4 Forces in One Dimension

Transcription

1 CHAPTER 4 Forces in One Dimension Practice Problems 4.1 Force and Motion paes pae 89 For each of the followin situations, specify the system and draw a motion diaram and a free-body diaram. Label all forces with their aents, and indicate the direction of the acceleration and of the net force. Draw vectors of appropriate lenths. 1. A flowerpot falls freely from a windowsill. (Inore any forces due to air resistance.) 3. A cable pulls a crate at a constant speed across a horizontal surface. The surface provides a force that resists the crate s motion. v v v v a 0 System x System F friction on crate 0 F pull on crate y F Earth s mass on flowerpot v a 4. A rope lifts a bucket at a constant speed. (Inore air resistance.) y 2. A sky diver falls downward throuh the air at constant velocity. (The air exerts an upward force on the person.) System y v v v a 0 F air resistance on diver 0 F Earth s mass on diver System 5. A rope lowers a bucket at a constant speed. (Inore air resistance.) System y v v v v v v v v a 0 a 0 F rope on bucket 0 F Earth s mass on bucket F rope on bucket 0 F Earth s mass on bucket Physics: Principles and Problems Solutions Manual 61

2 pae Two horizontal forces, 225 N and 165 N, are exerted on a canoe. If these forces are applied in the same direction, find the net horizontal force on the canoe. 225 N 165 N N in the direction of the two forces 7. If the same two forces as in the previous problem are exerted on the canoe in opposite directions, what is the net horizontal force on the canoe? Be sure to indicate the direction of the net force. 225 N 165 N N in the direction of the larer force 8. Three confused sleih dos are tryin to pull a sled across the Alaskan snow. Alutia pulls east with a force of 35 N, Seward also pulls east but with a force of 42 N, and bi Kodiak pulls west with a force of 53 N. What is the net force on the sled? Identify east as positive and the sled as the system. F Alutia on sled F Seward on sled F Kodiak on sled 35 N 42 N 53 N 24 N 24 N east Section Review 4.1 Force and Motion paes pae Force Identify each of the followin as either a, b, or c: weiht, mass, inertia, the push of a hand, thrust, resistance, air resistance, sprin force, and acceleration. a. a contact force b. a field force c. not a force weiht (b), mass (c), inertia (c), push of a hand (a), thrust (a), resistance (a), air resistance (a), sprin force (a), acceleration (c) 10. Inertia Can you feel the inertia of a pencil? Of a book? If you can, describe how. Yes, you can feel the inertia of either object by usin your hand to ive either object an acceleration; that is, try to chane the objects velocity. 11. Free-Body Diaram Draw a free-body diaram of a ba of suar bein lifted by your hand at a constant speed. Specifically identify the system. Label all forces with their aents and make the arrows the correct lenths. System 12. Direction of Velocity If you push a book in the forward direction, does this mean its velocity has to be forward? No, it could be movin backward and you would be reducin that velocity. 13. Free-Body Diaram Draw a free-body diaram of a water bucket bein lifted by a rope at a decreasin speed. Specifically identify the system. Label all forces with their aents and make the arrows the correct lenths. System Suar F hand on ba a 0 F Earth s mass on ba F rope on bucket a F Earth s mass on bucket 62 Solutions Manual Physics: Principles and Problems

3 14. Critical Thinkin A force of 1 N is the only force exerted on a block, and the acceleration of the block is measured. When the same force is the only force exerted on a second block, the acceleration is three times as lare. What can you conclude about the masses of the two blocks? Because m F/a and the forces are the same, the mass of the second block is one-third the mass of the first block. 18. In Fiure 4-8, the block has a mass of 1.2 k and the sphere has a mass of 3.0 k. What are the readins on the two scales? (Nelect the masses of the scales.) Practice Problems 4.2 Usin Newton s Laws paes pae You place a watermelon on a sprin scale at the supermarket. If the mass of the watermelon is 4.0 k, what is the readin on the scale? The scale reads the weiht of the watermelon: m (4.0 k)(9.80 m/s 2 ) 39 N 16. Kamaria is learnin how to ice-skate. She wants her mother to pull her alon so that she has an acceleration of 0.80 m/s 2. If Kamaria s mass is 27.2 k, with what force does her mother need to pull her? (Nelect any resistance between the ice and Kamaria s skates.) ma (27.2 k)(0.80 m/s 2 ) 22 N 17. Taru and Reiko simultaneously rab a 0.75-k piece of rope and bein tuin on it in opposite directions. If Taru pulls with a force of 16.0 N and the rope accelerates away from her at 1.25 m/s 2, with what force is Reiko pullin? Identify Reiko s direction as positive and the rope as the system. F Rieko on rope F Taru on rope ma F Rieko on rope ma F Taru on rope Fiure 4-8 Bottom scale: Identify the sphere as the system and up as positive. F scale on sphere F Earth s mass on sphere ma 0 F scale on sphere F Earth s mass on sphere m sphere (3.0 k)(9.80 m/s 2 ) 29 N Top scale: Identify the block as the system and up as positive. F top scale on block F bottom scale on block F Earth s mass on block ma 0 F top scale on block F bottom scale on block F Earth s mass on block F bottom scale on block m block 29 N (1.2 k) (9.80 m/s 2 ) 41 N (0.75 k)(1.25 m/s 2 ) 17 N 16.0 N Physics: Principles and Problems Solutions Manual 63

4 pae On Earth, a scale shows that you weih 585 N. a. What is your mass? The scale reads 585 N. Since there is no acceleration, your weiht equals the downward force of ravity: m 585 N so m 59.7 k 9.80 m/s 2 b. What would the scale read on the Moon ( 1.60 m/s 2 )? On the moon, chanes: m Moon (59.7 k)(1.60 m/s 2 ) 95.5 N 20. Use the results from Example Problem 2 to answer questions about a scale in an elevator on Earth. What force would be exerted by the scale on a person in the followin situations? a. The elevator moves at constant speed. Constant speed, so a 0 and 0. F scale m (75.0 k)(9.80 m/s 2 ) 735 N b. It slows at 2.00 m/s 2 while movin upward. Slowin while movin upward, so a 2.00 m/s 2 F scale ma m m(a ) (75.0 k)(2.00 m/s N 9.80 m/s 2 ) c. It speeds up at 2.00 m/s 2 while movin downward. Acceleratin downward, so a 2.00 m/s 2 F scale ma m m(a ) (75.0 k)(2.00 m/s N 9.80 m/s 2 ) d. It moves downward at constant speed. Constant speed, so a 0 and 0 F scale m (75.0 k)(9.80 m/s 2 ) 735 N e. It slows to a stop at a constant manitude of acceleration. Constant acceleration a, thouh the sin of a depends on the direction of the motion that is endin. F scale ma m (75.0 k)(a) (75.0 k)(9.80 m/s 2 ) (75.0 k)(a) 735 N 64 Solutions Manual Physics: Principles and Problems

