WEBVTT
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Professor Ramamurti
Shankar: Ok,
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so today, it's again a brand
new topic, so if you like
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relativity you would be in grief
and mourning.
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If it got to be really nasty,
you'll be relieved that that's
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behind us now.
And let me remind you one more
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time that the problem set for
relativity was quite challenging
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because I had to make up most of
the problems.
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And they're not from the book,
so they're a little more
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difficult.
The exam will be much more
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lenient compared to the
homework.
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That's generally true,
but particularly true in this
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case.
Okay.
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So, today I'm going to
introduce you to some
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mathematical tricks.
00:44.510 --> 00:48.210
As you've probably noticed by
now, a lot of physics has to do
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with mathematics,
and if you're not good in one,
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you're not going to be good in
the other.
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And I just thought I would
spend some time introducing you
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to some tricks of the trade,
and then we'll start using
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them.
First important trick is to
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know what's called a Taylor
series [laughter].Okay,
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I see this is greeted by boos
and hissing sounds.
01:14.939 --> 01:19.489
I'm not sure what your past
experience with Taylor series
01:19.487 --> 01:23.707
was, and why it leaves you so
scarred and unhappy.
01:23.709 --> 01:26.469
My experience with Taylor
series has been positively
01:26.473 --> 01:29.623
positive, and I don't think I
can carry on any of my things
01:29.616 --> 01:31.726
without knowledge of Taylor
series.
01:31.730 --> 01:34.030
But I'll tell it to you,
Taylor series,
01:34.034 --> 01:37.314
as we guys in the Physics
Department tend to use it.
01:37.310 --> 01:43.620
The philosophy of the Taylor
series is the following:
01:43.620 --> 01:49.690
That if some function
f(x)--and I'm going to
01:49.688 --> 01:53.928
draw that function,
something like this--but I'm
01:53.933 --> 01:56.793
going to imagine that you don't
have access to the whole
01:56.787 --> 01:58.727
function.
You cannot see the whole thing.
01:58.730 --> 02:03.810
You can only zero-in on a tiny
region around here.
02:03.810 --> 02:08.230
And the question is,
as you try to build a good
02:08.227 --> 02:13.177
approximation to the function,
how will you set it up so that
02:13.176 --> 02:16.676
you can write an approximation
for this whole function,
02:16.680 --> 02:21.690
valid near the point x =
0, where you know something.
02:21.689 --> 02:24.859
So, suppose I block out the
whole function--you've got to
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imagine mentally.
02:25.939 --> 02:28.209
I don't show you anything
except what's happening here.
02:28.210 --> 02:31.230
In other words,
I show you only f(0).
02:31.230 --> 02:34.920
02:34.920 --> 02:36.550
That's all you're told.
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The value of the function is 92.
02:39.250 --> 02:42.590
What should we do away from
x = 0?
02:42.590 --> 02:44.950
Well, you've got no information
about this function,
02:44.945 --> 02:47.435
you don't know if it's going
up, if it's going down.
02:47.440 --> 02:50.440
All I tell you is one number,
what it's doing now.
02:50.440 --> 02:53.620
It's clear the best
approximation you can make is a
02:53.617 --> 02:55.737
flat line.
There's no reason to tilt it
02:55.735 --> 02:58.345
one way or the other,
given the information you have.
02:58.349 --> 03:01.409
So, the first approximation of
the function you will say is,
03:01.410 --> 03:03.590
f(x) equal to that
[f(0)].
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It's not equal to;
you can put some symbol that
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means approximately equal to.
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It's like saying,
"Temperature today is 92,
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now what's it going to be
tomorrow?"
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Well, if you don't know
anything else,
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you cannot tell what it's going
to be tomorrow.
03:26.939 --> 03:30.009
But if you know that this is
fall, that the temperature's
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always going down,
and somebody tells you,
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"I know the rate of change of
temperature.
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I know that as of today,
the rate of change of
03:37.420 --> 03:39.070
temperature is something."
03:39.069 --> 03:42.839
Then you can use that
information to make a prediction
03:42.837 --> 03:44.967
on what it will be tomorrow.
03:44.970 --> 03:47.480
That is to say,
for values of x not
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equal to 0.
And let's denote by this
03:50.538 --> 03:54.578
following symbol:
f prime of 0 is the
03:54.583 --> 03:59.573
derivative, which is
df/dx at x = 0.
03:59.569 --> 04:03.319
And you'll multiply it by
x, and that's your best
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bet for what the function is,
away from x = 0.
04:07.069 --> 04:11.129
What that does is to
approximate the function by a
04:11.132 --> 04:16.192
straight line with the right
intercept and the right slope.
04:16.189 --> 04:19.259
By "right" I mean,
matches what you know about the
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exact function.
And if it turns out the
04:22.119 --> 04:26.109
function really was a straight
line, you are done.
04:26.110 --> 04:29.330
It's not even an approximation,
it'll track the function all
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the way to eternity.
04:30.529 --> 04:33.179
But it can happen,
of course, that the function
04:33.182 --> 04:36.352
decides to curve upwards,
as I've shown in this example,
04:36.354 --> 04:38.954
and this will not work if you
go too far.
04:38.949 --> 04:40.429
For a while,
you'll be tangent to the
04:40.433 --> 04:42.333
function, but then it'll bend
away from you.
04:42.329 --> 04:44.759
So, it's really good for a very
small x and you can say,
04:44.764 --> 04:47.164
"Well, I want to do a little
better when I go further out."
04:47.160 --> 04:50.570
Well, what this doesn't tell
you, this approximation,
04:50.571 --> 04:54.311
is that the rate of change
itself has a rate of change.
04:54.310 --> 04:57.100
This assumes the rate of change
is a fixed rate of change,
04:57.097 --> 04:59.247
which is the rate of change at
the origin.
04:59.250 --> 05:01.740
And the rate of change of the
rate of change is the second
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derivative.
So, if somebody told you,
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"Look, here's one more piece of
information."
05:06.220 --> 05:09.690
I know the second derivative of
the function,
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which I'm going to write as
f double prime at 0.
05:14.180 --> 05:16.540
Well, with that information,
you can build a better
05:16.539 --> 05:19.089
approximation to the function,
and you will put that in
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x^(2).
But the key thing is you've got
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to divide by 2,
but if you like 2 factorial.
05:25.900 --> 05:30.350
05:30.350 --> 05:34.100
And the way you divide by the
2, because your goal is take
05:34.100 --> 05:38.120
this approximation and make sure
that that's whatever you know
05:38.115 --> 05:40.545
about the function built into
it.
05:40.550 --> 05:42.890
Right?
Let's check this approximation
05:42.889 --> 05:44.889
and see if it satisfies those
properties.
05:44.889 --> 05:47.209
First of all,
at x = 0,
05:47.205 --> 05:49.595
let's compare the two sides.
05:49.600 --> 05:52.150
At x = 0,
the left hand side is
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f(0).
On the right hand side,
05:54.335 --> 05:57.345
when you put x = 0,
you kill this and you kill this
05:57.348 --> 05:59.438
and it matches.
So, you've certainly got the
05:59.439 --> 06:00.619
right value of the function.
06:00.620 --> 06:03.590
Then you say,
"What if I take the derivative
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of the function at x =
0?"
06:06.000 --> 06:08.570
Let's take the derivative of
this trial function,
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or the approximate function on
both sides.
06:10.920 --> 06:15.000
When I take the derivative,
that being a constant,
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does not contribute.
06:16.829 --> 06:21.369
This derivative of x is
1.
06:21.370 --> 06:24.130
In the next one,
the derivative of x^(2)
06:24.126 --> 06:26.336
is 2x,
2x cancels the 2,
06:26.343 --> 06:29.643
so I get f double prime
of 0 times x.
06:29.639 --> 06:33.239
But now, evaluate the
derivative at x = 0.
06:33.240 --> 06:36.970
That gets rid of this guy,
and the derivative of my test
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function matches the derivative
of the actual function,
06:40.829 --> 06:43.219
because this is the actual
derivative of the actual
06:43.223 --> 06:45.743
function.
Now, it follows that if you
06:45.742 --> 06:49.582
want to say, "How about the
second derivative of this
06:49.580 --> 06:52.670
function?"
let's take the test function I
06:52.665 --> 06:54.645
have, or the approximation.
06:54.649 --> 06:57.579
Take a second derivative,
and make sure that comes out
06:57.578 --> 06:59.368
right.
I don't feel like writing this
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out, but try to do this in your
head.
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Take two derivatives on the
left-hand side and see what
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happens on the right hand side.
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Well, if you take one
derivative, this guy's gone.
07:07.959 --> 07:10.299
If you take two derivatives,
this guy's gone.
07:10.300 --> 07:14.390
And then, if you take the first
derivative, you get 2x,
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you take the second derivative
you get 2, the 2 cancels the 2.
07:18.690 --> 07:20.920
Now, if you put x = 0.
07:20.920 --> 07:24.410
If you do, you find that
f(0) here matches the
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second derivative of the
function on the left-hand side.
07:28.310 --> 07:31.860
So, this 2 factorial is there
to make sure that the function
07:31.863 --> 07:35.483
you have cooked up has the right
value of the function at the
07:35.477 --> 07:37.617
origin,
has the right slope at the
07:37.622 --> 07:40.762
origin, has the right rate of
change of the slope at the
07:40.759 --> 07:42.739
origin.
Well, it's very clear what we
07:42.736 --> 07:45.506
should do.
If you had a whole bunch of
07:45.507 --> 07:49.897
derivatives, then the
approximation we will write will
07:49.898 --> 07:53.128
look like, f n-th
derivative.
07:53.129 --> 07:56.519
I cannot draw n primes
there, so n means
07:56.520 --> 07:59.720
n such primes,
x to the n over
07:59.718 --> 08:01.188
n factorial.
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And you go as far as you can.
08:03.310 --> 08:06.730
If you know 13 derivatives,
put 13 derivatives here.
08:06.730 --> 08:09.210
That's an approximation.
08:09.209 --> 08:11.699
That's not called a Taylor
series, that's the approximation
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to the function.
Now, it may happen that
08:14.260 --> 08:18.150
sometimes you hit the jackpot,
and you know all the
08:18.152 --> 08:20.752
derivatives.
Someone tells you all the
08:20.754 --> 08:23.284
derivatives of the function,
then why stop?
08:23.280 --> 08:27.040
Add them all up.
Then, you will get the sum
08:27.040 --> 08:31.760
going from the 0 derivative all
the way up to infinity.
08:31.759 --> 08:35.129
And if you do the summation of
every value of x,
08:35.133 --> 08:37.833
you will get a value,
and if that summation is
08:37.829 --> 08:39.929
meaningful and gives you a
finite number,
08:39.928 --> 08:42.698
that, in fact,
is exactly the function you
08:42.703 --> 08:46.863
were given.
That is the Taylor series.
08:46.860 --> 08:50.430
The Taylor series is a series
of infinite number of terms
08:50.433 --> 08:54.453
which, if you sum up -- and sum
up to something sensible -- will
08:54.454 --> 08:57.394
actually be as good as the
left-hand side.
08:57.390 --> 08:59.900
So, let me give you one example.
08:59.900 --> 09:01.610
Here's a famous example.
09:01.610 --> 09:05.850
1 over 1 minus x is the
real function.
09:05.850 --> 09:09.820
You and I know this function,
we know how to put it in a
09:09.821 --> 09:12.061
calculator;
we know how to plot it.
09:12.059 --> 09:14.469
You give me an x and I
stick it in the denominator,
09:14.468 --> 09:15.818
subtract it from 1,
invert it;
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that's the function.
09:17.529 --> 09:20.779
But instead,
suppose this function was
09:20.784 --> 09:23.164
revealed to us in stages.
09:23.159 --> 09:27.039
We were told f(0)--what
is f(0) here?
09:27.039 --> 09:30.699
Put the x equal to 0,
f(0) is 1.
09:30.700 --> 09:34.200
Now, let's take the derivative
of this function,
09:34.202 --> 09:38.062
df/dx.
df/dx has a minus 1
09:38.061 --> 09:44.601
because it's 1 minus x to
the minus 1, times a square;
09:44.600 --> 09:48.370
then the derivative of what's
inside this, that gives you a
09:48.370 --> 09:50.640
minus 1.
This is simple calculus,
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how to take the derivative of
this fellow here.
09:53.889 --> 09:58.229
Having taken the derivative,
evaluate it at x = 0,
09:58.228 --> 10:00.478
this vanishes,
that becomes 1,
10:00.475 --> 10:03.575
and you get 1.
If you now take the second
10:03.584 --> 10:07.544
derivative of this function,
which I don't feel like doing,
10:07.539 --> 10:13.339
and I evaluate that at x
= 0, then I'll find it is equal
10:13.339 --> 10:16.139
to 2 factorial.
In fact, I'll find the
10:16.139 --> 10:19.549
nth derivative at the
origin to be n factorial.
10:19.550 --> 10:21.940
Well, that's very nice.
10:21.940 --> 10:25.490
Because then,
my approximation to the
10:25.489 --> 10:30.319
function begins as 1 plus
x times the first
10:30.321 --> 10:33.661
derivative,
which happens to be 1 plus
10:33.656 --> 10:37.866
x^(2) over 2 factorial
times the second derivative,
10:37.870 --> 10:42.480
which happens to be 2 factorial
plus x^(3) over 3
10:42.477 --> 10:47.287
factorial times 3 factorial,
so you can see where this guy
10:47.287 --> 10:51.517
is going.
