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YEN-JIE LEE: So
welcome, everybody.
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00:00:26,370 --> 00:00:27,740
My name is Yen-Jie Lee.
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00:00:27,740 --> 00:00:32,210
I am a assistant professor
of physics in the physics
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00:00:32,210 --> 00:00:36,500
department, and I will
be your instructor
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00:00:36,500 --> 00:00:39,860
of this semester on 8.03.
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00:00:39,860 --> 00:00:44,930
So of course, one first
question you have is,
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00:00:44,930 --> 00:00:50,240
why do we want to learn
about vibrations and waves?
15
00:00:50,240 --> 00:00:52,460
Why do we learn about this?
16
00:00:52,460 --> 00:00:55,550
Why do we even care?
17
00:00:55,550 --> 00:00:58,350
The answer is really, simple.
18
00:00:58,350 --> 00:01:00,920
If you look at
this slide, you can
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00:01:00,920 --> 00:01:05,720
see that the reason you
can follow this class
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00:01:05,720 --> 00:01:11,270
is because I'm producing sound
wave by oscillating the air,
21
00:01:11,270 --> 00:01:14,180
and you can receive
those sound waves.
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00:01:14,180 --> 00:01:17,120
And you can see me--
23
00:01:17,120 --> 00:01:19,790
that's really pretty
amazing by itself--
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00:01:19,790 --> 00:01:22,880
because there are
a lot of photons
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00:01:22,880 --> 00:01:25,580
or electromagnetic waves.
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00:01:25,580 --> 00:01:27,700
They are bouncing
around in this room,
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00:01:27,700 --> 00:01:34,790
and your eye actually receive
those electromagnetic waves.
28
00:01:34,790 --> 00:01:37,430
And that translates
into your brain waves.
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00:01:37,430 --> 00:01:41,075
You obviously, start to think
about what this instructor is
30
00:01:41,075 --> 00:01:43,400
trying to tell you.
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00:01:43,400 --> 00:01:47,030
And of course, all those
things we learned from 8.03
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00:01:47,030 --> 00:01:50,870
is closely connected
to probability density
33
00:01:50,870 --> 00:01:57,500
waves, which you will learn
from 8.04, quantum physics.
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00:01:57,500 --> 00:01:59,660
And finally, it's
also, of course,
35
00:01:59,660 --> 00:02:05,030
related to a recent discovery
of the gravitational waves.
36
00:02:05,030 --> 00:02:08,050
When we are sitting
here, maybe there
37
00:02:08,050 --> 00:02:12,800
are already some space-time
distortion already passing
38
00:02:12,800 --> 00:02:16,370
through our body and
you don't feel it.
39
00:02:16,370 --> 00:02:18,550
When I'm moving
around like this,
40
00:02:18,550 --> 00:02:22,370
I am creating also the
gravitational waves,
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00:02:22,370 --> 00:02:25,430
but it's so small
to be detected.
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00:02:25,430 --> 00:02:26,960
So that's actually really cool.
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00:02:26,960 --> 00:02:32,570
So the take-home message is
that we cannot even recognize
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00:02:32,570 --> 00:02:37,310
the universe without using
waves and the vibrations.
45
00:02:37,310 --> 00:02:40,810
So that's actually why we
care about this subject.
46
00:02:40,810 --> 00:02:42,575
And the last is actually
why this subject
47
00:02:42,575 --> 00:02:49,170
is so cool even without quantum,
without any fancy names.
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00:02:49,170 --> 00:02:55,420
So what is actually the
relation of 8.03 to other class
49
00:02:55,420 --> 00:02:57,280
or other field of studies?
50
00:02:57,280 --> 00:03:00,410
It's closely related to
classical mechanics, which
51
00:03:00,410 --> 00:03:03,050
I will use it
immediately, and I hope
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00:03:03,050 --> 00:03:05,060
you will still
remember what you have
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00:03:05,060 --> 00:03:07,480
learned from 8.01 and 8.02.
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00:03:07,480 --> 00:03:13,310
Electromagnetic force is
actually closely related also,
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00:03:13,310 --> 00:03:16,640
and we are going to use
a technique we learned
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00:03:16,640 --> 00:03:21,270
from this class to understand
optics, quantum mechanics,
57
00:03:21,270 --> 00:03:24,800
and also there are many
practical applications, which
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00:03:24,800 --> 00:03:29,200
you will learn from this class.
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00:03:29,200 --> 00:03:31,390
This is the concrete goal.
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00:03:31,390 --> 00:03:36,100
We care about the future
of our space time.
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00:03:36,100 --> 00:03:40,450
We would like to predict
what is going to happen
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00:03:40,450 --> 00:03:42,790
when we set up an experiment.
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00:03:42,790 --> 00:03:47,290
We would like to design
experiments which can improve
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our understanding of nature.
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00:03:50,110 --> 00:03:53,850
But without using the
most powerful tool
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00:03:53,850 --> 00:03:58,030
is very, very difficult
to make progress.
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00:03:58,030 --> 00:04:03,900
So the most powerful tool
we have is mathematics.
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00:04:03,900 --> 00:04:07,840
You will see that it
really works in this class.
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00:04:07,840 --> 00:04:09,610
But the first thing
we have to learn
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00:04:09,610 --> 00:04:15,610
is how to translate physical
situations into mathematics
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00:04:15,610 --> 00:04:19,959
so that we can actually include
this really wonderful tool
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00:04:19,959 --> 00:04:22,960
to help us to solve problems.
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00:04:22,960 --> 00:04:25,180
Once we have done
that, we will start
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00:04:25,180 --> 00:04:29,180
to look at single
harmonic oscillator,
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00:04:29,180 --> 00:04:32,590
then we try to couple all
those oscillators together
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00:04:32,590 --> 00:04:36,590
to see how they interact
with each other.
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00:04:36,590 --> 00:04:41,860
Finally, we go to an infinite
number of oscillators.
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00:04:41,860 --> 00:04:46,080
Sounds scary, but it's
actually not scary after all.
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00:04:46,080 --> 00:04:49,810
And we will see waves
because waves are actually
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00:04:49,810 --> 00:04:52,990
coming from an infinite number
of oscillating particles,
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00:04:52,990 --> 00:04:55,360
if you think about it.
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00:04:55,360 --> 00:04:57,880
Then we would do Fourier
decomposition of waves
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00:04:57,880 --> 00:04:59,890
to see what we can
learn about it.
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00:04:59,890 --> 00:05:04,390
We learn how to put
together physical systems.
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00:05:04,390 --> 00:05:08,800
That brings us to the issue
of boundary conditions,
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00:05:08,800 --> 00:05:12,520
and we will also enjoy
what we have learned
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00:05:12,520 --> 00:05:14,620
by looking at the
phenomenon related
88
00:05:14,620 --> 00:05:18,610
to electromagnetic waves
and practical application
89
00:05:18,610 --> 00:05:21,530
and optics.
90
00:05:21,530 --> 00:05:22,240
Any questions?
91
00:05:22,240 --> 00:05:26,080
If you have any questions,
please stop me any time.
92
00:05:26,080 --> 00:05:29,040
So if you don't stop me, I'm
going to continue talking.
93
00:05:31,600 --> 00:05:33,950
So that gets started.
94
00:05:33,950 --> 00:05:37,090
So the first example,
the concrete example
95
00:05:37,090 --> 00:05:43,570
I'm going to talk about is a
spring block, a massive block
96
00:05:43,570 --> 00:05:45,520
system.
97
00:05:45,520 --> 00:05:50,150
So this is actually what
I have on that table.
98
00:05:50,150 --> 00:05:54,740
So basically, I have a
highly-idealized spring.
99
00:05:54,740 --> 00:05:58,480
This is ideal spring
with spring constant, k,
100
00:05:58,480 --> 00:06:01,480
and the natural length L0.
101
00:06:01,480 --> 00:06:04,150
So that is actually what I have.
102
00:06:04,150 --> 00:06:13,830
And at t equal to 0, what I am
going to do is I am going to--
103
00:06:13,830 --> 00:06:16,480
I should remove this
mass a little bit,
104
00:06:16,480 --> 00:06:22,900
and I hold this mass still and
release that really carefully.
