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Current time:0:00Total duration:14:36

Relating angular and regular motion variables

AP.PHYS: NT‑3.A.1.1 (LO)

Video transcript

so in the previous video we defined all the new angular motion variables and we made an argument that those are more useful in many cases to use than the regular motion variables for things that are rotating in a circle since every point on this string and tennis ball let's say this is a tennis ball you tied a string to and your whirling and around in a circle every point on the string including the tennis ball will have the same angular displacement angular velocity and angular acceleration but even though using angular motion variables is more convenient for these rotational motion problems it's also really important to know how to translate those angular motion variables back into the regular motion variables so that's what I want to show you in this video how to translate angular motion variables back into regular motion variables so let's do this the simplest possible angular motion variable was the angular displacement because that just represented how much angle an object has rotated through so let's say it rotated through this much we represented the angular displacement with a Delta Theta and we called it the angular displacement and in physics we typically choose to measure this in radians for a reason that I'll show you in just a second now how would we convert this into a regular motion variable and what regular motion variable would that be well if I were coming at this for the first time I'd be like alright this is the angular displacement let's figure out how to relate it to the regular displacement but that'd be weird because think about it the regular displacement for the ball that started over here and made it over here would be from this point to that point that would be the regular displacement of the ball the regular linear displacement of the ball that's a little weird and I don't want to show you how to find that for one you'd have to use the law of cosines and that's a little more in depth than I want to get to in this video and for to the better reason is this isn't all that useful it turns out there's a much more useful quantity that would tell you how far the ball went and that's the arc length of the ball so the ball traced out a path through space around this circle we call this the arc length and it turns out this is much more useful in a variety of problems and good news is it's much easier to find than that regular displacement so this is the arc length people vary on what letter to use here I've seen L but most math books use s so we'll just use s as well you might think this is hard to find but it's not in fact if we use radians and this is why we use radians it's extremely easy to find if we wanted to find the arc length of this tennis ball we're just going to take the radius of the circular path that tennis ball is tracing out so in this case it'd be the length of this string we take that radius and if we're in radians we just multiplied by the angular displacement this is why radians are so convenient you just take that measurement in radians multiplied by the radius of the circular path the object is tracing out and you get the arc length which is the number of meters along this path that the object is traveled if that seems miraculous it really isn't I mean the reason why this works so well is because this is how the Radian was defined one Radian is defined to be the angle through which you have to travel so that the arc length is equal to the radius of that circle so this isn't a surprise this was selected and defined strategically so that we can use this unit and we get a really easy way to convert between the angular displacement how many radians something is rotated through and how many meters it's actually traveled through its arc so this arc length is going to have units of meters as long as we measure the radius in meters all right so that's one relationship between angular displacement how much angle something is rotated through and how many meters it's actually traveled the next relationship I want to talk about relate to the angular velocity to the regular velocity so remember in the previous video we defined the angular velocity to be the angular displacement per time so this is the rate at which something is rotating through a certain amount of angle and the letter we use to represent angular velocity was the Greek letter Omega so this angular velocity represented the rate at which something is rotating in a circle so if it's rotating slowly it's going to have a small angular velocity and if it's rotating quickly it's going to have a large angular velocity so obviously the speed and the angular velocity-- are going to be related because the higher the angular velocity the higher the speed but what is that relationship how would we get from the angular velocity to the regular velocity well it's not actually that hard at all because all we need to do is turn this number of radians per second into meters per second and I can do that if I multiply both sides of this equation down here by our I'll get R times Omega is going to equal R times Delta Theta and then I still have to divide by delta T so I just multiply both sides of this equation by R but look what I get R times Delta Theta is just the arc length so this whole side over here is just how many meters that object has traveled around the edge of this circle divided by the time that it took but that's just the speed this arc length is just the distance the object has traveled and the time is the time that it took and distance per time is just speed so this is the speed of the object I'm going to write that as V even though it's not velocity this is not a vector and it's not velocity because think about it this arc length isn't displacement this was the distance the object traveled distance per time is the speed displacement per time is the velocity we didn't use displacement displacement was this weird one we didn't want to deal with that so since we're choosing to deal with arc length which is distance what we're going to do is relate the angular velocity to the speed and now we have that relationship look at this is R the radius times the angular velocity equals the speed of the object so this is the relationship between the angular velocity and the speed the speed of the object is going to equal the radius of the circular path the object is traveling in times the angular velocity I should box these these are important this arc length formula was how you relate the number of radians an object has rotated through to how much arc length that's traveled ie how much distance it's gone through in this formula down here relate to the angular velocity Omega the number of radians per second something is rotating with to how many meters per second it's traveling in other words how many meters per second is tracing out along this arc length so this is good now we know how to relate the angular displacement to the distance the object's traveled and we know how to relate the angular velocity to the speed of the object so you probably know what's coming next we have to relate the angular acceleration to the regular acceleration so recall that the angular acceleration which we represented with a Greek letter alpha was defined to be the change in the angular velocity per time so it's the rate at which your angular velocity was changing so if you're just moving at a constant rate you've got no angular acceleration because there's no change in omega but if omega starts off slow and then it gets faster and faster you do have angular acceleration and it's probably not a surprise that if