5 Section Review 4.2 Usin Newton s Laws paes pae Lunar Gravity Compare the force holdin a 10.0-k rock on Earth and on the Moon. The acceleration due to ravity on the Moon is 1.62 m/s 2. To hold the rock on Earth: F Earth on rock F hold on rock 0 F hold on rock F Earth on rock m Earth (10.0 k)(9.80 m/s 2 ) 98.0 N To hold the rock on the Moon: F Moon on rock F hold on rock 0 F hold on rock F Moon on rock m Moon (10.0 k)(1.62 m/s 2 ) 16.2 N Constant Velocity F scale apparent weiht real weiht Slowin While Risin/ Speedin Up While Descendin F scale apparent weiht real weiht Speedin Up While Risin/ Slowin While Descendin F scale 22. Real and Apparent Weiht You take a ride in a fast elevator to the top of a tall buildin and ride back down while standin on a bathroom scale. Durin which parts of the ride will your apparent and real weihts be the same? Durin which parts will your apparent weiht be less than your real weiht? More than your real weiht? Sketch free-body diarams to support your answers. Apparent weiht and real weiht are the same when you are travelin either up or down at a constant velocity. Apparent weiht is less than real weiht when the elevator is slowin while risin or speedin up while descendin. Apparent weiht is reater when speedin up while risin or slowin while oin down. 23. Acceleration Tecle, with a mass of 65.0 k, is standin by the boards at the side of an ice-skatin rink. He pushes off the boards with a force of 9.0 N. What is his resultin acceleration? Identify Tecle as the system and the direction away from the boards as positive. The ice can be treated as a resistance-free surface. F boards on Tecle ma a F boards on Tecle m 9.0 N 65.0 k apparent weiht real weiht 0.14 m/s 2 away from the boards Physics: Principles and Problems Solutions Manual 65

6 24. Motion of an Elevator You are ridin in an elevator holdin a sprin scale with a 1-k mass suspended from it. You look at the scale and see that it reads 9.3 N. What, if anythin, can you conclude about the elevator s motion at that time? If the elevator is stationary or movin at a constant velocity, the scale should read 9.80 N. Because the scale reads a lihter weiht, the elevator must be acceleratin downward. To find the exact acceleration: identify up as positive and the 1-k mass as the system. F scale on 1 k a 0.5 m/s 2 F Earth s mass on 1 k ma 0.5 m/s 2 downward 25. Mass Marcos is playin tu-of-war with his cat usin a stuffed toy. At one instant durin the ame, Marcos pulls on the toy with a force of 22 N, the cat pulls in the opposite direction with a force of 19.5 N, and the toy experiences an acceleration of 6.25 m/s 2. What is the mass of the toy? Identify the toy as the system and the direction toward his cat as the positive direction. F Marcos on toy F cat on toy ma m F scale on 1 k F Earth s mass on 1 k m 9.3 N 9.80 N 1 k F Marcos on toy F cat on toy a 22 N 19.5 N 6.25 m/s k 26. Acceleration A sky diver falls at a constant speed in the spread-eale position. After he opens his parachute, is the sky diver acceleratin? If so, in which direction? Explain your answer usin Newton s laws. Yes, for a while the diver is acceleratin upward because there is an additional upward force due to air resistance on the parachute. The upward acceleration causes the driver s downward velocity to decrease. Newton s second law says that a net force in a certain direction will result in an acceleration in that direction ( ma). 27. Critical Thinkin You have a job at a meat warehouse loadin inventory onto trucks for shipment to rocery stores. Each truck has a weiht limit of 10,000 N of caro. You push each crate of meat alon a low-resistance roller belt to a scale and weih it before movin it onto the truck. However, riht after you weih a 1000-N crate, the scale breaks. Describe a way in which you could apply Newton s laws to fiure out the approximate masses of the remainin crates. Answers may vary. One possible answer is the followin: You can nelect resistance if you do all your maneuverin on the roller belt. Because you know the weiht of the 1000 N crate, you can use it as your standard. Pull on the 1000 N crate with a particular force for 1 s, estimate its velocity, and calculate the acceleration that your force ave to it. Next, pull on a crate of unknown mass with as close to the same force as you can for 1 s. Estimate the crate s velocity and calculate the acceleration your force ave to it. The force you pulled with on each crate will be the net force in each case N crate unknown crate (1000 N)(a 1000-N crate ) (m unk )(a unk ) m unk (1000 N)(a 1000-N crate ) aunk 66 Solutions Manual Physics: Principles and Problems

