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Work, Energy, and Energy Bar Charts By: Jane Wang, Amber Sweeton, and Diane Zhou Period: 5-6 -1- Introduction Energy is all around us: it is what makes cars run, what makes objects move, what lets forests grow, and what operates all our electronics. You can not see it, but its still there. Energy enables you to perform actions and to move. In this chapter, we will explore the concepts of energy and work using free-body diagrams and mathematical representations. Objectives for this chapter: By the end of this chapter, you should… o Understand what work and energy are in a basic sense o Comprehend the concepts of negative, positive, and 0 work and be able to give examples of each o Know what the basic types of energy are (KE, GPE, EPE, IE) o Know that energy is conserved in a system and that energy can not be created or destroyed o Be able to both interpret and draw energy bar charts o Understand how the equations for GPE and KE are derived o Be able to mathematically represent energy o Be able to solve basic work/energy problems Experiments That You Can Do at Home Chalk Experiments: These experiments will model various energy conversions and show you that energy is conserved in all cases. With common household materials, we will investigate exactly how energy conversions occur as well as the various types of energy. Some of these are gravitational potential energy (GPE), kinetic energy (KE), elastic potential energy (EPE), and internal energy (IE). These concepts will also be explained later in the chapter. Materials: Sticks of chalk A heavy object (ex. a book) A slingshot/rubber band A large toy car Procedure: Modeling a Change from GPE to KE 1. Place one piece of chalk flat on a smooth surface. This might be a messy experiment. Get parental permission before trying. 2. Lift your heavy object a short distance above the stick of chalk Observe that it requires a little work (effort) to lift the object. -2- 3. Drop the object. The chalk should not break or should only crack slightly. This is because you gave the object only a little bit of GPE when you lifted it is slight distance. 4. Replace the chalk with a new piece. Lift the heavy object once again, only much higher above the chalk. Drop the object. Observe that it requires more work (effort) to lift the object this time, and that the object gained more speed when falling. The chalk should crack or break. What did we observe? It takes more work to lift an object a higher distance and give it more GPE. When an object has more GPE, it has more capacity for motion when it is dropped and applies a greater force on the chalk, causing it to break You could represent this experiment using an energy bar chart: KE GPE EPE IE KE GPE EPE IE At the left is an energy bar chart depicting this scenario (from the moment the object is dropped to the moment before it hits the chalk). Assume negligible air resistance. Notice that the heights of the bars before and after are the same. This shows that no energy is created or destroyed. All energy has been changed from GPE to KE. Modeling a Change from EPE to KE 1. Take your rubber band and stretch it between two of your fingers. 2. Place a piece of chalk on the rubber band and stretch it back a bit; release chalk. (Do not aim chalk at others, animals, or fragile objects; get parental permission before firing) Observe that this does not take much work. The chalk does not go far, because it started out with very little EPE 3. Repeat step two, but stretch the rubber band father Observe that this takes more work. The chalk goes farther and faster, because you originally gave it more EPE. What did we observe? It takes more work to stretch a rubber band farther and to give it more EPE. However, when you do so, the EPE transfers to KE and more motion is achieved. EPE is energy that an object has when it is stretched or compressed. You could also represent this using an energy bar chart: This energy bar chart shows the change in types of energy from right when the chalk is released from the rubber band to when the chalk is at the peak of its flight. Assume that there is negligible air resistance and that the chalk is released from a height off the ground and angled upwards when fired. KE GPE EPE IE KE GPE EPE IE -3- The energy converts from being GPE/EPE to KE/GPE. The total amount of energy remains the same before and after. Modeling a Change from Work to KE and KE → IE 1. Stand a piece of chalk against a wall. (Get parental permission and do this experiment on an easy to clean surface) 2. Place your toy car a fixed distance away from the chalk. Give it a small push. You have just done a little work on the car (you are not a part of the system). This work changed to KE and resulted in a little motion. Notice how the car slows down after a while. This is because the KE is changing into IE (heat, sound, etc.) because of friction. 