It looks like 1 + x + x^(2),
10:51.518 --> 10:56.638
et cetera.
The Taylor series is its
10:56.640 --> 10:59.820
infinite sum.
In practice,
10:59.823 --> 11:03.843
you may be happy to just keep a
couple of terms.
11:03.840 --> 11:07.140
So, let's get a feeling for
what those couple of terms can
11:07.144 --> 11:12.654
do for us.
So, let me take x = 0.1.
11:12.650 --> 11:18.400
x equal to point 1;
the real answer is 1 over 1
11:18.398 --> 11:22.098
minus point 1,
which is 1 over point 9,
11:22.097 --> 11:24.917
which is 1.1111,
etcetera.
11:24.920 --> 11:29.040
That's the target.
11:29.040 --> 11:30.600
What do you do with the series?
11:30.600 --> 11:34.470
The series starts at 1,
times one-tenth,
11:34.472 --> 11:37.752
plus 1 over 100,
plus 1 over 1000,
11:37.749 --> 11:40.109
and so on.
And you can see,
11:40.107 --> 11:43.617
as I keep more and more terms,
I keep just filling up these
11:43.616 --> 11:46.806
ones.
If you stop at 1 over 1000,
11:46.811 --> 11:49.021
you stop right there.
11:49.019 --> 11:53.509
So, this is the exact answer,
this is the approximation.
11:53.509 --> 11:55.569
But it's clear to you,
perhaps in a simple example,
11:55.571 --> 11:57.511
that if you kept all the terms
of the series,
11:57.509 --> 12:02.149
you really will get this
infinite number of recurring
12:02.145 --> 12:04.945
1's.
So, that is the function.
12:04.950 --> 12:07.640
That is the Taylor
approximation.
12:07.639 --> 12:10.789
We'll just chop it off whenever
you want, and it's a good
12:10.785 --> 12:13.365
approximation,
and that's the Taylor series.
12:13.370 --> 12:15.820
The series means sum everything.
12:15.820 --> 12:20.230
Now, summing an infinite number
of numbers is a delicate issue.
12:20.230 --> 12:22.970
I don't want to go there at
all, I discuss that in this math
12:22.965 --> 12:24.795
book.
But sometimes a sum makes no
12:24.802 --> 12:26.532
sense, then you've got to quit.
12:26.530 --> 12:29.220
For example, put x = 2.
12:29.220 --> 12:35.530
The correct function is 1 - 2,
which is -1.
12:35.529 --> 12:44.579
Our approximation for x
= 2 looks like 1 + 2 + 4 + 8.
12:44.580 --> 12:46.580
Fist of all,
this sum is going to grow to
12:46.580 --> 12:49.680
infinity, because the numbers
are getting bigger and bigger.
12:49.679 --> 12:51.269
This sum seems to be all
positive;
12:51.269 --> 12:52.999
that is the correct answer is
negative.
12:53.000 --> 12:55.850
Obviously, the series doesn't
work.
12:55.850 --> 12:58.720
So, the next lesson of our
Taylor series is,
12:58.723 --> 13:02.333
you can write down the series,
but it may not sum up to
13:02.332 --> 13:05.342
anything sensible beyond a
certain range.
13:05.340 --> 13:08.110
So, if you're doing a Taylor
series at x = 0,
13:08.105 --> 13:10.865
and you go to x = 2,
it just doesn't work.
13:10.870 --> 13:13.220
So, you can ask,
"How far can I go from the
13:13.220 --> 13:15.390
origin?"
Well, in this simple example,
13:15.394 --> 13:18.294
we know that x = 1,
the function just going to
13:18.289 --> 13:20.849
infinity, that's why you
couldn't go there.
13:20.850 --> 13:24.410
And you cannot go to the
right-hand side of that point.
13:24.409 --> 13:26.659
The function is well defined on
the other side,
13:26.662 --> 13:28.912
but this series,
this knowledge of the function
13:28.914 --> 13:31.514
here, is not enough to get you
on the other side.
13:31.509 --> 13:36.109
So, this is a case where there
are obvious problems at x
13:36.114 --> 13:38.504
= 1.
But if I wrote a function like
13:38.502 --> 13:41.292
1 over 1 plus x^(2),
that's a nice function,
13:41.286 --> 13:43.096
got no troubles anywhere.
13:43.100 --> 13:45.590
And yet, if you took the Taylor
series for it,
13:45.594 --> 13:48.984
you will find if you go beyond
absolute value at x = 1,
13:48.976 --> 13:50.636
the series makes no sense.
13:50.639 --> 13:54.789
So, I don't want to do that
mathematical theory of series.
13:54.789 --> 13:58.299
I just want to tell you that
functions can be approximated by
13:58.303 --> 14:00.103
series.
And if you're lucky,
14:00.098 --> 14:03.648
you can do the whole sum if you
know all the derivatives,
14:03.650 --> 14:07.330
and the whole sum may converge
to give a finite answer.
14:07.330 --> 14:09.530
In which case,
it's as good as the function.
14:09.529 --> 14:11.819
One guy can use 1over 1 minus
x;
14:11.820 --> 14:13.890
the other one can use the
infinite series,
14:13.893 --> 14:16.523
and they're morally and
mathematically in every sense
14:16.524 --> 14:18.694
equal,
as long as they don't stray
14:18.692 --> 14:22.082
outside the region of validity
of the infinite series.
14:22.080 --> 14:27.720
Ok, so the most popular example
that I've been using in the
14:27.724 --> 14:31.524
class, remember,
is 1 + x^(n).
14:31.519 --> 14:34.949
That's a function we can do
with Taylor series.
14:34.950 --> 14:37.160
f(0) is 1.
14:37.159 --> 14:40.719
What's the derivative of the
function?
14:40.720 --> 14:43.580
It's n times 1 plus
x^(n) minus 1.
14:43.580 --> 14:50.140
The x is n times
1 plus x^(n) minus 1;
14:50.139 --> 14:53.959
if I evaluate it at x =
0, it is just n.
14:53.960 --> 14:56.980
That's why we get this famous
result we've been using all the
14:56.977 --> 14:59.077
time.
If x is small enough you
14:59.080 --> 15:02.300
stop there, because the next
time it's going to involve an
15:02.297 --> 15:04.327
x^(2) and an
x^(3),
15:04.330 --> 15:06.350
then if x is tiny,
we have no respect for
15:06.352 --> 15:08.742
x^(2) and x^(3),
we just cut it off.
15:08.740 --> 15:10.740
But if you want the next
term--one of you guys even asked
15:10.740 --> 15:11.670
me, "What happens next?"
15:11.669 --> 15:13.359
we'll take the second
derivative.
15:13.360 --> 15:17.520
You can see it's n times n
minus 1, times 1 plus x^(n)
15:17.515 --> 15:19.875
minus 2.
If you put x = 0,
15:19.882 --> 15:22.652
you'll get n times
n minus 1,
15:22.652 --> 15:24.962
x^(2),
and don't forget the 2
15:24.960 --> 15:28.080
factorial.
So, what I've been doing is
15:28.078 --> 15:30.918
I've been saying 1 plus
x^(n),
15:30.916 --> 15:33.356
approximately equal to this.
15:33.360 --> 15:36.140
But even if you keep that,
it's still approximate.
15:36.140 --> 15:39.140
But it's a good approximation;
how many terms you want to keep
15:39.142 --> 15:41.042
depends on how tiny x is.
15:41.039 --> 15:44.519
Ok, so it's good to know
there's an infinite series,
15:44.517 --> 15:48.677
but it's also good to know you
can chop it off and do business
15:48.677 --> 15:50.107
with a few terms.
15:50.110 --> 15:54.480
In fact, all of relativity we
reduce as follows.
15:54.480 --> 15:58.930
The energy of a particle is
this, which we can write as
15:58.931 --> 16:03.631
mc^(2) times 1 minus
v^(2) over c^(2)
16:03.631 --> 16:07.281
to the minus ½.
If you expand this in a power
16:07.284 --> 16:10.394
series, you're going to get
mc^(2) + ½
16:10.394 --> 16:13.194
mv^(2),
plus stuff involving v
16:13.190 --> 16:16.320
to the 4th over c to the
4th times c^(2).
16:16.320 --> 16:19.440
We dropped all this,
and for 300 years we did
16:19.444 --> 16:21.934
mechanics keeping just this
term.
16:21.929 --> 16:25.169
We just kept that first
non-trivial term,
16:25.169 --> 16:28.969
and all of our collisions and
so on that we did,
16:28.974 --> 16:31.894
used only that up to that
point.
16:31.889 --> 16:34.009
So, the approximations have
really been useful,
16:34.010 --> 16:35.900
and they describe nature
approximately.
16:35.899 --> 16:38.509
If you say, "Well,
I want to be exact," you can go
16:38.506 --> 16:39.566
back and use this.
16:39.570 --> 16:41.420
Unfortunately,
generally somebody tells you,
16:41.415 --> 16:42.355
"That's not exactly.
16:42.360 --> 16:44.620
There in quantum mechanics,
tells you the whole thing is
16:44.620 --> 16:45.730
wrong."
At every stage,
16:45.725 --> 16:47.485
you will have to give up
something.
16:47.490 --> 16:50.370
So, I have a lot of respect for
approximations.
16:50.370 --> 16:52.690
If you could not describe the
world approximately,
16:52.686 --> 16:54.716
you couldn't have come where
you've come.
16:54.720 --> 16:58.140
Because no one knows the exact
answer to a single question you
16:58.142 --> 16:59.752
can pose.
Ask me a question,
16:59.751 --> 17:01.681
and the answer is,
"I don't know."
17:01.679 --> 17:04.389
Second part of the answer is,
"Nobody knows."
17:04.390 --> 17:07.440
Because if your question says
give an answer to arbitrary
17:07.444 --> 17:10.284
precision, any question asked,
we just don't know.
17:10.279 --> 17:14.299
Newtonian mechanics works for
small velocities.
17:14.299 --> 17:18.229
All of relativistic mechanics
works for any velocity,
17:18.230 --> 17:20.800
but not for really tiny
objects.
17:20.799 --> 17:22.769
Then you've got to use quantum
mechanics.
17:22.769 --> 17:26.259
So, always theories give way to
new theories,
17:26.259 --> 17:29.669
and so approximations are very
important.
17:29.670 --> 17:30.880
So, you have to learn this.
17:30.880 --> 17:36.760
Okay.
So now, I'm going to consider
17:36.755 --> 17:41.685
the following function:
e^(x).
17:41.690 --> 17:48.660
17:48.660 --> 17:51.430
Now ,this guy is something you
all know and love,
17:51.432 --> 17:54.092
because it's one nice thing
about the function,
17:54.089 --> 17:56.399
is every derivative is
e^(x).
17:56.400 --> 17:59.350
Right?
Every child knows e^(x)
17:59.349 --> 18:02.029
has got e^(x) as its
derivative.
18:02.030 --> 18:03.760
Why do we like that here?
18:03.759 --> 18:06.419
That means all the derivatives
are sitting in front of you.
18:06.420 --> 18:09.110
They're e^(x),
and at x = 0,
18:09.111 --> 18:12.461
e to the 0 is 1,
so every derivative is 1.
18:12.460 --> 18:15.060
And the function is very easy
to write down.
18:15.059 --> 18:19.829
It is really 1 + x +
x^(2) over 2 factorial,
18:19.826 --> 18:24.586
plus x to the 3 over 3
factorial, x to the 4
18:24.593 --> 18:26.353
over 4 factorial.
18:26.349 --> 18:28.919
You can go on like this,
and if you like to use a
18:28.915 --> 18:31.475
compact notation,
it's x^(n) over n
18:31.481 --> 18:34.651
factorial,
and going from 0 to infinity,
18:34.651 --> 18:37.741
that 0 factorial is defined to
be 1.
18:37.740 --> 18:40.530
That's e^(x).
18:40.529 --> 18:45.269
If you ever forget the value of
e, and I need to know the
18:45.273 --> 18:48.603
value of e,
because when I lock my suitcase
18:48.602 --> 18:51.782
and check it into the airport,
I use either e or
18:51.775 --> 18:54.005
π,
because they're the only two
18:54.010 --> 18:56.690
numbers I can remember--So,
if I forget the value of
18:56.689 --> 19:00.379
e,
I just say e is the same
19:00.375 --> 19:05.295
as e to the 1,
so I've got 1 + 1 + 1 + 2 + 1
19:05.304 --> 19:08.734
over 6,
and pretty soon I'm telling the
19:08.732 --> 19:12.122
guy scanning my luggage,
"That's my code."
19:12.119 --> 19:16.009
And it's much better than any
other code because I cannot
19:16.005 --> 19:18.915
reconstruct, say,
my grandfather's name.
19:18.920 --> 19:19.930
I may not get it right.
19:19.930 --> 19:22.630
I may not construct my house
address, or my phone address.
19:22.630 --> 19:23.980
This doesn't change.
19:23.980 --> 19:26.150
I've been moving around all
over the world,
19:26.147 --> 19:27.847
this is a very reliable number.
19:27.849 --> 19:30.569
Now, π is a good
number, but rules for computing
19:30.565 --> 19:32.535
π are somewhat more
difficult.
19:32.539 --> 19:34.999
π also can be computed
as an infinite series,
19:35.003 --> 19:36.333
but this guy is very easy.
19:36.329 --> 19:40.079
There's 2 point in 7-7
something, that is e.
19:40.080 --> 19:42.840
Okay.
Now, here is the very nice
19:42.842 --> 19:44.242
property of this series.
