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Anderson Video - Conservation of Angular Momentum

Professor Anderson
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>> Hello, class. Professor Anderson here. Let's talk about angular momentum and specifically, let's talk about the conservation of angular momentum. When we talk about momentum, we said that it is one of the sort of the foundational principles, this idea of conservation of momentum, right? In elastic collisions or in inelastic collisions, momentum is conserved. It turns out that angular momentum is also conserved, namely, if things are spinning, they tend to stay spinning, okay. So angular momentum is just this. Angular momentum -- -- L is conserved. And we have the exact same caveat that we had with linear momentum if there are no external -- not forces in this case but torques. Angular momentum L is conserved if there are no external torques. All right, so what do we mean by this. What we mean by this is the following. L which is a vector, of course, initial has to equal L final, all right. This is the idea of conservation of momentum. Within each of those L's, you have to account for the entire system. Okay, so you have to look at the system initially. You have to look at the system finally. And if there are no external torques acting on the system, then it's conserved and you should get the same quantity. All right, so let's try a very simple example. Let's say that we take our good old ball on a string. Okay, we love this example because it's easy to do. Here's our ball on the string. It's going around like this in a circle. Okay and this is the top view. Okay, so we're spinning the ball around in a horizontal circle. And we're looking at it from the top. Let's ask the question, what happens to the speed of the ball if we suddenly shrink the cord to half its original size? And you can imagine doing this the following way like if I'm holding up a string, and I'm swinging it around. And I'm swinging the ball around in a circle and now I pull the string down through my hand -- I can change the length of the string as the ball is still spinning. Okay, so that's sort of the picture that we're going to look at here. So we'll take this example and we're going to say, "As r goes to r over two, what happens to v?" All right, let's see if we can figure this out. The ball's spinning around at some particular v. Initially, we know that conservation of the angular momentum says the following. Li equals Lf. But we also know what the angular momentum is for a ball on a string. It is mvr. And so, if it's initial, I need to put subscripts on there m v initial, r initial. What about the final? Same mass so we don't have to worry about that. Again, we just have mvr. So when the ball gets to half its original length and it's going around, we had v initial before. Now we have some v final. And let's see if that's bigger or smaller, okay. All right, we want to solve this thing for v final. V final is equal to what? Well, the m's cancel out. That's really nice. And we're just left with r initial divided by r final, all of that times v initial. In our case, we said that r initial was just r. R final was r over two. The r's drop out. The half on the bottom flips up to be a two up on the top. And we get twice the speed, okay. As that object is pulled in by pulling on the string, it starts going faster, and faster, and faster. So when you're at halfway in, it's going twice as fast. If you went to a third, it would be going three times as fast. And in fact, in the limit as this thing goes to zero, it would be going infinitely fast, okay. It would be orbiting like this. Just kind of like what happens when something gets sucked into a black hole, right? It starts spinning faster, and faster, and faster as it goes around. It's kind of a cool idea. Okay, hopefully that one is clear. If not, definitely come see me at office hours. Cheers!