 Hello, Physics20s. Today we're taking a look at the analysis for the trio of energy labs that you complete in unit C of Physics20. We're going to take a look at the conservation of energy in the hot wheels, conservation of energy in the mass spring system on the air track, as well as the conservation of energy in a pendulum. And we'll look at all three of the energy labs. I'll go through the whole lab. My emphasis is going to be on doing the analysis in each one. You can also find keys for the labs posted to Moodle that you can check out as well. Let's start with the hot wheels conservation of energy lab. In this lab, and in fact in every lab, our hypothesis is more or less the same. What we're doing is comparing the theoretical speed, and in this case it's the theoretical speed of the car, to the experimental speed of the car. So we're going to measure these experimental speeds with photo gates, and we actually measured two in this lab. We measured a speed at the top of the loop. We call that one photo gate one, and we also measure the speed at the bottom of the loop, and we call that photo gate two. So those are our experimental speeds. We have two of them in this lab. We're going to compare those to the theoretical speeds that we calculate using our conservation of energy calculations. And those calculations are going to be just like the ones you did in class and in your notes. If those speeds are really, really similar or the same, that means it's an isolated system, and that energy was conserved, and if they're different, we'll have to explain why by looking at some sources of error. Let's take a quick look at our variables that we have for this lab. The mass of the car does not change during the lab. That is a controlled variable, as well as what I'm calling height two, which is just the height of that roller coaster loop, and that's going to stay the same in the lab as well. You'll get to manipulate height one, which is the height that you drop the car from. And in the lab, you actually have to decide how high to drop it from so the car will go through the loop properly. And then the responding variable, depending on what height you drop the car from, is the speed of the car. And again, there's two of those. There's one at the top of the loop, and there's one at the bottom of the loop. The data that I have here today is data that I've taken from the back of the lab manual. And if you look in the back of your lab manual, if you're away today, you're going to see what's called the sample data. And you can use this anytime you're away from class. So there's the sample data from the back of the lab manual. And you can look that up right now, and you can copy it out into your notebook. I didn't record multiple trials. I just recorded the average trials for the two speeds. But if you were here in class, you'd have had four different speeds that you averaged out for each of the trials at the top of the loop. That's photo gate one. And at the bottom of the loop, that's photo gate two. We also had the height one. This is our height one here. This is height two. And we don't know what the mass of the car is. So that's one of the things we'll have to deal with in the analysis. So let's take a look here at that analysis now. The first thing that we want to do is draw and label a diagram showing the system. And there's a diagram in the previous page, so we'll flip back to that. So let's think about what types of energy we have at each of these different positions. So I'm going to start off by thinking about the sort of initial position. I'll label that with an I for initial, where I start the car. And at that moment in time, the car only has gravitational potential energy. Because initially I hadn't released the car yet, so it was at rest, doesn't have kinetic energy because it's not moving, but it has gravitational potential energy because it has height. Next thing I'm going to do is label the energy I have at what I'm going to call the final position. I'm going to start off by finding the speed of the photogate one at the top of this loop. And there's going to be two types of energy here. There's going to be kinetic energy because the car is moving. It has some sort of speed. And there's also going to be some gravitational potential energy because the car also has height. That's what we're calling height two. Now, when I do laws of conservation of energy calculations, I need to make these two amounts of energy or these two types of energy at the initial and final position equal to each other. So here's what that calculation looks like. There's going to be two calculations I'm going to do. This is the first one. So this is for the top of the loop. The top of the loop, the sum of the initial energy equals the sum of the final energy. So I start with that. That's called my conservation of energy statement. My initial energy was all gravitational potential energy, EP. And in the final situation when the car was at the top of the loop, there were two types of energy, EP and EK. So there's all of my types of energy substituted in. The next thing I'm going to do is I'm going to replace the E's with the formulas. So in place of gravitational potential energy, MGH. And I'll put