Hey guys, so in this video, we're going to start talking about rotational motion, also known as rotational kinematics. I'm going to take you back all the way to the beginning of physics and we're going to look at basic motion, 1-dimensional motion, and we're going to contrast and compare the two. You're going to see that a lot of things are similar, but there are some differences. So let's check it out. All right. So rotational motion is when you have motion around a central point, around the central point. So imagine you have a tiny object that spins like this around a central point. It forms a circular path. Okay, that's one type. The other type is when you have a cylinder that spins around itself. Okay? Now you may remember that we described your location using position, which is variable x. Right? We used to call this simply position. But now that we have linear and rotational motion, we may want to specify that this is the linear position. Now if you don't see the word linear, you assume it's linear. The other one is going to be rotational position, which is where you are in a circle. Okay? So for example, if you are moving in a circle, think of this as a track and you can only move around that track either this way or this way, you're trapped, right? And there are two ways you could describe your position. You could do it by saying, well, this is coordinates X, Y. And then if I move over here, I have a new X and a new Y. Right? It's a 2-dimensional grid. It's a surface, so you could do that. The problem is that's more complicated than it needs to be because now I have x's and y's changing. What's actually easier is to use a single variable, a single number to describe where you are. And we do this using angles. Right? So for example, you may remember, that this is 0 degrees and this would be 90 right there. So we're going to say that this is let's make it up something 80 degrees. Okay? So that's easier because I'm using a single number to represent where you are around the circle. So that's what rotational position is, and it uses the variable theta. You might remember that theta is what we use to represent angles or degrees. Okay? Now notice here that I have the words rotational and angular. And I need you to know that these words basically mean the same thing. They're used interchangeably. So you see a lot of words like angular velocity just means rotational velocity. Okay? So these words are used interchangeably. All right. So whereas in linear motion, you use x, in rotational motion, we use theta. So what I'm going to say here is that x becomes theta. The x equivalent, in rotation is theta. Okay. Later on, you're going to get some equations where old equations, but instead of using x, we're going to use theta. All right? So let's look a little deeper into the differences between the two. So position is defined as how far you are from the origin. You may remember this. It's your distance from the origin. Rotational position is the same thing. It's how far you are from the origin. The difference is that the first one we measure using meters and the second one we measure using angles. Now you could do either radians or degrees, but we're going to use radians most of the time. Okay. So we're going to use radians, which is abbreviated rad. All right? Now origin, if you remember, origin is simply where x equals 0. So for example, let's say we got a line here and you are here, Okay. And then there are two points. Let's draw 3 points here. Just 2 points. Whatever. That's fine. And let's say that these two points are 10 meters apart. So this would be 0 and then this would be 10 and maybe you are at 7. Okay? So if this is x equals 0, this is where the origin is. Okay? But we could have done this a little bit differently. And then we say that your position x u, is plus 7. But we could have put the origin right here. We could have arbitrarily said, I want this to be x equals 0, and then this distance here is 3, so your x, x u would have been negative 3. And the point that I'm trying to make here is that origin in linear position is arbitrary. Arbitrary meaning up to you. You can change it and sometimes the problem will tell you but it could change. It does not have to be a fixed thing. In rotation, it's a little bit different. In rotation, the origin is still where position, in this case, theta equals 0. Okay. 0 degrees or 0 radians. Let me put a little meter here, 0 degrees or 0 radians. The difference is that whereas here, it's arbitrary, it's up to you, okay, up to you unless the problem tells you. In rotational position, the origin is always fixed. Okay? And it's always fixed. It's fixed at the positive x axis. Okay. 0 is always here. Remember the unit circle, this is always the origin. Okay? That's nonnegotiable. Whereas here, you could put it wherever you want if you're given that kind of liberty in the problem. Okay. The last thing is direction is also arbitrary. Direction is also arbitrary. You could say, up to you, you could say that this is positive or you could say that this is the direction of positive. Okay? Either or works and then you adjust accordingly. If you're in rotation, the direction is fixed. So clockwise, which follows the clock, goes this way, is negative, and counterclockwise, which goes this way, is positive. Okay? Direction here is also fixed. It's not up to you. All right? Now one quick note here, which is it might seem backward, right, and I like to think of this as backward. Why couldn't they have made the direction of the clock positive? Right? Why is it that the clock is backward? Well, it's because this stuff actually follows the unit circle and you might remember that the unit circle grows like this. The angles grow like this. The unit circle and the clock are backward from each other and we use the unit circle, and that's it. So those are the key differences between linear position and rotational position. We're going to quickly talk about the displacement now. So the rotational equivalent of linear displacement, so the position is x, displacement is changing position, which is delta x. Okay. Rotational position was theta, so rotational displacement is simply delta theta. Okay? So instead of delta x, the equivalence is delta theta. So if you're moving this way, we measure your delta x. If you're moving this way, we measure your delta theta. And these two quantities here, delta x and delta theta, are linked. They're connected. They can be converted from one to the other using the