Chapter 5 Problems Determine (a) the work done by the force of gravity, (b) the work done by the friction 1, 2, 3 = straightforward, intermediate, force between block and incline, and (c) the challenging work done by the normal force. = full solution available in Student Solutions Manual/Study Guide 6. A horizontal force of 150 N is used to = coached solution with push a 40.0-kg packing crate a distance of hints available at www.cp7e.com 6.00 m on a rough horizontal surface. If the = biomedical application crate moves at constant speed, find (a) the work done by the 150-N force and (b) the Section 5.1 Work coefficient of kinetic friction between the crate and surface. 1. A weight lifter lifts a 350-N set of weights from ground level to a position 7. A sledge loaded with bricks has a over his head, a vertical distance of 2.00 m. total mass of 18.0 kg and is pulled at How much work does the weight lifter do, constant speed by a rope inclined at 20.0° assuming he moves the weights at constant above the horizontal. The sledge moves a speed? distance of 20.0 m on a horizontal surface. The coefficient of kinetic friction between 2. If a man lifts a 20.0-kg bucket from a the sledge and surface is 0.500. (a) What is well and does 6.00 kJ of work, how deep is the tension in the rope? (b) How much the well? Assume that the speed of the work is done by the rope on the sledge? (c) bucket remains constant as it is lifted. What is the mechanical energy lost due to friction? 3. A tugboat exerts a constant force of 5.00 × 103 N on a ship moving at constant 8. A block of mass 2.50 kg is pushed speed through a harbor. How much work 2.20 m along a frictionless horizontal table does the tugboat do on the ship if each by a constant 16.0-N force directed 25.0° moves a distance of 3.00 km? below the horizontal. Determine the work done by (a) the applied force, (b) the normal 4. A shopper in a supermarket pushes force exerted by the table, (c) the force of a cart with a force of 35 N directed at an gravity, and (d) the net force on the block. angle of 25° downward from the horizontal. Find the work done by the shopper as she Section 5.2 Kinetic Energy and the moves down a 50-m length of aisle. Work–Energy Theorem

5. Starting from rest, a 5.00-kg block 9. A mechanic pushes a 2.50 × 103-kg slides 2.50 m down a rough 30.0° incline. car from rest to a speed of v, doing 5 000 J of The coefficient of kinetic friction between work in the process. During this time, the the block and the incline is μk = 0.436. car moves 25.0 m. Neglecting friction between car and road, find (a) v and (b) the angle of 30.0°. What is the kinetic energy of horizontal force exerted on the car. the ball at the highest point of its motion?

