight has a mass of 0.420 kg, and the pulley is a hollow cylinder with a mass of 0.350 kg, an inner radius of 0.020 0 m, and an outer radius of 0.030 0 m. The coefficient of kinetic friction between the block and the horizontal surface is 0.250. The pulley turns without friction on its axle. The light cord does not stretch and does not slip on the pulley. The block has a velocity of 0.820 m/s toward the pulley when it passes through a photogate. (a) Use energy methods to predict its speed after it has moved to a second photogate, 0.700 m away. (b) Find the angular speed of the pulley at the same moment. Figure P10.36 37. Two blocks, as shown in Figure P10.37, are connected by a string of negligible mass passing over a pulley of radius 0.250 m and moment of inertia I. The block on the frictionless incline is moving up with a constant acceleration of 2.00 m/s2. (a) Determine T1 and T2, the tensions in the two parts of the string. (b) Find the moment of inertia of the pulley. at one end as shown in Figure 10.9. The rod is released from rest in the horizontal position. What are the initial angular acceleration of the rod and the initial translational acceleration of the right end of the rod? 39. An object with a weight of 50.0 N is attached to the free end of a light string wrapped around a reel of radius 0.250 m and mass 3.00 kg. The reel is a solid disk, free to rotate in a vertical plane about the horizontal axis passing through its center. The suspended object is released 6.00 m above the floor. (a) Determine the tension in the string, the acceleration of the object, and the speed with which the object hits the floor. (b) Verify your last answer by using the principle of conservation of energy to find the speed with which the object hits the floor. Section 10.8 Angular Momentum 40. Heading straight toward the summit of Pikes Peak, an airplane of mass 12 000 kg flies over the plains of Kansas at nearly constant altitude 4.30 km with constant velocity 175 m/s west. (a) What is the airplane’s vector angular momentum relative to a wheat farmer on the ground directly below the airplane? (b) Does this value change as the airplane continues its motion along a straight line? (c) What is its angular momentum relative to the summit of Pikes Peak? Figure P10.37 38. A uniform rod of length L and mass M is free to rotate about a frictionless pivot 41. The position vector of a particle of mass 2.00 kg is given as a function of time by r (6.00iˆ 5.00tˆj) m. Determine the angular momentum of the particle about the origin as a function of time. 42. Big Ben (Fig. P10.16), the Parliament tower clock in London, has hour and minute hands with lengths of 2.70 m and 4.50 m and masses of 60.0 kg and 100 kg, respectively. Calculate the total angular momentum of these hands about the center point. Treat the hands as long, thin, uniform rods. 43. A particle of mass 0.400 kg is attached to the 100-cm mark of a meter stick of mass 0.100 kg. The meter stick rotates on a horizontal, frictionless table with an angular speed of 4.00 rad/s. Calculate the angular momentum of the system when the stick is pivoted about an axis (a) perpendicular to the table through the 50.0-cm mark and (b) perpendicular to the table through the 0-cm mark. 44. A space station is constructed in the shape of a hollow ring of mass 5.00 × 104 kg. Members of the crew walk on a deck formed by the inner surface of the outer cylindrical wall of the ring, with radius 100 m. At rest when constructed, the ring is set rotating about its axis so that the people inside experience an effective free-fall acceleration equal to g. (Fig. P10.44 shows the ring together with some other parts that make a negligible contribution to the total moment of inertia.) The rotation is achieved by firing two small rockets attached tangentially to opposite points on the outside of the ring. (a) What angular momentum does the space station acquire? (b) How long must the rockets be fired if each exerts a thrust of 125 N? (c) Prove that the total torque on the ring, multiplied by the time interval found in part (b), is equal to the change in angular momentum, found in part (a). This equality represents the angular impulse–angular momentum theorem. Figure P10.44 Problems 10.44 and 10.50. Section 10.9 Conservation of Angular Momentum 45. A cylinder with moment of inertia I1 rotates about a vertical, frictionless axle with angular speed ωi. A second cylinder, this one having moment of inertia I2 and initially not rotating, drops onto the first cylinder (Fig. P10.45). Because of friction between