# Physics. Chapter 7: Potential Energy, Gravity Rebecca Mallory

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Work
The product of the force and the distance moved by the force: (More generally, work is the component of force in the direction of motion times the distance moved.)W=Fd
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Power
The time rate of work: (More generally, power is the rate at which energy is expended)Power = work done/time interval
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Energy
The property of a system that enables it to do work.
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Mechanical Energy
Energy due to the position of something or the movement of something.
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Potential Energy
Energy that something possesses because of its position. (Stored Energy)PE = mgh
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Kinetic Energy
Energy that something possesses because of its motion, quantified by the relationship: (Energy in Motion) Kinetic Energy = ½ mv²
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Work-Energy Theorem
The work done on an object equals the change in the kinetic energy of the object. (Work can also transfer other form of energy to a system) Work = ∆KE
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Law of Conservation of Energy
Energy cannot be created or destroyed: it may be transformed from one form into another, but the total amount of energy never changes.
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Machine
A device, such as a lever or pulley, that increases (or decreases) a force or simply changes the direction of a force.
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Conservation of Energy
The work output of any machine cannot exceed the work input. In an ideal machine, where no energy is transformed into thermal energy, work (input) = work (output); (Fd) input = (Fd) output
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Lever
A simple machine consisting of a rigid rod pivoted at a fixed point called the fulcrum.
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Efficiency
The percentage of the work put into a machine that is converted into useful work output. (More generally, useful energy output divided by total energy input)
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1. When is energy most evident?
1. Energy is most evident when it is changing.
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2. A force sets an object in motion. When the force is multiplied by the time of its application, we call the quantity impulse, and an Impulse changes the momentum of that object. What do we call the quantity force multiplied by distance?
2. Force multiplied by distance is work.
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3. Cite an example in which a force is exerted on an object without doing work on the object.
3. No work is done in pushing on a stationary wall, as in Figure 7.4.
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4. Which requires more work: a) lifting a 50kg sack a vertical distance of 2m or b) lifting a 25kg sack a vertical distance of 4m?
4. It is the same, for the product of each is the same; (50 kg)(2 m) = (25 kg)(4 m).
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5. Exactly what is it that enables an object to do work?
5. Energy enables an object to do work.
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6. If both sacks are lifted their respective distances in the same time, how does the power required for each compare? How about for the case in which the lighter sack is moved its distance in half the time? a) lifting a 50kg sack a vertical distance of 2m or b) lifting a 25kg sack a vertical distance of 4m?
6. The same power when both are raised in the same time; Twice the power for the lighter sack raised in half the time.
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7. A car is raised a certain distance in a service station lift and therefore has potential energy relative to the floor. If it were raised twice as high, how much more potential energy would it have?
7. It would have twice because distance raised is twice.
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8. Two cars are raised to the same elevation. If one car is twice as massive as the other, compare their gains of potential energy.
8. Twice-as-massive car has twice the PE.
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9. When is the potential energy of something significant?
9. PE is significant when it changes, does work or transforms to energy of another form.
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10. When the speed of a moving car is doubled, how much more kinetic energy does it have?
10. Four times as much (as 22 = 4).
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11. Compared with a car moving at some original speed, how much work must the brakes of a car supply to stop a car that is moving twice as fast? How will the stopping distance compare?
11. Four times as much work; 4 times as much stopping distance (as 22 = 4).
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12. If you push a crate horizontally with 100N across a 10m floor and the friction between the crate and floor is a steady 70N, how much kinetic energy does the crate gain?
12. ΔKE = work done = (100 N - 70 N)(10 m) = (30 N)(10 m) = 300 Nm = 300 J.
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13. How does speed affect the friction between a road and skidding tire?
13. Speed has little or no effect on friction.
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14. What will be the KE of a pile driver ram that starts from rest and undergoes a 10kj decrease in PE?
14. Its gain in KE will equal its decrease in PE, 10 kJ.
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15. An apple hanging from a limb has PE because of its height. If it falls, what becomes of this energy just before it hits the ground? When it hits the ground?
15. Immediately before hitting the ground its initial PE becomes KE. When it hits the ground its energy becomes thermal energy.
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16. What is the source of energy in sunshine?
16. The source of the energy of sunshine is fusion power in the Sun.
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17. What is recycled energy?
17. Recycled energy is the reemployment of energy that otherwise would be wasted.
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18. Can a machine multiply input force? Input distance? Input energy? (If your three answers are the same, seek help; the last question is especially important.
