The stage is set. We started building the energy transfer model (ETM), and we’ve talked about the flavors of energy. We are ready for a new representation to help us start thinking about energy storage in a system. In a day or two, we’ll be using energy bar charts, but first, we’ll get used to thinking about energy storage with a simpler, stepping stone diagram.
I will run through the typical way I introduce the diagram in class in just a minute. First, here’s the teacher preview of some key energy pie chart features.
Before doing anything else, list the objects in your system. The diagram will only make sense in the context of your chosen system, so you need to be explicit about that. Teacher note: if you want students to pick up the habit of defining a system first, you need to be very careful to always do it that way yourself.
The size of the pie matters. The size of the pie determines the total amount of energy stored in the system. So, if the pie becomes larger, there is now more energy stored in the system. If the pie becomes smaller, there is now less energy stored in the system.
The size of a slice doesn’t matter on its own. Whether the slice has grown, shrunk, or stayed the same between snapshots is what matters. Students don’t have energy equations yet. They know what each flavor of energy depends on (per the flavors of energy discussion immediately before this activity), so they should have ideas like, for example, kinetic energy will be bigger if the object is moving faster. They don’t have a way of reasoning, yet, about whether the kinetic energy should take up half of the pie or a quarter of the pie at any particular snapshot. We’re looking to show how the energy storage changes, not what it necessarily is. Energy is boring. Change in energy is interesting.
For the situations we’re going to encounter, represent, and analyze in this class, we would be surprised to see energy being stored as Etherm in one snapshot, then stored as Kinetic or Gravitational energy in a subsequent snapshot. For our purposes, when there is a ∆Etherm, the energy is essentially “stuck” that way. We’ll never get to use that slice of the pie again. That slice of the pie is dead to us. It might get larger, but it will never get smaller. Since it’s a dead part of the pie, instead of writing ∆Etherm over and over again, we can just shade it in. We’ll all know what that means.
Introduction of diagram
Right. Ready for a new representation to add to our arsenal? When’s the last time you drew a pie chart? Third grade? (Students: Yep, that sounds about right. Back with the bar charts.) Okay, but these pie charts are going to be a little more sophisticated than what you were drawing back in the day.
Just inside the packet there’s a page with some energy pie chart practice. We’re going to do the first one together. Actually, we’re going to do the first one together twice. After that, I think you’ll get the idea, and I’ll let you argue the rest of them out without me.
So the problem is a ball being dropped to the ground, right? I think it looks like this? [I'm drawing the diagram on the board, but having done this problem at least 15 or 20 times by now, I don't have the packet in my hands.] And it has a sort of motion map next to it, right? The arrows are supposed to be showing how fast the ball is moving at that snapshot. The dot means it isn’t moving.
The first thing we need to do is define what we’re saying is in our system. Let’s put all the objects that store energy into our system. We could put the entire universe in our system, but that seems a little overkill. Can we narrow it down to just the relevant objects for this problem? (Likely answers: The ball. The ground. The air.) I think you’re missing something. Something really big. (Oh, the Earth.) Okay. Does the ground store energy? (Yes. Wait, it doesn’t hit the ground. So no?) So we probably don’t need the ground. It won’t change anything to include it, if it doesn’t store energy, but we don’t really need it. Why does the air store energy? (Air resistance.) Okay, so question—from how high above the ground is this ball being dropped? (It doesn’t say.) Let’s do it twice, then. Let’s do it the first time with it being dropped in this room. Then we can come back to it a second time and consider it being dropped from the Sears Tower. (I don’t think it’s called that anymore.) Okay, the second time we’ll do it being dropped from the Sears Tower in 1992, when I know for sure it was still called the Sears Tower. When we are dropping it in this room, what do you think about the air resistance? Will it be significant? (Probably not.) So we don’t need the air in our system right now, but we will need it on the second time through.
Okay. We’ve got our system now! Ball and Earth. There are three snapshots shown in the picture, so we’ll draw three pies. We put all of the objects that would store energy into our system, so the total energy should stay the same. In the future, we could have the total energy changing, but while we are practicing this diagram for the first time, let’s try to keep it so the total energy stays the same. So we don’t have to keep track of too many new things all at once. So if the total energy is the same, then the size of the pie should always be the same.