105
00:06:22,900 --> 00:06:24,910
So that is actually
the experiment,
106
00:06:24,910 --> 00:06:27,070
which I am going to do.
107
00:06:27,070 --> 00:06:33,390
And we were wondering what
is going to happen afterward.
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00:06:33,390 --> 00:06:36,940
Well, the mass as you
move, will it stay there
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00:06:36,940 --> 00:06:40,790
or it just disappear,
I don't know
110
00:06:40,790 --> 00:06:42,220
before I solved this question.
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00:06:44,890 --> 00:06:49,690
Now I have put together a
concrete question to you,
112
00:06:49,690 --> 00:06:52,900
but I don't know how
to proceed because you
113
00:06:52,900 --> 00:06:54,040
say everything works.
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00:06:54,040 --> 00:06:55,190
What I am going to do?
115
00:06:55,190 --> 00:06:58,540
I mean, I don't know.
116
00:06:58,540 --> 00:07:03,160
So as I mentioned before, there
is a pretty powerful tool,
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00:07:03,160 --> 00:07:04,780
mathematics.
118
00:07:04,780 --> 00:07:08,140
So I'm going to use
that, even though I don't
119
00:07:08,140 --> 00:07:11,587
know why mathematics can work.
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00:07:11,587 --> 00:07:12,670
Have you thought about it?
121
00:07:15,340 --> 00:07:19,180
So let's try it and see
how we can make progress.
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00:07:19,180 --> 00:07:25,540
So the first thing which you
can do in order to make progress
123
00:07:25,540 --> 00:07:29,170
is to define a
coordinate system.
124
00:07:29,170 --> 00:07:32,950
So here I define a
coordinate system, which
125
00:07:32,950 --> 00:07:34,630
is in the horizontal direction.
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00:07:34,630 --> 00:07:36,580
It's the x direction.
127
00:07:36,580 --> 00:07:39,640
And the x equal
to 0, the origin,
128
00:07:39,640 --> 00:07:44,800
is the place which the
spring is not stressed,
129
00:07:44,800 --> 00:07:47,330
is at its natural length.
130
00:07:47,330 --> 00:07:51,490
That is actually what I
define as x equal to 0.
131
00:07:51,490 --> 00:07:56,020
And once I define
this, I can now
132
00:07:56,020 --> 00:08:01,720
express what is actually the
initial position of the mass
133
00:08:01,720 --> 00:08:03,990
by these coordinates is x0.
134
00:08:03,990 --> 00:08:08,020
It can be expressed
as x initial.
135
00:08:08,020 --> 00:08:11,830
And also, initially, I said
that this mass is not moving.
136
00:08:11,830 --> 00:08:16,810
Therefore, the
velocity at 0 is 0.
137
00:08:16,810 --> 00:08:20,000
So now I can also formulate
my question really concretely
138
00:08:20,000 --> 00:08:21,460
with some mathematics.
139
00:08:21,460 --> 00:08:25,330
Basically, you can see
that at time equal to t,
140
00:08:25,330 --> 00:08:28,690
I was wondering
where is this mass.
141
00:08:28,690 --> 00:08:31,060
So actually, the question
is, what is actually
142
00:08:31,060 --> 00:08:34,210
x as a function of t?
143
00:08:34,210 --> 00:08:38,150
So you can see that once I have
the mathematics to help me,
144
00:08:38,150 --> 00:08:41,490
everything becomes
pretty simple.
145
00:08:41,490 --> 00:08:44,770
So once I have those
defined, I would
146
00:08:44,770 --> 00:08:48,460
like to predict what is going
to happen at time equal to t.
147
00:08:48,460 --> 00:08:54,370
Therefore, I would like to make
use physical laws to actually
148
00:08:54,370 --> 00:08:56,540
help me to solve this problem.
149
00:08:56,540 --> 00:09:01,200
So apparently what we are
going to use is Newton's law.
150
00:09:01,200 --> 00:09:05,700
And I am going to go
through this example
151
00:09:05,700 --> 00:09:10,100
really slowly so that
everybody is on the same page.
152
00:09:10,100 --> 00:09:12,300
So the first thing
which I usually do
153
00:09:12,300 --> 00:09:18,610
is now I would like to do
a force diagram analysis.
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00:09:18,610 --> 00:09:19,930
So I have this mass.
155
00:09:19,930 --> 00:09:25,080
This setup is on Earth,
and the question is,
156
00:09:25,080 --> 00:09:28,240
how many forces are
acting on this mass?
157
00:09:28,240 --> 00:09:32,185
Can anybody answer my question.
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00:09:32,185 --> 00:09:33,151
AUDIENCE: Two.
159
00:09:33,151 --> 00:09:35,566
We got the--
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00:09:35,566 --> 00:09:38,685
So acceleration and
the spring force.
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00:09:38,685 --> 00:09:41,150
YEN-JIE LEE: OK, so
your answer is two.
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00:09:41,150 --> 00:09:43,210
Any different?
163
00:09:43,210 --> 00:09:44,850
Three.
164
00:09:44,850 --> 00:09:45,450
Very good.
165
00:09:45,450 --> 00:09:48,120
So we have two and three.
166
00:09:48,120 --> 00:09:52,750
And the answer
actually is three.
167
00:09:52,750 --> 00:09:54,260
So look at this scene.
168
00:09:54,260 --> 00:09:57,120
I am drawing in and
I have product here.
169
00:09:57,120 --> 00:10:00,105
So this is actually the most
difficult part of the question,
170
00:10:00,105 --> 00:10:01,220
actually.
171
00:10:01,220 --> 00:10:05,140
So once you pass this step,
everything is straightforward.
172
00:10:05,140 --> 00:10:06,660
It's just mathematics.
173
00:10:06,660 --> 00:10:09,150
It's not my problem any more,
but the math department,
174
00:10:09,150 --> 00:10:12,300
they have problem, OK?
175
00:10:12,300 --> 00:10:13,320
All right.
176
00:10:13,320 --> 00:10:16,952
So now let's look at this mass.
177
00:10:16,952 --> 00:10:17,910
There are three forces.
178
00:10:17,910 --> 00:10:21,915
The first one as you mentioned
correctly is F spring.
179
00:10:21,915 --> 00:10:26,190
It's pulling the mass.
180
00:10:26,190 --> 00:10:31,020
And since we are
working on Earth,
181
00:10:31,020 --> 00:10:33,240
we have not yet
moved the whole class
182
00:10:33,240 --> 00:10:36,810
to the moon or somewhere
else, but there
183
00:10:36,810 --> 00:10:39,945
would be gravitational
force pointing downward.
184
00:10:42,720 --> 00:10:49,440
But this whole setup is on a
table of friction, this table.
185
00:10:49,440 --> 00:10:51,555
Therefore, there will
be no more force.
186
00:10:54,708 --> 00:10:56,369
So don't forget this one.
187
00:10:56,369 --> 00:10:57,535
There will be no more force.
188
00:11:00,080 --> 00:11:02,995
So the answer is that
we have three forces.
189
00:11:05,740 --> 00:11:08,265
The normal force is, actually,
a complicated subject,
190
00:11:08,265 --> 00:11:09,640
which you will
need to understand
191
00:11:09,640 --> 00:11:11,440
that will quantum physics.
192
00:11:15,160 --> 00:11:18,790
So now I have three force,
and now I can actually
193
00:11:18,790 --> 00:11:24,670
calculate the total
force, the total force,
194
00:11:24,670 --> 00:11:35,510
F. F is equal to
Fs plus Fn plus Fg.
195
00:11:38,830 --> 00:11:43,000
So since we know
that the mass is
196
00:11:43,000 --> 00:11:45,310
moving in the
horizontal direction,
197
00:11:45,310 --> 00:11:49,900
the mass didn't suddenly
jump and disappear.
198
00:11:49,900 --> 00:11:51,760
So it is there.
199
00:11:51,760 --> 00:11:59,020
Therefore, we know that the
normal force is actually
200
00:11:59,020 --> 00:12:07,690
equal to minus Fg, which is
actually Ng in the y direction.
