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16. Angular Momentum

Intro to Angular Momentum


Intro to Angular Momentum

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Hey, guys. So in this field, I'm gonna introduce the idea of angular momentum, which is the kind of momentum that you have. If you have rotation, let's check it out. So remember, if you have linear speed, linear speed is V You're gonna have linear momentum. Now, we used to call this just momentum because we only had one type. But now that we're gonna have two types, we have to make a distinction between them. So, Linda, momentum is good old little p, and it's just mass times velocity. So the units are kilograms for mass and meters per second for velocity. So you just multiply and objects massive velocity, and that it's and that's its momentum. If you have rotational speed, that's W. That's Omega. You have angular momentum or rotational momentum, and it's gonna take the letter l instead right. So instead of P, it's L. Now, instead of em, you're going to use the angular equivalent of mass, Which is I. Hopefully you get that and the angle equivalent of V, which is omega. So you can think of this equation as perfectly translating into angular into rotational variables. Okay, the units are gonna be a little bit different kilograms per meter, times meters squared, divided by seconds. Okay. And that's because of the makeup of the equation. Okay, so I moment of inertia is made up of Let's say, the moment of inertia of a point mass is m r square. But this works for all of them Masses kilograms times, meter square. Okay. And that's I Omega is radiance per second. So if I combine these two Okay, sorry. I meant to write this over here. Omega is in radiance per second. If I combine these two, you end up with kilograms times meter square times, radiance per second, but a radiant you might have seen when I talked about the fact that a radiant is really a meter over a meter. So it's a ratio of 2 m. So what we do is from this year, we just get rid of the radiance because we can think of as the meters canceling out and you're left with kilograms meters square over second. So that's how this comes about. Um, so the units are different for these two guys. Okay, Um, so another difference between them is that linear momentum is absolute. If you're masses 10 your masses. 10. It doesn't matter if your velocity is 10. It's always 10 unless you no long as it doesn't change. And then you just multiply those two. However angular momentum is relative, and what that means is that it depends on the axis of rotation, just like torque. So if you remember torque, if you push a door here with a force of 10 you get a different torque than if you push here. Same thing with angular momentum you could you could have the same object spinning a same speed. But if it's spinning at a different distance from the center, it's gonna have a different momentum. Okay, so momentum depends on the axis of rotation changes with the axis of rotation, which is something that doesn't happen with linear momentum. Last point I wanna make here is not to confuse angular momentum, which is what we just talked about, and it's l equals Iomega with moments of inertia. Okay, So even though you have angular Momenta, um, it's not the same as moment of inertia. These are two different things. In fact, moment of inertia is part of the momentum equation. It's this guy right here. Hi. Okay. So that these similar terms Moment of inertia, obviously, is I, which is the angular equivalent of mass. Okay, so don't confuse those two. Let's do a quick example and show you how to calculate. Um, angular momentum, foreign objects. I have a solid cylinder solid cylinder. This tells me I'm supposed thio. I'm supposed to use the moments of inertia equation of I equals half M R Square. It says that the masses five and the radius is true. So if you want, you can actually calculate this. I equals half mass is five two squared. So this is gonna be the moment of inertia of 10. It's as it rotates about a perpendicular access through its center with rpm. So here's a solid cylinder. Okay, On access to the center of the solid cylinder is a disc, But imagine this was a long cylinder. Anak says That's perpendicular to it. It's just an access through the cylinder like this perpendicular. So 90 degrees with the face of the cylinder, and it rotates about its center, which means it just does this right. So they're just rotates around itself. Um, and the equation for that when you have a rotation like this is is this right here? Okay, so it's rotating with an rpm of 1. 20. I want to know what is the angular momentum about? It's central axis. So, basically, what is l Well, l is I Omega. I know. I we just got that It's 10 but we don't have omega. But, you know, hopefully by now, you're tired of doing this. You know that you can convert RPM into Omega Omega is two pi f. Um, and then frequency can change into our PM, So frequency is our PM over 60 so I can replace this with our PM over 60. Okay, so it's gonna be two pi 1 20/60. So it's gonna be four pi. Okay, so I'm gonna put four pi here, which means l is going to be 40 pie, which is 126 kilograms meter square seconds. All right. Very straightforward. Just plug it into the equation. The only thing you have to do is convert rpm until makeup. That's it for this one. Hopefully, this makes sense. Let me know if you have any questions and let's keep going

When solid sphere 4 m in diameter spins around its central axis at 120 RPM, it has 1,000 kg m2 / s in angular momentum. Calculate the sphere’s mass.


A composite disc is built from a solid disc and a concentric, thick-walled hoop, as shown below. The inner disc has mass 4 kg and radius 2 m. The outer disc (thick-walled) has mass 5 kg, inner radius 2 m, and outer radius 3 m. The two discs spin together and complete one revolution every 3 s. Calculate the system’s angular momentum about its central axis.