Questions and Problems 189

In a few problems, you are given more data than you actually need; in a few other problems, you are required to supply data from your general knowledge, outside sources, or informed estimate.

Interpret as significant all digits in numerical values that have trailing zeros and no decimal points. For all problems, use g = 9.80 m>s2 for the free-fall acceleration due to gravity.• Basic, single-concept problem•• Intermediate-level problem, may require synthesis of con-cepts and multiple steps••• Challenging problemSSM Solution is in Student Solutions Manual

Conceptual Questions1. •Complete the sentence: The static frictional force between two surfaces is (never/sometimes/always) less than the normal force. Explain your answer.

2. •You want to push a heavy box of books across a rough floor. You know that the maximum value of the coefficient of static friction (ms) is larger than the maximum value of the coefficient of kinetic friction (mk). Should you push the box for a short distance, rest, push the box another short distance, and then repeat the process until the box is where you want it, or will it be easier to keep pushing the box across the floor once you get it moving?

3. •If the force of friction always opposes the sliding of an object, how then can a frictional force cause an object to in-crease in speed? SSM

4. •A solid rectangular block has sides of three different areas. You can choose to rest any of the sides on the floor as you apply a horizontal force to the block. Does the choice of side on the floor affect how hard it is to push the block? Explain your answer.

5. •You’re trying to press a book against a spot on the wall with your hand. As you get tired, you exert less force, but the book remains in the same spot on the wall. Do each of the follow-ing forces increase, decrease, or not change in magnitude when you reduce the force you are applying to the book: (a) weight, (b) normal force, (c) frictional force, and (d) maximum static frictional force?

6. •For an object moving in a circle, which of the follow-ing quantities are zero over one revolution: (a) displacement, (b) average velocity, (c) average acceleration, (d) instantaneous velocity, and (e) instantaneous centripetal acceleration?

7. •Why does water stay in a bucket that is whirled around in a vertical circle? Contrast the forces acting on the water when the bucket is at the lowest point on the circle to when the bucket is at the highest point on the circle. SSM

8. •An antilock braking system (ABS) prevents wheels from skidding while drivers stomp on the brakes in emergencies. How far will a car with an ABS move before finally stopping, as compared to a car without an ABS?

9. •Give two examples of two situations in which the nor-mal force acting on an object is not equal to the object’s weight.

10. •Medical A fluidlike substance called synovial fluid lubri-cates the surfaces where bones meet in joints, making the coeffi-cient of friction between bones very small. Why is the minimum force required for moving bones in a typical knee joint different

for different people, in spite of the fact that their joints have the same coefficient of friction? For simplicity, assume the surfaces in the knee are flat and horizontal.

11. •As a skydiver falls faster and faster through the air, does his acceleration increase, decrease, or remain the same? Explain your answer. SSM

12. •Why do raindrops fall from the sky at different speeds? Explain your answer.

13. •Why might your car start to skid if you drive too fast around a curve?

14. •What distinguishes the forces that act on a car driving over the top of a hill from those acting on a car driving through a dip in the road? Explain how the forces relate to the sensations passengers in the car experience during each situation.

15. •Explain how you might measure the centripetal accelera-tion of a car rounding a curve. SSM

16. •An object of mass M1 rests on a horizontal table; friction exists between the object and the table (Figure 5-21). A ring of mass MR is tied by massless strings both to the wall and to the object as shown. A second object of mass M2 hangs from the ring. Draw free-body diagrams for the situation.

Figure 5-21Problem16

Friction M1 MR

M2

17. •Explain why curves in roads and cycling velodromes are banked. SSM

18. •At low speeds, the drag force on an object moving through a fluid is proportional to its velocity. According to Newton’s second law, force is proportional to acceleration. As accelera-tion and velocity aren’t the same quantity, is there a contradic-tion here?

19. •James Bond leaps without a parachute from a burning airplane flying at 15,000 ft. Ten seconds later his assistant, who was following behind in another plane dives after him, wearing her parachute and clinging to one for her hero. Is it possible for her to catch up with Bond and save him?