201
00:12:07,690 --> 00:12:13,480
And here I define y is
actually pointing up,
202
00:12:13,480 --> 00:12:16,660
and the x is pointing
to the right-hand side.
203
00:12:16,660 --> 00:12:18,760
Therefore, what
is going to happen
204
00:12:18,760 --> 00:12:22,730
is that the total force
is actually just Fs.
205
00:12:27,740 --> 00:12:31,660
And this is equal
to minus k, which
206
00:12:31,660 --> 00:12:35,290
is the spring
constant and x, which
207
00:12:35,290 --> 00:12:43,230
is the position of the little
mass at time equal to t.
208
00:12:46,350 --> 00:12:55,670
So once we have those
forces and the total force,
209
00:12:55,670 --> 00:12:57,770
actually, we can
use Newton's law.
210
00:13:03,674 --> 00:13:09,240
So F is equal to m times a.
211
00:13:09,240 --> 00:13:16,760
And this is actually equal
to m d squared xt dt squared
212
00:13:16,760 --> 00:13:20,060
in the x direction,
and that is actually
213
00:13:20,060 --> 00:13:26,880
equal to mx double dot t x.
214
00:13:26,880 --> 00:13:28,190
So here is my notation.
215
00:13:28,190 --> 00:13:32,810
I'm going to use each of the dot
is actually the differentiation
216
00:13:32,810 --> 00:13:34,630
with respect to t.
217
00:13:37,760 --> 00:13:45,600
So this is actually equal to
minus kxt in the x direction.
218
00:13:45,600 --> 00:13:49,730
So you can see that here is
actually what you already
219
00:13:49,730 --> 00:13:51,330
know about Newton's law.
220
00:13:51,330 --> 00:13:55,250
And that is actually coming
from the force analysis.
221
00:13:55,250 --> 00:13:58,550
So in this example,
it's simple enough such
222
00:13:58,550 --> 00:14:00,830
that you can write
it down immediately,
223
00:14:00,830 --> 00:14:03,110
but in the later
examples, things
224
00:14:03,110 --> 00:14:06,740
will become very complicated
and things will be slightly more
225
00:14:06,740 --> 00:14:08,960
difficult. Therefore,
you will really need help
226
00:14:08,960 --> 00:14:12,080
from the force diagram.
227
00:14:12,080 --> 00:14:15,320
So now we have everything
in the x direction,
228
00:14:15,320 --> 00:14:18,140
therefore, I can drop the x hat.
229
00:14:18,140 --> 00:14:25,010
Therefore, finally, my equation
of motion is x double dot t.
230
00:14:25,010 --> 00:14:32,720
And this is equal to
minus k over n x of t.
231
00:14:32,720 --> 00:14:37,280
To make my life
easier, I am going
232
00:14:37,280 --> 00:14:43,460
to define omega equal to
square root of k over n.
233
00:14:43,460 --> 00:14:45,390
You will see why afterward.
234
00:14:45,390 --> 00:14:48,810
It looks really weird why
professor Lee wants to do this,
235
00:14:48,810 --> 00:14:53,680
but afterward, you will see that
omega really have a meaning,
236
00:14:53,680 --> 00:14:58,000
and that is equal to
minus omega squared x.
237
00:15:00,710 --> 00:15:06,110
So we have solved this problem,
actually, as a physicist.
238
00:15:06,110 --> 00:15:08,420
Now the problem is
what is actually
239
00:15:08,420 --> 00:15:11,180
the solution to
this differential
240
00:15:11,180 --> 00:15:13,440
second-order
differential equation.
241
00:15:13,440 --> 00:15:15,570
And as I mentioned,
this is actually
242
00:15:15,570 --> 00:15:17,530
not the content
of 8.03, actually,
243
00:15:17,530 --> 00:15:20,930
it's a content of 18.03, maybe.
244
00:15:20,930 --> 00:15:24,650
How many of you actually
have taken 18.03?
245
00:15:24,650 --> 00:15:27,380
Everybody knows the
solution, so very good.
246
00:15:27,380 --> 00:15:30,475
I am safe.
247
00:15:30,475 --> 00:15:31,475
So what is the solution?
248
00:15:35,730 --> 00:15:46,940
The solution is x of t equal
to a cosine of omega t plus b
249
00:15:46,940 --> 00:15:49,820
sine omega t.
250
00:15:49,820 --> 00:15:53,120
So my friends from
the math department
251
00:15:53,120 --> 00:15:58,160
tell me secretly that this
is actually the solution.
252
00:15:58,160 --> 00:16:00,470
And I trust him or her.
253
00:16:04,340 --> 00:16:07,130
So that's very nice.
254
00:16:07,130 --> 00:16:09,950
Now I have the
solution, and how do I
255
00:16:09,950 --> 00:16:12,450
know this is the only solution?
256
00:16:12,450 --> 00:16:14,850
How do I know?
257
00:16:14,850 --> 00:16:17,330
Actually, there
are two unknowns,
258
00:16:17,330 --> 00:16:20,964
just to remind you
what you have learned.
259
00:16:20,964 --> 00:16:23,810
There are two unknowns.
260
00:16:23,810 --> 00:16:29,860
And if you plug this
thing into this equation,
261
00:16:29,860 --> 00:16:32,000
you satisfy that equation.
262
00:16:32,000 --> 00:16:34,880
If you don't trust me,
you can do it offline.
263
00:16:34,880 --> 00:16:37,610
It's always good
to check to make
264
00:16:37,610 --> 00:16:39,290
sure I didn't make a mistake.
265
00:16:39,290 --> 00:16:40,980
But that's very good news.
266
00:16:40,980 --> 00:16:46,250
So that means we will
have two unknowns,
267
00:16:46,250 --> 00:16:49,760
and those will
satisfy the equation.
268
00:16:49,760 --> 00:16:54,900
So by uniqueness
theorem, this is actually
269
00:16:54,900 --> 00:17:01,080
the one and the only one
solution in my universe,
270
00:17:01,080 --> 00:17:07,800
also yours, which satisfy
the equation because
271
00:17:07,800 --> 00:17:09,700
of the uniqueness theorem.
272
00:17:09,700 --> 00:17:12,690
So I hope I have
convinced you that we
273
00:17:12,690 --> 00:17:16,810
have solved this equation.
274
00:17:16,810 --> 00:17:22,349
So now I take my
physicist hat back and now
275
00:17:22,349 --> 00:17:24,400
it is actually my job again.
276
00:17:24,400 --> 00:17:27,030
So now we have the
solution, and we
277
00:17:27,030 --> 00:17:29,290
need to determine
what is actually
278
00:17:29,290 --> 00:17:32,440
these two unknown coefficients.
279
00:17:32,440 --> 00:17:37,770
So what I'm going to use is to
use the two initial conditions.
280
00:17:44,840 --> 00:17:49,180
The first initial condition
is x of 0 equal to x initial.
281
00:17:52,770 --> 00:17:57,540
The second one is that since
I released this mass really
282
00:17:57,540 --> 00:18:03,180
carefully and the initial
velocity is 0, therefore,
283
00:18:03,180 --> 00:18:09,160
I have x dot 0 equal to 0.
284
00:18:09,160 --> 00:18:11,610
From this, you can solve.
285
00:18:11,610 --> 00:18:16,000
Plug these two conditions
into this equation.
286
00:18:16,000 --> 00:18:20,370
You can actually figure out
that a is equal to x initial.
287
00:18:26,400 --> 00:18:29,690
And b is equal to 0.
288
00:18:33,000 --> 00:18:34,110
Any questions so far?
289
00:18:39,830 --> 00:18:41,460
Very good.
290
00:18:41,460 --> 00:18:43,530
So now we have the solution.
291
00:18:43,530 --> 00:18:46,640
So finally, what is
actually the solution?
292
00:18:46,640 --> 00:19:03,660
The solution we get is x of t
equal to x initial cosine omega
293
00:19:03,660 --> 00:19:04,160
t.
