Tangential and Normal Components of Acceleration by @mes on 3Speak
▶️ Watch on 3Speak - YouTube - Telegram - Summary - Notes - Sections playlist - Vector Functions playlist - MES Links --- In this video, I write the vector equation of acceleration in terms of its ...
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Tangental and normal components of acceleration. And also note that I just quickly Google searched and pronunciation of this tangential. But apparently it's pronounced tangential. So tangential, it's not tangential, but I've been always calling it tangential this time, so I'll use whatever fits my boat. So, and a comment below how you pronounce this. So again, tangential is what I've been... pronouncing it, but apparently it's tangential. Okay, anyways. So, when we study the motion of a particle, it is often useful to resolve the acceleration into two components, one in the direction of the tangent, and the other in the direction of the normal. So first, recall the unit tangent, unit-tangent, unit-tangor, and unit-by-normal vectors from my earlier video. So here's the one I went over, so the unit tangent, and again, it says unit is because all these are length one, because you're taking the vector, and you're just dividing it by its magnitude. so then basically it's always going to be the length one. When you find the magnitude of this, always one, because you're dividing this vector by it, and then just so happens to, when you do that, it's always going to be length one. So all these are length one. They're all unit. That's why it's called unit. All right, and then there's the binormal. This is the normal one. So anyways, so the tangent one, the T, is basically in a direction of the velocity. This is the derivative of the position vector. and then divide by its magnitude, so T is like that. And then the unit normal is going to be, well, we take this one here, and then divide it, the derivative of this one. This is the unit normal one, actually. The normal one is here, the derivative of this T, and then it ends up going downwards like that. It's pretty interesting. Like that, and they're all right angles to each other. So again, watch my earlier video on that. And then there's the, yeah, this one is just the, yeah, derivative of the unit tangent divided by its magnitude. And then unit binormal is this setup here. Yes, unit is just to take the cross product of these two. So cross product, and then you're going to get a perpendicular vector here. So let's take cross product of the unit tangent, unit normal vectors. Anyways, so you could see that video to learn more. All right, so if we write now, if we write... v, just put in italics like that, equals to the magnitude of the velocity, and then so we're going to denote this for the speed of the particle, as I've done in my earlier examples above, then the unit tangent vector is, yeah, so if we write this down, so we have the unit tangent vector is this setup, so what we have is, well, unit tangent vector of T, this is a capital T of lowercase T equals 2, the derivative of the position vector, so this is a tangent vector, divided by the magnitude of it, of the position vector, and now remember, velocity is just the derivative of the position vector. So this is just a velocity vector, so this is just equals to velocity like that of T, divided by, now this is just going to be divided by the magnitude of this position vector derivative In other words, the magnitude of the velocity vector. Just put it like that. All right, like that. And now this equals to, well, this one is just our low-case v. I mean, just our italics v. So this just equals to, yes, the V velocity vector divided by V, the speed. And now what we could do is, well, we could just move this over. Yes, move this over to the other side. And so, we get the velocity vector of t is equal to the speed of the particle times it by its unit tangent vector. So it's pretty great stuff there. And this is expected because, again, this isn't, the velocity is going in the direction of the particle motion, and that's just the derivative of the position vector, which is just a tangent vector. and then basically we take a unit tangent of it, just get the vector in direction with the length 1, and then just multiply by the full speed. The full speed is going to be, let's say, this much. So it just means it's pointing this much, vectors are going to be just extended. So all you're doing is multiplying it up. So, yeah, it's pretty straightforward stuff. All right, and now, if we differentiate both sides of this equation with respect to T, we get, well, we get acceleration. Well, we're going to get the acceleration, the A vector. is equal to the derivative of the velocity vector. So now this equals to, well, this is just the speed of it. Well, the speed is going to be varying depending on whatever the magnitude of the velocity at that time t. So this is going to be using just product rules, it's derivative of v prime, yeah, it's the derivative of the velocity, I'm going to have the speed, and then times it times by t like that. Let's put t like that, separate. Yeah, just ignore that lowercase, you know, now plus up, again, using product rule, and now this is going to be now V, the speed times it by the derivative of the tangent vector, is just going to be T prime like that. All right, so that's what we have. Yeah, I'm actually going to box this out. So this is our speed. I mean, this is acceleration. This is pretty interesting. So that's acceleration in terms of the tangent vector, and it's also the