3. Place your toy car the same distance away from the chalk. Give it a larger push. You have added more work into the system, so therefore it changes into more KE and the car moves at a faster speed. The chalk might break this time or crack more than when only a little work was added. What did we observe? Energy is always conserved in a system. This is shown when more work resulted in more KE. Internal energy is energy that results from friction and turns into heat, light, sound, etc. This can also be represented using an energy bar chart: + KE GPE EPE IE W KE GPE EPE IE KE GPE EPE IE This energy bar chart shows the change in types of energy in three stages: when the car is first pushed, when you release the car and it begins moving, and when the car begins to slow down. Assuming that the system includes the table, the chalk, and the car , your work is first transferred all into KE. This KE is later converted into internal energy because of friction. Super Happy Fun Ball Experiment: This experiment will be discussed more in depth later in the section. This explanation will be mainly for observational purposes. You will observe that energy is conserved in a system. The procedure of this experiment will be explained later in this chapter; this is just a brief overview of the experiment. The super happy fun ball snowman is a set of 3 rubber balls stuck together with a pole sticking out of the top. There is also a separate smaller ball with a hole through its center. This ball can be placed on the rod and the snowman. Refer to the figure at the right. -4- Can come off Stuck Together What can you observe? When you drop the snowman from a certain height, it falls to the ground. When it hits the ground, all of the kinetic energy is transferred into the small ball at the top. The “super happy fun ball” flies off and jumps to a very high height. This demonstrates a conservation of energy. When you were holding the snowman above the ground, it had a certain amount of GPE. When the small ball was at the highest point in its jump, it had the same amount of GPE. This aptly demonstrates a property of the equation for GPE. GPE=mgh. Since the mass of the small ball is significantly less than that of the snowman, and the GPE and g remain the same, the height varies. When the mass is great, the height is small and when the mass is small, the height is great. These variables are inversely proportional. Simple Observations that You Can Make in Your Everyday World: Energy is all around you and many of its properties can be observed by just watching things and actions occur. For example, if you drop your pencil during class one day, you can observe an energy conversion. How, you ask? Right before your pencil leaves your hand, it has gravitational potential energy (it has the potential to move/fall). When you let go of your pencil, it falls and all of the GPE that it originally had is changed into KE. When it hits the ground, it stops moving. Why is this? It’s because all of the KE that your pencil had previously had has been changed to internal energy (i.e. sound, heat, etc.) You can also make simple observations about work. When you have a book sitting on the table, it has no kinetic energy since it is not moving. When you push it (add work to the system), that work changes into kinetic energy and the object starts to move. Elastic potential energy is also easy to observe. In addition to the rubber band and chalk experiment explained above, you could also jump on your bed to demonstrate the concept. At the lowest point of any jump (when your feet are on the bed and it is about to spring you up again), all of the energy in your system (the bed and you) is elastic potential energy. You are not moving at that time. Soon after that point, all of that elastic potential energy turns into gravitational potential and kinetic energy and you are moving again. Work, Energy, and Energy Bar Charts What is energy? Energy is the capacity to do work. Energy is what you have. Work is what you do. (Work is explained in more detail in “What is Work?”) Both work and energy are measured in joules. KE GPE EPE IE Figure 1 -5- KE GPE EPE IE This is an energy bar chart. Energy bar charts are used to represent the changing of energy. The left part represents the energy that the object started out with. The right part represents how the energy changed. Sometimes more than two energy bar charts are needed to represent all the energy transfers in a situation. In an energy bar chart, the energy amounts on both sides of the arrow must add up. In the example above, the initial KE is the sum of the resulting KE and GPE. This is because energy can not be created or destroyed. The amount of energy in a system will always stay constant. The fact that the energy in a system will always remain constant is known as the Work-Energy Theorem. To draw an energy bar chart, draw two perpendicular lines and draw the bars of energy inside. Label them so that their type can be identified. Most of the time, energy bar charts are