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  • 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,