Conservation of Angular Momentum

by Patrick Ford
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Hey, guys. So in this video, we're going to start talking about the conservation of angular momentum. Let's check it out. So remember that 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? Um, 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 any little mental problems are actually 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 want to 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 Iomega. Moments of inertia and angular speed. Linear momentum is conserved. If there no external forces. Yeah, and angular momentum is conservative there no external torques, right? And this should make sense. Um, 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. Um, it's just that there are no external forces or, um, that there are that if there are external forces, they cancel, they at least cancel each other out. So even better definition is if there's some off external forces again, they could exist. Jason. They just have to add up to zero. Something here does. Some of the external torques has to be zero. This is the condition, the condition for conservation of linear momentum in 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 involved two objects. Pretty much all of them involve two objects collide 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 to object. Soapy initial becomes P initial one plus p initial two equals p initial one plus p initial too. So it's gonna be this very familiar equation. M one v one initial plus and to be to initial equals m one v one final plus m two. The two final Now angular momentum is a little bit different, that there's a lot of momentum angular momentum problems that involved just a single object, right? So the most classic, probably the most classic, angular momentum conservation of angular momentum. Question is when you have a nice skater. So let's say you have ah, girl ice skating. And she is spinning with her arms open and she closed her arms that she's gonna spin faster. This is a conservation of angular momentum. Question. We're going to solve this later, and it's just one object that'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 Iomega. 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 wanna expand this equation a little bit to show you something. So moment of inertia I for a point mass is something like M R Square for a shape. It's something like, Let's say, for a for a solid cylinder, you'd be half M r square for another object for like, a solid sphere would be to fifth M R 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 here have to fits here. There's, like a one that hides in there, right, That's implicit. We don't have to write. So I'm going to 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 Iomega to show this, and I'm going to say that this is a constant. This is a constant constant. Okay, meaning this number doesn't change. So really, the kinds of problems you're gonna have. There's two basic types of problems in one type. The mass will change. I'm gonna put a little delta here on top of the mass will change, which will cause a change in omega on the other type of problem. Um, the R will change and cause the change in omega. So if the mass of a 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 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 are right. You can see that these things are going away from the axis of rotation, so the are grows. Therefore the omega becomes smaller. Okay, So the two types of changes we're gonna have for one objects is that either, um either Emma or are will change. And those will cause a change in omega. Okay, change maker. Now, when we have to objects when we have two objects, we have problems where you're essentially adding or removing mass. So the classic example here there's a disc that's spinning. You add a little block to it. What happens while the disks Now I'm going to spend a little bit slower, and we can calculate that. Okay. When we had linear momentum, the two big groups of of big groups of problems we had were push away problems where two things would like. When you shoot a gun, the bullet goes this way. The gun goes this way or collision problems. So push away. Two things are going away from each other. Collision two things. Air coming into each other. Okay. And we also had We also had these types of problems we're adding or removing a mass, adding or removing a mass, um, 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 escape or something, All right, so anyway, that's it for that. Let's do I have, uh, 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 way. Want to figure out whether the angular speed omega will increase or decrease? All right, so a nice skater. We just mentioned this nice skater spins and frictionless ice. What happens to her angular momentum if she closes or 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 I'm l is a constant L, which is Iomega is a constant mhm. I'm gonna expand Iomega into something m r squared Omega. And this is a constant. What's gonna write to see and look what's happening here is that by closing her arms by closing her arms, her are is decreasing. Therefore, her omega is going to increase. So the answer is that omega increases Omega will increase. Alright, be ah, large horizontal disk spins around itself. What happens to discs? Angular speed if you land on it. So there's a disk 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 Iomega is constant. I'm gonna expand Iomega to be something m r squared. Okay, 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 someone right here that Omega will decrease. All right, Uh, see, a knob checked 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 gonna spin horizontally around the string. So the 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 Iomega is a constant. See, Iomega, I'm gonna expand to be something, um, 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. Um, 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 spend something a really long cable the second you pull the cable in, um, it's gonna stand Instead of being like this, it's gonna get faster like this. Okay? And in the last one, a star like the sun spins around itself. Cool. Um, 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 billions 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 sighs, um, in volume and in mass. Okay, so that's gonna happen to our son, like in 10 billion years. You're safe. Don't worry. So what happens if it collapses and loses half of its mass and half of its radio? So l equals Iomega? See, it's an object that spins, but it's not going to, uh, it's angular. Momentum is conserved even though this thing is blowing up. Right. So I have This is gonna 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 two, yet that and then this goes down by a factor of two. Notice that our ISS squared, Um, so I'm gonna actually square the factor of to get out of the way. So the net result of this going down by a factor of two is that it actually goes down by the whole thing, goes down by a factor of four. So I have this going down by a factor of two that's going down by a factor of four. I multiply those two and I have a I have this thing growing by a factor of eight. Okay, so two times four is AIDS. If the if these two variables here become eight times smaller, this variable has to become eight times greater so that the whole thing is a constant. So this Starwood then spin a time faster eight 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 going to solve most of these later on. But that's it. Let me do you have any questions and let's get going