Problems

Note to Instructors: Most of the homework problems in this chapter are available for assignment via an on-line homework management program such as Wiley PLUS or WebAssign, and those marked with the icon are presented in a guided tutorial format that provides enhanced interactivity. See Preface for additional details.
ssm Solution is in the Student Solutions Manual.
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Section 6.1 Work Done by a Constant Force

| 1 | ssm The brakes of a truck cause it to slow down by applying a retarding force of to the truck over a distance of 850 m. What is the work done by this force on the truck? Is the work positive or negative? Why?

| 2 | A cable lifts a 1200-kg elevator at a constant velocity for a distance of 35 m. What is the work done by (a) the tension in the cable and (b) the elevator’s weight?

| 3 | A 75.0-kg man is riding an escalator in a shopping mall. The escalator moves the man at a constant velocity from ground level to the floor above, a vertical height of 4.60 m. What is the work done on the man by (a) the gravitational force and (b) the escalator?

| 4 | A person pulls a toboggan for a distance of 35.0 m along the snow with a rope directed 25.0° above the snow. The tension in the rope is 94.0 N. (a) How much work is done on the toboggan by the tension force? (b) How much work is done if the same tension is directed parallel to the snow?

| 5 | ssm During a tug-of-war, team A pulls on team B by applying a force of 1100 N to the rope between them. How much work does team A do if they pull team B toward them a distance of 2.0 m?

| 6 | Consult Interactive LearningWare 6.1 at for background pertinent to this problem. The drawing shows a plane diving toward the ground and then climbing back upward. During each of these motions, the lift force acts perpendicular to the displacement , which has the same magnitude, , in each case. The engines of the plane exert a thrust , which points in the direction of the displacement and has the same magnitude during the dive and the climb. The weight of the plane has a magnitude of . In both motions, net work is performed due to the combined action of the forces , , and . (a) Is more net work done during the dive or the climb? Explain. (b) Find the difference between the net work done during the dive and the climb.

| 7 | A person pushes a 16.0-kg shopping cart at a constant velocity for a distance of 22.0 m. She pushes in a direction 29.0° below the horizontal. A 48.0-N frictional force opposes the motion of the cart. (a) What is the magnitude of the force that the shopper exerts? Determine the work done by (b) the pushing force, (c) the frictional force, and (d) the gravitational force.

* | 8 | A crate is being pushed across a horizontal floor by a force that makes an angle of 30.0° below the horizontal. The coefficient of kinetic friction is 0.200. What should be the magnitude of , so that the net work done by it and the kinetic frictional force is zero?

* | 9 | ssm A husband and wife take turns pulling their child in a wagon along a horizontal sidewalk. Each exerts a constant force and pulls the wagon through the same displacement. They do the same amount of work, but the husband’s pulling force is directed 58° above the horizontal, and the wife’s pulling force is directed 38° above the horizontal. The husband pulls with a force whose magnitude is 67 N. What is the magnitude of the pulling force exerted by his wife?

* | 10 | A 55-kg box is being pushed a distance of 7.0 m across the floor by a force whose magnitude is 150 N. The force is parallel to the displacement of the box. The coefficient of kinetic friction is 0.25. Determine the work done on the box by each of the four forces that act on the box. Be sure to include the proper plus or minus sign for the work done by each force.

** | 11 | A 1200-kg car is being driven up a 5.0° hill. The frictional force is directed opposite to the motion of the car and has a magnitude of . A force is applied to the car by the road and propels the car forward. In addition to these two forces, two other forces act on the car: its weight and the normal force directed perpendicular to the road surface. The length of the road up the hill is 290 m. What should be the magnitude of , so that the net work done by all the forces acting on the car is +150 kJ?
Section 6.2 The Work–Energy Theorem and Kinetic Energy

| 12 | Refer to Concept Simulation 6.1 at for a review of the concepts with which this problem deals. A 0.075-kg arrow is fired horizontally. The bowstring exerts an average force of 65 N on the arrow over a distance of 0.90 m. With what speed does the arrow leave the bow?

| 13 | ssm www Two cars, A and B, are traveling with the same speed of 40.0 m/s, each having started from rest. Car A has a mass of , and car B has a mass of . Compared to the work required to bring car A up to speed, how much additional work is required to bring car B up to speed?