derivative. All right, so now going further, so if we use the expression for the curvature given by my earlier video on curvature, then we have, also we don't need to do this, yeah, given by my earlier video, again, on curvature, it's just given, then we have, so here's again, recall from my early video, there's link to it, and again, this is a link to the notes, and then if you scroll all the bottom, you get all the thumbnails for the sections, playlists. So there's the arc length, and there is the curvature. So if we have the arc length like this, Yeah, this is the arc length, there's their S value there, and I remember the S is just going to be the integral of, yeah, just the integral of a tangent vector there, and then I show that it's equals to this giant thingy there. Yeah, basically, these are just, think about it just a 3D distance from, and so on, and what it ends up being is, and then basically the derivative of this, is this equal to the integrative part there. So again, it's using fundamental theorem with calculus, so DS over DT equals to this. And then what I showed is, again, this is the key part that I'm interested in is this formula right here. Again, this arc length is just secondary, but there's the equation for curvature. So we have the tangent vector, and then I show that the curvature is this setup here. So curvature, this is a kappa symbol, and then kappa of t equals to the derivative, the magnitude of the derivative of the unit tangent vector divided by the... yeah, the unit tangent, I'm divided by the magnitude of the tangent vector, or the derivative of the position vector anyways. So that's that, and also I just did a quick recap on the arc length, but again, watch your videos, it's pretty extensive, and so on. So there's curvature, and this is, basic curvature is defined as D, yeah, this is the derivative of the unit tangent vector with respect to the arc length, so that's why, instead of DT, do that, but then if I rearrange it in terms of T, you get this set up here. All right, and we'll deal with this other one later, but anyways, so that's what we have. All right, so thus, in our case, we're going to have, well, we're going to have Kappa of, yeah, so Kappa is going to equal to the magnitude of the unit tangent vector derivative, like that, this part right here, just put that over here, yes, that's what we're looking at. and then divided by the magnitude of the position, derivative of the position vector, in other words, that's just a tangent vector. All right, so like that, and now this equals to, well, what do we have here? This right here, that is just this part here. This just equals to the magnitude of velocity, in other words, that's just the speed, so that's just equals to V. So this equals to, yeah, this equals to... the derivative of the unit tangent vector magnitude divided by the speed. And again, so yeah, that's what we have. So then we'll move this over to the other side. So then we have a value for this part right here. And that's going to be this part here. Well, we're going to have a setup there. Well, actually, yeah, that's the magnitude. This is just normal one, but then we'll use the other normal vector setup. But anyways, so we have that. I'm going to move this over to the other side. So what we're going to get is the magnitude of the unit tangent vector. is equal to the Kappa times it by the speed. So that's what we have, and I will box this in. I'll put, you know, I'll smooth it over there, and instead of putting a narrow, I'll put so, because the calculus book does that. So we have this, and this is going to be K, K, Kappa times V there for the speed. Alright, so now what this all means is, well, we can go further. So now the unit normal vector was defined as capital N equals to, yeah, this is the unit tangent vector derivative, divided by the magnitude of the unit tanger vector derivative. And again, that's over here. This is the unit normal. So we'll take this one right here. And again, we have this, and then we have this one in terms of Kappa V. So, again, that's why we want this normal there. So that because we're going to write the acceleration in terms of normal and tangent vector, but then we have this thing here, so we get rid of this derivative there. That's what we're going to reuse the normal vector. So we have, so we have capital and vector. This equals to, yeah, so this equals to, I'll spread it out fully, just like that, derivative of the unit tangent vector, divided by the magnitude of the unit tangent vector derivative. and so that's what we have, and we also have, yeah, so then basically we have that, so then we can just rearrange this thing here, or just plug that in, so we have this right here, this equals to unit tangent vector derivative, divided by, well, this is going to be, yeah, just throw this in there, just throw that in there, so we just throw that in there, so we just throw that in there, so we're going to throw that in there and see what happens, so this is going to be Kappa. and now we're going to move this over to the other side, so what we get is, yeah, yeah, so I'll put a, let's put it like this, arrow, arrows is good enough. Yes, we throw that, we throw it over to the end side, throw it in there, and then, so thus, yeah, we'll just leave it like that, so this is going to be unit tangent vector derivative is equal to Kappa times the speed, times it by the normal, unit tangent vector, or the unit normal vector. So that is what we have there. Yeah, so that is a pretty epic, I'll race this and fit this bit nicer. All