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PPT Work, Potential Energy, and Kinetic Energy Developer Notes Check Hewitt's definition of energy! Interesting. Do a worksheet with practice of PE to KE? I'd still like to tie d = at2/2 to KE = mv2/2 in some understandable way. Why are they the same form? They're obviously related. Other suggestions for work-PE-KE-work - there are two main things I haven't been able to get cleanly around - some of the PE going into rotational energy, and measuring the sliding force of the receptor cup at the end. dynamics cart - but steel wheels still take a large amount of rotational energy pinewood derby cart - the wheels are light, it might work, didn't try it (Could these possibly be a cheap source of dynamics carts? How to mount a spring?) croquet balls, bigger ball bearings, or pool balls - still have rotational energy, but the sliding force should be bigger and easier to measure than for a 1" ball bearing pendulum - no friction, but how do you capture the KE and convert it to measurable work at the end? Or how could you measure its velocity? ice on wax paper - the ice sticks, besides it being messy Hot Wheels cars - the wheels are light, but measuring the sliding force is tough because they're so small, and you also should have Hot Wheels track to guide the cars. Besides, the axles bend easily so the wheels rub on the bodies. Launch a ball bearing up off of a spring. Check the compression constant on the spring, check how far it is compressed, multiply to get PE, then see how high it goes. Calculate the velocity of the ball when it leaves the spring based on ht=at2/2. This doesn't give work, however. Another exercise - how high will a ball go given its initial velocity? Version 04 05 06 Date 2004/04/20 2004/04.21 2004/06/24 Who dk dk Sc Revisions Updated to new format Revised the exercises Minor editing- including measuring at the 0.100 m mark, & a note about measuring the height Goals Students should understand that work can transform energy. Students should know that potential energy is Fd, or wtht, or mgh. Students should understand that potential energy can be converted into kinetic energy and back. Students should know that kinetic energy is (mv2)/2. Students should know that work, PE, and KE are all measured in Joules. Students should know that energy is the ability to do work. Concepts & Skills Introduced Area 769831670 dk Concept Page 1 of 10 PPT Physics Physics Physics Physics Physics Work, Potential Energy, and Kinetic Energy Potential energy Kinetic energy Joule Energy is the ability to do work Energy can be converted into different forms Time Required Warm-up Question Presentation First, you need to make the transition from work to potential energy and kinetic energy. In the work activity we demonstrated that the work to raise an object is the same (ignoring friction) whether you lift it straight up or take a longer path with less force. Fd = fD. The final summary question introduced the idea that the work you put in can allow you to get work out. Dropping a bowling ball onto a soft drink can demonstrates the idea. "Stored work" is called energy. If you do work to lift an object, that object now has energy, called potential energy, or energy of position, due to its relative position compared to something else. The potential energy is equal to the work done on the object. The Fd to lift it is equal to its weight times its height. Weight is a force (mass times gravity), and height is a distance. So potential energy is also weight times height. If you drop the object, it will fall, so that its potential energy decreases as its height decreases. At the same time, however, its speed increases. The increase in speed makes up for the decrease in height. The energy of speed, or motion, is called kinetic energy. The loss of potential energy is made up for by the increase in kinetic energy (ignoring friction). Do the activity at this point. Prior to the activity, the students now know about potential vs. kinetic energy, but they don't know the equations. The goal of the activity is to find the relationships between potential energy, kinetic energy, and work. When the activity is done, students should have found that: the work an object can do is directly proportional to its potential energy, and the work an object can do is directly proportional to the square of its velocity. They won't have been able to find the exact relationships, however, because they are hidden in some of the effects of the lab setup. Friction, of course, always gets in the way. In the case of a ball rolling down a ramp, it is only about 