18. A machine can multiply input force or input distance, but NEVER input energy.
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19. If a machine multiplies force by a force of 4, what other quantity is diminished, and by how much?
19. As force is increased, distance is decreased by the same factor.
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20. A force of 50N is applied to the end of a lever, which is moved a certain distance. If the other end of the lever moves one-third as far, how much force can it exert?
20. The end moving 1/3 as far can exert 3 times the input force, 150 N.
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21. What is the efficiency of a machine that miraculously converts all the input energy to useful output energy?
21. Efficiency would be 100%.
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22. If an input of 100J in a pulley system increases the PE of a load by 60J, what is the efficiency of the system?
22. Efficiency will be 60%.
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23. What is the ultimate source of energy for fossil fuels, dams, and windmills?
23. The Sun is the source of these energies.
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24. What is the ultimate source of geothermal energy?
24. Radioactivity is the source of geothermal energy.
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25. Can we correctly say that hydrogen is a new source of energy? Why/why not.
25. Like electricity, hydrogen is a carrier of energy, not a source. That's because it takes energy to separate hydrogen from molecules.
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28. Calculate the work done when a force of 5N moves a book 1.2m (remember that 1N*m = 1J) W = Fd
28. W = Fd = (5 N)(1.2 m) = 6 N.m = 6 J.
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29. Show that 2.4J of work is done when a force of 2.0N moves a book 1.2m. W = Fd
29. W = Fd = (2.0 N)(1.2 m) = 2.4 N.m = 2.4 J.
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30. Calculate the work done when a 20N force pushes a cart 3.5m. W = Fd
30. W = Fd = (20 N)(3.5 m) = 70 N.m = 70 J.
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31. Calculate the work done in lifting a 500N barbell 2.2M above the floor. (What is the gain in PE of the barbell when it is lifted to this height?) W = Fd
31. W = Fd = (500 N)(2.2 m) = 1100 N.m = 1100 J, which is also the gain in PE.
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32. Shoe that 50W of power is required to give a brick 100J of PE in a time of 2s. P= W/t
32. P = W/t = (100 J)/(2 s) = 50 W.
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33. Show that about 786W of power is expended when a 500N barbell is lifted 2.2m above the floor in 1.4s. P= W/t
33. P = W/t = Fd/t = (500 N)(2.2 m)/(1.4 s) = 786 W.
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34. Shoe that when a 3.0kg book is lifted 2.0m its increase in gravitational PE is 60J. PE = mgh
34. PE = mgh = (3.0 kg)(10 N/kg)(2.0 m) = 60 N.m = 60 J.
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35. Show that the gravitational PE of a 100kg boulder raised 5m above ground level is 50,000J. (You can express g in units of N/kg because m/s² is = to N/kg) PE = mgh
35. PE = mgh = (1000 kg)(10 N/kg)(5 m) = 50,000 N.m = 50,000 J.
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36. Show that the KE of a 1.0kg book tossed across the room at a speed of 3.0m/s is 4.5J. (1J is = to 1 kg*m/s²) KE = ½mv²
36. KE = ½ mv2 = ½(1.0 kg)(3.0 m/s)2 = 4.5 kg(m/s)2 = 4.5 J.
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37. Calculate the KE of an 84kg scooter moving at 10 m/s. KE = ½mv²
37. KE = ½ mv2 = ½(84 kg)(10 m/s)2 = 4200 kg(m/s)2 = 4200 J.
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38. Show that 24J of work is done when a 3.0kg block of ice is moved from rest to a speed of 4.0m/s. Work = ∆KE
38. W = ΔKE = Δ½ mv2 = ½(3.0 kg)(4.0 m/s)2 = 24 J.
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39. Show that a 2,500,000J change in KE occurs for an airplane that is moved 500m in takeoff by a sustained force of 5000N. Work = ∆KE
39. From W = ΔKE, ΔKE = Fd = (5000 N)(500 m) = 2,500,000 J.
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40. Show that a machine that has an input of 100J and an output of 40J is 40% efficient. Efficiency = useful energy output/total energy input
40. Efficiency = energy output/energy input x 100% = (40 J)/(100 J) = 0.40 or 40%
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41. The second floor of a house is 6m above the street level. How much work is required to lift a 300kg refrigerator to the second story level?