| 14 | A fighter jet is launched from an aircraft carrier with the aid of its own engines and a steam-powered catapult. The thrust of its engines is . In being launched from rest it moves through a distance of 87 m and has a kinetic energy of at lift-off. What is the work done on the jet by the catapult?

| 15 | When a 0.045-kg golf ball takes off after being hit, its speed is 41 m/s. (a) How much work is done on the ball by the club? (b) Assume that the force of the golf club acts parallel to the motion of the ball and that the club is in contact with the ball for a distance of 0.010 m. Ignore the weight of the ball and determine the average force applied to the ball by the club.

| 16 | As background for this problem, review Conceptual Example 6. A 7420-kg satellite has an elliptical orbit, as in Figure 6-9b. The point on the orbit that is farthest from the earth is called the apogee and is at the far right side of the drawing. The point on the orbit that is closest to the earth is called the perigee and is at the left side of the drawing. Suppose that the speed of the satellite is 2820 m/s at the apogee and 8450 m/s at the perigee. Find the work done by the gravitational force when the satellite moves from (a) the apogee to the perigee and (b) the perigee to the apogee.

| 17 | ssm The hammer throw is a track-and-field event in which a 7.3-kg ball (the “hammer”), starting from rest, is whirled around in a circle several times and released. It then moves upward on the familiar curving path of projectile motion. In one throw, the hammer is given a speed of 29 m/s. For comparison, a .22 caliber bullet has a mass of 2.6 g and, starting from rest, exits the barrel of a gun with a speed of 410 m/s. Determine the work done to launch the motion of (a) the hammer and (b) the bullet.

| 18 | Multiple-Concept Example 4 and Interactive LearningWare 6.2 at review the concepts that are important in this problem. A space probe is traveling at a speed of 11 000 m/s through deep space. Retrorockets are fired along the line of motion to reduce the probe’s speed. The retrorockets generate a force of over a distance of 2500 km. What is the final speed of the probe?

* | 19 | ssm A sled is being pulled across a horizontal patch of snow. Friction is negligible. The pulling force points in the same direction as the sled’s displacement, which is along the axis. As a result, the kinetic energy of the sled increases by 38%. By what percentage would the sled’s kinetic energy have increased if this force had pointed 62° above the axis?

* | 20 | The concepts in this problem are similar to those in Multiple-Concept Example 4, except that the force doing the work in this problem is the tension in the cable. A rescue helicopter lifts a 79-kg person straight up by means of a cable. The person has an upward acceleration of 0.70 m/s2 and is lifted from rest through a distance of 11 m. (a) What is the tension in the cable? How much work is done by (b) the tension in the cable and (c) the person’s weight? (d) Use the work–energy theorem and find the final speed of the person.

* | 21 | ssm www A 6200-kg satellite is in a circular earth orbit that has a radius of . A net external force must act on the satellite to make it change to a circular orbit that has a radius of . What work must the net external force do?

* | 22 | Consult Multiple-Concept Example 5 for insight into solving this problem. A skier slides horizontally along the snow for a distance of 21 m before coming to rest. The coefficient of kinetic friction between the skier and the snow is . Initially, how fast was the skier going?

** | 23 | The model airplane in Figure 5.7 is flying at a speed of 22 m/s on a horizontal circle of radius 16 m. The mass of the plane is 0.90 kg. The person holding the guideline pulls it in until the radius of the circle becomes 14 m. The plane speeds up, and the tension in the guideline becomes four times greater. What is the net work done on the plane?

** | 24 | Multiple-Concept Example 5 reviews many of the concepts that play a role in this problem. An extreme skier, starting from rest, coasts down a mountain slope that makes an angle of 25.0° with the horizontal. The coefficient of kinetic friction between her skis and the snow is 0.200. She coasts down a distance of 10.4 m before coming to the edge of a cliff. Without slowing down, she skis off the cliff and lands downhill at a point whose vertical distance is 3.50 m below the edge. How fast is she going just before she lands?
Section 6.3 Gravitational Potential Energy,
Section 6.4 Conservative versus Nonconservative Forces

| 25 | Relative to the ground, what is the gravitational potential energy of a 55.0-kg person who is at the top of the Sears Tower, a height of 443 m above the ground?

| 26 | A shot-putter puts a shot that leaves his hand at a distance of 1.52 m above the ground. (a) Find the work done by the gravitational force when the shot has risen to a height of 2.13 m above the ground. Include the correct sign for the work. (b) Determine the change in the gravitational potential energy of the shot.