5%, so it's not too bad. Much worse is the effect of rotational kinetic energy (RKE) on a rolling ball. If a ball were to just slide down a ramp with no friction, all of the PE would be converted to KE. However, the ball is initially not spinning, and some of the PE has to go into making it spin, so that 769831670 dk Page 2 of 10 PPT Work, Potential Energy, and Kinetic Energy energy can't be used to accelerate the ball down the ramp. It turns out that 29% of the PE goes to RKE. See the note at the end of this section for the derivation. Now you can lead the students through the derivation of KE = mv2/2, which is shown in their reading. This could be done as a group exercise. One of the exercises asks the students to compare their lab data using KE = mv2/2 and KE = 0.71PE to see if it works out. It should. An interesting thing to note is that Ft is related to the change in momentum. Fd, on the other hand is related to the change in energy. (This is related to acceleration, and dt2.) Work and PE are straightforward - they're both Fd. KE is not quite so obvious. The square of velocity is significant. If you double the mass of an object, its PE and KE will both double. On the other hand, if you double its velocity, its energy will quadruple. The students have probably all heard of Einstein's famous equation, E = mc2. This would be a good time to point out the similarity to the formula for KE. Einstein's equation says energy equals mass the velocity of light squared, just like KE is mass velocity squared/2. KE is divided by 2 because it's the average. The speed of light is always the same, so it is not averaged. Energy can take many forms. A good class discussion is to try to list as many as possible: potential, kinetic, chemical, heat, light, nuclear, gravitational, sound, wind, solar, wave, gas, batteries, Work transforms energy. Energy gives you the ability to do work. It's a circular definition. But it can't go forever because friction gets in the way. Note on derivation of rotational kinetic energy (RKE) vs. kinetic energy (KE). You don't need to know this derivation, but it's here for reference. The derivation is based on rotational inertia and velocity, I2. We've used revolutions in this course, but if you use radians in the derivation, it works out very nicely. We want to know how KE and RKE are related. Just like KE = mv2/2, RKE = I2/2. For a sphere, the rotational inertia is I = 2/5(mr2). Substitute for I, RKE = 2/5(mr2)2/2 Linear and rotational velocity are related by v = r, or = v/r. Substitue , RKE = 2/5(mr2)(v/r)2/2 Re-arrange, RKE = 2/5(mr2v2/r2)/2 2 The r s cancel, and RKE = 2/5(mv2)/2 Since (mv2)/2 = KE, RKE = 2/5KE = 0.4KE In summary, for a sphere rolling down a ramp (and ignoring friction), PE = KE + RKE (+ friction) 769831670 dk Page 3 of 10 PPT Work, Potential Energy, and Kinetic Energy PE = KE + 0.4 KE PE = 1.4 KE 0.71PE = KE The translational linear kinetic energy is only 71% of the potential energy. 29% goes to rotational kinetic energy. And a little bit more is lost to friction, 5% by experiment. Here is an alternate derivation of kinetic energy Remember that d = (at2)/2. Substitute (at2)/2 for d. Fd = ma(at2)/2 Fd = ma2t2/2 Fd = m(at)(at)/2 Remember that a = ∆v/t, so ∆v = at. Since we're starting from rest, ∆v is the final speed, which we can then call simply v. Substitute v for at. Fd = mv2/2 Assessment Writing Prompts Relevance Answers to Exercises Answers to Challenge/ extension 769831670 dk Page 4 of 10 PPT Work, Potential Energy, and Kinetic Energy Background In the last activity you saw that the work done on an object is the same whether you lift it straight up or push it up a ramp. You've been told that the raised object has potential energy and you've seen that it can do work, like a bowling ball crushing a can. Finally, you've been told that as an object falls, it gains speed, and the potential energy turns to kinetic energy. Problem What is the relationship between work, potential energy, and kinetic energy? Materials 1 steel ball bearing about 2.5 cm diameter (or pool ball, or croquet ball) 1 track about 1 m long 1 prop for the track, about 10 cm high 1 smooth surface (press board) about 30 cm long, and props to raise it even with the end of the track 1 meter stick 1 metric ruler 1 stop watch 1 paper cup, cut out and flattened on one side as shown Side view End view Procedure Set up the equipment as shown. Ensure that the lower end of the track is just above the lip of the cup so there is a minimal drop-off and no bump. 769831670 dk Page 1 of 10 PPT Work, Potential Energy, and Kinetic Energy 1) Roll the ball down the