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Video transcript
let's have some object this traveling in a circular path just like this and what I've drawn here is its velocity vector at different points along that path and so this right over here is going to be the one to vector wanted this is going to be velocity vector too and this right over here is going to be velocity vector 300 going to assume in this video is that the magnitude of these velocity vectors is constant or another way to think that the speed is constant so I'll just say lower case of you without the Erawan tops is going to be scalar quantity I'll call this the speed or you could call this the magnitude of these vectors and this is going to be constant so this is going to be equal to the magnitude of vector one which is equal to the magnitude of actor to the direction is clearly changing but the magnitude is going to be the same which is equal to the magnitude the magnitude of the vector 3 I'm going to assume those travelling in a path in a circle with radius r what I'm going to do is i'ma drop position vector at each point so let's call this call are one actual student pink let's call our one that right over there that's position vector one or one that is position vector are too so the position is clearly changing its position vector are too and that is position vector are three but the magnitude of our position vectors are clearly the same and I'm going to call the magnitude of our position vectors are that's just the radius of the circle is this distance right over here so are as you go to the magnitude of our 10 to the magnitude of our to which is equal to the magnitude magnitude of our three know what I want to do in this video is proved to you visually that given this radius and given this speed that the magnitude of the centripetal acceleration and I'll just write that is a subsea I don't have an error on top of this is a scalar quantity so the magnitude of the centripetal acceleration is going to be easy will to our speed squared are constant speed squared divided by divided by the radius of the circle this is what I want you I want you to feel good that this is indeed the case by the end of this video and to understand that what I want to do is I want to reap lot these these velocity vectors on another circle and just think about how the vectors themselves are changing so let's copy and paste this so let me copy and paste of you on so copy and paste so that is actually do it from the center so that his view on that we do the same thing for you to let me copy and paste it that is easy to let me do it also 433 let's get back to Parliament get the label so copy and paste it and that right over there is Victor the three let me clean this up a little bit too so that we don't so that's clearly v2 to lay blame or we know that v2 is orange v2 is an orange and what is the radius of the circle going to be right over here with the radius of the circle is going to be the magnitude of the velocity vectors and we are do you know the magnitude of the velocity vectors is this quantity V the scalar quantity so the radius of the circle is the the radius of the circle we are the party know is equal to arm and just as the velocity vectors what's giving us the change in position over time the change in position vector overtime what's the vector that's going to give us the change in our velocity vector overtime well that's going to be our acceleration vectors so you will have some acceleration will call this a one will call this a 282 and I'll call this a 383 and I want to make sure that you get the analogy that's going on here as we go around the circle displays the position vectors first they point out to the left then in the upper kind of a kind of eleven o'clock position I guess that the top of the top left then to the top it's pointing in these different directions like a hand in a clock and what's moving it along there is the change in position vector overtime which is are these velocities factors over here the velocity vectors are moving around like the hands of a clock and what it what is it doing the moving around is are these acceleration vectors and over here the velocity vectors they are tangential to the radius or shine their tangential to the path which is a circle their perpendicular to a radius and you learn that in geometry that a that a line that is tangent to a circle is perpendicular to a radius is also going to be the same thing right over here and just going back to what we learned when we learn about the intuition of centripetal acceleration if you look at a one writer year and you translate this vector it'll be going just like that it is going towards the center a to once again is going towards the centre 83 once again if you translate that that is going towards the center so all of these are actually center seeking vectors and you see that right over here these are all these are actually centripetal acceleration vectors right over here here time I just the magnitude of it I would assume that all of these have the same magnitude so we're going to assume that our sincere but all they all have a magnitude of will call a subsea so that's the magnitude it's equal to the magnitude of a one and that vector is equal to the magnitude of a two and it's equal to the magnitude of a three now what I want to think about is how long is it gonna take for this thing to get from this point on the circle to that point on that circle right or they're so the way to think about it is what's the length of the arc that it traveled the length of this arc that it traveled right over there that's one-fourth or around the circle it's going to be one fourth of the circumference the circumference is too high are it is going to be one-fourth of that so that is the length of the arc that is the length of the arc and then how long would take it to go that well you would you divide the length of your path divided by the actual speed actual thing that's not get along that path so you want to divide that by your actual the magnitude of your velocity or your speed this is a magnitude of a city not velocity is not a vector right over here this is a scalar so this is going to be the time the time to travel along that path now that the time of travel along this path is going to be the exact same amount of time it takes to travel along this path for the velocity vector so this is for the position vector travel like that this is for the velocity vector of travel like that it's going to be the exact same tea and what is the length of this path and now think of it in the purely geometrical sense we look we're looking at the circle year the radius of the circle is v so the length of this path right over here is going to be one fourth is going to be be doing that same color CCD analogy it's equal to 14 times of the circumference of the circle the circumference of the circle is to buy times the radius of the circle which is now what is nudging it along this circle what is it not doing a long as pat what is the analogy for speed right over here is what's module along the path over here it is the magnitude of the velocity vector so it's a nudging along this arc right over here is the magnitude of the acceleration vector so it is going to be it is going to be a sun sea and these times are going to be the exact same thing the amount of time it takes to go for this vector to go like that for the position vector is the same amount of time it takes the velocity vector to go like that so we can set these two things equal each other so we get on this side we get one-fourth to buy our overview is equal to 1 4th to päivi to Ivy over over the magnitude of our acceleration vector and now we can simplify little bit we can divide both sides by one-fourth care that we can divide both sides by Tupac I get rid of that let me rewrite it so that we get our over the is equal to the over the centripetal acceleration and I could cross multiply and so you get V times so I'm just multiplying I'm cross multiplying right over here V time zev squared is equal to AC times are and cross multiplying remember is really just the same thing is multiplying both sides by both denominators by multiplying both sides times viene si times the AC so such a magical thing if you multiply both sides viene see these views cancel out these AC's cancel out you get these times easy squared is equal to a subsea times are is equal do a subsea times are and how to solve for the magnitude of our centripetal acceleration you just divide both sides by are you divide both sides by our and you are left with and I i guess we we've earned a drum roll now you're left with the magnitude of our centripetal acceleration is equal to the magnitude are constant magnitude of our velocity so this right here is our speed divided by the radius of the circle and we're done