qualitative, not quantitative, so the size of the bars doesn’t have to be exact. As long as they’re roughly correct and both sides of the arrow add up to each other, it’s fine. The dotted lines on the energy bar chart are like interval markers on a graph. They help to make your chart more exact, but they’re not always necessary. There are many kinds of energy that can be represented in Energy Bar Charts. In the graph above, the object starts out with KE. KE represents the object’s Kinetic Energy. Kinetic energy is the energy an object has as a result of its movement. GPE represents Gravitational Potential Energy, which an object has as long as it is above the “0” point. The 0 point is whatever you determine it to be. For example, the 0 point can be defined as sea level, or it can be defined as the ground on which a person is standing, depending on the situation. Some other things you might encounter in energy bar charts include Work (W) and Elastic Potential Energy (EPE). Work is explained below, in the section “What is work?”. EPE is the energy an object has as a result of being stretched or compressed, like a rubber band. In its original, un-stretched state, a rubber band has no EPE. However, when you tug on both sides and stretch it out, it has the potential to snap back, so it has Elastic Potential Energy. Also, there is something known as Electrical Potential Energy (ELPE). ELPE is an object’s potential to move based on its location, position, and charge in relation to another object. If two objects that are held next to each other have a positive charge, they have ELPE because they want to repel each other (likewise if they both have negative charges). If one object has a positive charge and one has a negative charge and they’re next to each other, they won’t have any ELPE because they attract each other anyway. If they’re already next to each other, they don’t have any potential to move any further. You may also see something called Internal Energy (IE). Internal energy is all the other types of energy, such as heat, light, sound, etc. It can be a result of friction. For example, if you rub your hands together very quickly, the friction will create heat and they’ll feel warmer. Now that you understand how to read the Energy Bar Chart, you can understand the motion of the object. -6- How is the object in Figure 1 moving? It’s moving at a high speed because it has a lot of kinetic energy at first. Then the KE goes down and the GPE goes up, so it must have slowed down, but also somehow moved above the zero point. For example: In the diagram the watermelon is running fast, and then he slows down as he goes up a hill, just like the previous description. There are many different situations that the chart in Figure 1 could represent. The running watermelon is only one example. As you can see, Energy Bar Charts are useful to understanding the transfer of energy. Based on that information, you can get a rough idea of the object’s motion. What is work? As previously mentioned, work is what you do when you have energy. The more energy you have, the more work you’re capable of doing. In science, it’s only considered regular positive work when the object is moving in the same direction of the force. So, if you stand still and hold a pineapple, it is not considered work, because you’re applying force upward against gravity, but the pineapple is not moving up. It is staying still even though your force goes up, so it isn’t work. There are three types of work: positive, negative, and zero. Positive work is when the work gives more energy to the object, such as pushing a shopping cart forward. Negative work is when the force causes the object to have less energy, such as a person catching a baseball that is thrown at them. Zero work is done when there is no change in energy. What is a system? A system is the group of objects that a problem is taking into consideration. Objects outside of the system can do work on objects within a system, but they are still not considered a part of it. Therefore, Energy Bar Charts only take into consideration how much energy is in the system. -7- Derivations for KE and GPE: Fstring → box Use the force diagram at the right as a starting point for the following derivation. A person is pulling a string attached to a box, while the earth is pulling down on the box with an equal and opposite force. For the following derivation, remember that: Fearth → box W = KE + GPE Fs box Fe box m F box mg a s m a F am mg Start with the basic Newton’s second law equation. In this example, a force (Fs) is being applied on an object upwards while the earth is applying a force on it downwards (Fe). The force of the earth on an object is equal to the mass of the object x g (acceleration of gravity or 9.8 m/s2) Multiply by m on both sides and move the mg to the other side. h( F ) h(am mg ) Recall that: F d W W mah mgh Multiply by h on both sides of the equation. Recall that force x distance/height = work V f Vi 2ah 2 2 V f Vi 2 a 2h mh(V f Vi ) 2 W W W Recall the equation listed at the right. 