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N. Giordano. College Physics: Reasoning and Relationships, Brooks Cole, 2010

 

Chapter 6

1. A crate of mass 50 kg is pushed across a level floor by a person. If the person exerts a force of 25 N in the horizontal direction and moves the crate a distance of 10 m, what is the work done by the person?

2. A sphere of mass 35 kg is attached to one end of a rope as shown in Figure P6.2. It is found that the rope does an amount of work W = 550 J in pulling the sphere upward through a distance 1.5 m. Find the tension in the rope. 

3. For the sphere in Problem 2, find the work done by the force of gravity on the sphere.

4. A hockey puck of mass 0.25 kg is sliding along a slippery frozen lake, with an initial speed of 60 m/s. The coefficient of friction between the ice and the puck is μk = 0.030. Friction eventually causes the puck to slide to a stop. Find the work done by friction.

5. A crate of mass 24 kg is pushed up a frictionless ramp by a person as shown in Figure P6.5. Calculate the work done by the person in pushing the crate a distance of 20 m as measured along the ramp. Assume the crate moves at a constant velocity.

6. Consider again the crate in Problem 5, but now include friction. Assume the coefficient of kinetic friction between the crate and the ramp is μk = 0.25. What is the work done by the person as he pushes the crate up the ramp?

7. Find the work done by gravity on the crate in Problem 6.

8. A block of mass m = 5.0 kg is pulled along a rough horizontal surface by a rope as sketched in Figure P6.8. The tension in the rope is 40 N, and the coefficient of kinetic friction between the block and the surface is μk = 0.25. (a) If the block travels a distance of 4.5 m along the surface, what is the work done by the rope? (b) Find the work done by friction on the block.

9. A car (m = 1200 kg) is traveling at an initial speed of 15 m/s. It then slows to a stop over a distance of 80 m due to a force from the brakes. How much work is done on the car? Assume the acceleration is constant.

10. A snowboarder of mass 80 kg slides down the trail shown in Figure P6.10. The first part of the trail is a ramp that makes an angle of 20° with the horizontal, and the final portion of the trail is fl at. Find the work done by gravity on the snowboarder as she travels from the beginning to the end of this trail.

11. Two railroad cars, each of mass 2.0 × 104 kg, are connected by a cable to each other, and the car in front is connected by a cable to the engine as shown in Figure P6.11. The cars start from rest and accelerate to a speed of 1.5 m/s after 1 min. (a) Find the work done by cable 1 on the car in the back. (b) Find the work done by cable 1 on the car in front. (c) Find the work done by cable 2 on the car in front.

12. A person pushes a broom at an angle of 60° with respect to the floor (Fig. P6.12). If the person exerts a force of 30 N directed along the broom handle, what is the work done by the person on the broom as he pushes it a distance of 5.0 m?

13. A car is pushed along a long road that is straight, flat, and parallel to the x direction. (a) The horizontal force on the car varies with x as shown in Figure P6.13. What is the work done on the car by this force? (b) What is the work done by gravity on the car?

14. The force on an object varies with position as shown in Figure P6.14. Estimate the work done on the object as it moves from the origin to x = 5.0 m. Assume the motion is one dimensional.

15. A tomato of mass 0.22 kg is dropped from a tall bridge. If the tomato has a speed of 12 m/s just before it hits the ground, what is the kinetic energy of the tomato?

16. Two objects have the same kinetic energy. One has a speed that is 2.5 times greater than the speed of the other. What is the ratio of their masses?

17. A car of mass 1500 kg is initially traveling at a speed of 10 m/s. The driver then accelerates to a speed of 25 m/s over a distance of 200 m. Calculate the change in the kinetic energy of the car.

18. A rock of mass 0.050 kg is thrown upward with an initial speed of 25 m/s. Find the work done by gravity on the rock from the time it leaves the thrower’s hand until it reaches the highest point on its trajectory. Hint: You do not have to calculate the rock’s maximum height.

19. A horizontal force of 15 N pulls a block of mass 3.9 kg across a level floor. The coefficient of kinetic friction between the block and the floor is μk = 0.25. If the block begins with a speed of 8.0 m/s and is pulled for a distance of 12 m, what is final speed of the block?

20. You are pushing a refrigerator across the floor of your kitchen. You exert a horizontal force of 300 N for 7.0 s, during which time the refrigerator moves a distance of 3.0 m at a constant velocity. (a) What is the total work (by all forces) done on the refrigerator? (b) What is the work done by friction?

21. An archer is able to fire an arrow (mass 0.020 kg) at a speed of 250 m/s. If a baseball (mass 0.14 kg) is given the same kinetic energy, what is its speed?

22. A baseball pitcher can throw a baseball at a speed of 50 m/s. If the mass of the ball is 0.14 kg and the pitcher has it in his hand over a distance of 2.0 m, what is the average force exerted by the pitcher on the ball?

23. A softball pitcher can exert a force of 100 N on a softball. If the mass of the ball is 0.19 kg and the pitcher has it in her hand over a distance of 1.5 m, what is the speed of the ball when it leaves her hand?

24. Consider a skydiver of mass 70 kg who jumps from an airplane flying at an altitude of 1500 m. With her parachute open, her terminal velocity is 8.0 m/s. (a) What is the work done by gravity and (b) what is the average force of air drag during the course of her jump?

25. Consider a small car of mass 1200 kg and a large sport utility vehicle (SUV) of mass 4000 kg. The SUV is traveling at the speed limit (v = 35 m/s). The driver of the small car travels so as to have the same kinetic energy as the SUV. Find the speed of the small car.

26. A truck of mass 9000 kg is traveling along a level road at an initial speed of 30 m/s and then slows to a final speed of 12 m/s. Find the total work done on the truck.

27. You use an elevator to travel from the first floor (h = 0) to the fourth floor of a building (h = 12 m). If you start from rest and you end up at rest on the fourth floor, what is the work done by gravity on you? Assume you have a mass of 70 kg.

28. A car of mass m = 1500 kg is pushed off a cliff of height h = 24 m (Fig. P6.28). If the car lands a distance of 10 m from the base of the cliff, what was the kinetic energy of the car the instant after it left the cliff?

29. The force on an object varies with position as shown in Figure P6.29. The object begins at rest from the origin and has a speed of 5.0 m/s at x = 4.0 m. What is the mass of the object?

30. Suppose the object in Figure P6.29 has a mass of 2.5 kg and a speed of 9.0 m/s at x= 0. What is its speed at x = 5.0 m?

31. A skier of mass 110 kg travels down a frictionless ski trail. (a) If the top of the trail is a height 200 m above the bottom, what is the work done by gravity on the skier? (b) Find the velocity of the skier when he reaches the bottom of the ski trail. Assume he starts from rest.

32. Suppose the ski trail in Problem 31 is not frictionless. Find the work done by gravity on the skier in this case.

33. Consider a roller coaster that moves along the track shown in Figure P6.33. Assume all friction is negligible. (a) Is the mechanical energy of the roller coaster conserved? (b) Add a coordinate system to the sketch in Figure P6.33. (c) If the roller coaster starts from rest at location A, what is its total mechanical energy at point A? (d) If vB is the speed at point B, what is the total mechanical energy at point B? (e) Find the speed of the roller coaster when it reaches locations B and C.

34. Consider again the roller coaster in Figure P6.33, but now assume the roller coaster starts with a speed of 12 m/s at point A. Find the speed of the roller coaster when it reaches locations B and C.

35. Mount McKinley (also called Denali) is the tallest mountain in North America, with a height of 6200 m above sea level. If a person of mass 120 kg walks from sea level to the top of Mount McKinley, how much work is done by gravity on the person?

36. The roller coaster in Figure P6.36 starts with a velocity of 15 m/s. One of the riders is a small girl of mass 30 kg. Find her apparent weight when the roller coaster is at locations B and C. At these two locations, the track is circular, with the radii of curvature given in the figure. The heights at points A, B, and C are hA = 25 m, hB = 35 m, and hC = 0 m.

37. A roller coaster (Fig. P6.37) starts at the top of its track (point A) with a speed of 12 m/s. If it reaches point B traveling at 16 m/s, what is the vertical distance (h) between A and B? Assume friction is negligible and ignore the kinetic energy of the wheels.

38. A skateboarder starts at point A on the track (Fig. P6.38) with a speed of 15 m/s. Will he reach point B? Assume friction is negligible and ignore the kinetic energy of the skateboard’s wheels.

39. Frontside air. A skateboarder is practicing on the “half-pipe” shown in Figure P6.39, using a special frictionless skateboard. (You can also ignore the kinetic energy of the skateboard’s wheels.) (a) If she starts from rest at the top of the half-pipe, what is her speed at the bottom? (b) If the skateboarder has a mass m = 55 kg, what is her apparent weight at the bottom of the half-pipe? (c) What speed does she then have when she reaches the top edge on the other side of the half-pipe? (d) Now suppose she has a speed of 11 m/s at the bottom of the half-pipe. What is the highest point she can reach? Hint: This point may be above the edge of the half-pipe.

40. Consider again the skateboarder in Figure P6.39. Assuming she starts at the top of the half-pipe, make qualitative sketches of her kinetic energy and potential energy as a function of height in the half-pipe.

