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, \( \Delta x \), and angular or rotational displacement, \( \Delta \theta \), using a tiny equation. Well, there are 2 more equations that we can use to connect velocity and acceleration between linear and rotational. Okay. So let's check it out.

Alright. So we have these tiny equations that are going to link, that are going to connect, that are going to allow us to convert from one to the other between linear and rotational. Now linear, we're also going to refer to linear as tangential. Right? Linear, tangential, both of these are going in a straight line. Going to connect from linear or tangential to rotational, which is also referred to as angular. So it's important that you know that these words are, have to mean the same thing. Okay. So the linear variable is \( x \) and the rotation equivalent is \( \Delta \theta \). Okay. Now from that, we get that \( \Delta x \) is the change in position, the change in \( x \), and \( \Delta \theta \) is a change in angular or rotational position \( \theta \). And the way that \( \Delta x \) and \( \Delta \theta \) connect is by this equation right here. We've used this. Similarly, \( v \) connects to its angular equivalent, \( \omega \), using a very similar equation. So \( \Delta x = r \Delta \theta \) and \( v \text{ tangential} \) (this \( t \) here means tangential velocity) is \( r \omega \).

And I want to point out that there's a pattern here. This is the linear, the linear variable \( r \) and the rotational variable. The same thing here, linear variable \( r \) rotational variable. I'm going to remove all these little circles so it's not messy. The same thing with \( a \) and \( \alpha \). \( A \) is going to be \( r \), and if you see the pattern, the equivalent of \( v \) is \( \omega \), the equivalent of \( a \) is \( \alpha \). Okay. And this is also the tangential acceleration. So these are the 2 new equations that we're going to be able to use.

Now when do they come up? Usually, it's on a problem like this. You have a disc and this disc spins 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, \( dt \). You might remember that when I have a point going around the circle, the point has tangential velocity and it also has centripetal acceleration. Well, it turns out that \( v_t \) is connected to \( \omega \) by this equation, \( v_t = r \omega \), which is a very, very useful relationship equation. Okay? So let's keep going. I want to quickly mention that there are 4 types of acceleration. I already mentioned 2 here. We have \( a_c \) and actually, I already mentioned 3. We have \( a_c \), we have \( a_t \), and we have \( \alpha \). There's a fourth one, but we're going to talk about that later. I want to just be very clear here that this equation right here, \( a_t = r \alpha \) refers to the tangential acceleration. It doesn't refer to the centripetal acceleration. It doesn't refer to the angular or rotational acceleration.

So there are 4 types of acceleration. Most of them have two names. So it's going to 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 itself, okay. All rotational quantities, \( \Delta \theta \), \( \omega \), and \( \alpha \), are the same at every point. So let me show you this real quick, illustrate this a little bit. So let's imagine a line here and then there's point, there's a little imagine this is a huge disc and there's people on top of it or whatever, right? So you have a guy a over here on that point and guy b is over here. Now imagine that this disc 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 you're here and this spins, we're still in the same place, right? We're moving together. Okay. So our \( \Delta \theta \), our change in angle will be the same, alright. Because and by the way, this happens even if we're not in the same line. It's just easier to see if it's in the same line. \( \Delta \theta \) is the same. And because \( \omega \) is defined in terms of \( \Delta \theta \), \( \frac{\Delta \theta}{\Delta t} \), \( \omega \) is also going to be the same. And since \( \omega \) is the same, \( \alpha \) 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 going to experience the same \( \alpha \) and the same \( \omega \). So all of the rotational quantities will be the same. Okay? However, the linear speeds might be different since they depend on \( r \) which is radial 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 an example and do a very straightforward one. So I have a wheel of radius 8. So let's draw this here. Just put a little radius here. Radius of this wheel is 8 meters. It spins around its central axis. So what that means is that imagine a circle and imagine a sort of an invisible line through the circle, right. An invisible line through the circle and it's free to spin around that invisible line. So I'm going to draw this here. You don't have to draw it. So I'm going to delete it. Imagine the imaginary line that goes through this thing, almost as if you stuck a thing through it and then it's free to spin around that. Okay. That's what that means. Let's get this out of here. Basically, it spins around at center, which is how these things always work, at 10 radians per second. So that's our \( \omega \) is going to be, 10 radians per second. We want to know the angular and linear speeds at different points. So I want to know at a point in the middle of the wheel on the central axis, so. So we're going to call this point 1 at a distance 4 meters from the center. If the radius is 8 meters, 4 meters is halfway in. I'm going to draw this here. This is point 2 and at the edge of the wheel, point 3. Okay? So what we want to know is we want to know \( V_1 \), \( V_2 \), and \( V_3 \). And and I want to know \( \omega_1 \), \( \omega_2 \) and \( \omega_3 \). Okay? That's what it says. I want the angular, which is \( \omega \), and linear, \( v \), speeds at these three points. K? So first thing is to realize that all these points have the same because they're on 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. \( \omega_1 = \omega_2 = \omega_3 = \omega_{\text{disk}} \). So this is more of a conceptual to know that you to know if you know that. So all of these will be 10