 Hi, I'm Zor. Welcome to a new Zor education. Today we will talk about ropes again, waves on the rope, again as a preliminary to talking about light waves. Now ropes again are very important for the purpose of basically talking about transverse waves. Because light is a transverse ways of electromagnetic field. So we're talking about, generally speaking, about transverse waves, but rope is basically a very convenient way of analyzing how all these waves are propagating. Now what's important about rope is we have derived some differential equations in the previous lecture and today I would like to talk about the energy, which is carried by the waves on the rope. Now, before actually talking about energy, I would like to change the model. So rope is kind of a complicated physical object. Spring, on the other hand, is much easier to handle. And if you noticed, I have started the whole chapter of energy carried by waves with springs, because it's easier. Regardless of the fact that the springs are actually making longitudinal oscillations versus rope, which makes transverse oscillations. But what I'm going to do today is I will kind of reduce the complexity of transverse waves to simplicity of longitudinal waves. It will allow us, in the next lecture, not in this one, sorry, it will allow us to basically research to analyze transverse waves using the apparatus of the longitudinal waves, which I consider to be simpler. Okay. Now, this lecture is part of the course called Physics for Genius. I suggest you to watch it on unisor.com. If you found it somewhere else, like on YouTube or anywhere else, you just have to understand that this lecture is a part of the whole course of lectures. And the whole course contains, well, maybe, mainly lectures, hundreds probably. And they're all organized on the unisor.com. They are all organized using the menus. So we have some, you know, some kind of chapters, parts, whatever. Also, on the same website, you can find Math 14's course, which I consider to be a prerequisite. Whatever is in that course is necessary to know before you approach physics. You might actually learn mathematics somewhere else. I don't care. But I would like you to understand that without mathematics, this physics course cannot be addressed at all. Okay. Now, the website unisor.com is completely free. There are no advertisement, no strings attached. You don't even have to sign on if you don't want to. You can just use it, listen to the lectures, solve the problems. And even there are exams on the website, which you can take as many times as you want. It's for your own good, for your own verification, how you basically master the knowledge presented in that website. All right. So back to the ropes. So ropes carry, well, no, the waves basically on the rope carry energy. We know that, obviously. But because, let's say, the waves in the ocean are lifting the ocean liners up and down. So it needs energy. So the waves transverse waves in this particular case. It's basically the way how we propagate energy. Okay. Now, the previous lecture was dedicated to basically analyzing the movement of every individual piece of the rope when it waves. So our model was, this is the rope, and this end is basically forced to go up and down in some kind of harmonic oscillation up and down. So this is my y-axis. This is my x-axis. And the end of the rope was actually forced to do something like this. d of y is equal to a times sin omega t, where a was the amplitude, obviously, of movements up and down. Omega is angular speed of oscillations. And that forces the rope, which is here. It's a long one. You can consider it's an infinite rope. It doesn't really matter. To go in waves, a sinusoidal waves. And this particular condition, if y of x and, sorry, and t is a function. Now, this is x. So this is a function, which describes movements up and down, depending on the time t of this end of the rope, which is at the length x from the origin of oscillations. So we assume this origin has x equal to 0. Now, this piece of the rope, which is basically in question, which we are analyzing, has certain distance x from this. And this describes oscillations up and down. Now, this particular condition can be actually expressed as y of x is equal to 0 and t. Let's put it this way. It's actually d of t. So this describes my initial oscillation, which somebody is forcing up and down. And then it propagates using what we have derived with, we have derived the differential equation in the previous lecture, which is called wave equation. And the solution to this wave equation, I mean there are many solutions, by the way, but the one which we have suggested was this one. So if you substitute x is equal to 0 beginning, you will have exactly this sign of omega t. But as the x goes further, now the whole movements would be basically repeated, but with a certain time delay. Now, before going any further, I would actually like to slightly change our previous lecture's initial condition from this to cosine. And you will see why a little later. Now, it doesn't really matter whether it's sine or cosine. Both describe the harmonic oscillations. The question is where is our time t is equal to 0. Now, before when it was sine, when t was equal to 0, my oscillations were actually initial in the neutral position. If I start with cosine at t is equal to 0, it means my y, my deviation from the neutral position is a, which means I am in the upper position of this rope. So it's the same oscillation. I just slightly change the beginning of time. So whenever my time was with a sine, my time was when my rope was in a y is equal to 0 position. But if it's a cosine, my time starts when I moved up to the amplitude. It doesn't really matter. Just a matter of when we start t is equal to 0. Now, in this particular case, the solution will be this. It's the same thing. And it's more convenient for me, which you will understand why. So this is a given oscillation of the