7 Chapter 4 continued Practice Problems 4.3 Interaction Forces paes pae You lift a relatively liht bowlin ball with your hand, acceleratin it upward. What are the forces on the ball? What forces does the ball exert? What objects are these forces exerted on? The forces on the ball are the force of your hand and the ravitational force of Earth s mass. The ball exerts a force on your hand and a ravitational force on Earth. All these forces are exerted on your hand, on the ball, or on Earth. 29. A brick falls from a construction scaffold. Identify any forces actin on the brick. Also identify any forces that the brick exerts and the objects on which these forces are exerted. (Air resistance may be inored.) The only force actin on the brick is the ravitational attraction of Earth s mass. The brick exerts an equal and opposite force on Earth. 30. You toss a ball up in the air. Draw a freebody diaram for the ball while it is still movin upward. Identify any forces actin on the ball. Also identify any forces that the ball exerts and the objects on which these forces are exerted. F Earth s mass on ball The only force actin on the ball is the force of Earth s mass on the ball, when inorin air resistance. The ball exerts an equal and opposite force on Earth. 31. A suitcase sits on a stationary airport luae cart, as in Fiure Draw a free-body diaram for each object and specifically indicate any interaction pairs between the two. Suitcase Fiure 4-13 pae You are helpin to repair a roof by loadin equipment into a bucket that workers hoist to the rooftop. If the rope is uaranteed not to break as lon as the tension does not exceed 450 N and you fill the bucket until it has a mass of 42 k, what is the reatest acceleration that the workers can ive the bucket as they pull it to the roof? Identify the bucket as the system and up as positive. F rope on bucket a F cart on suitcase F Earth s mass on suitcase ma 0.91 m/s 2 F Earth s mass on cart F Earth s mass on bucket F rope on bucket F Earth s mass on bucket m F rope on bucket m m Cart 450 N (42 k)(9.80 m/s 2 ) 42 k F surface on cart F suitcase on cart Physics: Principles and Problems Solutions Manual 67

8 33. Dieo and Mika are tryin to fix a tire on Dieo s car, but they are havin trouble ettin the tire loose. When they pull toether, Mika with a force of 23 N and Dieo with a force of 31 N, they just barely et the tire to bude. What is the manitude of the strenth of the force between the tire and the wheel? Identify the tire as the system and the direction of pullin as positive. F wheel on tire F Mika on tire F Dieo on tire ma 0 F wheel on tire F Mika on tire F Dieo on tire 23 N 31 N 54 N Section Review 4.3 Interaction Forces paes pae Force Hold a book motionless in your hand in the air. Identify each force and its interaction pair on the book. The forces on the book are downward force of ravity due to the mass of Earth and the upward force of the hand. The force of the book on Earth and the force of the book on the hand are the other halves of the interaction pairs. 35. Force Lower the book from problem 34 at increasin speed. Do any of the forces or their interaction-pair partners chane? Explain. Yes, the force of the hand on the book becomes smaller so there is a downward acceleration. The force of the book also becomes smaller; you can feel that. The interaction pair partners remain the same. 36. Tension A block hans from the ceilin by a massless rope. A second block is attached to the first block and hans below it on another piece of massless rope. If each of the two blocks has a mass of 5.0 k, what is the tension in each rope? For the bottom rope with the positive direction upward: F bottom rope on bottom block F Earth s mass on bottom block ma 0 F bottom rope on bottom block F Earth s mass on bottom block m (5.0 k)(9.80 m/s 2 ) 49 N For the top rope, with the positive direction upward: F top rope on top block F bottom rope on top block F Earth s mass on top block ma 0 F top rope on top block F Earth s mass on top block F bottom rope on top block m F bottom rope on top block (5.0 k)(9.80 m/s 2 ) 49 N 98 N 37. Tension If the bottom block in problem 36 has a mass of 3.0 k and the tension in the top rope is 63.0 N, calculate the tension in the bottom rope and the mass of the top block. For the bottom rope with the positive direction upward: F bottom rope on bottom block F Earth s mass on bottom block ma 0 F bottom rope on bottom block F Earth s mass on bottom block (3.0 k)(9.80 m/s 2 ) 29 N 68 Solutions Manual Physics: Principles and Problems

9 For the top mass with the positive direction upward: F top rope on top block F bottom rope on top block F Earth s mass on top block ma 0 F Earth s mass on top block m F top rope on top block Chapter Assessment Concept Mappin pae Complete the followin concept map usin the followin term and symbols: normal, F T,. force m F bottom rope on top block tension normal ravity F top rope on top block F bottom rope on top block 63.0 N 29 N 9.80 m/s k 38. Normal Force Poloma hands a 13-k box to 61-k Stephanie, who stands on a platform. What is the normal force exerted by the platform on Stephanie? Identify Stephanie as the system and positive to be upward. F platform on Stephanie F box on Stephanie F Earth s mass on Stephanie F platform on Stephanie F box on Stephanie F Earth s mass on Stephanie m box m Stephanie (13 k)(9.80 m/s 2 ) (61 k)(9.80 m/s 2 ) N 39. Critical Thinkin A curtain prevents two tu-of-war teams from seein each other. One team ties its end of the rope to a tree. If the other team pulls with a 500-N force, what is the tension? Explain. The tension would be 500 N. The rope is in equilibrium, so the net force on it is zero. The team and the tree exert equal forces in opposite directions. F T F N Masterin Concepts pae A physics book is motionless on the top of a table. If you ive it a hard push with your hand, it slides across the table and slowly comes to a stop. Use Newton s laws to answer the followin questions. (4.1) a. Why does the book remain motionless before the force of your hand is applied? An object at rest tends to stay at rest if no outside force acts on it. b. Why does the book bein to move when your hand pushes hard enouh on it? The force from your hand is reater than any opposin force such as friction. With a net force on it, the book slides in the direction of the net force. c. Under what conditions would the book remain in motion at a constant speed? The book would remain in motion if the net force actin on it is zero. 42. Cyclin Why do you have to push harder on the pedals of a sinle-speed bicycle to start it movin than to keep it movin at a constant velocity? (4.1) A lare force is required to accelerate the mass of the bicycle and rider. Once the desired constant velocity is reached, a much smaller force is sufficient to overcome the ever-present frictional forces. Physics: Principles and Problems Solutions Manual 69

