▶️ Watch on 3Speak - YouTube - Telegram - Summary - Notes - Sections playlist - Vector Functions playlist - MES Links --- In this video, I go over the vector version of Isaac Newton's 2nd Law of Mo...
Transcript
Isaac Newton's second law of motion. So if the force that acts on a particle is known, then the acceleration can be found from Newton's second law of motion, and the vector version of this law states that if... at any time T, a force, this capital F of T, acts on an object of mass M producing an acceleration a vector of T, then, then basically news and second law becomes, and vector form becomes, I'll put this like here, the vector F of T equals to mass M, so mass like that, times it by, A of T like that, so a vector of T. So yes, that is fascinating stuff. Alright, so now to illustrate this, we're going to actually use an example. We're going to use an example, example 4 on centripetal force. So an object with mass M. Yeah, that moves in a circular path with constant angular speed omega. This is a W, but in Greek, this is an omega symbol, has position vector R of T equals A times cosine omega T and then the I plus A sine omega T J vector. Right, and now we're asked, find the force acting on the object and show that it is directed towards the origins. In other words, basically show that the force is always going to be pointing inwards. In other words, we got a centribital force. So to find the force, we first need to know the acceleration. Because, remember, F equals MA is the force acting on the object. So you have the object with mass M and moves in basically this position vector. So we need to find the acceleration of this position vector. And the first thing to do is to find the velocity. So velocity is going to be, well, V of T is equal to the derivative. of the position vector, so r prime of t, and this equals 2, we'll take derivative of a cosine omega-a's constant, omega's constant, so what we'll do is derivative cosine is going to be negative, so negative sine, sine, um, omega-t, and then using chain rule, derivative of the inside omega-t is just going to be omega, or w like that, yeah, and this is the i-vector. and the i component so that's i like that and now the next setup is going to be well derivative of a sine omega t that's just going to be well a omega then sign is going to be just positive cosine t omega t yeah and then j and so that's j vector and now we'll take the acceleration is going to be the derivative of the velocity so i of t is equal to the derivative of the velocity or the second derivative of, or the second derivative of the position factor. So we should put that derivative of the velocity. And now derivative of this one here, sine omega-t, well, also is going to be cosine, and so negative stays there. And then we have to use chain rule, put the W in there, or the omega. So we're going to have, take it out, squared, just be cosine. Omega T, and then I vector, plus, now, well, this one, cosine is going to be negative, sine. So it would be the same thing, but now we have an omega. squared, and then sine Omega T, like this. And then this is a J vector. So that is our acceleration. Alright, so therefore, Newton's second law gives the force as, yeah, this is just going to be a force, so F of T is equal to M's mass times acceleration. So there's an acceleration vector. and now this equals to, well, M, and what also do is I'll factor out this W squared, well, keep the A there, and I'll see why, just so that it matches up with this, with this part there, so we'll keep that in there. So we're going to go, I'm going to take the negative omega, then we're going to have a mass times it by omega, so squared, and then times it by a cosine omega T, i, is i component. and then plus this is going to be a sine omega t a j component like that all right and now just move it over there so now note that this part right here this equals to r of t so this is just a position vector so what this means is yeah that's just this part right here so that means so thus the force is equal to F of T is equal to negative m omega squared and then times it by the R vector. So in other words, whatever the, so this is just going to be negative of the, with a constant, yeah, two constant times it by a position vector. So if the position vector is going this way, then this is just going to be negative of that. So this will be exactly the same, basically exactly the same parallel to it, but it's meaning reverse direction. So in other words, it's always going inwards, actually. Yes, it's always going away from this. So if you have a position like this, RFT, like that, and then, well, if the position vector is like this, so there's the curve, it always goes towards the origin. It's the origin. So, I mean, yeah, this is, that's the vector, the position vector that always points towards the order. That's pretty fascinating stuff. All right, yeah, so going further, so yeah, this shows that, and I'll show a box that sound is pretty important, so this shows that the force acts in the direction opposite to the radius vector. Yeah, this is radius vector, and that's because this is actually a circle, but anyways, this is a position vector, R of T, and therefore points towards origin, see vigor below, I'll illustrate that. So such a force is called a centripetal force or center-seeking force, and the angular speed of the object moving with position P is... Omega equals 2 D Theta over DT, where Theta is the angle