41. A hockey puck of mass 0.25 kg starts from rest and slides down the frictionless ramp shown in Figure P6.41. The bottom end of the ramp is horizontal. After it leaves the ramp, the puck travels as a projectile and lands a distance L from the ramp. (a) Is the puck’s mechanical energy at the top of the ramp equal to its mechanical energy when it leaves the ramp and is in the air? Explain why or why not. (b) Add a coordinate system to the sketch in Figure P6.41. Where is a convenient place to put the origin of the vertical (y) axis? Do you have to put it there? (c) What is the initial mechanical energy of the puck? (d) What is the mechanical energy of the puck the instant it leaves the ramp? (e) What is the speed of the puck the instant it leaves the ramp? (f) How far from the base of the ramp does the puck land?

42. A rock of mass 12 kg is tied to a string of length 2.4 m, with the other end of the string fastened to the ceiling of a tall room (Fig. P6.42). While hanging vertically, the rock is given an initial horizontal velocity of 2.5 m/s. (a) Add a coordinate system to the sketch in Figure P6.42. Where is a convenient place to choose the origin of the vertical (y) axis? (b) What are the initial kinetic and potential energies of the rock? (c) If the rock swings to a height h above its initial point, what is its potential energy? What is its total mechanical energy at that point? (d) How high will the rock swing? Express your answer in terms of the angle u that the string makes with the vertical when the rock is at its highest point. (e) Make qualitative sketches of the kinetic energy and potential energies as functions of height h.

43. Consider the rock and string in Figure P6.43. The string is fastened to a hinge that allows it to swing completely around in a vertical circle. If the rock starts at the lowest point on this circle and is given an initial speed vi, what is the smallest value of vi that will allow the rock to travel completely around the circle without the string becoming slack at the top? Assume the length of the string is 1.5 m.

44. A rock climber (m = 90 kg) starts at the base of a cliff and climbs to the top (h = 35 m). He then walks along the plateau at the top for a distance of L = 40 m. Find the work done by gravity and the change in the gravitational potential energy of the rock climber.

45. Two crates of mass m1 = 40 kg and m2 = 15 kg are connected by a massless rope that passes over a massless, frictionless pulley as shown in Figure P6.45. The crates start from rest. (a) Add a coordinate system to Figure P6.45. (b) What are the initial kinetic and potential energies of each crate? (c) If the crate on the left (m1) moves downward through a distance h, what is the change in the potential energy of m1 and m2? Express your answers in terms of h. (d) What is the speed of the crates after they have moved a distance h = 2.5 m?

46. Consider again the crates in Problem 45, but now assume the rope has a mass of 2 kg. If the tops of the crates start at the same initial height and the rope has a length of 10 m, what is the final speed?

47. Sketch how the potential energy, kinetic energy, and total mechanical energy vary with time for an apple that drops from a tree.

48. Sketch how the potential energy, kinetic energy, and total mechanical energy vary with time for a cannon shell that is fired from a large cannon. Ignore air drag.

49. A rock of mass 3.3 kg is tied to a string of length 1.2 m. The rock is held at rest as shown in Figure P6.49 so that the string is initially tight, and then it is released. (a) Find the speed of the rock when it reaches the lowest point of its trajectory. (b) What is the maximum tension in the string?

50. A ball of mass 1.5 kg is tied to a string of length 6.0 m as shown in Figure P6.50. The ball is initially hanging vertically and is given an initial velocity of 5.0 m/s in the horizontal direction. The ball then follows a circular arc as determined by the string. What is the speed of the ball when the string makes an angle of 30° with the vertical?

51. A bullet is fired from a rifle with a speed v0 at an angle θ with respect to the horizontal axis (Fig. P6.51) from a cliff that is a height h above the ground below. (a) Use conservation of energy principles to calculate the speed of the bullet when it strikes the ground. Express your answer in terms of v0, h, g, and θ, and ignore air drag. (b) Explain why your result is independent of the angle θ.

52. Calculate the escape velocity for an object on the Moon.

53.  An engineer at NASA decides to save money on fuel by launching satellites from a very tall mountain. (a) Calculate the escape velocity for a satellite launched from an altitude of 8000 m above sea level (a mountain comparable to Mount Everest). (b) If the amount of fuel required is proportional to the initial kinetic energy, by what fraction is the fuel load reduced?

54. Calculate the velocity needed for an object starting at the Earth’s surface to just barely reach a satellite in a geosynchronous orbit. Ignore air drag.

55. Sketch how the potential energy, kinetic energy, and total mechanical energy vary with position for a projectile that is fired from the Earth’s surface. Assume the projectile’s initial velocity is equal to the escape velocity and ignore air drag.

56. Consider a spacecraft that is to be launched from the Earth to the Moon. Calculate the minimum velocity needed for the spacecraft to just make it to the Moon’s surface. Ignore air drag from the Earth’s atmosphere. Hint: The spacecraft will not have zero velocity when it reaches the Moon.

57. A space station is orbiting the Earth. It moves in a circular orbit with a radius equal to twice the Earth’s radius. A supply satellite is designed to travel to the station and dock smoothly when it arrives. If the supply satellite is fired as a simple projectile from the Earth’s surface, what is the minimum initial speed required for it to reach the space station and have a speed equal to the station’s speed when it arrives? Ignore both air drag and the rotational motion of the Earth.

58. A mass and spring are arranged on a horizontal, frictionless table as shown in Figure P6.58. The spring constant is k = 500 N/m, and the mass is 4.5 kg. The block is pushed against the spring so that the spring is compressed an amount 0.35 m, and then it is released. Find the velocity of the mass when it leaves the spring.

59. A block is dropped onto a spring with k = 30 N/m. The block has a speed of 3.3 m/s just before it strikes the spring. If the spring compresses an amount 0.12 m before bringing the block to rest, what is the mass of the block?

60. A block of mass 35 kg is sitting on a platform as shown in Figure P6.60. The platform sits on a spring with k = 2000 N/m. The mass is initially at rest. (a) Add a coordinate system to this sketch. Where is a convenient place to choose the origin of the vertical (y) axis? (b) If the platform is depressed a distance 0.50 m and held there, what is the mechanical energy of the system? (c) The platform is then released, and the mass moves upward and eventually flies off the platform. Assume the spring returns to its relaxed state (unstretched and uncompressed) after the mass leaves the platform. What are the kinetic energy, the gravitational potential energy, and the elastic potential energy of the system when the mass reaches its highest point? (d) How high will the block go? Measure this distance from the height of the uncompressed platform.

61. An archer’s bow can be treated as a spring with k = 3000 N/m. If the bow is pulled back a distance 0.12 m before releasing the arrow, what is the kinetic energy of the arrow when it leaves the bow?

62. For the arrow in Problem 61, what is the work done on the bow as the bow string is pulled back into position?

63. The shock absorbers on a car are fancy springs. Because one is attached to each wheel, your car is supported by four of these springs. Estimate the spring constant for a shock absorber.

64. A tennis racket can be treated as a spring. The displacement of the spring (what we commonly call x) is the displacement of the strings at the center of the racket. Estimate the spring constant of this spring. Assume a typical tennis ball has a mass of about 57 g and a speed of 50 m/s when it leaves the racket.

65. A diving board acts like a spring and obeys Hooke’s law (Fig. 6.21D). Estimate the spring constant of a diving board.

66. A tennis ball is a flexible, elastic object. If a person of average size stands on a tennis ball, the ball will compress to about half its original (non-compressed) diameter. What is the approximate spring constant of a tennis ball?

67. A tennis ball (m = 57 g) is projected vertically with an initial speed of 8.8 m/s. (a) If the ball rises to a maximum height of 3.7 m, how much kinetic energy was dissipated by the drag force of air resistance? (b) How much higher would the ball have gone in a vacuum?

68. A snowboarder of mass 80 kg travels down the slope of height 150 m shown in Figure P6.68. If she starts from rest at the top and has a velocity of 12 m/s when she reaches the bottom, what is the work done on her by friction?

69. Big bounce. A Super Ball is a toy ball made from the synthetic rubber polymer polybutadiene vulcanized with sulfur. Manufactured by Wham-o since 1965, these toys have a “super” elastic property such that they bounce to 90% of the height from which they are dropped. A Super Ball of mass 100 g is dropped from a height of 2.0 m. How much kinetic energy is dissipated by the collision with the ground? A lump of clay of equal mass is dropped from a similar height and sticks to the ground without a bounce. Where did the kinetic energy go in the case of the clay? What about in the case of the Super Ball?

70. Consider the skateboarder in Figure P6.70. If she has a mass of 55 kg, an initial velocity of 20 m/s, and a velocity of 12 m/s at the top of the ramp, what is the work done by friction on the skateboarder? Ignore the kinetic energy of the wheels of the skateboard.

71. A skier of mass 100 kg starts from rest at the top of the ski slope of height 100 m shown in Figure P6.71. If the total work done by friction is 3.0×104 J, what is the skier’s speed when he reaches the bottom of the slope?

72. The main frictional force on a good bicycle is the force of air drag (Eq. 3.20). Consider a bicyclist coasting at a constant velocity on a road that is sloped at 5.0° with respect to the horizontal. Find her speed. How much work is done by air drag as she travels a distance of 5.0 km?

73. Every year, there is a foot race to the top of the Empire State Building. The vertical distance traveled (using the stairs!) is about 440 m. The current record for this race is about 9 min 30 s. If the record holder has a mass of 60 kg, what was his average power output during the race?

74. A bicyclist travels on the hill shown in Figure P6.74. She begins from rest at the top of the hill, coasts all the way to the bottom, and finds that she gets to a point approximately 60% of the way up the other side before the bicycle comes (instantaneously) to rest. The total mass of the bicyclist plus the bicycle is 100 kg. (a) Estimate the total work done on the bicycle by all frictional forces (including air drag). (b) Assuming the main frictional force is due to air drag, find the approximate magnitude of this force throughout the motion. Hint: Calculate the power due to air drag by assuming her speed during the entire time is a constant and approximately equal to vave.