10. A 7.00-kg bowling ball moves at 3.00 15. A 2.0-g bullet leaves the barrel of a m/s. How fast must a 2.45-g Ping-Pong ball gun at a speed of 300 m/s. (a) Find its move so that the two balls have the same kinetic energy. (b) Find the average force kinetic energy? exerted by the expanding gases on the bullet as it moves the length of the 50-cm- 11. A person doing a chin-up weighs 700 long barrel. N, exclusive of the arms. During the first 25.0 cm of the lift, each arm exerts an 16. A 0.60-kg particle has a speed of 2.0 upward force of 355 N on the torso. If the m/s at point A and a kinetic energy of 7.5 J upward movement starts from rest, what is at point B. What is (a) its kinetic energy at the person’s velocity at that point? A? (b) its speed at point B? (c) the total work done on the particle as it moves from 12. A crate of mass 10.0 kg is pulled up a A to B? rough incline with an initial speed of 1.50 m/s. The pulling force is 100 N parallel to 17. A 2 000-kg car the incline, which makes an angle of 20.0° moves down a level with the horizontal. The coefficient of highway under the actions of two forces: a kinetic friction is 0.400, and the crate is 1 000-N forward force exerted on the drive pulled 5.00 m. (a) How much work is done wheels by the road and a 950-N resistive by gravity? (b) How much mechanical force. Use the work–energy theorem to find energy is lost due to friction? (c) How much the speed of the car after it has moved a work is done by the 100-N force? (d) What distance of 20 m, assuming that it starts is the change in kinetic energy of the crate? from rest. (e) What is the speed of the crate after being pulled 5.00 m? 18. On a frozen pond, a 10-kg sled is given a kick that imparts to it an initial 13. A 70-kg base runner begins his slide speed of v0 = 2.0 m/s. The coefficient of into second base when he is moving at a kinetic friction between sled and ice is μk = speed of 4.0 m/s. The coefficient of friction 0.10. Use the work–energy theorem to find between his clothes and Earth is 0.70. He the distance the sled moves before coming slides so that his speed is zero just as he to rest. reaches the base. (a) How much mechanical energy is lost due to friction acting on the Section 5.3 Gravitational Potential runner? (b) How far does he slide? Energy Section 5.4 Spring Potential Energy 14. An outfielder throws a 0.150-kg baseball at a speed of 40.0 m/s and an initial 19. Find the height from which you helper spring constant is 3.60 × 105 N/m, would have to drop a ball so that it would and y0 = 0.500 m. (a) What is the have a speed of 9.0 m/s just before it hits the compression of the leaf spring for a load of ground. 5.00 = 105 N? (b) How much work is done in compressing the springs? 20. A flea is able to jump about 0.5 m. It has been said that if a flea were as big as a human, it would be able to jump over a 100- story building! When an animal jumps, it converts work done in contracting muscles into gravitational potential energy (with some steps in between). The maximum force exerted by a muscle is proportional to its cross-sectional area, and the work done by the muscle is this force times the length Figure P5.22 of contraction. If we magnified a flea by a factor of 1 000, the cross section of its 23. A daredevil on a motorcycle leaves muscle would increase by 1 0002 and the the end of a ramp with a speed of 35.0 m/s length of contraction would increase by 1 as in Figure P5.23. If his speed is 33.0 m/s 000. How high would this “superflea” be when he reaches the peak of the path, what able to jump? (Don’t forget that the mass of is the maximum height that he reaches? the “superflea” increases as well.) Ignore friction and air resistance.

21. An athlete on a trampoline leaps straight up into the air with an initial speed of 9.0 m/s. Find (a) the maximum height reached by the athlete relative to the trampoline and (b) the speed of the athlete when she is halfway up to her maximum Figure P5.23 height. 24. A softball pitcher rotates a 0.250-kg 22. Truck suspensions often have ball around a vertical circular path of radius “helper springs” that engage at high loads. 0.600 m before releasing it. The pitcher One such arrangement is a leaf spring with exerts a 30.0-N force directed parallel to the a helper coil spring mounted on the axle, as motion of the ball around the complete shown in Figure P5.22. When the main leaf circular path. The speed of the ball at the spring is compressed by distance y0, the top of the circle is 15.0 m/s. If the ball is helper spring engages and then helps to released at the bottom of the circle, what is support any additional load. Suppose the its speed upon release? leaf spring constant is 5.25 × 105 N/m, the 25. The chin-up is one exercise that can be used to strengthen the biceps muscle. This muscle can exert a force of approximately 800 N as it contracts a distance of 7.5 cm in a 75-kg male. How much work can the biceps muscles (one in each arm) perform in a single contraction? Compare this amount of work with the energy required to lift a 75-kg person 40 cm in performing a chin-up. Do you think the biceps muscle is the only muscle involved Figure P5.28 (Problems 28 and 36) in performing a chin-up? 29. A 5.00-kg steel ball is dropped onto a Section 5.5 Systems and Energy copper plate from a height of 10.0 m. If the Conservation ball leaves a dent 3.20 mm deep in the plate, what is the average force exerted by the 26. A 50-kg pole vaulter running at 10 plate on the ball during the impact? m/s vaults over the bar. Her speed when she is above the bar is 1.0 m/s. Neglect air 30. A bead of mass m = 5.00 kg is resistance, as well as any energy absorbed released from point and slides on the by the pole, and determine her altitude as frictionless track shown in Figure P5.30. she crosses the bar. Determine (a) the bead’s speed at points and and (b) the net work done by the 27. A child and a sled with a combined force of gravity in moving the bead from mass of 50.0 kg slide down a frictionless to . slope. If the sled starts from rest and has a speed of 3.00 m/s at the bottom, what is the height of the hill?