19:44.240 --> 19:46.420
It is good for any x.
19:46.420 --> 19:50.170
You remember the series for 1
over 1 minus x crashed
19:50.166 --> 19:52.036
and burned at x = 1.
19:52.040 --> 19:53.930
This series -- always good.
19:53.930 --> 19:57.760
You put x = 37 million,
you've got 37 million,
19:57.757 --> 20:01.287
37 million cubed,
37 million at the 4th power;
20:01.290 --> 20:03.870
don't worry.
These factorials downstairs
20:03.865 --> 20:06.885
will tame it down and make it
converge, and will give you
20:06.885 --> 20:09.145
e to the whatever number
I give.
20:09.150 --> 20:12.180
That's something I'm not
proving, but the series for
20:12.175 --> 20:14.425
e^(x),
no limits on validity.
20:14.430 --> 20:17.590
All right.
Then, we take the function,
20:17.590 --> 20:20.740
cos x.
Now, cos x,
20:20.737 --> 20:25.377
we all know and love as this
guy.
20:25.380 --> 20:27.000
That's got a period of
2π.
20:27.000 --> 20:31.750
But we can write a series for
it because it's a function.
20:31.750 --> 20:34.370
And what do I need to know to
write the series for it?
20:34.369 --> 20:37.149
I need to know the value of the
function at the origin.
20:37.150 --> 20:39.620
So, we know cosine of 0 is 1.
20:39.619 --> 20:45.059
If you take the derivative you
get minus sine,
20:45.064 --> 20:48.214
and its value at 0 is 0.
20:48.210 --> 20:51.720
You take one more derivative,
you get minus cosine 0,
20:51.721 --> 20:54.891
which is minus 1,
and I think by now you get the
20:54.894 --> 20:57.314
point.
Every other derivative will
20:57.311 --> 21:00.791
vanish, and the remaining
derivatives will alternate in
21:00.793 --> 21:02.473
sine from 1 to minus 1.
21:02.470 --> 21:04.870
That means, if you crank it
out, you will get 1,
21:04.868 --> 21:07.568
you won't have anything linear
in x because the
21:07.573 --> 21:10.383
coefficient of that is
sin 0 which vanishes;
21:10.380 --> 21:13.820
then you'll get x^(2)
over 2 factorial plus x
21:13.818 --> 21:16.128
to the 4 over 4 factorial,
and so on.
21:16.130 --> 21:21.060
21:21.060 --> 21:22.700
This is cos x.
21:22.700 --> 21:27.390
If you cut it off after some
number of terms,
21:27.393 --> 21:30.063
it's not going to work.
21:30.059 --> 21:31.929
In the beginning,
1 minus x^(2) looks very
21:31.927 --> 21:33.627
good.
This is 1 minus x^(2)
21:33.626 --> 21:34.896
over 2, looks like this.
21:34.900 --> 21:37.190
But eventually,
this approximation will go bad
21:37.194 --> 21:40.564
on you, because x to the
4th is just x to the 4th.
21:40.560 --> 21:41.890
It'll grow without limit.
21:41.890 --> 21:44.620
No one's telling you this
approximation has a property
21:44.624 --> 21:46.124
that cosine is less than 1.
21:46.119 --> 21:49.419
It doesn't satisfy that,
because you're not supposed to
21:49.423 --> 21:50.773
use it that far out.
21:50.769 --> 21:53.809
But if you keep all the terms,
then in fact,
21:53.813 --> 21:57.213
even though x to the 4th
is taking off,
21:57.210 --> 21:58.930
x to the minus 6 is
saying hey, come down,
21:58.925 --> 22:00.315
x to the 8th is saying,
go up.
22:00.319 --> 22:03.829
You add them all together,
remarkably, you will reproduce
22:03.827 --> 22:07.097
this function.
Very hard to imagine that the
22:07.101 --> 22:10.861
cosine that we know as
oscillatory is really this
22:10.860 --> 22:13.530
function.
And I hope all of you know
22:13.528 --> 22:16.118
enough to know how to find the
series.
22:16.119 --> 22:19.209
You should know the derivative
of cosine is minus sine,
22:19.214 --> 22:21.224
the derivative of sine is
cosine.
22:21.220 --> 22:24.360
And evaluate them at the origin;
I say you can build up the
22:24.359 --> 22:27.129
series.
Then, if you do similar tricks
22:27.130 --> 22:29.550
for sin x,
sin of 0 is 0,
22:29.549 --> 22:32.829
derivative of sin is cosine,
and of course,
22:32.827 --> 22:34.697
n of 0 is 1.
22:34.700 --> 22:37.300
So, the first derivative is 1;
that gives you this term.
22:37.299 --> 22:39.669
Next one won't give you
anything, the one after that
22:39.666 --> 22:41.846
will give you this,
and this guy will go on like
22:41.846 --> 22:42.446
this one.
22:42.450 --> 22:47.800
22:47.800 --> 22:50.210
Okay.
So, these are series for
22:50.206 --> 22:51.736
e^(x),
cos x,
22:51.741 --> 22:54.281
and sin x,
good for all x.
22:54.279 --> 22:56.319
So, there's no restriction on
this.
22:56.320 --> 23:02.210
23:02.210 --> 23:05.440
One of the really interesting
things to do when you are bored,
23:05.437 --> 23:08.237
or stranded somewhere in an
airport, is to take cosine
23:08.242 --> 23:09.832
squared plus sine squared.
23:09.829 --> 23:13.199
Squared the whole thing,
add the squared of the whole
23:13.196 --> 23:14.746
thing and add them up.
23:14.750 --> 23:16.840
You'll find,
by miracle, you will get the
23:16.842 --> 23:19.882
one from squaring the one and
every other power of x
23:19.877 --> 23:21.287
will keep on vanishing.
23:21.289 --> 23:23.789
Of course, you'll have to
collect powers of x by
23:23.789 --> 23:24.899
expanding the bracket.
23:24.900 --> 23:26.140
But it will all cancel.
23:26.140 --> 23:28.800
That's what I meant by saying
this series is as good as this
23:28.803 --> 23:31.613
function.
It will obey all the identities
23:31.608 --> 23:34.558
known.
Okay.
23:34.560 --> 23:36.540
Now, for the punch line.
23:36.539 --> 23:39.599
I think everyone--Some of you
may know what the punch line is,
23:39.596 --> 23:41.296
the punch line is the
following.
23:41.299 --> 23:45.189
Let us introduce,
without any preamble right now,
23:45.188 --> 23:49.078
the number i,
which is the squared root of
23:49.077 --> 23:52.147
minus 1.
All I expect of i is
23:52.148 --> 23:55.178
that when I squared it,
I get minus 1.
23:55.180 --> 23:59.300
If I cube it,
i^(3) is i^(2)
23:59.304 --> 24:03.634
times i,
so that's minus i,
24:03.630 --> 24:07.610
and i to the 4th is back
to plus 1 because i to
24:07.607 --> 24:10.587
the 4th is i^(2) times
i^(2),
24:10.590 --> 24:12.320
and i^(2) is minus 1.
24:12.319 --> 24:14.739
So, if you take powers of
i, it'll keep jumping
24:14.739 --> 24:15.789
between these values.
24:15.789 --> 24:17.289
It'll be i,
it'll become minus 1,
24:17.292 --> 24:19.452
it'll become minus i,
then it'll become plus 1.
24:19.450 --> 24:22.600
That's the set of possible
values for powers of i.
24:22.599 --> 24:24.249
It'll be one of these four
values.
24:24.250 --> 24:29.050
Well.
Then we are told,
24:29.052 --> 24:36.442
if you now consider the
following rather strange object,
24:36.437 --> 24:40.017
e^(ix).
Now, e^(ix),
24:40.019 --> 24:42.619
you have to agree,
is really bizarre.
24:42.619 --> 24:45.559
e is some number,
you want to raise it to 2,
24:45.557 --> 24:47.837
that's fine.
Number times number.
24:47.839 --> 24:50.439
i is a strange number,
right?
24:50.440 --> 24:54.500
It's the squared root of minus
1, and you wanted me to raise
24:54.497 --> 24:56.557
e to a complex power.
24:56.560 --> 24:58.260
What does that even mean?
24:58.259 --> 25:01.409
Multiply e by itself
ix times?
25:01.410 --> 25:03.880
Well, that definition of powers
is no good.
25:03.880 --> 25:08.880
But the series for e^(x)
defines it for all x,
25:08.878 --> 25:13.108
then we boldly define
e^(x) for even complex
25:13.113 --> 25:17.183
values of x as simply
this thing too,
25:17.180 --> 25:19.670
with ix plugged in place
of x.
25:19.670 --> 25:23.240
And by that fashion,
we define exponential.
25:23.240 --> 25:27.370
So, the exponential function is
simply defined as,
25:27.372 --> 25:31.592
suppose you write e
raised to dog.
25:31.589 --> 25:35.959
e raised to dog
is 1 plus dog plus
25:35.958 --> 25:37.898
dog^(2) over 2.
25:37.900 --> 25:41.670
e raised to anything is
a code for this series.
25:41.670 --> 25:43.910
And now we raise e to
various things.
25:43.910 --> 25:45.410
Real numbers,
complex numbers,
25:45.414 --> 25:46.974
matrices, whatever you want.
25:46.970 --> 25:48.880
Your pet, you can put that up.
25:48.880 --> 25:50.550
Of course, you've got to be a
little careful.
25:50.549 --> 25:52.329
You cannot raise e
raised to dog,
25:52.325 --> 25:53.555
because the units don't match.
25:53.559 --> 25:56.369
But there's a dog here
and dog^(2),
25:56.365 --> 25:59.345
so you should divide it by some
standard dog,
25:59.346 --> 26:01.446
like President's dog
[laughter].
26:01.450 --> 26:02.560
Take some standard.
26:02.559 --> 26:04.069
Napoleon's dog,
divide it, then you've got
26:04.073 --> 26:05.443
something dimensionless
[laughter].
26:05.440 --> 26:06.860
Then it will work.
26:06.859 --> 26:09.899
So, as long as it's some
dimensionless object,
26:09.901 --> 26:13.281
this is what we mean by
e raised to that.
26:13.279 --> 26:17.079
That's a fantastic leap of
imagination.
26:17.079 --> 26:20.069
How to raise e to
whatever power you like;
26:20.070 --> 26:21.970
take the series.
Let's do the series.
26:21.970 --> 26:29.210
So, I'm going to get 1 plus
ix, plus (ix)^(2)
26:29.208 --> 26:33.098
over 2 factorial,
plus (ix)^(3) over 3
26:33.097 --> 26:36.187
factorial, and I'm going to stop
after this last term,
26:36.190 --> 26:38.730
(ix)^(4) over 4
factorial.
26:38.730 --> 26:41.920
Keep going, then what do you
get?
26:41.920 --> 26:47.320
Let's take the 1 from here,
leave this alone for a minute.
26:47.319 --> 26:50.789
You write i times
x, go here,
26:50.788 --> 26:54.998
i^(2) is minus 1,
so I get minus x^(2)
26:54.999 --> 26:58.219
over 2 factorial,
then I come here.
26:58.220 --> 27:01.470
I cubed is minus i,
so I write here,
27:01.469 --> 27:04.409
minus x^(3) over 3
factorial.
27:04.410 --> 27:07.100
Then, I go back here and write
x to the 4th over 4
27:07.102 --> 27:09.342
factorial.
By now, everybody knows what's
27:09.337 --> 27:12.237
going on.
Everybody knows that this is
27:12.238 --> 27:14.668
just cos x + i sin x.
27:14.670 --> 27:19.300
27:19.299 --> 27:22.269
Okay, this is a super duper
formula.
27:22.270 --> 27:24.620
Life cannot go on without this.
27:24.619 --> 27:28.339
You want to draw a box around
this one, draw it.
27:28.339 --> 27:32.279
Because without this formula,
we cannot do so many things we
27:32.276 --> 27:32.606
do.
27:32.610 --> 27:37.960
27:37.960 --> 27:41.690
It says that the trigonometric
functions and the exponential
27:41.693 --> 27:44.923
functions are actually very
intimately connected.
27:44.920 --> 27:47.380
And if they're all defined by
the power series,
27:47.379 --> 27:48.769
you're able to prove it.
27:48.769 --> 27:50.839
So, this is the formula proven
by Euler.
27:50.839 --> 27:54.549
And it's considered a very
beautiful formula,
27:54.545 --> 27:59.425
and here is the particularly
beautiful special case of this
27:59.429 --> 28:02.489
formula.
If you put x = π,
28:02.488 --> 28:06.998
I get the result e^(iπ)
equal the cos π,
28:06.995 --> 28:10.105
which is minus 1,
plus i sin π,
28:10.108 --> 28:15.948
which is 0.
I get the result e^(iπ)
28:15.945 --> 28:19.285
+ 1 = 0.
Now, everybody agrees this has
28:19.287 --> 28:22.727
got to be one of the most
beautiful formulas you can
28:22.733 --> 28:25.913
imagine involving numbers or in
mathematics.
28:25.910 --> 28:28.200
Why is it such a great formula?
28:28.200 --> 28:29.920
Look at this formula here.
28:29.920 --> 28:33.030
π, defined from the
Egyptian times,
28:33.029 --> 28:35.619
is a ratio of circle to
diameter.
28:35.619 --> 28:38.219
i is square root to
minus 1.
28:38.220 --> 28:39.980
1 is the basis for all integers.
28:39.980 --> 28:42.200
e is the basis for the
logarithm.