that on either side. And then in place of kinetic energy, one half MV squared. Because there's a mass in every single term, so there's a mass here in the first term, a mass in the second term, and a mass in the third term, I can divide both sides of this equation by M and that cancels out the masses in every single term. Which is one of the reasons why you do not need to know the mass of the car for this calculation. I'm going to substitute in the acceleration due to gravity. I'm going to substitute in the height and I need to make sure this is the initial height since I'm on the initial side of the equation. So for me, that was 0.845 meters. You'll probably have a different initial height and that's just fine. Then on the right hand side of the equation, the acceleration due to gravity again. My second height, this is the height at the top of the loop and then one half V squared. So here's my substitution in. Notice I'm not putting the negative sign in on the acceleration due to gravity since I'm dealing with scalars here. Energy is a scalar so we don't put the negatives in. Here's how I type that into my calculator. So I'd like to start off by just working out what the left hand side is. 9.81 times 0.845. That gives my energy that I would have on the right left hand side. Then I'm going to subtract what 9.81 multiplied by 0.29 is. So I have to subtract to move that to the other side. So on the left hand side, I have about 6.12 and on the right hand side, I have one half V squared. So to get rid of the one half, I'll take the 6.12 and divide it by one half or divide it by 0.5 and then I have to make sure I square root in the last step to get V by itself. Now in my measurements, I had three significant digits so I can record my answer to three sig digs and it'll work out to 3.50 meters per second. So that's how fast the car should have been going at the top of the loop. That's its theoretical speed. And you can see that my theoretical speed at the top or my experimental speed at the top of the loop was much, much, much slower. So this is not an isolated system. Energy was not conserved. The car should have been going at 3.5 meters per second and it was going at a little under 1 meter per second. So we'll have to discuss why that wasn't a moment. Now we're going to do the second calculation to figure out how fast the car should have been going at the bottom of the loop. So I'll do bottom of loop now. It's going to look really similar. I'm still going to do a sum of the initial energy equals sum of the final energy calculation. My initial energy is still the gravitational energy that I had in my diagram when I first released the car. So that's right here. This is my initial situation still. But now I'm going to change what my final situation is or my final point because I want to know the speed going through photo gate 2. So I'm now going to call that the final. The only kind of energy the car has at this final position is kinetic energy because now the car doesn't have any height above the reference level. So this is a nice easy calculation. I can do gravitational potential E p equals kinetic E k. So all of the final energy was just kinetic energy. And this is a calculation you've done lots of times before. So I'm not going to spend lots of time discussing it. The mass is still cancel. The height that I started off with my initial height is still 0.845 meters. And the algebra is pretty similar to the last one except there's not the subtraction step. And I'm getting that this car at the bottom of the loop to three sig digs should be going at 4.07 meters per second. Which again, if you compare it, that's my theoretical. And if you can compare it to the experimental, here's the experimental. Yeah, there's a lot of speed lost, which means there is a lot of energy lost. This is not an isolated system. There are non-conservative forces from outside of the system that are doing work on the system, causing that speed to be much lower because there's not as much kinetic energy as there should be. So in your evaluation, you're always going to sort of summarize what happened in the lab. And I've got a little summary here that you can take a look at on the key on Moodle. But then you also need to be able to explain a little bit about what was happening in terms of the experimental and theoretical speeds. And here I'm stating that they're not the same. They're very different. The experimental was much smaller than theoretical. And the reason for that, that you need to have in your evaluation, is because energy was not conserved. That's why those speeds are not the same. Why was energy not conserved? Well, because there are non-conservative forces acting. And the major non-conservative force I would expect you to know about in this lab is the force of friction. There is a force of friction between the wheels and the track. And that is going to remove energy from the system. It's going to do work on the system. So that would be the major error I would want you to identify. And you may have other errors as well you include, which is just fine. But that's the one I would want everyone to know. So what's an improvement you can have to get rid of some of that friction? It's not exactly