75. In Example 6.10, we compared the gravitational potential energy of a 100-W lightbulb to the electrical energy consumed when it is turned on. Repeat that calculation, but now assume the lightbulb is turned on for one day. Find the height h through which the lightbulb would have to fall for the change in gravitational potential energy to equal the electrical energy consumed by the lightbulb. Compare this answer to the height of Mount Everest. Assume you can use the relation PEgrav = mgh.

76. The rate of energy use in a typical house is about 2.0 kW. If the kinetic energy of a car (mass 1400 kg) is equal to the total energy used by the house in 1 min, what is the car’s speed?

77. An electric motor is rated to have a maximum power output of 0.75 hp. If this motor is being used to lift a crate of mass 200 kg, how fast (i.e., at what speed) can it lift the crate? Hint: Assume the only other force on the crate is the force of gravity.

78. For a car moving with a speed v, the force of air drag on a car is proportional v2. If the power output of the car’s engine is doubled, by what factor does the speed of the car increase?

79. A typical car has a power rating of 150 hp. Estimate the car’s maximum speed on level ground.

80. Consider a molecular motor that has an efficiency of 50%. Find the maximum force the motor can produce. Assume a step size of 4 nm. Also assume the energy consumed for each step is 5 × 10–20 J (as given for an ATP- powered motor in Section 6.8).

81. Suppose a molecular motor consumes 3 ATP molecules in every step. If this motor has a step size of 8.0 nm, what is the maximum force the motor could exert?

82. Molecular motors are often studied by attaching them to small plastic spheres. These spheres are much larger than the motor and enable the motor’s motion to be observed with a microscope. Consider a molecular motor to which a plastic sphere of radius 20 nm is attached. As the motor moves through water, a drag force given by Equation 3.23 acts on the sphere. If the molecular motor is able to produce a power output of 1 × 10–17 W, what is the speed of the motor as it pulls the sphere through water?

83. Consider a molecular motor that consumes the energy from 100 ATP molecules per second. What is the power output of this motor?

84. If the motor in Problem 83 takes a step of length 6.0 nm for each ATP molecule it consumes, what are the speed of the motor and the force it is able to produce?

85. SSM In Section 6.8, we calculated the force produced by a myosin molecular motor and found that it is approximately 10 piconewtons. Compare this force to the weight of a typical amino acid.

86. Conceptualizing units. A small frozen burrito, like those bought at your local convenience store, has a mass of approximately 100 g, which gives it a weight of roughly 1 newton. (a) From what height would you need to drop the burrito to give it 1 J of kinetic energy when it hits the ground? How many burritos would you have to drop from that height to equal the kinetic energy of (b) the arrow in example 6.7 and (c) the snowboarder (m = 65 kg) in Figure 6.12?

87. Counting calories. The chemical potential energy in foods is measured in units called Calories (notice the capital C). This “food Calorie” is equal to 1000 “physics” calories. For example, a typical apple contains approximately 75 Calories, which is actually 75 kilocalories. If all the energy in an apple were converted to gravitational potential energy, how high would you be lifted?

88. Repeat Problem 87, but replace the apple with a typical fast-food hamburger (approximately 400 C).

89. A doughnut (Fig. P6.89) contains roughly 350 C (1.5 × 106 J) of potential energy locked in chemical bonds. Find the ratio of the potential energy in the doughnut to that in an equivalent volume of TNT (trinitrotoluene), which has a density of 1.65 g/cm3 and releases 2.7 × 106 J per kg of explosive. Aren’t you glad the doughnut does not release all its potential energy at once?

90. One gallon of gasoline contains 3.1 × 107 calories of potential energy that are released during combustion. If 1 gal of gasoline can provide the force that moves a car though a displacement of 25 mi, what is the average force produced by the gasoline?

91. Tarzan (m = 74 kg) commutes to work swinging from vine to vine. He leaves the platform of his tree house and swings on the end of a vine of length L = 8.0 m. (a) If the platform is 1.9 m above the lowest point in the swing, what is the tension in the vine at the lowest point in the swing? (b) Tarzan again takes to his morning commute, but this time a monkey of mass mM = 23 kg hitches a ride by jumping onto Tarzan’s back. If a vine can withstand a maximum tension of 1200 N, will it snap under the tension of the added passenger? If so, at what angle with respect to the vertical does the vine break?

92. A toy gun shoots spherical plastic projectiles by means of a spring. A typical projectile has a mass of m = 25 g. The spring used has a spring constant of k = 15 N/m, and when put under load, it is displaced 6.0 cm as shown in Figure P6.92. The barrel of the gun exerts a slight frictional force of magnitude Ffriction = 0.074 N on the pellet as it moves down the barrel from its starting point a total length of L = 15 cm. If the toy gun is fired in a horizontal position, (a) at what position measured from the starting point does the pellet reach maximum velocity? Hint: Consider an equilibrium condition between the spring force and the friction force on the pellet before it leaves the spring. (b) What is the maximum speed achieved by the pellet as it is fired from the gun? (c) At what speed does the pellet leave the barrel?

93. A pole vaulter of mass 70 kg can run horizontally with a top speed of 10.0 m/s. The current record height for the pole vault is about 6.2 m. (a) Approximately how much energy must be stored in the pole just before the vaulter leaves the ground? Assume the vaulter’s speed is zero just prior to taking off. (b) The energy in part (a) is the maximum energy stored in the pole, so we denote it by PEmax. How does PEmax compare to the maximum kinetic energy KErun of the pole vaulter before he takes off? (c) Explain how PEmax can be greater than KErun.

94. A spring is mounted at an angle of θ = 35° on a frictionless incline as illustrated in Figure P6.94. The spring is compressed to 15 cm where it is allowed to propel a mass of 5.5 kg up the incline. (a) If the spring constant is 550 N/m, how fast is the mass moving when it leaves the spring? (b) To what maximum distance from the starting point will the mass rise up the incline?

95. Consider the system in Problem 94 and assume the coefficient of kinetic friction between the mass and the incline is 0.17. (a) At what position, measured from the starting point, does the maximum velocity occur? (b) What is the maximum velocity attained by the mass at this point? (c) What is now the maximum distance from the starting point that the mass will rise up the incline?

96. Spring shoes. Tae-Hyuk Yoon of South Korea is the inventor of a novel method of human locomotion: the Poweriser spring boot shown in Figure P6.96. Promotional materials claim that a 180-lb man can jump up to 6 ft high while wearing these devices. Estimate the spring constant of the leaf springs on the Poweriser. The maximum compression can be estimated from the figure.

97. A chairlift rises in elevation 2000 ft up a ski slope and has a total of 200 chairs evenly spaced 40 ft apart. On a busy weekend, the ski lift is at full capacity (two people per seat), and skiers come off the lift every 10 s. Assuming a skier weighs 160 lb on average, what is the power output required for the ski lift to operate? State your answer in both watts and horsepower.

98. A 70-kg skydiver reaches a terminal velocity of 50 m/s. At what rate is the drag force dissipating his kinetic energy? How does this rate compare with his rate of change in potential energy?

99. The world record for the highest-altitude skydive was made by Joseph Kittinger in 1960. Kittinger jumped from a high- altitude balloon (Fig. P6.99) at a height of 102,800 ft (31,330 m) above sea level, and according to reports, attained speeds up to 624 mi/h allowed by the thin air of the stratosphere. Calculate his maximum speed (i.e., neglect air drag) for his 14,500 ft of free fall. How does this speed compare with the reported value?

100. The NEAR spacecraft flew by the asteroid Eros in 1997 (Fig. P6.100). The flyby allowed the craft to measure the asteroid’s density at 2.7 g/cm3. Assume most asteroids have similar composition and that you personally can jump 1.0 m high on the Earth. (a) What is the radius of the largest spherical asteroid you could literally jump off of (in other words, that you could jump from with an initial velocity equal to the escape velocity of the asteroid)? (b) Eros has a mass of 6.7 × 1015 kg. Could you jump off Eros? If not, how far from the surface would you go before descending back to the asteroid?

101. Aerobic workout. After a 5-min workout on a Climb Max stair machine, the readout panel (Fig. P6.101) reports that the 75-kg user burned 19.7 Calories (see Problem 87) and climbed a total of 180 steps. (a) If each step is 15 cm in height, how much would the potential energy of the user have changed if she were actually climbing a stairway? (b) What was her rate of change in potential energy? (c) At what rate was her metabolism consuming energy? (d) What is her efficiency in converting the energy in the chemical bonds in her food into potential energy? (e) She really didn’t change her  vertical displacement on this machine, so where did that energy go?

102. The expression for the force due to air drag described in Chapter 3 (Eq. 3.20) ignores the aerodynamic shape of an object. Some objects, such as a race car, are shaped so as to minimize the drag force. This can be accounted for by adding a factor called the drag coefficient CD. The drag force is then Fdrag = -1/2CDρv2. A boxy car might have a drag coefficient CD ≈ 1.0, whereas a racecar might have CD ≈ 0.25. If all else is the same (the power produced by the engine, etc.) and there is no other source of friction or drag, what is the ratio of the race car’s top speed to the boxy car’s top speed?


Chapter 7


1. What is the magnitude of the momentum of a baseball (m = 0.14 kg) traveling at a speed of 100 mi/h? Express your answer in SI units.