10 43. Suppose that the acceleration of an object is zero. Does this mean that there are no forces actin on it? Give an example supportin your answer. (4.2) No, it only means the forces actin on it are balanced and the net force is zero. For example, a book on a table is not movin but the force of ravity pulls down on it and the normal force of the table pushes up on it and these forces are balanced. 44. Basketball When a basketball player dribbles a ball, it falls to the floor and bounces up. Is a force required to make it bounce? Why? If a force is needed, what is the aent involved? (4.2) Yes, its velocity chaned direction; thus, it was accelerated and a force is required to accelerate the basketball. The aent is the floor. 45. Before a sky diver opens her parachute, she may be fallin at a velocity hiher than the terminal velocity that she will have after the parachute opens. (4.2) a. Describe what happens to her velocity as she opens the parachute. Because the force of air resistance suddenly becomes larer, the velocity of the diver drops suddenly. b. Describe the sky diver s velocity from when her parachute has been open for a time until she is about to land. The force of air resistance and the ravitational force are equal. Their sum is zero, so there is no loner any acceleration. The sky diver continues downward at a constant velocity. 46. If your textbook is in equilibrium, what can you say about the forces actin on it? (4.2) If the book is in equilibrium, the net force is zero. The forces actin on the book are balanced. 47. A rock is dropped from a bride into a valley. Earth pulls on the rock and accelerates it downward. Accordin to Newton s third law, the rock must also be pullin on Earth, yet Earth does not seem to accelerate. Explain. (4.3) The rock does pull on Earth, but Earth s enormous mass would undero only a minute acceleration as a result of such a small force. This acceleration would o undetected. 48. Ramon pushes on a bed that has been pushed aainst a wall, as in Fiure Draw a free-body diaram for the bed and identify all the forces actin on it. Make a separate list of all the forces that the bed applies to other objects. (4.3) F wall on bed Fiure 4-17 Forces that bed applies to other objects: F bed on Ramon, F bed on Earth, F bed on floor, F bed on wall F floor on bed F Ramon on bed 70 Solutions Manual Physics: Principles and Problems

11 49. Fiure 4-18 shows a block in four different situations. Rank them accordin to the manitude of the normal force between the block and the surface, reatest to least. Specifically indicate any ties. (4.3) 52. Baseball A sluer swins his bat and hits a baseball pitched to him. Draw free-body diarams for the baseball and the bat at the moment of contact. Specifically indicate any interaction pairs between the two diarams. (4.3) F ball on bat F batter on bat F bat on ball Fiure 4-18 from left to riht: second > fourth > third > first 50. Explain why the tension in a massless rope is constant throuhout it. (4.3) If you draw a free-body diaram for any point on the rope, there will be two tension forces actin in opposite directions. F up F down ma 0 (because it is massless). Therefore, F up F down.accordin to Newton s third law, the force that the adjoinin piece of rope exerts on this point is equal and opposite to the force that this point exerts on it, so the force must be constant throuhout. 51. A bird sits on top of a statue of Einstein. Draw free-body diarams for the bird and the statue. Specifically indicate any interaction pairs between the two diarams. (4.3) F statue on bird, bird F Earth on statue F bird on statue Applyin Concepts paes Whiplash If you are in a car that is struck from behind, you can receive a serious neck injury called whiplash. a. Usin Newton s laws, explain what happens to cause such an injury. The car is suddenly accelerated forward. The seat accelerates your body, but your neck has to accelerate your head. This can hurt your neck muscles. b. How does a headrest reduce whiplash? The headrest pushes on your head, acceleratin it in the same direction as the car. 54. Space Should astronauts choose pencils with hard or soft lead for makin notes in space? Explain. A soft lead pencil would work better because it would require less force to make a mark on the paper. The manitude of the interaction force pair could push the astronaut away from the paper., statue Physics: Principles and Problems Solutions Manual 71

12 55. When you look at the label of the product in Fiure 4-19 to et an idea of how much the box contains, does it tell you its mass, weiht, or both? Would you need to make any chanes to this label to make it correct for consumption on the Moon? 58. You toss a ball straiht up into the air. a. Draw a free-body diaram for the ball at three points durin its motion: on the way up, at the very top, and on the way down. Specifically identify the forces actin on the ball and their aents. On the way up F Earth s mass on ball At the top Fiure 4-19 The ounces tell you the weiht in Enlish units. The rams tell you the mass in metric units. The label would need to read 2 oz to be correct on the Moon. The rams would remain unchaned. 56. From the top of a tall buildin, you drop two table-tennis balls, one filled with air and the other with water. Both experience air resistance as they fall. Which ball reaches terminal velocity first? Do both hit the round at the same time? The lihter, air-filled table tennis ball reaches terminal velocity first. Its mass is less for the same shape and size, so the friction force of upward air resistance becomes equal to the downward force of m sooner. Because the force of ravity on the water-filled table-tennis ball (more mass) is larer, its terminal velocity is larer, and it strikes the round first. 57. It can be said that 1 k equals 2.2 lb. What does this statement mean? What would be the proper way of makin the comparison? It means that on Earth s surface, the weiht of 1 k is equivalent to 2.2 lb. You should compare masses to masses and weihts to weihts. Thus 9.8 N equals 2.2 lb. b. What is the velocity of the ball at the very top of the motion? 0 m/s c. What is the acceleration of the ball at this same point? Because the only force actin on it is the ravitational attraction of Earth, a 9.80 m/s 2. Masterin Problems 4.1 Force and Motion pae 113 Level What is the net force actin on a 1.0-k ball in free-fall? m (1.0 k)(9.80 m/s 2 ) 9.8 N On the way down F Earth s mass on ball F Earth s mass on ball 72 Solutions Manual Physics: Principles and Problems

13 60. Skatin Joyce and Efua are skatin. Joyce pushes Efua, whose mass is 40.0-k, with a force of 5.0 N. What is Efua s resultin acceleration? F ma a F m 0.12 m/s A car of mass 2300 k slows down at a rate of 3.0 m/s 2 when approachin a stop sin. What is the manitude of the net force causin it to slow down? F ma 5.0 N 40.0 k (2300 k)(3.0 m/s 2 ) N 65. Three objects are dropped simultaneously from the top of a tall buildin: a shot put, an air-filled balloon, and a basketball. a. Rank the objects in the order in which they will reach terminal velocity, from first to last. balloon, basketball, shot put b. Rank the objects accordin to the order in which they will reach the round, from first to last. shot put, basketball, balloon c. What is the relationship between your answers to parts a and b? They are inverses of each other. 66. What is the weiht in pounds of a N wooden shippin case? 1 k 9.80 N 2.2 lb 1 k (100.0 N) 22 lb 62. Breakin the Wishbone After Thanksivin, Kevin and Gamal use the turkey s wishbone to make a wish. If Kevin pulls on it with a force 0.17 N larer than the force Gamal pulls with in the opposite direction, and the wishbone has a mass of 13, what is the wishbone s initial acceleration? a 13 m/s Usin Newton s Laws paes Level What is your weiht in newtons? m (9.80 m/s 2 )(m) Answers will vary. 64. Motorcycle Your new motorcycle weihs 2450 N. What is its mass in kilorams? m m F m 0.17 N k 2450 N 9.80 m/s N 67. You place a 7.50-k television on a sprin scale. If the scale reads 78.4 N, what is the acceleration due to ravity at that location? m m 78.4 N 7.50 k 10.5 m/s 2 Level Dra Racin A 873-k (1930-lb) draster, startin from rest, attains a speed of 26.3 m/s (58.9 mph) in 0.59 s. a. Find the averae acceleration of the draster durin this time interval. a v t 45 m/s 2 b. What is the manitude of the averae net force on the draster durin this time? F ma (26.3 m/s 0.0 m/s) 0.59 s (873 k)(45 m/s 2 ) N Physics: Principles and Problems Solutions Manual 73

