 Now we can look at angular and translational quantities. And we're going to start by reminding ourselves of the definition of radiance. It was an angular unit where it was defined as the ratio of the path length to the radius of my circle. The path length is shown here in red as s, and my radius from the center out to the edge is shown here in blue as r. Putting this into an equation, what it says is that my angle in radiance, theta, is equal to the ratio of s to r, the path length to the radius. And I can rearrange this equation to give me s equals theta r. In other words, if I know my radius of the circle and my angle in radiance, I can find my path length. This only works though if the angle is in radiance. If you try to put it in any other units, this equation doesn't work because the angle was defined as the angle in radiance. That leads us to the velocity. Specifically here I'm talking about the tangential velocity on the edge of a rotating object. So again, I've got the same circle with the same radius marked. Now when I think about the speed at this moment, it's moving upwards. Of course, it's going to move around in terms of the angular velocity. Well, that tangential velocity is the derivative with respect to time of my position or my s. And I just found that as being related to theta and r. Plugging that in and doing the derivative, I recognize that r doesn't have any change so it comes out of the derivative and I have just d theta dt. And d theta dt, the derivative with respect to time of the angle, is omega, the angular velocity. So I can relate the tangential velocity and the angular velocity. I can do the same sort of thing with acceleration. Again, talking about the tangential acceleration on the edge of a rotating object. The acceleration is the time derivative of the velocity. I just found the velocity in the last slide. And that means I have d omega dt times r. Because again, r's not changing. And that means I've got the tangential acceleration is equal to the angular acceleration alpha times the radius. In summary, I've got three equations here that are very similar in form. Where I've got my translational quantities on my left hand side and my angular quantities on the right hand side. And they all have a r radius next to them. Let's take a look at units because it can get confusing on these ones. I started with an angle defined in radians. This does not work in degrees or revolutions. And as a ratio unit, we said when we first defined radians that the rad unit symbol is not always listed. It's considered a pure number as opposed to a normal measurement unit. So when I've got my equation, s equals theta r, theta is measured in radians. But I put it in brackets to let you know that it may or may not actually be listed. Our radius though has to be in meters and our path length is in meters. I find it helpful to remind students to think of radians as a placeholder unit. They're there to let us know that we're working with these ratio type equations and that we have to use radians. But it doesn't actually affect the units of the rest of the equation. So now let's look at units. And we're going to look at velocity and then we're going to look at acceleration. For velocity, I start with my omega being radians per second. And of course, the radius is still in meters. And thinking of that radians as a placeholder unit, we see that our tangential velocity is just meters per second, which is what we expect for a tangential velocity. I do the same sort of thing with the accelerations where alpha is going to be radians per second squared, the angular acceleration. Radius is still in meters. And that gives me meters per second squared for the tangential acceleration. Now be careful here to not confuse the radius, which is a number that has units of meters, with radian, which is a unit placeholder unit for our angular quantities. So that's how you convert back and forth between the angular and the translational quantities.