2. Consider two cars, one a compact model (m = 800 kg) and the other an SUV (m = 2500 kg). The SUV is traveling at 10 m/s, while the compact car is traveling at an unknown speed v. If the two cars have the same momentum, what is v?

3. Consider two cars that are on course for a head-on collision. If they have masses of m1 = 1200 kg and m2 = 1800 kg and are both traveling at 30 m/s, what is the magnitude of the total momentum?

4. Two particles of mass m1 = 1.2 kg and m2 = 2.9 kg are traveling as shown in Figure P7.4. What is the total momentum of this system? Be sure to give the magnitude and direction of the momentum.

5. Slap shot! A hockey player strikes a puck that is initially at rest. The force exerted by the stick on the puck is 1000 N, and the stick is in contact with the puck for 5.0 ms (0.0050 s). (a) Find the impulse imparted by the stick to the puck. (b) What is the speed of the puck (m = 0.12 kg) just after it leaves the hockey stick?

6. A constant force of magnitude 25 N acts on an object for 3.0 s. What is the magnitude of the impulse?

7. A rubber ball (mass 0.25 kg) is dropped from a height of 1.5 m onto the floor. Just after bouncing from the floor, the ball has a speed of 4.0 m/s. (a) What is the magnitude and direction of the impulse imparted by the floor to the ball? (b) If the average force of the floor on the ball is 18 N, how long is the ball in contact with the floor?

8. For the cue ball in Figure Q7.12, make a qualitative sketch of how the force exerted by the rail on the ball varies with time. Be sure to plot both the x and y components of the force and indicate when the ball is in contact with the rail. 9. Consider again the cue ball in Question 12 and Figure Q7.12. (a) What is the impulse imparted to the ball? Be sure to give the magnitude and direction. (b) What is the change in momentum of the ball? Be sure to give the magnitude and direction.

10. A rubber ball is dropped and bounces vertically from a horizontal concrete floor. If the ball has a speed of 3.0 m/s just before striking the floor and a speed of 2.5 m/s just after bouncing, what is the average force of the floor on the ball? Assume the ball is in contact with the floor for 0.12 s and the mass of the ball is 0.15 kg.

11. SSM A baseball hits a baseball (m = 0.14 kg) as shown in Figure P7.11. The ball is initially traveling horizontally with speed of 40 m/s. The batter hits a fly ball as shown, with a speed vf = 55 m/s. (a) What is the magnitude and direction of the impulse imparted to the ball? (b) If the ball (m = 0.14 kg) and bat are in contact for a time of 8.0 ms, what is the magnitude of the average force of the bat on the ball? Compare this answer to the weight of the ball. (c) What is the impulse imparted to the bat?

12. The baseball in Problem 11 is replaced by a very flexible rubber ball of equal mass. (a) Does the contact time increase or decrease? (b) If the contact time changes by a factor of 50 but the initial and final velocities are the same, by what factor does the force of the bat on the ball change?

13. An egg and a tomato of equal mass are dropped from the roof of a three-story building onto the sidewalk below. The sidewalk imparts an impulse to both as they splatter. (a) Is the impulse imparted to the egg greater than, less than, or equal to the impulse imparted to the tomato? (b) Is the average force exerted on the egg greater than, less than, or equal to the force on the tomato?

14. Consider an archer who fires an arrow. The arrow has a mass of 0.030 kg and leaves the bow with a horizontal velocity of 80 m/s. (a) What is the impulse imparted to the arrow? Give both the magnitude and direction of the impulse. (b) What is the approximate average force of the bow string on the arrow?

15. Consider a tennis player who hits a serve at 140 mi/h (63 m/s). Estimate the average force exerted by the racket on the ball (m = 57 g).

16. Consider the problem of returning a serve in tennis. A serve is hit at a speed of 63 m/s and is hit back to the server with a speed of 40 m/s. (a) What is the impulse imparted to the ball? (b) What is the approximate average force on the ball?

17. Suppose the force on an object varies with time as shown in Figure P7.17. Estimate the impulse imparted to the object. Assume the motion is one dimensional.

18. Uppercut! A boxer hits an opponent on the chin and imparts an impulse of 500 N∙s. Estimate the average force.

19. SSM A golf ball is hit from the tee and travels a distance of 300 yards. Estimate the magnitude of the impulse imparted to the golf ball. Ignore air drag in your analysis.

20. Two objects of mass m and 3m undergo a completely inelastic collision in one dimension. If the two objects are at rest after the collision, what was the ratio of their speeds before the collision?

21. Two particles of mass m1 = 1.5 kg and m2 = 3.5 kg undergo a one-dimensional head-on collision as shown in Figure P7.21. Their initial velocities along x are v1i = 12 m/s and v2i = -7.5 m/s. The two particles stick together after the collision (a completely inelastic collision). (a) Find the velocity after the collision. (b) How much kinetic energy is lost in the collision?

22. Two skaters are studying collisions on an ice-covered (frictionless) lake. Skater 1 (m1 = 85 kg) is initially traveling with a speed of 5.0 m/s, and skater 2 (m2 = 120 kg) is initially at rest. Skater 1 then “collides” with skater 2, and they lock arms and travel away together. (a) Identify a “system” whose momentum is conserved. (b) Draw a sketch of your system in part (a) before the collision. Include a coordinate system and identify the initial velocities of the different parts of the system. (c) Express the conservation condition for your system. (d) Does your system undergo an elastic collision or an inelastic collision? (e) Solve for the final velocity of the two skaters.

23. The particles in Figure P7.21 (m1 = 1.5 kg and m2 = 3.5 kg) undergo an elastic collision in one dimension. Their velocities before the collision are v1i = 12 m/s and v2i = -7.5 m/s. Find the velocities of the two particles after the collision.

24. Two hockey pucks approach each other as shown in Figure P7.24. Puck 1 has an initial speed of 20 m/s, and puck 2 has an initial speed of 15 m/s. They collide and stick together. (a) If the two pucks form a “system,” is the momentum of this system along x or y conserved? (b) Find the components along x and y of the initial velocities of both particles. (c) Express the conservation of momentum condition for motion along x and y. (d) Is this collision elastic or inelastic? (e) Find the final velocity of the two pucks after the collision. (f) What fraction of the initial kinetic energy is lost in the collision?

25. Consider again the collision between two hockey pucks in Figure P7.24, but now they do not stick together. Their speeds before the collision are v1i = 20 m/s and v2i = 15 m/s. It is found that after the collision one of the pucks is moving along x with a speed of 10 m/s. What is the final velocity of the other puck?

26. In Example 7.4, we considered an elastic collision in one dimension in which one of the objects is initially at rest. Notice that mass 2 is the one that is initially at rest. Use the results for v1f and v2f to answer the following problems. (a) Assume m1 = 10 m2 so that the incoming object is much more massive than the one at rest. Find v1f /v1i and v2f /v1i. Explain why the final velocity of the smaller object is much larger (in magnitude) than the final velocity of the bigger object. (b) Assume m1 = m2/10. Find v1f /v1i and v2f /v1i. Explain why the final velocity of the smaller object is opposite its initial direction.

27. Two billiard balls collide as shown in Figure P7.27. Ball 1 is initially traveling along x with a speed of 10 m/s, and ball 2 is at rest. After the collision, ball 1 moves away with a speed of 4.7 m/s at an angle θ = 60°. (a) Find the speed of ball 2 after the collision. (b) What angle does the final velocity of ball 2 make with the x axis?

28. Consider an elastic collision in one dimension that involves objects of mass 2.5 kg and 4.5 kg. The larger mass is initially at rest, and the smaller one has an initial velocity of 12 m/s. Find the final velocities of the two objects after the collision.

29. Two hockey players are traveling at velocities of v1 = 12 m/s and v2 = -18 m/s when they undergo a head-on collision. After the collision, they grab each other and slide away together with a velocity of - 4.0 m/s. Hockey player 1 has a mass of 120 kg. What is the mass of the other player?

30. Two cars of equal mass are traveling as shown in Figure P7.30 just before undergoing a collision. Before the collision, one of the cars has a speed of 18 m/s along +x, and the other has a speed of 25 m/s along +y. The cars lock bumpers and then slide away together after the collision. What are the magnitude and direction of their final velocity?

31. Two cars collide head-on and lock bumpers on an icy (frictionless) road. One car has a mass of 800 kg and an initial speed of 12 m/s and is moving toward the north. The other car has a mass of 1200 kg. If the speed of the cars after the collision is 4.5 m/s and they are moving north, what is the initial velocity of the large car?

32. A railroad car (m = 3000 kg) is coasting along a level track with an initial speed of 25 m/s. A load of coal is then dropped into the car as sketched in Figure 7.18. (a) Treat the railroad car plus coal as a system. Is the momentum of this system along the horizontal (x) or vertical (y) conserved? (b) What are the velocities of the car and the coal before the collision along the direction found in part (a)? (c) Express the conservation of momentum condition for the system. (d) If the final speed of the car plus coal is 20 m/s, what is the mass of the coal?

33. A baseball pitcher (m = 80 kg) is initially standing at rest on an extremely slippery, icy surface. He then throws a baseball (m = 0.14 kg) with a horizontal velocity of 50 m/s. What is the recoil velocity of the pitcher?

34. A military gun is mounted on a railroad car (m = 1500 kg) as shown in Figure P7.34. There is no frictional force on the car, the track is horizontal, and the car is initially at rest. The gun then fires a shell of mass 30 kg with a velocity of 300 m/s at an angle of 40° with respect to the horizontal. Find the recoil velocity of the car.