| 27 | ssm A bicyclist rides 5.0 km due east, while the resistive force from the air has a magnitude of 3.0 N and points due west. The rider then turns around and rides 5.0 km due west, back to her starting point. The resistive force from the air on the return trip has a magnitude of 3.0 N and points due east. (a) Find the work done by the resistive force during the round trip. (b) Based on your answer to part (a), is the resistive force a conservative force? Explain.

| 28 | A 0.60-kg basketball is dropped out of a window that is 6.1 m above the ground. The ball is caught by a person whose hands are 1.5 m above the ground. (a) How much work is done on the ball by its weight? What is the gravitational potential energy of the basketball, relative to the ground, when it is (b) released and (c) caught? (d) How is the change in the ball’s gravitational potential energy related to the work done by its weight?

| 29 | A 75.0-kg skier rides a 2830-m-long lift to the top of a mountain. The lift makes an angle of 14.6° with the horizontal. What is the change in the skier’s gravitational potential energy?

| 30 | A 55.0-kg skateboarder starts out with a speed of 1.80 m/s. He does +80.0 J of work on himself by pushing with his feet against the ground. In addition, friction does −265 J of work on him. In both cases, the forces doing the work are nonconservative. The final speed of the skateboarder is 6.00 m/s. (a) Calculate the change in the gravitational potential energy. (b) How much has the vertical height of the skater changed, and is the skater above or below the starting point?

| 31 | ssm “Rocket man” has a propulsion unit strapped to his back. He starts from rest on the ground, fires the unit, and is propelled straight upward. At a height of 16 m, his speed is 5.0 m/s. His mass, including the propulsion unit, has the approximately constant value of 136 kg. Find the work done by the force generated by the propulsion unit.
Section 6.5 The Conservation of Mechanical Energy

| 32 | Consult Interactive LearningWare 6.3 at for a review of the concepts on which this problem is based. A gymnast is swinging on a high bar. The distance between his waist and the bar is 1.1 m, as the drawing shows. At the top of the swing his speed is momentarily zero. Ignoring friction and treating the gymnast as if all of his mass is located at his waist, find his speed at the bottom of the swing.

| 33 | Interactive Solution 6.33 at presents a model for solving this problem. A slingshot fires a pebble from the top of a building at a speed of 14.0 m/s. The building is 31.0 m tall. Ignoring air resistance, find the speed with which the pebble strikes the ground when the pebble is fired (a) horizontally, (b) vertically straight up, and (c) vertically straight down.

| 34 | The skateboarder in the drawing starts down the left side of the ramp with an initial speed of 5.4 m/s. If nonconservative forces, such as kinetic friction and air resistance, are negligible, what would be the height h of the highest point reached by the skateboarder on the right side of the ramp?

| 35 | ssm A 2.00-kg rock is released from rest at a height of 20.0 m. Ignore air resistance and determine the kinetic energy, gravitational potential energy, and total mechanical energy at each of the following heights: 20.0, 10.0, and 0 m.

| 36 | Consult Conceptual Example 9 in preparation for this problem. Interactive LearningWare 6.3 at also provides useful background. The drawing shows a person who, starting from rest at the top of a cliff, swings down at the end of a rope, releases it, and falls into the water below. There are two paths by which the person can enter the water. Suppose he enters the water at a speed of 13.0 m/s via path 1. How fast is he moving on path 2 when he releases the rope at a height of 5.20 m above the water? Ignore the effects of air resistance.

| 37 | A 47.0-g golf ball is driven from the tee with an initial speed of 52.0 m/s and rises to a height of 24.6 m. (a) Neglect air resistance and determine the kinetic energy of the ball at its highest point. (b)What is its speed when it is 8.0 m below its highest point?

* | 38 | The drawing shows a skateboarder moving at 5.4 m/s along a horizontal section of a track that is slanted upward by 48° above the horizontal at its end, which is 0.40 m above the ground. When she leaves the track, she follows the characteristic path of projectile motion. Ignoring friction and air resistance, find the maximum height H to which she rises above the end of the track.

* | 39 | Review Interactive Solution 6.39 at for background on this problem. A wrecking ball swings at the end of a 12.0-m cable on a vertical circular arc. The crane operator manages to give the ball a speed of 5.00 m/s as the ball passes through the lowest point of its swing and then gives the ball no further assistance. Friction and air resistance are negligible. What speed vf does the ball have when the cable makes an angle of 20.0° with respect to the vertical?

