Work and Energy 6-1
Work and Energy / 6Responses to Questions
1.“Work” as used in everyday language generally means “energy expended,” which is similar to the way “work” is defined in physics. However, in everyday language, “work” can involve mental or physical energy expended and is not necessarily connected with displacement, as it is in physics. So a student could say she “worked” hard carrying boxes up the stairs to her dorm room (similar in meaning to the physics usage), or that she “worked” hard on a problem set (different in meaning from the physics usage).
2.No, not if the object is moving in a circle. Work is the product of force and the displacement in the direction of the force. Therefore, a centripetal force, which is perpendicular to the direction of motion, cannot do work on an object moving in a circle.
3.No work is done on the wall (since the wall does not undergo displacement), but internally your muscles are converting chemical energy to other forms of energy, which makes you tired.
4.Yes. The normal force is the force perpendicular to the surface an object is resting on. If the object moves with a component of its displacement perpendicular to this surface, the normal force will do work. For instance, when you jump, the normal force does work on you in accelerating you vertically. And it is the normal force of the elevator floor on you that accelerates you in an elevator.
5.(a)If the force is the same, then so The work done on spring 1 will be The work done on spring 2 will be Since work so more work is done on spring 2.
(b)If the displacement is the same, then and Since work and more work is done on spring 1.
6.The kinetic energy increases by a factor of 9, since the kinetic energy is proportional to the square of the speed.
7.Friction is not conservative; it dissipates energy in the form of heat, sound, and light. Air resistance is not conservative; it dissipates energy in the form of heat and the kinetic energy of fluids. “Human” forces, for example, the forces produced by your muscles, such as pushing a box across the floor, are also not conservative. They dissipate energy in the form of heat and also through chemical processes.
8.The speed at point C will be less than twice the speed at point B. The force is constant and the displacements are the same, so the same work is done on the block from A to B as from B to C. Since there is no friction, the same work results in the same change in kinetic energy. But kinetic energy depends on the square of the speed, so the speed at point C will be greater than the speed at point B by a factor of not a factor of 2.
9.The two forces on the book are the applied force upward (nonconservative) and the downward force of gravity (conservative). If air resistance is not negligible, it is nonconservative.
10.(a)The speed at the bottom of the hill does not depend on the angle of the hill if there is no friction. If there is no friction, then gravity is the only force doing work on the sled, and the system is conservative. All of the gravitational potential energy of the sled at the top of the hill will be converted into kinetic energy. The speed at the bottom of the hill depends on only the initial height h, not on the angle of the hill. and
(b)The speed at the bottom of the hill does depend on the angle of the hill if there is friction. If friction is present, then the net force doing work on the sled is not conservative. Only part of the gravitational potential energy of the sled at the top of the hill will be converted into kinetic energy; the rest will be dissipated by the frictional force. The frictional force is proportional to the normal force on the sled, which will depend on the angle θ of the hill. so which does depend on the angle of the hill and will be smaller for smaller angles.
11.At the top of the pendulum’s swing, all of its energy is gravitational potential energy; at the bottom of the swing, all of the energy is kinetic.
(a)If we can ignore friction, then energy is transformed back and forth between potential energy and kinetic energy as the pendulum swings.
(b)If friction is present, then during each swing energy is lost to friction at the pivot point and also to air resistance. During each swing, the kinetic energy and the potential energy decrease, and the pendulum’s amplitude decreases. When a grandfather clock is “wound up,” the amount of energy that will eventually be lost to friction and air resistance is stored as potential energy (either elastic or gravitational, depending on the clock mechanism), and part of the workings of the clock is to put that stored energy back into the pendulum at the same rate that friction is dissipating the energy.
12.For each of the balloons, the initial energy (kinetic plus potential) equals the final energy (all kinetic). Since the initial energy depends on only the speed and not on the direction of the initial velocity, and all balloons have the same initial speed and height, the final speeds are all the same.
13.The initial potential energy of the water is converted first into the kinetic energy of the water as it falls. When the falling water hits the pool, it does work on the water already in the pool, creating splashes and waves. Additionally, some energy is converted into heat and sound.