28. A 0.400-kg bead slides on a curved wire, starting from rest at point in Figure P5.28. If the wire is frictionless, find the speed of the bead (a) at and (b) at . Figure P5.30

31. Tarzan swings on a 30.0-m-long vine initially inclined at an angle of 37.0° with the vertical. What is his speed at the bottom of the swing (a) if he starts from rest? (b) if he pushes off with a speed of 4.00 m/s? 32. Three objects with masses m1 = 5.0 kg, m2 = 10 kg, and m3 = 15 kg, respectively, are attached by strings over frictionless pulleys, as indicated in Figure P5.32. The horizontal surface is frictionless and the system is released from rest. Using energy concepts, find the speed of m3 after it moves down a distance of 4.0 m.

Figure P5.32 (Problems 32 and 89) Figure P5.33

33. The launching 34. A projectile is launched with a speed mechanism of a toy of 40 m/s at an angle of 60° above the gun consists of a spring of unknown spring horizontal. Use conservation of energy to constant, as shown in Figure P5.33a. If the find the maximum height reached by the spring is compressed a distance of 0.120 m projectile during its flight. and the gun fired vertically as shown, the gun can launch a 20.0-g projectile from rest 35. A 0.250-kg block is placed on a light to a maximum height of 20.0 m above the vertical spring (k = 5.00 × 103 N/m) and starting point of the projectile. Neglecting pushed downwards, compressing the all resistive forces, determine (a) the spring spring 0.100 m. After the block is released, constant and (b) the speed of the projectile it leaves the spring and continues to travel as it moves through the equilibrium upwards. What height above the point of position of the spring (where x = 0), as release will the block reach if air resistance shown in Figure P5.33b. is negligible?

36. The wire in Problem 28 (Fig. P5.28) is frictionless between points and and rough between and . The 0.400-kg bead starts from rest at . (a) Find its speed at . (b) If the bead comes to rest at , find the loss in mechanical energy as it goes 40. An airplane of mass 1.5 × 104 kg is from to . moving at 60 m/s. The pilot then revs up the engine so that the forward thrust by the 37. (a) A child slides down a water slide air around the propeller becomes 7.5 × 104 at an amusement park from an initial height N. If the force exerted by air resistance on h. The slide can be considered frictionless the body of the airplane has a magnitude of because of the water flowing down it. Can 4.0 × 104 N, find the speed of the airplane the equation for conservation of mechanical after it has traveled 500 m. Assume that the energy be used on the child? (b) Is the mass airplane is in level flight throughout this of the child a factor in determining his motion. speed at the bottom of the slide? (c) The child drops straight down rather than 41. A 2.1 × 103-kg car starts from rest at following the curved ramp of the slide. In the top of a 5.0-m-long driveway that is which case will he be traveling faster at inclined at 20° with the horizontal. If an ground level? (d) If friction is present, how average friction force of 4.0 × 103 N impedes would the conservation-of-energy equation the motion, find the speed of the car at the be modified? (e) Find the maximum speed bottom of the driveway. of the child when the slide is frictionless if the initial height of the slide is 12.0 m. 42. A 25.0-kg child on a 2.00-m-long swing is released from rest when the ropes 38. (a) A block with a mass m is pulled of the swing make an angle of 30.0° with along a horizontal surface for a distance x  the vertical. (a) Neglecting friction, find the by a constant force F at an angle θ with child’s speed at the lowest position. (b) If respect to the horizontal. The coefficient of the actual speed of the child at the lowest kinetic friction between block and table is position is 2.00 m/s, what is the mechanical μk. Is the force exerted by friction equal to energy lost due to friction? μkmg? If not, what is the force exerted by friction? (b) How much work is done by the  43. Starting from rest, a 10.0-kg block friction force and by F ? (Don’t forget the slides 3.00 m down to the bottom of a signs.) (c) Identify all the forces that do no frictionless ramp inclined 30.0° from the work on the block. (d) Let m = 2.00 kg, x = floor. The block then slides an additional 4.00 m, θ = 37.0°, F = 15.0 N, and μk = 0.400, 5.00 m along the floor before coming to a and find the answers to parts (a) and (b). stop. Determine (a) the speed of the block at the bottom of the ramp, (b) the coefficient of 39. A 70-kg diver steps off a 10-m tower kinetic friction between block and floor, and drops from rest straight down into the and (c) the mechanical energy lost due to water. If he comes to rest 5.0 m beneath the friction. surface, determine the average resistive force exerted on him by the water. 44. A child slides without friction from a the diver is constant at 50.0 N with the height h along a curved water slide (Fig. parachute closed and constant at 3 600 N P5.44). She is launched from a height h/5 with the parachute open, what is the speed into the pool. Determine her maximum of the diver when he lands on the ground? airborne height y in terms of h and the (b) Do you think the skydiver will get hurt? launch angle θ. Explain. (c) At what height should the parachute be opened so that the final speed of the skydiver when he hits the ground is 5.00 m/s? (d) How realistic is the assumption that the total retarding force is constant? Explain.

