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Conservation of angular momentum

How angular momentum is conserved within a system when there is no external torque.

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Video transcript

- [Instructor] Let's talk a little bit about the conservation of angular momentum. And this is going to be really useful, because it explains diverse phenomena in the universe. From why an ice skater's angular speed goes up when they tuck their arms or their legs in, all the way to when you have something orbiting around a star, and the closer and closer it spirals in, it seems like its rotational velocity's angular speed is picking up. And it starts to rotate faster and faster around the body. You'll see that if you see simulations of astronomical phenomena. So the big picture here is, is if we have our initial angular momentum for a system, and we'll think about that a little bit. As long as the system has no external torque applied to it, then your final angular momentum is going to be the exact same thing. And so, one way to think about it, let's imagine that I have some type of spinning thing over here. I have some type of mass, let's say it's on a table, I'm looking from above, and a point on the outside of this disk is spinning in this direction, with a velocity of magnitude v. And let's say that there's a clump of clay, of orange... Maybe it's Play Doh, or clay or something. And it has a velocity going in this direction. It's on a collision course with this object. And let's say it has a velocity of 5v. And so, if we think about this disk-clay clump system, and so you always have to specify what system you are talking about, so if you think about this entire system, how does the angular momentum change before these two things collide, and then after these two things collide? So actually, let me draw the after the collision scenario. So the after the collision scenario, it looks something like this, where the clay has now clumped onto this, and now they are going to be rotating together. And I haven't even told you the mass of this disk, or the mass of this clay, so it would be unclear in which direction they would now be rotating. But how is the angular momentum going to change from this state, from the initial state, to the final state? Pause this video and try to think about it. Well, you might have guessed. Since we said, look, we have this whole system, and we're not applying any external torque to the system, our angular momentum is going to stay exactly the same. Now, we have to be careful. If I told you the system was just this disk, not the clay clump that's on a collision course with it, then the angular momentum for the disk would change, but why is that? Does that defy the conservation of angular momentum? No. Because this clay clump, when it collides, would be providing an external torque to the system, if we defined the system to just be the disk. But since it's the disk plus the clay clump, and we have no external torque to that combined system, then our angular momentum is not going to change. Now that we can appreciate that angular momentum is constant as long as that there is no net torque applied to the system, let's think about the famous situation where an ice skater's angular speed goes up as they tuck in their arms. And you can do a less graceful version of this on an office chair, where if you sit on the office chair and you begin spinning, this is my office chair, and if you stick your legs out, at first you're going to spin slowly, but then if you tuck your legs in, you're gonna start spinning faster. You're gonna have a higher angular speed. Now, why is that? Well, to appreciate that, we can think about the formula for angular momentum. So the formula for angular momentum, L, there's a couple of ways we can, or several ways that we can write that. We can write that as our moment of inertia, I, times our angular speed. Times omega. And this might look a little bit foreign at first to you, but it has a complete analog when we're dealing in the linear world. Here we're rotating. In the linear world, we say that linear momentum is equal to mass, is equal to mass times velocity. And the reason why we have an analog here, is mass can tell you about inertia of an object. How much force do you need to apply to accelerate that object? f equals ma. Well, moment of inertia, you have something similar going on. But instead of thinking about how to just linearly accelerate something, this tells you how hard is it to get angular acceleration, how much torque do you need to apply? Instead of just how much linear force you need to apply. And instead of velocity, you have angular speed, and sometimes this is called angular velocity as well. But this by itself, you might say, well this doesn't help me on tucking in my knees when I'm on an office chair, or the ice skater tucking in her arms. Well, to think about that, we just have to appreciate that the moment of inertia can be expressed as m times radius squared, m r squared. And then we still have our omega right over here. So this is another way of writing angular momentum. And so, when a skater tucks in her arms, her mass is not changing, that is staying constant. But remember, the radius, one way to think about it, it's a little complicated when you're thinking about a human body system. In a simple sense you just have a point mass rotating around a point like that, then this is the radius. But if we're dealing with a more complicated structure, like a human body, you can imagine the radius as being indicative of the average distance of the mass, from the center of rotation. So when the figure skater tucks in her arms, that average distance goes down. And so when she tucks in her arm, this goes down, but if this part goes down, but our angular momentum stays constant, because we have no torque being applied to the system, no net torque being applied to the system, well then, this needs to go up in order to keep that angular momentum constant. And that's exactly what's happening. The angular speed picks up, just as the radius goes down. Now, this also explains why if you have, let's say that's some type of planet, and you have a rock, or something orbiting it. As this gets closer and closer to the planet, it's angular speed is going to go higher, and higher, and higher. Why is that? Well, because when you have a high radius, so here your radius is higher, and so your angular speed might be a little bit lower, but then when you're closer in, your radius has gone down. And so your angular speed has to go up to make up for it. I'll leave you there. I encourage you to have fun spinning on office chairs.