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Angular momentum

CHA‑4.D (EU)
CHA‑4.D.2 (EK)
CHA‑4.D.2.1 (LO)
CON‑5.E (EU)
CON‑5.E.1 (EK)
CON‑5.E.1.1 (LO)
Introducing angular momentum conceptually starting from linear momentum. Also covers some real-life examples. Created by Sal Khan.

Video transcript

If we have some mass, m, and it is moving with some velocity, let's say the magnitude of that velocity we say is v, we know that this object right over here has momentum. Translational momentum. And that momentum, and we use the Greek letter rho to represent momentum. Translational momentum is defined as being equal to the mass times the velocity. This is all a review, we have other videos where we talk about translationial momentum, and one way to think about it is, "Well how hard is it to stop this thing?" Literally in everyday language you think, "Well how much momentum does something have? "The more momentum this has the harder it's going to, "the harder it is to stop it in some way." And so we know if we wanna get a little bit more mathematical, that if we wanna change momentum, we have to apply force for some amount of time. And so the magnitude of our force times the duration of the time that we apply it for, force times time and this is called impulse. And this is once again, all review. This is equal to change in momentum. Let me do this in that yellow color. That is equal to change in momentum. So if you don't have any impulse, especially if you don't have any net force acting on an object, its momentum is going to be constant. You have a conservation of momentum. And we use that idea in all sorts of interesting physics applications in the world, and especially a lot of cases using billiard balls and whatever else. So now let's try to take a similar idea, but go into the rotational world. So let's imagine you have a mass, for the sake of this we're gonna assume it's a point mass. So you have a mass there. And let's just say it's attached by, essentially a massless wire, to you know, it's just nailed down right over here. And so this right over here would be its center of rotation, and so you could imagine if someone applied a torque to this mass, this mass could start rotating in a circle. And you can just assume that maybe it's sitting on a, you know the screen, this is kind of a frictionless surface, there's no air resistance. And so then it will, if you apply a torque here, it will start rotating. And so you could think about, "Well there might be an idea, "just as momentum is this idea of, "well how hard is it to stop something?" You might say, "Well how?" And this is, stop translating something from moving. You might think, "Well maybe there's a similar idea "of how hard is it for something, "or how hard is it to stop rotating something?" And you could imagine that that idea has been defined and it has been defined as angular momentum. So let me make this clear, this right over here is momentum. And over here we'll talk about angular momentum. And actually both momentum and angular momentum are vector quantities. So here I just wrote kind of the magnitudes of velocity and momentum. But momentum is a vector and it could be defined, the momentum vector could be defined as equal to the mass which is a scalar quantity times the velocty. Times the velocity vector. Now the same thing is true for angular momentum, but I'm gonna stay focused on the magnitude of angular momentum. Angular momentum can have direction as you can imagine you could rotate in two different ways, but that gets a little bit more complicated when you start thinking about taking the products of vectors because as you may already know or you may see in the future, there's different ways of taking products of vectors. But just to get the intuition of angular momentum, I'll focus on the magnitudes. So angular momentum is defined and the letter used is L. I did a lot of research to try to figure out why it is called L, and I could not find a good reason. So in the message board below if anyone has a good reason I would like to know why angular momentum is called L. A lot of the best arguments I saw is that almost everything else was, all the other letters were used up for other ideas in physics. But anyway, angular momentum is defined, and it's defined very similarly. Just as kind of torque is the thing that can change how something rotates, and force is the way that something changes how something translates it, and torque is force times distance from the center of rotation, everything in kind of the rotational world is defined in a similar way. You kind of take the analogue in the translational world, and you multiply it times the distance from your center of rotation. So angular momentum is defined as mass times velocity times distance from the center of rotation so let's call this distance right over here, r. r for radius 'cause you could imagine if this was traveling in a circle that would be the radius of the circle. m, v, r. Actually let me be a little bit more careful here. It's the magnitude of the velocity that is perpendicular to the radius. Sometimes it might be called the tangential velocity. So this symbol right over here, this is the magnitude of the velocity that is perpendicular to the radius. So it would be that magnitude right over here. This is what we define as angular momentum. Now what I will tell you here is, just as in the absence of a net force, momentum is constant. We know, and I haven't shown it to you yet, I haven't proven it to you yet mathematically, but in the absence of torque, so if torque is equal to zero, we'll do torque in pink. If torque is equal to zero, if there's no net torque going on here, if the magnitude of torque is equal to zero, then we will have no change. No change in angular momentum. And we will look at that mathematically in a few seconds. But just from this there's a very interesting thing that arises. And something that you might have observed at even the Olympics or in other things. And this is the idea that you can, by changing your radius, you could actually change your tangential velocity. And as we've seen in previous videos, tangential velocity is closely related to angular velocity. So let's explore that a little bit. So when we write it in the world where, well actually you see it straight out of this, if L is constant, if r went down, so let me write this down. So let me rewrite it over here. So L, whoops. L is equal to mass times tangential velocity, or actually well yeah, tangential velocity, or the velocity that's perpendicular to the radius, times the radius. Now what happens, if we assume that this is constant, if we assume that there's no torque, so we're in this world. So this over here is going to be constant. So what happens if we were to reduce r? Somehow this wire started to reel in a little bit or started to wrap around here, and that's actually a reasonable thing, you could imagine as it rotates it starts to wrap around this thing so the wire gets shorter. So if r goes down, and this is constant, the mass isn't going to change, Well if L is constant, mass isn't changing, r is going down, tangential velocity, or the velocity that's perpendicular to the radius is going to go up. And if we wanna think about it, we can think about it in terms of angular velocity, we know that angular velocity, which we would measure in radians per second, we would use the symbol omega, omega is defined, and we go into much more depth in this in other videos, as tangential velocity, the magnitude of the velocity that is perpendicular to the radius, divided by the radius. Or if you solve for tangential velocity, you get v is equal to is equal to omega r. And so if you substitute back into this, really this definition for angular momentum, you get angular momentum is equal to mass times this times r. So mass times, I'm just substituting for velocity here, times omega r, times r. Which of course is just omega r squared. So once again we do the same exercise. If our radius goes down, what happens to our angular velocity? Remember we could measure this in angles per second or radians per second, well if this is constant, remember we're assuming that there's no net torque being applied to the system, so we're still in this world right over here. If we assume that this thing isn't changing, but the radius were to change, what's going to happen to omega? Well omega is going to go, omega is going to go up. Now likewise, if the radius got longer, So the radius got longer, what's going to happen to omega? Omega is going to go, omega is going to go down. So if you reduce the radius you're going to start spinning faster if you increase that radius, you're gonna start spinning slower. And you have seen this, or I think there's a high likelihood that you have seen this probably at the Olympics when you have seen figure skaters. Where they might start spinning and they have their arms out. So when their arms are out you could say that their radius is further out. And obviously a figure skater is a much more complex system than a point mass. You could imagine a figure skater is a bunch of point masses. Well you could just model a figure skater as a huge number of point masses at different radii, and you would wanna sum up their angular momentum, but the essence of what is happening is is when her arm is out the average radius when you're calculating all of the point masses in her arms and all the rest, the average radius is higher. And then when she pulls them in, when she pulls them in that radius goes down, and her angular velocity goes up. And you see that. They start spinning, and then without applying any torque, when they pull their arms in they start spinning faster. And then if they push their arms out, once again without applying any torque, they start spinning slower.