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Connecting period and frequency to angular velocity

Definition of period and frequency for an object moving in a circle. Connecting period and frequency to the magnitude of the angular velocity.

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Video transcript

- [Instructor] What we're going to do in this video is continue talking about uniform circular motion. And in that context, we're gonna 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 lower case 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 lower case omega. Since I don't have a little arrow on top, you could view it, just the lower case omega, as the magnitude of angular velocity. But first, what is period and what is frequency? Well, period is how long does it take to complete a cycle? And if we're talking about uniform circular motion, a cycle is how long does it take, 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, well, 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 one over, let me write that one a little bit neater, one over the period. And one way to think about it is well, how many cycles can you complete in a second? Period is how many seconds does it take to complete a cycle, while frequency is how many cycles can you 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. And 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. And Hertz is sometimes substituted with cycles per second. So 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, 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. So, 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? 'Cause remember, one complete rotation is two pi radians. Well, if it's going pi radians per second, it's gonna take it two seconds to go two pi radians. And 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 is one entire cycle is going to be two pi radians. And then I'm gonna divide it by how fast, what my angular velocity is going to be. So I'm gonna divide it by, in this case I'm gonna divide it by pi radians, pi, and I could write it out pi radians per second. I'm saying how far do I have to go to complete a cycle and that I'm dividing it by how fast I am going through the angles. And that's where I got the two seconds from. And so, already you can think of a formula that connects period and angular velocity. Period is equal to, remember, two pi radians is an entire cycle. And so you just want to divide that by how quickly you're going through the angles. And so that there 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 one over the period. So the frequency is, we've already said it's one over the period, and so the reciprocal of two pi over omega is going to be omega over two pi. And in this situation where the period was two seconds, you don't even know what omega is, and someone says the period is two seconds, then you know that the frequency is going to be one over two seconds. Or, you could view this as being equal to 1/2. You could sometimes see units like that, which is kind of per second. But I like to use Hertz, and in my brain I say this means 1/2 cycles per second. So one way to think about it, it takes two 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. And you see, just like that, my period is indeed two seconds. And you also see that in each second, remember in each second I cover pi radians. Well, pi radians is half a cycle. I complete half a cycle per second.