Everyone’s favorite—drawing circles! We need to draw three circles that are all about the same size. [I start drawing my totally inadequate circles on the board. The flavors of energy chart is still up on the board next to the circles, for easy reference for the next part.]
Oh! Big warning! Do NOT write this next part down yet! Trust me, you’ll just get really angry with me if you write it down now. You’ll be able to write it down at the end.
Alright! Finally! Now, it’s time to play the Energy Pie Chart Game! Who wants to go first? [Choose an arbitrary kid who is excited to play the "game" even though they don't know what it is yet. Let's call the first kid Brandon.]
Great! Brandon, what’s your favorite flavor of energy, besides thermal? [You want to do thermal energy last. Trust me. You'll see in a minute.] (Uh, gravitational interaction energy, I guess?) Super! In the first snapshot, is there any energy being stored as Ug? (Yes.) [I section off a corner of the pie and label it Ug on the board.]
Okay, second snapshot. Does the amount of energy stored as Ug get bigger, smaller, or stay the same? (It’s smaller.) [I section off a smaller chunk of the pie for Ug in the second pie.]
Final snapshot. Does the amount of energy stored as Ug get bigger, smaller, or stay the same? (Smaller.) [I start sectioning off a still smaller slice for the third pie. If the class (or Brandon himself) hasn't steered us off toward needing a y = 0 m line yet, it usually happens at this point as some think there should be no Ug in the third pie, but others think there should. At whatever point seems reasonable, there's a discussion of needing to agree where Ug will be zero and the drawing of a y = 0 m line on the picture. Most classes decide that the ball is above the ground, that they want the y = 0 m line on the ground, and that there is therefore still some Ug left at the end. It really doesn't matter what they decide as long as it is consistent and makes sense to them. It's probably not a good idea to put the y = 0 m line above the object because, while a negative amount of Ug is fine, it's really tough to draw a negative slice of a pie (and hurts your brain a bit to even think about).]
Thank you for playing! who wants to play next? Zoe? Great! What’s your favorite flavor of energy (not thermal)? (Okay, spring energy.) Fantastic! In the first pie, is there any energy stored as spring interaction energy? (No.) Second pie: does the amount of energy stored as Us get bigger, smaller, or stay the same? (There’s still none.) Third pie: does the amount of energy stored as Us get bigger, smaller, or stay the same? (None again.) Excellent, thank you for playing!
Next! Kirstin. What’s your favorite flavor of energy (not thermal)? (I guess it must be kinetic energy, then.) A fine choice. Okay, first pie. Is there any energy stored as kinetic energy? (Well, it’s not moving, so no.)
Okay, second pie. Does the amount of energy stored as kinetic energy increase, decrease, or stay the same? (It increases.) [I add a section of Kinetic energy to the second pie. It doesn't matter how much. Students might get concerned about how big the kinetic slice should be compared to the Ug slice, but they have no way of making that comparison yet, so it really doesn't matter. It just matters that the K slice has gotten larger and the Ug slice has gotten smaller since the first pie.]
Third pie. Does the amount of energy stored as kinetic increase, decrease, or stay the same? (Increase.) [I draw a larger slice of K in the final pie.]
Thank you so much for playing! Next! Victor. What’s your favorite flavor of energy? (That would be change in thermal energy.) Aces.
Is there any interaction that would cause energy to be stored as thermal energy between the first pie and the first pie? (No. Wait, what?) Remember that we’re always talking about ∆Etherm, not Etherm itself, so we will always have to keep comparing the current snapshot back to the first snapshot. (Aha.) So, could there ever be ∆Etherm in the first pie? (No.) Sweet.
Okay, second pie. Has there been any interaction that would cause energy to be stored as thermal energy between the first pie and the second pie? (Well, no.) How about between the second pie and the third pie? (Does it hit the ground? Oh, it doesn’t. So, no.) Alright! Thanks for playing!
So that’s it, right? We’re finished? [Look at the board and see the weird-looking pies with blank slices.] Okay, there’s clearly something weird going on. When we started, we didn’t know how much of the pie each slice would take, so we didn’t end up using the whole pie. Is there anything outside the system that would take energy out or put energy in? (No, we put everything that mattered in the system.) Okay, so the total energy must stay the same. So we just need to redistribute the slices so that we take up the whole pie each time.