14 c. Assume that the driver has a mass of 68 k. What horizontal force does the seat exert on the driver? F ma (68 k)(45 m/s 2 ) N 69. Assume that a scale is in an elevator on Earth. What force would the scale exert on a 53-k person standin on it durin the followin situations? a. The elevator moves up at a constant speed. F scale m (53 k)(9.80 m/s 2 ) N b. It slows at 2.0 m/s 2 while movin upward. Slows while movin up or speeds up while movin down, F scale F slowin m ma m( a) (53 k)(9.80 m/s m/s 2 ) N c. It speeds up at 2.0 m/s 2 while movin downward. Slows while movin up or speeds up while movin down, F scale F speedin m ma m( a) (53 k)(9.80 m/s m/s 2 ) N d. It moves downward at a constant speed. F scale m (53 k)(9.80 m/s 2 ) N e. It slows to a stop while movin downward with a constant acceleration. Depends on the manitude of the acceleration. F scale F slowin m ma m( a) (53 k)(9.80 m/s 2 a) 70. A rocery sack can withstand a maximum of 230 N before it rips. Will a ba holdin 15 k of roceries that is lifted from the checkout counter at an acceleration of 7.0 m/s 2 hold? Use Newton s second law ma. If roceries 230, then the ba rips. roceries m roceries a roceries m roceries m roceries (a roceries ) (15 k)(7.0 m/s m/s 2 ) 250 N The ba does not hold. 71. A 0.50-k uinea pi is lifted up from the round. What is the smallest force needed to lift it? Describe its resultin motion. F lift m (0.50 k)(9.80 m/s 2 ) 4.9 N It would move at a constant speed. Level Astronomy On the surface of Mercury, the ravitational acceleration is 0.38 times its value on Earth. a. What would a 6.0-k mass weih on Mercury? m(0.38) (6.0 k)(9.80 m/s 2 )(0.38) 22 N 74 Solutions Manual Physics: Principles and Problems

15 Chapter 4 continued b. If the ravitational acceleration on the surface of Pluto is 0.08 times that of Mercury, what would a 7.0-k mass weih on Pluto? m(0.38)(0.08) (7.0 k)(9.80 m/s 2 )(0.38)(0.08) 2.1 N 73. A 65-k diver jumps off of a 10.0-m tower. a. Find the diver s velocity when he hits the water. v f 2 v i 2 2d v i 0 m/s so v f 2d 2(9.80 (10.0 m/s 2 ) m) 14.0 m/s b. The diver comes to a stop 2.0 m below the surface. Find the net force exerted by the water. v f 2 v i 2 2ad v f 0, so a and F ma N 74. Car Racin A race car has a mass of 710 k. It starts from rest and travels 40.0 m in 3.0 s. The car is uniformly accelerated durin the entire time. What net force is exerted on it? 1 d v 0 t at 2 2 Since v 0 0, a F 2d t 2 2md t 2 mv i 2 2d v i 2 2d (65 k)(14.0 m/s) 2 2(2.0 m) and F ma, so 4.3 Interaction Forces pae 114 Level A 6.0-k block rests on top of a 7.0-k block, which rests on a horizontal table. a. What is the force (manitude and direction) exerted by the 7.0-k block on the 6.0-k block? N m F N F 7-k block on 6-k block m (6.0 k)(9.80 m/s 2 ) 59 N; the direction is upward. b. What is the force (manitude and direction) exerted by the 6.0-k block on the 7.0-k block? equal and opposite to that in part a; therefore, 59 N downward 76. Rain A raindrop, with mass 2.45 m, falls to the round. As it is fallin, what manitude of force does it exert on Earth? F raindrop on Earth m ( k)(9.80 m/s 2 ) N 77. A 90.0-k man and a 55-k man have a tu-of-war. The 90.0-k man pulls on the rope such that the 55-k man accelerates at m/s 2. What force does the rope exert on the 90.0-k man? same in manitude as the force the rope exerts on the 55-k man: F ma (55 k)(0.025 m/s 2 ) 1.4 N (2)(710 k)(40.0 m) (3.0 s) N Physics: Principles and Problems Solutions Manual 75

16 Level Male lions and human sprinters can both accelerate at about 10.0 m/s 2. If a typical lion weihs 170 k and a typical sprinter weihs 75 k, what is the difference in the force exerted on the round durin a race between these two species? Use Newton s second law, ma. The difference between F lion and F human is F lion F human m lion a lion m human a human (170 k)(10.0 m/s 2 ) (75 k)(10.0 m/s 2 ) N F middle block on top block F Earth s mass on top block ma 0 F middle block on top block F Earth s mass on top block m top block (4.6 k)(9.80 m/s 2 ) 45 N The normal force is between the bottom and middle block; the middle block is the system; upward is positive. F bottom block on middle block F top block on middle block 79. A 4500-k helicopter accelerates upward at 2.0 m/s 2. What lift force is exerted by the air on the propellers? ma F appl F appl m so F appl ma m m(a ) (4500 k)((2.0m/s 2 ) (9.8 m/s 2 )) N Level Three blocks are stacked on top of one another, as in Fiure The top block has a mass of 4.6 k, the middle one has a mass of 1.2 k, and the bottom one has a mass of 3.7 k. Identify and calculate any normal forces between the objects. F Earth s mass on middle block ma 0 F top block on middle block F middle block on top block F bottom block on middle block F middle block on top block F Earth s mass on middle block F middle block on top block m middle block 45 N (1.2 k)(9.80 m/s 2 ) 57 N The normal force is between the bottom block and the surface; the bottom block is the system; upward is positive. F surface on bottom block F middle block on bottom block F Earth s mass on bottom block ma 0 Fiure 4-20 The normal force is between the top and middle blocks; the top block is the system; upward is positive. 76 Solutions Manual Physics: Principles and Problems