35. An open railroad car of mass 2500 kg is coasting with an initial speed of 14 m/s on a frictionless, horizontal track. It is raining, and water begins to accumulate in the car. After some time, it is found that the speed of the car is only 11 m/s. How much water (in kilograms) has accumulated in the car?

36. The M79 grenade launcher was first used by the U.S. Army in 1961. It fires a grenade with an approximate mass m = 7.0 kg at a speed of 75 m/s. (a) If a soldier fires the M79 horizontally while standing on a very slippery surface, what is the recoil speed of the soldier? Take the mass of the soldier plus M79 to be 100 kg. (b) The barrel of the M79 is 36 cm long. Estimate the average recoil force on the soldier.

37. A skateboarder (m = 85 kg) takes a running jump onto a friend’s skateboard that is initially at rest as sketched in Figure P7.37. The friend is standing on the skateboard (mass of friend plus skateboard = 110 kg). After landing on the skateboard, the velocity of the board plus the two skateboarders is 3.0 m/s. What was the horizontal component of the velocity of the jumping   skateboarder just before he landed on the skateboard?

38. A thief is traveling with stolen goods on his bicycle when he spots a police officer approaching. The thief wants to throw the stolen goods away before the officer arrives and wants the goods to land as far away as possible from him. The thief is traveling along the +x direction and throws the goods with a certain speed v0 relative to the bicycle and an angle θ = 45° relative to the x axis. Should he throw the goods (a) in the +x direction or (b) in the -x direction as sketched in Figure P7.38? Does it matter? Assume the bicycle is coasting throughout.

39. An airplane is sent on a rescue mission to help a stranded explorer near the South Pole. The plane cannot land, but instead drops a supply package of mass m = 60 kg while it is flying horizontally with a speed of vp = 120 m/s. The supply package drops onto a waiting sled that is initially at rest (Fig. P7.39). After the package lands in the sled, the speed of the sled plus package is found to be 30 m/s. What is the mass of the sled?

40. A railroad car containing explosive material is initially traveling south at a speed of 5.0 m/s on level ground. The total mass of the car plus explosives is 3.0 × 104 kg. An accidental spark ignites the explosive, and the car breaks into two pieces, which then roll away along the same track. If one piece has a mass of 2.0 × 104 kg and a final speed of 2.5 m/s toward the south, what is the final speed of the other piece?

41. Find the center of mass of the two particles in Figure P7.41.

42. Find the center of mass of the four particles in Figure P7.42. Their masses are m1 = 12 kg, m2 = 18 kg, m3 = 7.9 kg, and m4 = 23 kg.

43. Two objects are moving along the x axis with velocities of 35 m/s (object 1) and -25 m/s (object 2). (a) If the center of mass has a velocity of +10 m/s, which object has the greater mass? (b) What is the ratio of their masses?

44. Consider the motion of the two ice skaters in Figure 7.27 and assume they have masses of 60 kg and 100 kg. If the larger skater has moved a distance of 12 m from his initial position, where is the smaller skater?

45. Estimate the location of the center of mass of the thin metal plate in Figure P7.45.

46. Consider the Sun and the Earth as a “system” of two “particles”. The center of mass of this system lies a distance L from the center of the Sun. Find L and compare it to the radius of the Sun.

47. Finding the barycenter. How far is the center of mass of the Earth-Moon system from the center of the Earth? Is it below the surface of the Earth?

48. A person of mass 90 kg standing on a horizontal, frictionless surface fires a bullet of mass 40 g in a horizontal direction. After the bullet has traveled a distance of 50 m, what distance has the person moved?

49. Consider the motion of an apple and the Earth as the apple falls to the ground. If the apple begins at the top of a tree of height 15 m, how much does the Earth move?

50. A bomb that is initially at rest at the origin explodes and breaks into two pieces of mass m1 = 1500 kg and m2 = 2500 kg as shown in Figure P7.50. If piece 1 has a velocity of +35 m/s directed along x, what is the velocity of piece 2 after the explosion?

51. A hand grenade is thrown with a speed of v0 = 30 m/s, as sketched in Figure P7.51, just prior to exploding. It breaks into two pieces of equal mass after the explosion. One piece has a final velocity of 40 m/s along x. Find the velocity of the other piece after the explosion.

52. The ballistic pendulum. A bullet of mass m = 30 g is fired into a wooden block of mass M = 5.0 kg as shown in Figure P7.52. The block is attached to a string of length 1.5 m. The bullet is embedded in the block, causing the block to then swing as shown in the figure. If the block reaches a maximum height of h = 0.30 m, what was the initial speed of the bullet?

53. A rock of mass 30 g strikes a car windshield while the car is traveling on a highway. Estimate the force of the rock on the windshield. Hint: You will need to estimate the collision time (i.e., the time that the rock is in contact with the windshield). For comparison, the collision time for a baseball-bat collision is approximately 0.001 s, while the time for a tennis racket-ball collision is about 0.01 s.

54. Two particles of mass 2.3 kg and mass 4.3 kg that are free to move on a horizontal track are initially held at rest so that they compress a spring as shown in Figure P7.54. The spring has a spring constant k = 400 N/m and is compressed 0.12 m. Find the final velocities of the two particles.

55. A tennis ball of mass 0.12 kg is dropped vertically onto a hard concrete floor from a height of 1.5 m. The ball then bounces up to some maximum height h. If the ball bounces up to a height h = 1.0 m, what is the approximate average force of the floor on the ball?

56. Consider a reaction that involves the creation of two elementary particles as in Figure P7.56. One particle is a proton of mass mp, and the other has a mass of m2 = mp/1800. If the proton leaves the reaction with a speed vp, what is the final speed of the other particle? Express your answer in terms of vp.

57. A bullet of mass 15 g is fired with an initial speed of 300 m/s into a wooden block that is initially at rest. The bullet becomes lodged in the block, and the two slide together on the floor for a distance of 1.5 m before coming to rest (Fig. P7.57). If the coefficient of friction between the block and the floor is 0.40, what is the mass of the block?

58. A father (mF = 75 kg) and his daughter (mD = 35 kg) stand on a fl at, frozen lake of negligible friction. They hold a 10-mlong rope stretched between them. The father and daughter then pull the rope to bring them together. If the father is initially standing at the origin, how far from the origin will they meet?
59. Twin brothers, each of mass 50 kg, sit in a symmetric canoe of mass 40 kg. Each boy sits 1.5 m away from the center of the canoe as they toss a basketball (mass = 0.60 kg) back and forth. (a) How much does the center of the canoe move as the ball moves from one brother to the other? (b) How much would the canoe shift on each throw if instead they used a medicine ball of mass 11 kg?

60. A block of mass 3.5 kg is initially at rest on a wedge of mass 20 kg, height 0.30 m, and an incline angle of θ = 35° as shown in Figure P7.60. There is no friction between the wedge and the floor. Starting at the top of the incline, the block is released and slides toward the bottom of the wedge. At the same time, the wedge “recoils” and slides some distance L to the right. Find L when the block has reached the bottom of the wedge. Hint: The center of mass of a right triangle of height h is a distance h/3 above the base of the triangle.

61. Consider three scenarios for a one-dimensional elastic collision with a stationary target. For each scenario, determine the final velocities of both the projectile mass, v1f, and the target mass, v2f, in terms the projectile’s initial velocity, v0. Assume the masses are sliding on a horizontal surface of negligible friction. Case (a): Both the projectile and target are bowling balls; m1 = m2 = 7.25 kg. Case (b): The target is one of the bowling balls, but the projectile is a Styrofoam ball of the same diameter; m1 = 140 g, m2 = 7.25 kg. Case (c): Now make the Styrofoam ball the target and the bowling ball the projectile; m2 = 140 g, m1 = 7.25 kg. Compare parts (b) and (c) to the following extremes: (d) when the target is the Earth and the projectile is a tennis ball, and (e) when the target is a frozen pea and the projectile is a truck’s windshield.

62. Fun with a Super Ball. A Super Ball has a coefficient of restitution of a = 0.90. This means that the y components of the ball’s velocity just before and after hitting the floor are related by vfy = -αvif . It is dropped from a height of 1.8 m such that it goes under a table of height 70 cm. The ball bounces back and forth from the underside of the table and the ground (Fig. P7.62). How many total times will the ball bounce off the bottom of the table?

63. Consider again the discussion in Section 7.2 concerning air bags and impulse for a car colliding with a tree. The car’s velocity of 40 mi/h is reduced to zero very quickly, but the head of a passenger is still moving at 40 mi/h (about 18 m/s). Calculate the average force exerted on the passenger’s head coming to rest for the case (a) when it is stopped by the dashboard in 5.0 ms and (b) when it is stopped by an air bag that compresses in 45 ms. (c) Air bags must deploy quickly. If the distance between the passenger’s head and the dashboard is 60 cm and it takes the air bag 45 ms to safely bring the passenger to rest, how much time is there for the air bag to inflate?

64. Crumple zones. A life-saving development in automobile manufacture is the invention of crumple zones, areas of the body and frame of a car deliberately made to collapse in a collision (Fig. P7.64) such that the passenger compartment will not. Introduced in 1955 on the Heckflosse made by  Mercedes, crumple zones became mandatory on all cars sold in the United States in 1967. For a head-on collision with a tree at 40 mi/h (about 18 m/s), find the ratio of the average force on a car with a stiff frame (stop time of 0.010 s) to that on a car with a crumple zone (stop time of 0.25 s).