Section 5.6 Power Figure P5.44 48. A skier of mass 70 kg is pulled up a 45. A skier starts from slope by a motor-driven cable. (a) How rest at the top of a much work is required to pull him 60 m up hill that is inclined 10.5° with respect to the a 30° slope (assumed frictionless) at a horizontal. The hillside is 200 m long, and constant speed of 2.0 m/s? (b) What power the coefficient of friction between snow and must a motor have to perform this task? skis is 0.075 0. At the bottom of the hill, the snow is level and the coefficient of friction 49. Columnist Dave Barry poked fun at is unchanged. How far does the skier glide the name “The Grand Cities,” adopted by along the horizontal portion of the snow Grand Forks, North Dakota, and East before coming to rest? Grand Forks, Minnesota. Residents of the prairie towns then named a sewage 46. In a circus performance, a monkey is pumping station for him. At the Dave Barry strapped to a sled and both are given an Lift Station No. 16, untreated sewage is initial speed of 4.0 m/s up a 20° inclined raised vertically by 5.49 m in the amount of track. The combined mass of monkey and 1 890 000 liters each day. With a density of 1 sled is 20 kg, and the coefficient of kinetic 050 kg/m3, the waste enters and leaves the friction between sled and incline is 0.20. pump at atmospheric pressure through How far up the incline do the monkey and pipes of equal diameter. (a) Find the output sled move? power of the lift station. (b) Assume that a continuously operating electric motor with 47. An 80.0-kg skydiver jumps out of a average power 5.90 kW runs the pump. balloon at an altitude of 1 000 m and opens Find its efficiency. In January 2002, Barry the parachute at an altitude of 200.0 m. (a) attended the outdoor dedication of the lift Assuming that the total retarding force on station and a festive potluck supper to which the residents of the different Grand Forks sewer districts brought casseroles, period? (b) How does this amount of power Jell-O® salads, and “bars” (desserts). compare with its power during an upward trip with constant speed? 50. While running, a person dissipates about 0.60 J of mechanical energy per step Section 5.7 Work Done by a Varying per kilogram of body mass. If a 60-kg Force person develops a power of 70 W during a race, how fast is the person running? 55. The force acting on (Assume a running step is 1.5 m long.) a particle varies as in Figure P5.55. Find the work done by the 51. The electric motor of a model train force as the particle moves (a) from x = 0 to accelerates the train from rest to 0.620 m/s x = 8.00 m, (b) from x = 8.00 m to x = 10.0 m, in 21.0 ms. The total mass of the train is 875 and (c) from x = 0 to x = 10.0 m. g. Find the average power delivered to the train during its acceleration.