* | 40 | A particle, starting from point A in the drawing, is projected down the curved runway. Upon leaving the runway at point B, the particle is traveling straight upward and reaches a height of 4.00 m above the floor before falling back down. Ignoring friction and air resistance, find the speed of the particle at point A.

* | 41 | ssm www Two pole-vaulters just clear the bar at the same height. The first lands at a speed of 8.90 m/s, and the second lands at a speed of 9.00 m/s. The first vaulter clears the bar at a speed of 1.00 m/s. Ignore air resistance and friction and determine the speed at which the second vaulter clears the bar.

* | 42 | A skier starts from rest at the top of a hill. The skier coasts down the hill and up a second hill, as the drawing illustrates. The crest of the second hill is circular, with a radius of . Neglect friction and air resistance. What must be the height h of the first hill so that the skier just loses contact with the snow at the crest of the second hill?

** | 43 | Conceptual Example 9 provides background for this problem. A swing is made from a rope that will tolerate a maximum tension of without breaking. Initially, the swing hangs vertically. The swing is then pulled back at an angle of 60.0° with respect to the vertical and released from rest. What is the mass of the heaviest person who can ride the swing?

** | 44 | A person starts from rest at the top of a large frictionless spherical surface, and slides into the water below (see the drawing). At what angle does the person leave the surface? (Hint: When the person leaves the surface, the normal force is zero.)
Section 6.6 Nonconservative Forces and the Work–Energy Theorem

| 45 | A hot-air balloon takes off from rest at the surface of the earth. The nonconservative wind and lift forces take the balloon up, doing of work on the balloon in the process. At what height above the surface of the earth does the balloon have a speed of 8.00 m/s?

| 46 | The surfer in the photo is catching a wave. Suppose she starts at the top of the wave with a speed of 1.4 m/s and moves down the wave until her speed increases to 9.5 m/s. The drop in her vertical height is 2.7 m. If her mass is 59 kg, how much work is done by the (nonconservative) force of the wave?

(Pierre Tostee/Allsport Australia/Allsport/Getty Images Sport Service)

| 47 | ssm A roller coaster (375 kg) moves from A (5.00 m above the ground) to B (20.0 m above the ground). Two nonconservative forces are present: friction does of work on the car, and a chain mechanism does of work to help the car up a long climb. What is the change in the car’s kinetic energy, , from A to B?

| 48 | One of the new events in the 2002 Winter Olympics was the sport of skeleton (see the photo). Starting at the top of a steep, icy track, a rider jumps onto a sled (known as a skeleton) and proceeds—belly down and head first—to slide down the track. The track has fifteen turns and drops 104 m in elevation from top to bottom. (a) In the absence of nonconservative forces, such as friction and air resistance, what would be the speed of a rider at the bottom of the track? Assume that the speed of the rider at the beginning of the run is relatively small and can be ignored. (b) In reality, the best riders reach the bottom with a speed of 35.8 m/s (about 80 mi/h). How much work is done on an 86.0-kg rider and skeleton by nonconservative forces?

(Jacques Demarthon/AFP/Getty Images)

| 49 | ssm The (nonconservative) force propelling a car up a mountain road does of work on the car. The car starts from rest at sea level and has a speed of 27.0 m/s at an altitude of above sea level. Obtain the work done on the car by the combined forces of friction and air resistance, both of which are nonconservative forces.

* | 50 | In attempting to pass the puck to a teammate, a hockey player gives it an initial speed of 1.7 m/s. However, this speed is inadequate to compensate for the kinetic friction between the puck and the ice. As a result, the puck travels only one-half the distance between the players before sliding to a halt. What minimum initial speed should the puck have been given so that it reached the teammate, assuming that the same force of kinetic friction acted on the puck everywhere between the two players?

* | 51 | Refer to Interactive Solution 6.51 for a review of the approach taken in problems such as this one. A 67.0-kg person jumps from rest off a 3.00-m-high tower straight down into the water. Neglect air resistance during the descent. She comes to rest 1.10 m under the surface of the water. Determine the magnitude of the average force that the water exerts on the diver. This force is nonconservative.

* | 52 | A pitcher throws a 0.140-kg baseball, and it approaches the bat at a speed of 40.0 m/s. The bat does of work on the ball in hitting it. Ignoring air resistance, determine the speed of the ball after the ball leaves the bat and is 25.0 m above the point of impact.