17 F surface on bottom block F middle block on bottom block F Earth s mass on bottom block F middle block on bottom block m bottom block F thrust ma, so a m m F thrust m v f v i at and v i 0, so 57 N (3.7 k)(9.80 m/s 2 ) 93 N Mixed Review paes Level The draster in problem 68 completed a m ( mi) run in s. If the car had a constant acceleration, what was its acceleration and final velocity? 1 d f d i v i t at 2 2 d i v i 0, so a m/s 2 1 d f d i (v f v i )t 2 d i v i 0, so v f 2d f t 2 (2)(402.3 m) (4.936 s) 2 2d f t (2)(402.3 m) s m/s Level Jet A N catapult jet plane is ready for takeoff. If the jet s enines supply a constant thrust of N, how much runway will it need to reach its minimum takeoff speed of 285 km/h? 1 h v f (285 km/h)(1000 m/km) 3600s 79.2 m/s t v f a F th vfm F thrust 1 d f d i v i t at 2 2 d i v i 0, so 1 d f at F th 1 2 F v f rust m 139 m F rust m vf 2 thrm ust v 2 f Fthrust vfm 2 thrust (79.2 m/s) N 9.80 m/s N 83. The draster in problem 68 crossed the finish line oin m/s. Does the assumption of constant acceleration hold true? What other piece of evidence could you use to determine if the acceleration was constant? m/s is slower than found in problem 81, so the acceleration cannot be constant. Further, the acceleration in the first 0.59 s was 45 m/s 2,not m/s 2. Physics: Principles and Problems Solutions Manual 77

18 84. Suppose a 65-k boy and a 45-k irl use a massless rope in a tu-of-war on an icy, resistance-free surface as in Fiure If the acceleration of the irl toward the boy is 3.0 m/s 2, find the manitude of the acceleration of the boy toward the irl. 86. Baseball As a baseball is bein cauht, its speed oes from 30.0 m/s to 0.0 m/s in about s. The mass of the baseball is k. a. What is the baseball s acceleration? a v f v i tf t i 0.0 m/s 30.0 m/s s 0.0 s Fiure 4-21 F 1,2 F 1,2,so m 1 a 1 m 2 a 2 and a m/s Space Station Pratish weihs 588 N and is weihtless in a space station. If she pushes off the wall with a vertical acceleration of 3.00 m/s 2, determine the force exerted by the wall durin her push off. Use Newton s second law to obtain Pratish s mass, m Pratish.Use Newton s third law F A F B m A a A m B a B. m Pratish? 3.0 m/s 2 m 2 a 2 m1 (45 k)(3.0 m/s 2 ) (65 k) F wall on Pratish F Pratish on wall m Pratish a Pratish a Pratish (588 N)(3.00 m/s 2 ) 9.80 m/s N m/s 2 b. What are the manitude and direction of the force actin on it? F ma (0.145 k)( m/s 2 ) N (opposite direction of the velocity of the ball) c. What are the manitude and direction of the force actin on the player who cauht it? Same manitude, opposite direction (in direction of velocity of ball) Level Air Hockey An air-hockey table works by pumpin air throuh thousands of tiny holes in a table to support liht pucks. This allows the pucks to move around on cushions of air with very little resistance. One of these pucks has a mass of 0.25 k and is pushed alon by a 12.0-N force for 9.0 s. a. What is the puck s acceleration? F ma a F m 12.0 N 0.25 k 48 m/s 2 b. What is the puck s final velocity? v f v i at v i 0, so v f at (48 m/s 2 )(9.0 s) m/s 78 Solutions Manual Physics: Principles and Problems

19 88. A student stands on a bathroom scale in an elevator at rest on the 64th floor of a buildin. The scale reads 836 N. a. As the elevator moves up, the scale readin increases to 936 N. Find the acceleration of the elevator. 89. Weather Balloon The instruments attached to a weather balloon in Fiure 4-22 have a mass of 5.0 k. The balloon is released and exerts an upward force of 98 N on the instruments. F elevator F elevator ma m a,so ( ) F 5.0 k 98 N Fiure m/s 2 b. As the elevator approaches the 74th floor, the scale readin drops to 782 N. What is the acceleration of the elevator? F elevator F elevator ma m a,so m/s 2 c. Usin your results from parts a and b, explain which chane in velocity, startin or stoppin, takes the loner time. Stoppin, because the manitude of the acceleration is less and t v a (9.80 m/s 2 )(963 N 836 N) 836 N ( ) F (9.80 m/s 2 )(782 N 836 N) 836 N a. What is the acceleration of the balloon and instruments? F appl a F appl m 98 N (5.0 k)(9.80 m/s 2 ) 49 N (up) 9.8 m/s 2 b. After the balloon has accelerated for 10.0 s, the instruments are released. What is the velocity of the instruments at the moment of their release? v at m 49 N 5.0 k (9.8 m/s 2 )(10.0 s) 98 m/s (up) c. What net force acts on the instruments after their release? just the instrument weiht, 49 N (down) Physics: Principles and Problems Solutions Manual 79

