Guys, now that we've been introduced to uniform circular motion, in this video, we're going to talk about three more important variables you'll need to know: circumference, period, and frequency. So let's go ahead and check it out. Basically, they are all related to when objects complete a full rotation. We have this circular path, and once you complete a full lap around this circle, you've completed a full rotation. Some books will call this a revolution, and some will call it a cycle. Let's talk about the first one, which is basically the distance that you travel around the circular path. That's called the circumference. The circumference is given by the letter *c*, and it's really just related to the radius. We know that the radius of this is really just the distance from the edge of the path to the center of the circle. That's *r*. The circumference is basically going to be the distance all the way around the circle once you've completed that path. So this is our letter *c* here. And, really, *c* is just equal to 2πr. So if you know one of those variables, you can always figure out the other one. Alright?

Let's talk about the other two; they're very closely related to each other and they have to do with time. The first one we'll talk about is called period, and the symbol we use is this capital T. Basically, the period is just how long it takes for you to complete one cycle or one lap around that circle. And so, one way you can think about this is that it's the number of seconds divided by or per the number of cycles. And so the unit that we'll use for this, because it's seconds per cycle, is just the second. And so one really quick way to figure out the period is just if you take the number of seconds elapsed and you divide it by the number of cycles. So quick example here. If it took us 2 seconds to complete one lap, then our period is just 2 over 1, and that's just 2 seconds. Alright?

The other related variable is called the frequency. And the frequency is basically just the opposite of the period. Instead of it being seconds per cycle, it's the number of cycles that you complete per second. So notice how these are basically just opposites of each other. So the unit that we'll use for this is not going to be a second. It's actually going to be a hertz, and a hertz is essentially just an inverse second. It's 1/s. And so because we're basically just flipping these two things, what we can do is we can take this equation here, seconds per cycle, and we just flip it upside down. So the way you calculate the frequency is just by doing the number of cycles divided by the number of seconds. So if you take a look at these two variables here, notice how they're basically just opposites of each other. You just flip the fractions. And so, in general, you can always figure out the period from the frequency and then vice versa. And here are the equations to do that. If you want the period, you're just gonna take the inverse of the frequency. And if you want the frequency, you're just gonna take the inverse of the period. You're basically just gonna flip the fractions.

Let me show you how this works, and we'll do a couple of examples here. We're going to calculate the period and frequency of our motion if we complete 4 rotations in 2 seconds. Imagine you were walking in a circle and you did 1, 2, 3, 4 rotations. Right? So this is basically 4 cycles, and it took us 2 seconds to complete these 4 cycles. So how do we calculate the period? Remember, this period is really just going to be the number of seconds divided by the number of cycles or rotations or whatever word you're using. And so we're just going to use 2 seconds divided by 4 cycles, and you'll get 1 half. So if it takes you 2 seconds to complete 4 cycles, then basically each one of these is 1 half of a second and so on and so forth. Alright? So that's the period.

The frequency is going to be the opposite, the flipped fraction of that. We're going to do the number of cycles divided by the number of seconds. So we're just really going to do 4 cycles divided by 2 seconds, and then you'll get 2 hertz. So one way you could also have done this is you could just take these fractions. Right? You could have taken this 1/2 of a second and flip the fraction that becomes 2 over 1, and that's really just 2 hertz. Alright? So let's take a look at the next one. Now we're doing 0.5 rotations in 3 seconds. So what this means is that we got half a rotation, right, but the other half is kind of missing. So we have 0.5 cycles, and then we have 3 seconds to complete. So we'll do the same exact idea here. Right? The period is really just going to be the number of seconds divided by the number of cycles. So it's 3 seconds divided by 0.5 cycles, and then you get 6 seconds. Basically, if it takes you 3 seconds to complete half of a circle and it's going to take you another 3 seconds to complete the other half, then your total period will be 6 seconds. Alright? And so the frequency is just going to be 0.5 cycles divided by 3 seconds. Basically, you just flip the fraction, and then you'll get, as you expected, 1/6 of a hertz. Alright? So that was, again, how these two fractions are basically just flipped versions of each other. Right. So that's it for this one, guys. Let me know if you have any questions.