14.Stepping on top of a log and jumping down the other side requires you to raise your center of mass farther than just stepping over a log does. Raising your center of mass farther requires you to do more work and thereby use more energy.
15.(a)The golfer raises the club, giving the club potential energy. In swinging, the golfer does work on the club. This work and the change in potential energy increase the kinetic energy of the club. At the lowest point in the swing (when the club’s kinetic energy is a maximum), the club hits the ball, converting some of the kinetic energy of the club into kinetic energy of the ball.
(b)The tennis player throws the ball upward, giving the ball some initial kinetic energy. As the ball rises to its highest point, the kinetic energy is converted to potential energy. The player then does work on the racket to increase the racket’s kinetic energy. When the racket collides with the ball, the racket does work on the ball. The racket loses kinetic energy, and the ball gains kinetic energy, accelerating the ball forward.
(c)The player pushes the ball upward, doing work on the ball, which gives the ball an initial kinetic energy. As the ball rises, it slows down as its kinetic energy is converted to potential energy. At the highest point the potential energy is a maximum and the kinetic energy is a minimum. As the ball descends the kinetic energy increases.
16.The drawing shows water falling over a waterfall and then flowing back to the top of the waterfall. The top of the waterfall is above the bottom, with greater gravitational potential energy. The optical illusion of the diagram implies that water is flowing freely from the bottom of the waterfall back to the top. Since water won’t move uphill unless work is done on it to increase its gravitational potential energy (for example, work done by a pump), the water from the bottom of the waterfall would NOT be able to make it back to the top.
17.The faster arrow has the same mass and twice the speed of the slower arrow, so the faster arrow will have four times the kinetic energy Therefore, four times as much work must be done on the faster arrow to bring it to rest. If the force on the arrows is constant, the faster arrow will travel four times the distance of the slower arrow into the hay.
18.When the ball is released, its potential energy will be converted into kinetic energy and then back into potential energy as the ball swings. If the ball is not pushed, it will lose a little energy to friction and air resistance. It will return almost to the initial position but will not hit the instructor. If the ball is pushed, it will have an initial kinetic energy, and will, when it returns, still have some kinetic energy when it reaches the initial position, so it will hit the instructor on the chin. (Ouch!)
19.When a child hops around on a pogo stick, gravitational potential energy (at the top of the hop) is transformed into kinetic energy as the child moves downward, and then stored as spring potential energy as the spring in the pogo stick compresses. As the spring begins to expand, the energy is converted back to kinetic and gravitational potential energy, and the cycle repeats. Since energy is lost due to friction, the child must add energy to the system by pushing down on the pogo stick while it is on the ground to get more spring compression.
20.At the top of the hill, the skier has gravitational potential energy. If the friction between her skis and the snow is negligible, the gravitational potential energy is changed into kinetic energy as she glides down the hill, and she gains speed as she loses elevation. When she runs into the snowdrift, work is done by the contact force between her and the snow. The energy changes from kinetic energy of the skier to kinetic energy of the snow as it moves and to thermal energy from the friction between the skier and the snow.
21.The work done on the suitcase depends on only (c) the height of the table and (d) the weight of the suitcase.
22.Power is the rate of doing work. Both (c) and (d) will affect the total amount of work needed, and hence the power. The time the lifting takes, (b), will also affect the power. The length of the path (a) will affect only the power if different paths take different times to traverse.
23.When you climb a mountain by going straight up, the force needed is large (and the distance traveled is small), and the power needed (work per unit time) is also large. If you take a zigzag trail, you will use a smaller force (over a longer distance, so that the work done is the same) and less power, since the time to climb the mountain will be longer. A smaller force and smaller power output make the climb seem easier.
Responses to MisConceptual Questions
1.(b)A common misconception is that all forces do work. However, work requires that the object on which the force is acting has a component of motion in the direction of the force.
2.(c)Work is done when the force acting on the object has a component in the direction of motion. Gravity, which provides the centripetal force, is not zero but is always perpendicular to the motion. A common error is the notion that an object moving in a circle has no work done on it. This is true only if the object is moving at constant speed.
3.(c)The kinetic energy is proportional to the square of the speed. Therefore, doubling the speed quadruples the kinetic energy.