20 d. When does the direction of the instruments velocity first become downward? The velocity becomes neative after it passes throuh zero. Thus, use v f v i t, where v f 0, or 91. Two blocks, masses 4.3 k and 5.4 k, are pushed across a frictionless surface by a horizontal force of 22.5 N, as shown in Fiure t v i ( 98 m/ s) ( 9.80 m/ s2) s after release 90. When a horizontal force of 4.5 N acts on a block on a resistance-free surface, it produces an acceleration of 2.5 m/s 2. Suppose a second 4.0-k block is dropped onto the first. What is the manitude of the acceleration of the combination if the same force continues to act? Assume that the second block does not slide on the first block. F m first block a initial m first block F m both blocks a final (m first block m second block )a final so, a final F F m ain second block F ainitial F m first block m second block itial 4.5 N 4.5 N 2. 5 m/s k 0.78 m/s 2 Fiure 4-23 a. What is the acceleration of the blocks? Identify the two blocks toether as the system, and riht as positive. ma, and m m 1 m 2 a F m1 m N 4.3 k 5.4 k 2.3 m/s 2 to the riht b. What is the force of the 4.3-k block on the 5.4-k block? Identify the 5.4-k block as the system and riht as positive. F 4.3-k block on 5.4-k block ma (5.4 k)(2.3 m/s 2 ) 12 N to the riht c. What is the force of the 5.4-k block on the 4.3-k block? Accordin to Newton s third law, this should be equal and opposite to the force found in part b, so the force is 12 N to the left. 80 Solutions Manual Physics: Principles and Problems

21 92. Two blocks, one of mass 5.0 k and the other of mass 3.0 k, are tied toether with a massless rope as in Fiure This rope is strun over a massless, resistance-free pulley. The blocks are released from rest. Find the followin. a. the tension in the rope b. the acceleration of the blocks Hint: you will need to solve two simultaneous equations. 3.0 k 5.0 k Fiure 4-24 Equation 1 comes from a free-body diaram for the 5.0-k block. Down is positive. F Earth s mass on 5.0-k block F rope on 5.0-k block m 5.0-k block a (1) Equation 2 comes from a free-body diaram for the 3.0-k block. Up is positive. F rope on 3.0-k block F Earth s mass on 3.0-k block m 3.0-k block a (2) The forces of the rope on each block will have the same manitude, because the tension is constant throuhout the rope. Call this force T. F Earth s mass on 5.0-k block T m 5.0-k block a (1) T F Earth s mass on 3.0-k block m 3.0-k block a (2) Solve equation 2 for T and plu into equation 1: m 5.0-k block a F Earth s mass on 5.0-k block a (m 5.0-k block m 3.0-k block ) m 3.0-k block m 5.0-k block (5.0 k 3.0 k)(9.80 m/s 2 ) 3.0 k 5.0 k 2.4 m/s 2 F Earth s mass on 3.0-k block m 3.0-k block a F Earth s mass on 5.0-k block F Earth s mass on 3.0-k block m 5.0-k block m 3.0-k block Physics: Principles and Problems Solutions Manual 81

22 Solve equation 2 for T: T F Earth s mass on 3.0-k block m 3.0-k block a m 3.0-k block m 3.0-k block a m 3.0-k block ( a) (3.0 k)(9.80 m/s m/s 2 ) 37 N Thinkin Critically paes Formulate Models A 2.0-k mass, m A, and a 3.0-k mass, m B, are connected to a lihtweiht cord that passes over a frictionless pulley. The pulley only chanes the direction of the force exerted by the rope. The hanin masses are free to move. Choose coordinate systems for the two masses with the positive direction bein up for m A and down for m B. a. Create a pictorial model. x m A b. Create a physical model with motion and free-body diarams. x F T a m A a F T m A m B m B c. What is the acceleration of the smaller mass? ma where m is the total mass bein accelerated. m B For m A, m A a F T m A For m B, m B a F T m B F T m B m B a m B ( a) Substitutin into the equation for m A ives m A a m B m B a m A or (m A m B )a (m B m A ) m Therefore a B m A ma m B 3.0 k 2.0 k 2.0 k 3.0 k (9.80 m/s 2 ) 2.0 m/s 2 upward 94. Use Models Suppose that the masses in problem 93 are now 1.00 k and 4.00 k. Find the acceleration of the larer mass. m a B m A ma m B 4.00 k 1.00 k (9.80 m/s 2 ) 1.00 k 4.00 k 5.88 m/s 2 downward 95. Infer The force exerted on a k baseball by a bat chanes from 0.0 N to N in s, then drops back to zero in the same amount of time. The baseball was oin toward the bat at 25 m/s. a. Draw a raph of force versus time. What is the averae force exerted on the ball by the bat? Force (N) F ave F peak 1 ( N) N Time (ms) 82 Solutions Manual Physics: Principles and Problems

23 b. What is the acceleration of the ball? a m m/s 2 c. What is the final velocity of the ball, assumin that it reverses direction? v f v i at 25 m/s ( m/s 2 ) 43 m/s N k ( s) F T2 m 2 a F T1 (4.0 k)(3.0 m/s 2 ) 6.0 N 18 N 97. Critique Usin the Example Problems in this chapter as models, write a solution to the followin problem. A block of mass 3.46 k is suspended from two vertical ropes attached to the ceilin. What is the tension in each rope? 96. Observe and Infer Three blocks that are connected by massless strins are pulled alon a frictionless surface by a horizontal force, as shown in Fiure k T T 2.0 k F T1 Fiure 4-25 a. What is the acceleration of each block? Since they all move toether, the acceleration is the same for all 3 blocks. F ma a (m 1 m 2 m 3 )a 4.0 k 3.0 m/s 2 b. What are the tension forces in each of the strins? Hint: Draw a separate free-body diaram for each block. ma F F T2 m 3 a F T2 F m1 m 2 m 3 F T2 F T1 m 2 a 6.0 k m 1 m 2 m 3 F 36.0 N 36 N 2.0 k 4.0 k 6.0 k 1 Analyze and Sketch the Problem Draw free-body diarams for the block and choose upward to be positive Solve for the Unknown Known: m block 3.46 k Unknown: F rope1 on block F rope2 on block? 2 Solve for the Unknown Use Newton s second law to find the tension in the ropes 2F rope1 on block F Earth s mass on block ma 0 F rope1 on block F rope1 on block F Earth s mass on block 2 m 2 (3.46 k)(9.80 m/s 2 ) N F T1 m 1 a (2.0 k)(3.0 m/s 2 ) 6.0 N Physics: Principles and Problems Solutions Manual 83

