Chapter 7 Energy
SOLUTIONS TO CHAPTER 7
RANKING
1. a. B, A, C
b. C, B, A
c. C, B, A
2. a. C, B = D, A
b. C, B = D, A
c. A, B = D, C
3. a. D, B, C, E, A
b. D, B, C, E, A
c. A, E, C, B, D
4. A, B = C (same as two supporting ropes)
EXERCISES
1. Stopping a lightly loaded truck of the same speed is easier because it has less KE and will therefore require less work to stop. (An answer in terms of impulse and momentum is also acceptable.)
2. You do no work because you haven’t exerted more than a negligible force on the backpack in the direction of motion. Also, the energy of the backpack hasn’t changed. No change in energy means no work done.
3. Your friend does twice as much work (4 ´ 1⁄2 > 1 ´ 1).
4. Although no work is done on the wall, work is nevertheless done on internal parts of your body (which generate heat).
5. More force is required to stretch the strong spring, so more work is done in stretching it the same distance as a weaker spring.
6. Work done by each is the same, for they reach the same height. The one who climbs in 30 s uses more power because work is done in a shorter time.
7. Solar energy is merely energy from the Sun. Solar power, like power in general, is the rate at which energy is transferred. Solar power is therefore the same from hour to hour, whereas the amount of solar energy depends on the amount of time energy is transferred.
8. The PE of the drawn bow as calculated would be an overestimate (in fact, about twice its actual value) because the force applied in drawing the bow begins at zero and increases to its maximum value when fully drawn. It’s easy to see that less force and therefore less work is required to draw the bow halfway than to draw it the second half of the way to its fully-drawn position. So the work done is not maximum force ´ distance drawn, but average force ´ distance drawn. In this case where force varies almost directly with distance (and not as the square or some other complicated factor) the average force is simply equal to the initial force + final force, divided by 2. So the PE is equal to the average force applied (which would be approximately half the force at its full-drawn position) multiplied by the distance through which the arrow is drawn.
9. When a rifle with a long barrel is fired, more work is done as the bullet is pushed through the longer distance. A greater KE is the result of the greater work, so of course, the bullet emerges with a greater velocity. (Note that the force acting on the bullet is not constant, but decreases with increasing distance inside the barrel.)
10. Agree, because speed itself is relative to the frame of reference (Chapter 3). Hence 1/2mv2 is also relative to a frame of reference.
11. The KE of the tossed ball relative to occupants in the airplane does not depend on the speed of the airplane. The KE of the ball relative to observers on the ground below, however, is a different matter. KE, like velocity, is relative.
12. You’re both correct, with respect to the frames of reference you’re inferring. KE is relative. From your frame of reference she has considerable KE for she has a great speed. But from her frame of reference her speed is zero and KE also zero.
13. The energy goes mostly into frictional heating of the air.
14. Without the use of a pole, the KE of running horizontally cannot easily be transformed to gravitational PE. But bending a pole stores elastic PE in the pole, which can be transformed to gravitational PE. Hence the greater heights reached by vaulters with very elastic poles.
15. The KE of a pendulum bob is maximum where it moves fastest, at the lowest point; PE is maximum at the uppermost points. When the pendulum bob swings by the point that marks half its maximum height, it has half its maximum KE, and its PE is halfway between its minimum and maximum values. If we define PE = 0 at the bottom of the swing, the place where KE is half its maximum value is also the place where PE is half its maximum value, and KE = PE at this point. (By energy conservation: Total energy = KE + PE.)
16. If the ball is given an initial KE, it will return to its starting position with that KE (moving in the other direction!) and hit the instructor. (The usual classroom procedure is to release the ball from the nose at rest. Then when it returns it will have no KE and will stop short of bumping the nose.)
17. Yes to both, relative to Earth, because work was done to lift it in Earth’s gravitational field and to impart speed to it.
18. In accord with the theorem, once moving, no work is done on the satellite (because the gravitational force has no component parallel to motion), so no change in energy occurs. Hence the satellite cruises at a constant speed.
19. According to the work-energy theorem, twice the speed corresponds to 4 times the energy, and therefore 4 times the driving distance. At 3 times the speed, driving distance is 9 times as much.
20. The answers to both (a) and (b) are the same: When the direction of the force is perpendicular to the direction of motion, as is the force of gravity on both the bowling ball on the alley and the satellite in circular orbit, there is no force component in the direction of motion and no work is done by the force.
21. On the hill there is a component of gravitational force in the direction of the car’s motion. This component of force does work on the car. But on the level, there is no component of gravitational force along the direction of the car’s motion, so the force of gravity does no work in this case.
22. The string tension is everywhere perpendicular to the bob’s direction of motion, which means there is no component of tension along the bob’s path, and therefore no work done by the tension. The force of gravity, on the other hand, has a component along the direction of motion everywhere except at the bottom of the swing, and does work, which changes the bob’s KE.
23. The fact that the crate pulls back on the rope in action-reaction fashion is irrelevant. The work done on the crate by the rope is the horizontal component of rope force that acts on the crate multiplied by the distance the crate is moved by that force—period. How much of this work produces KE or thermal energy depends on the amount of friction acting.
24. The 100 J of potential energy that doesn’t go into increasing her kinetic energy goes into thermal energy—heating her bottom and the slide.