28:42.200 --> 28:46.480
Here is a formula in which all
the key numbers appear in one
28:46.484 --> 28:48.304
single compact formula.
28:48.299 --> 28:52.139
The fact that these numbers
would have a relationship at all
28:52.142 --> 28:54.942
is staggering,
and this is the nature of the
28:54.942 --> 28:58.552
relationship.
This is voted the formula most
28:58.554 --> 29:01.814
likely to be remembered,
this formula.
29:01.809 --> 29:04.309
Now, we are going to use not
only the special case,
29:04.309 --> 29:06.509
but we are going to use this
all the time.
29:06.509 --> 29:11.229
So, how many people have seen
this?
29:11.230 --> 29:15.450
Okay.
If you have not seen this,
29:15.451 --> 29:18.311
then you've got more work than
people who have seen this.
29:18.309 --> 29:21.089
You've got to go and you've got
to do all the intermediate steps
29:21.088 --> 29:22.498
so you get used to this thing.
29:22.500 --> 29:25.380
Now, let's do the following two
other variations,
29:25.380 --> 29:26.820
and then I'll move on.
29:26.819 --> 29:30.859
If I change x to minus
x, then I get
29:30.858 --> 29:33.888
e^(-ix).
Cosine of minus x is
29:33.891 --> 29:36.551
cosine x,
and sin of minus x is
29:36.550 --> 29:37.880
minus sin x.
29:37.880 --> 29:39.320
So, I can get that relation.
29:39.320 --> 29:43.810
29:43.809 --> 29:48.739
Then, I'm going to combine
those two to do the following:
29:48.736 --> 29:53.216
I'm going to say that cos
x = e^(ix) +
29:53.222 --> 29:57.372
e^(-ix) over 2,
you can check that.
29:57.369 --> 30:05.149
And sin x =
e^(ix) - e^(-ix)
30:05.153 --> 30:09.873
over 2i.
You just add and subtract these
30:09.869 --> 30:11.829
two formulas and you get that.
30:11.830 --> 30:19.800
30:19.800 --> 30:23.040
Okay.
So, what does this mean?
30:23.039 --> 30:26.729
It means that trigonometric
functions, you don't need them.
30:26.730 --> 30:29.140
If you've got exponential
functions, you can manufacture
30:29.136 --> 30:30.796
trigonometric functions out of
them,
30:30.799 --> 30:34.689
provided you're not afraid to
go to exponents with complex
30:34.686 --> 30:36.456
number i in them.
30:36.460 --> 30:41.330
And all the identities about
sines and cosines will follow
30:41.325 --> 30:43.065
from this.
For example,
30:43.068 --> 30:45.698
if you take cosine squared plus
sine squared,
30:45.703 --> 30:47.383
you're supposed to get 1.
30:47.380 --> 30:50.030
Well, you can square the right
hand side, you can square the
30:50.028 --> 30:51.688
left hand side,
and you will get 1.
30:51.690 --> 30:54.210
To get the 1,
you better remember the
30:54.210 --> 30:58.130
following: e^(ix) times
e^(-ix) is what?
30:58.130 --> 31:01.560
31:01.559 --> 31:03.319
You know how you come by an
exponent?
31:03.319 --> 31:07.659
You raise it to one power times
the same thing raised to another
31:07.659 --> 31:11.519
power, is e raised to ix minus
x, which is e raised to 0,
31:11.516 --> 31:14.456
which is 1.
So, when you raise the number
31:14.460 --> 31:18.280
to a power, you multiply it by
the number times a different
31:18.280 --> 31:22.430
power, the product is a number
to the sum of the two powers.
31:22.430 --> 31:25.910
Powers combine.
2 cubed times 2 to the 4th is 2
31:25.907 --> 31:28.197
to the 6th.
That's true for 2;
31:28.200 --> 31:29.860
it's true for e;
it's true for everything.
31:29.859 --> 31:33.199
Exponents add when you multiply
them, that's why e^(ix)
31:33.197 --> 31:35.547
and e^(-ix) combine to
give you 1.
31:35.549 --> 31:37.989
Once you know that,
you can prove this.
31:37.990 --> 31:44.880
Okay.
Now, I'm going to do a little
31:44.876 --> 31:49.036
more of complex numbers,
so maybe I'll leave this alone
31:49.041 --> 31:51.741
for now.
I introduced you the number
31:51.742 --> 31:55.312
i by saying it's the
square root of minus 1,
31:55.309 --> 31:59.839
and complex numbers entered our
life, even though we didn't go
31:59.839 --> 32:01.249
looking for them.
32:01.250 --> 32:04.410
You can write down equations
with real numbers with no
32:04.412 --> 32:06.742
intention of invoking anything
fancy,
32:06.740 --> 32:11.440
like this, and you find there
is no solution to this equation.
32:11.440 --> 32:13.730
Even though everything there is
completely real.
32:13.730 --> 32:17.270
So, you can say,
"Well, x^(2) is minus
32:17.266 --> 32:20.876
1," and you can manufacture a
number i,
32:20.883 --> 32:24.423
with a property i^(2) is
minus 1;
32:24.420 --> 32:26.680
then, of course,
you can have x equal to
32:26.677 --> 32:28.687
plus and minus i as your
answer.
32:28.690 --> 32:31.810
So, complex numbers arose by
trying to solve quadratic
32:31.812 --> 32:35.492
equations.
So, let me write you a slightly
32:35.488 --> 32:41.038
more interesting quadratic
equation, x^(2) + x + 1 =
32:41.041 --> 32:43.731
0.
So, there's no funny business,
32:43.732 --> 32:45.062
all real numbers.
32:45.059 --> 32:51.649
We want to solve this equation,
so we go back to our good old
32:51.650 --> 32:58.460
sandbox days when we knew what
the answer for this was--minus 3
32:58.461 --> 33:01.021
over 2.
And we already have a problem
33:01.024 --> 33:03.924
because we don't know what to do
with square root of minus 3.
33:03.920 --> 33:10.900
So, we will write it as square
root of minus 1 times square
33:10.896 --> 33:14.046
root of 3.
Square root of minus 1 we will
33:14.046 --> 33:16.486
call i,
and we will say this equation
33:16.492 --> 33:20.272
has two roots.
x plus/minus (there are
33:20.271 --> 33:25.711
) two roots, which are minus 1
plus or minus square root of
33:25.705 --> 33:27.575
3i over 2.
33:27.579 --> 33:35.159
So, one root is 1 minus 1,
plus root 3i over 2.
33:35.160 --> 33:40.410
And the other root is minus 1
minus root 3i over 2.
33:40.410 --> 33:47.150
These are solutions formerly to
this equation in the following
33:47.146 --> 33:49.226
sense.
Take this x that I'm
33:49.227 --> 33:51.397
giving you;
put that into that equation,
33:51.400 --> 33:54.100
square the x and add the
x to it,
33:54.095 --> 33:56.375
and add the 1,
it'll in fact satisfy the
33:56.379 --> 33:58.849
equation.
All you will have to know when
33:58.851 --> 34:01.661
you manipulate it is that
i^(2) is minus 1.
34:01.660 --> 34:04.800
Using the one property,
you can now solve the quadratic
34:04.800 --> 34:07.100
equation.
So, people realize if you
34:07.096 --> 34:09.876
enlarge numbers to include
complex numbers,
34:09.878 --> 34:13.258
then you can solve any
polynomial equation with that
34:13.257 --> 34:14.977
many number of roots.
34:14.980 --> 34:16.610
If it's quadratic,
it'll have two roots,
34:16.609 --> 34:18.279
if it's cubic,
it'll have three roots.
34:18.280 --> 34:21.650
They may not all be real;
they may involve the complex
34:21.646 --> 34:25.506
number i.
Now, a very important point to
34:25.513 --> 34:30.893
notice is that this whole thing,
minus 1 plus root 3i
34:30.889 --> 34:35.809
over 2, the whole thing is a
single complex number.
34:35.809 --> 34:37.279
Don't think of it as the sum of
two numbers;
34:37.280 --> 34:38.870
it's a single complex number.
34:38.869 --> 34:42.339
For example,
if this had been plus instead
34:42.336 --> 34:47.486
of minus, you'd get something
like 1 plus or minus root 3 over
34:47.492 --> 34:49.272
2.
Take the positive root,
34:49.270 --> 34:51.370
you don't think of it as two
numbers.
34:51.370 --> 34:52.980
This is a single number.
34:52.980 --> 34:56.760
And that continues to be true
even if it's a complex number.
34:56.760 --> 35:00.900
So, you should think of the
whole combination as a single
35:00.904 --> 35:05.124
entity which you can add to
others as entities and square,
35:05.123 --> 35:07.073
and so on.
Okay.
35:07.070 --> 35:08.960
So, that's what we are going to
do now.
35:08.960 --> 35:14.030
We're going to generalize this
particular case and introduce
35:14.030 --> 35:16.780
now a complex number z.
35:16.780 --> 35:19.970
And we're going to write every
complex number z as a
35:19.972 --> 35:23.442
part that has no i in it,
and a part that has an i
35:23.439 --> 35:26.349
in it.
The example I had here,
35:26.350 --> 35:32.300
x is equal to--in this
example, x was minus 1
35:32.304 --> 35:37.954
over 2 and y was plus or
minus root 3 over 2.
35:37.949 --> 35:41.099
And we're going to visualize
that complex number as a point
35:41.101 --> 35:42.461
in the xy plane.
35:42.460 --> 35:45.130
We just measure x
horizontally and y
35:45.131 --> 35:48.131
vertically, put a dot there,
that's your complex number,
35:48.131 --> 35:48.841
z.
35:48.840 --> 35:54.300
35:54.300 --> 35:58.130
Then, we say the new complex
number called z star [*]
35:58.128 --> 36:02.398
and it's defined to be x -
iy, is called the complex
36:02.399 --> 36:03.429
conjugate;
36:03.430 --> 36:07.610
36:07.610 --> 36:09.600
complex conjugate of z.
36:09.599 --> 36:12.669
It's obtained by changing the
sign of i.
36:12.670 --> 36:14.910
So, if you like,
if z is sitting here,
36:14.908 --> 36:17.958
z star is sitting there,
reflected around the x
36:17.962 --> 36:18.422
axis.
36:18.420 --> 36:27.240
36:27.239 --> 36:31.129
So, how does one work with
complex numbers?
36:31.130 --> 36:32.980
You add them,
you subtract them,
36:32.977 --> 36:35.657
and you multiply them,
and you divide them.
36:35.659 --> 36:38.289
If you know how to do that,
you can do with complex numbers
36:38.290 --> 36:40.060
everything you did with real
numbers.
36:40.059 --> 36:44.119
So if z_1 is a
complex number that is equal to
36:44.120 --> 36:46.740
x_1 +
iy_1,
36:46.739 --> 36:49.459
and z_2 is a
complex number that is
36:49.461 --> 36:51.971
x_2 +
iy_2,
36:51.969 --> 36:55.189
we will define
z_1 +
36:55.187 --> 36:59.477
z_2 to be
x_1 +
36:59.477 --> 37:03.047
x_2 times
i times
37:03.052 --> 37:07.792
y_1 plus
y_2.
37:07.789 --> 37:11.219
If you draw pictures,
then if z_1 is
37:11.217 --> 37:14.047
that number and
z_2 is that
37:14.052 --> 37:17.242
number [pointing to the board],
then z_1 +
37:17.235 --> 37:19.015
z_2 it really
is like adding vectors.
37:19.019 --> 37:22.879
This number is z_1
+ z_2.
37:22.880 --> 37:25.270
It's exactly like adding
vectors.
37:25.269 --> 37:30.599
But you don't think of this
plus that as two disjoined
37:30.597 --> 37:34.817
numbers, but as forming a single
entity.
37:34.820 --> 37:41.170
In a complex number z,
x is called a real part
37:41.171 --> 37:47.201
of z, and is denoted by
symbol "real z."
37:47.199 --> 37:50.179
If you want to get z,
if you want to get the real
37:50.176 --> 37:53.206
part of a complex number
z, you add to its complex
37:53.206 --> 37:54.826
conjugate and divide by 2.
37:54.829 --> 37:58.489
Because you are adding x +
iy to x - iy,
37:58.485 --> 38:02.205
you get 2x and you
divide by 2, you get that.
38:02.210 --> 38:06.760
If you want to get the
y, which is called the
38:06.759 --> 38:11.839
imaginary part of z,
you take z - z star and divide
38:11.843 --> 38:15.843
by 2i.
It's like saying find the
38:15.838 --> 38:19.008
x component of a vector.
38:19.010 --> 38:20.440
It's whatever multiplies
i.
38:20.440 --> 38:22.690
y component of a vector
is whatever multiplies j.
38:22.690 --> 38:25.550
Similarly, a complex number can
be broken down into the part
38:25.549 --> 38:28.359
which is real and the part that
multiplies the i.
38:28.360 --> 38:30.850
To extract the real part,
add to it the conjugate divided
38:30.852 --> 38:33.612
by 2, to extract the imaginary
part, subtract the conjugate and
38:33.611 --> 38:34.681
divide by 2i.
38:34.679 --> 38:39.669
Again, if you have seen this,
you don't need this.
38:39.670 --> 38:41.810
If you haven't seen it,
it may look a little fast.
38:41.809 --> 38:45.319
But that's why I posted all the
notes on the website,
38:45.322 --> 38:49.512
chapter 5 from this math book,
which has got all this stuff.