65. A projectile of mass 10 kg is launched at an angle of 55° and an initial speed of 87 m/s. Just as the object reaches the maximum height in its trajectory, a small explosion along the horizontal blows off the projectile’s back portion equal to one-fourth its original mass. The smaller piece lands exactly at the launch point as shown in Figure P7.65. How far from the launch point does the larger portion land?

66. A cart of mass m1 = 10 kg slides down a frictionless ramp and is made to collide with a second cart of mass m2 = 20 kg, which then heads into a vertical loop of radius 0.25 m as shown in Figure P7.66. (a) Determine the height h from which cart 1 would need to start to make sure that cart 2 completes the loop without leaving the track. Assume an elastic collision. (b) Find the height needed if instead the more massive cart is allowed to slide down the ramp into the smaller cart.

67. Again consider the track and carts in Figure P7.66 with m1 = 10 kg and m2 = 20 kg, but this time the carts stick together after the collision. (a) Find the height h from which cart 1 would need to start to make sure that both carts complete the loop without leaving the track. (b) Find the height needed if
instead the more massive cart is made to collide with the smaller cart.

68. A block of wood (Mblock = 5.2 kg) rests on a horizontal surface for which the coefficient of kinetic friction is 0.28. A bullet (mbullet = 32 g) is shot along the horizontal direction and embeds in the block, displacing the block 87 cm. What was the initial velocity of the bullet before impact?

69. A firefighter directs a horizontal stream of water toward a fire at an angle of 40° measured from the horizontal. The fire hydrant and hose deliver 95 gallons per minute (1 gal/min = 6.31 × 10-4 m3/s), and the nozzle projects the water with a speed of 7.5 m/s. How much horizontal force must the firefighter exert on the hose to keep it stationary?

70. A machine gun directs a flow of bullets at a steel target mounted on a cart with negligible friction and of mass 600 kg. In this case, the gun is an Uzi, which uses 9-mm (8.0-g) bullets and can fire at a rate of 10 rounds/s, and the bullets have an average muzzle velocity of 400 m/s. (a) Assuming each bullet’s ricochet off the target can be approximated by an elastic collision, with what final velocity would the cart be moving if all 50 rounds in the Uzi’s magazine were fired at the maximum rate at the target? (b) How fast would the cart be moving if the target were made of soft wood such that all 50 bullets became embedded? (c) The Uzi is now firmly attached to the cart while at rest. The trigger is then pulled, and all 50 rounds are discharged horizontally into the air. What is the final velocity of the cart?

71. Rocket engines work by expelling gas at a high speed as illustrated in Figure P7.71. We wish to design an engine that expels an amount of gas mg = 50 kg each second, and we want the engine to exert a force F = 20,000 N on a rocket. At what speed vg should the gas be expelled? Assume the mass of the rocket is very large.

72. Comet Shoemaker-Levy struck Jupiter in 1994. Just before the collision, the comet broke into several pieces. The total mass of all the pieces was estimated to be 3.8 × 108 kg, and they all had a velocity relative to Jupiter of 60 km/s. How much energy was deposited on Jupiter?

 

Chapter 8


1. An angle has a value of 85°. What is the angle in radians?

2. The shaft on an engine turns through an angle of 45 rad. (a) How many revolutions does the shaft make? (b) What is this angle in degrees?

3. An old-fashioned (vinyl) record rotates at a constant rate of 33 rpm. (a) What is its angular velocity in rad/s? (b) If the record has a radius of 18 cm, what is the linear speed of a point on its edge?

4. The angular speed of the shaft of a car’s engine is 360 rad/s. How many revolutions does it complete in 1 hour?

5. Consider again the record in Problem 3. If it plays music for 20 min, how far does a point on the edge travel in meters?

6. The tachometer in the author’s car has a maximum reading of 8000 rpm. (A tachometer is a gauge that shows the angular velocity of a car’s engine.) Express this angular velocity in rad/s.

7. A car engine is initially rotating at 200 rad/s. It is then turned off and takes 3.0 s to come to a complete stop. What is the angular acceleration of the engine? Assume a is constant.

8. What is the angular velocity of the Earth’s center of mass as it orbits the Sun?

9.What is the angular velocity of a person standing (a) at the equator? (b) At a latitude of 45°?

10. Mercury spins about its axis with a period of approximately 58 days. What is the rotational angular velocity of Mercury?

11. Mercury orbits the Sun with a period of approximately 88 days. What is the angular velocity of Mercury’s orbital motion?

12. An adult is riding on a carousel and finds that he is feeling a bit nauseated. He then decides to move from the outer edge of the carousel to a point a distance r2 from the center so that his linear speed is reduced by a factor of 2.5. If the carousel has a constant angular velocity of 1.0 rad/s and a radius of 3.3 m, what is r2?

13. What is the angular velocity and the period of the second hand of a clock? If the linear speed of the end of the hand is 5.0 mm/s, what is the clock’s radius?

14. Construct a graph of the angular displacement of the minute hand of a clock as a function of time. The axes of your graph should include quantitative scales (i.e., numbers and units). What is the slope of the relation between θ and t?

15. Consider a race car that moves with a speed of 200 mi/h. If it travels on a circular race track of circumference 2.5 mi, what is the car’s angular velocity?

16. A diver stands at rest at the end of a massless diving board as shown in Figure P8.16. (a) If the mass of the diver is 120 kg and the board is 4.0 m long, what is the torque due to gravity on the diving board with a pivot point at the fixed end of the board? (b) According to our convention for positive and negative torques, is the torque on the diving board positive or negative?

17. Consider again the diver in Figure P8.16. Assume the diving board now has a mass of 30 kg. Find the total torque due to gravity on the diving board.

18. SSM Consider the clock in Figure 8.17. Calculate the magnitude and sign of the torque due to gravity on the hour hand of the clock at 4 o’clock. Assume the hand has a mass of 15 kg, a length of 1.5 m, and the mass is uniformly distributed.

19. A rod of length 3.8 m is hinged at one end, and a force of magnitude F = 10 N is applied at the other (Fig. P8.19). (a) If the magnitude of the torque associated with this force is 18 N∙m, what is the angle f? (b) What is the sign of the torque?

20. A person carries a long pole of mass 12 kg and length 4.5 m as shown in Figure P8.20. Find the magnitude of the torque on the pole due to gravity.

21. If the length of the pole in Problem 20 is increased by a factor of three and its mass is increased by a factor of two, by what factor does the torque change?

22. A difficult maneuver for a gymnast is the “iron cross,” in which he holds himself as shown in Figure P8.22. Estimate the torque on the gymnast due to gravity. Choose a rotation axis that passes through one of his hands and assume each hand must support half of his weight.

23. A tree grows at an angle of 50° to the ground as shown in Figure Q8.5. If the tree is 25 m from its base to its top, and has a mass of 500 kg, what is the approximate magnitude of the torque on the tree due to the force of gravity? Take the base of the tree as the pivot point. (The answer is one reason trees need roots.)

24. A baseball player holds a bat at the end of the handle. If the bat is held horizontally, what is the approximate torque due to the force of gravity on the bat, with a pivot point at the batter’s hands?

25. Suppose a crate of mass 7.5 kg is placed on the plank in Figure P8.25 at a distance 3.9 m from the left end. Find the forces exerted by the two supports on the plank.

26. A uniform wooden plank of mass 12 kg rests on two supports as shown in Figure P8.25. The plank is at rest, so it is in translational equilibrium and rotational equilibrium. (a) Make a sketch showing all the forces on the plank. Be sure to show where the forces act. (b) Choose the left-hand support for the pivot point and calculate the torque from each of the forces in part (a). (c) Express the conditions for translational and rotational equilibrium. (d) Solve to find the forces exerted by the two supports on the plank.

27. Consider the flagpole in Figure P8.27. If the flagpole has a mass of 20 kg and length 10 m and the angle the cable makes with the pole is θ = 25°, what are the magnitude and direction of the force exerted by the hinge (at point P) on the flagpole? Assume the mass of the pole is distributed uniformly.

28. The seesaw in Figure P8.28 is 4.5 m long. Its mass of 20 kg is uniformly distributed. The child on the left end has a mass of 14 kg and is a distance of 1.4 m from the pivot point while a second child of mass 30 kg stands a distance L from the pivot point, keeping the seesaw at rest. (a) Is the seesaw in rotational equilibrium? (b) Make a sketch showing all the forces on the seesaw. Be sure to show where each force acts. (c) Express the conditions for translational and rotational equilibrium for the seesaw. (d) Solve to find L, the location of the child on the right.

29. Consider the ladder in Example 8.5 and assume there is friction between the vertical wall and the ladder, with μS = 0.30. Find the angle φ at which the ladder just begins to slip.

30. A painter is standing on the ladder (mass 40 kg and length 2.5 m) in Figure P8.30. There is friction between the bottom of the ladder and the floor with μS = 0.30, but there is no friction between the ladder and the wall. A painter of mass of 70 kg is standing a distance of 0.60 m from the top. (a) Make a sketch showing all the forces on the ladder. Be sure to show where each force acts. (b) Express the conditions for translational and rotational equilibrium for the ladder. (c) If θ = 60°, use the results from part (b) and determine whether or not this ladder will slip.

31. Consider again the ladder in Problem 30. What is the sign of the torque on the ladder due to the force from the wall?

32. A solid cube of mass 40 kg and edge length 0.30 m rests on a horizontal floor as shown in Figure P8.32. A person then pushes on the upper edge of the cube with a horizontal force of magnitude F. At what value of F will the cube start to tip? Assume the frictional force from the floor is large enough to prevent the cube from sliding.