ramp and into the cup, and let the cup and ball slide to a stop. Start the ball from five different distances, 0.100 m, 0.200 m, 0.400 m, 0.600 m, 0.800 m, and 1.000 m. Start the ball so that its center is on the starting distance. Align with start distance Down 2) For each distance, record a) the distance the ball rolled, b) the vertical distance the ball descended (the height- be careful and think carefully about the actual height the ball descends), c) how far the cup and ball slid, and d) the time it took for the ball to roll down the ramp. You don't need to record the time at the same time you are measuring sliding distance. Summary Find how potential energy, kinetic energy, and work are related. The height (vertical distance) the ball descended represents potential energy. The velocity of the ball represents kinetic energy. The sliding distance represents work. 1) Make three graphs. a) Graph the vertical distance the ball descended vs. the distance the cup and ball slid. b) Graph the vertical distance the ball descended vs. the final velocity of the ball. c) Graph the final velocity of the ball vs. the distance the cup and ball slid. 2) Manipulate the numbers and graph until you find the relationships. You don't need to find exact numbers. Use the terms direct, inverse, linear, and exponential. a) How are potential energy and work related? b) How are potential energy and kinetic energy related? c) How are kinetic energy and work related? d) How are work and velocity related? 769831670 dk Page 2 of 10 PPT Work, Potential Energy, and Kinetic Energy Reading What do we call the ability to do work? We could call it stored work. Instead we use the word energy. Work and energy are two sides of the same coin. It takes energy to do work. Doing work on an object changes its energy. If you want to pound a stake into the ground by dropping a big rock onto it, you have to pick the rock up first. You do work on the rock to lift it, and the rock can then do work on the stake. If the rock were left on the ground, it couldn't do much pounding. When you lift the rock, you increase its energy, giving it the ability to do work. The energy it has when it has been lifted is called energy of position, or potential energy. The higher the rock is above the stake, the more potential energy it has - it can pound the stake harder. But when you drop the rock, it falls and, the closer it gets to the stake, the lower it is, and the less potential energy it has. Where does the potential energy go? The rock gains speed. The potential energy changes into energy of motion, or kinetic energy. What the rock loses in potential energy, it gains in kinetic energy. For example, a pendulum hanging straight down has neither potential nor kinetic energy - it is at its lowest energy position. But if you put some work into the pendulum and raise it, it gains potential energy. When you let it go, it loses height but gains speed until it reaches the bottom, then it loses speed but gains height up the other side. Potential energy changes to kinetic energy, then potential energy, then kinetic energy… back and forth. Work can create potential energy. Potential energy and kinetic energy are interchangeable and can do work. It makes sense that they are all measured in the same units, Joules. Work is Fd. Potential energy is weight times height, but weight is a force and height is a distance, so it is also Fd. Since a Joule is Fd, in SI base units it is (kgm/s2)m, or kgm2/s2. Kinetic energy is also measured in Joules, of course. Look at the units for Joules - kgm2/s2. Those can be re-arranged as kg(m/s)(m/s). In symbols they are mvv, or mv2, mass times velocity squared. The units are right, but there's a little more to it. Here's the derivation, starting with work and potential energy. Begin with Newton: Start with F = ma Multiply both sides by d. Fd = mad (You can remember it as “work makes me mad.”) a = vf/t, substitute Fd = mdvf/t Rearrange Fd = mvfd/t d/t = vav, substitute Fd = mvfvav vav = vf/2, substitute Fd = mvfvf/2 Rearrange Fd = mvf2/2 vf is the final speed. We can call vf simply v. Fd = mv2/2 So kinetic energy = mv2/2. This is a nice equation, because acceleration is taken out of it - all you need to know is an object's mass and velocity. If you get hit in the head with a coconut, it doesn't matter whether it fell from a tree or was thrown at the same speed; it will hurt just as much either way. 769831670 dk Page 3 of 10 PPT Work, Potential Energy, and Kinetic Energy Sometimes it seems like you can do some work to create energy, then the energy can do an equal amount of work, which creates energy, which can do work, on and on. But don't forget our old friend friction! Since nothing operates without friction, there is always loss, and you can't create a machine that goes on forever - a perpetual motion machine. All of the rock's energy doesn't go into the stake - some is lost to air resistance, sound, and heat. 769831670 dk Page 4 of 10 PPT Work, Potential Energy, and Kinetic Energy Exercises 1) If you do work on an object, does its energy change? 