294
00:19:08,120 --> 00:19:12,620
So this is actually the
amplitude of the oscillation,
295
00:19:12,620 --> 00:19:17,240
and this is actually
the angular velocity.
296
00:19:17,240 --> 00:19:20,420
So you may be
asking why angular?
297
00:19:20,420 --> 00:19:22,970
Where is the
angular coming from?
298
00:19:22,970 --> 00:19:25,910
Because this is actually
a one-dimensional motion.
299
00:19:25,910 --> 00:19:28,490
Where is the angular
velocity coming from?
300
00:19:28,490 --> 00:19:34,250
And I will explain that
in the later lecture.
301
00:19:34,250 --> 00:19:38,220
And also this is actually
a harmonic oscillation.
302
00:19:38,220 --> 00:19:40,700
So what we are
actually predicting
303
00:19:40,700 --> 00:19:46,490
is that this mass is going
to do this, have a fixed
304
00:19:46,490 --> 00:19:48,830
amplitude and it's
actually going
305
00:19:48,830 --> 00:19:54,920
to go back and forth with the
angular frequency of omega.
306
00:19:54,920 --> 00:20:00,230
So we can now do an experiment
to verify if this is actually
307
00:20:00,230 --> 00:20:01,940
really the case.
308
00:20:01,940 --> 00:20:04,400
So there's a small difference.
309
00:20:04,400 --> 00:20:08,870
There's another spring here,
but essentially, the solution
310
00:20:08,870 --> 00:20:10,290
will be very similar.
311
00:20:10,290 --> 00:20:15,860
You may get this
in a p-set or exam.
312
00:20:15,860 --> 00:20:20,670
So now I can turn on the air
so that I make this surface
313
00:20:20,670 --> 00:20:23,880
frictionless.
314
00:20:23,880 --> 00:20:29,120
And you can see that now
I actually move this thing
315
00:20:29,120 --> 00:20:33,390
slightly away from the
equilibrium position,
316
00:20:33,390 --> 00:20:36,410
and I release that carefully.
317
00:20:36,410 --> 00:20:42,030
So you can see that really it's
actually going back and forth
318
00:20:42,030 --> 00:20:43,080
harmonically.
319
00:20:46,290 --> 00:20:50,880
I can change the amplitude
and see what will happen.
320
00:20:50,880 --> 00:20:53,540
The amplitude is
becoming bigger,
321
00:20:53,540 --> 00:20:59,120
and you can see that the
oscillation amplitude really
322
00:20:59,120 --> 00:21:03,365
depends on where you put
that initially with respect
323
00:21:03,365 --> 00:21:04,750
to the equilibrium position.
324
00:21:04,750 --> 00:21:08,330
I can actually make a small
amplitude oscillation also.
325
00:21:08,330 --> 00:21:11,570
Now you can see that now the
amplitude is small but still
326
00:21:11,570 --> 00:21:14,660
oscillating back and forth.
327
00:21:14,660 --> 00:21:16,401
So that's very encouraging.
328
00:21:20,730 --> 00:21:25,340
Let's take another example,
which I actually rotate
329
00:21:25,340 --> 00:21:28,520
the whole thing by 90 degrees.
330
00:21:28,520 --> 00:21:32,150
You are going to get a
question about this system
331
00:21:32,150 --> 00:21:33,510
in your p-set.
332
00:21:33,510 --> 00:21:38,500
The amazing thing is that
the solution is the same.
333
00:21:38,500 --> 00:21:41,330
What is that?
334
00:21:41,330 --> 00:21:45,470
And you don't believe me,
let me do the experiment.
335
00:21:45,470 --> 00:21:48,680
I actually shifted the position.
336
00:21:48,680 --> 00:21:53,220
I changed the position, and I
release that really carefully.
337
00:21:53,220 --> 00:21:56,460
You see that this mass is
oscillating up and down.
338
00:21:56,460 --> 00:22:00,410
The amplitude did not change.
339
00:22:00,410 --> 00:22:03,890
The frequency did not change
as a function of time.
340
00:22:03,890 --> 00:22:08,960
It really matched with the
solution we found here.
341
00:22:11,690 --> 00:22:14,060
It's truly amazing.
342
00:22:14,060 --> 00:22:16,040
No?
343
00:22:16,040 --> 00:22:19,940
The problem is that we are
so used to this already.
344
00:22:19,940 --> 00:22:25,250
You have seen this maybe
100 times before my lecture,
345
00:22:25,250 --> 00:22:28,040
so therefore, you
got so used to this.
346
00:22:28,040 --> 00:22:30,680
Therefore, when I say,
OK, I make a prediction.
347
00:22:30,680 --> 00:22:34,120
This is what happened, you
are just so used to this
348
00:22:34,120 --> 00:22:36,740
or you don't feel
the excitement.
349
00:22:36,740 --> 00:22:41,510
But for me, after I teach
this class so many times,
350
00:22:41,510 --> 00:22:46,030
I still find this
thing really amazing.
351
00:22:46,030 --> 00:22:48,110
Why is that?
352
00:22:48,110 --> 00:22:53,120
This means that
actually, mathematics
353
00:22:53,120 --> 00:22:56,520
really works, first of all.
354
00:22:56,520 --> 00:23:03,560
That means we can use the same
tool for the understanding
355
00:23:03,560 --> 00:23:11,300
of gravitational waves, for the
prediction of the Higgs boson,
356
00:23:11,300 --> 00:23:13,920
for the calculation
of the property
357
00:23:13,920 --> 00:23:17,810
of the quark-gluon plasma
in the early universe,
358
00:23:17,810 --> 00:23:22,280
and also at the
same time the motion
359
00:23:22,280 --> 00:23:25,400
of this spring-mass system.
360
00:23:25,400 --> 00:23:30,350
We actually use always the
same tool, the mathematics,
361
00:23:30,350 --> 00:23:32,840
to understand this system.
362
00:23:32,840 --> 00:23:35,360
And nobody will understands why.
363
00:23:35,360 --> 00:23:37,845
If you understand
why, please tell me.
364
00:23:37,845 --> 00:23:38,720
I would like to know.
365
00:23:38,720 --> 00:23:39,886
I will be very proud of you.
366
00:23:43,780 --> 00:23:49,400
Rene Descartes said
once, "But in my opinion,
367
00:23:49,400 --> 00:23:54,190
all things in nature
occur mathematically."
368
00:23:54,190 --> 00:23:55,340
Apparently, he's right.
369
00:23:58,570 --> 00:24:03,670
Albert Einstein also once said,
"The most incomprehensible
370
00:24:03,670 --> 00:24:08,810
thing about the universe is
that it is comprehensible."
371
00:24:08,810 --> 00:24:13,870
So I would say this
is really something
372
00:24:13,870 --> 00:24:17,470
we need to appreciate
the need to think
373
00:24:17,470 --> 00:24:21,250
about why this is the case.
374
00:24:21,250 --> 00:24:22,325
Any questions?
375
00:24:26,790 --> 00:24:29,990
So you may say, oh, come on.
376
00:24:29,990 --> 00:24:33,990
We just solved the problem
of an ideal spring.
377
00:24:33,990 --> 00:24:35,380
Who cares?
378
00:24:35,380 --> 00:24:39,120
It's so simple, so easy,
and you are making really
379
00:24:39,120 --> 00:24:41,760
a big thing out of this.
380
00:24:41,760 --> 00:24:44,490
But actually, what
we have been solving
381
00:24:44,490 --> 00:24:47,110
is really much more than that.
382
00:24:47,110 --> 00:24:53,850
This equation is much more
than just a spring-mass system.
383
00:24:53,850 --> 00:24:58,800
Actually, if you think about
this question carefully,
384
00:24:58,800 --> 00:25:02,130
there's really no
Hooke's law forever.
385
00:25:02,130 --> 00:25:05,820
Hooke's law will give you
a potential proportional
386
00:25:05,820 --> 00:25:07,800
to x squared.
387
00:25:07,800 --> 00:25:15,970
And if you are so far away, you
pull the spring so really hard,
388
00:25:15,970 --> 00:25:19,080
you can store the energy
of the whole universe.
