ï»¿ >> All right, let's talk a little bit about rolling motion and the idea behind rolling motion is this. We're going to take a wheel and we're going to send it rolling on down the road. It's going to be spinning around and it's also going to be moving in that direction. And that speed is the velocity of the center of mass. Now let's see if we can figure out the relationship between the two. Okay? So when the wheel rolls along and it goes one full circumference, how far has it gone in terms of linear distance? Namely if I mark the tire here with a spot, then I go until that spot comes back down to the bottom of the tire. How far have we gone? Well it's just the circumference of the wheel. Right? And the circumference of the wheel is just 2 Pi r. Right here is the radius, r. So that's if we go one full circle. One full rotation. Okay? But let's say we don't go one full rotation. Let's say we go some smaller amount. This thing's going to roll along and now we've gone some other amount, which we don't know exactly what it is yet, but let's call it Theta. How far has that wheel gone? It has gone a distance r times Theta. A full rotation is 2 Pi, so we would get back to 2 Pi r. If it's a partial rotation then we just go a distance r times Theta, which is also known as the arc length. All right. Can we somehow figure out now what the velocity, Vcm, is for this case? Well yeah, I think so because how I increase my distance as a function of time, that's my velocity. Velocity is just going to be ds/dt. But we know what s is. It's right there, r Theta. So this is d/dt of r Theta. R is a constant. That's just the radius of the wheel, so that can come right on out, and we get our relationship between V and Omega, like we knew before. V is equal to r times Omega. Specifically, this V is the velocity of the center of mass. If I look at the center of mass of the wheel, which is right in the center of the wheel, what's it doing? It's rolling along with speed Vcm. What about acceleration? Well if velocity is that, then we can calculate acceleration because acceleration is a derivative of velocity. So the acceleration of that center of mass is going to be the derivative of the velocity with respect to time, but now we know what that is. It's right there, r times Omega. Again, r is a constant, and so it comes out of the derivative. And we have r d Omega dt, but we know exactly what that is. That is Alpha. And so the center of mass acceleration is equal to r times Alpha. Okay? So all these things relate to each other through the radius of the object. And now let's think about how much energy you might have in a rolling wheel. So let's talk about the energy in pure rolling. Pure just means no slip. Okay? The wheel isn't sliding relative to ground. It's always sticking to the ground. Okay? So here's our object. There's a wheel that's rolling along. And it has some velocity center of mass as it goes. What is the energy in this thing? Well the energy is going to consist of two things. It's the kinetic energy of that center of mass moving plus the rotational energy of the object rotating on its axis. We know what this is. This is 1/2 m V squared. What about this guy? That guy is 1/2 I Omega squared, where this rotational angular speed is Omega. But if you are pure rolling there is a relationship between the two. Namely, if the wheel has radius r, then we know that Omega is Vcm over r. And now we have a bunch of terms that are the same. One-half is the same in both of them. Vcm squared is the same in both of them. And so we can write down the total energy now as the following. One-half times the quantity m plus I over r squared. All of that times Vcm squared. This is the total energy in an object rolling along. If it's not rolling then you drop out this last term. There's no relevant moment of inertia. And so you just get back to 1/2 mV squared. But if the object is rolling along then there's energy, not only in the translational kinetic energy but in the rotational kinetic energy. And so wheels that are rolling along -- At Vcm have more energy in them than a box sliding along at Vcm. This energy in the wheel is bigger than the energy in the box sliding. And this should make sense to you. Why should it make sense? Because pretend you started moving along with the wheel. Right? You saw this wheel rolling along, and you were in a car driving next to it. To you it doesn't look like it's moving anywhere, but it's certainly rotating. Right? And so it looks like a stationary wheel that is rotating. Anytime you have things that are moving relative to you, they have kinetic energy in them. It doesn't matter if those things in the wheel are rotating in a circle. They still have some speed relative to you, whereas if you drive along next to the box and you look at it, the box looks stationary. It has zero kinetic energy because nothing is moving relative to you. Okay? So that's a nice way to think about this and realize that there has to be more energy in the wheel than in the box, if they're the same mass and Vcm is the same in both those cases.