easy. You can sometimes find better or worse Hot Wheels cars that seem to lose less energy to friction. But one of the things you can do that's very practical is you can think about where you want to put photo gate two. So if I look at the diagram, in the lab you had an option of where to put photo gate two. You could put it pretty close to the bottom of the loop, or you could put it way further out to the right here. So the car had to go flat along the table for a few moments before I went through the photo gate. In order to reduce error, you want to have photo gate two as close to the bottom of the loop as possible. That gives friction less time to act and it will remove less energy from the system. If photo gate two is a really long ways away from the bottom of the loop, that gives more time for friction to act and more kinetic energy gets taken out of the system. So your experimental speed and theoretical speeds are even further apart. So in this first lab we saw there was not much conservation of energy. It was a very non isolated system because of the outside force of friction. The second lab that we did was dealing with a mass and a spring on an air track. Very similar hypothesis we want to compare the experimental speed and the theoretical speed if they're the same energies conserved. In terms of our variables in this lab, the elastic constant, which is called K, sometimes we call that the spring constant stays the same for the lab. So that's a controlled variable as is the mass of the cart. You get to manipulate X in this lab. You decided how far to stretch the spring out. And then as a result of stretching the spring, the spring applied a force, which is called FS, force in the spring. That's your responding variable. The procedure if you weren't here for the lab was really pretty simple. You're going to pull the spring back by a certain value of X displacement, you're going to measure the force in the spring at that point using a spring scale. So that's going to generate a bunch of values of X displacements and forces for you that you recorded in a data table. You did this five times for five different values of X. Once you had that data down, you then pulled the cart all the way back to your maximum value of X, and you released it. And then that cart would go and zoom back to its equilibrium position with lots of speed. And you would measure the speed at equilibrium by putting a little photo gate here. Alright, so we'd have a photo gate right here that allowed us to measure how fast the cart was going at that point in time. Let's take a look at our diagram and label the types of energy that we had to deal with here. So when the cart was pulled all the way back in the initial position, cart wasn't moving at that moment in time. So we would say that the initial speed was zero. And in this situation, all of the energy is elastic potential energy. So this is the energy that stored in the spring or some sort of elastic object. When the cart is released and the spring comes back towards equilibrium, then it will have some sort of final speed, which means that all the energy will turn into kinetic energy. So in our evaluation, or pardon me, in our analysis, we're going to be doing an EP equals EK calculation. So let's take a look at the data. This is again data from the back of the lab manual. So it might look different from the data you collected in class, which is fine. But if you weren't here in class, I want you to flip to the back of the lab manual now where you can get this data. So here are all my displacements. These are my values of X. And here's all of my forces that the spring applied. And I graph those on a force versus displacement graph. This is very similar to what you've seen in class and in your notes. I also recorded down the mass the cart was 0.3 kilograms of the cart I used. And the final speed of the cart when I released it was 2.8 meters per second. I released the car from 0.5 meters. So that's how far it moved. That's how much the spring was stretched when I measured the speed of 2.8 meters per second. So we're now going to go through and do our analysis. We're going to start off in the same sort of way as we did the last analysis. We're going to say that the sum of the initial energy equals the sum of the final energy. This is the conservation of energy statement. And that's how you start every conservation of energy problem. Initially, all of the energy was gravitational pardon me elastic potential energy, which we call EP. Afterwards, in the final situation, all of the energy had turned into kinetic energy, EK. We know how to calculate EK. That's the formula one half MV squared. There's the theoretical speed that V is what I want to solve for in this question. And I know what the mass of the cart is. So the right hand side is looking good. Now for the left hand side, there's two ways you can do this. One way you can do this is you can find the area under the graph. And so here's my graph makes a triangle. Well, it makes pretty close to a triangle. If I was to extend this line, a best fit just a bit. There we go. So there's the triangle it would make. And one method you can use for finding the energy. When you've got a force versus