every infinitesimal piece of rope, which is a distance x from the beginning and at time t. So every individual one, every individual piece of rope for every x fixed, as the time goes on, it oscillates up and down with the same amplitude a as the origin of oscillations. But this introduces some kind of a time delay. Now, what is known about this? Well, omega and a are basically set by originator. Whoever is originates these oscillations, he knows what kind of amplitude and what kind of angular speed omega he is using or she. So these are parameters given by, now what is k? k is an interesting parameter. Let me just slightly change it. A cosine omega of t minus k over omega. That's the same thing. I just took omega out of the parenthesis. Now, this is actually, if this, if function of t is a cosine omega t, then function of t minus k x over omega cosine omega t. What's the difference between these two functions? Well, again, back to mathematics, the graph of this function is basically shifted to the right from this one by this value. If you will graph it as a function of time t, you will see that this, which means this is basically a time delay. That's what I want to say. This is a time delay. Now, it's not just that. Again, from the previous lecture, we were actually talking about speed of propagation of the waves. And we have come up with the formula that v is equal to omega over k. Now, again, in the previous lecture, if you have these oscillations of the rope, the speed of propagation of the wave front is dependent on the parameters of this particular equation. It was very easily derived. And it was in the previous lecture, if you don't remember, you can always go back and either watch the lecture or read the contents of the lecture, which is on the website, Unison.com, which means, okay, if omega over k is a speed, then k over omega is divided by v. Now, what is this? Now, this is a really very simple thing. If x is a distance from the origin of oscillations, v is the speed of propagation of oscillations, x over v is exactly the time. Sorry, I was too fast. Omega t minus. Then t minus x over v. What is x over v? That's the time which takes for the wave, which is propagating with the speed v, to cover the distance x. So, it has a very physical sense. It's not just abstract coefficient, like in this particular case. In this form, the solution to the wave equation, which describes how our waves are propagating, is very, very clear. Now, omega is really just the angular speed. And x over v is time delay until my next wave will basically reach the point x. Okay, so that's given. Now, let's talk about what I was talking about before, about how to model the wave propagation on the rope with springs. Okay, I think the picture is worth a thousand words. So, if this is your wave, this is your wave at any particular moment. Okay, what I will do, I will introduce a little spring here, a little spring here, a little spring here, here, here. Every infinitesimal piece of the rope I will cut from the rope and attach to a spring. Now, what I am just staging right now, that I can make such springs that will basically oscillate up and down with exactly the same type of equation. Now, if I will be able to do it, I will model basically the movement of every piece of the rope, infinitesimal piece of the rope, with my spring, corresponding spring, and whatever the machinery about the spring, and primarily potential and kinetic energy of the spring, which we have addressed in the first lecture of this particular chapter of the course. Chapter was called Light Energy, and I started this Light Energy from the first lecture, which was basically about springs. And the purpose was that first I was talking about springs, the second lecture was about waves on the rope, and now this I will just combine them together. I will model my waves on the rope with oscillations of infinite number of infinitesimal tiny springs, each attached to a corresponding piece of the rope. So, let's consider that's my goal to arrange it. Now, can I arrange it this way? Can I arrange all these springs in such a way that they will basically oscillate in a way which basically the tops of each spring will exactly be oscillating according to the same equation? Well, the answer is yes, we can, and now let's talk about how. Okay, now, first of all, let's just arrange these springs in such a way that their neutral state, they're all the same, by the way, all springs are the same. So, their neutral position is exactly at y is equal to zero. Now, they can either go up, stretch, or go down, squeeze. Now, if neutral position is at zero, and I will stretch it initially by A, each spring would actually oscillate up and down from plus A to minus A, which is exactly like the oscillation of the rope. So, A is satisfied by initial stretching of all springs to the position A. From this position, they will start oscillating according to which law? Well, A times cosine omega t, right? That's basically our when t is equal to zero, it would be A, and that's my initial stretch. As the t goes, the spring goes up and down. Now, that's why I decided to change sine to cosine here, because I would like to basically have this exactly the same as this one. So, we have satisfied amplitude. Now, omega is something which we have to really make exactly the same as oscillations of this. Now, omega is known from the oscillations of the rope. So, the question is, how can I make such omega actually here? Well, whenever we do S spring oscillations, in spring oscillations, my omega is equal to, okay, omega square is equal to k divided by m, where k is elasticity coefficient, and m is mass attached to the end of the spring. So, how can I make this omega? Now, m, I know what m actually is. m is an infinitesimal piece, which is, if my infinitesimal length is dx, mu is my linear density of the mass. We were talking about this in the previous lecture. So, m times dx, mu times dx is the mass, which