Converting Between Linear & Rotational

by Patrick Ford
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Hey, guys. So you may remember that one of the very first things I showed you in rotation is how we can connect linear displacement, Delta X, an angular or rotational displacement. Delta data using a tiny equation. Well, there's two more equations that we can use to connect velocity and acceleration between linear and rotational. Okay, so let's check it out. All right, So we have these tiny equations that air going toe link they're going to connect that are going to allow us to convert from one to the other between linear and rotational. Now, linear, uh, we're also gonna refer to linear as tangential, right? Linear, tangential. Both of these are going in a straight line. Um, gonna connect on linear or tangential to rotational, which is also referred to as angular. So it's important, you know, that these words are have to mean the same thing. Okay, so the linear variable is X in the rotation equivalent is Delta Theta. Okay, now, from that we get that Delta X is the change in position to change in X and Delta data is a change in angular or rotational position. Fada and the way that Delta X notated connect is by this equation right here. We've used this similarly. The connect to it so make to its angular equivalent Omega using a very similar equation. So Delta X is our delta data and V tangential. This t here means, um tangential velocity is our omega. Okay? And I want to point out that there's a pattern here. This is the linear of the linear variable. Our and the rotational variable. Same thing here. Linear variable are rotational variable. I'm gonna remove all these little circles. So it's not messy simply with a an Alfa Hey is going to be our And if you see the pattern, the equivalent of es w the equivalent of a is Alfa. Okay? And this is also the tangential acceleration. Okay, so these are the two new equations that we're going to be able to use now. When did they come up? Usually it's on a problem like this. You have a disk and this disc spends with angular speed omega. Well, if you pick a point in this disc, right, a point here. And I want to know what is the velocity, the linear velocity of this point? Well, this point moves with linear velocity or tangential velocity that looks like this D t. You might remember that when I have a point going around the circle. The point has tangential velocity, and it also has a centripetal acceleration. Well, it turns out that VTs connected Tau Omega by this equation, V t equals R omega, which is a very, very, very useful relationship equation. Okay, so let's get going. I want to quickly mention that there are four types of acceleration I already mentioned to. Here we have a C and actually mentioned three. We have a C, we have a T, and we have Alfa. There's 1/4 1, but we're gonna talk about that later. Um, I wanna just be very clear here that this equation right here 80 equals R Alfa refers to the tangential acceleration. Okay, It doesn't refer to the centripetal acceleration. It doesn't refer to the angular or rotational acceleration. Okay, so there's four types of acceleration. Most of them have two names, so it's gonna be a mess. But I'll show you pretty soon. Okay, A few more points here, whenever you have a rigid body or a shape. So let's say this is a cylinder, right? Let's say this is a cylinder that spins around self. Okay, All rotational quantities. Delta, Fada, Omega and Alfa are the same at every point. So let me show you this real quick. Um, illustrate this a little bit. So let's imagine a line here. And then there's point. Um, there's a little Imagine this is a huge disk, and there's people on top of it or whatever. Right, So you have a guy a over here on that point and guy bees over here. Now imagine that this disk spins from here to here. Okay? To that point right there. Now, Guy A is going to be here, and Guy B is going to be here. Notice how they all spin on the same line, right? So if I'm here and your here and this spins, we're still in the same place, right? We're moving together. Okay, so our delta feta are changing. Angle will be the same. All right, because and And by the way this happens, even if we're not in the same line, it's just easier to sit if it's in the same line. Delta there is the same. And because Omega is defined in terms of Delta Theta Delta theta over Delta T Omega is also going to be the same. And since Omega is the same, Alfa depends on Omega. All these three things are the same. Okay, long story short. If you're in a circle, all the objects on top of a circle have the same delta theta as they move. They're gonna experience the same Alfa and the same w. So although the rotational quantities will be the same Okay. However, the linear speeds might be different since they depend on our which is radio distance okay or distance to the center. That's another way to think about it. Okay, they might be different. So the best way to illustrate this is by doing example do a very straightforward one. So I have a wheel of Radius eight. So let's draw this here. Um, put a little radios here radius of this week. It was 8 m. It spins around its central axis. Eso What that means is that imagine a circle and imagine a sort of an invisible line through the circle. Right? Um, invisible line through the circle, and it's free to spin around that invisible line, So I'm gonna draw this here. You don't have to draw it. Um, I'm gonna delete it. Imagine imaginary line that goes through this thing almost as if you stuck a thing through it. And then it's free to spend around that. Okay, that's what that means. Let's get this out of here. Um, basically, it spins around its center, which is how this thing is Always work. A 10 radiance per second. So that's our omega. He's going to be, um, 10 radiance per second. We wanted all the angular and linear speeds at different points. So I wanna know at a point in the middle of the wheel in the central axis. So we're gonna call this 0.0.1 at a distance 4 m from the center. Um, if the radius is 8 m, 4 m is halfway in. I'm gonna draw this here thistles point to and at the edge of the wheel, 0.3. Okay, so the but we want to know is we want to know V one V two and V three, and I wanna know Mega One Omega to Omega three. Okay, that's what it says. I want the angular, which is Omega in linear V speeds at these three points. Okay, so first thing is to realize that all these points have the same because they're all the same disk. They have the same omega and that Omega is the same omega as the disk. Okay, so that's the first part. Omega one equals omega two, which equals omega three, which equals omega disk. So this is more of a conceptual to know that you to know if you know that So all of these will be 10 radiance per second. So please remember, all of them are the same, and they are the same as the disk. If you're on top of a disk you're spending with the disk, you have the same omega. What about view on V two v three? This is gonna be a little bit different. The point, the tangential or linear velocity of an object on a disk is given by V tangential. So V one tangential, which is our omega in this case are one omega one. Now all these objects have V two t. I'm gonna write this for all of them are two mega two and V three t is our three Omega three. Now, all these objects have the same omega, but they have different ours, which means they're going to have different V's. Okay, so let's calculate this real quick. Um, the first are here is how far from the center is that point. Remember, R is the distance to the center. The first point is at the center, at the middle of the wheel. What's the difference? The distance from the center to the center zero. So our one is actually zero, so it doesn't really matter what this is. Thea answer will be zero. Okay, we'll talk about that in a second. Let's go to the next one are two is at a distance of 4 m from the center. So this is four and Omega is 10. So the answer is 40 m per second and this is at a distance. Eight. It's at the edge, W s 10. So this is 80 meters per second. Okay, so these are the answers. Let me talk about real quick. Um, if you're at the edge, you move faster, right? So think about you and a friend in inside of a carousel. That's spinning. And if you're at the edge, You're going to feel faster. You are, in fact, moving faster on the linear direction. Anyway, you have a harder force, a stronger force pulling into the middle. Okay, so if you're at the edge of a spin, you are faster. If you are dead at the center, right. If you could be in the center of carousel in the spinning, you're basically doing this right. You're rotating in place and you have no V. Velocity is when you're moving sideways. When you're spinning the place, that's w Okay, so if you're spinning in place, um, as opposed to spinning sort of. You're just sort of doing this spinning in place as opposed to spinning like this. Okay, so you spin around yourself, so you have no velocity on Lee W Okay. And if you're at the edge, you are faster. Okay, That's it. Let's try the next example