52. An electric scooter has a battery capable of supplying 120 Wh of energy. [Note that an energy of 1 Wh = (1 J/s)(3600 s) = 3600 J] If frictional forces and other losses account for 60.0% of the energy usage, what change in altitude can a rider achieve when driving in hilly terrain if the rider and scooter have a combined weight of 890 N? Figure P5.55

3 53. A 1.50 × 10 -kg car starts from rest 56. An object is subject to a force Fx that and accelerates uniformly to 18.0 m/s in varies with position as in Figure P5.56. Find 12.0 s. Assume that air resistance remains the work done by the force on the object as constant at 400 N during this time. Find (a) it moves (a) from x = 0 to x = 5.00 m, (b) the average power developed by the engine from x = 5.00 m to x = 10.0 m, and (c) from x and (b) the instantaneous power output of = 10.0 m to x = 15.0 m. (d) What is the total the engine at t = 12.0 s, just before the car work done by the force over the distance x stops accelerating. = 0 to x = 15.0 m?

54. A 650-kg elevator starts from rest and moves upwards for 3.00 s with constant acceleration until it reaches its cruising speed, 1.75 m/s. (a) What is the average power of the elevator motor during this comes to rest, how far has the spring been compressed?

61. (a) A 75-kg man steps out a window and falls (from rest) 1.0 m to a sidewalk. What is his speed just before his feet strike the pavement? (b) If the man falls with his knees and ankles locked, the only cushion for his fall is an approximately 0.50-cm give Figure P5.56 in the pads of his feet. Calculate the average force exerted on him by the ground in this 57. The force acting on an object is given situation. This average force is sufficient to by Fx = (8x – 16) N, where x is in meters. (a) cause damage to cartilage in the joints or to Make a plot of this force versus x from x = 0 break bones. to x = 3.00 m. (b) From your graph, find the net work done by the force as the object 62. A toy gun uses a spring to project a moves from x = 0 to x = 3.00 m. 5.3-g soft rubber sphere horizontally. The spring constant is 8.0 N/m, the barrel of the Additional Problems gun is 15 cm long, and a constant frictional force of 0.032 N exists between barrel and 58. A 2.0-m-long pendulum is released projectile. With what speed does the from rest when the support string is at an projectile leave the barrel if the spring was angle of 25° with the vertical. What is the compressed 5.0 cm for this launch? speed of the bob at the bottom of the swing? 63. Two objects are connected by a light string passing over a light, frictionless 59. An archer pulls her bowstring back pulley as in Figure P5.63. The 5.00-kg object 0.400 m by exerting a force that increases is released from rest at a point 4.00 m above uniformly from zero to 230 N. (a) What is the floor. (a) Determine the speed of each the equivalent spring constant of the bow? object when the two pass each other. (b) (b) How much work does the archer do in Determine the speed of each object at the pulling the bow? moment the 5.00-kg object hits the floor. (c) How much higher does the 3.00-kg object 60. A block of mass 12.0 kg slides from travel after the 5.00-kg object hits the floor? rest down a frictionless 35.0° incline and is stopped by a strong spring with k = 3.00 × 104 N/m. The block slides 3.00 m from the point of release to the point where it comes to rest against the spring. When the block hemispherical bowl of radius R = 30.0 cm (Fig. P5.65). Calculate (a) its gravitational potential energy at A relative to B, (b) its kinetic energy at B, (c) its speed at B, (d) its potential energy at C relative to B, and (e) its kinetic energy at C.

