Connecting period and frequency to angular velocity | AP Physics 1 | Khan Academy

What we're going to do in this video is continue talking about uniform circular motion. In that context, we're going to talk about the idea of period, which we denote with a capital T, or we tend to denote with a capital T, and a very related idea, and that's of frequency, which we typically denote with a lowercase f.

So you might have seen these ideas in other contexts, but we'll just make sure we get them, and then we'll connect it to the idea of angular velocity, in particular, the magnitude of angular velocity, which we've already seen we can denote with a lowercase omega. Since I don't have a little arrow on top, you could view it just the lowercase omega as the magnitude of angular velocity.

But first, what is period and what is frequency? Well, period is how long it takes to complete a cycle. If we're talking about uniform circular motion, a cycle is how long it takes. If this is, say, some type of a tennis ball that's tethered to a nail right over here and it's moving with some uniform speed, a period is how long does it take to go all the way around once.

So for example, if you have a period of one second, this ball would move like this: one second, two seconds, three seconds, four seconds. That would be a period of one second. If you had a period of two seconds, well, it would go half the speed: you would have one second, two seconds, three seconds, four seconds, five seconds, six seconds.

And if you went the other way, if you had a period of half a second, well then it would be one second, two seconds, and so your period would be half a second. It would take you half a second to complete a cycle. The unit of period is going to be the second, the unit of time, and it's typically given in seconds.

Now what about frequency? Well, frequency literally is the reciprocal of the period. So frequency is equal to ( \frac{1}{\text{period}} ). One way to think about it is, well, how many cycles can you complete in a second? Period is how many seconds it takes to complete a cycle, while frequency is how many cycles you can do in a second.

So for example, if I can do two cycles in a second: one second, two seconds, three seconds, then my frequency is two cycles per second. The unit for frequency is sometimes you'll hear people say just "per second." So the unit sometimes you'll see people just say an inverse second like that or sometimes they'll use the shorthand Hz, which stands for Hertz. Hertz is sometimes substituted with cycles per second.

This you could view as seconds or even seconds per cycle, and this is cycles per second. Now, with that out of the way, let's see if we can connect these ideas to the magnitude of angular velocity. So let's just think about a couple of scenarios.

Let's say that the magnitude of our angular velocity, let's say it is ( \pi ) radians per second. So if we knew that, what is the period going to be? Pause this video and see if you can figure that out.

So let's work through it together. This ball is going to move through ( \pi ) radians every second, so how long is it going to take for it to complete two ( \pi ) radians? Because remember, one complete rotation is two ( \pi ) radians. Well, if it's going ( \pi ) radians per second, it's going to take it two seconds to go two ( \pi ) radians. So the period here—let me write it—the period here is going to be equal to two seconds.

Now, I kind of did that intuitively, but how did I actually manipulate the omega here? Well, one way to think about it: the period I said, look, in order to complete one entire rotation, I have to complete two ( \pi ) radians. So that entire cycle is going to be two ( \pi ) radians. Then I'm going to divide it by how fast my angular velocity is going to be.

So I'm going to divide it by, in this case, I'm going to divide it by ( \pi ) radians per second. I'm saying how far do I have to go to complete a cycle, and I'm dividing it by how fast I am going through the angles, and that's where I got the two seconds from.

So already you can think of a formula that connects period and angular velocity. The period is equal to, remember, two ( \pi ) radians is an entire cycle, and so you just want to divide that by how quickly you are going through the angles. That will connect your period and angular velocity.

Now, if we know the period, it's quite straightforward to figure out the frequency. So the frequency is just ( \frac{1}{\text{period}} ). So the frequency is—we've already said it's ( \frac{1}{\text{period}} ), and so the reciprocal of ( \frac{2 \pi}{\omega} ) is going to be ( \frac{\omega}{2 \pi} ).

In this situation where the period was 2 seconds, if you don't even know what omega is and someone says the period is 2 seconds, then you know that the frequency is going to be ( \frac{1}{2} ) seconds, or you could view this as being equal to ( \frac{1}{2} ); you could sometimes see the units like that, which is kind of per second. But I like to use Hertz, and in my brain, I say this means ( \frac{1}{2} ) cycles per second.

So one way to think about it: it takes 2 seconds to complete. If I'm doing ( \pi ) radians per second, my ball here is going to go: one second, two seconds, three seconds, four seconds. You see just like that, my period is indeed 2 seconds, and you also see that in each second—remember, any second I cover ( \pi ) radians—well, ( \pi ) radians is half a cycle. I complete half a cycle per second.