17. Periodic Motion

Energy in Pendulums

# Energy in Pendulums

Patrick Ford

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Hey, guys, remember what we talked about? The energy of mastering systems were gonna do the same thing for pendulums and this one. So remember that just a little refresher about mastering systems. You have this object that's moving back and forth and oscillating between do different kinds of energies. We've got the elastic potential energy that's the maximum at the end points, and then you've got the kinetic energy that's maximum at the equilibrium position. But for pendulums, it's the same thing. So as you pull this thing back and you're swinging back and forth, it's got two types of energy. It's got kinetic energy, depending on how fast it's moving and because it's going up and then coming back down again. And it's changing this vertical height. It's also got gravitational potential energy, and those are the two things are gonna be looking at in this video. So as this thing is swinging back and forth, right, you're gonna take this pendulum that's length L. And you're gonna pull on it so that it's on some feta angle and you're gonna release it. But when you release it, that data is equal to its maximum value because it's just going to swing back and forth between those angles. So what is the gravitational potential look like? Well, the gravitational potential. We always have to establish, like, a zero point. And for the zero point in pendulums, it's gonna be at the bottom of the swing. So that's hte, not equal zero. So now what happens is that when you pull this thing all the way out to theta maximum now you have the maximum height right here where the height is measured relative to that zero point. So that means that the very, very top, the gravitational potential energy is maximum and the kinetic energy is equal to zero. It's not moving, so let's take a look. The equilibrium position, the equilibrium is the exact opposite. So now you're at the bottom of the swing right here at the bottom of the swing. So your gravitational potential energy, when data is zero, is just zero and then your kinetic energy is at its maximum value. So that means the total mechanical energy for these things in the first case is just mg when h is maximum. And then for this guy, it's just one half m v maximum square and then at any other point in between, I've got this point right here that api So this is state api, and the energy is just gonna be a combination of both of them. So I'm just gonna have whatever the height is in that specific point, plus whatever the kinetic energy is at that specific point, and that's it. But a lot of the times that happens in problems is we don't have what the height is. Specifically, we're actually going to see that in the example, they're gonna get Thio. So how do we figure out what that height is if we're on Lee just given l and the data angle itself? So we're really what we're looking for is if we're ever looking for the height at a specific point, and this works for any theta, we're really looking for this distance here between the zero point and where however high we are, So let's figure that out. If you've got the total entire length of the pendulum and that's equal toe l and I'm trying to figure out what this highlighted distance is, then all I need to do is just figure out what this little distance is in the triangle. So I've got what l is and I've got with data is and I'm trying to figure out what this adjacent side of this triangle is so using. So, kata, I'm gonna have that l A Times co sign of data is equal to that little vertical piece right there. So that means that this little highlighted distance is actually gonna be the difference between the length and the elk assigned data. So that means that for any theta, the height is just equal to L minus. L cosine theta. This is sometimes called the pendulum equation. And the other way you might see it is l one minus cosine theta and that's it. So let's rewrite our mechanical energy formula. So if all of these things air conserved between all these cases, that means that MGH Max is just equal to one half MV Max squared. And that's just equal to M. G. H. At a certain point. And then plus one half m V at a certain point squared. And that's basically the energy conservation formula for pendulums. So that's it. Let's get take a look at an example. So, like I said before A lot of times and problems, you won't see what the actual height of this thing is, because then you could just use normal energy conservation. Um so notice how in our energy conservation for pendulums it's exactly the same thing is what we did for energy conservation for, like, gravity and kinetic energy and stuff. Okay, so we've got this mass and it's attached to this pendulum. It's some length else, and they go and start feeling stuff in. So I've got this is l. And it's pulled open angle theta, and then it's released. So that means that that angle is already at its maximum. It's just gonna swing back and forth. That's the amplitude. And we're supposed to figure out what the maximum speed is. We're supposed to derive an expression for the max speed. So here's what I'm gonna dio. I'm gonna go up here and I'm just gonna copy down my energy conservation formula. I'm gonna put that guy like right over here. So that's my energy conservation formula. And what am I looking for? So this is equal is equal, and this is a plus sign, So I'm actually looking for what the V Max is. And what am I given? Um, I given the energy at a specific point, Or am I given the amplitude energy or the energy of the amplitude? Well, let's see when theta is equal to its maximum Now, we just have to establish what our zero potential energies are. So let me just move this over a second. And so I've got the zero energy is gonna be the bottom of the swing, and the maximum energy is gonna be at H max. So if I've got a Chmara ax because I've got theta Max, I'm gonna use these two equations right here. The problem is that I'm not asked for H Max, and I'm not given what h. Max is. I'm supposed to get this in terms of l and theta. So here's gonna do. I'm gonna take my pendulum equation that I just arrived when h is equal to l one minus cosine theta. And when Fada is maximum. So H Max is gonna be when one minus cosine theta is at its maximum value. So now I'm just gonna take this expression here and plug it in for H max. So let's go ahead and do that. So I've got mg and this is gonna be l one minus cosine. Theta Max equals one half M fi Max squared. And that's what I'm looking for, V Max. So if we're looking for this V maximum right here, take a look. I've got these ems that actually cancel, and then I could move one half to the other side. So that means I'm gonna get to G L one minus cosine theta max his equal to v max squared. Now I'll have to do is just take the square root. So if I move that over, I get that V max is equal to the square root of two G l one minus cosine theta maximum. So this is how problems will usually go. They'll they'll tell you the length of the pendulum and they'll tell you the angle that they pull it back and you can find out what the energy is based on you just using the energy conservation formula. So let me know if you guys have any questions. Let's keep moving on for now.

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