389
00:25:19,080 --> 00:25:21,730
Does that make sense?
390
00:25:21,730 --> 00:25:22,980
No.
391
00:25:22,980 --> 00:25:25,620
At some point, it
should break down.
392
00:25:25,620 --> 00:25:27,990
So there's really no Hook's law.
393
00:25:27,990 --> 00:25:31,860
But there's also
Hook's law everywhere.
394
00:25:31,860 --> 00:25:37,230
If you look at this system,
it follows the harmonic
395
00:25:37,230 --> 00:25:38,280
oscillation.
396
00:25:38,280 --> 00:25:40,950
If you look at this
system I perturb this,
397
00:25:40,950 --> 00:25:42,150
it goes back and forth.
398
00:25:44,940 --> 00:25:46,560
It's almost like everywhere.
399
00:25:46,560 --> 00:25:48,540
Why is this the case?
400
00:25:48,540 --> 00:25:51,580
I'm going to answer this
question immediately.
401
00:25:54,280 --> 00:25:58,860
So let's take a
look at an example.
402
00:25:58,860 --> 00:26:01,060
So if I consider
a potential, this
403
00:26:01,060 --> 00:26:03,870
is an artificial
potential, which
404
00:26:03,870 --> 00:26:07,110
you can find in
Georgi's book, so v
405
00:26:07,110 --> 00:26:11,780
is equal to E times L
over x plus x over L.
406
00:26:11,780 --> 00:26:14,640
And if you practice as
a function of x, then
407
00:26:14,640 --> 00:26:16,800
basically you get
this funny shape.
408
00:26:16,800 --> 00:26:20,300
It's not proportional
to x squared.
409
00:26:20,300 --> 00:26:23,130
Therefore, you
will see that, OK,
410
00:26:23,130 --> 00:26:28,710
the resulting motion for the
particle in this potential,
411
00:26:28,710 --> 00:26:31,950
it's not going to
be harmonic motion.
412
00:26:31,950 --> 00:26:35,370
But if I zoom in,
zoom in, and zoom in
413
00:26:35,370 --> 00:26:38,550
and basically, you will see
that if I am patient enough,
414
00:26:38,550 --> 00:26:44,440
I zoom in enough, you'll
see that this is a parabola.
415
00:26:44,440 --> 00:26:50,100
Again, you follow Hooke's law.
416
00:26:50,100 --> 00:26:52,590
So that is actually really cool.
417
00:26:52,590 --> 00:26:59,100
So if I consider an
arbitrary v of x,
418
00:26:59,100 --> 00:27:05,740
we can do a Taylor
expansion to this potential.
419
00:27:05,740 --> 00:27:10,920
So basically v of x
will be equal to v of 0
420
00:27:10,920 --> 00:27:16,890
plus v prime 0 divided
by 1 factorial times
421
00:27:16,890 --> 00:27:24,900
x plus v double prime
0 over 2 factorial
422
00:27:24,900 --> 00:27:30,670
x squared plus v
triple prime 0 divided
423
00:27:30,670 --> 00:27:37,110
by 3 factorial x to the third
plus infinite number of terms.
424
00:27:37,110 --> 00:27:47,010
v 0 is the position of where
you have minimum potential.
425
00:27:47,010 --> 00:27:50,550
So that's actually where
the equilibrium position
426
00:27:50,550 --> 00:27:54,020
is in my coordinate system.
427
00:27:54,020 --> 00:27:55,770
It's the standard,
the coordinate system
428
00:27:55,770 --> 00:28:02,070
I used for the solving
the spring-mass question.
429
00:28:02,070 --> 00:28:06,900
So if I calculate the
force, the force, f of x,
430
00:28:06,900 --> 00:28:12,690
will be equal to
minus d dx v of x.
431
00:28:12,690 --> 00:28:15,150
And that will be
equal to minus v
432
00:28:15,150 --> 00:28:24,000
prime 0 minus v double
prime 0 x minus 1
433
00:28:24,000 --> 00:28:34,320
over 2 v triple prime 0 x
squared plus many other terms.
434
00:28:34,320 --> 00:28:39,980
Since I have mentioned
that v of 0--
435
00:28:46,108 --> 00:28:50,160
this will be x.
436
00:28:50,160 --> 00:28:55,260
v of 0 is actually the
position of the minima.
437
00:28:55,260 --> 00:29:01,002
Therefore, v prime of
0 will be equal to 0.
438
00:29:03,540 --> 00:29:05,100
Therefore.
439
00:29:05,100 --> 00:29:07,610
This term is gone.
440
00:29:11,660 --> 00:29:16,490
So what essentially is left over
is the remaining terms here.
441
00:29:19,910 --> 00:29:26,450
Now, if I assume that x is very
small, what is going to happen?
442
00:29:26,450 --> 00:29:31,090
Anybody know when x is very
small, what is going to happen?
443
00:29:31,090 --> 00:29:32,140
Anybody have the answer?
444
00:29:32,140 --> 00:29:34,720
AUDIENCE: [INAUDIBLE].
445
00:29:34,720 --> 00:29:36,110
YEN-JIE LEE: Exactly.
446
00:29:36,110 --> 00:29:41,630
So when x is very small, he said
that the higher order terms all
447
00:29:41,630 --> 00:29:43,095
become negligible.
448
00:29:43,095 --> 00:29:43,910
OK?
449
00:29:43,910 --> 00:29:45,290
So that is essentially correct.
450
00:29:45,290 --> 00:29:50,060
So when x is very
small, then I only
451
00:29:50,060 --> 00:29:55,130
need to consider the
leading order term.
452
00:29:55,130 --> 00:29:59,030
But how small is the question.
453
00:29:59,030 --> 00:30:00,900
How small is small?
454
00:30:03,760 --> 00:30:07,630
Actually, what you can
do is to take the ratio
455
00:30:07,630 --> 00:30:12,517
between these two terms.
456
00:30:12,517 --> 00:30:14,350
So if you take the
ratio, then basically you
457
00:30:14,350 --> 00:30:19,300
would get a condition
xv triple dot
458
00:30:19,300 --> 00:30:29,940
0, which will be much smaller
than v double prime 0.
459
00:30:29,940 --> 00:30:32,050
So that is essentially
the condition
460
00:30:32,050 --> 00:30:37,630
which is required to satisfy it
so that we actually can ignore
461
00:30:37,630 --> 00:30:39,340
all the higher-order terms.
462
00:30:39,340 --> 00:30:42,160
Then the whole
question becomes f
463
00:30:42,160 --> 00:30:48,910
of x equal to minus
v double prime 0 x.
464
00:30:48,910 --> 00:30:52,100
And that essentially,
Hooke's law.
465
00:30:52,100 --> 00:30:54,760
So you can see that
first of all, there's
466
00:30:54,760 --> 00:30:57,190
no Hooke's law in general.
467
00:30:57,190 --> 00:31:01,240
Secondly, Hook's law
essentially applicable
468
00:31:01,240 --> 00:31:06,490
almost everywhere when you
have a well-behaved potential
469
00:31:06,490 --> 00:31:10,860
and if you only perturb
the system really slightly
470
00:31:10,860 --> 00:31:14,960
with very small amplitude,
then it always works.
471
00:31:14,960 --> 00:31:19,120
So what I would like
to say is that after we
472
00:31:19,120 --> 00:31:23,620
have done this exercise, you
will see that, actually, we
473
00:31:23,620 --> 00:31:28,960
have solved all the
possible systems, which
474
00:31:28,960 --> 00:31:33,630
have a well-behaved potential.
475
00:31:33,630 --> 00:31:39,830
It has a minima, and if I have
the amplitude small enough,
476
00:31:39,830 --> 00:31:46,400
then the system is going to do
simple harmonic oscillation.
477
00:31:46,400 --> 00:31:47,860
Any questions?
478
00:31:54,540 --> 00:31:56,065
No question, then
we'll continue.
479
00:32:02,060 --> 00:32:07,040
So let's come back to
this equation of motion.
480
00:32:07,040 --> 00:32:13,360
x double dot plus omega
squared x, this is equal to 0.
