**Concept:** Intro to Angular Momentum

Hey guys! 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. Remember, if you have linear speed, linear speed is V, you're going to have linear momentum. We used to call this just momentum because we only had one type, but now that we're going to have two types, we have to make a distinction between them. 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. You just multiply an object's mass to its velocity and thatÕs its momentum. If you have rotational speed that's w, that's omega, you have angular momentum or rotational momentum and it's going to take the letter L instead. Instead of p, it's L. Instead of m, you're going to use the angular equivalent of mass which is I, hopefully you get that. The angular equivalent of v which is w. You can think of this equation as like perfectly translating into angular into rotational variables. The units are going to be a little bit different, itÕd be kg*m^2/s. That's because of the makeup of the equation. 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 kg*m^2 and that's I. w is rad/s. If I combine these two, w is in rad/s. If I combine these two, you end up with kg*m^2*rad/s. But a radian, you might have seen what I talked about the fact that a radian is really a meter/meter. It's a ratio of two meters. What we do is from this here, we just get rid of the radian because we can think of as the meters canceling out and you're left with kg m^2 / s. That's how this comes about. The units are different for these two guys. 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 as long as it doesn't change and then you just multiply those two. However, angular momentum is relative. What that means is that it depends on the axis of rotation just like torque. 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 object spinning at the same speed but if it's spinning at a different distance from the center, it's going to have a different momentum. Momentum depends on the axis of rotation, changes with the axis of rotation, which is something that doesn't happen with linear momentum. The last point I want to make here is not to confuse angular momentum, which is what we just talked about and itÕs L = Iw with moment of inertia. Even though you have angular momentum, 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, I. I got these similar terms. Moment of inertia obviously is I which is the angular equivalent of mass. Don't confuse those two. Let's do a quick example and show you how to calculate angular momentum for an object. I have a solid cylinder. This tells me I'm supposed to use the moment of inertia equation of I = _ MR^2. It says here that the mass is 5 and the radius is 2. If you want, you can actually calculate this. I = _ mass is 5, 2 squared. This is going to be the moment inertia of 10. It says it rotates about a perpendicular axis through its center with 120 rpm. Here's a solid cylinder, an axis through the center of a solid cylinder to the disk but imagine this was a long cylinder, an axis that's perpendicular to it is just an axis 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. Cylinder just rotates around itself and the equation for that when you have a rotation like this is this right here. 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. L is Iw. I know I, we just got that, itÕs 10 but we don't have w. But you know hopefully by now, you're tired of doing this, you know that you can convert RPM into w. w is 2¹f and then frequency can change into rpm. Frequency is RPM/60, so I can replace this with RPM/60. It's going to be 2¹(120/60) so it's going to be 4¹. I'm going to put 4¹ here which means L is going to be 40¹ which is 126 kg m^2 / s. Very straightforward. Just plug it into the equation. The only thing we had to do is convert RPM into w. That's it for this one. Hopefully this makes sense. Let me know if you have any questions and let's keep going.

**Problem:** When solid sphere 4 m in diameter spins around its central axis at 120 RPM, it has 1,000 kg m^{2} / s in angular momentum. Calculate the sphere’s mass.

**Problem:** 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.