Multiple-Choice Questions20. •If a sport utility vehicle (SUV) drives up a slope of 45°, what must be the minimum coefficient of static friction between the SUV’s tires and the road?

A. 1.0B. 0.5C. 0.7D. 0.9E. 0.05

Questions and Problems

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190 Chapter 5 Forces and Motion II: Applications

21. •A block of mass m slides down a rough incline with con-stant speed. If a similar block that has a mass of 4m were placed on the same incline, it would

A. slide down at constant speed.B. accelerate down the incline.C. slowly slide down the incline and then stop.D. accelerate down the incline with an acceleration four

times greater than that of the smaller block.E. not move. SSM

22. •A 10-kg crate is placed on a horizontal conveyor belt mov-ing with a constant speed. The crate does not slip. If the coef-ficients of friction between the crate and the belt are ms equal to 0.50 and mk equal to 0.30, what is the frictional force exerted on the crate?

A. 98 NB. 49 NC. 29 ND. 9.8 NE. 0

23. •Biology The Escherichia coli (E. coli) bacterium propels itself through water by means of long, thin structures called flagella. If the force exerted by the flagella doubles, the velocity of the bacterium

A. doubles.B. decreases by half.C. does not change.D. increases by a factor of four.E. cannot be determined without more information.

24. •A 1-kg wood ball and a 5-kg lead ball have identical sizes, shapes, and surface characteristics. They are dropped simulta-neously from a tall tower. Air resistance is present. How do their accelerations compare?

A. The 1-kg wood ball has the larger acceleration.B. The 5-kg lead ball has the larger acceleration.C. The accelerations are the same.D. The 5-kg ball accelerates at five times the acceleration

of the 1-kg ball.E. The 1-kg ball accelerates at five times the acceleration

of the 5-kg ball.

25. •A skydiver is falling at his terminal speed. Immediately after he opens his parachute

A. his speed will be larger than his terminal speed.B. the drag force on the skydiver will decrease.C. the net force on the skydiver is in the downward

direction.D. the drag force is larger than the skydiver’s weight.E. the net force on the skydiver is zero. SSM

26. •Two rocks that are of equal mass are tied to massless strings and whirled in nearly horizontal circles at the same speed. One string is twice as long as the other. What is the tension T1 in the shorter string compared to the tension T2 in the longer one?

A. T1 = 14T2B. T1 = 12T2C. T1 = T2D. T1 = 2T2E. T1 = 4T2

27. •You are on a Ferris wheel moving in a vertical circle. When you are at the bottom of the circle, how does the magnitude

of the normal force N exerted by your seat compare to your weight mg?

A. N = mgB. N 7 mgC. N 6 mgD. N = 12mgE. N = 2mg SSM

28. •Two rocks are tied to massless strings and whirled in nearly horizontal circles so that the period of motion is the same for both. One string is twice as long as the other. The tension in the longer string is twice the tension in the shorter one. What is the mass m1 of the rock at the end of the shorter string compared to the mass m2 of the rock at the end of the longer one?

A. m1 = 14m2B. m1 = 12m2C. m1 = m2D. m1 = 2m2E. m1 = 4m2

Estimation/Numerical Analysis29. •Estimate the numerical value of the coefficient of static friction between a car’s tires and dry pavement.

30. •Estimate the value of the coefficient of kinetic friction for a box of books on a carpeted floor.

31. •Sports Make a rough estimate of the value of the coeffi-cient of kinetic friction between a baseball player’s uniform and the infield surface as he slides into second base. SSM

32. •Give a numerical estimate of the minimum radius of cur-vature of an unbanked on-ramp for a typical freeway.

33. •Sports Estimate the size of the coefficient of kinetic friction between a hockey puck and the ice of a rink.

34. •(a) Estimate the magnitude of the coefficient of kinetic friction of a book as it slides across a tabletop. (b) Estimate the magnitude of the coefficient of static friction for the same book on the same tabletop.

35. •Estimate the magnitude of the coefficient of kinetic fric-tion between you and the surface of a waterslide. SSM

36. •Estimate the magnitude of the coefficient of kinetic fric-tion for a mug of root beer as it slides across a wooden bar.

