We're talking about rotational motion, but we do want to connect it to circular motion, because that comes up in a lot of problems and everyday occurrences. We're going to take our wheel, and we're going to set it into rotational motion. Definitely rotational. But we're going to think about a point on the wheel, and you'll realize that any little point on the wheel it's actually going through circular motion. The wheel is rotating, but that little point, the yellow piece of tape is going circular. Circular is a special translational motion. If you zoom in really tight on that yellow piece of tape and watch it fly by, if you just look really narrow, it looks like it's just flowing by in 1D motion, with a little bit of curvature. These are the connections we want to think about. Here it is. Here is our wheel going around and our object might be here at one point, our piece of tape might be there, and then later maybe it's there. If I wanted to do this with proper vectors, I would say the motion is going this way, therefore the angular velocity vector points out like that. If I use the right - hand rule, my fingers go around with the motion, my thumb sticks out in the omega vector. Let's think about the wheel having a radius R. We're going to think about the difference between these two positions. We'll call them 1 and 2, and let's say it goes around by an angle theta. There's a little brief part of that circular motion. The first relationship we want to look at is not the velocity. That's usually where books and people start. But let's go all the way back to the distance. This is the simplest fundamental one that often gets skipped. When we first started in a regular translational motion, the first thing we thought about was distance and displacement and position. Let's look at the relationship here. If we were going to talk about distance in a rotational motion, it's theta. It goes around in theta. But we can connect that with S. S equals theta times R. This is the first one, and S is the arc length. It's this path. It's the length of that path going around the angle. It is a curved path, it's not even a straight path. It is not the displacement from 1 - 2. We're not going that far into translational motion back to day one. But in translational motion we talked about the distance of a path, so it's like the distance of this curved path. That's a very simple relationship that we often don't write down when beginning physics. It really, if you want to think about it, it just comes from the formula for the circumference. You know that if you go all the way around, that's a circumference, and that equals, well, what's theta all the way around, is 2 pi R. You can think of it that way. It's really just the fundamental definition of an arc length. Those two positions went through an angle theta, and through an arc length theta R. Let's see. Now, let's look at velocity. Well, how we get from distance to velocity is take a derivative. We're going to do a time derivative of this. Time derivative of moving around the circle, ds dt would basically be the speed v, and then R is a constant, and theta is changing, so we say d theta dt. But d theta dt we know is omega. We can also say the speed, v, is R omega, for this object on the edge of the wheel. This is just a speed. If we wanted to get technical and call in a vector, we know this doesn't really tell us the direction. This just tells the magnitude. The direction of v, we know from our circular motion is tangent to the circle. This would be the vector v, but this quick formula just gives us the speed. It clearly doesn't give us the direction because the direction of omega and v are not the same. I'll say the direction if you need it, is tangent to the circle. Now you can see it depends on where we put the little object. For now I've put the object all the way at R. What if I put a second object here? I'll find that yellow piece, and now I'll put a little piece of green tape closer in like that, and I'll give it a spin here. Now we can zoom in and watch both pieces of tape go by, and see which one is going faster? As you can see, the yellow is going faster, and the reason is the speed you get is proportional to the radius. If we have the yellow piece of tape at big radius and the green piece of tape at smaller radius, the yellow is going faster than the green, but they're both going at omega. What do we do in kinematics? If we don't know what to do, we take a derivative. Let's take another derivative to get the acceleration. Yes, derivative of speed would be the acceleration or the magnitude of the acceleration. The derivative here, R is a constant, we're up to d2 theta dt2 or d omega dt. That is just the angular acceleration that we talked about, alpha. A equals R times alpha. Now, like here, this is just a magnitude. We've got to add a little bit more detail to the story. I need to put a t here before I box this equation. Whoops, I just invented a letter. This is only the tangential acceleration. The tangential acceleration you can think of as the component due to speeding up or slowing down. If you have your wheel spinning at a constant omega, then this is zero, there is no tangential acceleration. But if you have a case where the wheel is speeding up because you maybe have a weight pulling on it and creating a torque and speeding it up or if you're slowing it down like we did in the other demo when we stop the wheel, then you'll have a tangential acceleration. But you want to remember that there's also that other. When we're thinking about circular motion, there is also the centripetal acceleration. Remember that there was a_c is the speed squared over R. This isn't replacing this, is just another component. This is still true. While the object goes around, it still has centripetal acceleration. We could even though write it in terms of rotational quantities if we want. You can see if we just plug this in here, we get R omega squared. There you go. We get both components of acceleration. Just to draw them and to try to really clarify what it will look like in a diagram, because people get very concerned about tangential and radial acceleration. If we have our circular motion, and we have an object here, and it's got omega here, because it's going around like that, it's got speed like that. Let's say omega and v are increasing. Lets say the magnitude of omega is increasing. Let's think about the total acceleration then. Well, if it's going that way at a speed v at this moment in time, it has centripetal acceleration pointing in like that, and it also has a_c equals R omega squared, but it also has tangential acceleration, because it's speeding up around the curve. That's equal to what we just got, R times the angular acceleration. That is also center to v, it's increasing, it's along its tangent to the circle. We'll draw that one like that. There's the at part. The total acceleration is just the vector, sum of those two. Since they're always perpendicular to each other, they really just make vector components, and a little Cartesian coordinate system it spins around the circle. If I were to draw the total acceleration vector, and there it is, a. For this motion, we always break it into those components. Sometimes you'll see this called the radial acceleration. If its radial, that just means the positive direction is out, and you say it's R squared omega, but you put a negative sign, because centripetal by definition points in, but radial by definition points out, so that little sign might throw you off every now and then.
