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12. Rotational Kinematics

1

concept

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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

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example

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All right. So here we have a small object that rotates at the end of a light string. So here's a string, and you've got a small objects. I'm gonna do this. Andi. It's spinning like this. Okay, imagine that the, um if you spin a string, it forms a circular path. Right. Theology. It reaches 120 rpm from rest. In just four seconds, I will give you a ton of information here. First, you start arrest. So, Megan, issue zero. Um, you reach the final rpm of 1. 20. And you did this in just four seconds. Okay? It also says you that the engine show acceleration. Tangential acceleration, remember, is a T after the four seconds is, um, 15 m per second square. And we want to know what is the length of the strength. Okay, we haven't talked about the length of string yet, but I hope you can figure out that the length of the string is the It's this distance here, right? It's the radius of the circle that forms the circular path you get. Or it is the radial distance from the center of rotation of which is your hand and the edge of rotation, which is where the object is. So little are the distance to the middle is your length. So essentially, what we're looking for is little are think of it as little are not as l because there's no l's in any of these equations, so you're not gonna find out. Okay, so this is a little bit of a mess, because we have we're gonna have to use a combination of equations here. Okay, So if you look through all the equations with you so far, you might first think about one of the three or four, um, motion equations, one of three or four emotion equations. And you might think of that because I gave you Omega initial zero I gave you Delta T. I gave you RPM, which you can convert to a make a final. And if you do that, you're gonna have three out of five variables once you convert. Okay, however, notice that if you look through all of those four equations, there are no ours in them. So you're not going to be able to solve for R by doing this. Okay, Now, if you look a little further, um I do have a nickel Asian that I give that I gave you recently that links up 80 with our 80 with our and that equation is 80 equals R Alfa and I know 18. So all I have to do is find Alfa r equals which is 15 right there, divided by Alfa. So we're gonna have to do is find Alfa and plug in here. Now, how do I find Alfa? Well, Alfa is one of my five variables of motion, so I'm going to be able to use thes three guys to find Alfa. Okay? And that's what we gonna do now. So first I'm gonna convert from rpm into frequency. I'm sorry into w final. So remember that w final is two pi f and F is our PM over 60 so 20 divided by 60. It's true. Therefore, Omega Final is two pi, And instead of that, I'm gonna put it to which is four pi. Okay, so I'm going to rewrite this year just to clean it up. Omega initial zero Omega Final is for pie. Delta T is four. We're looking for Alfa and they ignore variable is delta theta notice how I know three things. Um, ignore variables. Delta theta the Onley equation that doesn't have Delta. Fate is the first one out of the four Omega Final equals Omega initial plus 18. And if we're looking for Alfa, we just gotta move everything out of the way. Initial zero. So Alfa is Omega final, divided by time Or make a final. Is we found here? It's two pi divided by time. Time is four. I'm sorry for pie four cancels with four and Alpha's 3. 15 radiance per second square. Okay, let me disappear here so you can see. All right, Um, now all we gotta do is plug in this number here, and we're good, So 15 divided by 3.14. And if you divided to you get that are is 4.77 meters. Okay, 4.77 m. And that is the final answer. Cool. So whips. So that's it for this one. It's interesting question that combined sort of these two equations and the basic idea here is just that old school physics hustle. You got stuck in one and you're gonna have to go find the other and just kind of work your way through it. All right. There is not a very clear path. There's a few different ways you could have done this. But the most important thing is, you know, try to figure out what's an equation that has my variable and and then sort of look for all different ways to find all the letters, all the variables you have to solve for cool. Alright, that's it. Finished one. Let me know if you guys have any questions.

3

Problem

A disc of radius 10 m rotates around itself with a constant 180 RPM. Calculate the linear speed at a point 7 m from the center of the disc.

A

18.8 m/s

B

21.0 m/s

C

66.0 m/s

D

132 m/s

4

Problem

A rock rotates around a light, 4-m long string. The rock is initially at rest, but reaches 150 RPM in 3 seconds. Calculate its tangential acceleration after 3 s.

BONUS:Calculate its tangential speed after 3 s.

A

1.31 m/s^{2}

B

15.8 m/s^{2}

C

20.9 m/s^{2}

D

62.7 m/s^{2}

5

Problem

A 4 m long blade initially at rest begins to spin with 3 rad/s^{2} around its axis, which is located at the middle of the blade. It accelerates for 10 s. Find the tangential speed of a point at the tip of the blade 10 s after it starts rotating.

A

0 m/s

B

15 m/s

C

30 m/s

D

60 m/s

E

188 m/s

Additional resources for Converting Between Linear & Rotational

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