37. •The following table lists the forces applied to a 1.00-kg crate at particular times. The crate begins at rest on a rough, horizontal surface but is then pushed with a constant force in the horizontal direction. Using the data, estimate the coefficient of static friction and the coeffi-cient of kinetic friction between the crate and the surface.

Problems5-2 The static friction force changes magnitude to offset other applied forces

38. •What is the minimum horizontal force that will cause a 5.00-kg box to begin to slide on a horizontal surface when the coefficient of static friction is 0.670?

t (s) F (N) t (s) F (N)

0 0 0.25 8.26

0.01 1.33 0.30 7.84

0.05 3.28 0.35 5.17

0.10 8.11 0.40 5.21

0.15 8.20 0.45 5.22

0.20 8.24 0.50 5.37

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Questions and Problems 191

39. •A 7.60-kg object rests on a level floor with a coefficient of static friction of 0.550. What minimum horizontal force will cause the object to start sliding? SSM

40. •Draw a free-body diagram for the situation shown in Fig-ure 5-22. An object of mass M rests on a ramp; there is friction between the object and the ramp. The system is in equilibrium.

Figure 5-22 Problem 40

M

Friction

5-3 The kinetic friction force on a sliding object has a constant magnitude

41. •A book is pushed across a horizontal table at a constant speed. If the horizontal force applied to the book is equal to one-half of the book’s weight, calculate the coefficient of kinetic friction between the book and the tabletop. SSM

42. •An object on a level surface experiences a horizontal force of 12.7 N due to kinetic friction. If the coefficient of kinetic friction is 0.37, what is the mass of the object?

43. •A 25.0-kg crate rests on a level floor. A horizontal force of 50.0 N accelerates the crate at 1.00 m>s2. Calculate (a) the nor-mal force on the crate, (b) the frictional force on the crate, and (c) the coefficient of kinetic friction between the crate and the floor.

44. •A block of mass M rests on a block of mass M1 = 5.00 kgwhich is on a tabletop (Figure 5-23). A light string passes over a frictionless peg and connects the blocks. The coefficient of kinetic friction mk at both surfaces equals 0.330. A force of 60.0 N pulls the upper block to the left and the lower block to the right. The blocks are moving at a constant speed. Determine the mass of the upper block.

Figure 5-23Problem 44

MF

M1

45. •A mop is pushed across the floor with a force F of 50.0 N at an angle of u = 50.0° (Figure 5-24). The mass of the mop head is 3.75 kg. Calculate the acceleration of the mop head if the coef-ficient of kinetic friction between the head and the floor is 0.400.

Figure 5-24Problem 45

u

F

µk

46. •If the coefficient of kinetic friction between an object with mass M = 3.00 kg and a flat surface is 0.400, what force will

cause the object to accelerate at 2.50 m>s2? The force is applied at an angle of u = 30.0° (Figure 5-25).

Figure 5-25Problem 46

µk

M

F = ?

θ

5-4 Problems involving static and kinetic friction are like any other problem with forces

47. ••A taut string connects a crate with mass M1 = 5.00 kg to a crate with mass M2 = 12.0 kg (Figure 5-26). The coefficient of static friction between the smaller crate and the floor is 0.573; the coefficient of static friction between the larger crate and the floor is 0.443. What is the minimum horizontal force F required to start the crates in motion? SSM

Figure 5-26 Problem 47

M2

M1

F

48. ••A box of mass Mbox = 2.00 kg rests on top of a crate with mass Mcrate = 5.00 kg (Figure 5-27). The coefficient of static friction between the box and the crate is 0.667. The coefficient of static friction between the crate and the floor is 0.400. Cal-culate the minimum force F that is required to move the crate to the right and the corresponding tension T in the rope that connects the box to the wall when the crate is moved.

