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# Linear velocity comparison from radius and angular velocity: Worked example

## Video transcript

let's say that we have two pumpkin catapults so let me just draw the ground here and so the first pumpkin catapult let me just draw it right over here that's its base and then this is the part that actually caught up catapults the pumpkin so that's what it looks like it holds the pumpkin right over here that's the pumpkin that's about to be shot away and what it does is once it releases I guess someone presses a button or pulls a lever the pumpkin catapult is going to release and so this arm is going to turn a certain amount and then just stop immediately right over there and then that pumpkin that pumpkin is going to be released with some some type of initial velocity so your pumpkin is just gonna go like that so this is our small pumpkin catapult this is our small one but let's say we also have a large pumpkin catapult let me draw that so our large pumpkin catapult right over here similar similar mechanism but let's say it's arm is four times as long so that looks about four times as long right over there so this is the larger one the large pumpkin catapult let me make sure to draw the pumpkin to remember what we are catapulting and then it will go through actually the exact same angle it'll go through the exact same angle and then let go of its pumpkin this is gonna be a very useful video because you will find yourself making many pumpkin catapults in your life so and then it will let it go and you will have some linear velocity now we know a few things about these pumpkin catapults let's say this small pumpkin catapult the radius between the center of the pumpkin the radius between the center of the pumpkin and the center of rotation right over there let's say that this is our well for the large one this one is this distance right over here is 4 R we also know the angular velocity when this wildest thing is moving so we know that the angular velocity here let's say the magnitude of the angular velocity is Omega it would actually be negative if we were to write it as a vector because we're going in the clockwise direction because that's the convention but this right here is the magnitude of the angular velocity and just to make that tangible for you we could say let's say that this is I don't know two pi radians per second and let's say this thing while it's while it's in motion it also has the same magnitude of its angular velocity so this thing right over here is also the magnitude of angular velocities once again two pi radians per second so my question to you is how would the velocity the magnitude of the velocity of the pumpkin being released from the small catapult so V sub small if I put a vector if I put a arrow on top of it we'd be talking about velocity since I didn't put an arrow we're talking about just the magnitude of velocity you could think about this is the speed how does this compare to V sub large we have the same angular velocity but we have different radii pause the video and see if you can figure that out well the key thing to realize here and we've seen this in multiple videos is the relationship between the magnitude of angular velocity and the magnitude of velocity linear velocity the magnitude of angular velocity times your radius is going to give you the magnitude of your linear velocity so for V small right over here we could write this we could write this as V small is going to be equal to Omega these are the same Omega that Omega and that Omega is the same in fact we don't even have to know what this is we could say V sub small is equal to Omega times our radius which is R which is R and what's V sub large going to be well V sub large is going to be equal to that same Omega so I'm talking about this particular Omega right over here so it's that same Omega but now our radius isn't our it is for our so we're talking about so times for our or if we were to rewrite this this would be equal to four times Omega times R four times Omega times R and what is this right over here Omega times R that is the magnitude of the velocity of our smaller catapult or the pumpkin being released from the smaller smaller catapult so just like that you see by having the same angular velocity but if you increase your arm length by a factor of four your velocity is going to increase by a factor of four and so you have the magnitude of velocity of the pumpkin being released from the large catapult is going to be equal to four times the magnitude of the velocity of the pumpkin being released from the smaller catapult
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