the surfaces, the two eventually reach the same angular speed ωf. (a) Calculate ωf. (b) Show that the kinetic energy of the system decreases in this interaction and calculate the ratio of the final to the initial rotational energy. with an angular speed of 0.750 rad/s. The moment of inertia of the student plus stool is 3.00 kg · m2 and is assumed to be constant. The student pulls the weights inward horizontally to a position 0.300 m from the rotation axis. (a) Find the new angular speed of the student. (b) Find the kinetic energy of the rotating system before and after he pulls the weights inward. Figure P10.45 46. A playground merry-go-round of radius R = 2.00 m has a moment of inertia I = 250 kg · m2 and is rotating at 10.0 rev/min about a frictionless vertical axle. Facing the axle, a 25.0-kg child hops onto the merrygo-round and manages to sit down on the edge. What is the new angular speed of the merry-go-round? 47. A 60.0-kg woman stands at the rim of a horizontal turntable having a moment of inertia of 500 kg · m2 and a radius of 2.00 m. The turntable is initially at rest and is free to rotate about a frictionless, vertical axle through its center. The woman then starts walking around the rim clockwise (as viewed from above the system) at a constant speed of 1.50 m/s relative to the Earth. (a) In what direction and with what angular speed does the turntable rotate? (b) How much work does the woman do to set herself and the turntable into motion? 48. A student sits on a freely rotating stool holding two weights, each of mass 3.00 kg (Fig. P10.48). When his arms are extended horizontally, the weights are 1.00 m from the axis of rotation and he rotates Figure P10.48 49. A puck of mass 80.0 g and radius 4.00 cm slides along an air table at a speed of 1.50 m/s as shown in Figure P10.49a. It makes a glancing collision with a second puck of radius 6.00 cm and mass 120 g (initially at rest) such that their rims just touch. Because their rims are coated with instant-acting glue, the pucks stick together and spin after the collision (Fig. P10.49b). (a) What is the angular momentum of the system relative to the center of mass? (b) What is the angular speed about the center of mass? Determine the work done on the puck. (Suggestion: Consider the change of kinetic energy.) Section 10.10 Precessional Motion of Gyroscopes (a) (b) Figure P10.49 50. A space station shaped like a giant wheel has a radius of 100 m and a moment of inertia of 5.00 × 108 kg · m2. A crew of 150 is living on the rim, and the station’s rotation causes the crew to experience an apparent free-fall acceleration of g (Fig. P10.44). When 100 people move to the center of the station for a union meeting, the angular speed changes. What apparent free-fall acceleration is experienced by the managers remaining at the rim? Assume that the average mass for each inhabitant is 65.0 kg. 51. The puck in Figure 10.24 has a mass of 0.120 kg. The distance of the puck from the center of rotation is originally 40.0 cm, and the puck is sliding with a speed of 80.0 cm/s. The string is pulled downward 15.0 cm through the hole in the frictionless table. 52. The angular momentum vector of a precessing gyroscope sweeps out a cone as shown in Figure 10.25b. Its angular speed, called its precessional frequency, is given by ωp = τ/L, where τ is the magnitude of the torque on the gyroscope and L is the magnitude of its angular momentum. In the motion called precession of the equinoxes, represented in Figure P10.52, the Earth’s axis of rotation precesses about the perpendicular to its orbital plane with a period of 2.58 × 104 yr. Model the Earth as a uniform sphere and calculate the torque on the Earth that is causing this precession. NASA Figure P10.52 (a) At present, the spin axis of the Earth points toward the North Star. (b) Torque on the spinning Earth will cause it to precess, so the spin axis will no longer be pointing in this direction in the future. Section 10.11 Rolling Motion of Rigid Objects 53. A cylinder of mass 10.0 kg rolls without slipping on a horizontal surface. At a certain instant its center of mass has a speed of 10.0 m/s. Determine (a) the translational kinetic energy of its center of mass, (b) the rotational kinetic energy about its center of mass, and (c) its total energy. 