 Hello and welcome to the screencast for trigonometry where we will discuss and solve an angular velocity problem. For this problem, there are some things with which you should be familiar before we start trying to solve the problem. And here's a summary of some of those things that we will need to know in solving an angular velocity problem. The diagram there on the upper right kind of shows the picture there where we have something that we call the arc length, which is designated by S on a circumference of a circle of radius R. The angle theta is the central angle that intercepts the arc that we have there. These are related by the formula S equals R theta. We probably won't be using that formula directly when we solve the problem, but one thing we do have here is what we mean by the linear velocity of a point on the circle. That's kind of a misnomer. The term linear velocity kind of implies it's moving in a straight line. It's not. It's moving around the circle. And so sometimes I just refer to that as the velocity of the object moving around in a circle. And here's the formula that relates the arc length S to the time t, which the object travels, and of course S is equal to R theta. And then lastly we have the angular velocity. Basically how fast is the object rotating about the circle? The general, the usual letter that's used for that is the Greek letter omega, a lowercase omega. And the definition of it is basically the angle that it sweeps out theta divided by time. And what we use more often, and I think the one we will use mostly in this next problem, is this one which says the velocity is the radius times the angular velocity. The other thing to notice in this, which is important in the whole thing, is that the angle is always measured in radians. So you always see the radian measure indicated in here. And so oftentimes the angular velocity can be given in other units, rather than like radians per second or radians per minute. And sometimes revolutions per minute. So in order to use this formula, we first have to determine the angular velocity in radians per time unit. So here's the problem we're going to tackle. This involves two pulleys connected by a belt. And that is kind of an important thing is that they are connected by the belt, so that in effect when something is moving around that belt and then moving around portions of the circle, the velocity, if you want, the linear velocity is always the same, no matter which pulley it happens to be on, or whether it's in the middle between the two pulleys. The important thing we have to watch if you remember the formulas is the radius of the pulley. And this is where we start reading carefully. The smaller pulley has a diameter of 6 centimeters. So what we have to do is use the radius. So what we will use then is r1 equal to 3. And we're going to use an r sub 1 here, basically because we have two pulleys, and this will keep the values for each pulley separate. And again, the larger pulley has a diameter of 15, so its radius is 7.5. And of course these are all measured in centimeters. The one thing that we really have to contend with here is the angular velocity. And here it is at 120 revolutions per minute. So in order to work with this problem, we do have to convert that to radians per minute. And that can sometimes be a bit of a challenge, but here's a nice way to sometimes do that, and we'll probably kind of do it in a string along the bottom of the screen here. We would have omega equals 120 revolutions per minute. Notice I did not use the RPM, but basically wrote the units as a rate of change in revolutions per minute. And what we want to do now is convert this to radians per minute. So we will start off with revolutions per minute. And the question then is that we work with is revolutions. And what I like to think of then as a conversion fraction, I'm going to put one revolution here in the denominator and then two pi radians in the numerator. And then when we do the conversion, you can see in effect we can say the revolutions cancel. Probably not completely mathematically correct, but it is a nice way to think about these. And what we end up with is radians per minute. And as you can see, if we do the calculation there, we get 240 pi radians per minute. So that's our angular velocity, and that, of course, is omega sub 1. That's the angular velocity for the small pulley, because basically this was a small pulley that we were working with there. So now we have both radii, the angular velocity. And as we said, what we want to do is what conclusions can we make about linear velocity and angular velocity of the larger pulley? So here's our problem again, and some of the given information we have. And since we know most of our information now about the small pulley, we will start to work with that. And in particular, we can use v sub 1. The velocity, linear velocity, is r sub 1 times omega. And now we know both of those values. And as you can see, what we will get is 3 centimeters times, and I forgot to write down the angular velocity in radians per minute. So let's do that. Now it was 240 pi, and that would be in radians per minute. And when we do that computation, you can see we get 3 times 240, so we get 270, I'm sorry, 720 centimeters per minute. And that's our linear velocity. And of course what that means is that's also the linear velocity for the larger pulley. And so for the larger pulley, we'll see that we have v2, which is v1, which is also 720 centimeters per minute. And now we can go to work and find the angular velocity of the larger pulley. As we can see now, we've got, for the larger pulley, we have a radius and a velocity, or a linear velocity. So we can use that information to determine the angular velocity of the larger pulley. So let's again quickly write down what we have here. We have v1 equals v2 equals 720 centimeters per minute. We also know the radius of each of the pulleys, the smaller radius is 3 centimeters. The larger pulley has a radius of 7.5 centimeters. So now we're going to use the formula that relates linear velocity to angular velocity, but we're going to do that for the larger pulley. So we would have v2 equals r2 times omega2. And with that then, you can see we know the value for v2. We know the value for r2. We can, in essence, solve for omega2 and we get the velocity divided by the radius. And now we carry out that computation. And we get 720 pi centimeters per minute divided by 7.5 centimeters. And so our angular velocity comes out to be, when we do that computation, 720 divided by 7.5, we get 96. And again, this is going to be in radians per minute. That now tells us what the angular velocity is. And again, if we want to kind of relate that back to revolutions per minute, we can do, in effect, the same thing. We take the, that we did before, kind of in reverse, take the radians per minute, and we want to convert that to revolutions per minute. And so, again, we want, in essence, the radians to cancel. So we put the radians down here, 2 pi radians for every one revolution. And again, now when we do the computations now, you can see, I left out a pi here. So it's 96 pi radians per minute. And that's important because now, in essence, the pi's will cancel. And when we do the division, we end up with 48 revolutions per minute. And that's our final answer for the angular velocity of the larger pulley. Let's just take final summary of this and look at what we had here. We had R1 is 3 centimeters. Its angular velocity of the smaller pulley was 120 revolutions per minute. And it had a linear velocity of 720 pi centimeters per minute. And of course, the thing that connected the two pulleys was basically the fact that they're connected by a belt. So their linear velocities are the same. With that, then, we were able to use the radius of the larger pulley, 7.5 centimeters, to determine that the angular velocity was 48 revolutions per minute. So there you have it. Notice again, we had to use that formula for two different pulleys. And we had to find something that connected the pulleys. And of course, physically, that's the belt. But in terms of the formulas, that meant that the two linear velocities were the same. And there you have it. See you next time.