displacement graph is you can find the area of the triangle. So if you want one way to do this analysis is to find the area. And that will equal the energy. And it's easy to find the area of a triangle, you just do one half base times height. Now you might recognize there's a little bit of a problem with that, though, a little bit of error if you use that method. And it's still okay to do the method as long as you know what the error is. You might notice that right here, my triangle goes a little bit off of the graph. So that means the base if you're to measure the base here, I think my base I would say is 0.4 meters. That's not quite exactly what the base is because the base would have to be a little tiny bit longer. So that's fine. It'll mean your answer is a little bit off of what it should be, but it's still a valid way of analyzing the data in the analysis, or pardon me, in the evaluation, you would just say one of the sources of error was the triangle went a little off the graph. If you don't want to deal with that, if you want a little bit of a better result, then the other thing you can do is a slope calculation. Because the slope of this graph, if I was to take that line at best fit, and if I was to find the slope, it will be equal to the spring constant or the elastic constant. The reason you would want the elastic constant is because the elastic potential energy in a spring is equal to one half kx squared. And I know what x is, 0.5 meters. And to find k, I'd have to get the slope of the graph. So I'm going to do the slope method here. If you do area, that's fine. It's not wrong. But I'm going to show you how to do the slope method. To do slope, I'm going to do y2 minus y1 over x2 minus x1, just like we have always done slopes. I actually have some points already plotted on my graph that I'm going to go and put into my slope calculation. Just like we did back in unit A, you want to make sure that whenever you're doing slope, you're using points off of the graph, not your plotted points. So that's what I've done here. So 2.8 Newtons was my y2 minus 1.6 Newtons. That was my y1. 0.25 meters was my x2 minus 0.125 meters. That was my x1. And you can use different points if you want. Your slope will be more or less the same. So I'll do this calculation to see what my slope gives me. So that's going to give me a slope of 9.6 Newtons per meter. The units are really important because if you look at the slope calculation, you have Newtons on the top and meters in the bottom, and that's the same units you get for the value of k for your elastic constant. So that means that this 9.6 Newtons per meter, this is k. That's a really important thing you need to know for your exams and quizzes is how to get the value of k from a graph. Now that I have that, I can substitute it into my formula. So 1.5k, which is 9.6 Newtons per meter, times x. Now the x I need to use here is the amount I pulled the mass back by, the amount I stretched the spring by when I found the speed of 2.8 meters per second because I want to compare it to 2.8 meters per second. So I had to stretch the spring by 0.5 meters to get the speed of 2.8 meters per second. So let's see what that's going to give me then as a theoretical speed. Again, your numbers might be different and that's fine. You're just going to follow along with the same steps as I'm doing with different numbers. So now I'm going to go through and do the calculation. Here's what the calculation looks like. I'm going to evaluate the left-hand side first, so 0.5 times 9.6 times 0.5 squared, don't forget to square, then I'm going to divide by the 0.5, divide by 0.3 and square root the answer. So what I get here is 2.8, it goes on and on and on. So it's just a little over 2.8, but if I was surrounded to two sig digs, I get 2.8 meters per second. So I actually get a speed which is really, really close to, not exactly the same, 0.03 meters per second off, but by the time I round it, really close to what my theoretical and experimental speed was, very similar to the same number. Now you may have got different results. You may have found that your theoretical and experimental speeds were different. That doesn't mean that you've done the calculation wrong or even that you did the measurements wrong. Maybe on that particular day with the setup you were using, there was just a little bit less energy conservation and we could talk about why that was. So this is closer to being an isolated system. In fact, I'd say this one in my data does look like an isolated system because it does not seem like there's any energy lost. So there weren't any outside forces doing work on the system. So energy is conserved. But I know lots of times when students do this lab, they don't get exactly the same theoretical and experimental speed. So what could be errors? One possible error is that there still could be some friction against the track. Those air tracks aren't perfect. There might have been a bit of friction still. So we can also think about only pulling that spring back through a small distance, again, so that friction doesn't have as much time to act on the cart, and that'll reduce the amount of energy that's lost to the system. Another source of error and improvement