2 Subtract Vi2 from both sides and divide both sides by 2h. 2 2h 2 2 m(V f Vi ) 2 mgh mgh Plug this expression in for a. Cancel out the h from the numerator and denominator of the fraction. mv 2 mgh 2 Arrive at the final equation for work. Notice a few key things about this equation: 1. This equation looks a lot like our previous equation, W = KE + GPE 2. Therefore, GPE = mgh mv 2 3. KE would then equal 2 -8- Important Concepts to Grasp from this Section: Work is equal to the sum of kinetic energy and gravitational potential energy. Using equations that we had previously learned, you can derive and arrive at equations for kinetic energy and gravitational potential energy: KE mv 2 2 GPE mgh Using these equations, we will work to solve problems in later sections of this chapter. In future sections, you will learn how to create generalized equations for a given scenario as well as how to solve for specific variables. The Super Happy Fun Ball Experiment The super happy fun ball experiment can be used to enhance one’s knowledge on the various equations for acceleration, forces, and energy. The super happy fun ball snowman is a set of three rubber balls stacked one on top of another. A rod runs through the center of the three balls and all of the balls are immobile. A smaller ball is also with a hole through it is also slipped onto the rod. When dropped from a certain height, the balls will fall together. However, when it hits the floor and bounces back up, most of the energy will get transferred to the little ball on top and it will shoot up to a great height. Using the equations for acceleration, forces, and Newton’s second law, the exact height that the small can reach can be determined. Can come off Stuck Together Materials Needed: o Super Happy Fun Ball Snowman o Scale o Meter stick o Calculator Procedure: 1. Measure the mass of your snowman using the scale 2. Measure the mass of just the little ball that can come off 3. Hold your snowman at a specific height (the height that you will drop it from); measure this height with a meter stick 4. Drop the super happy fun ball snowman 5. Using the equation for GPE, figure out the final height of the small ball. -9- The method for finding the maximum height: In this example, we are using the following: msmall ball = 4.2 g msnowman = 87 g hdrop = 1.25 m Assume that the height of the snowman, friction, and air resistance are negligible to the final height of the super happy fun ball. Solving the Equation: Remember that energy is conserved in a system. Therefore, the GPE of the snowman (GPE1) when first dropped will equal the GPE of the super happy fun ball at the height of its jump (GPE2). GPE1 mgh GPE1 (0.087 kg )(9.8 m s 2 )(1.25m) GPE1 1.066J GPE1 GPE2 GPE2 mgh 1.066J (0.004kg) (9.8 m s 2 )( h) h 27.19m The final height of the small ball (if dropping it from 1.25 m) is 27.19 meters. Model Problems Sample Problem 1: A 3 kg pineapple is thrown directly up into the air by a person who does 250 J of work on it. How high will the pineapple fly? Assume negligible air resistance. Energy Bar Chart… + KE GPE EPE IE KE GPE EPE IE W - 10 - Solving the Problem… The pineapple has no energy in the beginning of the problem (it has no height and is not moving). Work is then done on it, causing it to shoot up. At the highest point of its flight, it stops for a very brief moment, and all of its energy is GPE. Therefore, the equation for this situation would be: W mgh To solve for height, just plug in numbers for W, m, and g: W mgh 200J (3kg) 9.8 m s 2 (h) h 6.8m Sample Problem 2: A man drops a 4 kg bowling ball from the edge of a building. Assume negligible air resistance. If the ball’s final velocity is 196 m/s, what is the height of the building from which the bowling ball was dropped? Energy Bar Chart… KE GPE EPE IE KE GPE EPE IE Solving the problem… When first dropping the ball, the only type of energy that the ball had was gravitational potential energy (it was really high up, but was not moving). When it fell, that GPE turned into KE. If its final velocity was 196 m/s, then all of the energy that it had at that point would have been KE. Therefore, the equation for this would be mgh= ½m(vf - vi)2 mv 2 KE 2 (4kg)(196m/s ) 2 KE 2 KE 392J KE GPE GPE mgh 392J (4kg)(9.8m /s 2 )( h) h 10m - 11 - Sample Problem 3: Mary weighs 50 kg and is swinging on a vine. At the lowest point of her swing, she is going at a speed of 20 m/s. Considering that energy is conserved and assuming that there is no negligible friction, what is the highest point of Mary’s swing? For this problem, this is the energy bar chart if you consider Mary’s swing in three stages: the top of her swing, when she’s going the fastest, and the top of her swing on the other side. KE GPE EPE IE KE GPE EPE IE KE GPE EPE IE Solving the Problem… Mary had all kinetic energy at the point in which she was swinging the fastest (in a pendulum-like situation, the