41. Work = ΔE = Δmgh = 300 kg × 10 N/k × 6 m = 18,000 J.64
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42. a) How much work is done when you push a crate horizontally with 100N across a 10m factory floor? b) If the force of friction on the crate is a steady 70N, show that the KE gained by the crate is 300J c) Show that 700J is turned into heat.
42. (a) You do F x d = 100 N x 10 m = 1000 J of work. (b) Because of friction, net work on the crate is less. ΔKE = Net work = net force x distance = (100 N -70 N)(10 m) = 300 J. (c) So the rest, 700 J, goes into heating the crate and floor.
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43. A car that was moving at 50 km/h kids 15m with locked brakes. How far will the car skid with locked brakes if it was moving at 150km/h?
43. At three times the speed, it has 9 times (32) the KE and will skid 9 times as far—135 m. Since the frictional force is about the same in both cases, the distance has to be 9 times as great for 9 times as much work done by the pavement on the car.
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44. Bernie dives from atop a tall flagpole into a swimming pool. His PE at the top is 10,000J (relative to the surface of the pool). What is his KE when his PE is reduced to 1000J?
44.PE + KE = Total E; KE = 10,000 J - 1000 J = 9000 J.
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45. Nellie applies a force of 50N to the end of a lever, which is moved a certain distance. If the other end of the lever moves 1/3rd as far, show that the force it exerts is 150N
45. From F x d = F' x d/4, we see F' = 4F = 200 N.
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46. Consider an ideal pulley system. If you pull one end of the rope 1m downward with a 50N force, show that you can lift a 200N load ¼ of a meter high.
46. Your input work is 50 J, so 200-N x h = 50 J. h = 50/200 = 0.25 m.
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47. In raising a 5000N piano with a pulley system, the workers note that for every 2m of rope pulled downward, the piano rises 0.2m. Ideally, show that 500N is required to lift the piano.
47.(F x d)in=( F x d)out F x 2 m = 5000 N x 0.2 m F = [(5000 N)(0.2 m)]/2 m = 500 N
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48. In the hydraulic machine shown, you observe that when the small piston is pushed down 10cm, the large piston is raised 1cm. If the small piston is pushed down with a force of 100N, what is the most weight that the large piston can support?
48.(F x d)in = (F x d)out (100 N x 10 cm)in = (? x 1 cm)out So we see that the output force and weight held is 1000 N (less if efficiency < 100%).
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49. How many watts of power do you expend when you exert a force of 50N that moves a crate 8m in a time interval of 4s?
49. Power = Fd/t = (50N)(8m)/(4s) = 100J/1s = 100 watts.
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50. Emily holds a banana of mass m over the edge of a bridge of height h. She drops the banana and it falls to the river below. Use conservation of energy to show that the speed of the banana just before hitting the water is v=√2gh.
50. The initial PE of the banana is transformed to KE as it falls. When the banana is about to hit the water, all of its initial PE becomes KE. From PE₀=KEƒ=>mgh=½mv²=2gh=>v⁰=√2gh
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51. The mass and speed of the three vehicles A,B,C, are shown. Rank them from greatest to least for the following: a) 1.0m/s →→ 800kg b) 2.0m/s →→ 1000kg c) 8.0m/s →→ 90kg
51. a) B, A, C b) C, B, A c) C, B, A
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52. A ball is released from rest at the left of the metal track. Assume it has only enough friction to roll, but not to lessen its speed. Rank these quantities from greatest to least at each point. a) Momentum b) KE c) PE
52. a) C, B=D, A b) C, B=D, A c) A, B=D, C
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53. The roller coaster ride starts from rest at point A. Rank these quantities from greatest to least at each point. a) Speed b) KE c) PE
53. a) D, B, C, E, A b) D, B, C, E, A c) A, E, C, B, D
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54. Rank the scale reading from highest to lowest.
54. B=C, A (same as two supporting ropes) Think and Explain
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55. Why is it easier to stop a lightly loaded truck than a heavier one that has equal speed?
55. 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.)
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56. Why do you do no work on a 25kg backpack when you walk a horizontal distance of 100m?
56. 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.
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57. If your friend pushes a lawnmower four times as far as you do while exerting only half the force, which one of you does more work? How much more?
57. Your friend does twice as much work (4 x 1/2 > 1 x 1).
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58. Why does one get tired pushing against a stationary wall when no work is done on the wall?
58. Although no work is done on the wall, work is nevertheless done on internal parts of your body (which generate heat).