And now you see why I told you that you probably didn’t want to write this down while we did it, right? [At which point, the kid who didn't listen to that is grumbling while reaching for an eraser, but also can only really blame himself.]
Okay, we just need to make sure that we keep the idea of whether the slices get bigger or smaller. The first pie has only Ug, so that must be the entire pie. The second pie has Ug and K. We don’t know whether more energy is stored as Ug or as K yet, but we know that Ug is smaller than it was before. [Usually, they want to make it half and half. Even though they wouldn't agree with that in a couple of days, it really doesn't matter here, so I go with what they want.] Final pie, we know Ug has to be even smaller and the rest has to be K. [They usually want a very little slice because the ball is just barely above the ground (which is usually their y = 0 m line).]
That looks a lot better, huh?
And—we’re finished! Okay, should we try that one again?
One more time
[The second time through is much faster. They have a second space to write the new version of the problem, but I usually just modify what was on the board—that makes more sense for the class conversation, anyway. The new version of the problem is to drop the ball from much, much, much higher (the Sears Tower in 1992) so that we have to take the effects of air into account.
We quickly decide that the main thing that will be different is that we now have an interaction between snapshots that causes energy to be stored as thermal. That is, there is now some ∆Etherm. Let's join the conversation back at that point, when they've decided what is different is that there is now some ∆Etherm.]
Okay, let’s give Victor another shot at playing the pie chart game.
Victor, is there any interaction between the first pie and the first pie that would cause some energy to be stored as thermal? (Yes. No, wait, no. Not in the first pie.) Can you ever have a change in Etherm in the first pie? (No, that doesn’t make sense.)
Okay, is there any interaction between the first pie and the second pie that would cause some energy to be stored as thermal? (Yes. Air resistance.) Why does that cause a change in Etherm? (The ball is hitting the air.) [Reference back to the matter model and look at what happens to the particles when you hit the matter models together. Faster random motion. ∆Etherm.] Great, so we need to take a slice of the pie back and make it ∆Etherm. Do you think we should take it out of Ug or out of K? [Good, brief discussion usually happens here. We settle in on taking it out of K because with air resistance, it wouldn't be moving as fast as it would have without air resistance. I section off a part of the K slice, but I don't label it yet. I pause.]
Remember earlier when we were first talking about thermal energy? We said that we never see a box start suddenly sliding across a table while everything gets colder. In fact, once the energy ends up stored as thermal, it’s usually kind of stuck there. At least, in most of the simple examples we’re thinking about right now, that’s true. Right? It’s tough to get the energy that is stored as Etherm to be stored as another flavor of energy. Basically, if any slice of the pie represents ∆Etherm, that part of the pie is going to stay that way for the rest of the problem. It sort of becomes dead to us. So knowing that, it will be pretty easy to just shade in any part of the pie that represents ∆Etherm. We’ll all know that it means it’s the dead part of the pie. [So now I shade in the ∆Etherm slice instead of labeling it.]
Last one. Third pie. Is there any interaction that happens between the second and third pie that would cause even more energy to be stored as thermal energy? (Yes, it keeps hitting the air.) So is the amount of ∆Etherm bigger, smaller, or the same as the second pie? (It’s bigger.) Could it be smaller? (No, it has to stay ∆Etherm.) Right—there might be some cases where that happens, but not in the situations with simple objects that we’re looking at right now. [We decide to slice out the kinetic energy again, and I shade in the last slice. On the whiteboard, I just draw diagonal lines to "shade" the area instead of using all of that ink, but they tend to color it in when they are using pencil.]
That’s great, everyone! Do you get the general idea so far with pie charts? (Yep.) Wonderful. There are a bunch more problems on this page and the next one to get some pie chart practice and a deeper understanding of what’s going on with energy storage. My recommendation would be to work in groups and to do the problems on the whiteboards at your tables first so that you can try out your division of slices and erase to make revisions without getting frustrated about having to erase so much on the paper. You can do whatever works for you, though. Let’s work on these for a bit, then we’ll whiteboard.
Okay, you’re ready for me to stop talking now, right? [At this point, most of them probably aren't listening to me anymore anyway. They're grabbing markers or fighting out the next problem with their groups, already.]