 Now we can look at rotational kinetic energy. We'll start by reminding ourselves about kinetic energy in general. It was the energy of motion. And in my linear equations, it was 1 half m v squared, where m is the mass and v is the velocity. If I have more than one mass, I find the kinetic energy for each one and then add them up. So now we got circular motion. So if I start with this kinetic energy equation, I realize that my linear speed, my v, is related to the angular speed where v was equal to r omega. And that means my equation becomes 1 half m r squared omega squared where r is the radius and omega is the angular speed or angular velocity. Because I'm squaring the velocities, it doesn't really matter what direction they're in at that point, so I can call them speeds if I need to. Now for a rigid body. In this case, my mass is spread out over the whole object. So here I've just marked a couple of specific little points. Of course the entire thing is full of different masses. But what I want to note is that as I'm going through here, each of these little masses is at a different radius out from the center. They are rotating with the same angular speed, however. So as it starts to rotate, what I see is some of them have kinetic energy that's smaller and some is larger. And I want to add up all these little kinetic energies. So the way I rate that is that the sum of all the little kinetic energies is equal to the sum of each little one is an Mi or an Ri. Now since the one-half and the omega squared part aren't changing, I see all I'm really having to sum are these parts. These parts are multiplied by the whole thing. And that's going to give me the rotational kinetic energy. Now if I take that equation, what I recognize is that in this chunk of it, I've seen that before. That's actually my moment of inertia. And some people actually use this rotational kinetic energy equation as the justification for defining the moment of inertia this way. And so that means the more common form of writing the kinetic rotational energy is one-half i omega squared, the moment of inertia times the angular velocity squared. Now let's just compare these two equations here real quick. I've got my linear kinetic energy and my rotational kinetic energy. My M is my mass, which is like a linear inertia. My I is like a rotational inertia. So I've got the linear and the rotational quantities. Similarly, my V is my linear speed and my omega is my rotational speed. So these two equations are parallel each other. One for linear, one for rotation. Now I can take a look at the units. And starting with moment of inertia, that's kilogram per meter squared. Not kilogram per meter squared. Students make that mistake often. Kilogram times meter squared. And my omega is in radians per second. But it's squared, so it's really radians squared per second squared. Then if I remember that radians is really just a placeholder unit, I can see that all of this put together is really a kilogram meter squared per second squared. And going way back to our early discussions, that is the joule, which is the unit I expect for any kind of energy, including kinetic energy. So that shows me my rotational kinetic energy.