字幕列表 影片播放 列印英文字幕 I would ideally like to have a class with maybe just ten students so that we can have a lot of dialogue and I can find out what you're thinking about, address your questions. That'll make a lot of fun but that's not possible in an introductory course. But I think you should feel free to interrupt or ask questions or discuss any reasonably related issue that I didn't get to. Okay? But that'll make it interesting for everybody because I told you many times the subject is not new to me. It's probably--it's new to you. So what makes it interesting for me is the fact that it's a different class with a different set of people with different questions. And trust me, your ability to surprise me is limitless because even on a subject which I thought is firmly fixed in my mind, some of you guys come up with some point of view or some question that's always of interest to me, of people who've been practicing this business for a long time. So I welcome that, and that'll make it more lively. All right, so I think last time I got off on a rant about all the different pedagogical techniques, some of which I don't endorse fully. I'm going to go back now 300 years in time, to one of the greatest laws that we have because--look at the power of this law, right? Here is the equation, and all the mechanical phenomena that we see, around the world, can be understood with this law. And I was starting to give you examples on how to put this law to work, because I think I at least made you realize that simply writing down the law does not give you a good feeling for how you actually use it. So, maybe you have understood it, but I'm going to remind you one more time on how you're supposed to use this law. So, I'm going to take a concrete example. The use of any law of physics is to be able to predict something about the future, given something about the present. So, all problems that we solve can be categorized that way. So I'm going to take a very simple problem, a problem to which I will return in great detail later on; but let me first start with--let me start with this problem, which I will do very quickly now and we'll come back and do it more slowly later. But at least it's a concrete problem. The problem looks like this. So, here is a table and here's a spring and here's the mass m. There's a force constant k. I want to pull it by some amount A, and let it go. So that's the knowledge of the present. The question is, when I let it go, what's this guy going to do? That's the typical physics problem. It can get more and more complex. You can replace the mass by a planet; you can replace the spring by the Sun, which is attracting the planet; you can put many planets, you can make it more and more complicated. But they all boil down to a similar situation. I know some information now and I want to be able to say what'll happen next. So, here I pull the mass, when I want A and I want to know what'll it do next. Remember, when we go back to the laws of Newton, the laws of Newton only tell you this--and we've been talking about this is useful information. The first thing you have to know in order to use the laws of Newton, you have to separately know the left-hand side. You have to know what force is going to act on a body. You cannot simply say, "Oh, I know the force on the body, it is m times a; ma is not a force acting on a body; a is the response to a force; you got to have some other means of finding the force. And in this case, the force on the mass is due to the spring. So, I pull the spring by various amounts and I see what force it exerts. Now, I think you know now in practice how I know what force it exerts, right? I pull it by some amount, attach, say, the one kilogram standard mass. I see what acceleration it experiences, and m times that acceleration, or 1 times the acceleration, is the force the spring exerts. So I pull it by various amounts and I study the spring. And I've learned, by studying the spring, that the force it exerts is some number k, called a force constant, times the amount by which I pull it. If I start off the mass in a position where the spring is neither expanded nor contracted, that's what we like to call x = 0. So I pulled it to x = A. Now, what I'm told is when you pull it to any point x, that's the force the spring exerts. So, this is part of an independent study. People who work in spring physics will study springs and they will find out from you, find out and tell you that any time you buy a spring from me it'll exert this force. And they have done--they figured that out by pulling the spring and attaching it to various entities and seeing what acceleration it produces. There, the masses are taken to be known, because you can always borrow the mass from the Bureau of Standards, or we discussed last time how if you have an unknown mass you can then compare it to this mass and find out what its value is. So every object's mass can be measured. And then the guys making the spring have studied what it does to different masses and figured this out. Now, you come with this mass and you say, what happens when I connect it? Well, I'm assuming the mass of this guy can also be found by comparisons, the way I described to you last time. So we can always find a mass of any object, as we went into in some length. Then, Newton's law says this is equal to ma, but I want to write a as the second derivative of x. So, you now go from a physical law, which is really a postulate. There is no way to derive F = ma. You cannot just think about it and get it. So, whenever I do physics I will sometimes tell you this is a law; that means don't even try to derive it. It just summarizes everything we know in terms of some new terms, but it cannot be deduced. On the other hand, the fate of this mass can now be deduced by applying Newton's law to this equation. Now, this is a new equation, you may not have seen this equation before. For example, if I told you -- forget the left-hand side -- if I told you the right-hand side is 96, I think you guys know how to solve that, right? You have to find a function whose second derivative is 96 divided by m, and you all know how to do that; it's t^(2) times a number, and you can fudge the number so it works. This is more complicated. The time derivative of this unknown function is not a given number but the unknown function itself; in other words, x itself is a function of time. This is called a differential equation. A differential equation is an equation that tells you something about an unknown function in terms of its derivatives. You can have a differential equation involving the second derivative or the first derivative or the fourteenth derivative, whatever it is. You are supposed to find out what x (t) is, given this information. So, one thing is, you can go to the Math Department and say, "Hey look, I got this equation, what's the solution?" and they will tell you. Now, sometimes we have to do our own work and we can solve this equation by guessing. In fact, the only way to solve a differential equation is by guessing the answer; there is no other way. You can make a lot of guesses and every time it works you keep a little table; then you publish it, called Table of Integrals. So, I have in my office a huge table, Mark Caprio has got his own integral, we don't leave home without our Table of Integrals. I got one at home, I got one at work, I may want to keep one in the car because you just don't know when you will need an integral. Okay? So people have tabulated them over hundreds of years. But how do they find them? They're going to find them in the way I'm going to describe to you now. You look at the equation and you guess the answer. Let's make our life simple by taking a case where the forced constant is just 1, okay? It takes 1 Newton per meter to stretch that spring. I let the mass of the object be 1 kilogram. This is just to keep the algebra simple. Later on, you can put any m and k, and we'll do all that. Then what am I saying? I'm saying, find me a function whose second derivative is minus that function. Now, as a word problem maybe it rings a bell,