Figure 5-27 Problem 48

Box

CrateF

T

49. •Two blocks are connected over a massless, frictionless pulley (Figure 5-28). The mass of block 2 is 8.00 kg, and the coefficient of kinetic friction between block 2 and the incline is 0.220. The angle u of the incline is 28.0°. Block 2 slides down the incline at constant speed. What is the mass of block 1? SSM

Figure 5-28Problems 49 and 50

1

2

Friction

q

50. •Two blocks are connected over a massless, frictionless pulley (Figure 5-28). The mass of block 2 is 10.0 kg, and the

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192 Chapter 5 Forces and Motion II: Applications

coefficient of kinetic friction between block 2 and the incline is 0.200. The angle u of the incline is 30.0°. If block 2 moves up the incline at constant speed, what is the mass of block 1?

51. ••An object with mass M1 of 2.85 kg is held in place on an inclined plane that makes an angle u of 40.0° with the horizon-tal (Figure 5-29). The coefficient of static friction between the plane and the object is 0.552. A second object that has a mass M2 of 4.75 kg is connected to the first object with a massless string over a massless, frictionless pulley. Calculate the initial acceleration of the system and the tension in the string once the objects are released.

Figure 5-29Problem 51

M2

M1

u

Static friction

52. •Draw free-body dia-grams for the situation shown in Figure 5-30. An object of mass M2 rests ona frictionless table, and an object of mass M1 sits on it; there is friction between the objects. A horizontal force Fs is applied to the lower object as shown.

53. ••A horizontal force F of 10.0 N is applied to a stationary block with a mass M of 2.00 kg as shown in Figure 5-31. The coefficient of static friction between the block and the floor is 0.750; the coefficient of kinetic friction is 0.450. Find the accel-eration of the box. SSM

Figure 5-31 Problem 53

FrictionM F

5-5 An object moving through air or water experiences a drag force

54. •Biology A single-celled animal called a paramecium pro-pels itself quite rapidly through water by using its hairlike cilia.A certain paramecium experiences a drag force of magnitude Fdrag = cv2 in water, where the drag coefficient c is approxi-mately 0.310. What propulsion force does this paramecium generate when moving at a constant (terminal) speed v of 0.150 * 103 m>s?55. •Biology The bacterium Escherichia coli propels itself with long, thin structures called flagella. When its flagella exert a force of 1.50 * 10213 N, the bacterium swims through water at a speed of 20.0 mm>s. Find the speed of the bacterium in water when the force exerted by its flagella is 3.00 * 10213 N.

56. ••We model the drag force of the atmosphere as propor-tional to the square of the speed of a falling object, Fdrag = cv2,

where the value of c for a 70.0-kg person with a parachute is 18.0 kg>m. (a) What is the person’s terminal velocity? (b) With-out a parachute, the same person’s terminal velocity would be about 50.0 m>s. What would be the value of the proportional-ity constant c in that case?

57. ••A girl rides her scooter on a hill that is inclined at 10° with the horizontal. The combined mass of the girl and scooter is 50.0 kg. On the way down, she coasts at a constant speed of 12.0 m>s, while experiencing a drag force that is proportional to the square of her velocity. What force, parallel to the surface of the hill, is required to increase her speed to 20.0 m>s? Ne-glect any other resistive forces.

5-6 In uniform circular motion, the net force points toward the center of the circle

58. •A hockey puck that has a mass of 170 g is tied to a light string and spun in a circle of radius 1.25 m (on frictionless ice). If the string breaks under a tension that exceeds 5.00 N, what is the maximum speed of the puck without breaking the string?

59. •A 1500-kg truck rounds an unbanked curve on the highway at a speed of 20.0 m>s. If the maximum frictional force between the surface of the road and all four of the tires is 8000 N, calculate the minimum radius of curvature for the curve to prevent the truck from skidding off the road. SSM

60. •A 25.0-g metal washer is tied to a 60.0-cm-long string and whirled around in a vertical circle at a constant speed of 6.00 m>s. Calculate the tension in the string (a) when the wash-er is at the bottom of the circular path and (b) when it is at the top of the path.

61. •A centrifuge spins small tubes in a circle of radius 10.0 cm at a rate of 1200 rev>min. What is the centripetal force on a sample that has a mass of 1.00 g? SSM

62. •Astro What centripetal force is exerted on the Moon as it orbits about Earth at a center-to-center distance of 3.84 * 108 m with a period of 27.4 days? What is the source of the force? The mass of the Moon is equal to 7.35 * 1022 kg.

