Hey, guys. So now that we've seen how rotational position and displacement relate to linear position displacement, we're going to look into velocity and acceleration. Let's check it out. So the rotational equivalents of linear velocity and linear acceleration are rotational velocity and rotational acceleration or angular velocity and angular acceleration. Similar to how x becomes theta and delta x becomes delta theta in rotation, v and a will take different letters as well. Average velocity, if you remember, is simply delta x over delta t, and the units are meters per second. Average acceleration, or I should say acceleration, is delta v over delta t. So velocity, change in velocity over change in time, and it's measured in meters per second squared. That's if you are dealing with linear problems. Now if you have rotational problems, angular motion problems, instead of v we're going to use omega, ω. Now ω is a Greek w, so it's like a curly w. It's essentially a w. Now instead of delta x over delta t, remember instead of delta x, we don't have delta theta. So ω is delta theta over delta t. V is how quickly I can get from here to here. It's a measurement of how quickly I move between two points. ω is how quickly you spin in a circle. Okay? Remember also that this was in meters and these are in radians. So instead of meters, you have radians. So instead of meters per second, you're going to have radians per second. And the acceleration is very similar. Instead of a, we're going to have alpha, which is, a Greek a, and same thing here, acceleration is velocity over time, acceleration will be angular or rotational velocity over time. So it's delta ω over delta t, and it's radians per second squared. So you might start to see the pattern. The variables are x or delta x and then v and a, and they become theta, ω, and alpha. And the pattern is that English letters are representing linear motion and Greek letters are representing angular or rotational motion. These are all Greek letters, Theta, ω, and Alpha. Alright. So ω is a way to indicate how quickly something spins. Well, there are actually 3 additional variables that will help us describe how something moves, and they're related to ω, in fact, all related to each other. So you might remember I showed you, right here we just did ω, which is delta theta over delta t. And you might remember that we talked about if you have a complete revolution, then your delta theta is 2 pi. Well, the entire angle for a complete revolution is 2 pi. Now the time that it takes for a full cycle, the time for a full cycle is called the period, T. T is period. So one way that you can rewrite ω is not just delta theta over delta t, but also 2 pi over t. And remember also that period and frequency are inverse of each other. Okay? So instead of 2 pi over t, I could also write this as 2 pi f. Okay. So you have ω is a measurement of how quickly something spins, the period is a measurement of how quickly something spins, and frequency is also a measure of how quickly you sb.spin, and they're all related by this equation.
Last one we're going to talk about is RPM. Now RPM stands for revolutions per minute. So 1 RPM is 1 revolution per minute. Hertz, which is a unit of frequency, okay, frequency is measured in hertz, is 1 revolution per second. So you can see how these 2 are related. So for example, if I tell you something spins with a 120 RPM or at a rate of 120 RPM, RPM is simply 120 revolutions per minute. And what I can do is I can put a minute up here and convert this into seconds by dividing by 60. One minute equals 60 seconds. I can do this and look what I end up with, 120 divided by 60 is 2. So I end up with 2 revolutions per second. Revolutions per second is frequency. So if you have RPM, you can convert to frequency by dividing by 60.
So I'm going to quickly write another equation here, which is that frequency is RPM over 60. Now, we have a way to connect all of these guys. ω, T, f, and rpm are all connected. Typically, you're going to convert from any of these 3, T, f, or RPM back into ω using these equations. And the idea, you'll see more of this later, is most of the equations I give you will have ω, but not any of the other letters. And I have a little diagram to connect all of these things. So if you have RPM, you will want to convert it to frequency. And the way you do this is by using f equals RPM over 60. And then from frequency, you can convert into either the period using the fact that the period is the inverse of frequency or you can convert into ω using the fact that ω is 2 pi f. And obviously, you can convert backwards in any direction. Generally, you want to end up here, but you might have to go from, let's say, ω to RPM. We'll do some of this stuff. Alright, cool. So these are the 4 units that tell you how quickly something spins and you may have to convert among them. Trying to highlight this. There you go. Alright. One last point before I do an example and I've mentioned this briefly before. Rotational equations, which is what I'm showing you a few already by now, they work for 2 different situations. One is when you have a point mass. A point mass is a tiny object of negligible size that spins around the circular path. We call it a point mass because we just represent it by a point that has no volume. Okay. I'm going to say that the radius of this object is 0. Imagine a sphere with a radius of 0; it has no radius, it has no volume. Okay. Radius is 0. You could also write if you want volume equals 0. You can actually write out volume, so you don't think it's velocity. It's a point. It could be a small object that we simplified down to a point. So that's a point mass. The other one is when you have a rigid body, which is something where the radius is not 0, the radius is greater than 0. So it has volume. Okay. And I refer to this as either a rigid body, that's sort of the classic textbook name, or a shape. The reason why I like to think of it as a shape is that in these problems, usually, you will be told what the shape is. So if I tell you you have a small object, that's a point mass, And if I tell you have a cylinder, usually I'll tell you what the radius is, and then you treat that a little bit separately. Okay. So you can either have a point mass, I'm going to draw a tiny little m going around the circle, and the circle has a radius r. In this case, little r is the distance from the circle to the edge of the circle, and you're going around at the edge, and r is the radius of the circle, and then little r is how far you are from the center. Those are the same thing. Or you can have a cylinder, for example, spinning around itself. So let me draw like a little cylinder here. Looks like a cake and a cylinder that rotates around itself and that cylinder has a radius of r. Okay. So you can have either one of these two situations. We'll look at both a lot. Cool. So that's a quick intro, getting some equations, how to connect things together between these 4 different variables, and we're not going to do a problem. So I have a 30-kilogram disc.]
