17. Periodic Motion
Energy in Pendulums
Energy in Pendulums
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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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Hey guys, let's get some more practice with these energy and pendulum questions. So we're told the mass of this some object that's hanging from a 2 m pendulum and then as it's making a five degree angle with the vertical, it has a speed of something, and we're supposed to figure out what the maximum height is. So let's just start writing some stuff out. Mass is equal to 0.4. The length is equal to two. Now, what does the second this third? Yeah, seconds. I mean, so it makes a five degree angle with the vertical, and it has a speed of 1.5 m per second. So if I just go ahead and draw on my little pendulum diagram, so what happens is so this is gonna be my pendulum. And at this specific point here, where it makes five degrees, we're told that this thing already has a velocity in this direction or speed of 1.5 m per second. So what this is this telling us is this is not the amplitude. This is actually a very, very specific point where the amplitude is actually somewhere out here, probably. So that's the That's the rial amplitude and this is state API. So where they're telling us is that Data P is equal to five degrees, and at that specific point, the velocity as is 1.5 m per second. And now we're supposed to find out what the maximum height is. So what does that mean? That represents H Max. So let's go through our energy conservation equations for pendulum. And really the H Max Onley pops up in one place this MGH max. So we know we're using energy conservation, but which equations that we're gonna use. Let's take a look. So I've got MGH Max and I've got the M and I've got the g. So I've got em. But I don't have what V Max squared is I'm not told what the's speed is at the bottom of the swing not told V Max. So I'm not sure I'd be able to figure that out. And then let's look at the second part. So I've got em n g. So let's go ahead and check that off M and G, I've got em and the velocity at a point squared. But what about this hp? What about this height at a specific point. Well, if I've got the angle and I've got the length of the pendulum, I can probably figure out what the height is. So let's go ahead and use that relationship. So let me set up that equation. I've got M G H. Max is equal to M G. Whoops. M g h at a point plus one half M v at a point squared. So what I'm really looking for here is what is this? Maximum height. So again, I've got all of these variables. All I have to do is just figure out what the height at a specific point is. So how do I do that? Let's go ahead and bring that over here. So how do I figure out what the height is given some masses lengths and then theta angles? Well, I can use the pendulum equation. Remember that the height at any point is given as the length of the pendulum one minus co sign of theta P. We just have to remember that all of this stuff we're using coastline has to mean radiance. Okay, so if you take a look here, I've got the length of the pendulum. And I know what they Tapie is in terms of degrees at least. So that means I could actually figure out what this HP is. I can figure out that height at that specific point is, so let's go ahead and figure that out. The first thing I'm gonna do is I'm gonna convert this five degrees over the radiance, so it means that a P is five times pi over 80. That's what I get is an angle of zero point 087 and that's gonna be in rads. So now I'm just gonna put that in there and then make sure that my calculator is in degrees mode. So I've got that the height at a point. So HP is the length of the pendulum, which is two. And then I've got one minus the co sign of 0. 87. What you're gonna get is H P is equal to something very, very small. So 0.76 it's important to keep those extra decimal places because I don't want to round off too early. Okay, so now that we've got this number, I could just plug it into this formula and now I'm good to go. The rest of it is just plugging in numbers and then dividing stuff over. Okay, so if I write all this stuff out, I get 0.4. Then I get 9.8 and then I get H. Max is equal to 0.4. Then I get 9.8. The height at that specific point is 0.0.76 Then I got plus one half m on Sorry, em is equal. 2.4 again. And then I've got V P squared. So that is the 1.5 m per second right? That's the velocity at a specific point. So if you go ahead and just plug all of this stuff into your calculator, what you're gonna get is you're gonna get 0.4 times 9. h max is equal to 0.48. So if you just divide this stuff over to the other side, you're just gonna get that H Max is equal to 0. m. So that's the maximum height that this pendulum will reach. Alright, guys, let me know if you have any questions. That's it for this one
A mass swinging at the end of a pendulum has a speed of 1.32m/s at the bottom of its swing. At the top of its swing, it makes a 9° with the vertical. What is the length of the pendulum?