Rotational Position & Displacement - Video Tutorials & Practice Problems

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Rotational Position & Displacement

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Hey, guys. So in this video, we're going to start talking about rotational motion, also known as rotational Kinnah Matics. I'm gonna take you back all the way to the beginning of physics, and we're gonna look at basic motion one dimensional motion, and we're going to contrast and compare the two. You're gonna see a lot of things. They're similar, but there's some differences, so let's check it out. Alright. So rotational motion is when you have motion around a central points around a 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. Um, the other type is when you have a cylinder that spins around itself. Okay, Now, you may remember that we describe 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 describing where you are in a circle. Okay, So, for example, if you are moving in a circle, think this is a track and you can only move around that track either this way or this way, you're trapped right, and there's two ways you could describe your position. You could do it by saying, Well, this is coordinates X comma y, and then if I move over here, I have a new X in a new I right. It's a two dimensional grid. Um, it's a surface, so you could do that. The problem is, that's more complicated than uh than it needs to be, because now I have access and wise 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 zero degrees and this would be 90 right there. So we're going to say that this is just making 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 you uses the variable fada you might remember. That data is what used 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 the same thing there used interchangeably. So you see a lot of words like angular velocity just means rotational velocity. Okay, so these words are used interchangeably. Alright. So whereas in a linear rotation linear motion, you used X in rotational motion, we used data. So what I'm gonna say here is that X becomes feta. The X equivalent in rotation is theta. Okay. Later on, you're gonna get some equations. Where? Old equations. But instead of using X, we're gonna use feta. All right, so let's let's look a little deeper into the differences between the two. So position is the finest. 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 radiance or degrees, but we're going to use radiance most of the time. Okay, so we're going to use radiance, which is abbreviated rat. Alright, now, origin, If you remember, origin is simply where X equals zero. So, for example, let's say we got a line here and you are here. Okay? And then there's two points. Let's draw three points here. Just two points. Whatever. That's fine. And let's say that the these two points are 10 m apart. So this would be zero, and then this would be 10. And maybe you are at seventh. Okay, So if this is X equals zero, this is where the origin is. Okay? But we could have done this a little bit different. Then We say that your position x you is plus seven. But we could have put the origin right here. We could have arbitrarily said, I want this to be X equals zero and then this distance. Here's three. So your ex ex, you would have been negative three. 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 doesn't. It doesn't have to be Ah, fixed thing in rotation is a little bit different in rotation. Origin is still where position in this case data equals zero. Okay, zero degrees. There's your radiance. When you put a little meters here zero degrees or zero radiance The difference is that whereas here, it's arbitrary. It's up to you, OK, up to you. Unless the problem tells you, uh, in rotational position that our origin is always fixed. Okay. And it's always fixed. It's fixed at the positive X axis. Okay, zero is always here. Remember the unit circle? This is always the origin. Okay, that's non negotiable. Whereas here, you could put it Whatever you want. If you're given that kind of liberty in the problem, Okay? The last thing is direction is also arbitrary. Um directions 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, direction is fixed. So clockwise clockwise which follows oclock The goes this way is negative en counter clockwise 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 backwards, right? And I like to think of this is backwards. Why couldn't they have made the direction of the clock positive? Right? Why is it the clock is backwards? Well, it's because this stuff actually follows the unit circle. And you might remember that the unit circle grills like this. The angles grow like this. The unit circle and the clock are backwards from each other, and we use the unit circle. Um, and that's it. So those were the key differences between linear position and rotational position. We're gonna quickly talk about the displacement