right, and then box this in. Yeah, so now we could throw that in here. And then we're going to have the acceleration in terms of the unit tangent vector and the unit normal vector. All right. Yeah, so let's put that in here. So thus, the acceleration. So again, I'll just put the acceleration first, the full thing we had here. Thus, acceleration, this equals to v, yeah, the derivative of the speed, so v prime, and then times it by unit tangent vector, and then plus the speed times it by the derivative of the unit tangent vector, plus this one, just v, the speed, times by unit tangent vector derivative, So in other words, this equals 2, and now we throw this inside here. Yeah, so this is going to equal to v prime t plus v, yeah, there's going to be v squared, there's going to be kappa, I'll put the kappa better, kappa, v squared, and then this is going to be the normal vector, so we wrote the acceleration in terms of, the unit tangent and unit normal vectors, which is quite astonishing if you think about it. All right, you're not going further. So now if we write a subscript capital T and a subscript capital N for the tangential and normal components of acceleration, what we end up having is we get A is equal to a tangential. Yeah, and that's just this velocity, and the speed derivative, so that's going to be a tangential times by T tangent, the unit tangent vector, plus, now we have this is going to be a normal, A-N, and this is going to be a normal like that, times by unit normal vector there. Where, and I'll just write this here, where, what should I write this, I think, where, where, A, uh, subscripte is equal to just a speed, the derivative of the speed, and, yeah, and A n, capital, uh, a subscript, uh, capital N is equal to Kappa, the curvature times by the speed squared. All right, so yes, that's pretty, uh, fascinating. I'm gonna put this in the center. Let's, uh, write this like this, and put this in the center. Just fascinating stuff there. So there, that's pretty epic. Alright, now going further, let's draw this out. So this resolution is illustrated in the figure below. So let's say we had a curve. The position is like this. Let's say it looks like that. Yeah, something like that. Or make it a bit more like this. All right, so it looks like that. So now let's say we have, at this point here, we're going to have a tangent vector. So this is going to be our tangent vector. What I'll draw is actually, I'll just leave it like this. So this is going to be our tangent vector, T, t, prime, like that. And let's say we have, the acceleration is somewhere like this. It's accelerating this way, and it has a big acceleration vector like that. There, and now it has a unit normal vector. So it has unit normal vector like this. This is going to be N. But number, these are all length 1. So you have to keep that in mind. So this right here is length 1. It's a unit, yeah, it's a unit tangent vector. This is unit normal vector. This is going to be length 1. 1 like that. So these are all length 1. And now, like there. So now what I'm going to do is this is the vector, this is the acceleration vector, but now if you draw its components, it's going to be relative to the normal and the tangent. This is why it's pretty useful, so this extend all the way across like that. So now this part right here, this full magnitude, is going to be equal to a T, that's just the V speed derivative. And now this setup here, the horizontal, or the one on the normal side, this is going to be, yeah, this is all going to be like this. This is equal to the A, N. So that's this part. So that's, and again, you're putting these into components, and these are just unit vectors. Same thing as IJ, IJ, K vectors, these are standard basis vectors. And these are also have magnitude 1. I mean, they're also length 1. So it's exactly the same thing, but now we're, instead of using the XYX, XYZ axis, we're looking at only the tangent and normal. of the curves. It's actually quite actually fascinating here. So it doesn't matter where, what the X and Y axes are. So it looks like that. And what I'll do now is, I'm going to just actually, so I'm actually just use the graph from the calculus book. I'll just screenshot from that. I'll just screenshot from that, just because it looks a bit nicer than this one. And yeah, here, I just pause video and, yeah, I remove my drawing. Now we have this drawing, so it looks, again, much, much nicer. And again, the length of these are one. 1. So this length is 1. And instead of ruining the drawing, what I'll do is, I'll just write this here, the magnitude of T, like this, yeah, the magnitude, because it's a unit tangent vector, and that one is unit normal, because we're dividing by the same vector, but the magnitude of it. This right here, the magnitude equals to this. The length, I'm actually going to keep going, this extended further, because it's pretty interesting, it's the same thing as the length of the I equals to J, and again, I put the means all these here. Yeah, just go finally, I was like driving the point home, because it's, sometimes I need a reminder, it's unit, yeah, they're all unit vectors, so they're all equals to one, so all of them equal to ones. Keep that always in mind. So then this one here always has to be the same length. So then this one here, yeah, the T and the N have the same exact length. It's pretty fascinating. So whenever you draw it, you have to draw it exactly the same length, and so on. Yeah, so anyways, let's erase all these scribbles I did on this, and that's fascinating stuff.