Springs and Hooke's law
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- [Instructor] I wanna talk to you about LOL diagrams. That's right, I said LOL diagrams. These are a great ways to visualize conservation of energy, and even better, they force you to think about what's part of your energy system and what isn't part of your energy system. And if you don't know what an energy system is, maybe that's where we should start. An energy system is an object or a collection of objects whose energies were gonna keep track of, and we're gonna keep track of them in these two charts. And notice, it looks like an L, and then an O, and then an L. That's where they get their name from. This isn't laugh out loud. This is an energy chart, a circle where we're gonna define our system, and then another chart. So to understand what these LOL diagrams mean, let's just look at an actually example. Let's say you took a mass M, and you released it from rest from a height H. And this mass falls down. The first thing we should do is choose what is gonna be part of our energy system, whose energies are we gonna keep track of, and whose are we not gonna keep track of. The way people typically do this problem when they're doing conservation of energy, they would say it starts with potential energy, and then it turns into kinetic energy. And if you're doing that, what you're really doing is you're saying, "I'm taking my mass. "That's gonna be part of my system." So I'm gonna take this mass here, and I'm gonna make that part of my system. So we'll keep track of the energy of this mass, and then if you're talking about gravitational potential energy, here's the weird thing, you're also talking about the earth. That earth is part of your system. So this is the earth right here. It's got continents, and it's got California, and Mexico, and South America, and there's Florida, and then there's all kinds of stuff here. So now that we've selected our system, we can go ahead and starting charting what the energies look like. So what kind of energy was there initially in this system? There was gravitational potential energy, because this mass M started at height H above the earth. So I'm gonna represent that with this bar. This is gonna be a bar chart. Let me just go up four units. So I'm gonna say there were four units of gravitational potential energy. You might be like, "How do you know there were four? "How do you know there weren't like three, or five, "or 4.2?" Well, it doesn't really matter too much. The real value in this LOL diagram is being able to conceptualize what's happening to the energies. No matter what you draw over here, you just wanna be consistent when you then draw this diagram over here, and I'll show you what that means in a minute. So for now let's just say there were four units of gravitational potential energy to start with. Was there any kinetic energy? No, because it was released from rest. So if we drop this mass, it starts with no kinetic energy. And this US here is gonna be any spring energy. We won't have that for this example, because there aren't any springs, so there won't be any spring or elastic energy. That's it, we just had gravitational potential energy to start. And then what did that energy turn into? If this was the initial position, what kind of energy do we have when the mass gets right above the surface of the earth? Right before it's gonna hit the earth, what kind of energy is there? There's gonna be kinetic energy, so this mass is gonna have kinetic energy, and we know it's gonna have kinetic energy because it's gonna be moving right before it hits the earth, and kinetic energy is the energy an object has because of its motion. So it's gonna have kinetic energy. How much kinetic energy is it gonna have? Well, energy is gonna be conserved in our system. And since the energy is gonna be conserved, if I started with four units of gravitational potential energy, I should end with four units of kinetic energy. Now, you might be confused. You might wonder why wasn't there any gravitational potential energy. Well, I was assuming that this ground was the H equals zero line. And we know that the gravitational potential energy is mgh. And where H equals zero, the potential energy is zero. So when this mass gets down to the final point, which I'm representing in this diagram, it no longer has any potential energy since the H would be zero. And since it has no potential energy, all of this gravitational potential energy it had initially has to turn into kinetic energy in order for the energy of our system to be conserved. Now, if I were you, I might object and say, "Wait a minute. "We learned earlier "that when work is done, "energy gets transferred, "and the total energy of a object might change. "And this earth is pulling down on this mass, right? "So the earth is exerting a gravitational force downward. "The objects moving downward through some displacement, "that means the earth is doing some positive "amount of work on this mass, giving it energy. "Doesn't that mean "the energy of the system is gonna change?" And the answer is no. And the reason is this earth is also part of our energy system. So the earth did do work on the mass, and this mass gained kinetic energy. But since the earth and the mass were both part of the system, this work was internal to the system, and internal work never changes the total energy of the system. And if that was confusing, think about it this way. Let's say you and your friend, instead of the earth and the mass, there is you and your friend. And instead of energy, we'll talk about money. You, the earth, give your friend, the mass, $10. Instead of 10 