now, So the rotation equivalent of linear displacement. So positions X displacement is changing position, Which is Delta X. Okay. Rotational position was data. So rotational displacement is simply delta theta. Okay, So instead of Delta X, the equivalent is Delta data. So if you're moving this way, we measure Delta X. If you're moving this way, measure your delta Theta and these two quantities here Delta X, delta theta are linked. They're connected. They can be converted from one to the other using the following equation. Delta X equals R. Delta fatum this are here. You can loosely refer to it as radius. I'll talk about this a little bit more, but what it really is is radio distance, which is distance to the center. So I'm gonna write distance to center. Okay. Radius would be the radius of a cylinder. But if it's a distance, appoint 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 equals R theta. 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 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 radiance. What do I mean by speaks? Radiance. Well, if you're plugging in a Delta fade into this equation will do an example just now. But if you're plugging in of Delta theta, that number that you're plugging into the equation has to be in radiance. Otherwise, the equation doesn't work. Also, if you're instead of plugging in Delta data you're solving for Delta data, the answer will be in radiance. So either you're giving the equation radiance or if the equation is giving you an angle, it's giving that in radius. That's why I say here that the input must be in radiance. You have to plug in in the radiance for the equation toe work, and the output will be in radiance if you get an answer out of that equation, If you get a Delta state out of the equation, you will be in radiance. Okay, Now what the hell is the radiant one radiance? Approximately 57 degrees, right, So 57 degrees is somewhere around here somewhere in the first circle first quadrant. So that's what roughly one radiant is. It's just a different way. You're measuring angles, right? Um, and to convert between radiance and degrees, you just have to remember that 360 degrees equals two pi. Now most people remember that. What a lot of people don't realize is that the unit for pie is radiance. That's why this conversion works. So pie is 3.1415 radiance. Okay. Another way you could do this is just by saying pie radiance equals 80 degrees. Okay, quote. I'm gonna do a quick example. Um, 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 around a circle of radius of m. Which means that the radio distance if you go around the circle of Radius 10 and these their radio distance to the middle is 10. Um, and it says here that you starts, um, you starts at 30 above the x axis, and then you go all the way to 1 20 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 1 20 will be somewhere here. Okay, So you're going from something like this from here to here. And we wanna know What is your angular displacement? Angular means rotational. I'm asking, What is your Delta theta? Very straightforward. Definition of Delta theta Delta theta is fatal. Final minus state initial aan den. The angles are 1 20 minus 30. So this is just 90 degrees e. I have to be very careful. Um, if I had a negative here, like 45 down here, I'd have to plug it in a negative. Okay? And that makes things a little bit different. You just have to be careful to negatives. So that's the answer for party. Um, it just asked for angular displacement. It didn't say if I wanted a radio or degrees, So degrees is fine. And then for part B, it wants to linger. Displacement linger. Displacement is Delta X. And I just showed you how I can connect Delta X Delta theta. So we're basically converting from one to the other. Delta X is our delta theta I have are ours. 10 m and delta theta is degrees now. Here, I hope you're saying no. It's not. This is wrong and you're supposed to use radiance. 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 being rad. Okay, So what we're gonna do is we're gonna quickly converted to 90 degrees. I converted using this ratio here, which means I'm going to put him in a fraction. So I'm gonna say over here I have degrees at the tops. I want degrees at the bottom 1 80 degrees and then up top. I have pie Radiance. Then what happens is the degree symbol cancels and I'm left with just radiance. And then you just multiply this in the calculator, gonna put 90 times pi divided by 180. And if you do that, I have it here. Um, actually have it. Here is pi over two, right? It's a little cleaner way of doing. And then the other version is 1.7. They're both radiant. Okay, so now I can plug this in here 10 times one point 57 and the answer will be 15. m. Why is it meters? Because meters disappear. Why is it meters? Because meters is the unit of Delta X. Okay, so that's it for this one, Hopefully make sense in future. And I mean, if you have any questions