2) How can you increase the potential energy of an object? 3) What happens to your potential energy as you go up in an airplane? Compared to what? 4) If an object weighs 500 N and you lift it 2 m, a) how much work have you done on it? b) how much potential energy does it have relative to its starting point? 5) If an object has 100 J of PE, how much work can it do (with that PE)? Ignore friction. 6) How much energy does a 2 kg ball sitting on a 1m high table have? 7) If a 1,000 N rock falls 10 m, how much work will it do when it hits? 8) What is the PE (relative to the moon's surface) of a 1 kg rock that is 1 m above the moon's surface? The acceleration of gravity on the moon is 1.6 m/s2. 9) How can you increase the kinetic energy of an object? 10) What happens to your kinetic energy as you up in an airplane? Compared to what? 11) If a car is traveling at a steady speed, but you double its mass, a) what happens to its momentum? b) what happens to its kinetic energy? 12) If you double a car’s speed, a) what happens to its momentum? b) what happens to its kinetic energy? 13) Two cars are traveling with the same momentum – one is light and fast, and the other is heavy and slow. Assuming you apply the same braking force to each, a) which takes more time stop? b) which takes more distance to stop? 14) When you throw a ball thrown up in the air, when is its kinetic energy greatest? When is its potential energy greatest? 15) Draw a picture of a 20 N rock falling from a 100 m high cliff. Give its PE and KE when it is 100 m high, 75 m, 50 m, 25 m, and 0 m. Ignore air resistance. 16) Galileo (supposedly) dropped two balls from the leaning tower of Pisa. If the bigger ball had twice the mass of the smaller one, what was its KE compared to the smaller one just before they hit? 17) By what factor does the KE of a small plane change if it cuts its speed in half? 18) Static friction (the amount of friction before something starts sliding) is greater than sliding friction, and sliding friction stays the same regardless of speed. That's one reason why braking your car so hard that you skid is a bad idea, and it's why anti-lock brakes are a good safety feature. So, if you skid your tires, how much farther does it take you to stop if you're going 50 mph as opposed to 25 mph? 19) Two identical rocks are dropped from different heights. The higher one starts out four times as high as the lower one. How much faster is the higher one going just before they hit? Ignore air resistance. 20) A 1 kg ball on a simple pendulum is lifted so that it is 1 m above the bottom point of the pendulum. What is its PE compared to the bottom point? What will its velocity be at the bottom point? 21) In the activity, we didn't compare potential energy, kinetic energy and work directly. That's because some of the PE is lost as the ball goes down the ramp. About 5% of the PE is lost to friction. Making the ball spin (rotational kinetic energy) takes up another 29% of the PE, 769831670 dk Page 5 of 10 PPT Work, Potential Energy, and Kinetic Energy making 34% total. Check your data. Compare the PE of the ball at the starting position, less 34%, with the KE of the ball. (Hint: vf = 2vav) 22) In the diagram, the ball's velocity is 2 m/s, and the height of the ramp is 0.2 m. Will the ball make it to the top of the ramp? Use 9.8 for g. Ignore friction. 23) For a ball thrown straight up in the air on Earth, if you are given the ball's KE just as it leaves the hand, can you calculate how high the ball will go? Derive and show one formula that will give you the answer. Ignore air resistance. d = ? 24) If a 1 kg ball is thrown straight down on Earth with a KE of 50 J, what will its velocity be after 5 s? Ignore air resistance. Challenge/ extension Glossary Joule - A Joule is the unit for work and energy. It is abbreviated J. The SI base units for Joules are kgm2/s2. Energy - Energy is the ability to do work. Potential energy - Potential energy is energy of position. It is Fd, measured in Joules. Most commonly potential energy is weightheight. Kinetic energy - Kinetic energy is energy of motion. It is (mv2)/2, measured in Joules. 769831670 dk Page 6 of 10