following equation, Δx=rΔθ. This r here, you can loosely refer to it as the radius. I'll talk about this a little bit more. But what it really is is a radial distance which is distance to the center. So I'm going to write the distance to the center. Okay. Radius would be the radius of a cylinder, but if it's a distance, a point spinning around a circle, then you're talking about distance to the center. That's a technicality. Don't worry about that too much. You might have seen this. You might remember this equation. You've seen this before. In math, this looks like this, s=rθ. In fact, most textbooks, I think every textbook actually, talks about this equation like this. But I like to use delta x instead of s because that's what you're used to, and delta theta because we're looking at the displacement. This is the arc length equation, and that's where this stuff comes from. Okay? So I'm going to use this version right here, and it should be fine. So quick points about this equation, really important equation, This equation speaks radians. What do I mean by speaks radians? Well, if you're plugging in a delta theta into this equation, we'll do an example just now. But if you're plugging in a delta theta, that number that you're plugging into equation has to be in radians. Otherwise, the equation doesn't work. Also, if you're instead of plugging in delta theta, you're solving for delta theta, the answer will be in radians. So either you're giving the equation radians or if the equation is giving you an angle, it's giving that in radians. That's why I say here that the input must be in radians. You have to plug in in radians for the equation to work, and the output will be in radians. If you get an answer out of that equation, if you get a delta theta out of that equation, you will be in radians. Okay? Now what the hell is a radian? One radian is approximately 57 degrees. Right? So 57 degrees is somewhere around here, somewhere in the first circle, first quadrant. So that's what roughly what a radian is. It's just a different way of measuring angles. Right? And to convert between radians and degrees, you just have to remember that 360 degrees equals 2 pi. Now most people remember that. What a lot of people don't realize is that the unit for pi is radians. That's why this conversion works. So pi is 3.1415 radians. Okay. Another way you can do this is just by saying pi radians equals 180 degrees. Okay? Cool. I'm going to do a quick example. We have an object that moves along a circus of radius 10. I'm sorry, a circle, not a circus. So let's draw this. You got a tiny little object. It spins around a circle here, and it has a radius of 10 meters, which means that the radial distance, if you go around a circle with a radius of 10, it means your radial distance to the middle is 10. And it says here that you start at 30 above the x-axis and then you go all the way to 120 above the x-axis. So let me draw another circle here just so I can put the angles. So 30 is somewhere here. You start here. And then remember this is 0, 30, this is 90, so 120 will be somewhere here. Okay? So you're going from something like this, from here to here. And we want to know what is your angular displacement? Angular means rotational. I'm asking what is your delta theta. Very straightforward. The definition of delta theta is delta theta is theta final minus theta initial. And then the angles are 120 minus 30, so this is just 90 degrees. I have to be very careful. If I had a negative here, like 45 down here, I'd have to plug it into as a negative. Okay? And that makes things a little bit different, just have to be careful with the negatives. So that's the answer for part a. It just asked for angular displacement. It didn't say if I wanted any radians or degrees, so degrees is fine. And then for part b, it wants linear displacement. Linear displacement is delta x. And I just showed you how I can connect delta x to delta theta, so we're basically converting from one to the other. Delta x is r delta theta. I have r, r is 10 meters, and delta theta is 90 degrees. Now here, I hope you're saying, no, it's not. This is wrong. And you're supposed to use radians. Okay. So I want you to actually write this out and then cross it out so you remember not to do this. Right? It has to be in rad. Okay. So what we're going to do is we're going to quickly convert the two. 90 degrees, I convert it using this ratio here, which means I'm going to put them in a fraction. So I'm going to say over here, I have degrees at the top, so I want degrees at the bottom, 180 degrees. And then up top, I have pi radians. And then what happens is the degree symbol cancels, and I'm left with just radians. And then you just multiply this in the calculator. You're going to put 90 times pi divided by 180. And if you do that, I have it here. Actually, I have it here as pi over 2. Right? That's a little cleaner way of doing. And then the other version is 1.57. They're both radian. Okay. So now, I can plug this in here, 10 times 1.57. And the answer will be 15.7 meters. Why is it meters? Because meters is the unit of delta x. Okay? So that's it for this one. Hopefully, it makes sense. Let me know if you guys have any questions.

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# Rotational Position & Displacement - Online Tutor, Practice Problems & Exam Prep

Rotational motion involves movement around a central point, described using angular position (θ) instead of linear position (x). Key concepts include angular displacement (Δθ), linked to linear displacement (Δx) through the equation ${\mathrm{\Delta x}}^{=}r\Delta \theta $, where r is the radial distance. A full revolution equals 360 degrees or 2π radians. Understanding these relationships is crucial for analyzing motion in circular paths, emphasizing the fixed nature of rotational origin and direction.

### Rotational Position & Displacement

#### Video transcript

### Displacement in Multiple Revolutions

#### Video transcript

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.

While you drive, your tires, all of radius 0.40 m, rotate 10,000 times. How far did you drive, in meters?

An object moves a total distance of 1,000 m around a circle of radius 30 m. How many degrees does the object go through?

BONUS:How many complete revolutions does it make?

A car travels a total of 2,000 m and 1140° around a circular path, starting from 0° . What is the radius of the circular path?

BONUS:How far (in degrees) from 0° does the car end up?