24 3 Evaluate the Answer Are the units correct? N is the correct unit for a tension, since it is a force. Does the sin make sense? The positive sin indicates that the tension is pullin upwards. Is the manitude realistic? We would expect the manitude to be on the same order as the block s weiht. 98. Think Critically Because of your physics knowlede, you are servin as a scientific consultant for a new science-fiction TV series about space exploration. In episode 3, the heroine, Misty Moonlow, has been asked to be the first person to ride in a new interplanetary transport for use in our solar system. She wants to be sure that the transport actually takes her to the planet she is supposed to be oin to, so she needs to take a testin device alon with her to measure the force of ravity when she arrives. The script writers don t want her to just drop an object, because it will be hard to depict different accelerations of fallin objects on TV. They think they d like somethin involvin a scale. It is your job to desin a quick experiment Misty can conduct involvin a scale to determine which planet in our solar system she has arrived on. Describe the experiment and include what the results would be for Pluto ( 0.30 m/s 2 ), which is where she is supposed to o, and Mercury ( 3.70 m/s 2 ), which is where she actually ends up. Answers will vary. Here is one possible answer: She should take a known mass, say 5.00-k, with her and place it on the scale. Since the ravitational force depends upon the local acceleration due to ravity, the scale will read a different number of newtons, dependin on which planet she is on. The followin analysis shows how to fiure out what the scale would read on a iven planet: Identify the mass as the system and upward as positive. F scale on mass ma 0 F scale on mass F scale on mass m Pluto: F scale on mass (5.00 k)(0.30 m/s 2 ) 1.5 N Mercury: F scale on mass (5.00 k)(3.7 m/s 2 ) 19 N 99. Apply Concepts Develop a CBL lab, usin a motion detector, that raphs the distance a free-fallin object moves over equal intervals of time. Also raph velocity versus time. Compare and contrast your raphs. Usin your velocity raph, determine the acceleration. Does it equal? Student labs will vary with equipment available and desins. p-t raphs and v-t raphs should reflect uniform acceleration. The acceleration should be close to. Writin in Physics pae Research Newton s contributions to physics and write a one-pae summary. Do you think his three laws of motion were his reatest accomplishments? Explain why or why not. Answers will vary. Newton s contributions should include his work on liht and color, telescopes, astronomy, laws of motion, ravity, and perhaps calculus. One arument in favor of his three laws of motion bein his reatest accomplishments is that mechanics is based on the foundation of these laws. His advances in the understandin of the concept of ravity may be suested as his reatest accomplishment instead of his three laws of motion. 84 Solutions Manual Physics: Principles and Problems

25 101. Review, analyze, and critique Newton s first law. Can we prove this law? Explain. Be sure to consider the role of resistance. Answers will vary. Newton s first law of motion involves an object whose net forces are zero. If the object is at rest, it remains at rest; if it is in motion, it will continue to move in the same direction at a constant velocity. Only a force actin on an object at rest can cause it to move. Likewise, only a force actin on an object in motion can cause it to chane its direction or speed. The two cases (object at rest, object in motion) could be viewed as two different frames of reference. This law can be demonstrated, but it cannot be proven. Taro s house to yours, when should you expect the skier to pass your house? (Chapter 2) d vt, or t d 5.2 km m v (8.0 km/h) 2.2 m/s t s 39 min d v m 2.2 m/s 1000 m 1 km 1h 3600 s The skier should pass your house at 8:25 0:39 9:04 A.M Physicists classify all forces into four fundamental cateories: ravitational, electromanetic, stron nuclear, and weak nuclear. Investiate these four forces and describe the situations in which they are found. Answers will vary. The stron nuclear force has a very short rane and is what holds protons and neutrons toether in the nucleus of an atom. The weak nuclear force is much less stron than the stron nuclear force and is involved in radioactive decay. The electromanetic force is involved in holdin atoms and molecules toether and is based on the attraction of opposite chares. Gravity is a lon-rane force between two or more masses. Cumulative Review pae Cross-Country Skiin Your friend is trainin for a cross-country skiin race, and you and some other friends have areed to provide him with food and water alon his trainin route. It is a bitterly cold day, so none of you wants to wait outside loner than you have to. Taro, whose house is the stop before yours, calls you at 8:25 A.M. to tell you that the skier just passed his house and is plannin to move at an averae speed of 8.0 km/h. If it is 5.2 km from 104. Fiure 4-26 is a position-time raph of the motion of two cars on a road. (Chapter 3) Distance (m) Fiure 4-26 a. At what time(s) does one car pass the other? 3 s, 8 s b. Which car is movin faster at 7.0 s? car A c. At what time(s) do the cars have the same velocity? 5 s d. Over what time interval is car B speedin up all the time? none e. Over what time interval is car B slowin down all the time? 3 s to 10 s Position of Two Cars A Time (s) B Physics: Principles and Problems Solutions Manual 85

26 105. Refer to Fiure 4-26 to find the instantaneous speed for the followin: (Chapter 3) a. car B at 2.0 s 0 m/s b. car B at 9.0 s 0 m/s c. car A at 2.0 s 1 m/s Challene Problem pae 100 An air-track lider passes throuh a photoelectric ate at an initial speed of 0.25 m/s. As it passes throuh the ate, a constant force of 0.40 N is applied to the lider in the same direction as its motion. The lider has a mass of 0.50 k. 1. What is the acceleration of the lider? F ma a F m 0.40 N 0.50 k 0.80 m/s 2 2. It takes the lider 1.3 s to pass throuh a second ate. What is the distance between the two ates? 1 d f d i v i t at 2 2 Let d i position of first ate 0.0 m d 0.0 m (0.25 m/s)(1.3 s) 1.0 m 1 (0.80 m/s 2 )(1.3 s) The 0.40-N force is applied by means of a strin attached to the lider. The other end of the strin passes over a resistance-free pulley and is attached to a hanin mass, m. How bi is m? m mass m mass k 4. Derive an expression for the tension, T, in the strin as a function of the mass, M, of the lider, the mass, m, of the hanin mass, and. T m Ma 0.40 N 9.80 m/s 2 86 Solutions Manual Physics: Principles and Problems