shown below. All right, so what I'll do now is I'm going to draw a circle. I'll put out a circle right here. Before I got a circle, actually, I just realized this OneNote has this built-in graph thingy there. So I could move this in there. We have this built in, all right, and now let's put the circle in there. And I'll put the circle. If I hold shift, it makes it a circle. Okay, so we got this. and I'll just move these around, move this over here, get some more, make it bit bigger, okay, and now we could even make it even bigger, alright, this is amazing, okay, now, this is gonna save me a lot of, just makes it just much easier now to draw now, because I've had people complaining about my drawings, okay, now we have perfect drawings, all right, so we got this, All right, now that we have this, what I'm going to do next is I'm going to draw our position vector. It's going to be like this. And this is going to be our R of T. I'll move it outward like this. RT. All right, so that's R of T, and this is the point P. And the reason I'm doing all this is just so we can make sense of this omega, this D-theta over D-T, and this theta there. All right, yeah, I just fix that up. But now, let's see we have an angle theta. So, yeah, that's our angle theta. So what I'll do now is I'm going to actually draw our, actually, I'll draw our typical trigonomic setup here, because again, this is just a circled equation of it. So let's say we have that, but now next step I'm going to draw this in red, actually, to make it easier to see. And now this full length, this full length of this position vector, let's say that's A. So let's say this equals to, this equals to move it out here, actually. So let's say that that's just equals to A, and that just equals to the length of the magnitude, or just the position vector at time T. All right, so that is what our A is, all right, and if it's A, that means basically that's going to be, well, a constant, that means that basically the magnitude of this is constant, so you're always going to have just a radius, that's just the radius A. So A is a radius. So now you could use our setup here. So this component here, is going to be our, well, this is our X, this is our Y setup. And again, using basic trigonometry, so we have sine y, and sine theta is equal to opposite of our hypotenous y ray. So this, move this to the other side. What we're going to get this is this equals to, this is going to be equal to a, yeah, a sine theta. and that's the Y component, and now this part right here, X is going to equal to cosine theta equals adjacent of our button, so X over A. All right. And then move to the other side. So we're going to get a A cosine theta. All right, so in other words, where now the position vector now is going to equal to R of T. So it's a function of T, so that means theta has to be a function of T. So in other words, this equals to A. cosine theta i component plus and then this is going to be a a sine theta j like that all right here i just moved it over to here so uh yeah basically now this is the function of t but then we have theta there so theta has to be a function of t and basically what the calculus book does is Yeah, it uses this setup here, so basically it has omega equals to the rate change of theta, so that's how we get the theta, and then it also multiplies by T there, so we have this right here. So omega times T, and I'll show you why it does this. So now what does is, so basically where, in our case, the angle, Theta is equal to, again, this equals to D, the rate change of theta, D over DT times T. where this is just omega, or w, or however it wants to use it. All right, this is going to equal to omega T. Just keep that in mind. So essentially, we're just making the variable theta slightly, yeah, it has more variables within it. All right, so then that means, well, our R of T position vector is equal to, well, the radius is going to be A. and I'll show you that when I graph it on G-Gibros, so this is going to be cosine, now omega-T. So that's a very change of it, i vector, plus a sine omega-ti vector. All right, going further, and before you get to the Geogibra, here's just the figure from my calculus book, and again, it shows you the inwards force there. So, everywhere you go, force can be opposite of this vector, so it's also going to be inwards. So this is basically centribal force center seeking. And I'll illustrate that with GeoGibra. So let's also graph in the GeoGibra 2D -Graphing calculators, a link to it, and I also add all the variables there. So there's A, that's actually the radius, remember that. Keep that in mind. it just gets bigger and bigger. For example, A is one here, actually this is 1.4, in this graph, so 1 .4, and then that's the radius, 1.4. Yeah, that's quite amazing. So A is just the radius, and you can see as you slide it, it will get bigger and bigger, or smaller, as you move it. Now, B, this is just time T, this is T. I put B, because we already have the time T variable here, and I rather not make this B, I just leave it like up. So there's the curve R. and this curve is equal to, well, A, cosine, omega, and then A sine omega T, and then this is negative 10, 10, but you can just leave it from 0 to 2 pi. That's all you need. This is just excess, because this is going on a circle. So that is our T, and that curve is this whole thing. That's the curve, so that's what the T is used for, in this case, so then B is, I just move it