joules of energy, you give him $10, and ask you, "How much total energy is there "between you and your friend?" Well, there's still the total amount. You lost $10. Your friend gained $10. But between the two of you, you still have the total amount of money you had when you started. And the same is true with energy. Whenever work is done internally, between objects in your system, there's no change in the total energy from one chart to another. That's why I had to draw the kinetic energy as four units, because no energy entered or exited our system. And something that's really useful about these LOL diagrams is that you can translate them straight into a conservation of energy equation. Because whenever initial energy that you start with, plus any external work that's done has to equal the final energy that you end with, and the reason is external work is how much energy gets transferred into the system. So if you start with 10 joules of energy, when you transfer five joules of energy into the system, you've gotta end with 15 joules of energy. So we can just plug straight into this. What do we have for our particular scenario? We started with potential energy. So our initial energy was mgh plus external work. There was no external work. There's only internal work, so this would be zero, and that's gotta equal the final energy. The only type of energy we had to end with was kinetic energy, 1/2 mv squared. And then if we had numbers, we can plug in to this formula and just solve for the height, or the speed, or whatever we wanted to solve for. So what would an example look like where energy wasn't conserved, in other words, where there was external work done on the system? Well, we could use this same example. I just need to take the earth. I wanna stick this earth outside of our system. So now our system is just the mass M. What that means is that the work done by the earth on the mass is now gonna be external work. And since there's external work done on this system, this system's energy has gotta change now. It's not gonna stay the same. You're not gonna see the energy of this system remained conserved, since something is giving it energy. The earth is giving this mass energy this time because it's doing external work. So how would that change our LOL diagram? This mass M still ends with kinetic energy. I mean just by imagining the earth as not part of the system, that doesn't change what actually happened. The mass still has to get down here with four units of kinetic energy. It's still moving just as fast as it did before. But our system, which is this mass, has to be gaining energy. And a way that's possible is that the mass does not start with any gravitational potential energy. So, people don't like this. People are like, "Wait, what? "Of course this mass starts with "gravitational potential energy. "It started up here." But technically speaking, gravitational potential energy is an energy that exists between two masses. It takes two to tango, and it takes two masses to have gravitational potential energy. If I stick one of these masses out here, it can do work on the other mass. We wouldn't say that it started with any gravitational potential energy, so this would now be zero. But we would say that there was external work done, so that's where this energy comes from. The earth is now doing how much work? Well, if the mass gained four units of kinetic energy? Then the work done had to be four units of work. So if you stick the earth outside of the system, the system no longer has gravitational potential energy. The earth is doing external work on the mass, and this might bother you. You might be like, "Wait. "But it's the same situation that's happening. "All we did was change "whether we considered the earth part of our system, "so how can that change the math?" It didn't really. I mean if you think about it, instead of the four units of energy existing initially as this potential energy and having now work, we're just saying that there was no potential energy to start, and there were for units of external work done. It's just a different story. The numbers will come out the same, and it's all gonna be consistent. But depending on whether you stick that earth outside of your system or inside of your system will determine whether you say that there was initially gravitation potential energy, or whether you say there was external work done on the mass. So let's look at another example. Let me get rid of this. Let's instead consider this example, where a mass starts compressed against the spring from rest. That'll be our initial point. And it fires the mass, and the mass goes up this incline, and comes over to here, and then it's moving with some velocity over here. We'll make this our final point. So if we wanna make a LOL diagram, we first have to pick what's part of our energy system. Because if we don't know what's part of our system, we don't even know what's gonna be eligible to have an energy in here. So again, let's make the mass part of our system. So the mass will be part of our system. Let's make the spring part of our system, so we'll put the spring in here. And let's again put the earth into our system. So what would this LOL diagram look like for this scenario? Well, since we start from rest, we don't start with any kinetic energy. And even though the earth is part of our system, which means that our system could have gravitational potential energy. I'm just gonna assume that the mass starts at H equals zero, so let's make the ground H equals zero. That way this potential energy term mgh, starts as zero, so there'd be no gravitational potential energy to start with. And the only