38:49.510 --> 38:51.720
So, you can go home and you can
read it.
38:51.719 --> 38:58.039
Ok, the next question is what
is z_1 times
38:58.035 --> 39:00.535
z_2?
39:00.539 --> 39:03.779
Well, that's very easy to
calculate.
39:03.780 --> 39:06.600
x_1 +
iy_1 times
39:06.602 --> 39:09.762
x_2 +
iy_2.
39:09.760 --> 39:12.880
Just open all the brackets and
remember i^(2) is minus
39:12.881 --> 39:14.531
1.
That's all you've got to do.
39:14.530 --> 39:15.820
So, that gives me
x_1x
39:15.819 --> 39:17.829
_2;
then, let me multiply the
39:17.827 --> 39:20.657
iy_1 times the
iy_2.
39:20.659 --> 39:23.249
And that gives me minus
y_1y
39:23.253 --> 39:26.373
_2,
because i^(2) is minus 1
39:26.365 --> 39:28.695
plus i times
x_1y
39:28.699 --> 39:30.969
_2 +
x_2y
39:30.969 --> 39:32.589
_1.
39:32.590 --> 39:34.810
This is how you multiply two
complex numbers.
39:34.810 --> 39:39.440
39:39.440 --> 39:43.020
That's something very nice that
happens when you multiply a
39:43.023 --> 39:45.683
number z by its complex
conjugate.
39:45.679 --> 39:49.679
What happens when you multiply
z by its conjugate,
39:49.675 --> 39:52.095
you are saying,
take x + iy,
39:52.100 --> 39:55.240
multiply it by x minus
iy;
39:55.239 --> 39:59.039
a + b times a - b =
a^(2) - b^(2).
39:59.039 --> 40:01.459
But remember,
b is iy.
40:01.460 --> 40:05.220
So, when you take the square of
that, you should keep track of
40:05.220 --> 40:08.920
all the signs and you will find
it's x^(2) + y^(2).
40:08.920 --> 40:15.280
We denote that as this,
and this thing is called the
40:15.278 --> 40:18.768
modulus of z [|z|].
40:18.769 --> 40:20.399
It's the length of the complex
number z,
40:20.396 --> 40:21.666
just given by Pythagoras'
theorem.
40:21.670 --> 40:30.790
40:30.789 --> 40:32.949
You should know,
whenever you take a complex
40:32.945 --> 40:34.795
number, multiply by each
conjugate,
40:34.800 --> 40:37.700
by its complex conjugate,
the result will be a real
40:37.695 --> 40:40.815
number equal to the square of
the length of the complex
40:40.822 --> 40:43.472
number.
Because I'm going to use that
40:43.473 --> 40:46.183
now to do the last of the
manipulations,
40:46.181 --> 40:49.511
which is, what is
z_1 divided by
40:49.514 --> 40:51.324
z_2?
40:51.320 --> 40:53.720
How do you divide these crazy
numbers?
40:53.719 --> 40:56.509
So, on the top I write
x_1 +
40:56.510 --> 40:59.160
iy_1,
the bottom, I write
40:59.160 --> 41:02.090
x_2 +
iy_2.
41:02.090 --> 41:04.400
If I only had
x_2,
41:04.404 --> 41:05.974
I know how to divide.
41:05.970 --> 41:07.260
It's just a number.
41:07.260 --> 41:08.580
Divide that by
x_2 and that by
41:08.579 --> 41:09.239
x_2.
41:09.239 --> 41:11.239
But I've got the sum of these
two numbers in the bottom,
41:11.237 --> 41:12.977
and you can ask,
"How do you do the division?"
41:12.980 --> 41:15.980
So, the trick,
always, when you run into this
41:15.980 --> 41:19.800
problem, is to multiply the top
and bottom by the complex
41:19.800 --> 41:21.710
conjugate of the bottom.
41:21.710 --> 41:23.950
Multiply it by x_2
- iy_2,
41:23.948 --> 41:24.678
top and bottom.
41:24.680 --> 41:28.610
41:28.610 --> 41:30.920
Then, something nice happens to
the denominator,
41:30.920 --> 41:33.770
because the denominator then
becomes x_2^(2) +
41:33.771 --> 41:36.571
y_2^(2),
which is an ordinary real
41:36.566 --> 41:38.746
number, nothing complex about
that.
41:38.750 --> 41:40.890
The numerator,
you can open out the brackets.
41:40.889 --> 41:42.159
I don't know if I want to do
that;
41:42.159 --> 41:47.009
it's just going to be
x_1x_2 +
41:47.011 --> 41:52.331
y_1y_2 +
i times y_1x
41:52.328 --> 41:55.588
_2 -
y_2x
41:55.594 --> 41:57.744
_1.
41:57.739 --> 41:59.509
Now, don't worry about all the
details.
41:59.510 --> 42:01.920
All it means is,
if you know how to multiply two
42:01.917 --> 42:03.967
complex numbers,
you can also divide by a
42:03.966 --> 42:06.246
complex number.
That's the key point.
42:06.250 --> 42:08.670
Why?
Because if you've got a complex
42:08.672 --> 42:10.302
number denominator,
you don't like it,
42:10.297 --> 42:13.107
multiply top and bottom by the
complex conjugate of this guy,
42:13.110 --> 42:16.360
the denominator turns into a
purely real number.
42:16.360 --> 42:19.700
So, this whole thing could be
36, for example.
42:19.699 --> 42:22.259
Well, we know how to divide the
numerator by 36,
42:22.255 --> 42:23.845
right?
x_1x_2 +
42:23.848 --> 42:25.828
y_1y_2
some number, maybe 9;
42:25.830 --> 42:27.310
then you divide by 36.
42:27.310 --> 42:30.440
One-fourth.
This could be i times 18.
42:30.440 --> 42:33.260
It would be 36,
you get i over 2.
42:33.260 --> 42:37.490
So, this number could have been
one fourth plus i over 2.
42:37.489 --> 42:39.399
So, dividing by a complex
number is not a problem.
42:39.400 --> 42:43.500
42:43.500 --> 42:50.590
Now, why did I do all that
stuff today about e^(ix)?
42:50.590 --> 42:52.460
You're going to see that now.
42:52.460 --> 42:56.710
The rationale for going through
that math is the following.
42:56.710 --> 43:02.380
Let's take this complex number
that goes from here to here,
43:02.375 --> 43:04.715
that is x + iy.
43:04.719 --> 43:08.309
And let's introduce,
just like we would for an
43:08.307 --> 43:11.897
ordinary vector,
this angle θ and that
43:11.895 --> 43:15.095
length r.
Whenever you have a vector,
43:15.098 --> 43:18.138
you can talk about the
Cartesian components x
43:18.136 --> 43:20.186
and y,
or you can talk about the
43:20.186 --> 43:22.156
length of the vector and the
angle it makes.
43:22.159 --> 43:26.669
Then, it's clear that x = r
cos θ, and y = r sin
43:26.668 --> 43:31.018
θ, where θ is now
an angle associated with a
43:31.020 --> 43:35.840
complex number which tells you
at what angle it's located,
43:35.840 --> 43:37.840
and r is the length of
the complex number.
43:37.840 --> 43:40.940
It's called the polar form of
the complex number.
43:40.940 --> 43:45.360
Then, you can see that
z, which is x +
43:45.356 --> 43:50.806
iy, is equal to r
times cos θ + i sin θ
43:50.807 --> 43:56.537
which you can now write as
r times e^(iθ).
43:56.539 --> 44:00.069
That's why we did all that
work, to tell you that a complex
44:00.068 --> 44:02.318
number can be written in this
form.
44:02.320 --> 44:06.070
You can either write this x
+ iy, or else
44:06.065 --> 44:09.035
re^(iθ).
They both talk about the same
44:09.044 --> 44:10.924
number.
One talks about how much real
44:10.920 --> 44:14.070
part and how much imaginary part
it has, other way of writing it
44:14.070 --> 44:15.970
talks about how long the vector
is,
44:15.970 --> 44:18.190
and at what angle it is located.
44:18.190 --> 44:20.070
It contains the same
information.
44:20.070 --> 44:23.680
So, that r is equal to--
the inverse of this formula is
44:23.679 --> 44:27.289
that r is square root of
x squared plus y
44:27.289 --> 44:30.349
squared,
and θ is tan
44:30.352 --> 44:33.142
inverse y over x.
44:33.139 --> 44:35.009
In other words,
tan θ is y over
44:35.012 --> 44:36.542
x.
This is y,
44:36.542 --> 44:37.932
and this is x.
44:37.930 --> 44:43.160
44:43.159 --> 44:46.399
Now, the advantage of writing a
complex number in polar form is
44:46.403 --> 44:49.073
the following.
Suppose I give you two complex
44:49.065 --> 44:51.615
numbers, z_1 is
r_1e
44:51.621 --> 44:54.861
^(iθ)_1,
and z_2 is
44:54.864 --> 44:56.514
r_2
e^(iθ
44:56.510 --> 44:57.470
)_2.
44:57.469 --> 45:00.719
The product z_1z
_2 is very easy to
45:00.724 --> 45:03.234
calculate in this form because
r_1r
45:03.232 --> 45:05.742
_2 multiply to
give you this,
45:05.740 --> 45:08.800
and the exponentials combine.
45:08.800 --> 45:13.700
It's a lot easier to multiply
them in this form than when I
45:13.701 --> 45:16.491
multiplied them somewhere here.
45:16.489 --> 45:18.119
See, if I write it in this
Cartesian form,
45:18.123 --> 45:20.823
it's a big mess.
In polar form, it's very easy.
45:20.820 --> 45:23.320
So, the rule is,
if you want to multiply two
45:23.315 --> 45:27.025
numbers, multiply the lengths to
get the length of the new number
45:27.028 --> 45:30.218
and add the angles to get the
angle of the product.
45:30.219 --> 45:33.929
So, what I'm telling you is
that if z_1
45:33.928 --> 45:37.298
looks like this guy with some
angle θ 1,
45:37.300 --> 45:40.250
and z_2 is
that guy with an angle θ
45:40.253 --> 45:43.263
2, the number z_1z
_2 has a length
45:43.257 --> 45:45.557
equal to the product of these
two lengths,
45:45.559 --> 45:48.049
and that angle equal the sum of
these two angles,
45:48.050 --> 45:50.800
so it will look like that
[writing graph on board].
45:50.800 --> 45:54.340
To go from this to that,
you multiply the lengths and
45:54.340 --> 45:55.770
you add the angles.
45:55.769 --> 45:59.939
So, the polar form is very well
suited for multiplying,
45:59.939 --> 46:03.259
and it's even better suited for
dividing.
46:03.260 --> 46:05.450
If I say, "Give me
z_1 over
46:05.446 --> 46:08.236
z_2," well,
you can all do that in your
46:08.242 --> 46:09.842
head.
It's r_1,
46:09.839 --> 46:11.329
e^(iθ
)_1,
46:11.329 --> 46:12.689
divided by
r_2,
46:12.687 --> 46:14.437
e^(iθ
)_2.
46:14.440 --> 46:17.560
The modulus of the new number
is just r_1
46:17.558 --> 46:19.088
over r_2.
46:19.090 --> 46:20.280
And how about this?
46:20.280 --> 46:21.820
e^(iθ
)_1 divided by
46:21.822 --> 46:23.122
e^(iθ
)_2.
46:23.119 --> 46:25.499
Dividing by
e^(iθ) is the
46:25.500 --> 46:28.430
same as multiplying by
e^(-iθ).
46:28.429 --> 46:31.509
So, this is really
e^(iθ
46:31.510 --> 46:34.590
)_1 - θ 2.
46:34.590 --> 46:39.340
So, I'm assuming you guys can
figure this part out,
46:39.337 --> 46:43.327
1 over e^(iθ) is
the same as
46:43.326 --> 46:45.886
e^(-iθ).
46:45.889 --> 46:49.109
Well, if you doubt me,
multiply--cross multiply,
46:49.114 --> 46:53.374
and all I'm telling you is that
1 = e^(iθ) times
46:53.367 --> 46:56.777
e^(-iθ),
which you know is true when you
46:56.776 --> 46:57.776
combine the angles.
46:57.780 --> 46:59.660
When you've got
e^(iθ)
46:59.664 --> 47:02.914
downstairs, you can take it
upstairs with the reverse angle.
47:02.910 --> 47:05.950
And that's a useful trick;
I hope you will remember that
47:05.949 --> 47:06.489
useful trick.
47:06.490 --> 47:09.660
47:09.659 --> 47:12.649
So, there is one part of this
thing that I want you to carry
47:12.647 --> 47:14.217
in your head,
because it's very,
47:14.217 --> 47:17.847
very important.
When you take a complex number,
47:17.850 --> 47:21.640
it's got a length and it's got
a direction.
47:21.639 --> 47:24.039
Then, you multiply by a second
complex number,
47:24.039 --> 47:26.599
you're able to do two things at
the same time.
47:26.599 --> 47:29.859
You're able to rescale it,
and you're able to rotate it.
47:29.860 --> 47:32.520
You rescale by the length of
the second factor,
47:32.523 --> 47:35.943
and you rotate by the angle
carried by the second factor.
47:35.940 --> 47:40.270
47:40.269 --> 47:43.639
The fact that two operations
are done in one shot is the
47:43.640 --> 47:47.190
reason complex numbers play an
incredibly important role in
47:47.194 --> 47:49.294
physics,
and certainly in engineering
47:49.293 --> 47:50.563
and mathematical physics.
