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Current time:0:00Total duration:14:22

Video transcript

what I want to do in this video is a calculus proof of the famous centripetal acceleration formula that tells us the magnitude of centripetal acceleration the actual direction will change it's always going to be pointing inwards but the magnitude of centripetal acceleration is equal to the magnitude of the velocity squared divided by the radius I want to be very clear this is a scalar formula right over here it's talking about the magnitude of the acceleration and the magnitude of the velocity if these were vectors we would have arrows drawn over so this really I don't want people to get confused because this is a V this is really referring to the speed squared and this is the magnitude so these are all these are all scalar quantities so to do that let's imagine some object maybe it's some object in orbit around a planet or something so let's say that that's the planet and that you have some object that is in orbit around the planet and it is going in a counterclockwise direction and so let's specify its position vector as a function of time so let's specify its position functor as a function of time so that is its position vector and it's going to change as a function of time as this thing spins around and what we're going to assume for the purposes of this for the purposes of this proof I guess you can call it let me draw some axes over here let me draw some axes so if that is our Y axis and this is our x axis we're going to define theta to be the angle the angle between the positive x axis and our vector so that is going to be theta and we're going to assume that this thing is in an orbit with a radius of R so the magnitude of our position vector even though the direction is going to change the magnitude of our position vector is not going to change it's always going to have length R so this is going in a circle of radius R so I can write it here the magnitude of our position vector which is changing as a function of time is going to be R so how can we write how can we actually express the position vector in terms of its components at any given time well we can write the position vector and I'll do it an engineering notation and so you might want to review those videos if some of this looks for and I will do a little bit of basic trigonometry and breaking down the vector to its components and I encourage you to review those videos if some of that looks a little bit daunting if you take the position vector at any time so if you take the position vector at any time the magnitude is R this angle is Theta right over here it's X component under the X component in blue it's X component this vector right over here is going to be the magnitude of its X component I should say is going to be R cosine of theta we've learned that this comes from basic trigonometry when we started actually we start I think a two-dimensional projectile motion we saw how to break these vectors down into its components and the y component of this vector the y component of this vector is going to be our sine of theta it's going to be the same thing as this vector the magnitude right over there so this is going to be our sine sine of theta so the position vector at any time so the position vector at any time can be written as the sum of its x and y components so it's the magnitude of its X component is going to be R cosine of theta and I could write theta as a function of time if I like but I'm just going to write R cosine of actually let me write it that way so that shows that theta is going to be a function of time this thing is moving and there's going to be that times the I unit vector we're going to do engineering notation right over here so that is the I unit vector this tells us that that is the X component it's going in the positive x Direction plus plus the magnitude of the Y component which is our sine of theta which is going to be a function of time so I'm going to be clear the function of time the function of time applies to the theta put some parentheses around it like that and that is going in the J direction that is going in the J direction so that is our J unit vector so that is going in the J direction so now we have position as a function of theta which is actually a function of time so let's take the derivative of this thing right over here so what is so what is d the derivative of our position vector the derivative of our position vector with respect to time well that's just going to be our velocity vector so that's just going to be our velocity vector as a function of time and it's going to be equal to we just have to take the derivative of each of these to H of these parts with respect to time and you just do the chain rule so you're going to have the R sit outside that's just a constant so you're going to have are the derivative of cosine of theta T with respect to theta T so I'm just doing the chain rule right over here that's going to be negative sine of theta T theta T and then it's the chain rule we also have to multiply that times the derivative of theta T I'm sorry the derivative of theta of T with respect to T so times D theta DT this is just the chain rule right over here and so that's going to be how it's changing in the X direction and in the Y direction we do something very similar in the Y direction we take the same derivative you have the our scalar out front our and then the derivative of sine of theta with respect to theta is going to be cosine of theta and I'll write it as a function of time and then do the chain rule you also have to multiply that times the rate at which data is changing with respect to T times D theta DT and this is going all going to be times our J this is all times our J unit vector now there's something that you might already realize and you should re-watch the video on angular velocity if this is foreign to you but D theta DT this is our angular velocity that's why I said to rewatch that video this right over here the rate at which the angle changes with respect to time that is angular velocity so this right over here is that right over there is angular velocity and for the sake of this video and this is an assumption that we'll have to make for this for this formula right over here we're going to assume that Omega which is the trait of change of our angle with respect to time we're going to assume that that is we're going to assume that this is constant so this is an assumption that we're making for this proof this is we are going to assume that Omega is constant and if Omega is constant that we can treat it literally as a constant and we can actually factor it out of this expression so let's actually factor out a negative Omega R from this expression right over here so we can rewrite our velocity as a