2

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Displacement in Multiple Revolutions

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5m

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Hey, guys. So now we're gonna 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. Ah, complete spin. There's all these words. You make a full one around the circle, you've gone a total change in angle off three hundred sixty degrees. Everyone knows that a full app is three hundred sixty degrees or two Pi radiance. Okay, therefore Delta X, Remember, the Delta X equation is Delta X equals R Delta theta. That's how you link these two variables. Okay, it becomes two pi r. And what I've done here is I replaced Delta Theta with two pi because that's what Delta theta is. If you make a revolution, um, you might recognize this equation that Delta X equals two 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've traveled, would give you a quantity equal to two pi r Okay, so that's the linear distance as you go in a rotational motion around the circle. Now, that's if you spend once. If you spend once you get to be sixty, what happens if you spend twice? Then you get to be sixty times, too. Eso If you spend any end times, then your delta theta is three sixty times and or two pi times and obviously radiance. So instead of having a delta x of two pi r, you get a delta x of two pi r times n where n is the number of rotations. Okay, two more things you may need to know if you wanna know how many revolutions you go through, and we'll do. Example An example of this just now. All you gotta do is divide your number of angles by either three sixty degrees or by two pi. Okay. For example, if I tell you, I spun, um, seven hundred and twenty degrees. Andi, I wanna know how many how many revolutions that is you divided by three. Sixty and you get to That means I spun twice. Okay, Something with if it's in radiance. If it's in pies, you could just divided by two pints. 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 gotta draw this. If I wanna know how far I end up, all you gotta do is you keep subtracting by three sixty until your angles less than three. Sixty. So, for example, if you, um, spun four hundred and ten degrees and I wanna know how far from zero you end up, four hundred ten is more than three. Sixty. So you made multiple revolutions. All I gotta do, subtract by three. Sixty, and you see that the answer is fifty. You keep doing this until your final answer is less than three sixty, which it is. So we're good to go. If it wasn't, you would subtract by three sixty again. Okay, Same thing with radiance. If it's in ratings, it just keeps attracting by two pi until the answer is less than two pi. Let's do a quick example here. All right, so starting from zero degrees of stroller ball circle starting from here, the two degrees is always the positive X axis, You make two point two revolutions. So we're using the letter N to represent the number of revolutions two point two around a circular path of radius. Um, twenty. You have a circular path of rate is twenty. That means that your radio distance from the middle little are is twenty you could use little are a big are interchangeably, little arch. Technically more correct because it's not big guards reserved for the radius of like a disc. Little R is the distance from the center. Okay, but the words were used kind of interchangeably. All right, so what is the rotational displacement in degrees? Hey, rotational displacement is delta theta, and we want that in degrees. I'm gonna put a little d e g here to indicate that we want to do this degrees and not in ratings. Well, if you spin once, you spend three hundred sixty degrees. But if you spend two point two times, you just multiply them, okay? And this is going to give you seven hundred and ninety two degrees. Seven hundred ninety two degrees is your How much is button Cool for? That's it for part B. How many degrees from zero are you again? We're just gonna subtract seven ninety two until we get to a number that's less than three hundred sixty. So I want to know how far from zero. Okay, so seven ninety two minus three. Sixty. That gives you four. Thirty two. We're gonna have to keep going because we're not below three. Sixty, minus three. Sixty. And then finally, that's the answer. Seventy two degrees. Okay, that's the final answer. And for part C, what is your linear displacement leader? Displacement. Remember, is Delta X. And if I wanna know Delta X, Delta X is our delta fatum R is the distance here. Twenty and Delta theta has to be in radiance. Delta Data has to be in radiance, so I cannot use seven ninety two. Okay, It's just a little gentle seventy two. I cannot use that. I'm gonna have to use in Radiance Radiance. This is going to be two pi times two point two right to piles of four rotation times two point two because we rotated two point two times. So if you multiply all of this, put in the calculator. Pious people in one four, one five. But you calculate has a button for that. Um, if you do all this, you get the distance is two hundred and seventy six meters. Okay, that's it for this. Hopefully makes sense. Let me know if you have any questions.

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Problem

Problem

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

A

5027 m

B

8667 m

C

12566 m

D

25132 m

4

Problem

Problem

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

33.3°

B

488°

C

987°

D

1908°

5

Problem

Problem

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?

A

1.754 m

B

100.5 m

C

780.8 m

D

2000 m

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