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,

right? Do you know such a function?

Student: Exponential. Professor Ramamurti

Shankar: Exponential is good.

Another one? Yes?

Student: Any trig functions.

Professor Ramamurti Shankar: Trigonometric

functions are good. So I'm going to temporarily

dismiss the exponential. I will tell you why.

An exponential function is actually a very good guess.

You can say x = e^(t), for example.

The trouble is, this is growing exponentially,

and we know this spring mass system, the x is not

growing exponentially. Or you might say,

well, let me fix that by making it minus t;

then it quickly comes to rest after a time and doesn't move.

We sort of feel we want this thing to be jiggling around for

a long time. So, the exponential function is

even better than this function because it says,

take one derivative and I reproduce myself.

But that's too much; we only want a function that

reproduces itself when you take two derivatives,

and that's where the trigonometric functions come in.

So, I make a guess, x (t) is cos t,

and you can check if you take two derivatives,

it obeys the equation, the second derivative of

x is –x itself. But do I like this solution or

is there something not fully satisfactory?

Yes? Student: But you can't

help the negative when you [inaudible]

Professor Ramamurti Shankar: Pardon me?

Student: [inaudible] Professor Ramamurti

Shankar: Oh it will be negative x.

So, let's do that. So let's take dx/dt = -sin

(t). Then, take one more derivative,

d²x over dt², derivative of sine is cosine.

So, if this is –cos (t), but x was cos

(t); so it's really minus of the

x I started with. Yes?

So you have a question or--? Student: I was thinking,

you wrote it up there [inaudible]

Professor Ramamurti Shankar: Oh,

this one? Student: Yes.

Professor Ramamurti Shankar: Caught on film.

Yes, that's a mistake. Okay?

That should be minus. Very good.

Look, it makes me immensely happy when you guys catch me.

Of course, it becomes an addiction, you try to catch me;

sometimes it doesn't work, but keep trying.

This time you're absolutely right.

That's why you were looking confused.

Anything else? Everyone follow the way this

very elementary problem is solved?

It's solved by making a guess. So, you're right.

So, the minus sign was wrong but now you agree everything's

working. This is the answer.

But no, I meant something else is not quite okay with this.

Yes? Student: Your function

also works if you added a factor of the sine, also?

Professor Ramamurti Shankar: Yes.

But first of all, I can add some sin (t)

and it'll still work. But more importantly,

if I put t = 0, I get x = 1.

Why should it be true that I pulled it by exactly one meter?