47:50.560 --> 47:59.080
47:59.079 --> 48:04.559
I'm going to now change gears,
but I want to stop a little bit
48:04.558 --> 48:08.418
and answer any questions any of
you have.
48:08.420 --> 48:12.300
So, how many people have seen
all of this before?
48:12.300 --> 48:14.540
So, if you saw something,
you saw the whole thing.
48:14.539 --> 48:18.129
But look, that's not such a big
number, so I am cognizant of the
48:18.126 --> 48:20.456
fact that not many of you have
seen it.
48:20.460 --> 48:24.570
But you will have to go and
learn this today.
48:24.570 --> 48:28.590
I'm going to post the problem
set for this sometime in the day
48:28.591 --> 48:30.571
so you can start practicing.
48:30.570 --> 48:33.780
Don't wait, the problem set has
nothing to do with next
48:33.782 --> 48:36.342
Wednesday, it has a lot to do
with today.
48:36.340 --> 48:39.360
When I assign a problem today,
I imagine you're going
48:39.357 --> 48:41.847
breathlessly to your room,
not able to wait,
48:41.852 --> 48:44.872
jumping into the problem set
and working it out.
48:44.869 --> 48:47.389
That's the only way you're
going to find out what these
48:47.386 --> 48:49.246
things mean if you've never seen
them.
48:49.250 --> 48:52.300
If you've seen them before in
high school, or taken a math
48:52.297 --> 48:55.447
course and you've got lots of
practice, then I'm not talking
48:55.451 --> 48:57.441
to you.
But I'm going to use this at
48:57.443 --> 49:00.153
some point, so you should
understand complex numbers.
49:00.150 --> 49:02.930
And I'm telling you,
there are very few branches of
49:02.931 --> 49:05.881
any science where complex
numbers will not be used.
49:05.880 --> 49:10.020
Now, you may not believe this
now, but that is true.
49:10.019 --> 49:13.009
If an electrical engineer,
it's an absolute must.
49:13.010 --> 49:16.400
If you solve any kind of
differential equation which can
49:16.395 --> 49:19.345
occur in biology and chemistry,
that's a must.
49:19.350 --> 49:22.670
So, it's very, very important.
49:22.670 --> 49:26.980
All right.
So now, I'm going back to
49:26.982 --> 49:29.652
physics.
Going back from mathematics to
49:29.651 --> 49:33.791
physics, and the physics of this
topic for today has to do with
49:33.787 --> 49:36.587
something very different from
the past,
49:36.590 --> 49:38.560
so forget all the relativity
now.
49:38.559 --> 49:41.579
You're going back to Newtonian
days.
49:41.579 --> 49:43.769
Kinetic energy is ½
mv^(2).
49:43.769 --> 49:47.139
It's a little difficult to go
back and forget what you
49:47.143 --> 49:48.483
learned.
On the other hand,
49:48.479 --> 49:50.189
for some of you,
it may not be so hard if you
49:50.188 --> 49:51.158
didn't learn anything.
49:51.159 --> 49:54.089
Well then, you are that much
ahead of the other guys.
49:54.090 --> 49:57.750
But remember now,
the truth is the relativistic
49:57.753 --> 49:59.913
theory.
We're going to go back to
49:59.907 --> 50:02.377
Newtonian days,
and the reason we do it that
50:02.380 --> 50:04.910
way is to give you something
interesting,
50:04.909 --> 50:08.279
hopefully, compared to what you
normally do.
50:08.280 --> 50:12.840
So, what we're going to study
now is what's called small
50:12.844 --> 50:16.334
oscillations,
or simple harmonic motion.
50:16.329 --> 50:19.489
It's a ubiquitous fact,
that if you took any mechanical
50:19.493 --> 50:22.953
system which is in a state of
equilibrium, and you give it a
50:22.950 --> 50:24.650
little kick, it vibrates.
50:24.650 --> 50:27.530
Now, if you've got a pillar,
ceiling to floor,
50:27.530 --> 50:30.090
you hit it with a hammer,
it vibrates.
50:30.090 --> 50:34.070
If you take a gong and hit it,
it also vibrates.
50:34.070 --> 50:38.550
If you've got a rod hanging
form the ceiling through a pivot
50:38.554 --> 50:43.574
-- it's hanging there -- if you
pull it, it goes back and forth.
50:43.570 --> 50:47.940
The standard example everyone
likes to use is if you've got a
50:47.940 --> 50:51.510
particle in a bowl,
it's very happy sitting in the
50:51.508 --> 50:54.728
bottom.
If you push it up a little bit,
50:54.729 --> 50:56.679
it'll go back and forth.
50:56.679 --> 50:59.529
And the example that we're
going to consider is the
50:59.534 --> 51:01.514
following.
If you have a mass,
51:01.505 --> 51:05.085
m, connected to a
spring, and the spring is not
51:05.091 --> 51:08.881
stretched or contracted,
it's very happy to be there.
51:08.880 --> 51:10.620
That's what I mean by
equilibrium.
51:10.619 --> 51:14.019
Equilibrium means the body has
all the forces on it adding up
51:14.022 --> 51:16.112
to 0;
it has no desire to move.
51:16.110 --> 51:18.320
The question is,
if you give it a little kick,
51:18.316 --> 51:21.066
what'll happen?
Well, there are two situations
51:21.072 --> 51:23.282
you can have.
You can have a situation where
51:23.279 --> 51:24.719
the particle's on top of a hill.
51:24.719 --> 51:27.799
That's called unstable
equilibrium because if you give
51:27.804 --> 51:31.474
that a kick, it's going to come
down and never return to you.
51:31.469 --> 51:34.159
That's equilibrium,
but unstable equilibrium.
51:34.159 --> 51:35.489
I'm talking about stable
equilibrium.
51:35.489 --> 51:38.089
That's because there are
restoring forces.
51:38.090 --> 51:41.440
If you stray away from the
equilibrium, there are forces
51:41.441 --> 51:42.661
bringing you back.
51:42.659 --> 51:45.869
And in the case of the mass and
spring system,
51:45.873 --> 51:47.733
the force, md 2..
51:47.730 --> 51:49.990
ma, is equal to
-kx.
51:49.989 --> 51:52.589
And what this equation tells
you is if you stray to the
51:52.587 --> 51:55.757
right, x is positive,
[I will apply a force to the
51:55.755 --> 51:57.845
left.]
Remember, this is ma,
51:57.850 --> 51:59.330
and this is F.
51:59.329 --> 52:03.159
F is such that it always
sends you back to your normal
52:03.158 --> 52:03.858
position.
52:03.860 --> 52:09.650
52:09.650 --> 52:13.720
So, we want to understand the
behavior of such a problem.
52:13.720 --> 52:15.140
We want to solve this problem.
52:15.140 --> 52:18.500
How do you solve this problem?
52:18.500 --> 52:21.760
Well, I did it for you along
back, when I gave you a typical
52:21.762 --> 52:24.142
paradigm for how you apply
Newton's laws;
52:24.140 --> 52:25.750
I gave you this example.
52:25.750 --> 52:27.960
So, I'm going to go through it
somewhat fast.
52:27.960 --> 52:31.710
Our job is to find the function
x that satisfies this
52:31.710 --> 52:34.540
equation.
And we would like to write it
52:34.535 --> 52:39.005
as follows: d^(2)x over
dt^(2) is equal to minus
52:39.007 --> 52:42.227
Ω^(2) x,
where Ω is the
52:42.229 --> 52:46.719
shorthand for squared root of
k over m.
52:46.719 --> 52:50.239
By the way, I'm using big
X and small x like
52:50.243 --> 52:52.533
crazy, so it's just small
x.
52:52.530 --> 52:56.200
I don't mean anything
significant between this and
52:56.204 --> 52:58.214
this.
They're all the same.
52:58.210 --> 53:02.020
If you want,
you can take this--pardon me?
53:02.019 --> 53:05.269
[inaudible question]
This is k,
53:05.269 --> 53:07.639
and this guy is k.
53:07.639 --> 53:11.809
And the other equations you use
to follow the lecture today is,
53:11.812 --> 53:14.372
[x=X]
these are all the same.
53:14.370 --> 53:17.060
Okay?
So, don't say what happened
53:17.063 --> 53:19.223
there, when you move from one to
the other.
53:19.219 --> 53:21.569
Alright, so what did I say we
should do?
53:21.570 --> 53:25.030
You can make it a word problem
and say, "I'm looking for a
53:25.034 --> 53:27.834
function which,
when I take two derivatives,
53:27.829 --> 53:32.429
looks like minus itself,
except for this number."
53:32.430 --> 53:34.490
And we just saw that today.
53:34.489 --> 53:37.209
Trigonometric functions have
the property that you take two
53:37.213 --> 53:39.423
derivatives, they return to
minus themselves.
53:39.420 --> 53:43.900
So, you can take a guess that
x looks like cos t
53:43.900 --> 53:47.280
but it won't work,
that I showed you before.
53:47.280 --> 53:48.930
So, I'm not going to go through
that again.
53:48.929 --> 53:52.519
If you took this guess--but
x is not a number,
53:52.517 --> 53:55.757
x is a function of time
[x(t)].
53:55.760 --> 53:58.670
If you took this function of
time, it, in fact,
53:58.665 --> 54:02.135
will obey this equation,
provided the Ω that you
54:02.138 --> 54:05.358
put in is the Ω that's
in the equation.
54:05.360 --> 54:10.030
Why?
Because, take two derivatives.
54:10.030 --> 54:13.580
First time you get -Ω,
sin Ω.
54:13.579 --> 54:16.029
The second time you get -
Ω ^(2) cos Ωt,
54:16.033 --> 54:18.133
which means it's minus
Ω ^(2),
54:18.130 --> 54:22.550
times your x itself,
and it is whatever you like.
54:22.550 --> 54:25.580
54:25.579 --> 54:28.299
Then I said,
"Let's plot this guy."
54:28.300 --> 54:30.160
When you plot this guy it looks
like this.
54:30.160 --> 54:31.940
And A is the amplitude.
54:31.940 --> 54:39.820
54:39.820 --> 54:41.520
What is Ω?
54:41.519 --> 54:45.479
Well, Ω is related to
the time of the frequency of
54:45.481 --> 54:49.101
oscillations as follows:
If I start with t = 0
54:49.096 --> 54:52.906
over the x = A,
how long do I have to wait
54:52.908 --> 54:54.718
'till I come back to A?
54:54.719 --> 54:58.289
I think everybody should know
that I have to wait at time
54:58.286 --> 55:01.526
capital T so that
Ω times T is
55:01.534 --> 55:03.974
2π.
Because that's when the cosine
55:03.969 --> 55:06.749
returns to 1.
That means the time that you
55:06.746 --> 55:09.946
have to wait is 2π over
Ω.
55:09.949 --> 55:13.229
Or you can say Ω is
2π over T,
55:13.228 --> 55:16.758
or you can also write it a
2π times frequency,
55:16.760 --> 55:19.590
where frequency is what you and
I would call frequency,
55:19.593 --> 55:21.853
how many oscillations it does
per second.
55:21.849 --> 55:24.449
That's the inverse of the time
period.
55:24.449 --> 55:30.839
So, if you pull a mass and you
let it go, it oscillates with a
55:30.842 --> 55:37.762
frequency which is connected to
the force constant and the mass.
55:37.760 --> 55:40.020
If the spring is very stiff and
k is very large,
55:40.022 --> 55:41.072
frequency is very high.
55:41.070 --> 55:44.250
If the mass is very big and the
motion is very sluggish,
55:44.246 --> 55:45.686
f is diminished.
55:45.690 --> 55:48.860
So, all that stuff comes out of
the equation.
55:48.860 --> 55:52.390
One really remarkable part of
the equation is that you can
55:52.386 --> 55:54.176
pick any A you like.
55:54.180 --> 55:56.660
Think about what that means.
55:56.660 --> 55:58.320
What is the meaning of A?
55:58.320 --> 56:00.920
A is the amount by which
you pulled it when you let it
56:00.923 --> 56:03.123
go.
You are told whether you pull
56:03.117 --> 56:07.207
the spring by one inch or by ten
inches, the time it takes to
56:07.209 --> 56:10.759
finish a full back and forth
motion is independent of
56:10.755 --> 56:13.345
A.
This frequency here is
56:13.348 --> 56:15.208
independent of A.
56:15.210 --> 56:16.850
Yes?
Student:
56:16.848 --> 56:18.688
[inaudible]
Professor Ramamurti
56:18.693 --> 56:20.653
Shankar: This t here?
56:20.650 --> 56:23.420
Here?
Student:
56:23.421 --> 56:24.021
[inaudible]
Professor Ramamurti
56:24.017 --> 56:24.557
Shankar: Ah,
very good.
56:24.560 --> 56:26.110
That's correct.
So, let's be careful.
56:26.110 --> 56:27.570
This is the t axis
[pointing at graph on board];
56:27.570 --> 56:30.400
this is a particular time,
capital T.
56:30.400 --> 56:36.070
56:36.070 --> 56:39.440
So, the remarkable property of
simple harmonic motion is that
56:39.442 --> 56:42.312
the amplitude does not determine
the time period.
56:42.309 --> 56:44.779
If you pull it by two inches,
compared to one inch,
56:44.777 --> 56:46.157
it's got a long way to go.
56:46.159 --> 56:49.889
But because you pulled it by
two inches, the spring is going
56:49.887 --> 56:52.977
to be that much more tense,
and it's going to exert a
56:52.978 --> 56:55.948
bigger force so that it'll go
faster for most of the time,
56:55.950 --> 56:57.750
that's very clear.
