Hey, guys. So we talked about some of the differences between mass spring systems and pendulums. We discussed the differences between f, a, and omega. So now you might be wondering what the other equations look like. I'm going to give them to you in this video. So remember that we have a pendulum here at some length l and you're going to take this mass and you're going to pull it out to some angle, which is theta. You're going to let it go and it's going to 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. It's the same exact thing for pendulums. So I'm going to 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 trig to involve here. I've got a triangle. I've got the hypotenuse. I've got the angle and I've got the opposite side. So from SOHCAHTOA, I can relate this x at any point during the pendulum swing by lsinθ. But remember that when we're talking about pendulums, sine of theta and theta are, like, just about the same thing. So I can simplify this by saying that x=lθ. And remember that we're going to be working with radians when we're doing this. Right? So x=lθ. So we 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 amplitudes. So now what happens, this it's the same thing for pendulums. So you're going to swing you're going to take this thing all the way back to some theta. And once theta 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 was. So it's the same thing. But because x=lθ, then that mean the amplitude x max is going to be at l times theta max. So that means that for pendulums, the maximum x is the amplitude and that is equal to l times theta max. Okay. So we also had a couple of other things about mass spring systems. We have the maximum values for x, v and a. So we said here at the very middle that the velocity is at its maximum, and at the end points, the on either 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=lθmmax. So what I'm going to do here is I'm going to just replace these a's with lθmmax. So v max is a omega, then we can also say that it's equal to lθmmaxω, just replacing the a with l theta max. And so a omega squared is a=lxmmaxorθmmaxω2. Okay. So the other thing is that the omegas are slightly different for pendulums as well. Remember that ω is g/l, whereas for mass springs, it's k/m. So 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're ever asked for what theta is and you're given what t is, you're going to be using this equation, theta t. It looks just like the XT, VT, and a at equations are. 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 it's hanging from the spring, so we've got this pendulum system here. So I'm going to go ahead 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 going to be like over here somewhere. Okay. So we're told a couple of things. We're given that the mass is equal to 500 grams, so that's equal to 0.5 kilograms. Just remember everything has to be in SI. The length of the string, which is the length of the pendulum, is equal to 0.4 meters. Again, we're given centimeters. And then we're told that the object has a speed of 0.25 meters 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 what they're really telling us about this 0.25 meters per second is that v max is equal to 0.25. 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 theta does this reach in degrees. So just remember that when we're working with all of these equations, we're going to get radians. And so I'm just going to 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 going to take a look at all of my equations that have theta max in them. So I don't want to go all the way up there. I'm just going to paste all of these equations right here. So I've got all my theta maxes are in these equations. I got theta max here, theta max, theta max, and theta 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 what the length is. I've got this length right here. Let's take a look. I do know what the maximum velocity is. So I'm going to go ahead and circle that. I've got the max velocity. I don't know what the amplitude is, so I'm just going to cross out all the As. And I also don't know what the a max is. The last thing is that I can only use this bottom equation if I'm given something about time. But I'm not given any time. And because I'm not given at a time and I'm supposed to find what theta max is, I have 2 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 going to 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 we've got vmax=lθmaxω. So if I wanted to figure out what this theta max is, let me just go ahead and rearrange that equation. So I've got θmax=vmax/lω. I'm just going to move these guys to the other side. Right? So I'm just going to 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 the last thing I just have to figure out is what omega is. So I'm going to go over here and do that. Well, omega is equal to what? Let me write out that big omega equation that I have. I've got 2 pi frequency, 2 pi divided by t, and then I've got that is equal to square root of g over l. Now, I'm not given anything about frequency or the period, so I'm just going to have to use this last one, square root of g over l. So I've got ω=gl. 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, and that's going to 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 rads per second. Great. So now I'm just going to plug that back into here. Okay. So I've got θmax=vmax/lω. Remember, theta max is going to be in radians, and that's going to equal, 0.126. That's in rads. So now the last step is just to convert it back to degrees. So how do I do that? So I've got θmax=0.126180degrees/piradians. So that's going to cancel out, and then we're going to get 7.2 degrees. That's the maximum angle that we make for the pendulum. So let me know if you if you guys had any questions. Let's keep going for now.
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17. Periodic Motion
Simple Harmonic Motion of Pendulums
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