54. A uniform solid disk and a uniform hoop are placed side by side at the top of an incline of height h. If they are released from rest and roll without slipping, which object reaches the bottom first? Verify your answer by calculating their speeds when they reach the bottom in terms of h. 55. A tennis ball is a hollow sphere with a thin wall. It is set rolling without slipping at 4.03 m/s on a horizontal section of a track as shown in Figure P10.55. It rolls around the inside of a vertical circular loop 90.0 cm in diameter and finally leaves the track at a point 20.0 cm below the horizontal section. (a) Find the speed of the ball at the top of the loop. Demonstrate that it will not fall from the track. (b) Find its speed as it leaves the track. (c) Suppose static friction between ball and track were negligible so that the ball slid instead of rolling. Would its speed then be higher, lower, or the same at the top of the loop? Explain. Figure P10.55 56. A metal can containing condensed mushroom soup has mass 215 g, height 10.8 cm, and diameter 6.38 cm. It is placed at rest on its side at the top of a 3.00-m-long incline that is at 25.0° to the horizontal and is then released to roll straight down. It takes 1.50 s to reach the bottom of the incline. Assuming mechanical energy conservation, calculate the moment of inertia of the can. Which pieces of data, if any, are unnecessary in calculating the solution? Section 10.12 Context Connection— Turning the Spacecraft 57. A spacecraft is in empty space. It carries on board a gyroscope with a moment of inertia of Ig = 20.0 kg · m2 about the axis of the gyroscope. The moment of inertia of the spacecraft around the same axis is Is = 5.00 × 105 kg · m2. Neither the spacecraft nor the gyroscope is originally rotating. The gyroscope can be powered up in a negligible period of time to an angular speed of 100 s–1. If the orientation of the spacecraft is to be changed by 30.0°, for how long should the gyroscope be operated? Additional Problems 58. Review problem. A mixing beater consists of three thin rods, each 10.0 cm long. The rods diverge from a central hub, separated from each other by 120°, and all turn in the same plane. A ball is attached to the end of each rod. Each ball has crosssectional area 4.00 cm2 and is so shaped that it has a drag coefficient of 0.600. Calculate the power input required to spin the beater at 1 000 rev/min (a) in air and (b) in water. 59. A long uniform rod of length L and mass M is pivoted about a horizontal, frictionless pin through one end. The rod is released from rest in a vertical position as shown in Figure P10.59. At the instant the rod is horizontal, find (a) its angular speed, (b) the magnitude of its angular acceleration, (c) the x and y components of the acceleration of its center of mass, and (d) the components of the reaction force at the pivot. Figure P10.59 60. A uniform, hollow, cylindrical spool has inside radius R/2, outside radius R, and mass M (Fig. P10.60). It is mounted so that it rotates on a fixed, horizontal axle. A counterweight of mass m is connected to the end of a string wound around the spool. The counterweight falls from rest at t = 0 to a position y at time t. Show that the torque due to the friction forces between spool and axle is 2y 5y f Rm g 2 M 2 t 4t Figure P10.60 61. The reel shown in Figure P10.61 has radius R and moment of inertia I. One end of the block of mass m is connected to a spring of force constant k, and the other end is fastened to a cord wrapped around the reel. The reel axle and the incline are frictionless. The reel is wound counterclockwise so that the spring stretches a distance d from its unstretched position and is then released from rest. (a) Find the angular speed of the reel when the spring is again unstretched. (b) Evaluate the angular speed numerically at this point taking I = 1.00 kg · m2, R = 0.300 m, k = 50.0 N/m, m = 0.500 kg, d = 0.200 m, and θ = 37.0°. 63. A common demonstration, illustrated in Figure P10.63, consists of a ball resting at one end of a uniform board of length ℓ, hinged at the other end, and elevated at an angle θ. A light cup is attached to the board at rc so that it will catch the ball when the support stick is suddenly removed. (a) Show that the ball will lag behind the falling board when θ is less than 35.3°. (b) Assume that the board is 1.00 m long and is supported at this limiting angle. Show that the cup must be 18.4 cm from the moving end. Figure P10.61 62. A block of mass m1 = 2.00 kg and a block of mass m2 = 6.00 kg are connected by a massless string over a pulley in the shape of a solid disk having radius R = 0.250 m and mass M = 10.0 kg. These blocks are allowed to move on a fixed block-wedge of angle θ = 30.0° as shown in Figure P10.62. The coefficient of kinetic friction is 0.360 for both blocks. Draw free-body