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what we're going to do in this video is look at a tangible example where we calculate angular velocity but then we're going to see if we can connect that to the notion of speed so let's start with this example where once again we have some type of a ball tethered to some type of centre of rotation right over here let's say this is a it's connected with a string and so if you were to move the ball around it would move along this blue circle in either direction and let's just say for the sake of argument the length of the string is 7 meters and we know that at time is equal to 3 seconds our angle is equal to theta is equal to PI over 2 radians which we've seen in previous videos we would measure from the positive x axis just like that and let's say that at T is equal to 6 seconds T is equal to 6 seconds theta is equal to PI radians and so after 3 seconds the ball is now right over here and so if we wanted to actually visualize how that happens let me see if I can rotate to this ball in 3 seconds so it would look like this one-mississippi two-mississippi three-mississippi let's do that again it would be one-mississippi two-mississippi three-mississippi so now that we can visualize or conceptualize what's going on see if you can pause this video and calculate two things so the first thing that I want you to calculate is what is the angular velocity of the ball and actually would be the ball and every point on that string what is that angular velocity which we denote with Omega and then I want you to figure out what is the speed of the ball so what is the speed see if you can figure out both of those things and four extra points see if you can figure out our relationship between the two all right well let's do angular velocity first I'm assuming you've had a go at it angular velocity you might remember is just going to be equal to our angular displacement which we could say is Delta Theta and it is a vector quantity and we are going to divide that by our change in time so delta T and so what is this going to be well this is going to be our angular displacement our final angle is PI pi radians minus our initial angle PI over 2 radians and then all of that is going to be over our change in time which is 6 seconds which is our final time minus our initial time minus 3 seconds and so we are going to get in the numerator we have been rotated in the positive direction PI over 2 radians because it's positive we know it's counter clockwise and that happened over 3 seconds and so we could rewrite this as this is going to be equal to PI over 6 and let's remind ourselves about the units our change in angle that's going to be in radians and then that is going to be per second so we're going PI over 6 radians per second and if you do that over 3 seconds well then you're going to go PI over 2 radians now with that out of the way let's see if we can calculate speed if you haven't done so already pause this video and see if you can calculate it well speed is going to be equal to the distance the ball travels and we've touched on that in other videos I encourage you to watch those if you haven't already the distance that we travel we could denote with s s is sometimes used to denote arc length or the distance traveled right over here so the speed is going to be our arc length divided by our change in time divided by our change in time but what is our arc length are going to be well we saw in a previous video where we related angular displacement to arc length or distance that our arc length is nothing more than the absolute value of our of our angular displacement of our angular displacement times the radius the radius and in this case our radius would be seven meters so if we substitute all of this up here what are we going to get we are going to get that our speed our speed I'm writing it out because I don't want to overuse well I am overusing s but I want people to get confused our speed is going to be equal to the distance we travel which we just wrote is our the magnitude of our angular displacement and this is all fancy notation but when you actually apply it it's pretty straightforward times the radius of the circle I guess you can say we are we are we are traveling along so let me write that in a different color so times the radius all of that over our change in time all of that over our change in time well we could put in the numbers right over here we know that this is going to be PI over two you take the absolute value of that is still going to be PI over two we know that our radius in this case is the length of that string it is seven meters and we know that our change in time here we know that this over here is going to be three seconds and we can calculate everything but what's even more interesting is to recognize that what is what is this right over here what is the absolute value of our angular displacement over change in time well this is just the absolute value of our angular velocity so we could say that speed speed is equal to the absolute value of our angular velocity absolute value of our angular velocity times our radius times our radius and now so this is super useful our speed in this case is going to be PI over six radius radians per second so PI over PI over six times the radius times seven meters times seven meters and so what do we get we are going to get seven PI over six meters per meters per second which will be our units for speed here and the reason why we're doing the absolute values because remember speed is a scalar quantity we're not specifying the direction in fact the whole time we're traveling our direction is constantly changing so there you have it there's multiple ways to approach these types of questions but the big takeaway here is one how we calculated angular velocity and then how we can relate angular velocity to speed and what's nice is there's a nice simple formula for it and all of this just came out of something that relates to what we learned in seventh grade around the circumference of the circle which we touch on in the video relating angular displacement to arc length or a distance traveled

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