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59. Which requires more work: stretching a strong spring a certain distance or stretching a weak spring the same distance? Why
59. More force is required to stretch the strong spring, so more work is done in stretching it the same distance as a weaker spring.
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60. Two people who weigh the same climb a flight of stairs. The first person climbs the stairs in 30s and the second person climbs them in 40s. Which person does more work? Which uses more power?
60. 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.
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61. In determining the PE of Tenny's drawn bow (figure 7.10), would it be an underestimate or an overestimate to multiply the force with the which she holds the arrow in it's drawn position by the distance she pulls it back? Why do we say the work done is average force x distance?
61. 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 x distance drawn, but average force x 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.
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62. When a rifle with a longer barrel is fired, the force of expanding gases acts on the bullet for a longer distance. What effect does this have on the velocity of the emerging bullet?
62. 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.)
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63. Your friend says that the KE of an object depends on the reference frame of the observer. Explain why you agree or disagree.
63. Agree, because speed itself is relative to the frame of reference (Chapter 3). Hence ½ mv2 is also relative to a frame of reference.
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64. You and a flight attendant toss a ball back and forth in an airplane in flight. Does the KE of the ball depend on the speed of the airplane? Carefully explain.
64. 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.
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65. You watch your friend take off in a jet plane, and you comment on the KE she has acquired. But she says she experiences no such increase in KE. Who is correct?
65. 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.
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66. When a jumbo jet slows and descends on the approach to landing, there is a decrease in both its KE and PE. Where does this energy go?
66. The energy goes mostly into frictional heating of the air.
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67. Explain how elastic PE dramatically changed the sport of pole vaulting when flexible poles replaced stiffer wooden poles.
67. 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.
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68. At what point in its motion is the KE of a pendulum bob at a maximum? At what point is its PE at a maximum? When its KE is at half its maximum value, how much PE does it have relative to its PE at the center of the swing?
68. 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.)
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69. A physics instructor demonstrates energy conservation by releasing a heavy pendulum bob, as shown in the sketch, and allowing it to swing to and fro. What would happen if, in his exuberance, he gave the bob a slight shove as it left his nose? Explain
69. 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.)
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70. Does the international space station have gravitational PE? KE? Explain
70. Yes to both, relative to Earth, because work was done to lift it in Earth's gravitational field and to impart speed to it.
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71. What does the work energy theorem say about the speed of a satellite in circular orbit?
71. 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.
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72. A moving hammer hits the nail and drives it into a wall. If the hammer hits the nail with twice the speed, how much deeper will the nail be driven? If the hammer hits with three times the speed?
72. 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.
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73. Why does the force of gravity do no work on: a) a bowling ball rolling along an alley and b) a satellite in circular orbit about earth?
73. 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, or parallel to, the direction of motion and no work is done by the force.
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74. Why does the force of gravity do work in a car that rolls down a hill but no work when it rolls along a level part of the road?
74. On the hill there is a component of gravitational force parallel to the car's motion. This component of force does work on the car. But on the level, there is no component of gravitational force parallel to the direction of the car's motion, so the force of gravity does no work in this case.
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75. Does the string that supports a pendulum bob do work on the bob as its swinging? Does the force of gravity do any work on the bob?
75. The string tension is everywhere perpendicular to the bob's direction of motion, which means there is no component of tension parallel to the bob's path, and therefore no work done by the tension. The force of gravity, on the other hand, has a component parallel to the direction of motion everywhere except at the bottom of the swing, and does work, which changes the bob's KE.
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76. A crate is pulled across a horizontal floor by a rope. At the same time, the crate pulls back on the rope, in accord with Newton's third law. Does the work done on the crate by the rope then equal zero? Why
76. 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.
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77. On a playground slide, a child has PE that decreases by 1000J while her KE increases by 900J. What other form of energy is involved, and how much?
77. 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.
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78. Someone who wants to sell you a superball claims that it will bounce to a height greater than the height from which it is dropped. Can this be?
78. 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.
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79. Why can't a superball released from rest reach its original height when it bounces from a rigid floor?
79. 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.
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80. Consider a ball thrown straight up in the air. At what position is its KE at a maximum? Where is its gravitational PE at a maximum?
80. Kinetic energy is a maximum as soon as the ball leaves the hand. Potential energy is a maximum when the ball has reached its highest point.
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82. Suppose that you and two classmates are discussing the design of a roller coaster. One classmate says that each summit must be lower than the proceeding one. Your other classmate says this is nonsense, for as long as the first one is the highest, it doesn't matter what height the other are. What do you say?
