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# Connecting period and frequency to angular velocity

## Video transcript

what we're going to do in this video is continue talking about uniform circular motion and 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 context 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 does it take to complete a cycle and if we're talking about uniform circular motion or 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 and so period is 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 1 over 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 hurts and hurts 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 because remember one complete rotation is two pi radians well if it if it's going pi radians per second it's gonna take it two seconds to go to 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 entire cycle is going to be two pi radians and then I'm going to 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 that period 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 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 1 over the period so the frequency is we've already said it's 1 over the period and so the reciprocal of 2 PI over Omega is going to be Omega over 2 pi over 2 pi and 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 the frequency is going to be 1 over 2 seconds 1 over 2 seconds or you could view this as being equal to 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 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 1 second 2 seconds 3 seconds 4 seconds and 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
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