is attached to this tiny spring, infinitesimal, yes. Now, k, I will put index k e, which is elasticity, I want to distinguish this of this k, which is completely different. So, this is elasticity coefficient. It's basically a characteristic of a spring. We can basically make a spring with any kind of elasticity coefficient. Now, since I know the mass and I know what my omega must be, I will choose k e equals to omega square times mu times dx. So, basically, by choosing the proper elasticity coefficient for all these springs, I will satisfy omega. So, now I have a bunch of springs. Each one, if attached the mass, infinitesimal mass mu times dx, would oscillate with exactly this angular speed of oscillations. Now, how can I do this? How can I make this spring to have a time delay based on this spring? Well, here is just plain physical thing. First, we will stretch all the springs to the height a. That's initial position. And we will fix them with some kind of plank, let's say. Then I will move plank to the right, releasing the spring. But I will move it to the right with the same speed, v, as my wave propagation. And v is known again. Since I know everything here, I know this. Now, what it allows me to do is it will release the spring one after another, and each one would be a little delayed relative to the previous one. And since I'm moving my releasing plank, or whatever mechanism is, with the same speed v as speed of propagation of waves, my waves of these will resemble the waves of the rope. And that's my purpose right now. So, I have replaced the oscillation of a rope with individual oscillations of infinitesimal masses attached to these springs. Now, let's go back to what my purpose was. My purpose was to find out what kind of energy the transverse waves are carrying with themselves. Now, if the oscillations of every little piece of mass, infinitesimal piece of mass, piece of the rope, actually, resembles, in my model, resembles these movements of these infinitesimal, the same infinitesimal pieces of mass attached to the spring. So, in this case, all these masses are moving up and down. And in this case, they're all moving up and down. But energy is transferred somehow. So, what I'm going to do is I'm going to find out how much energy is concentrated in one wavelength of the oscillations. Either this or this doesn't really matter, right? Because it's the same pieces of mass attached, in this case, to a rope, in this case, to individual springs. But they carry exactly the same energy because they have exactly the same motions, which means exactly the same forces are acting on these individual pieces of mass. And that's why I can say that I can actually summarize energy of these springs. And that would be exactly the same as the amount of energy, one particular wave of the length lambda, which is wavelengths, is carrying. So, again, I have built the model. That's good, right? Now, I'm basically stating that using this model, I can basically replace the up and down movements of the rope to up and down movements of the ends of these springs. And the same mu times dx individual pieces of mass is attached to each spring here as in each piece on the rope. So, they're moving in exactly the same fashion, which means they have exactly the same energy, potential energy and kinetic energy. And that's what I will actually do, this type of research in the next lecture. I just want to say that what I have to do is, since I know about potential and kinetic energy of every individual piece, every individual tiny string, all I have to do, I have to integrate them on any wavelengths, let's say from 0 to lambda or from 25 lambda to 26 lambda. It doesn't really matter because it's so periodic. So, this integration will give me the amount of energy one particular wave carries. Now, what else is important here? Well, basically that's it. I mean, I have expressed lambda, but that's a technology right now. I don't think we need it. Whenever we will address next lecture the wavelengths, I will derive the value of all the parameters based on individual characteristics of the wave, which we have on the rope. So, that's the explanation of my approach. How do I calculate the amount of energy in one wavelength? You see, it's not really easy because as far as kinetic energy it might be a little easier because we know how each point is moving up and down. As far as potential energy, it's not as easy because it's related to tension of the rope and the tension has all the different angles at different times. Here, the potential energy is much easier. It's basically we have already addressed it many times. If you stretch it by amount of whatever, I don't know why. The potential energy I think is one-half coefficient of elasticity times y square. We have already done this a few times and my first lecture in this chapter devoted to energy is basically about derivation of this formula as well. Now, the next lecture will basically summarize all three previous lectures which I had in this particular chapter. The first lecture was dedicated to spring. The second lecture was dedicated to wave equation for the rope. That's transverse. The third lecture is this one. It combines the spring model using this thing. It combines and uses for the transverse for the rope. The next lecture would actually be about energy of the rope using the model related to springs. Okay, that's it for today. I suggest you to read the notes for this lecture. They're probably a little bit better organized than this particular board. And there is a better picture than this one. But the idea is, again, as I was saying, first longitudinal the springs then transverse the rope. This one combines them together. It models transverse with a bunch of longitudinal oscillations of many, many infinite number of springs actually. And the next lecture will be about the energy and I will use this model to get energy for this. That's it. Thank you very much and good luck.