you have angular acceleration this ball is going to have regular acceleration too because if it's speeding up in its angular rotation it's going to be changing its velocity as well so how do we do this how do we relate the angular acceleration to the regular acceleration well the simplest thing to try is the GUI will work down here we multiplied both sides of our equation by the radius and we found a relationship that related speed to angular velocity so let's try it again let's multiply both sides of this equation by radius and see what we get on the left hand side we're going to get the radius times the angular acceleration and that's going to equal the radius times the change in angular velocity over the change in time so all I've done here is multiplied both sides of this equation this is a definition of angular acceleration by the radius so see what we get on the right hand side we've got R times Delta Omega so this is really R times the change in omega well that's just omega final minus omega initial and then divided by delta T so I can distribute this R I get that this would equal R times Omega final minus R times Omega initial divided by the time that it took but now look what happens we've got R times Omega final and R times Omega initial we know what r times omega is it's the speed not the velocity but the speed so i could rewrite this i could say that this is really the final speed minus the initial speed over the time that it took to change by that much speed so this is if I just rewrite the left-hand side this is what R times alpha is equal to now if I were you I'd be tempted to just be like all ago we did it that's the acceleration look at change in speed over time but you got to be careful acceleration the through acceleration vector is the change in velocity per time but these are not velocity vectors these were speeds so this isn't the true acceleration vector this is something different this is the change in speed per time so that's still an acceleration but it's not necessarily the entire acceleration because there's two ways to accelerate you can change your speed or change your direction basically this acceleration we just found doesn't take into account any acceleration that's coming from changing your direction this is only the acceleration that's going to be changing your speed so if I were you I'd probably be confused at this point so let me try to show you what this means so if this ball is rotating in a circle just by the mere fact that the ball is rotating in a circle it has to be accelerating even if the ball isn't speeding up or slowing down there's got to be an acceleration because this ball is changing the direction of its velocity that means there's got to be a force a centripetal force in this case it would be the tension and there's got to be a centripetal acceleration that's changing the direction of the velocity that is not this acceleration over here this is a different acceleration we know the centripetal acceleration is directed inward and we already know how to find the centripetal acceleration remember the formula for centripetal acceleration is the speed squared divided by the radius this component this isn't rip Atal acceleration is the component of the acceleration that changes the direction of the velocity so I'm going to say that again because this is important the centripetal acceleration which you can find with v squared over R is the component of the acceleration that changes the direction of the velocity and if something is going in a circle it must have centripetal acceleration but this acceleration component that we found out here is different this is what's changing the speed you don't have to have this if you're going in a circle you could imagine something going in a circle at a constant rate and if that's happening it's got centripetal acceleration but it doesn't have this thing down here because this thing we found R times alpha is the change in the speed of the object per time how would I draw that up here if I wanted to represent this a that we found out here visually up here I draw it tangential to the direction of motion ie tangential to the circle because components of acceleration that are directed perpendicular to the velocity changed the direction of the velocity but components of acceleration that are directed parallel to the direction of the velocity change the magnitude of the velocity ie the speed to change the magnitude of the velocity in other words to speed something up or to slow it down you need a component of that acceleration that's either in the direction of motion or opposite the direction of motion if it was opposite the direction of motion the acceleration would be slowing the object down and if the component of acceleration is in the direction of motion then it's speeding the object up and that's what we found down here that's what this component of acceleration is our alpha which is why it's often called the tangential acceleration so I'm going to write that up here the tangential acceleration which is equal to R times alpha the radius times the angular acceleration is the component of the acceleration that's changing the magnitude of the velocity ie it's changing the speed and in order to do that it's got to be directed tangential to the direction of motion and that's what this R times alpha represents only box that that's important this is the formula to find the tangential acceleration it doesn't give you the total acceleration we know that there's always a component of acceleration that's acting centripetal II if an object is going in a circle and you can find that with V squared over R but if the object going in a circle is also speeding up not only going in a circle but changing its speed it's going to also have this component of acceleration which is the tangential acceleration so this point you might be confused like wait we've got tangential acceleration we've got centripetal which one is the acceleration well they're both just components of the total acceleration which you could find if you really wanted to you could say that the total acceleration squared we could use the Pythagorean theorem because these are the two perpendicular components of the total acceleration we'd say the total acceleration squared would equal the tangential acceleration squared plus the centripetal acceleration squared and what we'd be finding is the total acceleration squared which if you wanted a direction the direction of that total acceleration would point this way somewhere if we you had centripetal acceleration inward and let's say the object was speeding up so I'll say it wasn't slowing down so you didn't have this component you've got this forwards component and an in Words component the total acceleration would be directed here somewhere and since you could form a right triangle out of this with these two sides you can imagine moving this centripetal acceleration over to this side and you can find the hypotenuse which would be the total acceleration by just taking the tangential acceleration squared plus the centripetal acceleration squared and then taking a square root give you the magnitude of the total acceleration so recapping there's two components of acceleration the tangential acceleration which is R times alpha either speeds an object up or slows it down the centripetal acceleration works to change the direction of the motion of the object you can relate the speed of an object to the angular velocity by multiplying by R and you can relate the arc length ie the distance the object traveled around this edge of the circle to the angular displacement by also multiplying by R so these three equations are how you relate the angular motion variable to its linear counterpart