Anderson Video - Pendulum Mechanics

Professor Anderson
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>> Hello class. Professor Anderson here. Let's say that you are a kid playing on the playground and you're on the monkey bars and you're swinging from one to the next. And we want to calculate what your speed is, your average sort of speed relative to the ground if you do that. So let's think about that problem for a second. All right. Here is the top of the monkey bars. And as you swing, you go from this position some theta to the next position also theta and then you grab onto the next bar and repeat that process. Okay. And if you do it again and again, you sort of map out this trajectory that looks something like that. Now how do we calculate your average speed in this direction? Well, let's just take one cycle of it and see if we can figure that out. So if I am at a distance L away because that's how long my arms are and I'm swinging through an angle theta, what I really want is this horizontal distance. How far is that theta? Okay. We can probably do that. Let's draw it a little bit bigger right here. If that's theta and this is L then this side of the triangle is L sine theta. But there's another one over on the other side and that one's also L sine theta. Okay. So the distance that you've gone, the horizontal distance is just 2L sine theta. Great. How long did it take you to do that? It to you a time, delta t. And how does that time relate to something that we know? So let's say something about how long this takes and see if we can relate it to the period of a pendulum. Right. A pendulum will swing there and back in a period capital T. And so if we want to go half of that, the delta t that we're interested in is the period of a pendulum over 2. So what is the period of a pendulum? Well, we know that it has to go like the length of the pendulum and gravity and we probably remember that it's got to have a square root in it and there is a 2pi floating around somewhere. And the 2pi is, in fact, where? It's right here. All right. What do we get now? We get delta t equals T over 2 so that's pi times square root L over g. And now we can calculate the speed because the speed is just how far you go over how long it takes and we know all those numbers. 2L sine theta divided by delta t which we just that was pi square root L over g. Now we can plug in some numbers. So let's take a arm length of .9 meters. Let's say that angle that you swing to is 20 degrees. And if you plug all those numbers in, you should get V is 0.65 meters per second. All right. Try that one out yourself. Hopefully this is clear. If not, come see me in my office. Cheers.
>> Hello class. Professor Anderson here. Let's say that you are a kid playing on the playground and you're on the monkey bars and you're swinging from one to the next. And we want to calculate what your speed is, your average sort of speed relative to the ground if you do that. So let's think about that problem for a second. All right. Here is the top of the monkey bars. And as you swing, you go from this position some theta to the next position also theta and then you grab onto the next bar and repeat that process. Okay. And if you do it again and again, you sort of map out this trajectory that looks something like that. Now how do we calculate your average speed in this direction? Well, let's just take one cycle of it and see if we can figure that out. So if I am at a distance L away because that's how long my arms are and I'm swinging through an angle theta, what I really want is this horizontal distance. How far is that theta? Okay. We can probably do that. Let's draw it a little bit bigger right here. If that's theta and this is L then this side of the triangle is L sine theta. But there's another one over on the other side and that one's also L sine theta. Okay. So the distance that you've gone, the horizontal distance is just 2L sine theta. Great. How long did it take you to do that? It to you a time, delta t. And how does that time relate to something that we know? So let's say something about how long this takes and see if we can relate it to the period of a pendulum. Right. A pendulum will swing there and back in a period capital T. And so if we want to go half of that, the delta t that we're interested in is the period of a pendulum over 2. So what is the period of a pendulum? Well, we know that it has to go like the length of the pendulum and gravity and we probably remember that it's got to have a square root in it and there is a 2pi floating around somewhere. And the 2pi is, in fact, where? It's right here. All right. What do we get now? We get delta t equals T over 2 so that's pi times square root L over g. And now we can calculate the speed because the speed is just how far you go over how long it takes and we know all those numbers. 2L sine theta divided by delta t which we just that was pi square root L over g. Now we can plug in some numbers. So let's take a arm length of .9 meters. Let's say that angle that you swing to is 20 degrees. And if you plug all those numbers in, you should get V is 0.65 meters per second. All right. Try that one out yourself. Hopefully this is clear. If not, come see me in my office. Cheers.