56:57.750 --> 57:00.740
But the fact that it goes
faster in exactly the right way
57:00.742 --> 57:02.882
to complete the trip at the same
time,
57:02.880 --> 57:06.820
is rather a miraculous property
of the fact that this equation
57:06.824 --> 57:08.574
has this particular form.
57:08.570 --> 57:12.120
If you tamper with this,
if you add to this a little
57:12.123 --> 57:14.843
bit, like 1 over 100 times
x^(3),
57:14.840 --> 57:17.070
then all these bets are off.
57:17.070 --> 57:21.310
It's got to be that equation
for that result to be true.
57:21.310 --> 57:23.980
Okay.
So, this is simple harmonic
57:23.977 --> 57:26.277
motion.
Then, you can do the following
57:26.278 --> 57:29.518
variant of this solution,
which I will write down now.
57:29.519 --> 57:36.019
Suppose I set my clock to 0,
right there.
57:36.019 --> 57:38.369
You set your clock to 0 here,
but I can say,
57:38.370 --> 57:40.610
"You know what?
I just got up and I'm looking
57:40.609 --> 57:42.189
at the mass, I set my clock to
0."
57:42.190 --> 57:44.370
When it hits 0,
it's not at the maximum.
57:44.369 --> 57:47.649
But it's the same physics,
it's the same equation.
57:47.650 --> 57:51.310
And that really comes from the
fact that we had one more
57:51.311 --> 57:54.841
latitude here of adding a
certain number φ,
57:54.840 --> 57:58.530
which is called a "phase," to
the oscillator.
57:58.530 --> 58:03.600
Whatever you pick for
φ, it'll still work.
58:03.599 --> 58:05.289
And whatever you pick for
capital A,
58:05.286 --> 58:06.046
it'll still work.
58:06.050 --> 58:08.470
So, whenever you have an
oscillator, namely,
58:08.469 --> 58:11.279
a mass and spring system,
and you want to know what
58:11.282 --> 58:13.592
x is going to be at all
times,
58:13.590 --> 58:18.640
you need to somehow determine
the amplitude and the phase.
58:18.639 --> 58:21.659
Once you know that,
you can calculate x at
58:21.656 --> 58:22.846
all future times.
58:22.850 --> 58:25.610
So, let me give you an example.
58:25.610 --> 58:32.820
Suppose an oscillator has
x = 5 and velocity equal
58:32.820 --> 58:35.910
to 0, at t equal to 0.
58:35.910 --> 58:36.830
So what does that mean?
58:36.829 --> 58:39.929
I pulled it by 5 and I let it
go.
58:39.929 --> 58:42.999
I tell you the spring constant
k and I tell you the
58:43.002 --> 58:45.592
mass, and I say,
"What's the future x?"
58:45.590 --> 58:47.320
You've got to come back to this
equation.
58:47.320 --> 58:54.100
And you've got to say,
5 = A cosine of 0 +
58:54.096 --> 58:57.046
φ.
And how about the velocity?
58:57.050 --> 59:02.750
Velocity is supposed to be 0;
that is minus ΩA sin
59:02.753 --> 59:05.143
Ωt + φ.
59:05.140 --> 59:06.910
But t is 0.
59:06.909 --> 59:11.349
So, that tells me 0 is minus
Ω a sin φ.
59:11.349 --> 59:13.649
Ω is not 0,
A is not 0.
59:13.650 --> 59:18.790
Therefore sin φ is 0,
that means φ is 0 if
59:18.792 --> 59:23.032
you get rid of that,
so the subsequent motion is
59:23.032 --> 59:25.922
x = 5 cos Ωt.
59:25.920 --> 59:28.440
This is a problem where we did
not need φ.
59:28.440 --> 59:32.240
But it could have been that
when you joined the experiment,
59:32.244 --> 59:35.084
you were somewhere here;
then, the mass is actually
59:35.081 --> 59:36.981
moving.
Then I will say at t =
59:36.975 --> 59:38.835
0, velocity was some other
number.
59:38.840 --> 59:41.310
Maybe 6.
Then, you've got to go back and
59:41.311 --> 59:42.161
write two equations.
59:42.159 --> 59:45.369
One for x:
take the derivative which looks
59:45.371 --> 59:48.651
like this, and instead of
putting 0, you put 6.
59:48.650 --> 59:51.420
You've got two unknowns,
A and φ.
59:51.420 --> 59:53.800
You've got to juggle them
around, and you've got to solve
59:53.797 --> 59:56.087
for A and φ,
given these two numbers.
59:56.090 --> 59:59.570
It's a simple exercise to solve
for them, and when you do,
59:59.570 --> 1:00:01.890
you have completely fit the
problem.
1:00:01.889 --> 1:00:04.269
In other words,
simply to be told a mass is
1:00:04.268 --> 1:00:07.668
connected to a spring of known
mass and force constant is not
1:00:07.665 --> 1:00:10.435
enough to determine the future
of the spring.
1:00:10.440 --> 1:00:13.720
You have to be told--For
example, if nobody pulled the
1:00:13.722 --> 1:00:16.512
mass, it's just going to sit
there forever.
1:00:16.510 --> 1:00:19.000
If you pulled it by 10,
it's going to go back and forth
1:00:19.003 --> 1:00:20.023
from 10 to minus 10.
1:00:20.019 --> 1:00:23.919
So, you have to be told further
information to nail down these
1:00:23.915 --> 1:00:26.345
two numbers.
They're called free parameters
1:00:26.354 --> 1:00:28.274
in the solution.
On the other hand,
1:00:28.272 --> 1:00:31.522
no matter how much you pulled
it and how you released it,
1:00:31.515 --> 1:00:34.005
this Ω,
and therefore the frequency,
1:00:34.005 --> 1:00:35.275
are not variable.
1:00:35.280 --> 1:00:37.090
Yes?
Student:
1:00:37.090 --> 1:00:39.400
[inaudible]
Professor Ramamurti
1:00:39.400 --> 1:00:42.200
Shankar: If you start it at
rest,
1:00:42.199 --> 1:00:45.419
indeed, it will be 0,
because the velocity at
1:00:45.418 --> 1:00:48.488
t = 0 is minus ΩA sin
φ.
1:00:48.489 --> 1:00:50.889
If you want to kill that,
the only way is φ = 0.
1:00:50.889 --> 1:00:53.639
You could put φ = π if
you like, but if you put 0,
1:00:53.642 --> 1:00:54.262
that'll do.
1:00:54.260 --> 1:00:58.950
1:00:58.950 --> 1:01:02.260
Okay.
So, this is the elementary
1:01:02.264 --> 1:01:04.154
parts of this thing.
1:01:04.150 --> 1:01:07.140
And I already told you the
following things.
1:01:07.139 --> 1:01:12.829
If x(t) is equal to
A cos Ωt--let's
1:01:12.828 --> 1:01:14.858
put φ = 0.
1:01:14.860 --> 1:01:17.250
In other words,
we'll agree that since there's
1:01:17.250 --> 1:01:19.690
only one oscillator,
it is perverse to set your
1:01:19.694 --> 1:01:22.784
clock to 0 at any time other
than when the oscillator is at
1:01:22.775 --> 1:01:24.045
one extreme point.
1:01:24.050 --> 1:01:27.070
If there are two oscillators
oscillating out of step,
1:01:27.065 --> 1:01:30.365
it's impossible to make
φ = 0 for both of them.
1:01:30.369 --> 1:01:32.779
Because if you can sync your
clock when one of them is at a
1:01:32.782 --> 1:01:35.322
maximum, that may not be when
the other one's at a maximum.
1:01:35.320 --> 1:01:38.390
But with one oscillator,
to choose φ other than
1:01:38.390 --> 1:01:40.150
0 is perverse.
It's like saying,
1:01:40.153 --> 1:01:41.813
"I'm doing a projectile
problem.
1:01:41.809 --> 1:01:46.029
I'll pick the place where I
launch the projectile to be x =
1:01:46.034 --> 1:01:48.044
y = 0."
You can pick some other crazy
1:01:48.039 --> 1:01:49.389
origin, but it doesn't help.
1:01:49.389 --> 1:01:52.379
So, we'll assume φ is 0
in this problem.
1:01:52.380 --> 1:01:55.780
If that is x,
what is velocity at time
1:01:55.780 --> 1:01:59.120
t?
Take the derivative of this,
1:01:59.117 --> 1:02:02.847
you get minus Ω A
sin Ωt.
1:02:02.849 --> 1:02:07.169
That means the velocity is also
oscillating sinusiodally but the
1:02:07.170 --> 1:02:11.010
amplitude for oscillation is
Ω times A.
1:02:11.010 --> 1:02:13.540
So, if A is the range of
variation for x,
1:02:13.543 --> 1:02:16.643
Ω times A is the
range of variation for velocity.
1:02:16.639 --> 1:02:19.609
Velocity will go all the way
from plus Ω A to minus
1:02:19.612 --> 1:02:21.492
Ω A.
The acceleration,
1:02:21.488 --> 1:02:24.688
which is one more derivative,
is minus Ω
1:02:24.690 --> 1:02:26.870
^(2)A,
cos Ωt.
1:02:26.869 --> 1:02:30.989
Which is really minus Ω
^(2) times x itself.
1:02:30.989 --> 1:02:35.459
So, the amplitude for
oscillation for the acceleration
1:02:35.455 --> 1:02:37.725
is Ω^(2) times A.
1:02:37.730 --> 1:02:40.880
So, I think you should
understand if Ω is very
1:02:40.875 --> 1:02:42.915
large;
then, the velocity will have
1:02:42.915 --> 1:02:45.935
very big excursions and the
acceleration will have even
1:02:45.935 --> 1:02:48.285
bigger excursions,
because the range of,
1:02:48.292 --> 1:02:50.942
this A really means if
you plot x,
1:02:50.938 --> 1:02:54.068
it's going to look like this,
going from A to minus
1:02:54.073 --> 1:02:56.293
A.
This says if you plot the
1:02:56.288 --> 1:02:59.068
velocity, it's going to look
like this.
1:02:59.070 --> 1:03:03.410
The range will go from Ω
A to minus Ω A.
1:03:03.409 --> 1:03:05.369
If you plot the acceleration,
it'll go from Ω
1:03:05.369 --> 1:03:07.139
^(2)A to minus Ω
^(2)A.
1:03:07.140 --> 1:03:11.400
1:03:11.400 --> 1:03:15.460
Last thing you want to verify
is the Law of Conservation of
1:03:15.460 --> 1:03:17.640
Energy.
I think I mentioned this to you
1:03:17.643 --> 1:03:20.053
when I studied energy,
but let me repeat one more
1:03:20.049 --> 1:03:21.539
time.
If you take,
1:03:21.544 --> 1:03:24.834
in this problem,
½ mv^(2) +½
1:03:24.826 --> 1:03:30.566
kx^(2)--I did that for
you--You can also verify this.
1:03:30.570 --> 1:03:34.670
Okay?
½ kx^(2) is going to
1:03:34.671 --> 1:03:39.491
involve A^(2) cosine
squared Ωt.
1:03:39.489 --> 1:03:41.839
½ mv^(2) is going to
involve something,
1:03:41.842 --> 1:03:43.992
something, something sin^(2)
Ωt.
1:03:43.989 --> 1:03:45.419
And by magic,
sin^(2) and
1:03:45.418 --> 1:03:47.628
cos^(2) will have the
same coefficient,
1:03:47.630 --> 1:03:50.650
so you can use the identity to
set them equal to 1,
1:03:50.645 --> 1:03:53.595
and get an answer that does not
depend on time.
1:03:53.600 --> 1:03:54.860
That's the beauty.
1:03:54.860 --> 1:03:57.500
Even though position and
velocity are constantly
1:03:57.502 --> 1:04:00.152
changing, this combination,
if you crank it out,
1:04:00.145 --> 1:04:01.715
will not depend on time.
1:04:01.720 --> 1:04:04.140
And what can it possibly be?
1:04:04.139 --> 1:04:06.979
If it doesn't depend on time,
I can calculate it whenever I
1:04:06.977 --> 1:04:09.227
like.
So, let me go to the instant
1:04:09.233 --> 1:04:13.333
when the mass has reached one
extremity and is about to swing
1:04:13.333 --> 1:04:15.653
back.
At that instant it has no
1:04:15.651 --> 1:04:17.651
velocity;
it only has an x,
1:04:17.646 --> 1:04:19.786
which is equal to amplitude,
or motion.
1:04:19.789 --> 1:04:23.219
So, that's the energy of an
oscillator, ½ kA^(2).
1:04:23.219 --> 1:04:26.409
When v = 0,
x will be plus or minus
1:04:26.408 --> 1:04:29.988
A, but any other
intermediate point you can find
1:04:29.994 --> 1:04:33.054
the velocity,
if you give me the x.
1:04:33.050 --> 1:04:37.860
We did that too.
So, this is all I want to say
1:04:37.855 --> 1:04:41.275
about the linear oscillator.
1:04:41.280 --> 1:04:45.090
But let me mention to you
another kind of oscillation,
1:04:45.088 --> 1:04:47.098
which is very interesting.
1:04:47.099 --> 1:04:53.829
Suppose you suspend a rod from
the ceiling like this.
1:04:53.830 --> 1:04:55.680
It's hanging there.
1:04:55.679 --> 1:05:00.049
If you give it a twist,
then it'll go twisting back and
1:05:00.051 --> 1:05:01.591
forth, like that.