25. A Superball will bounce higher than its original height if thrown downward, but if simply dropped, no way. Such would violate the conservation of energy.
26. When a Superball hits the floor some of its energy is transformed to heat. This means it will have less kinetic energy after the bounce than just before and will not reach its original level.
27. Kinetic energy is a maximum as soon as the ball leaves the hand. Potential energy is a maximum when the ball has reached its zenith.
28. The design is impractical. Note that the summit of each hill on the roller coaster is the same height, so the PE of the car at the top of each hill would be the same. If no energy were spent in overcoming friction, the car would get to the second summit with as much energy as it starts with. But in practice there is considerable friction, and the car would not roll to its initial height and have the same energy. So the maximum height of succeeding summits should be lower to compensate for friction.
29. You agree with your second classmate. The coaster could just as well encounter a low summit before or after a higher one, so long as the higher one is enough lower than the initial summit to compensate for energy dissipation by friction.
30. Both will have the same speed. This is easier to see here because both balls convert the same PE to KE. (Think energy when solving motion problems!)
31. Yes, a car burns more gasoline when its lights are on. The overall consumption of gasoline does not depend on whether or not the engine is running. Lights and other devices are run off the battery, which “runs down” the battery. The energy used to recharge the battery ultimately comes from the gasoline.
32. Except for the very center of the plane, the force of gravity acts at an angle to the plane, with a component of gravitational force along the plane—along the block’s path. Hence the block goes somewhat against gravity when moving away from the central position, and moves somewhat with gravity when coming back. As the object slides farther out on the plane, it is effectively traveling “upward” against Earth’s gravity, and slows down. It finally comes to rest and then slides back and the process repeats itself. The block slides back and forth along the plane. From a flat-Earth point of view the situation is equivalent to that shown in the sketch.
33. If KEs are the same but masses differ, then the ball with smaller mass has the greater speed. That is, 1⁄2Mv2 = 1⁄2mV2. Likewise with molecules, where lighter ones move faster on the average than more massive ones. (We will see in Chapter 15 that temperature is a measure of average molecular KE—lighter molecules in a gas move faster than same-temperature heavier molecules.)
34. A car with windows open experiences more air drag, which causes more fuel to be burned in maintaining motion. This may more than offset the savings from turning off the air conditioner.
35. A machine can multiply force or multiply distance, both of which can be of value.
36. Sufficient work occurs because with each pump of the jack handle, the force she exerts acts over a much greater distance than the car is raised. A small force acting over a long distance can do significant work.
37. Einstein’s E = mc2. (More on this in Chapters 34 and 35.)
38. Your friend may not realize that mass itself is congealed energy, so you tell your friend that much more energy in its congealed form is put into the reactor than is taken out from the reactor. About 1% of the mass that undergoes fission is converted to energy of other forms.
39. The work that the rock does on the ground is equal to its PE before being dropped, mgh = 100 joules. The force of impact, however, depends on the distance that the rock penetrates into the ground. If we do not know this distance we cannot calculate the force. (If we knew the time during which the impulse occurs we could calculate the force from the impulse-momentum relationship—but not knowing the distance or time of the rock’s penetration into the ground, we cannot calculate the force.)
40. When we speak of work done, we must understand work done on what, by what. Work is done on the car by applied forces that originate in the engine. The work done by the road in reacting to the backward push of the tires is equal to the product of the applied force and the distance moved, not the net force that involves air resistance and other friction forces. When doing work, we think of applied force; when considering acceleration, we think of net force. Actually, the frictional forces of the internal mechanisms in the car, and to some extent the road itself are doing negative work on the car. The zero total work explains why the car’s speed doesn’t change.
41. When air resistance is a factor, the ball will return with less speed (discussed in Exercise 58 in Chapter 4). It therefore will have less KE. You can see this directly from the fact that the ball loses mechanical energy to the air molecules it encounters, so when it returns to its starting point and to its original PE, it will have less KE. This does not contradict energy conservation, for energy is transformed, not destroyed.
42. The ball strikes the ground with the same speed, whether thrown upward or downward. The ball starts with the same energy at the same place, so they will have the same energy when they reach the ground. This means they will strike with the same speed. This is assuming negligible air resistance, for if air resistance is a factor, then the ball thrown upward will lose more energy to the air in its longer path and strike with somewhat less speed. Another way to look at this is to consider Figure 3.8 back on page 43; in the absence of air resistance, the ball thrown upward will return to its starting level with the same speed as the ball thrown downward. Both hit the ground at the same speed (but at different times).
43. The other 15 horsepower is supplied by electric energy from the batteries (which are ultimately recharged using energy from gasoline).
44. In a conventional car, braking converts KE to heat. In a hybrid car, braking charges up the batteries. In this way, braking energy can soon be transformed to KE.
45. The question can be restated; Is (302 – 202) greater or less than (202 – 102)? We see that (302 – 202) = (900 – 400) = 500, which is considerably greater than (202 – 102) = (400 – 100) = 300. So KE changes more for a given ∆v at the higher speed.
46. If an object has KE, then it must have momentum—for it is moving. But it can have potential energy without being in motion, and therefore without having momentum. And every object has “energy of being”—stated in the celebrated equation E = mc2. So whether an object moves or not, it has some form of energy. If it has KE, then with respect to the frame of reference in which its KE is measured, it also has momentum.