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# Relationship between angular velocity and speed

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Let's say we have some object that's moving in a circular path Let's say this is the center of the object path, the center of the circle So the object is moving in a circular path that looks something like that counterclockwise circular path--you could do that with clockwise as well I want to think about how fast it is spinning or orbiting around this center how that relates to its velocity? So let's say that this thing right over here is making five revolutions every second So in 1 second, 1 2 3 4 5. Every second it's making 5 revolutions So how could we relate that to how many radians it is doing per second? Remember radians is just one way to measure angles You could do with how degrees per second If we do it with radians, we know that each revolution is 2 pi radians If you go all the way around a circle, you have gone 2 pi radians which is really just you say you've gone 2 pi radii, whatever the radius of the circle is and that's where actually the definition of the radian comes from So if you go 5 revolutions per second and they're 2 pi per revolution then you can do a little bit of dimensional analysis. These cancel out and you get 5 times 2 pi which gets us to 5 times 2 pi gets us 10 pi radians per second And it works out the dimensional analysis and hopefully it also makes sense to you intuitively If you're doing five revolution a second, each of those revolutions is 2 pi radians so you're doing 10 pi radians per second. You're going 1 2 3 4 5, so that gives us 10, or 2 pi 2 pi 2 pi 2 pi 2 pi radians every time, you're doing it five times a second. So you're doing it 10 pi radians per second So this right here, either five revs per second or 10 pi radians per second they're both essentially measuring the same thing how fast are you orbiting around this central point? And this measure of how fast you're orbiting around a central point is called angular velocity It's called angular velocity because if you think about it this is telling us how fast is our angle changing, or speed of angle changing When you're dealing with it in two dimensions and this is typically when in a recent early physics course how we do deal with it Even though it's called the angular velocity it tends to be treated as angular speed It actually is a vector quantity and it's a little unintuitive that the vector's actually popping out of the page for this. It's actually a pseudo-vector and we'll talk more about that in the future So it is a vector quantity and the direction of the vector is dependant on which way it's spinning. So for example when it's spinning in a counterclockwise direction there is a vector, the real angular vector does pop out of the page We start thinking about operating in three dimensions And if it's going clockwise, the angular velocity vector would pop into the page The way you think about that, right-hand rule Curl your fingers of your right hand in the direction that it's spinning and then your thumb is essentially pointing in the direction that the actual vector or the pseudo-vector's gonna going We'll not think too much about that For our purposes, when we're just thinking about two-dimensional plane right over here we can really think of an angular velocity as a--the official term is a pseudo-scaler but we can include that as a scaler quantity, as long as we do specify which way it is rotating So this right over here, this 10 pi radians per second we could call this its angular velocity And this tends to be denoted by an omega a lower case omega right there Upper case omega looks like this So there's a couple of ways you could think about it You could say angular velocity is equal to change in angle over a change in time So for example, this is telling us 10 pi radians per second Or if you want to do in the calculus sense and take instantaneous angular velocity it would be the derivative of your angle with respect to time How the angle is changing with respect to time With that out of the way, I want to see if we can see how this relates to speed How does this relate to the actual speed of the object? So to get the speed of the object, we just have to think about how far is this object traveling every revolution that it's doing And what we can do right over here--let's say that this radius is r So in every revolution, it is traveling 2 pi r Let's say this is r meters. Give ourselves some units right over there So the circumference over here is going to be 2 pi r meters Let's say that the angular velocity is equal to omega radians per second And so how many revolutions is that per second? We can go backwards from what we did over here We have one revolution is equal to 2 pi radians Just to be clear, sometimes angular velocity is actually measured in revolutions per second but the SI unit is in radians per second So here I want to convert omega from radians per second into revolutions per second Radians cancel out. We are left with--we get omega over 2 pi revolutions per second We know how many meters we get for a revolution We have 2 pi r meters per revolutions So we copy and paste this So our angular velocity, if we want revolutions per second it's going to be omega over 2 pi revolutions per second Omega is in radians per second if we put it into revolutions per second omega / 2 pi revolutions per second And then let's multiply that times--we want to convert this into meters per second So how many meters do we have per revolution? Well, we're gonna travel a whole circumference per revolution, so 2 pi r meters per revolution So these two cancel out 2 pi cancels out with the 2 pi So you end up getting omega times r meters per second And just like that, we have the magnitude of the velocity or we could say the speed of the object as it goes around in a circle So what we can say is the magnitude of the velocity-- I'll specify that by v--I want to be clear. This is not vector quantity. It's not the velocity. It's the magnitude of velocity Or we can say this is the speed. It's going to be equal to omega times r So the speed is equal to the angular velocity times r I guess we could say the magnitude of the angular velocity times the radius I don't want you to be confused. I am not saying that this is a vector quantity If this was a vector, I would put an arrow right over there And if this was a vector, I would put an arrow over there then I'll be referring to the thing that's popping out of the page but here I'm talking about the magnitude of the angular velocity and so writing in words, you get speed is equal to angular velocity-- if you want to be particular, this is the magnitude of the angular velocity-- times the radius of the circle that you are going around and if you want to solve for angular velocity you divide both sides by radius and you get angular velocity Omega is equal to speed which we're using v for, divided by the radius So we can actually use this information to do other interesting things later on But hopefully this gives you a sense of how all of this stuff is related