1:05:01.590 --> 1:05:05.150
That's also a simple harmonic
motion, except what's varying
1:05:05.152 --> 1:05:07.672
with time is not the linear
coordinate,
1:05:07.670 --> 1:05:10.830
but the angle θ by
which the rod has been twisted.
1:05:10.829 --> 1:05:15.219
What's the equation in this
case?
1:05:15.219 --> 1:05:19.969
The equation here is I,
d 2θ over
1:05:19.972 --> 1:05:22.382
dt^(2).
You guys remember now,
1:05:22.378 --> 1:05:23.828
that's the analog of ma?
1:05:23.829 --> 1:05:25.799
That's going to be equal to the
torque.
1:05:25.800 --> 1:05:32.180
1:05:32.179 --> 1:05:35.349
Now, what'll happen in the
problems we consider is that
1:05:35.349 --> 1:05:39.239
θ is a small number,
and the torque will be such
1:05:39.238 --> 1:05:42.548
that it brings you back to
θ = 0.
1:05:42.550 --> 1:05:47.440
So, this number will be
approximately equal to minus
1:05:47.442 --> 1:05:51.282
some number κ times
θ.
1:05:51.280 --> 1:05:55.050
That's the analog of minus
kx.
1:05:55.050 --> 1:05:58.930
Little k was the
restoring force divided by the
1:05:58.928 --> 1:06:01.928
displacement,
which produced the restoring
1:06:01.928 --> 1:06:04.038
force.
Little κ is the
1:06:04.044 --> 1:06:07.844
restoring torque divided by the
angle at which you twisted it to
1:06:07.836 --> 1:06:12.126
get that torque.
So, this isn't always true.
1:06:12.130 --> 1:06:15.320
For small oscillations,
sin τ vanishes at
1:06:15.324 --> 1:06:17.694
θ = 0;
it will look like some number
1:06:17.688 --> 1:06:19.068
times θ.
If you like,
1:06:19.067 --> 1:06:21.897
it's the Taylor series for
τ for small angles.
1:06:21.900 --> 1:06:25.040
If you knew this κ you
are done, because
1:06:25.039 --> 1:06:27.909
mathematically,
this equation is the same as
1:06:27.911 --> 1:06:31.051
md 2 x over
dt^(2) is minus
1:06:31.051 --> 1:06:34.271
kx.
Because the Ω for this
1:06:34.267 --> 1:06:37.987
guy will be square root of
κ over I.
1:06:37.989 --> 1:06:40.679
You can steal the whole answer,
because mathematically,
1:06:40.682 --> 1:06:43.582
mathematicians don't care if
you're talking about θ
1:06:43.575 --> 1:06:45.565
or if you're talking about
x.
1:06:45.570 --> 1:06:49.150
Now, if you compare this
equation to this equation,
1:06:49.146 --> 1:06:52.936
md 2 x over
dt^(2) equal to minus
1:06:52.938 --> 1:06:55.538
kx,
if the Ω there was the
1:06:55.539 --> 1:06:58.629
root of k over m,
the Ω here would be
1:06:58.633 --> 1:07:00.503
root of κ over I.
1:07:00.500 --> 1:07:04.400
So, you may be thinking,
"Give me an example where
1:07:04.400 --> 1:07:08.380
τ looks like κ
times θ."
1:07:08.380 --> 1:07:09.320
I'm going to give you that
example.
1:07:09.320 --> 1:07:21.360
1:07:21.360 --> 1:07:25.060
So, that example is going to be
the following simple pendulum.
1:07:25.060 --> 1:07:31.760
1:07:31.760 --> 1:07:33.990
Take a pendulum like this.
1:07:33.990 --> 1:07:35.890
It's got a mass m.
1:07:35.890 --> 1:07:38.710
It's got some length l;
it's hanging from the ceiling.
1:07:38.710 --> 1:07:41.120
It's a rigid rod,
not a string,
1:07:41.115 --> 1:07:43.355
but a rigid,
massless rod.
1:07:43.360 --> 1:07:46.180
And the mass m is at the
bottom.
1:07:46.179 --> 1:07:50.769
So, it's happy to hang like
this, but suppose you give it a
1:07:50.770 --> 1:07:54.970
kick, so you push it over there
to an angle θ,
1:07:54.966 --> 1:07:57.416
which is whatever you like.
1:07:57.420 --> 1:08:04.390
What is the torque in this
particular case?
1:08:04.390 --> 1:08:07.170
Let's find out.
The force is mg,
1:08:07.165 --> 1:08:10.955
and the angle between the
force--and you're trying to do
1:08:10.955 --> 1:08:14.395
rotations around this
point--this angle here is the
1:08:14.401 --> 1:08:19.751
same θ.
So, torque is really minus
1:08:19.750 --> 1:08:23.720
mgl sin θ.
1:08:23.720 --> 1:08:27.410
That's the formula for torque.
1:08:27.409 --> 1:08:30.679
It's the force times the
distance of the lever arm or the
1:08:30.680 --> 1:08:34.420
distance over which the force is
from the rotation axis times the
1:08:34.418 --> 1:08:37.858
sine of the angle between the
direction of the force and the
1:08:37.864 --> 1:08:39.854
direction of the separation.
1:08:39.850 --> 1:08:44.460
So, the equation for the
pendulum really is the
1:08:44.456 --> 1:08:50.766
following: Id^(2) θ over
dt^(2) is equal to minus
1:08:50.765 --> 1:08:53.665
mgl sin θ.
1:08:53.670 --> 1:08:58.050
Now, we cannot do business with
this equation.
1:08:58.050 --> 1:09:00.650
This equation is a famous
equation, but you and I don't
1:09:00.647 --> 1:09:02.617
know how to solve it with what
we know.
1:09:02.619 --> 1:09:05.769
But we have learned,
even earlier today--look at the
1:09:05.771 --> 1:09:07.441
formula for sin x.
1:09:07.439 --> 1:09:10.269
Sin x begins with
x minus x^(3) over
1:09:10.267 --> 1:09:11.527
3 factorial, and so on.
1:09:11.529 --> 1:09:15.349
So, if θ is small,
then you may write
1:09:15.348 --> 1:09:18.298
mglθ.
So, for small angles,
1:09:18.303 --> 1:09:22.323
sin θ is
θ, and the restoring
1:09:22.317 --> 1:09:25.857
torque, in fact,
is linear in the angle of the
1:09:25.860 --> 1:09:28.920
coordinate.
So, by comparison now,
1:09:28.917 --> 1:09:32.427
we know that this fellow is our
κ.
1:09:32.430 --> 1:09:35.130
So, κ for the pendulum
is mgl.
1:09:35.130 --> 1:09:37.680
So, κ may not be
universal in every problem,
1:09:37.680 --> 1:09:40.030
in each problem you've got to
find κ.
1:09:40.029 --> 1:09:43.079
You find that by moving the
system off equilibrium,
1:09:43.079 --> 1:09:46.219
finding the restoring torque,
and if you're doing it right,
1:09:46.221 --> 1:09:48.381
restoring torque will always
look like something,
1:09:48.380 --> 1:09:50.350
something, something times
θ.
1:09:50.350 --> 1:09:53.110
Everything multiplying
θ is our κ.
1:09:53.109 --> 1:09:57.499
So, let's now calculate
Ω for the simple
1:09:57.502 --> 1:10:00.712
pendulum.
I've shown you this is
1:10:00.712 --> 1:10:06.172
κ over I,
and κ was mgl.
1:10:06.170 --> 1:10:07.720
And what's I?
1:10:07.720 --> 1:10:13.560
It's equal to moment of inertia
of this mass around this point.
1:10:13.560 --> 1:10:15.260
The rod is massless.
1:10:15.260 --> 1:10:17.590
For a single point mass,
we know the moment of inertia
1:10:17.589 --> 1:10:19.259
is just m times
l^(2).
1:10:19.260 --> 1:10:23.850
So, you find Ω is equal
to square root of g over
1:10:23.851 --> 1:10:27.481
l, and that is 2π
over T.
1:10:27.479 --> 1:10:30.659
So, you get back the famous
formula you learned,
1:10:30.663 --> 1:10:34.933
T is 2π square
root of l over g.
1:10:34.930 --> 1:10:40.810
This is the origin of the
formula you learned in school.
1:10:40.810 --> 1:10:43.970
It comes from small
oscillations of a pendulum.
1:10:43.970 --> 1:10:47.170
So, think about this problem.
1:10:47.170 --> 1:10:50.580
If you pulled a pendulum by an
angle so large that sin
1:10:50.579 --> 1:10:53.479
θ is no longer
approximated by θ but
1:10:53.478 --> 1:10:56.998
you need a θ^(3) over 6
term -- it can happen for large
1:10:57.001 --> 1:11:00.721
enough θ – then,
the equation to solve is no
1:11:00.723 --> 1:11:03.323
longer this.
What that means is,
1:11:03.320 --> 1:11:08.540
if you took a real pendulum
with a massless rod and a bob at
1:11:08.537 --> 1:11:11.357
the end,
the time it takes to finish an
1:11:11.361 --> 1:11:14.571
oscillation, in fact,
will depend on the amplitude.
1:11:14.569 --> 1:11:16.739
But only for small
oscillations,
1:11:16.735 --> 1:11:20.785
when sin θ can
be approximated by θ,
1:11:20.787 --> 1:11:24.907
you will find whether you do
that the time is the same.
1:11:24.909 --> 1:11:27.559
But if you go too far,
the other terms in sin
1:11:27.561 --> 1:11:30.161
θ will kick in,
and the result is no longer
1:11:30.162 --> 1:11:31.782
true.
The period of a pendulum,
1:11:31.776 --> 1:11:34.346
in fact, is not just a function
of l and g,
1:11:34.345 --> 1:11:36.175
but also a function of the
amplitude.
1:11:36.180 --> 1:11:40.080
Only for small amplitudes,
you get this marvelous result.
1:11:40.079 --> 1:11:42.759
There's one problem I want you
to think about.
1:11:42.760 --> 1:11:44.690
I may do it myself.
1:11:44.689 --> 1:11:48.089
I think you should try to read
up on it, and I will try to
1:11:48.088 --> 1:11:49.458
explain it next time.
1:11:49.460 --> 1:11:55.900
1:11:55.899 --> 1:11:58.279
I'll give the problem to you
and I will do it for you next
1:11:58.277 --> 1:12:00.017
time.
I want you to think about it.
1:12:00.020 --> 1:12:03.460
Here is a hoop,
like a hula hoop.
1:12:03.460 --> 1:12:06.140
It's hanging on the wall with a
nail.
1:12:06.140 --> 1:12:08.600
And it's very happy to be where
it is.
1:12:08.600 --> 1:12:10.770
Now, I come and give it a
little push.
1:12:10.770 --> 1:12:12.510
It'll start oscillating.
1:12:12.510 --> 1:12:15.230
It'll look like that,
and some time later it'll look
1:12:15.229 --> 1:12:17.309
like that, it's going back and
forth.
1:12:17.310 --> 1:12:21.030
Your job is to find the
Ω for that hoop.
1:12:21.030 --> 1:12:25.730
1:12:25.729 --> 1:12:28.759
So, you move it a little bit,
and you find the restoring
1:12:28.763 --> 1:12:30.543
torque.
And you're done.
1:12:30.539 --> 1:12:33.719
But you've got to do two things
right to get the right answer.
1:12:33.720 --> 1:12:36.670
You've got to find the moment
of inertia for circular loop not
1:12:36.667 --> 1:12:38.887
around the center,
but around the point and the
1:12:38.890 --> 1:12:41.000
circumference.
So, you guys know the magic
1:12:41.000 --> 1:12:42.150
words you've got to use.
1:12:42.150 --> 1:12:43.750
Parallel Axis theorem.
1:12:43.750 --> 1:12:46.470
And to find the restoring
torque, here is the part that
1:12:46.470 --> 1:12:47.730
blows everybody's mind.
1:12:47.729 --> 1:12:51.509
When the loop goes to that
position, where is gravity
1:12:51.514 --> 1:12:54.154
acting?
Gravity's acting at the center
1:12:54.154 --> 1:12:56.554
of mass.
The center of mass is somewhere
1:12:56.554 --> 1:13:00.114
here in the middle of the loop,
even though there is no matter
1:13:00.106 --> 1:13:03.386
there.
So, center of mass of a body
1:13:03.388 --> 1:13:05.948
need not lie on the body.
1:13:05.950 --> 1:13:07.890
The ring is a perfect example.
1:13:07.890 --> 1:13:11.080
So, if you remember the center
of mass of a loop is in the
1:13:11.078 --> 1:13:13.818
center of the loop and put the
mg there,
1:13:13.819 --> 1:13:16.289
you will get the right torque,
and everything will just
1:13:16.288 --> 1:13:20.268
follow.
By the way, my offer to you is
1:13:20.272 --> 1:13:23.642
always open.
If there's something about the
1:13:23.640 --> 1:13:26.310
class you want to change,
you find it's hard,
1:13:26.311 --> 1:13:28.861
this, that, you can always
write to me.
1:13:28.859 --> 1:13:31.929
I want your feedback,
because I just don't know which
1:13:31.929 --> 1:13:34.049
parts are easy,
which parts are hard,
1:13:34.054 --> 1:13:36.124
which are slow,
which are fast.
1:13:36.119 --> 1:13:39.469
But if you haven't had enough
math and you need some help,
1:13:39.471 --> 1:13:43.001
send me some e-mail and I will
try my best to answer them.