 In this final video for lecture two, I want to introduce an example that is an application of these angle measures and compare the difference between linear velocity and angular velocity. So first of all, linear velocity, this is probably what you're used to. Velocity is essentially a fancy word for speed. It's a little bit more than that. Velocity is a vector quantity. Something we'll talk about a little bit later on in this lecture series. It's a vector quantity, it means it has a magnitude and a direction. I think of like the billet vector from the Spickle meme. He's committing crimes with both magnitude and direction. So speed is just the magnitude, how fast are you going? But velocity as a vector quantity also has direction. So you could be driving 50 miles per hour north, 50 miles per hour south, 50 miles per hour northeast or whatever. So the direction matters a lot. And so what we're gonna see often in this course is that the direction can be measured with an angle. So velocity is a speed with an angle, which is why we'll talk about it a lot. And so linear velocity is the idea of velocity along a line. So linear velocity of a moving object is the change of position along a line with respect to time. And typically linear velocity, it will be measured with a distance divided by time. So you might get things like meters per second or miles per hour, right? And there's a direction involved in that as well. But you take a length divided by time. That's how you measure linear velocity. In contrast, you have the idea of angular velocity, which linear velocity is typically denoted with a V. Angular velocity is typically denoted using the Greek letter omega. This is a lowercase omega. It kind of looks like a W, but it's the Greek letter omega. Angular velocity is that of a spinning object is the change of the angle around a point with respect to time. So if you have some fixed line here, angular velocity is telling you how quickly is the angle changing. Is it slow like this? Or is it fast like this, right? The change of the angle over time is the angular velocity. And one often denotes angular velocity using the idea of revolutions per minute or we could talk about RPM. There are other ways of doing it, like radiance per second is probably the better measurement. But as we haven't introduced radiance yet in this lecture series, we'll work with RPMs for a moment, revolutions per minute. So for example, a constant angular velocity disk drive, like maybe plug a CD or DVD or a Blu-ray into the appropriate reader, that disk is gonna spin around at a constant speed. Suppose that DVD is spinning at a constant rate of 48 revolutions per minute. So this would tell us that the angular velocity of that DVD is 480 RPM, revolutions per minute. Through how many degrees will a point on the edge of the disk move in two seconds? So this question really is just a question about converting between different units. So one way of computing an angle is by revolutions. What fraction of an evolution, a revolution is there? Like a half angle is just, as the name suggests, half of a revolution, a right angle is just one fourth of a revolution. So we have to convert from revolutions to degrees. We also are measuring things in minutes, so we have to switch them to seconds. So let's think about that for a second. So if you have 480 revolutions per minute, well, let's first deal with the time. Minutes, as we know, we have to switch to seconds, right? So you have 60 seconds is equal to one minute. So if we were to multiply this by one minute, divided by 60 seconds, you'll notice that one minute and 60 seconds, that's the same thing, so that's just the quantity one, but I actually wanna cancel out the minutes right here, so we get 60 seconds in the bottom. So now this would give us revolutions per second, but we wanna get into degrees, right? How do we switch to degrees? Well, one revolution is equal to 360 degrees, so we're gonna do the same trick again. So we're gonna put one revolution in the bottom, and we're gonna get 360 degrees on the top, so that one revolution cancels with this, and so then we end up with 480 times 360, all over 60 is equal to degrees per minute, excuse me, degrees per second. So now we have the correct units for the object we're trying to measure here. Now notice that 60 goes into 488 times, we get something like that, and so you're gonna end up with eight times 360, that's gonna give you degrees per second, which gives us 2,880 degrees per second. So we take two seconds times by the rate in which it's spinning 2,880 degrees per second, and so we end up that in two seconds, our DVD would spin 5,760 degrees. Thus we found this using the angular velocity of the disc. Now you'll notice that this angle is in fact larger than 360 degrees. This is a situation in which we would not want to reduce the angle of something coterminal less than 360 degrees, because the fact that there were several revolutions of the disc in these two seconds is relevant, right? We discovered that this thing is rotating at eight times, eight revolutions per second, and so in two seconds it's gonna revolve 16. So we should expect a very large degree measure above 360. Don't reduce this down to something less than 360.