Rotational Velocity & Acceleration

by Patrick Ford
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Hey, guys. So now that we've seen how rotational position and displacement relate to linear position displacement, we're gonna 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. Angular acceleration similar to how X becomes Thatta 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 changing velocity of a change in time and it's measured in meters per second square. That's if you are dealing with linear problems. Now, if you have rotational problems, angular motion problems instead of the we're going to use Omega Omega. Now Omega is a Greek W, so it's like a curly W. It's essentially W now instead of Delta X over Delta T. Remember, instead of Delta X, we don't have Delta theta, So omega 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. Omega is a direction is a measurement of how quickly you spin in a circle. Okay, remember also that this was in meters and this is in radiance. So instead of meters, you have radiance instead of meters per second, you're going tohave radiance per second. Okay. And the acceleration is very similar. Instead of a we're going tohave Alfa, which is Ah, Greek A and same thing here. Accelerations velocity over time acceleration will be angular or rotational velocity over time. So it's Delta Omega over Delta T and its radiance per second squared. So you might start to see a pattern. The variables are X or Delta X and then v N w. I'm sorry, DNA, and they become Thatta w in Alfa. And the pattern is that English letters are representing linear motion and Greek letters are representing angular or rotational motion. Okay, thes air. All Greek letters Fada, Omega and Alfa. All right. So w ho Maiga is a way to indicate how quickly something, Spence Well, there are actually three additional variables that will help us describe how something moves and their related to W in fact, all related to each other. So you might remember I showed you right here with just a w eyes Delta data over Delta T. And you might remember that we talked about If you have a complete revolution, then your delta theta is two pi. Well, the entire angle for a complete revolution is two pi. Now, the time that it takes for a full cycle, The time for a full cycle is called period, period t t years, period. So one way that you can rewrite omega is not just Delta fade over Delta T, but also Delta. I'm sorry. Also to pi over tea. And remember also that period and frequency are inverse of each other. Okay, So instead of two pi over t, I could also right. This is two pi f. Okay, so you have Omega is a measurement of how quickly something spins. Period is a measurement of how quickly something spins and frequency is also a measure of how quickly you spin. And they're all related by this equation. Okay, last one we're gonna talk about is our PM now, rpm stands for revolutions per minute. So one rpm is one revolution per minute A hurts, which is a unit of frequency. Okay, frequencies emerging hurts is one revolution per second, so you can see how these two are related. So, for example, if I tell you something spins with 120 rpm or at a rate of 120 rpm rpm is simply revolutions per minute. And what I can do is I can say I can put a minute up here and convert us into seconds by divided by 61 minute equals 60 seconds. I could do this. And look what I end up with. 120 um, divided by 60 is too. So that with two revolutions per second revolutions per second is frequency. Okay, revolutions per second. Right here is frequency. So if you have RPM, you can convert to frequency by dividing by 60. Okay, so I'm gonna quickly write another equation here, which is that frequency is our PM over 60. Now, we have a way to connect all of these guys. Okay. W t f in our PM are all connected. Okay, typically, what you're going to do is we're going to convert from any of these three big T f or RPM back into Omega using these equations. Okay. And the idea you see more of this later is most of the equations I give you will have w 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 r p m over 60. Okay, that's I go from rpm to frequency, and then from frequency, you can convert into either period using the fact that period is the inverse of frequency. Or you can convert into Omega, using the fact that omega is two pi f. Okay. And obviously you can convert backwards in any direction. Um, generally, you wanna end up here, but you might have to go from, let's say, W two rpm. We'll do some of this stuff. All right, Cool. So these are the four units that tell you how quickly something spins and you may have to convert um, among them trying to highlight this. There you go. All right. One last point before doing example, I've mentioned this briefly before. Rotational equations. Which is what? I'm showing you a few already by now, um, they work for two different situations. One is when you have a point mass. Ah, point masses. A tiny object of negligible size that spins around these circular path We call our point mass because we just represented by a point that has no volume. Okay, I'm gonna say that the radius of this object zero imagine a sphere with a radius of zero has no radius. It has no volume. Okay, rate is 20 You could also right. If you want volume equals zero. I'm gonna actually write out volumes. I don't think its velocity, it's a point. It could be a small object that we simplified down to a point. So that's point mass. The other one is when you have a rigid body, which is something where the radius is not zero. The radius is greater than zero. So it has volume 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 there's a shape is because in these problems usually 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 that it's what the radius is on. Ben, you treat that a little bit separately, okay? So you can either have a point. Mass will draw a tiny little m going around a circle, and the circle has a radius are 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. Um, and R is the radius of the circle. And then little artist, how far you are from the center, Those are the same thing. Or you can have, like, a cylinder, for example spinning around itself. Let me drive like a little cylinder here, Um, looks like a cake and a cylinder that rotates around itself. And that cylinder has a radius of our okay, so you can have either one of these two situations will look at both lots. Cool. So that's a quick intro. Getting some equations, how to connect things together between these four different variables, and we're not gonna do a problem. So I have a 30 kg disk. So mass equals 30 um, radius, too. So let's draw this. It's just circle. The radius is two. This is a disc. It's a rigid body or shape. The radius of this thing is too, and he rotates at a constant 120 rpm. So I'm gonna right here that our PM equals 1 20 everyone I know it's period and angular speed. So for part A, I wanna know what is big team. And then for part B. I want to know what is Omega Omega? Which is W angular speed. Same thing as angular velocity. Same thing is rotational velocity. Rotational speed. Cool. So how do we tied together? Well, if you look at this little diagram here, we can convert from one to the other. I'm gonna convert rpm into frequency, and then I'm gonna convert from frequency into period and into w. Okay, So first we're gonna go Frequency equals R P m. Over 60. So it's 1 20 divided by 60 to the unit's air hurts. But I don't actually want frequency. I want period. So period will be won over the frequency. 1/2 five period is measured in seconds. Now to find angular speed. W I just use that w is two pi over tea, or I can use the W's two pi f I'm gonna use to P. F. Because it's more straightforward to pie. Frequency is too. So I get four pi, and if you multiply all of this, the answer is 12.6 radiance per second. Okay, so that's it for this one. Let's do the next one.