481
00:32:16,380 --> 00:32:19,600
There are two
important properties
482
00:32:19,600 --> 00:32:22,930
of this linear
equation of motion.
483
00:32:26,950 --> 00:32:48,130
The first one is that if x1 of t
and x2 of t are both solutions,
484
00:32:48,130 --> 00:32:54,280
then x12, which is
the superposition
485
00:32:54,280 --> 00:33:01,340
of the first and second
solution, is also a solution.
486
00:33:13,930 --> 00:33:17,690
The second thing, which is very
interesting about this equation
487
00:33:17,690 --> 00:33:27,235
of motion, is that there's a
time translation invariance.
488
00:33:31,990 --> 00:33:38,830
So this means that if
x of t is a solution,
489
00:33:38,830 --> 00:33:49,285
then xt prime equal to xt
plus a is also a solution.
490
00:33:52,190 --> 00:33:55,520
So that is really
cool, because that
491
00:33:55,520 --> 00:34:04,370
means if I change t equal to
0, so I shift the 0-th time,
492
00:34:04,370 --> 00:34:08,090
the whole physics
did not change.
493
00:34:10,989 --> 00:34:13,650
So this is actually
because of the chain law.
494
00:34:13,650 --> 00:34:19,929
So if you have chain
law dx t plus a dt, that
495
00:34:19,929 --> 00:34:35,690
is equal to d t plus a
dt, dx t prime dt prime
496
00:34:35,690 --> 00:34:39,949
evaluated at t prime
equal to t plus a.
497
00:34:39,949 --> 00:34:52,969
And that is equal to dx t prime
dt, t prime equal to t plus a.
498
00:34:52,969 --> 00:35:00,020
So that means if I have
changed the t equal to 0
499
00:35:00,020 --> 00:35:04,670
to other place, the
whole equation of motion
500
00:35:04,670 --> 00:35:06,860
is still the same.
501
00:35:06,860 --> 00:35:10,880
On the other hand, if the
k, or say the potential,
502
00:35:10,880 --> 00:35:14,640
is time dependent, then that
may break this symmetry.
503
00:35:17,970 --> 00:35:18,932
Any questions?
504
00:35:22,150 --> 00:35:27,670
So before we take a
five minute break,
505
00:35:27,670 --> 00:35:33,640
I would like to discuss further
about this point, this linear
506
00:35:33,640 --> 00:35:34,990
and nonlinear event.
507
00:35:34,990 --> 00:35:38,230
So you can see that
the force is actually
508
00:35:38,230 --> 00:35:42,760
linearly dependent on x.
509
00:35:42,760 --> 00:35:49,420
But what will happen
if I increase x more?
510
00:35:49,420 --> 00:35:50,350
Something will happen.
511
00:35:50,350 --> 00:35:54,850
That means the higher-ordered
term should also be
512
00:35:54,850 --> 00:35:57,370
taken into account carefully.
513
00:35:57,370 --> 00:36:01,210
So that means the
solution of this kind, x
514
00:36:01,210 --> 00:36:07,070
initial cosine omega t,
will not work perfectly.
515
00:36:07,070 --> 00:36:12,690
In 8.03, we only consider the
linear term most of the time.
516
00:36:12,690 --> 00:36:15,700
But actually, I would
like to make sure
517
00:36:15,700 --> 00:36:17,450
that everybody
can at this point,
518
00:36:17,450 --> 00:36:21,220
the higher-order
contribution is actually
519
00:36:21,220 --> 00:36:23,290
visible in our daily life.
520
00:36:23,290 --> 00:36:28,710
So let me actually give
you a concrete example.
521
00:36:28,710 --> 00:36:33,520
So here I have two pendulums.
522
00:36:33,520 --> 00:36:36,550
So I can now perturb
this pendulum slightly.
523
00:36:36,550 --> 00:36:42,970
And you you'll see that it goes
back and forth and following
524
00:36:42,970 --> 00:36:45,610
simple harmonic emotion.
525
00:36:45,610 --> 00:36:48,670
So if I have both
things slightly
526
00:36:48,670 --> 00:36:52,810
oscillating with
small amplitude, what
527
00:36:52,810 --> 00:36:58,140
is going to happen is that
both pendulums reach maxima
528
00:36:58,140 --> 00:37:00,800
amplitude at the same time.
529
00:37:00,800 --> 00:37:05,360
You can see that very clearly.
530
00:37:05,360 --> 00:37:08,916
I don't need to
do this carefully.
531
00:37:08,916 --> 00:37:14,150
You see that they always
reach maxima at the same time
532
00:37:14,150 --> 00:37:16,040
when the amplitude is small.
533
00:37:16,040 --> 00:37:17,250
Why?
534
00:37:17,250 --> 00:37:24,000
That is because the higher-order
terms are not important.
535
00:37:24,000 --> 00:37:26,350
So now let's do a experiment.
536
00:37:26,350 --> 00:37:29,010
And now I go crazy.
537
00:37:29,010 --> 00:37:33,840
I make the amplitude
very large so that I
538
00:37:33,840 --> 00:37:37,800
break that approximation.
539
00:37:37,800 --> 00:37:39,300
So let's see what will happen.
540
00:37:39,300 --> 00:37:41,805
So now I do this then.
541
00:37:41,805 --> 00:37:47,170
I release at the same time
and see what will happen.
542
00:37:47,170 --> 00:37:49,410
You see that originally
they are in phase.
543
00:37:49,410 --> 00:37:53,830
They are reaching
maxima at the same time.
544
00:37:53,830 --> 00:37:57,110
But if we are patient
enough, you see that now?
545
00:37:57,110 --> 00:38:01,020
They are is
oscillating, actually,
546
00:38:01,020 --> 00:38:02,920
at different frequencies.
547
00:38:02,920 --> 00:38:06,150
Originally, the
solution, the omega,
548
00:38:06,150 --> 00:38:09,810
is really independent
of the amplitude.
549
00:38:09,810 --> 00:38:11,970
So they should,
actually, be isolating
550
00:38:11,970 --> 00:38:13,810
at the same frequency.
551
00:38:13,810 --> 00:38:16,260
But clearly you
can see here, when
552
00:38:16,260 --> 00:38:19,380
you increase the
amplitude, then you
553
00:38:19,380 --> 00:38:24,600
need to consider also
the nonlinear effects.
554
00:38:24,600 --> 00:38:30,067
So any questions before we
take a five-minute break.
555
00:38:30,067 --> 00:38:32,150
So if not, then we would
take a five-minute break,
556
00:38:32,150 --> 00:38:34,850
and we come back at 25.
557
00:38:40,590 --> 00:38:43,290
So welcome back, everybody.
558
00:38:43,290 --> 00:38:45,630
So we will continue
the discussion
559
00:38:45,630 --> 00:38:51,170
of this equation of motion, x
double dot plus omega square x
560
00:38:51,170 --> 00:38:54,120
equal to 0.
561
00:38:54,120 --> 00:38:58,750
So there are three
possible way to like
562
00:38:58,750 --> 00:39:01,935
the solution to this equation.
563
00:39:01,935 --> 00:39:04,860
So the first one as
I mentioned before,
564
00:39:04,860 --> 00:39:15,960
x of t equal to a cosine
omega t plus b sine omega t.
565
00:39:15,960 --> 00:39:17,800
So this is actually
the functional form
566
00:39:17,800 --> 00:39:21,010
we have been using before.
567
00:39:21,010 --> 00:39:26,520
And we can actually also
rewrite it in a different way.
568
00:39:26,520 --> 00:39:39,600
So x or t equal to capital
A cosine omega t plus phi.
569
00:39:39,600 --> 00:39:41,640
You may say, wait a second.
570
00:39:41,640 --> 00:39:44,190
You just promised me that
this is the first one,
571
00:39:44,190 --> 00:39:47,190
the one is the one
and only one solution
572
00:39:47,190 --> 00:39:51,750
in the universe, which actually
satisfy the equation of motion.
573
00:39:51,750 --> 00:39:53,960
Now you write another one.
574
00:39:53,960 --> 00:39:56,210
What is going on?
