 This short video will be our last video for lecture eight in our lecture series and as a consequence our last video for chapter three about circle trigonometry in the radian measure and in this video I want to talk some more about the idea of angular velocity. Now remember that linear velocity is the speed that you're traveling along a line. So it's a change of position with respect to time so it'd be measured in something like miles per hour feet per second. Angular velocity is similar but angular velocity tells you how quickly your angle is changing with respect to time and the best way to measure that is probably going to be radians per second or something like that. That is the angle measurement ought to be in radians and let me give you one reason why the angle measurement of angular velocity ought to be in radians. Well consider the arc length formula that we have seen previously in chapter three here. S equals r theta so the length of an arc and so this is measured the linear length of an arc is equal to r theta where r is the radius of the circle or the radius of the motion you could say getting sort of getting ahead of here what we're trying to do and then theta is the measurement of the angle. Now when you do this formula S equals r theta is necessary that angle be measured in radians that's the only angle measurement that's appropriate. Kind of like various chemical formulas require that temperature be measured in Kelvin similar arc length must be measured in radians. Well if you took this formula divided everything by time some some unit of time T will S of T that you know since S is a linear distance distance divided by time that's a linear velocity. And so we're going to call that V for short for the second one though. Let's keep the radius fixed and let's take theta over T in that situation because theta over T we've seen this before this is our angular velocity as we previously called omega. So the arc length formula if you divide it by time this gives you the angular velocity formula this tells you that linear velocity is equal to the radius times angular velocity. And of course you can do variations of this that angular velocity omega is equal to V over r and things like that it's an equation you can use but be aware where this formula came from it came from the standard arc length formula. This formula requires things being radians therefore this formula V equals r omega must also be in radiant measure that is your angular velocity needs to be in radians per whatever unit of time seconds minutes hours. The time you have some flexibility but the angle does need to be in radians. So let's just remember how we could do such a thing. Suppose we have a water wheel that completes one rotation every five seconds what would be the angular speed in radians per second. So you know trying to convert these things over so we have one rotation one revolution every five seconds so we have this ratio one revolution per five seconds. Okay, well we want to convert that revolution into radians. So one revolution would be the same thing as two pi radians. So that one revolution turns into two pi you have five seconds right there. And so this would give us a two pi over five and this we measured in radians per second. Or you might just call it per seconds because radians themselves is really not a unit. And it measures the angle there but with these formulas radians is often a unit list quantity there. And so you get two pi over five radians per second or you could approximate that and you end up with 1.257 radians per second. And so we could then calculate now we do have the angular speed here this is this omega value and we could compute say the linear velocity using that information. So consider a bicycle whose wheels are 28 inches in diameter. If a tachometer determines the wheels are rotating at 180 rpm rpm here represents revolutions per minute. Let's find the speed that the bike is traveling down the road in miles per hour. Alright, so what is what is being asked in the story problem here we want to find the speed of the bicycle. Well, there's two types of speeds we talked about there's linear speed and there's angular speed. This tells us the speed down the road this thing is looking for the linear velocity we need to find V and we need to do it in miles per hour. Okay, so in order to do that we're going to utilize the formula that V equals R times omega so we can compute the linear speed if we know the radius and we know the angular velocity. Now we do know about the radius right we told that the diameter of the bicycle wheel is 28 inches. What tells us that the diameter will equal 28 inches and taking half of that the radius is equal to 14 inches so we do have the radius that's good. What about the angular velocity omega what we're told that the wheels are spinning at 100 revolution 180 revolutions per minute. So that gives us that omega equals 180 rpm. But in order to use this formula, we have to be measuring the angles in not revolutions, not in degrees, but in radians. And so we have to convert the angles over like we saw just a little bit ago, revolutions per minute means 180 revolutions, how many times it rotates per one minute of time. And like we've seen what to do one revolution is equal to two pi radians like so so we stick that together 180 times two of course is 360. So we see omega is equal to 360 pi radians per minute. So now we have an appropriate angular velocity, but it turns out that's just half the fight now we could take our times omega to get V, but the units don't really match up right we need to be in miles per hour. Okay, so our distance for the race was measured inches we got to cover that over and our measurement of time wasn't in hours it's in minutes. Okay, so let's let's think about that for a little bit V is going to equal our times omega which are was 14 inches and omega is 360 pi radians per one minute of time. Okay, so let's first figure out how we deal with the time right we don't want minutes we want hours. So we want to have one hour in the denominator. Well, one hour is equivalent to 60 minutes. So we have to take 60 divided by one to convert from minutes to hours, so that the minutes cancel out. Now we're in hours as our timing what about distance here though, well we have to convert from inches to miles and that you can just, you know, you can Google it to try to figure it out. But you can do the same type of strategy right we can switch from inches to feet right you have per one foot we want this in the numerator right we want the units in the top to be miles you have one foot is equal to 12 inches so that these inches cancel out. Now we're measuring feet per hour, which isn't quite right yet. So we have to convert from feet to miles for which one mile is equal to 5,280 feet for which now the feet cancel out and now looking at the units here because again you kind of ignore the radians when it comes to these things, you're going to have miles per hour. This is now the correct unit that we want here so what does this quantity look like. Well we have a 14 times a 360 pi. Going forward we have a 60. This is over 12. And then a 5,280, like so. And so then we try to simplify this best we can, you know, and we could go through all the details of this like 12 goes into 60. You know five times whatever but without without boring you with all of the arithmetic here this thing would simplify to be 105 pi over 22. There's a lot of common factors in the numerator and denominator there that cancel out. So you're going to get 105 pi over 22. This of course is still in miles per hour. And again that's not a very useful number in the current expression. So let's approximate it. If you put that in a calculator you'd end up with 14.99 miles per hour. So I think it's fair to say that oh the bike is traveling at 15 miles per hour, which we'll be able to compute using our angular velocity formula. That concludes then lecture eight and thanks everyone for watching. If you learned anything please hit the like button. If you would like to see more videos like this in the future subscribe to the channel. 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