Intro to Angular Momentum: Study with Video Lessons, Practice Problems & Examples

Angular momentum, denoted as L = I⁢ω, is the rotational equivalent of linear momentum, which is p = m⁢v. Here, I represents the moment of inertia, and ω is the angular speed in radians per second. Unlike linear momentum, angular momentum is relative and depends on the axis of rotation. Understanding these concepts is crucial for analyzing rotational dynamics in physics.

Intro to Angular Momentum

Hey, guys. So in this video, I'm going to 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 going to have linear momentum. Now we used to call this just momentum because we only had one type. But now that we're going to have 2 types, we have to make a distinction between them. So linear 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 an object's mass, its velocity, and that's its momentum. If you have rotational speed, that's \( \omega \) that is omega, you have angular momentum or rotational momentum, and it's going to take the letter \( l \) instead. Right? So instead of \( p \), it's \( l \). Now instead of \( m \), you're going to use the angular equivalent of mass, which is \( I \). Hopefully, you get that. And the angular 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 going to be a little bit different. It's going to be 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 \( mr^2 \). But this works for all of them. Mass is kilograms times meter squared. Okay. And that's \( I \). Omega is radians per second. So if I combine these 2, okay, sorry, I meant to write this over here. Omega is in radians per second. If I combine these two, you end up with kilograms times meter squared times radians per second. But a radian, you might have seen when I talked about the fact that a radian is really a meter over a meter. So it's a ratio of 2 meters. So what we do is from this here, we just get rid of the radian because we can think of it as the meters cancelling out. And you're left with kilograms meter squared over second. So that's how this comes about. So the units are different for these 2 guys. Okay. So another difference between them is that linear momentum is absolute. If your mass is 10, your mass is 10. It doesn't matter. If your velocity is 10, it's always 10 unless, you know, as long as it doesn't change. And then you just multiply those 2. 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 have the same objects spinning, at the same speed but if it's spinning at a different distance from the sensor, it's going to 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 want to make here is not to confuse angular momentum, which is what we just talked about. And it's \( l = I\omega \) with moments of inertia. Okay. So even though you have angular momentum, it's not the same as moment of inertia. These are 2 different things. In fact, moment of inertia is part of the momentum equation. It's this guy right here, \( I \). Okay? So don't confuse those two. Let's do a quick example and show you how to calculate angular momentum for an object. So I have a solid cylinder solid cylinder. This tells me supposed to I'm supposed to use the moments of inertia equation of \( I = \frac{1}{2}mr^2 \). It says here that the mass is 5 and the radius is 2. So if you want, you can actually calculate this, \( I = \frac{1}{2}(5)(2^2) \). So this is going to be the moment of inertia of 10. It says it rotates about a perpendicular axis through its center with a 120 RPM. So here's a solid cylinder. Okay. An axis through the center of a solid cylinder, this is a disk, but imagine this was a long cylinder. An axis that's perpendicular to it, it's just an axis through the cylinder like this, perpendicular. So 90 degrees to the face of the cylinder and it rotates about its center which means it just does this. Right? Cylinder just rotates around itself, 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 120. I want to know what is the angular momentum about its central axis. So basically, what is \( l \)? Well, \( l = I\omega \). 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 \( 2\pi f \), and then the frequency can change into RPM. So frequency is RPM over 60. So I can replace this with RPM over 60. Okay. So it's going to be \( 2\pi(120/60) \). So it's going to be \( 4\pi \). Okay. So I'm going to put \( 4\pi \) here, which means \( l \) is going to be \( 40\pi \), which is \( 126 \) kilograms meter squared seconds. All right. Very straightforward. Just plug it into the equation. The only thing you have to do is convert RPM into omega. That's it for this one. Hoping 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. 7.46 kg
• B. 12.4 kg
• C. 29.8 kg
• D. 49.7 kg

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.

• A. 64 kg•m²/s
• B. 85 kg•m²/s
• C. 102 kg•m²/s
• D. 148 kg•m²/s

Here’s what students ask on this topic:

What is the difference between linear momentum and angular momentum?

Linear momentum (p) is the product of an object's mass (m) and its velocity (v), given by the equation $p=mv$. It is measured in kilograms meters per second (kg·m/s). Angular momentum (L), on the other hand, is the rotational equivalent and is given by $L=I\omega$, where I is the moment of inertia and ω is the angular velocity in radians per second. Angular momentum depends on the axis of rotation, unlike linear momentum, which is absolute and does not change with the axis.

How do you calculate the moment of inertia for a solid cylinder?

The moment of inertia (I) for a solid cylinder rotating about its central axis is given by the formula $I=\frac{1}{2}m{r}^2$, where m is the mass of the cylinder and r is its radius. For example, if a solid cylinder has a mass of 5 kg and a radius of 2 meters, its moment of inertia would be $I=\frac{1}{2}\left(5\right)\left(2^2\right)=10$ kg·m².

What are the units of angular momentum?

The units of angular momentum are derived from its formula $L=I\omega$. The moment of inertia (I) has units of kilograms meter squared (kg·m²), and angular velocity (ω) has units of radians per second (rad/s). Since radians are dimensionless, the units of angular momentum are kilograms meter squared per second (kg·m²/s).

How does the axis of rotation affect angular momentum?

The axis of rotation significantly affects angular momentum because it determines the distribution of mass relative to the axis. Angular momentum (L) is given by $L=I\omega$, where I is the moment of inertia. The moment of inertia depends on the mass distribution relative to the axis of rotation. For example, the same object spinning at the same speed will have different angular momentum values if the axis of rotation changes, as the moment of inertia will change accordingly.

How do you convert RPM to angular velocity (ω)?

To convert revolutions per minute (RPM) to angular velocity (ω) in radians per second, use the formula $\omega =\frac{2}{\pi }\left(\frac{\mathrm{RPM}}{60}\right)$. For example, if an object rotates at 120 RPM, its angular velocity would be $\omega =\frac{2}{\pi }\left(\frac{120}{60}\right)=4\pi$ radians per second.