82. 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.
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83. When the girl in figure 7.17 jacks up a car, how can applying so little force produce sufficient force to raise the car?
83. 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. KE is increased by a factor of four. Momentum is proportional to speed, KE to speed squared.
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84. What famous equation by Albert Einstein describes the relationship between mass and energy?
84. Einstein's E = mc2. (More on this in Chapters 34 and 35).
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85. When the mass of a moving object is doubled with no change in speed, by what factor is its momentum changes? By what factor is its KE changed?
85. When the mass is doubled with no change in speed, both momentum and KE are doubled.
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86. When the velocity of an object is doubled, by what factor is its momentum changed? By what factor is its KE changed?
86. When the velocity is doubled, the momentum is doubled and the KE is increased by a factor of four. Momentum is proportional to speed, KE to speed squared.
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87. Which, if either, has greater momentum: a 1kg ball moving at 2m/s or a 2kg ball moving at 1m/s? Which has greater KE?
87. Both have the same momentum, but the faster 1-kg one has the greater KE.
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88. A car has the same KE when traveling north as when it turns around and travels south. Is the momentum of the car the same in both cases?
88. The momentum of the car is equal in magnitude but opposite in direction in the two cases—not the same since momentum is a vector quantity.
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89. If an objects KE is zero, what is its momentum?
89. Zero KE means zero speed, so momentum is also zero.
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90. If your momentum is zero, is your KE necessarily zero also?
90. Yes, if we're talking about only you, which would mean your speed is zero. But a system of two or more objects can have zero net momentum, yet have substantial total KE.
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91. If two objects have equal KE, do they necessarily have the same momentum? Defend
91. Not at all. For two objects of the same KE, the one of greater mass has greater momentum. (The mathematical relationship is p2 = 2m x KE.)
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92. Two lumps of clay with equal and opposite momenta have a head on collision and come to rest. Is momentum conserved? Is KE conserved? Why are you answers the same or different?
92. Net momentum before the lumps collide is zero and after collision is zero. Momentum is indeed conserved. Kinetic energy after is zero, but was greater than zero before collision. The lumps are warmer after colliding because the initial kinetic energy of the lumps transforms into thermal energy. Momentum has only one form. There is no way to \"transform\" momentum from one form to another, so it is conserved. But energy comes in various forms and can easily be transformed. No single form of energy such as KE need be conserved.
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93. Scissors for cutting paper have long blades and short handles, whereas metal cutting shears have long handles and short blades. Bolt cutters have very long handles and very short blades. Why
93. Scissors and shears are levers. The applied force is normally exerted over a short distance for scissors so that the output force is exerted over a relatively long distance (except when you want a large cutting force like cutting a piece of tough rope, and you place the rope close to the \"fulcrum\" so you can multiply force). With metal-cutting shears, the handles are long so that a relatively small input force is exerted over a long distance to produce a large output force over a short distance.
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94. An inefficient machine is said to waste energy. Does this mean that energy is actually lost? Why
94. Energy is transformed into nonuseful forms in an inefficient machine, and is \"lost\" only in the loose sense of the word. In the strict sense, it can be accounted for and is therefore not lost.
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SI unit for work (N.M)
SI unit for energy (Joule) I Nm = 1 Joule
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SI unit for power Watt 1000 watts=1 kilowatt (kw)
Mechanical Energy can be in the form of PE or KE or both.
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SI unit of PE (mgh)
PE can change to KE.
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PE is equal to KE
Types of PE: Compressed spring (elastic energy) Gravitational PE: Object at a certain height Stretched rubber band (elastic energy)
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PE due to chemicals: Gasoline (fossil fuels) Batteries Candy Bars
a) As height increases so does the velocity. b) Weight is a force
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a) Gravity, if not stated, is 10m/s². b) When PE is at it's highest height PE is higher c) PE is zero when on ground
a) KE is higher at it's lowest level b) KE is 4 times higher c) if speed is doubled, KE is quadrupled d) Work is a change in KE
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Driving 80km/hr, how much MORE DISTANCE is needed to stop compared to 40km/hr? Needs 4 times the distance
Driving 90km/hr, how much MORE DISTANCE is needed to stop compared to 30km/hr? Needs 9 times the distance
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Driving 120km/hr, how much MORE DISTANCE is needed to stop compared to 30km/hr? Needs 16 times the distance