Hey, everyone. So in this video, we're gonna talk about something called the moment of inertia, which you can think of as kind of like the rotational equivalent of mass. Remember that when we talked about linear motion, mass was the quantity of an object's resistance to changes in velocity. Well, similarly, what we're gonna see is that the moment of inertia is the same idea but for rotational motion. It's the amount of resistance that something has to changes in its rotational velocity. Alright? So I want to talk about the differences between these 2, and then we'll do a couple of examples together. So let's check it out.

Before we start, I just want to recap everything we've talked about with linear motion. Remember that when we talked about linear motion and did linear motion problems, we used our 3 to 4 equations of motion, and they did not depend on the mass. Right? We didn't have mass anywhere in those motion equations. Then when we started talking about energy and forces, mass was important. Right? We have 12mv² and then we also have F = ma. So similarly, before we start talking about rotational kinetic energy and rotational forces, we're gonna have to talk about what the equivalent of mass is for rotation, and that's what this moment of inertia is.

Remember, with linear motion, mass was the amount of resistance to linear acceleration. Right? If you have a higher mass, you have a higher resistance to changes in velocity. We call that property, that resistance to change, inertia. Remember, inertia just means that you want to keep doing whatever it is that you were doing before. If you're at rest, you stay at rest. If you're in motion, you stay in motion. That's inertia. We can actually see this directly from F = ma. Right? If mass is m, then if we have F = ma and you rearrange for acceleration, acceleration is just force divided by mass. So we can see here that if your mass increases, then your acceleration decreases. Right? If you have higher mass, your change in velocity is lower. And so we call that that resistance, that was basically what mass was. So a higher resistance just means that you have a higher inertia.

In rotation, what happens is that the moment of inertia is the amount of resistance to angular acceleration. With linear motion, mass was resistance to linear acceleration. With rotation, the moment of inertia is resistance to angular acceleration. It depends on 2 things, it depends on mass and also the distance to the axis. With linear motion, it only depended on mass, but with rotational motion, the moment of inertia depends on mass and also how far you are from the axis of rotation.

So I want you to picture a yo-yo that's sort of going around like this, on a string. And if it's at 10 centimeters versus 20 centimeters, this one has lower inertia, but at 20 centimeters because you're doing a bigger circle, you have a higher resistance to rotational motion. This combination of mass and distance to the axis is called the moment of inertia. Both of those things make up the moment of inertia. Now that moment of inertia takes the letter I, so you can think of I for inertia, and it's the rotational equivalent of mass as we've said before. You can think of it as rotational inertia. It's the resistance to change in angular or rotational velocity.

There are 2 kinds of objects that we're going to cover in your problems. You're going to see point masses and also rigid bodies or other shapes. For a point mass, it would be kind of like a yo-yo or a ball or marble on a string that goes around in a circle. So imagine that you have a little string like this and you've got a little point mass here that's on the end and it's going around in a circle. The radius of this circle that you're making is little r. For a point mass because it's so small, a lot of times your problems will say it's a very small object or something like that. We can say that its radius is equal to 0. This little r is a distance, but this big R is a radius. We're going to treat these objects as if they're really tiny; they don't have a lot of volume.

On the other hand, you might see something called a rigid body or a shape something like a cylinder or a sphere, and rigid bodies do not have zero radius. They actually do have some size. So a very common one that you're going to see is a cylinder like this. You have a cylinder like that, and you'll have an axis of rotation that goes through, and you're going to try to spin the cylinder on its axis. This object does have a radius because it does have finite size, and that's going to be the big R, okay? That's the difference between those 2. In one, you have a distance little r and in one you actually have a radius.

So let's talk about the moments of inertia for these 2. For a point mass, the moment of inertia equation is actually very straightforward. It's just m times little r squared, and again that little r is just going to be the distance to the axis of rotation. For rigid bodies or shapes, if you have a cylinder or a sphere, it's going to be different. You're going to find this moment of inertia by looking it up in a table. Your textbook will have a table. You'll see some pretty shapes and they'll have different formulas for the moment of inertia. But one thing I want to mention here is that the general format for a moment of inertia is going to be some kind of a fraction times mr². So you'll see that for point masses it's mr² and for something like spinning a rod through the center, you would have some distance squared.

Let's just go ahead and get into our problems here. We have a system of point masses that are basically on the ends of this rope. We want to calculate the moment of inertia of the system if it spins about the perpendicular axis through the center of the rod. So we're going to calculate the moment of inertia of the system. In other words, we're going to calculate I_{system}. That's really what we're looking for here.

The axis, if it spins about a perpendicular axis through the center of the rod. So this is the rod like this. We want perpendicular, remember that just means 90 degrees. There are 2 ways you could do this. If the rod is like this, you could have this as my axis or you could also have this going back and forth as your axis. So obviously, it's going to be this because it's going to spin around sort of horizontally like that. So you can kind of imagine that it's going to spin like this, right? Sort of on the same plane as you. That's how it's going to spin, through the center of the rod as shown.

How do we calculate this moment of inertia? Well, in general, the moment of inertia of a system of objects is just going to be the sum of each moment of inertia of the objects that make it up. What do I mean by this? In this problem, we have a rod, and by the way, the rod is massless. So what we have here is that m_{rod} = 0, but we also have these two point masses on the end. I'm going to call this m_{r}, because it's 4 kilograms, and this one's going to be m_{l}, which is 3. Both of these pieces, these little point masses here, are some distance away from the axis of rotation. And what we said here is that for point masses, there is a moment of inertia, m_{r}. So we actually have 2 of them in this problem. And we also have a rod that is also spinning. Everything is spinning in this problem, the