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# Relationship between angular velocity and speed

## Video transcript

let's say we have some object that's moving in a circular path so let's say this is the center of the object the center of the path right over here the center of the circle so the object is moving in a circular path that looks something like that a counter clockwise circular path we could do that with clockwise as well and what I want to do is think about how 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 five revs per second so in one second it goes 1 2 3 4 5 1 2 3 4 5 it's doing that 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 we could do it how many degrees per second if we did degrees each revolution would be 360 degrees if we do it with radians if we do it with radians we know that each revolution is 2 pi radians 2 pi if you go all the way around a circle if you go all the way around a circle you have gone 2 pi you have gone two pi radians which is really just you're saying 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're going 5 revolutions per second and there are 2 pi 2 pi radians radians per per revolution then you could do a little dimensional analysis these cancel out and you see you get 5 times 2 pi which gets us so 5 times 2 pi gets us 10 pi 10 pi radians radians radians per second and it works out with the dimensional analysis and hopefully it also makes sense to you intuitively if you're doing five revolutions a second each of those revolutions is 2 pi radians so you're doing 10 pi radians per second one you want you know you're just going 1 2 3 4 5 so that gives us 10 or 2 pi 2 pi 2 pi 2 pi radians every time you do that five times in two seconds you're doing 10 pi radians per second so this right here either 5 reps per second or 10 PI radians per second there essentially measuring the same thing how fast are you orbiting around this central point and this measure of how fast you are orbiting around a central point is called angular velocity angular angular velocity and it's called angular velocity because if you think about it this is telling us how fast is our angle changing how fast let me write it this way speed of angle changing and when you're dealing it with it in two dimensions and this is typically in at least in an early physics course how we do deal with it it even though it's called angular velocity tends to be treated as angular speed it actually is a vector quantity and it's a little unintuitive the vector is 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 it is dependent the direction of the vector is dependent on which way it's spinning so for example when it's spinning in a counterclockwise direction there is a vector the real angular velocity vector does pop out of the page once you start thinking about operating in three dimensions and if it was going clockwise if it was going that way right over here 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 is spinning and then your thumb is essentially pointing in the direction your thumb is pointing in the direction that the ank that the actual vector the pseudo vector is going to go in but we don't have to think too much about that for our purposes when we're just thinking about a two-dimensional plane right over here we can really think of angular velocity as a it's official term is a pseudo scalar but we can treat it as a scalar quantity as long as we 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 by an Omega a lowercase Omega right there an upper case Omega looks like this lower case Omega is what people tend to use for angular velocity so there's a couple of ways that you could think about it you could say angular velocity angular velocity is equal to change in angle change an angle over a change in time so for example this is telling us ten PI radians per second 10 PI radians per second or if you wanted to do it in the calculus sense if you want to say instantaneous angular velocity and don't worry about this if it doesn't make a ton of sense to you but this is just instantaneous angular velocity it would be the derivative of it would be the derivative of your angle with respect to time how the angle is changing with respect to time now 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 how far is this object traveling at every revolution that it's doing and what we can do right over here 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 2 pi r let's say this is our meters give ourselves some units right over there so the rate of the circumference over here is going to be 2 pi r 2 pi r 2 pi r meters and if the if the let's say that the angular velocity let's say that the angular velocity angular velocity let's say it is equal to omega omega radians per second radians radians per second and so how many how many revolutions is that per second well we could go backwards from what we did over here we have one revolution is equal to 2 pi is equal to 2 pi radians and sometimes just to be clear sometimes angular velocity is actually measured in revolutions per second but the SI unit is in radians per second and so if we want to convert Omega radians per second into revolutions per second radians cancel out we are left with we get omega over over 2 pi revolutions revolutions per second but we know how many meters we get per revolution we have 2 pi r meters per revolutions let's write that down right over here so we have let me copy and paste this so our angular velocity if we wanted in revolutions per second is going to be Omega over 2 pi revolutions per second Omega is in radians per second if we put it into into revolutions per second Omega divided by 2 pi revolutions per second and then let's multiply that and 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 going to travel our whole circumference per revolution so we're going to have two PI R two PI R meters per revolution so these two cancel out the two pi cancels out with the two pi so you end up getting omega Omega times R Omega times let me write the R and white Omega times are meters meters per second and just like that we have the magnitude of the velocity or I guess 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 a vector quantity it's not the velocity it's the magnitude of the velocity or we could say this is the speed is going to be equal to Omega times R so the speed so let me be is equal to the angular velocity times R I guess we could say the magnitude of the angular velocity times the radius let me write that out in words and I don't want you to be confused I'm not saying that this is a vector quantity if this was a vector I would I would put an arrow right over there and if this was a vector I would put an arrow over there and then I'd be referring to the thing that's popping out of the page but here I'm just talking about the magnitude of the angular velocity and so writing it in words you get speed speed is equal to is equal to angular velocity angular velocity and if you want to be particularly will say this is the magnitude of the angular velocity the magnitude of angular velocity times times the radius of the circle that you are going around and if you wanted if you wanted 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 so this is V is going to be equal to speed divided by 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