diagrams of both blocks and of the pulley. Determine (a) the acceleration of the two blocks and (b) the tensions in the string on both sides of the pulley. Figure P10.62 Figure P10.63 64. The hour hand and the minute hand of Big Ben, the Parliament tower clock in London, are 2.70 m and 4.50 m long and have masses of 60.0 kg and 100 kg, respectively (see Fig. P10.16). (a) Determine the total torque due to the weight of these hands about the axis of rotation when the time reads (i) 3:00, (ii) 5:15, (iii) 6:00, (iv) 8:20, and (v) 9:45. (You may model the hands as long, thin, uniform rods.) (b) Determine all times when the total torque about the axis of rotation is zero. Determine the times to the nearest second, solving a transcendental equation numerically. child? List the assumptions you make in solving this problem. The stove is supplied with a wall bracket to prevent the accident. 65. A string is wound around a uniform disk of radius R and mass M. The disk is released from rest with the string vertical and its top end tied to a fixed bar (Fig. P10.65). Show that (a) the tension in the string is one-third the weight of the disk, (b) the magnitude of the acceleration of the center of mass is 2g/3, and (c) the speed of the center of mass is (4gh/3)1/2 after the disk has descended through distance h. Verify your answer to (c) using the energy approach. Figure P10.66 Figure P10.65 66. A new General Electric stove has a mass of 68.0 kg and the dimensions shown in Figure P10.66. The stove comes with a warning that it can tip forward if a person stands or sits on the oven door when it is open. What can you conclude about the weight of such a person? Could it be a 67. (a) Without the wheels, a bicycle frame has a mass of 8.44 kg. Each of the wheels can be roughly modeled as a uniform solid disk with a mass of 0.820 kg and a radius of 0.343 m. Find the kinetic energy of the whole bicycle when it is moving forward at 3.35 m/s. (b) Before the invention of a wheel turning on an axle, ancient people moved heavy loads by placing rollers under them. (Modern people use rollers, too. Any hardware store will sell you a roller bearing for a lazy Susan.) A stone block of mass 844 kg moves forward at 0.335 m/s, supported by two uniform cylindrical tree trunks each of mass 82.0 kg and radius 0.343 m. No slipping occurs between the block and the rollers or between the rollers and the ground. Find the total kinetic energy of the moving objects. 68. A skateboarder with his board can be modeled as a particle of mass 76.0 kg, located at his center of mass. As shown in Figure P7.59 on page 219, the skateboarder starts from rest in a crouching position at one lip of a half-pipe (point ). The halfpipe forms one half of a cylinder of radius 6.80 m with its axis horizontal. On his descent, the skateboarder moves without friction and maintains his crouch so that his center of mass moves through one quarter of a circle of radius 6.30 m. (a) Find his speed at the bottom of the half-pipe (point ). (b) Find his angular momentum about the center of curvature. (c) Immediately after passing point , he stands up and raises his arms, lifting his center of gravity from 0.500 m to 0.950 m above the concrete (point ). Explain why his angular momentum is constant in this maneuver, whereas his linear momentum and his mechanical energy are not constant. (d) Find his speed immediately after he stands up, when his center of mass is moving in a quarter circle of radius 5.85 m. (e) What work did the skateboarder’s legs do on his body as he stood up? Next, the skateboarder glides upward with his center of mass moving in a quarter circle of radius 5.85 m. His body is horizontal when he passes point , the far lip of the half-pipe. (f) Find his speed at this location. At last he goes ballistic, twisting around while his center of mass moves vertically. (g) How high above point does he rise? (h) Over what time interval is he airborne before he touches down, facing downward and again in a crouch, 2.34 m below the level of point ? (i) Compare the solution to this problem with the solution to Problem 7.59. Which is more accurate? Why? (Caution: Do not try this maneuver yourself without the required skill and protective equipment, or in a drainage channel to which you do not have legal access.) 