energy we would start with is the fact that this spring is coiled up, ready to explode and fire this mass forward. All this energy is stored up as spring or elastic energy, so we will have elastic or spring energy to start with. Let's just say we started with five units of spring energy. Again, this is somewhat random. For a particular problem with numbers, you could solve for exactly how much spring energy you started with, because remember, spring energy can be found using the formula 1/2 k x squared, where x is the compression of the spring, and k is the spring constant. But to keep things simple, let's just say we start with five units of spring energy. So what would our final energies look like? If this mass is up here moving, if it's moving, we know it's got kinetic energy. So I can say there's kinetic energy. How much? Again, to figure out exactly how much, you'd have to solve for your particular problem, and know this height. But to give you an idea of what conservation of energy says, let's just say there were three units of kinetic energy that you ended with up there What are the kinds of energy do you have up here? We're gonna have gravitational potential energy because this mass starts above the H equals zero line. Since the mass is higher up in the air, it's got gravitational potential energy. So now, if I say there's three units of kinetic energy, and I started with five units of spring energy, if energy is conserved, I better end up with two units of gravitational potential energy. So I could draw two unit, and now I'd make sure that if I started with five units of energy, I've ended with a total amount of five units. That would say energy is conserved. Let's check. Is energy of our system gonna be conserved? We can determine that by just figuring out whether there were any external sources of work. Did anything external to our system do any work? If not, the total energy has got to stay the same. This five units has got to turn into five units over here. And if we assume that these surfaces here are frictionless, and there was no air resistance or no loss in energy due to dissipative forces, there will be no external work done because everything is part of our system. So, like, work was done in here, the spring did work on the mass, giving it energy. There was internal work done. But since there was no external work, the energy had to be conserved. We can write this in an equation again by saying that the total energy we start with, the spring energy, which would be 1/2 k x squared, plus any external work done, but the external work done was zero, since everything is internal, has to equal the final energy. And the final energy here is the kinetic energy, which is 1/2 m v squared, plus the potential energy, which is gonna be plus mgh. And now if you had numbers for a particular problem, you could just plug those into here, and solve for whatever variable you were looking for. But you might be wondering, "What if there is friction between the mass and the ramp? "What if there is friction over here? "What would that do?" Well, there's two ways we can handle it now. You know there's two ways. We can include these surfaces as part of our system, or we could say that those surfaces are outside of our system. If the surfaces are outside of our system, we would represent that as them taking energy away. Now there's external work done, but instead of the external work giving our system energy, the external work is taking energy away. This friction is taking energy and turning it into thermal energy. That's what this extra spot is here, for thermal energy. But since the surfaces are now not part of our system, we wouldn't show that over here. We would just include that as part of the external work done, so there would be some negative amount of work done. Maybe the surfaces did negative one unit of external work. What that would mean is that if we started with five units of energy, and there was negative one unit of external work done, we've got to only end up with four units of energy, but potential energy has got to be the same. I mean I ended up this high, so that can't really change. So that means my kinetic energy now is gonna be smaller, and that makes sense. The friction slowed this mass down. I end with four units of energy, even though is started with five, because there was negative one unit of external work done by the friction on these surfaces. That's one way to handle this problem, and that's all consistent. The other way to do it, we could just say that our surfaces, which we said were external, are now internal to our system. That way, this external work done is no longer external. This work done is now gonna be internal. The surface took the energy. It still did negative work, but since it's inside of our system, that's now gonna be part of our energy system. So how would we represent that? It'd no longer be a negative one here. We wouldn't say that this is done. We'd essentially, conceptually move that negative one over by adding one to both sides. And that means we would gain some thermal energy. I'm gonna write that as delta E thermal, because there's gonna be some gain in the thermal energy of our system. And I would represent that now not as an external work done, but as one unit of thermal energy generated during this process. So recapping, LOL diagrams are a great way to visualize what we're talking about when we say conservation of energy. And since we define our system, they're a nice way to visualize what we mean when we say that the total initial energy of our system, plus any external work done on our system has to equal the total final energy of our system.