Pendulum Equations

by Patrick Ford
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Hey, guys. So we talked about some of the differences between mass spring systems and pendulums. We talked about the differences between F A and Omega. So now you might be wondered what the other equations look like. I'm gonna give them to you in this video. So remember that we have a pendulum here. It's some length l and you're gonna take this mass and you're gonna pull it out to some angle, which is data you're gonna let it go, and it's gonna keep going back and forth, right? And so, in a mass spring system, whenever you pulled something back away from the equilibrium position, that distance was X right. It's the same exact thing for pendulums. So I'm gonna draw this line here, and this X distance represents the distance away, the deformation from the equilibrium. But now, in order to solve that, I've got some trick to involve here. I've got a triangle, so I've got the high pot news, I've got the angle and I've got the opposite side. So from so Kyoto, uh, I can relate this X at any point during the pendulum swing by L Times, they've sign of data. But remember that when we're talking about pendulums sign of data and data are just about the same thing. So I could make this sort of simple simple by saying that X is equal to L. A Times data. And remember that we're gonna be working with radiance when we're doing this right? So X equals health data. So he said that in a mass spring system you're pulling this thing all the way back and its maximum displacement or its maximum deformation was equal to the amplitude. Right? Swings back and forth between the two amplitude. So now it happens. This is the same thing for pendulums. So you're gonna swing. You're gonna take this thing all the way back to some theta. And once data is at its maximum value, X is at its maximum value away from the equilibrium position. And that's what we said that the amplitude waas. So it's the same thing. But because X is equal to L Times theta thing, I mean the amplitude X max is going to be at l Times theta max. So that means that the for pendulums, the maximum X is the amplitude, and that is equal to L a Times data, Max. Okay, so we also had a couple of other things about mass spring systems. We have the maximum values for X V N A. So he said here at the very middle that the velocity is at its maximum and at the end points, the acceleration is at its maximum, and it points towards the center. Right? So that's a max on either side. It's the same thing for pendulums. Except for now. Now that what's happening is at the bottom of its swing, we've got a velocity in this direction and that is going to be the maximum. And then we've got a max over here that wants to pull it back towards the center. So it's just the same exact thing for pendulums except now. So our equation for mass spring systems was that V Max was a Omega and a max was a Omega squared. It's the same thing for pendulums. We've got a omega and a Omega squared. The only difference is that a is now something slightly different. Remember that now A is equal to l Time State a max. So what I'm gonna do here, I'm gonna just replace these ays with El Fattah Max So V Max is a Omega. Then we can also say that it's equal to L. A Times data Max Omega Just replacing the A with Al Qaeda Max. And so a Omega squared is a is L X max or theta Max Omega squared. OK, so the other thing is that the Omegas are slightly different for pendulums as well. Remember that this is G over L Square rooted, whereas for mass springs, it's k over m Just remember those two. Okay, so it's very rare for you, actually to be asked what the horizontal displacement is away from the equilibrium. You most likely won't be asked that What's more common is you'll be asked what the angle is at a certain time. And so if you ever asked for what data is and you're given what T is, you're gonna be using this equation. Tha tha t it looks just like the VT and 80 equations are, um that's it. So that's basically it. So let's go ahead and take a look at an example. So we've got this mass and a tanking from the spring, so we've got this pendulum system here so I'm gonna draw and draw that out. I've got the equilibrium positions down here and this thing is just going to swing back and forth until it reaches the other side. So that's gonna be, like over here somewhere. Okay, so we're told a couple of things were given that the mass is equal to g, so that's equal to 0.5 kg. Just remember, everything has to be a Nessie. The length of the string, which is the length of the pendulum, is equal to 0.4 m again were given centimeters and they were told that the object as a speed of 0.25 m per second as it passes through the lowest point. What does that mean? Remember we just said at the lowest point, this thing has its maximum speed. So they're really telling us about this. 0.25 m per second is that V Max is equal to 0.2 25. And we're supposed to figure out what the maximum angle is in degrees from the equilibrium position. So we're supposed to figure out basically how far in terms of data does this reach in degrees. So just remember that we're working with all of these equations were gonna get radiance. And so I'm just gonna have to do a conversion to get it back into degrees at the very end. So we're looking for Theta Max, right? So I'm gonna take a look at all of my equations that have theta max in them, so I don't wanna go all the way up there. I'm just gonna paste all of these equations right here. So I've got all my theta max. Is Aaron these equations? I got theta Max here. Betamax, Betamax and data Max. So let's take a look at all of these equations so I don't have anything about X max or the amplitude, But I do know what the length of the pendulum is, so I know that the length is I've got this length right here. Let's take a look. I do know what the maximum velocity is. Uh, I'm gonna go ahead and circle that I've got the max velocity. I don't know what the amplitude is, so we're just gonna cross it all the A's. And I also don't know what the Amax is. The last thing is that I could Onley use this bottom equation if I'm given something about time. But I'm not given any time. And because I'm not giving at a time and I'm supposed to find what data Max is, I have two unknowns, and I can't use that equation either. So let's look at the equation that I know the most about. I know what V Max is, and I know what l is. So in order to find Theta Max, I'm just gonna need to find out what this Omega is. So that's basically it. Let me write out that equation for V Max. So let's just try that. So you've got V. Max is equal to I've got l times theta max, then times omega. So if I wanted to figure out what this Theta Max is, let me just go ahead, rearrange that equation. So I've got Theta Max is equal to v Max, divided by L. Omega. I'm just gonna move these guys to the other side, right? So I'm just gonna divide them over to the other side. Okay, so I've got everything I know what this V Max is, and I've got l. The last thing I just have to figure out is what Omega is. So I'm gonna go over here and do that. Well, Omega is equal to once Let me write out that big omega equation that I have. I've got two pi frequency to pie divided by T and then I've got that is equal to square root of G over L. Now, I'm not giving anything about frequency or the period, so I'm just gonna have to use this last one squared of geo over l. So I've got omega equals square roots. Well, G is equal to 9.8 and then I've got l is equal to 0.4. So what I get is that Omega is equal to 4.39 that's gonna be rads per second. I think that's what I get. So I get 4 39 rads per second. So sorry, not 4.39. I get 4.95 so I got 4.95 rats per second. Great. So I'm just gonna plug that back into here. Okay, so I've got Theta Max now is equal to 0.25 then I've got divided by the length of the pendulum 0.4 and then multiply that by 4.95. So what I'm gonna get is Theta Max. Remember, Betamax is gonna be in radiance, and that's gonna equal 0126 That's new rads. So now the last step is just to convert it back to degrees. So how do I do that? So I've got data. Max is equal 2.126 and now in a multiplied by something that's gonna get rid of the rads. So I've got 180 degrees divided by Pi Radiance. So that's gonna cancel out, and then we're gonna get 7.2 degrees. That's the maximum angle that we make for the pendulum. So let me know if you guys have any questions, let's keep going on for now.