33. Consider again the cube in Problem 32 (Fig. P8.32), but now assume the force is applied along a horizontal line that is 0.20 m above the floor. At what value of F will the crate begin to tip?

34. Repeat Problem 32, but assume the force F is applied at the corner of the cube and at an angle of 30° above the horizontal direction. At what value of F will the crate begin to tip?

35. Wheelbarrows are designed so that a person can move a massive object more easily than if he simply picked it up. If the person using the wheelbarrow in Figure P8.35 is able to apply a total vertical (upward) force of 200 N on the handle, what is the approximate mass of the heaviest object he could move with the wheelbarrow?

36. One way to put a ladder in place on a wall is to “walk” the ladder to the wall. One end of the ladder is held fixed (perhaps at the base of the wall), and a person slides his hands along the ladder as he walks toward the fixed end as sketched in Figure P8.36. Anyone who has raised a ladder in this way knows that the amount of force the person must apply grows larger as the person moves toward the end that is fixed. Assuming the force from the person is perpendicular to the ladder, find the force the person must apply to hold the ladder at (a) an angle of 10° and (b) an angle of 20°.

37. Consider again the ladder in Figure P8.36. We saw in Problem 8.36 that the force supplied by the person increases with increasing angle φ as the ladder is initially lifted off the ground. When the ladder is sufficiently high (the angle φ is sufficiently large), though, it becomes easier to raise the ladder higher (i.e., the necessary applied force becomes smaller). Find the angle φ at which that happens.

38. A flagpole of length 12 m and mass 30 kg is hinged at one end, where it is connected to a wall as sketched in Figure P8.38. The pole is held up by a cable attached to the end. Find the tension in the cable.

39. A long, nonuniform board of length 8.0 m and mass m = 12 kg is suspended by two ropes as shown in Figure P8.39. If the tensions in the ropes are mg/3 (on the left) and 2mg/3 (on the right), what is the location of the board’s center of mass?

40. Consider again the problem of a tipping car in Example 8.6. This time, instead of applying a force F to the car, assume the car is traveling around a curve on a level road. Let the radius of curvature of the turn be r and assume the car’s speed v is just barely fast enough to make the car’s inside wheels lift off the ground. (a) Find v. Express your answer in terms of r and the variables w, d, and h from Figure 8.25. (b) Use realistic values to get a numerical estimate for v for r = 80 m. Hint: Use the car’s center of mass as your pivot point.

41. Two particles of mass m1 = 15 kg and m2 = 25 kg are connected by a massless rod of length 1.2 m. Find the moment of inertia of this system for rotations about the following pivot points: (a) the center of the rod, (b) the end at m1, (c) the end at m2, and (d) the center of mass. Assume the rotation axis is
perpendicular to the rod.

42. Four particles with masses m1 = 15 kg, m2 = 25 kg, m3 = 10 kg, and m4 = 20 kg sit on a very light (massless) metal sheet and are arranged as shown in Figure P8.42. Find the moment of inertia of this system with the pivot point (a) at the origin and (b) at point P. Assume the rotation axis is parallel to the z direction, perpendicular to the plane of the drawing.

43. Consider a merry-go-round that has the form of a disc with radius 5.0 m and mass 100 kg. If three children, each of mass 20 kg, sit on the outer edge of the merry-go-round, what is the total moment of inertia?

44. Which has a larger moment of inertia, a disc of mass 20 kg or a hoop of mass 15 kg with the same radius?

45. Estimate the moment of inertia of a bicycle wheel.

46. If the mass of a wheel is increased by a factor of 2 and the radius is increased by a factor of 1.5, by what factor is the moment of inertia increased? Model the wheel as a solid disc.

47. Consider two cylinders with the same density and the same length. If the ratio of their diameters is 1.5/1, what is the ratio of their moments of inertia?

48. In our examples with CDs (such as Example 8.7), we usually ignored the hole in the center. Calculate the moment of inertia of a CD, including the effect of the hole. For a CD of radius 6.0 cm, estimate the percentage change in the moment of inertia due to a hole of radius 7 mm.

49. The moment of inertia for a square plate of mass M and length L that rotates about an axis perpendicular to the plane of the plate and passing through its center is ML2/6 (Fig. P8.49A). What is the moment of inertia of the same plate when it is rotated about an axis that lies along one edge of the plate (Fig. P8.49B)?

50. Estimate the moment of inertia of a baseball bat for the following cases. (a) Assume the bat is a wooden rod of uniform diameter and take the pivot point to be (i) in the middle of the bat, (ii) at the end of the bat, and (iii) 5 cm from the end of the handle, where a batter who “chokes up” on the bat would place his hands. Hint: Consider the bat as two rods held together at the pivot point. (b) Assume a realistic bat, which is narrower at the handle than at the opposite end, and consider the three locations from part (a).

51. Figure P8.51 shows the angular displacement of an object as a function of time. What is the approximate angular velocity of the object?

52. Figure P8.52 shows the angular displacement of an object as a function of time. (a) What is the approximate angular velocity of the object at t = 0? (b) What is the approximate angular velocity at t = 0.10 s? (c) Estimate the angular acceleration at t = 0.050 s.

53. A ceiling fan of radius 45 cm runs at 90 rev/min. How far does the tip of the fan blade travel in 1 hour?

54. Construct a plot of the angular displacement of a CD as a function of time. Use information from this chapter to attach quantitative labels (numbers and units) to your graph. Take t = 0 to be the time when the CD player is first turned on. End your graph when the CD player is turned off and the CD stops spinning.

55. A potter’s wheel of radius 0.20 m is turned on at t = 0 and accelerates uniformly. After 60 s, the wheel has an angular velocity of 2.0 rad/s. Find the angular acceleration and the total angular displacement of the wheel during this time.

56. Construct a graph of the angular velocity of a car wheel as a function of time. (a) Assume the wheel starts from rest and moves with a constant (center of mass) velocity. (b) Assume the car starts from rest and accelerates uniformly. (c) Assume the car comes to a stop uniformly at a traffic light.

57. Two crates of mass 5.0 kg and 9.0 kg are connected by a rope that runs over a pulley of mass 4.0 kg as shown in Figure P8.57. (a) Make a sketch showing all the forces on both crates and the pulley. (b) Express Newton’s second law for the crates (translational motion) and for the pulley (rotational
motion). The linear acceleration a of the crates, the angular acceleration a of the pulley, and the tensions in the right and left portions of the rope are unknowns. (c) What is the relation between a and a? (d) Find the acceleration of the crates. (e) Find the tensions in the right and left portions of the rope.

58. Two crates of mass m1 = 15 kg and m2 = 25 kg are connected by a cable that is strung over a pulley of mass mpulley = 20 kg as shown in Figure P8.58. There is no friction between crate 1 and the table. (a) Make a sketch showing all the forces on both crates and the pulley. (b) Express Newton’s second law for the crates (translational motion) and for the pulley (rotational motion). The linear acceleration a of the crates, the angular acceleration a of the pulley, and the tensions in the right and left portions of the rope are unknowns. (c) What is the relation between a and a? (d) Find the acceleration of the crates. (e) Find the tensions in the right and left portions of the rope.

59. For the system in Figure P8.58, if the coefficient of friction between crate 1 and the table is μk = 0.15, what is the acceleration?

60. Consider the CD in Example 8.2. This CD player is turned on at t = 0 and very quickly starts to play music so that the CD has an angular velocity of 500 rpm just a few seconds after t = 0. The CD then plays for 1 hour, during which time it has a constant angular acceleration and a final angular velocity of 200 rpm. Estimate the total distance traveled by a point on the edge of the CD. Assume the CD has a radius of 6.0 cm.

61. Consider again the CD in Problem 60. As was explained in Example 8.2, the data on the CD are encoded in a long spiral “track,” where the spacing between each turn of the track is 1.6 μm and the inner and outer radii of the program area are about 25 mm and 58 mm, respectively. Estimate the length of this spiral.

62. You like to swim at a nearby lake. On one side of the lake is a cliff, and the top of the cliff is 6.5 m above the surface of the lake. You dive off the cliff doing somersaults so as to have an angular speed of 2.2 rev/s. How many revolutions do you make before you hit the water? Assume your initial center of mass velocity is horizontal and you begin from a standing position. Do you enter the water head first or feet first?

63. A Ferris wheel is moving at an initial angular velocity of 1.0 rev/30 s. If the operator then brings it to a stop in 3.0 min, what is the angular acceleration of the Ferris wheel? Express your answer in rad/s2. Through how many revolutions will the Ferris wheel move while coming to a stop?

64. A board of length 1.5 m is attached to a floor with a hinge at one end as shown in Figure P8.64. The board is initially at rest and makes an angle of 40° with the floor. If the board is then released, what is its angular acceleration?

65. A car starts from rest and then accelerates uniformly to a linear speed of 15 m/s in 40 s. If the tires have a radius of 25 cm, what are (a) the average linear acceleration of a tire, (b) the angular acceleration of a tire, (c) the angular displacement of a tire from t = 0 to 40 s, and (d) the total distance traveled by a point on the edge of a tire?

66. A bicycle wheel of radius 25 cm rolls a distance of 3.0 km. How many revolutions does the wheel make during this journey?

67. SSM Consider a tennis ball that is hit by a player at the baseline with a horizontal velocity of 45 m/s (about 100 mi/h). The ball travels as a projectile to the player’s opponent on the opposite baseline, 24 m away, and makes 25 complete revolutions during this time. What is the ball’s angular velocity?

68. A cylinder rolls without slipping down an incline that makes a 30° angle with the horizontal. What is the acceleration of the cylinder?