4.(d)A common misconception is that the stopping distance is proportional to the speed. However, for a constant stopping force, the stopping distance is proportional to the initial kinetic energy, which is proportional to the square of the speed. Doubling the initial speed will quadruple the initial kinetic energy and therefore quadruple the stopping distance.
5.(c)As the ball falls, gravitational potential energy is converted to kinetic energy. As the ball hits the trampoline, kinetic energy is converted to elastic potential energy. This energy is then transferred back to kinetic energy of the ball and finally gravitational potential energy. No energy is added to the ball during the motion, so it can’t rise higher than it started. Some energy may be lost to heat during the motion, so the ball may not rise as high as it initially started.
6.(e)The term “energy” is commonly misunderstood. In this problem energy refers to the sum of the kinetic and potential energies. Initially the kinetic energy is a maximum. During the flight kinetic energy is converted to potential energy, with the potential energy a maximum at the highest point. As the ball falls back down the potential energy is converted back into kinetic energy. Since no nonconservative forces act on the ball (there is no air resistance), the total energy remains constant throughout the flight.
7.(b)Since the changes in speed are equal, many students think that the change in energy will also be equal. However, the energy is proportional to the square of the speed. It takes four times as much energy to accelerate the car from rest to 60 km/h as it takes to accelerate the car from rest to 30 km/h. Therefore, it takes three times the energy to accelerate the car from 30 km/h to 60 km/h as it takes to accelerate it from rest to 30 km/h.
8.(d)Horsepower is not a unit of energy nor of force but a measure of the rate at which work is done.
9.(b)The two balls have the same initial kinetic energy and the same initial potential energy. When they hit the ground they will have the same final potential energy, so their final kinetic energies, and therefore speeds, will be the same. Even though they have the same initial and final speeds, it is a misconception to think they will spend the same time in the air. The ball thrown directly upward travels to a higher point, as all of the kinetic energy can be converted into potential energy, and therefore will spend a longer time in the air.
10.(e)A common misconception is that the steeper the slope, the faster the skier will be traveling at the bottom. Without friction, all of the skier’s initial gravitational potential energy is converted into kinetic energy. The skier starting from a higher initial position will have the greater speed at the bottom. On the steeper slope, the skier accelerates faster but over a shorter time period. On the flatter slope, the skier accelerates slower but over more time.
11.(c)Friction is a nonconservative force, which removes energy from the system. The work done by friction is related to the product of the force of friction and the distance traveled. For a given coefficient of friction, the force of friction on the steeper slope is smaller than on the flatter slope, as it has a smaller normal force. Also, on the steeper slope, the skier travels a shorter distance.
12.(c)The kinetic energy depends on the speed and not the position of the block. Since the block moves with constant speed, the kinetic energy remains constant. As the block moves up the incline its elevation increases, so its potential energy also increases.
13.(a)The speed of the crate is constant, so the net (total) work done on the crate is zero. The normal force is perpendicular to the direction of motion, so it does no work. Your applied force and a component of gravity are in the direction of motion, so both do positive work. The force of friction opposes the direction of motion and does negative work. For the total work to be zero, the work done by friction must equal the sum of the work done by gravity and by you.
14.(a)The kinetic energy is proportional to the square of the ball’s speed, and the potential energy is proportional to the height of the ball. As the ball rises, the speed and kinetic energy decrease while the potential energy increases. As the ball falls, the speed and kinetic energy increase while the potential energy decreases.
Solutions to Problems
1.The minimum force required to lift the firefighter is equal to his weight. The force and the displacement are both upward, so the angle between them is Use Eq. 6–1.
2.The maximum amount of work would be the work done by gravity. Both the force and the displacement are downward, so the angle between them is Use Eq. 6–1.
This is a small amount of energy. If the person adds a larger force to the hammer during the fall, then the hammer will have a larger amount of energy to give to the nail.
3.Draw a free-body diagram for the crate as it is being pushed across the floor. Since it is not accelerating vertically, Since it is not accelerating horizontally, The work done to move it across the floor is the work done by the pushing force. The angle between the pushing force and the direction of motion is