you can do is checking to see if the air track is level. If the air track is tilted a little bit, then if the cart is going downhill, it'll pick up some extra kinetic energy due to the gravitational potential energy it had from the tilt. So leveling the air track is also a good way to make an improvement if your results for your theoretical and experimental speeds weren't the same. The third lab that we did was the conservation of energy in the pendulum. And again, you've got about the same hypothesis here. We're comparing experimental and theoretical speeds. If they're the same, conservation energy worked out. If they aren't, it didn't. It's non isolated. Controlled variables, we control the mass of the pendulum, and we control what I'm calling height one, which is the height from the floor that the pendulum starts off with at equilibrium. You can manipulate height two, you can decide how high up from the ground you want to release the pendulum, and you're going to measure the speed at equilibrium that the pendulum moves through. I'm going to do my energy labeling again. So at my initial situation, when I first release the pendulum bob, it has just gravitational potential energy because it has height in the air. It doesn't have any kinetic energy though because it hasn't started moving yet. When the pendulum bob gets to height one at equilibrium, that's our final situation, it'll have some gravitational potential energy because it's still got height above the ground, but it'll also have kinetic energy because it's moving. So I'm going to do an EP equals EP plus EK calculation here. Same as before, data in the back of the lab manual for you to go and use if you were not here. And I'm going to go nice and quick here through this analysis, since this is the third time you've seen a conservation of energy calculation. Again, it's the same calculation as you saw in your notes. So if you do get stuck with these and you're looking for a little bit more detail on how to solve them, go and look in your notebook because you've done the same calculations there. I also have standalone videos for how to solve a pendulum problem like this that you can also watch. I think one of the things you have to be most careful of when you're doing these calculations is just make sure you've got the right height in the right spot. The left-hand side of the equation is always your initial, and so my initial height was 0.216. Oh, sorry, my initial was not that. My initial was here. Oops, you can't see it. That was when I was at height 2. I've got to be careful. So my initial was actually height 2, 0.223 meters. I almost made the mistake that I was describing. So that's the first height it started off with, the tallest height. And then the second height here, the final height, is where it ended up at equilibrium. There we go. The algebra is exactly the same as the last one we did. So I'm going to type through my calculator in the same way. I'm going 9.81 times 0.223, 9.81 times 0.223. I'm going to subtract 9.81 times 0.216 like so. I'm going to divide by 0.5 and I'm going to square root. So I get that the speed of the pendulum bob, the theoretical speed, how fast it should have been going, was 0.3, actually it works up to 0.3705, but to two sig-digs, actually I was measuring three sig-digs, pardon me, three sig-digs. I'm getting 0.371 meters per second, which is really close to my average velocity that I got from the photogate of about 0.370 meters per second. So almost exactly the same. So in the evaluation, I would again say this is a good example of a situation where the model of law of conservation of energy applies. We have a very similar, in fact almost exactly the same, experimental speed and theoretical speed, but you again may have found some differences. So what could have those differences have been? What are some sources of error you might want to talk about? Sometimes students have trouble measuring when they repeat the calculation or when they repeat the height measurements because you want to do more than one trial. Sometimes they have a trouble getting the same height every time, and an easy fix for making sure that your height 2 is repeatable is just to use a measuring tool like a ruler or even like a textbook and just always drop it from the top of that same height. Instead of eyeballing it and saying I think it was about here, just measure it every single time, you can also think a little bit about the fact that we're only accounting for sort of the horizontal and vertical motion of this pendulum, not its side-to-side, like third dimension motion. So sometimes when you release the pendulum it doesn't just move horizontally and vertically, it also goes kind of sideways and that's a bit of extra energy that we weren't accounting for. So if you repeat the trials, if you notice the pendulum is going kind of tilted to the side as well, that'll also give you better data that should give you closer to the conservation of energy. In all three of these labs, when you stop and think about all of them, you should be able to identify which ones are best modeled as an isolated or non-isolated system and that's one of the main ideas you're going to do on your lab quiz based on these labs.