575
00:39:56,210 --> 00:39:57,680
Why?
576
00:39:57,680 --> 00:39:59,060
But actually, they are the same.
577
00:40:01,770 --> 00:40:09,740
This is actually A
cosine phi cosine omega t
578
00:40:09,740 --> 00:40:16,250
minus A sine phi sine omega t.
579
00:40:19,250 --> 00:40:23,390
So the good thing
is that A and phi
580
00:40:23,390 --> 00:40:27,770
are arbitrary constant
so that it should be you
581
00:40:27,770 --> 00:40:31,040
can use two initial
conditions to determine
582
00:40:31,040 --> 00:40:32,570
the arbitrary constant.
583
00:40:32,570 --> 00:40:39,050
So you can see that one and
two are completely equivalent.
584
00:40:39,050 --> 00:40:44,390
So I hope that solves
some of the questions
585
00:40:44,390 --> 00:40:48,320
because you really
find it confusing
586
00:40:48,320 --> 00:40:53,960
why we have different
presentations of the solution.
587
00:40:53,960 --> 00:40:59,600
So there's a third one, which
is actually much more fancier.
588
00:40:59,600 --> 00:41:04,270
The third one is
that I have x of t.
589
00:41:04,270 --> 00:41:10,575
This is actually
a real part of A--
590
00:41:10,575 --> 00:41:16,650
again, the amplitude--
exponential i omega t
591
00:41:16,650 --> 00:41:24,030
plus phi, where i is equal to
the square root of minus 1.
592
00:41:27,800 --> 00:41:28,460
Wait a second.
593
00:41:28,460 --> 00:41:32,530
We will say, well, professor,
why are you writing
594
00:41:32,530 --> 00:41:34,975
such a horrible solution?
595
00:41:38,820 --> 00:41:40,150
Right?
596
00:41:40,150 --> 00:41:41,150
Really strange.
597
00:41:41,150 --> 00:41:42,400
But that will explain you why.
598
00:41:45,030 --> 00:41:48,120
So three is actually
a mathematical trick.
599
00:41:48,120 --> 00:41:53,220
I'm not going to prove anything
here because I'm a physicist,
600
00:41:53,220 --> 00:41:58,600
but I would like to share with
you what I think is going on.
601
00:41:58,600 --> 00:42:01,410
I think three is really
just a mathematical trick
602
00:42:01,410 --> 00:42:05,310
from the math department.
603
00:42:05,310 --> 00:42:10,780
In principle, I can drive it
an even more horrible way.
604
00:42:10,780 --> 00:42:24,486
x of t equal to a real part of
A cosine omega t plus phi plus i
605
00:42:24,486 --> 00:42:29,501
f of t.
606
00:42:29,501 --> 00:42:36,910
And f of t is a real function.
607
00:42:36,910 --> 00:42:38,730
In principle, I can do that.
608
00:42:38,730 --> 00:42:41,870
It's even more horrible.
609
00:42:41,870 --> 00:42:43,500
Why is that?
610
00:42:43,500 --> 00:42:46,470
Because I now have
this function.
611
00:42:46,470 --> 00:42:48,870
I take the real
part, and I actually
612
00:42:48,870 --> 00:42:56,200
take the two out
of this operation.
613
00:42:56,200 --> 00:42:59,900
So f of t is actually
the real function.
614
00:42:59,900 --> 00:43:02,170
It can be something arbitrary.
615
00:43:02,170 --> 00:43:11,690
And i can now plot the locus
of this function, the solution
616
00:43:11,690 --> 00:43:15,280
on the complex print.
617
00:43:15,280 --> 00:43:19,800
Now I'm plotting this solution
on this complex print.
618
00:43:19,800 --> 00:43:23,780
What is going to happen is
that you're going to have--
619
00:43:31,250 --> 00:43:33,310
That is what you
are going to get.
620
00:43:33,310 --> 00:43:36,520
If I am lucky, if this
f of t is confined
621
00:43:36,520 --> 00:43:39,250
in some specific
region, if I not lucky,
622
00:43:39,250 --> 00:43:41,160
then it goes out
of the print there.
623
00:43:41,160 --> 00:43:42,490
I couldn't see it.
624
00:43:42,490 --> 00:43:44,490
Maybe it go to the
moon or something.
625
00:43:47,150 --> 00:43:50,090
But if you are smart
enough, and I'm
626
00:43:50,090 --> 00:44:00,080
sure you are, if I choose f
of t equal to A sine omega
627
00:44:00,080 --> 00:44:06,410
t plus phi, can anybody tell
me what is going to happen?
628
00:44:09,314 --> 00:44:10,766
AUDIENCE: [INAUDIBLE].
629
00:44:15,122 --> 00:44:16,950
YEN-JIE LEE: Would
you count a circle?
630
00:44:19,920 --> 00:44:22,770
Very good.
631
00:44:22,770 --> 00:44:27,700
If I plot the locus
again of this function,
632
00:44:27,700 --> 00:44:33,970
the real axis, imaginary axis,
then you should get a circle.
633
00:44:33,970 --> 00:44:37,030
Some miracle happened.
634
00:44:37,030 --> 00:44:41,530
If you choose the
f of t correctly,
635
00:44:41,530 --> 00:44:50,400
wisely, then you can actually
turn all this mess into order.
636
00:44:50,400 --> 00:44:51,649
Any questions?
637
00:44:56,440 --> 00:44:59,590
So I can now follow
up about this.
638
00:45:09,380 --> 00:45:20,110
So now I have x of t is equal to
the real part of A cosine omega
639
00:45:20,110 --> 00:45:28,260
t plus phi plus iA
sine omega t plus phi.
640
00:45:31,390 --> 00:45:36,120
And just a reminder,
exponential i theta
641
00:45:36,120 --> 00:45:43,830
is equal to cosine
theta plus i sine theta.
642
00:45:43,830 --> 00:45:45,920
Therefore, I arrive this.
643
00:45:45,920 --> 00:45:54,020
This is a real part of A
exponential i omega t plus phi.
644
00:45:58,840 --> 00:46:01,090
So if I do this
really carefully,
645
00:46:01,090 --> 00:46:08,680
I look at this the position of
the point at a specific time.
646
00:46:08,680 --> 00:46:13,500
So now time is equal to t.
647
00:46:13,500 --> 00:46:16,915
And this is the real axis, and
this is the imaginary axis.
648
00:46:16,915 --> 00:46:21,100
So I have this circle here.
649
00:46:21,100 --> 00:46:24,520
So at time equal to t,
what you are getting
650
00:46:24,520 --> 00:46:27,400
is that x is actually--
651
00:46:27,400 --> 00:46:31,430
before taking the real part, A,
exponential i omega t plus phi,
652
00:46:31,430 --> 00:46:33,730
it's actually here.
653
00:46:33,730 --> 00:46:39,380
And this vector actually
shows the amplitude.
654
00:46:39,380 --> 00:46:46,540
Amplitude is A. And the angle
between this vector pointing
655
00:46:46,540 --> 00:46:54,460
to the position of this
function is omega t plus phi.
656
00:46:54,460 --> 00:47:00,100
So this is actually the
angle between this vector
657
00:47:00,100 --> 00:47:03,500
and the real axis.
658
00:47:03,500 --> 00:47:06,060
So that's pretty cool.
659
00:47:06,060 --> 00:47:06,770
Why?
660
00:47:06,770 --> 00:47:12,410
Because now I understand why
I call this omega angular
661
00:47:12,410 --> 00:47:15,830
velocity or angular frequency.
662
00:47:15,830 --> 00:47:22,140
Because the solution to
the equation of motion,
663
00:47:22,140 --> 00:47:26,325
which we have actually
derived before,
664
00:47:26,325 --> 00:47:35,610
is actually the real part of
rotation in a complex print.
665
00:47:35,610 --> 00:47:39,540
If you think about
it, that means now
666
00:47:39,540 --> 00:47:45,910
I see this particle
going up and down.
667
00:47:45,910 --> 00:47:49,160
I see this particle
going up and down.
668
00:47:49,160 --> 00:47:51,950
You can think about
that, this is Earth.