* | 53 | ssm At a carnival, you can try to ring a bell by striking a target with a 9.00-kg hammer. In response, a 0.400-kg metal piece is sent upward toward the bell, which is 5.00 m above. Suppose that 25.0% of the hammer’s kinetic energy is used to do the work of sending the metal piece upward. How fast must the hammer be moving when it strikes the target so that the bell just barely rings?

** | 54 | A 3.00-kg model rocket is launched straight up. It reaches a maximum height of above where its engine cuts out, even though air resistance performs of work on the rocket. What would have been this height if there were no air resistance? Section 6.7 Power

| 55 | ssm One kilowatt·hour (kWh) is the amount of work or energy generated when one kilowatt of power is supplied for a time of one hour. A kilowatt·hour is the unit of energy used by power companies when figuring your electric bill. Determine the number of joules of energy in one kilowatt·hour.

| 56 | Bicyclists in the Tour de France do enormous amounts of work during a race. For example, the average power per kilogram generated by Lance Armstrong is 6.50 W per kilogram of his body mass. (a) How much work does he do during a 135-km race in which his average speed is 12.0 m/s? (b) Often, the work done is expressed in nutritional Calories rather than in joules. Express the work done in part (a) in terms of nutritional Calories, noting that nutritional Calories.

| 57 | Interactive Solution 6.57 at offers a model for solving this problem. A car accelerates uniformly from rest to 20.0 m/s in 5.6 s along a level stretch of road. Ignoring friction, determine the average power required to accelerate the car if (a) the weight of the car is , and (b) the weight of the car is .

| 58 | You are trying to lose weight by working out on a rowing machine. Each time you pull the rowing bar (which simulates the “oars”) toward you, it moves a distance of 1.2 m in a time of 1.5 s. The readout on the display indicates that the average power you are producing is 82 W. What is the magnitude of the force that you exert on the handle?

* | 59 | Multiple-Concept Example 13 presents useful background for this problem. The cheetah is one of the fastest-accelerating animals, because it can go from rest to 27 m/s (about 60 mi/h) in 4.0 s. If its mass is 110 kg, determine the average power developed by the cheetah during the acceleration phase of its motion. Express your answer in (a) watts and (b) horsepower.

* | 60 | A motorcycle (mass of cycle plus ) is traveling at a steady speed of 20.0 m/s. The force of air resistance acting on the cycle and rider . Find the power necessary to sustain this speed if (a) the road is level and (b) the road is sloped upward at 37.0° with respect to the horizontal.

** | 61 | ssm The motor of a ski boat generates an average power of when the boat is moving at a constant speed of 12 m/s. When the boat is pulling a skier at the same speed, the engine must generate an average power of 8.30 × 104 W. What is the tension in the tow rope that is pulling the skier?

** | 62 | A 1900-kg car experiences a combined force of air resistance and friction that has the same magnitude whether the car goes up or down a hill at 27 m/s. Going up a hill, the car’s engine needs to produce 47 hp more power to sustain the constant velocity than it does going down the same hill. At what angle is the hill inclined above the horizontal?
Section 6.9 Work Done by a Variable Force

| 63 | ssm The drawing shows the force-versus-displacement graph for two different bows. These graphs give the force that an archer must apply to draw the bowstring. (a) For which bow is more work required to draw the bow fully from to ? Give your reasoning. (b) Estimate the additional work required for the bow identified in part (a) compared to the other bow.

| 64 | Review Example 14, in which the work done in drawing the bowstring in Figure 6-22 from to is determined. In part b of the figure, the force component reaches a maximum at . Find the percentage of the total work that is done when the bowstring is moved (a) from to 0.306 m and (b) from to 0.500 m.

| 65 | The graph shows the net external force component along the displacement as a function of the magnitude s of the displacement. The graph applies to a 65-kg ice skater. How much work does the net force component do on the skater from (a) 0 to 3.0 m and (b) 3.0 m to 6.0 m? (c) If the initial speed of the skater is 1.5 m/s when , what is the speed when ?

| 66 | The graph shows how the force component along the displacement varies with the magnitude s of the displacement. Find the work done by the force. (Hint: Recall how the area of a triangle is related to the triangle’s base and height.)

| 67 | A net external force is applied to a 6.00-kg object that is initially at rest. The net force component along the displacement of the object varies with the magnitude of the displacement as shown in the drawing. (a) How much work is done by the net force? (b) What is the speed of the object at ?

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