around, this is going to be the new T. So, for example, if you put B for the position vector, I mean, for the position P, equals to R, where T equals to B. And that's, you get this part right here. So you just plug that inside. All right. So I'll erase these, and so on. And okay, now the next setup is omega. And omega is, again, is rate change of this theta. So what that means is if you have it equals to 1, well, it just basically, it's just going once in a loop. But if you make it equal to 2, That means for every T value, for example, when you get to 2 pi, the full 360 degrees, you actually get 720. So in other words, you do two loops around. So again, that governs how fast you go around the circle. And then there's a mass am, it just changes the force vector F there. And force, the force vector here is just p, you know, it starts from p, so the point p, to the point P minus M omega square times P, which is, again, this part here. And then P is the initial position. This is right here. So where P is the initial position there. Or just so it graphs it out, that's the RFT. Yeah, so that's going to be the negative, the force there. So basically, it's the vector from here, this starting point, and then subtracting by the force vector. So again, that's how we all, and it's always aiming inwards. and again, we'll play this around with it here. So there it is. This is actually quite an amazing setup here. So here is the A and so on, and again, this is from 0 to 2 pi. I made it like that. If you just plug in 2 pi, I can go to 6.283, etc. So there's our equation. I'm going to move it over to this side. Yeah, so now I'm going to zoom in, zoom in. So now what we have is there's our A. So notice what happens with our A. A is the radius. If I go to 1, the radius is 1. M just affects the force, because you're just multiplying it by the acceleration. And now this is the omega one, so just keep it at 1, and now here's our B, goes from 0, there's a 0, that's at time T is 0, and if I go to 360, it makes a full 360 loop. But now, let's go back to 0, and change this as the angular speed. or the angular frequency of whatever you want to call it. Basically, this angle speed, this is the rate change of theta with time. So let's say we have double the speed. But also notice what happened here. Notice what happens. I'm going to make this 1 here. So let's say we have 1. So 1 becomes like similar that I got. So let's say we have 2. But number, this increases the acceleration because now this is the rate change of the theta. So in other words, the object's moving faster. So now look at the force. The force is huge now. And now, if I go back to zero, and so that it's zero, and now if I go over here, look at the position P, it goes over all the way here, and it goes there, okay, it goes here, it crosses it once, and if I go again all the way to 3, 60 degree, or pi is two radiance, 6.2, well, and there it is. So it makes two loops because this is two. So if you have it at 0.5, that means now, you actually get half a loop. So from zero all the way here, you only go here. And look at tiny the forces. So, but if you have a super heavy mass, the force is really big. So this is a heavy mass, super slow angular frequency. It only goes like that. It only goes an arc there. Hey, where's the rest of the thing here? Oh, that's because it's affected by sign. Okay, so I do need more than this. Okay, never mind. So this is going to be, I need to make this 100. Well, that's on a negative side, so zero, and this is going to be, let's make another zero there, I just make it, yeah, so just fill those up, negative 100, 100. Yeah, because this whole equation, it actually goes to the end, the limits there, because we have this omega, so yeah, you need more than just 2 pi, if you change the angular frequency to be super slow. So basically, that whole idea of it is adds, it adds that component, and does it make it thicker? I'm not trying to make it thicker, okay, anyways, I'll just leave it like out. Yeah, so that is that, and that is pretty cool. So again, this is pretty amazing. And then if you make it super fast, and look how giant the forces, because now it's going around the loop super, super, super fast. Unless you make this, it bigger, maybe. I guess you make it super, super giant. Okay, this super giant, I'll make it not as giant, because this laptop is it not as powerful as fast. Okay, yeah, but it still doesn't, okay, the mass, it makes the mass super tiny. Oh, if you have a negative goes like a binder, but anyways, it's pretty cool. Okay, there's angular frequency, and let's see what happens if I change this mess. Yeah, yeah, the force actually, it gets bigger. It gets bigger as you make it bigger because it's the same, oh yeah, that's because this is the angular with the theta one. But if you have a bigger circle, theta's going to be going, it's going to be going the same speed, but the actual object is going faster. So, so, for example, Yeah, it's covering more and more space. This is this circle right here. This point right here has to go super, super, super fast to be the same speed as someone super tiny. Yeah, that's pretty fast. So yeah, so notice that's like super tiny force here because it's going slow, even though the angular speed is the same. And then if you make it bigger, it has to be, it has to catch up with this little small one, so it's faster there. Anyways, that's pretty fastening stuff, so play around with that over there.