Figure P5.63 Figure P5.65 64. Two blocks, A and B (with mass 50 66. Energy is conventionally measured kg and 100 kg, respectively), are connected in Calories as well as in joules. One Calorie by a string, as shown in Figure P5.64. The in nutrition is 1 kilocalorie, which we define pulley is frictionless and of negligible mass. in Chapter 11 as 1 kcal = 4 186 J. The coefficient of kinetic friction between Metabolizing 1 gram of fat can release 9.00 block A and the incline is μ = 0.25. k kcal. A student decides to try to lose weight Determine the change in the kinetic energy by exercising. She plans to run up and of block A as it moves from to , a down the stairs in a football stadium as fast distance of 20 m up the incline if the system as she can and as many times as necessary. starts from rest. Is this in itself a practical way to lose weight? To evaluate the program, suppose she runs up a flight of 80 steps, each 0.150 m high, in 65.0 s. For simplicity, ignore the energy she uses in coming down (which is small). Assume that a typical efficiency for human muscles is 20.0%. This means that when your body converts 100 J from metabolizing fat, 20 J goes into doing mechanical work (here, climbing stairs). Figure P5.64 The remainder goes into internal energy. Assume the student’s mass is 50.0 kg. (a) 65. A 200-g particle is released from rest How many times must she run the flight of at point A on the inside of a smooth stairs to lose 1 pound of fat? (b) What is her average power output, in watts and in 70. A 5.0-kg block is pushed 3.0 m up a horsepower, as she is running up the stairs? vertical wall with constant speed by a constant force of magnitude F applied at an 67. In terms of saving energy, bicycling angle of θ = 30° with the horizontal, as and walking are far more efficient means of shown in Figure P5.70. If the coefficient of transportation than is travel by automobile. kinetic friction between block and wall is  For example, when riding at 10.0 mi/h, a 0.30, determine the work done by (a) F , (b) cyclist uses food energy at a rate of about the force of gravity, and (c) the normal force 400 kcal/h above what he would use if he between block and wall. (d) By how much were merely sitting still. (In exercise does the gravitational potential energy physiology, power is often measured in increase during the block’s motion? kcal/h rather than in watts. Here, 1 kcal = 1 nutritionist’s Calorie = 4 186 J.) Walking at 3.00 mi/h requires about 220 kcal/h. It is interesting to compare these values with the energy consumption required for travel by car. Gasoline yields about 1.30 × 108 J/gal. Find the fuel economy in equivalent miles per gallon for a person (a) walking and (b) bicycling.

68. An 80.0-N box is pulled 20.0 m up a 30° incline by an applied force of 100 N that points upwards, parallel to the incline. If the coefficient of kinetic friction between Figure P5.70 box and incline is 0.220, calculate the change in the kinetic energy of the box. 71. The ball launcher in a pinball machine has a spring with a force constant 69. A ski jumper starts from rest 50.0 m of 1.20 N/cm (Fig. P5.71). The surface on above the ground on a frictionless track and which the ball moves is inclined 10.0° with flies off the track at an angle of 45.0° above respect to the horizontal. If the spring is the horizontal and at a height of 10.0 m initially compressed 5.00 cm, find the above the level ground. Neglect air launching speed of a 0.100-kg ball when the resistance. (a) What is her speed when she plunger is released. Friction and the mass of leaves the track? (b) What is the maximum the plunger are negligible. altitude she attains after leaving the track? (c) Where does she land relative to the end of the track? Figure P5.71

72. The masses of the javelin, discus, and shot are 0.80 kg, 2.0 kg, and 7.2 kg, respectively, and record throws in the corresponding track events are about 98 m, 74 m, and 23 m, respectively. Neglecting air resistance, (a) calculate the minimum initial kinetic energies that would produce these throws, and (b) estimate the average force Figure P5.73 exerted on each object during the throw, assuming the force acts over a distance of 74. A hummingbird is able to hover 2.0 m. (c) Do your results suggest that air because, as the wings move downwards, resistance is an important factor? they exert a downward force on the air. Newton’s third law tells us that the air 73. Jane, whose mass is 50.0 kg, needs to exerts an equal and opposite force swing across a river filled with crocodiles in (upwards) on the wings. The average of this order to rescue Tarzan, whose mass is 80.0 force must be equal to the weight of the kg. However, she must swing into a  bird when it hovers. If the wings move constant horizontal wind force F on a vine through a distance of 3.5 cm with each that is initially at an angle of θ with the stroke, and the wings beat 80 times per vertical. (See Fig. P5.73.) In the figure, D = second, determine the work performed by 50.0 m, F = 110 N, L = 40.0 m, and θ = 50.0°. the wings on the air in 1 minute if the mass (a) With what minimum speed must Jane of the hummingbird is 3.0 grams. begin her swing in order to just make it to the other side? (Hint: First determine the 75. A child’s pogo stick (Fig. P5.75) potential energy that can be associated with stores energy in a spring (k = 2.50 × 104 the wind force. Because the wind force is N/m). At position (x1 = –0.100 m), the constant, use an analogy with the constant spring compression is a maximum and the gravitational force.) (b) Once the rescue is child is momentarily at rest. At position complete, Tarzan and Jane must swing back (x = 0), the spring is relaxed and the child is across the river. With what minimum speed moving upwards. At position , the child must they begin their swing? is again momentarily at rest at the top of the jump. Assuming that the combined mass of child and pogo stick is 25.0 kg, (a) calculate the total energy of the system if both potential energies are zero at x = 0, (b) determine x2, (c) calculate the speed of the child at x = 0, (d) determine the value of x for which the kinetic energy of the system is a maximum, and (e) obtain the child’s maximum upward speed.