69. A string is rolled around a cylinder (m = 4.0 kg) as shown in Figure P8.69. A person pulls on the string, causing the cylinder to roll without slipping along the table. If there is a tension of 30 N in the string, what is the acceleration of the cylinder?

70. A truck accelerates from rest to a speed of 12 m/s in 5.0 s. If the tires have a radius of 40 cm, how many revolutions do the tires make during this time?

71. A car is traveling with a speed of 20 m/s. If the tires have an angular speed of 62 rad/s, what is the radius of the tires?

72. A car has a top speed of 70 m/s (about 150 mi/h) and has wheels of radius 30 cm. If the car starts from rest and reaches its operating speed after 9.0 s, what is the angular acceleration of the wheels? Assume the car accelerates uniformly.

73. The wheels on a special bicycle have different radii Rf and Rb (Fig. P8.73). The bicycle starts from rest at t = 0 and accelerates uniformly, reaching a linear speed of 10 m/s at t = 9.0 s. (a) What is the bicycle’s linear acceleration? (b) At t = 4.5 s, the angular speed of the back wheel is 11 rad/s, whereas
the angular speed of the front wheel is greater by a factor of three. Find Rb and Rf.

74. Two uniform density bricks are stacked as shown in Figure P8.74. What is the maximum overhang distance d that can be achieved? Hint: Consider how the torque on each brick’s center of mass must be in equilibrium about the balance point (the edge of the ledge).

75. Consider again the problem of overhanging bricks in Figure P8.74. Find the maximum overhang distance, d, for systems of (a) three bricks and (b) four bricks. (c) How many bricks are needed for the top brick to have an overhang of d = 2L?

76. Three gears of a mechanism are meshed as shown in Figure P8.76 and are rotating with constant angular velocities. The ratio of the diameters of gears 1 through 3 is 3.5/1.0/2.0. A torque of 20 N·m is applied to gear 3. If gear 2 has a radius of 10 cm, what is the torque on gears 1 and 2?

77. A chain drive (Fig. P8.77) transfers the rotational motion of a tractor transmission to a grain elevator. The drive shaft places a counterclockwise torque on the small sprocket of radius 2.2 cm so that it can overcome the resistance in the mechanism of the auger, which puts a clockwise torque of 200 N·m on the larger sprocket of radius 17 cm. If the gears turn at a constant rate, (a) what is the tension in the chain and (b) what is the torque applied to the small sprocket by the drive shaft?

78. Bicycles like the one shown in Figure P8.78 typically have 24 possible gear ratios (three gears on the front and eight on the back). Suppose the diameters of the largest and smallest sprockets are 17 cm and 9.0 cm on the front cluster and 11 cm and 5.0 cm on the rear cluster. The crank measures 18 cm from the center of the pedal shaft axis to the center of the front sprocket, and the back wheel (including the tire) has a diameter of 68 cm. Assume the cyclist is traveling at constant speed. If a downward force of 320 N is applied to the pedal when the crank is in the horizontal position, what is the horizontal force the rear wheel exerts on the road for (a) the lowest and (b) the highest gear combinations? Also find the torque on the rear axle in both cases. Assuming rolling without slipping, how far does the bike move forward for each full pedal stroke (each full 360° rotation) for (c) the lowest and (d) the highest gear ratios?

79. A bodybuilder is in training and performs curls with a 10-kg barbell to strengthen her biceps muscle. At the top of the stroke, her arm is configured as in Figure P8.79, with her forearm at 35° with respect to the horizontal. Approximate her forearm and hand as a uniform rod of mass 1.4 kg and length 32 cm (from pivot point at the elbow to center of barbell). The biceps muscle is attached 3.1 cm from the pivot point at the elbow joint and produces a force that is directed 85° away from the forearm. What force must the biceps muscle exert to keep the weight at its current position?

80. Runner’s gait. The amount of torque required by an external applied force (other than gravity) to move a leg forward is less than that required to move it backward if the leg is bent as shown in Figure P8.80. The lengths of the vectors r1, r2, and r3 indicate the position of the center of mass of the upper
leg, lower leg, and foot from the axis of rotation. The ratio of the mass of the upper leg/lower leg/foot is approximately 5/3/1. Find the ratio of the moment of inertia for the leg moving forward to that of the leg moving backward. Approximate the three parts of the leg as point masses.

81. A wheel of mass 11 kg is pulled up a step by a horizontal rope as depicted in Figure P8.81. If the height of the step is equal to one-third the radius of the wheel, h = 1/3 R, what minimum tension is needed on the rope to start the wheel moving up the step? Hint: The wheel will start to move just as the normal force on the very bottom of the wheel becomes zero.

82. The end of a pencil of mass m and length L rests in a corner as the pencil makes an angle θ with the horizontal (x) direction (Fig. P8.82). If the pencil is released, it will rotate about point O with an acceleration α1. Suppose the length of the pencil is increased by a factor of two but its mass is kept the
same; the pencil would then have an angular acceleration α2 when released. Find the ratio α12.

83. A uniform wooden plank of length L and mass m is supported by an axle that passes through its center. A small package of mass mp is placed at one end (Fig. P8.83). Find the angular acceleration of the system. Express your answer in terms of L, m, and mp.


Chapter 9


1. A baseball (mass 0.14 kg, radius 3.7 cm) is spinning with an angular velocity of 60 rad/s. What is its rotational kinetic energy?

2. If the baseball in Problem 1 has a linear velocity of 45 m/s, what is the ratio of its rotational and translational kinetic energies?

3. Consider a quarter rolling down an incline. What fraction of its kinetic energy is associated with its rotational motion? Assume it rolls without slipping.

4. Estimate the maximum rotational kinetic energy of a yo-yo. Use a rotation axis that passes through the yo-yo’s center.

5. A wheel of mass 0.50 kg and radius 45 cm is spinning with an angular velocity of 20 rad/s. You then push your hand against the edge of the wheel, exerting a force F on the wheel as shown in Figure P9.5. If the wheel comes to a stop after traveling 1/4 of a turn, what is F?

6. It has been proposed that large flywheels could be used to store energy. Consider a flywheel made of concrete (density 2300 kg/m3) in the shape of a solid disk, with a radius of 10 m and a thickness of 2.0 m. If its rotational kinetic energy is 100 MJ (1.0 × 108 J), what is the angular velocity of the flywheel?

7. Consider a wheel on a racing bicycle. If the wheel has a mass of 0.40 kg and an angular speed of 15 rad/s, what is the rotational kinetic energy of the wheel?

8. An automobile wheel has a mass of 18 kg and a diameter of 0.40 m. What is the total kinetic energy of one wheel when the car is traveling at 20 m/s?

9. Estimate the rotational kinetic energy of an airplane propeller.

10. Estimate the kinetic energy of the spinning top in Figure P9.10. Assume it is made of wood and is rotating with v = 30 rad/s.

11. Estimate the fraction of a bicycle’s total kinetic energy associated with the rotational motion of the wheels.

12. What is the rotational kinetic energy of the Earth as it spins about its axis?

13. What is the ratio of the rotational kinetic energy of the Earth to the rotational kinetic energy of the Moon as they spin about their axes?

14. Two crates of mass m1 = 15 kg and m2 = 9.0 kg are connected by a rope that passes over a frictionless pulley of mass mp = 8.0 kg and radius 0.20 m as shown in Figure P9.14. (a) What force(s) can do work on the masses and the pulley? Identify a system whose mechanical energy will be conserved. (b) The crates are released from rest. Crate 1 falls a distance of 2.0 m, at which time its speed is vf. Make a sketch showing the initial and final states of the system. (c) What are the total initial and final kinetic energies of the system? Express your answers in terms of vf  (d) What are the initial and final potential energies of the system? (e) What is the final speed vf of the crates? (f) What is the final angular velocity of the pulley?

15. A marble (radius 1.0 cm and mass 8.0 g) rolls without slipping down a ramp of vertical height 20 cm. What is the speed of the marble when it reaches the bottom of the ramp?

16. For the system of two crates and a pulley in Figure P9.14, what fraction of the total kinetic energy resides in the pulley?

17. Consider again the system in Figure P9.14. Use the result from Problem 9.14 to find the acceleration of one of the crates. Hint: Is the acceleration constant?

18. Consider a hoop of mass 3.0 kg and radius 0.50 m that rolls without slipping down an incline. The hoop starts at rest from a height h = 2.5 m above the bottom of the incline and then rolls to the bottom. (a) What forces act on the hoop? Which of these forces do work on the hoop? (b) Is the mechanical energy of the hoop conserved? (c) Make a sketch showing the hoop at the top and the bottom of the incline. (d) What is the total initial mechanical energy of the hoop? Take the zero of potential energy to be at the bottom of the incline. (e) If the hoop’s speed at the bottom of the incline is vf, what is its mechanical energy? Express your answer in terms of vf. (f) Solve for vf.

19. A marble rolls on the track shown in Figure P9.19, with hB = 25 cm and hC = 15 cm. If the marble has a speed of 2.0 m/s at point A, what is its speed at points B and C?

20. A yo-yo falls through a distance of 0.50 m. If it starts from rest, what is its speed?

21. A sphere rolls down the loop-the-loop track shown in Figure P9.21, starting from rest at a height h above the bottom. The ball travels around the inside of the circular portion of the track (radius r = 5.0 m). Notice that the ball is upside down when it reaches the top of the circular part of the track. It is found that the ball is just barely able to travel around the inside of the track without losing contact at the top. Find h.

 

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