63. •Biology Very high-speed ultracentrifuges are useful devices to sediment materials quickly or to separate materials. An ul-tracentrifuge spins a small tube in a circle of radius 10.0 cm at 60,000 rev>min. What is the centripetal force experienced by a sample that has a mass of 0.00300 kg?

64. •At the Fermi National Accelerator Laboratory (Fermilab), a large particle accelerator, protons are made to travel in a circular orbit 6.3 km in circumference at a speed of nearly 3.0 * 108 m>s .What is the centripetal acceleration of one of the protons?

65. •In the game of tetherball,a 1.25-m rope connects a 0.750-kg ball to the top of a vertical pole so that the ball can spin around the pole as shown in Figure 5-32. What is the speed of the ball as it rotates around the pole when the angle u of the rope is 35.0° with the vertical? SSM

Figure 5-30 Problem 52

Friction

Frictionless

M1

M2

F

Figure 5-32 Problem 65

u1.25 m

0.75 kg

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Questions and Problems 193

66. •What is the force that a jet pilot feels against his seat as he completes a vertical loop that is 500 m in radius at a speed of 200 m>s? Assume his mass is 70.0 kg and that he is located at the bottom of the loop.

67. •The radius of Earth is 6.38 * 106 m, and it completes one revolution in 1 day. (a) What is the centripetal acceleration of an object located on the equator? (b) What is the centripetal acceleration of an object located at latitude 40.0° north?

68. ••A coin that has a mass of 25.0 g rests on a phonograph turntable that rotates at 78.0 rev>min. The center of the coin is 13.0 cm from the turntable axis. If the coin does not slip, what is the minimum value of the coefficient of static friction between the coin and the turntable surface?

69. •Sports In executing a windmill pitch, a fast-pitch softball player moves her hand through a circular arc of radius 0.310 m. The 0.190-kg ball leaves her hand at 24.0 m>s. What is the magnitude of the force exerted on the ball by her hand immedi-ately before she releases it?

General Problems70. ••A 150-kg crate rests in the bed of a truck that slows from 50.0 km>h to a stop in 12.0 s. The coefficient of static friction between the crate and the truck bed is 0.655. (a) Will the crate slide during the braking period? Explain your answer. (b) What is the minimum stopping time for the truck in order to prevent the crate from sliding?

71. ••The coefficient of static friction between a rubber tire and dry pavement is about 0.800. Assume that a car’s engine only turns the two rear wheels and that the weight of the car is uniformly distributed over all four wheels. (a) What limit does the coefficient of static friction place on the time required for a car to accelerate from rest to 60 mph (26.8 m>s)? (b) How can friction accelerate a car forward when friction opposes motion? SSM

72. ••Two blocks are connected over a massless, frictionless pul-ley (Figure 5-33). Block m1 has a mass of 1.00 kg and block m2 has a mass of 0.400 kg. The angle u of the incline is 30.0°. The coefficients of static friction and kinetic friction between block m1 and the incline are ms equal to 0.500 and mk equal to 0.400, respectively. What is the value of the tension in the string?

Figure 5-33Problems72and73

m2

m1

Friction

θ

73. ••Two blocks are connected over a massless, frictionless pulley (Figure 5-33). Block m1 has a mass of 1.00 kg and block m2 has a mass of 2.00 kg. The angle u of the incline is 30.0°. The coefficients of static friction and kinetic friction between block m1 and the incline are ms equal to 0.500 and mk equal to 0.400. What is the acceleration of block m1?

74. •A runaway ski slides down a 250-m-long slope inclined at 37.0° with the horizontal. If the initial speed is 10.0 m>s, how long does it take the ski to reach the bottom of the incline if

the coefficient of kinetic friction between the ski and snow is (a) 0.100 and (b) 0.150?

