16. Angular Momentum
Conservation of Angular Momentum
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Hey, guys. So in this video, we're gonna start talking about the conservation of angular momentum. Let's check it out. So remember when we talked about linear momentum, that the most important part about linear momentum was the fact that it was conserved or that it is conserved. Right? It's conserved in certain situations. We'll talk about that in just a bit. The same thing is gonna happen for angular momentum. Most angular momentum problems are going to be about the conservation of angular momentum. So they're gonna be about conservation. They're gonna be conservation problems. Okay? So what I wanna do here is do sort of a compare and contrast between linear momentum and its angular equivalent, angular momentum. So linear momentum little p is mass times velocity. Angular momentum big l is I omega. Moment of inertia and angular speed. Linear momentum is conservative. There are no external forces. And angular momentum is conservative. There are no external torques. Right? And this should make sense. Angular momentum is the rotation equivalent of linear momentum. Torques is the rotation equivalent of forces. Now, even better description is it's actually not that there are no forces. It's just that there are no external forces or, that there are, that if there are external forces, they cancel, they at least cancel each other out. So an even better definition is if the sum of external forces again, they could exist just as they just have to add up to 0. Same thing here. The sum of the external torques has to be 0. This is the condition, the condition for conservation of linear momentum and the conservation of angular momentum. In the vast majority, of physics problems, those quantities are conserved. Certainly, all the problems we're gonna look into from now on for angular momentum will have conservation. Alright. One difference between these is that most problems for linear momentum involve 2 objects. Pretty much all of them involve 2 objects colliding against each other. Okay. So the conservation equation will look like this. P initial equals p final. Right? So it's saying that momentum doesn't change. This is of the system. So I can expand this equation and I have 2 objects. So p initial becomes p initial 1 plus p initial 2 equals p initial 1 plus p initial 2. So it's going to be this very familiar equation. M1v1initialplusm2v2initial equalsm1v1finalplusm2v2final. Now angular momentum is a little bit different that there's a lot of momentum, angular momentum problems that involve just a single object. Right? So the most classic probably the most classic angular momentum, conservation of angular momentum question is when you have an ice skater. So let's say you have a girl ice skating and she is spinning with her arms open and she closes her arms and she's gonna spin faster. This is a conservation of angular momentum question. We're gonna solve this later. And it's just one object. It's one body that's spinning. Okay? Now, conservation equation will be similar. I'm gonna have that L initial equals L final. Right? Because L doesn't change. That's the whole deal. And l is I omega. So I'm going to say that I initial, omega initial is not going to change. It's a constant. Okay? But what I want to do here is I want to expand this equation a little bit to show you something. So moment of inertia, I, for a point mass is something like m r squared. For a shape, it's something like, let's say, for a for a solid cylinder to be half m r squared. For another object for, like a solid sphere, it'd be 2 5ths m r square. Square. The point that I want to make here is that it's something m r squared, something m r squared. Right? What changes is that here you have half, you have 2 5ths. Here there's like a one that hides in there. Right? That's implicit. We don't have to write. So I'm gonna say that this takes the shape, this I takes the shape of box, which is some fraction, m r squared. And then I have omega. So I'm just expanding I omega to show you this, and I'm gonna say that this is a constant. This is a constant. Constant. K? Meaning, this number doesn't change. So, really, the kinds of problems you're gonna have, there's 2 basic types of problems. In one type, the mass will change. I'm going to put a little Delta here on top of M, the mass will change, which will cause a change in Omega. And the other type of problem, the r will change and cause the change in omega. So if the mass of the system changes, the system will slow down. Right? You might be able to see here if this mass grows, the system will slow down or if the radius of the system, the effective total radius of the system increases, then the mass the the the velocity, of rotation will go down as well. So the opposite case of what I just mentioned with the girl spinning is if she's spinning like this and then she opens her arms, she slows down. And that's because her total r, right, you can see that these things are going away from the axis of rotation. So the r grows, therefore, the omega becomes smaller. Okay? So the two types of changes we're going to have for one object is that either, either m or r will change and those will cause a change in omega. Okay? Change in omega. Now when we have 2 objects, when when we have 2 