69. Two astronauts (Fig. P10.69), each having a mass M, are connected by a rope of length d having negligible mass. They are isolated in space, orbiting their center of mass at speeds v. Treating the astronauts as particles, calculate (a) the magnitude of the angular momentum of the system and (b) the rotational energy of the system. By pulling on the rope, one of the astronauts shortens the distance between them to d/2. (c) What is the new angular momentum of the system? (d) What are the astronauts’ new speeds? (e) What is the new rotational energy of the system? (f) How much work does the astronaut do in shortening the rope? Figure P10.69 70. When a person stands on tiptoe (a strenuous position), the position of the foot is as shown in Figure P10.70a. The total gravitational force on the body Fg is supported by the force n exerted by the floor on the toes of one foot. A mechanical model for the situation is shown in Figure P10.70b, where T is the force exerted by the Achilles tendon on the foot and R is the force exerted by the tibia on the foot. Find the values of T, R, and θ when Fg = 700 N. large forces exerted on the muscles and vertebrae. The spine pivots mainly at the fifth lumbar vertebra, with the principal supporting force provided by the erector spinalis muscle in the back. To see the magnitude of the forces involved and to understand why back problems are common among humans, consider the model shown in Fig. P10.71b for a person bending forward to lift a 200-N object. The spine and upper body are represented as a uniform horizontal rod of weight 350 N, pivoted at the base of the spine. The erector spinalis muscle, attached at a point two thirds of the way up the spine, maintains the position of the back. The angle between the spine and this muscle is 12.0°. Find the tension in the back muscle and the compressional force in the spine. (a) (b) Figure P10.70 71. A person bending forward to lift a load “with his back” (Fig. P10.71a) rather than “with his knees” can be injured by (a) friction and the position of the resultant normal force. (b) Taking F = 300 N, find the value of h for which the cabinet just begins to tip. (b) Figure P10.71 72. A wad of sticky clay with mass m and velocity v i is fired at a solid cylinder of mass M and radius R (Fig. P10.72). The cylinder is initially at rest and is mounted on a fixed horizontal axle that runs through its center of mass. The line of motion of the projectile is perpendicular to the axle and at a distance d < R from the center. (a) Find the angular speed of the system just after the clay strikes and sticks to the surface of the cylinder. (b) Is mechanical energy of the clay–cylinder system conserved in this process? Explain your answer. Figure P10.73 74. The following equations are obtained from a free-body diagram of a rectangular farm gate, supported by two hinges on the left-hand side. A bucket of grain is hanging from the latch. –A + C = 0 +B – 392 N – 50.0 N = 0 A(0) + B(0) + C(1.80 m) – 392 N(1.50 m) – 50.0 N(3.00 m) = 0 Figure P10.72 73. A force acts on a rectangular cabinet weighing 400 N as shown in Figure P10.73. (a) Assuming that the cabinet slides with constant speed when F = 200 N and h = 0.400 m, find the coefficient of kinetic (a) Draw the free-body diagram and complete the statement of the problem, specifying the unknowns. (b) Determine the values of the unknowns and state the physical meaning of each. 75. A stepladder of negligible weight is constructed as shown in Figure P10.75. A painter of mass 70.0 kg stands on the ladder 3.00 m from the bottom. Assuming that the floor is frictionless, find (a) the tension in the horizontal bar connecting the two halves of the ladder, (b) the normal forces at A and B, and (c) the components of the reaction force at the single hinge C that the left half of the ladder exerts on the right half. (Suggestion: Treat the ladder as a single object, but also treat each half of the ladder separately.) What are the force components on the sphere at the point P if h = 3R? Figure P10.76 77. Figure P10.77 shows a vertical force applied tangentially to a uniform cylinder of weight Fg. The coefficient of static friction between the cylinder and all surfaces is 0.500. In terms of Fg, find the maximum force P that can be applied that does not cause the cylinder to rotate. (Suggestion: When the cylinder is on the verge of slipping, both friction forces are at their maximum values. Why?) Figure P10.75 76. A solid sphere of mass m and radius r rolls without slipping along the track shown in Figure P10.76. It starts from rest with the lowest point of the sphere at height h above the bottom of the loop of radius R, much larger than r. (a) What is the minimum value of h (in terms of R) such that the sphere completes the loop? (b) Figure P10.77 © Copyright 2004 Thomson. All rights reserved.