fastest speeds are achieved when the object is closest to the ground). Therefore, she has all GPE at the highest point of her swing. Using that information, the equation would be… ½m(vf - vi)2= mgh From there, all you have to do is plug in numbers and solve the equation… mv 2 KE 2 (50kg)(10m/s) 2 KE 2 KE 2500J KE GPE GPE mgh 2500J (50kg)(9.8 m/s 2 )( h) h 5.1m Sample Problem 4: Make an energy bar chart for the energy that the skier will have at all three points: A, B, & C. - 12 - Energy Bar Charts… A. KE GPE EPE IE B. C. KE GPE EPE IE KE GPE EPE IE Explanation… A. For the energy bar chart, it is all Gravitational Potential Energy because the skier is at the top of the hill and isn’t moving, so there is no Kinetic Energy. B. It is two units of Kinetic Energy, one unit of Gravitational Potential and one unit of Internal Energy. It is like this because the skier is moving, still above the zero point, or ground, and there is friction beneath the skis on the snow. C. There is all Internal Energy because it is not moving and not in the air but there is still the heat on the skis that was made by the friction, which is Internal Energy. Sample Problem 5: Make an energy bar chart and a general Newton’s laws equation for a car traveling up a hill and halfway up revs its engine and it goes faster and higher. Energy Bar Charts… + KE GPE EPE IE KE GPE EPE IE W ½mv12 + mgh1 + W = ½mv22 + mgh2 First, there is Kinetic and Gravitational Potential Energy, then the engine, a force outside of the system, does work. This then comes to all Kinetic and Gravitational Energy. - 13 - Summary In this chapter, we have learned about the different types of energy and about work. Energy is the ability to do work. Energy is something you have, work is something you do. Energy in a system is always conserved. The different types of energy are once again: Gravitational Potential Energy (GPE): energy you have when you are above the ground or 0 point. Kinetic Energy (KE): energy as a result of movement. Elastic Potential Energy (EPE): energy something has when it can be stretched or compressed. Internal Energy (IE): all other types of energy like heat, light & sound. Work is what you do when you have energy. You can do positive, negative & zero work. Positive is when the work gives more energy to the object. Negative is when a force causes the object to have less energy. Zero is when there is no change in energy. Practice Questions A B D E C F 0 Point Diagram for Questions 1-6 1. Does the cart have GPE or KE at Point A? Why? 2. Draw an energy bar chart for the cart at Point B. 3. Does the cart have any GPE at point C? Why or why not? 4. Draw the energy bar chart for the cart at Point D. 5. How is the energy bar chart for Point E different from the one at Point D? Why? - 14 - 6. What happened to the cart at Point F? Does it have any GPE or KE? If not, what happened to all its energy? 7. A person has jumped off of a cliff, but he lands a conveniently placed obese pig and is saved. What type of force does the obese pig exert on the person? Draw energy bar charts for the person as soon as he jumped off and just before he hit the pig. 8. An object is hanging in the air from a pulley and staying still. Then someone pulls the other end of the pulley and the object moves up. Draw energy bar charts to represent before and after the pull. 9. By how much does the gravitational potential energy of a 64 kg pole vaulter change if his center of mass rises about 4 meters during the jump?\ 10. A 55 kg hiker starts at an elevation of 1600 meters and climbs to the top of a 3100 meter peak. (a) What is the hiker’s change in potential energy? (b) What is the minimum work required of the hiker? (c) Can the actual work done be more than this? 11. If a box is initially at rest and is pulled up, make an energy bar chart and Newton’s laws equation of the box ending up above the ground and moving. 12. How high will a 0.325 kg rock go if thrown straight up by someone who does 115 J of work on it? Neglect air resistance. - 15 - Answer Key 1. GPE, because it’s not moving and it’s high above the zero point. 2. KE GPE EPE IE 3. Yes, but not very much, because it’s just above the zero point. 4. KE GPE EPE IE 5. For Point E, it has the same amount of GPE as point D, but it’s moving more slowly, so it has less KE and more IE instead. 6. At Point F the cart has neither GPE nor KE because it’s at the zero point and it’s not moving. All its energy has been converted into IE. 7. Negative force. KE GPE EPE IE KE GPE EPE IE 8. + KE GPE EPE IE W KE GPE EPE IE - 16 - 9. 2.5 x 103 J GPE mgh GPE (64kg)(9.8 m s 2 )( 4m) GPE 2500 10. (a) 8.1 x 105 J GPE mgh GPE (55kg)(9.8 m s 2 )(1500m) GPE 81000 (b) 8.1 x 105 J; it would be the same amount as the energy done. (c) yes; it can account for internal energy as well. 11. + KE GPE EPE IE KE GPE EPE IE W W = ½mv22 + mgh2 12. 36.1 m GPE mgh 115 (0.325kg)(9.8 m s 2 )( m) GPE 36.1 Sources used: Physics: Principles with Applications, by Giancoli - 17 -