75. •In a mail-sorting facility, a 2.50-kg package slides down an inclined plane that makes an angle of 20.0° with the hori-zontal. The package has an initial speed of 2.00 m>s at the top of the incline, and it slides a distance of 12.0 m. What must the coefficient of kinetic friction between the package and the inclined plane be so that the package reaches the bottom with no speed? SSM

76. ••In Figure 5-34, two blocks are connected to each other by a massless string over a massless and frictionless pulley. The mass m1 is 6.00 kg. Assuming the coefficient of static fric-tion ms equals 0.542 for all surfaces, find the range of values of the mass m2 of the second block so that the system is in equilibrium.

Figure 5-34 Problem76

m2

m160° 35°

Static friction

Static friction

77. ••The terminal velocity of a raindrop that is 4.00 mm in diameter is approximately 8.50 m>s under controlled, wind-less conditions. The density of water is 1000 kg>m3. Recall that the density of an object is its mass divided by its vol-ume. (a) If we model the air drag as being proportional to the square of the speed, Fdrag = cv2, what is the value of c? (b) Under the same conditions as above, what would be the terminal velocity of a raindrop that is 8.0 mm in diameter? Try to use your answer from part (a) to solve the problem by proportional reasoning instead of just doing the same calcula-tion over again.

78. •Biomedical laboratories routinely use ultracentrifuges, some of which are able to spin at 100,000 rev>min about the central axis. The turning rotor in certain models is about 20.0 cm in diameter. At its top spin speed, what force does the bot-tom of the rotor exert on a 2.00-g sample that is in the rotor at the greatest distance from the axis of spin? Would the force be appreciably different if the sample were spun in a vertical or a horizontal circle? Why or why not?

79. ••An amusement park ride called the Rotor debuted in 1955 in Germany. Passengers stand in the cylindrical drum of the Rotor as it rotates around its axis. Once the Rotor reaches its operating speed, the floor drops but the riders remain pinned against the wall of the cylinder. Suppose the cylinder makes 25.0 rev>min and has a radius of 3.50 m. What is the coeffi-cient of static friction between the wall of the cylinder and the backs of the riders?

80. •An object of mass m1 undergoes constant circular motion and is connected by a massless string through a hole in a fric-tionless table to a larger object of mass m2 (Figure 5-35). If the larger object is stationary, calculate the tension in the string and the speed of the circular motion of the smaller object. Assume that the objects have masses of 0.225 and 0.125 kg and the

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194 Chapter 5 Forces and Motion II: Applications

radius R of the circular path of the smaller object is equal to 1.00 m.

Figure 5-35Problem80

m1

m2

R

81. •An object that has a mass M hangs from a support by a massless string of length L (Figure 5-36). The support is rotated so that the object follows a circular path at an angle u from the vertical as shown. The object makes N revolutions per second. Derive an expression for the angle u in terms of M, L, N, and any necessary physical constants. SSM

Figure 5-36 Problem81

θL

M

82. •Medical Occupants of cars hit from behind, even at low speed, often suffer serious neck injury from whiplash. During a low-speed rear-end collision, a person’s head suddenly pivots about the base of the neck through a 60° angle, a motion that lasts 250 ms. The distance from the base of the neck to the cen-ter of the head is typically about 20 cm, and the head normally comprises about 6.0% of body weight. We can model the motion of the head as having uniform speed over the course of its pivot. Compute your answers to the following questions

to two significant figures. (a) What is the acceleration of the head during the collision? (b) What force (in newtons and in pounds) does the neck exert on the head of a 75-kg person in the collision? (As a first approximation, neglect the force of gravity on the head.) (c) Would headrests mounted to the backs of the car seats help protect against whiplash? Why or why not?

83. •The wings of an airplane flying in a horizontal circle at a speed of 680 km>h are tilted 60.0° to the horizontal (Fig-ure 5-37). What is the radius of the circle? Assume that the required force is provided entirely by the wings’ lift, a force perpendicular to the surface of the wings.

Figure 5-37 Problem83

60°

84. •A curve that has a radius of 100 m is banked at an angle of 10.0° (Figure 5-38). If a 1000-kg car navigates the curve at 65 km>h without skidding, what is the minimum coefficient of static friction between the pavement and the tires?

Figure 5-38Problem84

10°

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