Figure P5.76

77. In the dangerous “sport” of bungee jumping, a daring student jumps from a hot-air balloon with a specially designed elastic cord attached to his waist, as shown in Figure P5.77. The unstretched length of the cord is 25.0 m, the student weighs 700 N, and the balloon is 36.0 m above the surface of a river below. Calculate the required force constant of the cord if the student is to stop safely 4.00 m above the river.

Figure P5.75

76. A 2.00-kg block situated on a rough incline is connected to a spring of negligible mass having a spring constant of 100 N/m (Fig. P5.76). The block is released from rest when the spring is unstretched, and the pulley is frictionless. The block moves 20.0 cm down the incline before coming to rest. Find the coefficient of kinetic friction between block and incline.

© Jamie Budge/Corbis Figure P5.77 Bungee jumping. (Problems 77 and 82) 78. An object of mass m is suspended 79. A truck travels uphill with constant from the top of a cart by a string of length L velocity on a highway with a 7.0° slope. A as in Figure P5.78a. The cart and object are 50-kg package sits on the floor of the back initially moving to the right at a constant of the truck and does not slide, due to a speed v0. The cart comes to rest after static frictional force. During an interval in colliding and sticking to a bumper, as in which the truck travels 340 m, what is the Figure P5.78b, and the suspended object net work done on the package? What is the swings through an angle θ. (a) Show that work done on the package by the force of the initial speed is v0  2gL1 cos  . (b) If gravity, the normal force, and the friction L = 1.20 m and θ = 35.0°, find the initial force? speed of the cart. (Hint: The force exerted by the string on the object does no work on 80. As part of a curriculum unit on the object.) earthquakes, suppose that 375 000 British schoolchildren stand on their chairs and simultaneously jump down to the floor. Seismographers around the country see whether they can detect the resulting ground tremor. (This experiment was actually based on a suggestion by the children themselves.) (a) Find the energy released in the experiment. Model the children as having average mass 36.0 kg and as stepping from chair seats 38.0 cm above the floor. (b) Most of the energy is converted very rapidly into internal energy within the bodies of the children and the floors of the school buildings. Assume that 1% of the energy is carried away by a seismic wave. The magnitude of an earthquake on the Richter scale is given by

log E  4.8 M  1.5

where E is the seismic wave energy in joules. According to this model, what is the magnitude of the demonstration quake? Figure P5.78 81. A loaded ore car has a mass of 950 at the moment when the vertical separation kg and rolls on rails with negligible friction. of the blocks is h. It starts from rest and is pulled up a mine shaft by a cable connected to a winch. The shaft is inclined at 30.0° above the horizontal. The car accelerates uniformly to a speed of 2.20 m/s in 12.0 s and then continues at constant speed. (a) What power must the winch motor provide when the car is moving at constant speed? (b) What maximum power must the motor provide? (c) What total energy transfers out of the motor by work by the time the car moves off the end of the track, which is of length 1 250 m?