objects, we have problems where you're essentially adding or removing mass. So the classic example here is there's a disc that's spinning. You add a little block to it. What happens? Well, the disc is now gonna spin a little bit slower and we can calculate that. Okay? When we had linear momentum, the 2 big groups of pro of big groups of of problems we had were push away problems where 2 things would like, when you shoot a gun, the bullet goes this way, the gun goes this way, or collision problems. So push away, 2 things are going away from each other. Collision, 2 things are coming into each other. Okay? And we also had, we also had these types of problems where you're adding or removing a mass adding or removing a mass, in linear motion, which if you think about it, adding a mass is a collision. Right? One mass joins the other. And, removing a mass is really a push away problem, is if you jump out of a of a, of escape or something. Alright? So anyway, that's it for that. Let's do I have an introductory example here talking about a bunch of different situations to see so we can discuss what happens in these situations. And we want, we want to figure out whether the angular speed, omega, will or decrease. Alright. So a nice skater, we just mentioned this nice skater spins in frictionless ice. What happens to her angular momentum if she closes her arms? If you close your arms, you spin faster. You might know this from class, from just watching TV, from doing it yourself, or we're gonna use the equation here. So what I'm gonna do is I'm gonna say, l is a constant. L, which is Iomega, is a constant. I'm going to expand, Iomega into something m r squared Omega and this is a constant. I'm just going to write c. And look, what's happening here is that by closing her arms, by closing her arms, her r is decreasing, therefore, her omega is going to increase. So the answer is that omega increases. Omega will increase. Alright. B, a large horizontal disc spins around itself. What happens to disc's angular speed if you land on it? So there's a disc spinning around itself like this. You land on it right here. So this is you. You got added to the disc. What happens to the disc's speed? Well, I equals I omega is constant. I'm gonna expand I omega to be something m r squared, k, times omega. I m r squared times omega is a constant. What's happening here is there's mass being added to the system. Therefore, the system will slow down. So I'm gonna write here that omega will decrease. Alright. C, an object is tied to a point via a string that spins horizontally around it. So here's an object and it's tied to a point here. It's connected by a string and it's going to spin horizontally around the string. So an object is going like this because it's connected to a string and what we want to know is what happens if you shorten the string. So again, I equals I omega is a constant. C I omega, I'm going to expand to be something, m r squared. It doesn't matter what this something is for these problems. We're just doing a quick, analysis of what would happen. If you remove, if you shorten the string, if you shorten the length of the string, you're shortening the radius of rotation of this object. Therefore, the omega will increase. Omega increases. Okay? You can imagine that if you spin something on a really long cable, the second you pull the cable in, it's gonna spin instead of spinning like this, it's gonna get faster like this. K. And then the last one, a star like the sun spins around itself. Cool. And I wanna know what happens if it collapses and loses half of its mass and half of its radius. Okay? So, you might know this, you may know this, stars live for obviously 1,000,000,000 of years. Eventually, they run out of star fuel and they collapse. And what that means is that they're going to significantly shrink in size, in in volume and in mass. K? So that's gonna happen to our sun, like, in 10,000,000,000 years. You're safe. Don't worry. So what happens if it collapses and loses half of its mass and half of its radius? So I l equals I omega c. It's an object that spins, but it's not going to, its angular momentum is conserved even though this thing is blowing up. Right? So, I have this is going to be something m r squared omega, and that's a constant. So here we actually have precise numbers half and half. So if this goes down by a factor of 2, get that and then this goes down by a factor of 2, Notice that r is squared. So I'm gonna actually square the factor of 2. Let me get out of the way. So the net result of this going down by a factor of 2 is that it actually goes down by the whole thing goes down by a factor of 4. So I have this going down by a factor of 2, this going down by factor of 4. I multiply those 2 and I have a I have, this thing growing by a factor of 8. Okay. So 2 times 4 is 8. If the if these two variables here, become 8 times smaller, this variable has to become 8 times greater so that, the whole thing is a constant. So this star would then spin 8 times faster, 8 times faster than it was before it collapsed. Okay? So that's it for this one. Some introduction in terms of what to expect in these different kinds of problems. We're gonna solve, most of these later on. But, that's it. Let me know if you have any questions and let's get going.
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