Hey guys. So now we're going to talk about rotational displacement. When you go around the circle multiple times, let's check it out. So if you make one full revolution, a revolution is a complete circle, a complete cycle, a complete rotation, a complete spin. There's all these words. If you make a full one around the circle, you've gone a total change in angle of 360 degrees. Everyone knows that a full app is 360 degrees, or \(2 \pi\) radians. Okay. Therefore, delta x, remember the delta x equation is \( \Delta x = r \Delta \theta \). That's how you link these two variables. Okay? It becomes \(2 \pi r\). And what I've done here is I replaced \(\Delta \theta\) with \(2 \pi\) because that's what \(\Delta \theta\) is if you make a revolution. And you might recognize this equation that \(\Delta x = 2 \pi r\). This is the circumference equation. The definition of circumference is the size, the length of the border of a circle, and that's linear distance. If you're driving your car around a circle, your odometer, which tells you how far you traveled, would give you a quantity equal to \(2 \pi r\). Okay? So that's the linear distance as you go in a rotational motion around the circle. Now that's if you spin once. If you spin once, you get 360. What happens if you spin twice? Then you get 360 times 2. So if you spin any n times, then your \(\Delta \theta\) is 360 times n or \(2 \pi\) times n, obviously, in radians. So instead of having a \(\Delta x\) of \(2 \pi r\), you get a \(\Delta x\) of \(2 \pi r \times n\), where n is the number of rotations. Okay. 2 more things, you may need to know. If you want to know how many revolutions you go through, and we'll do an example of this just now, all you got to do is divide your number of angles by either 360 degrees or by \(2 \pi\). Okay. For example, if I tell you I spun, 720 degrees, and I want to know how many how many revolutions that is, you divide by 360 and you get 2. That means I spun twice. Okay. Same thing with if it's in radians, it's in pi, so you can just divide it by \(2 \pi\). And the last thing is, let's say you go around the circle many times and you end up over here. Okay? If I want to know how far you end up, you don't got to draw this. If I want to know how far I end up, all you got to do is you keep subtracting by 360 until your angle is less than 360. So for example, if you, spun 410 degrees, and I want to know how far from 0 you end up. 410 is more than 360, so you made multiple revolutions. All I got to do is subtract by 360, and you see that the answer is 50. You keep doing this until your final answer is less than 360, which it is. So we're good to go. If it wasn't, you would subtract by 360 again. Okay? Same thing with, radians. If it's in radians, you just keep subtracting by \(2 \pi\) until the answer is less than \(2 \pi\). Let's do a quick example here. Alright. So starting from 0 degrees of the Strahler ball, circle, starting from here, Zero degrees is always the positive x axis. You make 2.2 revolutions. So we're using the letter n to represent the number of revolutions, 2 2.2, around a circular path of radius, 20. If you have a circular path of radius 20, that means that your radial distance from the middle, little r is 20. You could use little r or big r interchangeably. Little r is technically more correct because it's not, big r is reserved for the radius of like a disc. Little r is the distance from the center. Okay. But the words are used kind of interchangeably. Alright. So what is your rotational displacement in degrees? A. Rotational displacement is \(\Delta \theta\), and we want that in degrees. So I'm gonna put a little deg here to indicate that we want to do this in degrees and not in radians. Well, if you spin once, you spin 360 degrees. But if you spin 2.2 times, you just multiply them. Okay. And this is going to give you 792 degrees. 792 degrees is your, how much is spun. Cool, for that's it, for part b, how many degrees from 0 are you? Again, we're just going to subtract 792 until we get to a number that's less than 360. So I want to know how far from 0. Okay. So 792 minus 360, that gives you 432. We're gonna have to keep going because we're not below 360, minus 360. And then finally, that's the answer, 72 degrees. Okay? That's the final answer. And for part c, what is your linear displacement? Linear displacement, remember, is \(\Delta x\). And if I want to know \(\Delta x\), \(\Delta x\) is \(r \Delta \theta\). R is the distance here 20, and \(\Delta \theta\) has to be in radians. \(\Delta \theta\) has to be in radians. So I cannot use 792. Okay. It's just a little gentle 792. I cannot use that. I'm gonna have to use in radians. And in radians, this is going to be \(2 \pi \times 2.2\). Right? \(2 \pi\) is a full rotation times 2.2 because we rotated 2.2 times. So if you multiply all of this, put in the calculator pi is \(3.1415\), but your calculator has a button for that. If you do all this, you get the distance is 276 meters. Okay? That's it for this. I hope it made sense. Let me know if you guys have any questions.