function of time is equal to I'm going to factor out a negative Omega times R and if you factor out a negative Omega R what you're left with put some big brackets right over here is this first term the negative got factored out the R got factored out the Omega got factored out you just have a sine of theta T here sine of theta of T so let me an ident even after I didn't have to make it explicit that theta is a function of T but this makes it explicit that theta is a function of T theta is a function of T and then times our I unit vector times a unit vector plus plus so if we're factoring out a negative a negative Omega R this becomes negative cosine cosine of theta which is a function of T theta which is a function of T and that that's times our J unit vector times our J unit vector and let me close the parentheses in the right way so we factored out negative Omega R now let's take the derivative of this with respect to time so if we take if we take and let me give myself some real estate if we take the derivative of the velocity with respect to time this is clearly just what the acceleration is as a function of time and we're going to assume that the magnitude of this thing is constant but the actual direction is changing so this is the acceleration as a function of time is going to be equal to is going to be equal to let me get my colors right it's going to be equal to we have this negative Omega R open from the parentheses so what's the derivative of this thing right over here so the derivative of sine with respect to theta we're just doing the chain rule here derivative of sine with respect to theta is going to be cosine of theta so we get cosine of theta cosine of theta as a function of T and then chain rule we also have to take multiply that times the derivative of theta with respect to T and the derivative of theta with respect to T I could write D theta DT or I could write D theta DT here but that's once again is just another that is just Omega so that is just Omega and that's of course in the I direction and then from that we and then next to that we're going to have to take the derivative cosine of theta of T with respect to theta so that's going to be that would be negative sine of theta but we have a negative out front so it becomes positive sine positive sine of theta as a function of T and then we have to do the chain rule the derivative of theta with respect to T we have to multiply by this and that we could write d theta DT right here but that once again is the same thing the same thing as Omega and all of that two being multiplied times the J the J unit vector and then we can close our parentheses so now let's factor out this other Omega and we get something interesting we get the acceleration our acceleration vector as a function of time is equal to if we factor out another Omega we get negative Omega negative Omega squared R I'm just factoring out another Omega so that becomes negative Omega squared R times and I'll write it in parentheses here x times cosine of theta cosine of theta as a function of T times our I unit vector the color changing is the hard part here plus plus B the same color plus sine of theta which is a function of T times our J unit vector times our J unit vector now what is what is our what is this all of this business right over here what is all of this business if you just look at this part right over there well R times this especially if you distributed the R that's exactly this thing right over here in fact if you distribute the AR you get exactly R cosine of theta as a function of T times our unit vector plus R sine theta as a function of T times the J unit vector so everything that I squared off in orange right over here this is our position vector this is our position vector as a function of time so all of that work we did we just got a very interesting result we got that our acceleration vector as a function of time is equal to the negative of our constant angular velocity squared or I should say really the magnitude of our angular velocity which we're assuming is a constant the negative of that squared times our position vector and just to be clear angular velocity is kind of the pseudo vector it tends to be treated like a scalar especially when you're dealing in two-dimensional and in two dimensions like this it's actually formally a pseudo scalar but let's just go with this right here we're assuming that this right over here is a constant scalar quantity now how do we now what we want to do is we want to relate this we want to relate this essentially the scalar version of it so if we want to take the magnitude of both sides so let's take the magnitude so that we're saying the acceleration vector is equal to this constant times times the position vector so let's take the magnitude of both sides of this thing the magnitude of both sides the magnitude of both sides of this thing so then we get the magnitude of the acceleration vector which I'm just going to call a sub C is going to be equal to so you could say the magnitude of this negative Omega squared well when you take the magnitude that's like taking the absolute value in fact the absolute value is the one dimensional version of magnitude that's just going to be a positive Omega squared we just care we don't care about the direction sine gives us direction we just care about the actual size so this is going to be this is going to be Omega squared let me be clear this is equal to the magnitude of negative Omega squared times the magnitude of our position vector times the magnitude of our position vector the magnitude of Omega squared is just going to be Omega squared you can get rid of the sign and the magnitude of our position vector we saw at the beginning of this video is just R it's just our radius is just our radius so this right over here is just going to be equal to the radius of the circle that we are going around now we also know that angular velocity or if we want to be particularly the magnitude of angular velocity is equal to the magnitude of our velocity or another way to think about it is the speed of our object divided by the radius of the circle that it is going around so we can substitute that right over here so if we square it so this is going to be V over R squared and we saw that in the video on angular velocity times R and this is all going to be the magnitude of our acceleration which is really our centripetal acceleration our inward directed acceleration and so this is going to be equal to and I think you see where this is going this is equal to V squared over R squared times R this R cancels out with with the R squared so you're just left with V squared over R and you're done the magnitude of the centripetal acceleration is equal to your speed the magnitude of your velocity squared divided by your radius and we are done