82. A daredevil wishes to bungee-jump Figure P5.83 from a hot-air balloon 65.0 m above a carnival midway (Fig. P5.77). He will use a 84. A cafeteria tray dispenser supports a piece of uniform elastic cord tied to a stack of trays on a shelf that hangs from harness around his body to stop his fall at a four identical spiral springs under tension, point 10.0 m above the ground. Model his one near each corner of the shelf. Each tray body as a particle and the cord as having has a mass of 580 g and is rectangular, 45.3 negligible mass and a tension force cm by 35.6 cm, and 0.450 cm thick. (a) Show described by Hooke’s force law. In a that the top tray in the stack can always be preliminary test, hanging at rest from a at the same height above the floor, however 5.00-m length of the cord, the jumper finds many trays are in the dispenser. (b) Find that his body weight stretches it by 1.50 m. the spring constant each spring should have He will drop from rest at the point where in order for the dispenser to function in this the top end of a longer section of the cord is convenient way. Is any piece of data attached to the stationary balloon. (a) What unnecessary for this determination? length of cord should he use? (b) What maximum acceleration will he experience? 85. In bicycling for aerobic exercise, a woman wants her heart rate to be between 83. The system shown in Figure P5.83 136 and 166 beats per minute. Assume that consists of a light, inextensible cord, light her heart rate is directly proportional to her frictionless pulleys, and blocks of equal mechanical power output. Ignore all forces mass. Initially, the blocks are at rest the on the woman-plus-bicycle system, except same height above the ground. The blocks for static friction forward on the drive are then released. Find the speed of block A wheel of the bicycle and an air resistance m. Compute the power-per-area measure of force proportional to the square of the this energy use. (b) Consider a car 2.10 m bicycler’s speed. When her speed is 22.0 wide and 4.90 m long traveling at 55.0 mi/h km/h, her heart rate is 90.0 beats per using gasoline having a “heat of minute. In what range should her speed be combustion” of 44.0 MJ/kg with fuel so that her heart rate will be in the range economy 25.0 mi/gallon. One gallon of she wants? gasoline has a mass of 2.54 kg. Find the power-per-area measure of the car’s energy 86. In a needle biopsy, a narrow strip of use. It can be similar to that of a steel mill tissue is extracted from a patient with a where rocks are melted in blast furnaces. (c) hollow needle. Rather than being pushed Explain why the direct use of solar energy by hand, to ensure a clean cut the needle is not practical for a conventional can be fired into the patient’s body by a automobile. spring. Assume the needle has mass 5.60 g, the light spring has force constant 375 N/m, 88. In 1887 in Bridgeport, Connecticut, and the spring is originally compressed 8.10 C. J. Belknap built the water slide shown in cm to project the needle horizontally Figure P5.88. A rider on a small sled, of without friction. The tip of the needle then total mass 80.0 kg, pushed off to start at the moves through 2.40 cm of skin and soft top of the slide (point ) with a speed of tissue, which exerts a resistive force of 7.60 2.50 m/s. The chute was 9.76 m high at the N on it. Next, the needle cuts 3.50 cm into top, 54.3 m long, and 0.51 m wide. Along its an organ, which exerts a backward force of length, 725 wheels made friction negligible. 9.20 N on it. Find (a) the maximum speed of Upon leaving the chute horizontally at its the needle and (b) the speed at which a bottom end (point ), the rider skimmed flange on the back end of the needle runs across the water of Long Island Sound for into a stop, set to limit the penetration to as much as 50 m, “skipping along like a flat 5.90 cm. pebble,” before at last coming to rest and swimming ashore, pulling his sled after 87. The power of sunlight reaching each him. (a) Find the speed of the sled and rider square meter of the Earth’s surface on a at point . (b) Model the force of water clear day in the tropics is close to 1 000 W. friction as a constant retarding force acting On a winter day in Manitoba, the power on a particle. Find the work done by water 2 concentration of sunlight can be 100 W/m . friction in stopping the sled and rider. (c) Many human activities are described by a Find the magnitude of the force the water 2 power-per-footprint-area on the order of 10 exerts on the sled. (d) Find the magnitude 2 W/m or less. (a) Consider, for example, a of the force the chute exerts on the sled at family of four paying $80 to the electric point . company every 30 days for 600 kWh of energy carried by electric transmission to their house, with floor area 13.0 m by 9.50 Figure P5.88

89. Three objects with masses m1 = 5.0

kg, m2 = 10 kg, and m3 = 15 kg, respectively, are attached by strings over frictionless pulleys as indicated in Figure P5.32. The horizontal surface exerts a force of friction

of 30 N on m2. If the system is released from Engraving from Scientific American, July 1888. rest, use energy concepts to find the speed

of m3 after it moves down 4.0 m.

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