669
00:47:51,950 --> 00:47:55,370
If there is an extra
dimension mention,
670
00:47:55,370 --> 00:47:58,620
which you couldn't see.
671
00:47:58,620 --> 00:48:02,620
Actually, this particle
in the dimension
672
00:48:02,620 --> 00:48:06,200
where we can see into the extra
dimension, which is hidden
673
00:48:06,200 --> 00:48:09,180
is actually rotating.
674
00:48:09,180 --> 00:48:12,470
And while we see
that reality, it's
675
00:48:12,470 --> 00:48:16,040
a projection to the real axis.
676
00:48:16,040 --> 00:48:16,650
You see?
677
00:48:16,650 --> 00:48:25,210
So in reality, this particle
is actually rotating,
678
00:48:25,210 --> 00:48:29,450
if you add the image
and the extra dimension.
679
00:48:29,450 --> 00:48:33,620
So that is actually pretty
cool, but the purity artificial.
680
00:48:33,620 --> 00:48:37,580
So you can see that
I can choose f of t
681
00:48:37,580 --> 00:48:42,530
to be a different function,
and then this whole picture
682
00:48:42,530 --> 00:48:43,970
is different.
683
00:48:43,970 --> 00:48:46,490
But I also would
create a lot of trouble
684
00:48:46,490 --> 00:48:49,250
because then the mathematics
become complicated.
685
00:48:49,250 --> 00:48:50,880
I didn't gain anything.
686
00:48:50,880 --> 00:48:58,240
But by choosing this
functional form,
687
00:48:58,240 --> 00:49:02,280
you actually write a
very beautiful picture.
688
00:49:02,280 --> 00:49:05,920
Another thing, which
is very cool about this
689
00:49:05,920 --> 00:49:12,480
is that if I write this thing
in the exponential functional
690
00:49:12,480 --> 00:49:18,340
form, since we are dealing
with differential equations,
691
00:49:18,340 --> 00:49:22,170
there is a very good property
about exponential function.
692
00:49:22,170 --> 00:49:26,856
That is it is essentially
a phoenix function.
693
00:49:26,856 --> 00:49:30,480
Do you know what is a phoenix?
694
00:49:30,480 --> 00:49:36,060
Phoenix is actually some kind
of animal, a long-beaked bird,
695
00:49:36,060 --> 00:49:41,100
which is cyclically called
the regenerated or reborn.
696
00:49:41,100 --> 00:49:44,100
So basically, when
this phoenix die,
697
00:49:44,100 --> 00:49:48,820
you will lay the eggs in the
fire and you were reborn.
698
00:49:48,820 --> 00:49:50,670
This is actually the
same as this function.
699
00:49:54,240 --> 00:49:59,650
I can do differentiation,
still an exponential function,
700
00:49:59,650 --> 00:50:02,460
and differentiate,
differentiate, differentiate.
701
00:50:02,460 --> 00:50:05,460
Still exponential function.
702
00:50:05,460 --> 00:50:07,620
So that is very
nice because when
703
00:50:07,620 --> 00:50:13,350
we deal with
differential equation,
704
00:50:13,350 --> 00:50:16,590
then you can actually
remove all those dots
705
00:50:16,590 --> 00:50:20,940
and make them become just
exponential function.
706
00:50:20,940 --> 00:50:25,770
So essentially, a
very nice property.
707
00:50:25,770 --> 00:50:32,910
So the first property, which is
very nice is that it cannot be
708
00:50:32,910 --> 00:50:39,480
killed by differentiation.
709
00:50:39,480 --> 00:50:46,980
You will see how useful this
is in the following lectures.
710
00:50:49,620 --> 00:50:52,230
The second thing,
which is really nice
711
00:50:52,230 --> 00:50:55,170
is that it has a
very nice property.
712
00:50:55,170 --> 00:51:02,090
So basically the exponential
i theta 1 times exponential i
713
00:51:02,090 --> 00:51:07,410
theta 2, and that will give
you exponential i theta
714
00:51:07,410 --> 00:51:08,980
1 plus theta 2.
715
00:51:11,610 --> 00:51:13,420
So what does that mean?
716
00:51:13,420 --> 00:51:22,110
That means if I have a solution
in this form, A exponential i
717
00:51:22,110 --> 00:51:25,400
omega t plus phi.
718
00:51:29,040 --> 00:51:40,200
And I do a times translation,
t become t plus A. Then
719
00:51:40,200 --> 00:51:47,650
this become A exponential
i omega t plus A plus phi.
720
00:51:51,350 --> 00:51:59,430
So this means that times
translation in this rotation
721
00:51:59,430 --> 00:52:04,880
is just a rotation
in complex print.
722
00:52:04,880 --> 00:52:05,730
You see?
723
00:52:05,730 --> 00:52:08,900
So now t becomes t plus
A. Then you are actually
724
00:52:08,900 --> 00:52:16,190
just changing the angle between
this vector and the x-axis.
725
00:52:16,190 --> 00:52:18,140
So as time goes
on, what is going
726
00:52:18,140 --> 00:52:21,300
to happen is that this thing
will go around and around
727
00:52:21,300 --> 00:52:28,100
and around and the physics is
always the set, no matter when
728
00:52:28,100 --> 00:52:32,340
you start counting, and
the translation is just
729
00:52:32,340 --> 00:52:35,790
the rotation in this print.
730
00:52:35,790 --> 00:52:37,165
Any questions?
731
00:52:44,940 --> 00:52:50,160
So I think this is actually a
basic slide just to remind you
732
00:52:50,160 --> 00:52:52,710
about Euler's formula.
733
00:52:52,710 --> 00:52:55,460
So basically, the
explanation i phi
734
00:52:55,460 --> 00:52:58,920
is equal to cosine
phi plus i sine phi.
735
00:52:58,920 --> 00:53:03,300
And I think it will be useful if
you are not familiar with this.
736
00:53:03,300 --> 00:53:06,180
It is useful to actually
review a little bit
737
00:53:06,180 --> 00:53:10,230
about exponential
function, which will
738
00:53:10,230 --> 00:53:12,050
be very useful for this class.
739
00:53:17,350 --> 00:53:20,260
So I'm running a
bit faster today.
740
00:53:20,260 --> 00:53:25,930
So let's take a look at
what we have learned today.
741
00:53:25,930 --> 00:53:31,060
We have analyzed the physics
of a harmonic oscillator.
742
00:53:31,060 --> 00:53:37,830
So basically, we start by asking
really just a verbal question,
743
00:53:37,830 --> 00:53:40,650
what is going to
happen to this mass
744
00:53:40,650 --> 00:53:44,310
on the table
attached to a spring.
745
00:53:44,310 --> 00:53:49,620
And what we have learned is that
we actually use mathematics.
746
00:53:49,620 --> 00:53:57,750
Basically, we translate all what
we have learned about this mass
747
00:53:57,750 --> 00:54:03,480
into mathematics by first
define a coordinate system.
748
00:54:03,480 --> 00:54:09,130
Then I'd write everything
using that coordinate system.
749
00:54:09,130 --> 00:54:13,125
Then I use Newton's law to
help us to solve this question.
750
00:54:15,870 --> 00:54:19,710
And we have analyzed the physics
of this harmonic oscillator.
751
00:54:19,710 --> 00:54:23,700
And Hooke's law, we found
that he actually, not only
752
00:54:23,700 --> 00:54:30,380
works for this
spring-mass system,
753
00:54:30,380 --> 00:54:37,260
it also works for all kinds of
different small oscillations
754
00:54:37,260 --> 00:54:40,530
about a point of equilibrium.
755
00:54:40,530 --> 00:54:44,700
So basically, it's actually
a universal solution
756
00:54:44,700 --> 00:54:47,070
what we have been doing.
757
00:54:47,070 --> 00:54:51,990
And we have found out a
complex exponential function
758
00:54:51,990 --> 00:54:55,450
is actually a beautiful
way to present
759
00:54:55,450 --> 00:54:59,910
the solution to the equation of
motion we have been studying.
760
00:54:59,910 --> 00:55:02,240
So everything is nice and good.
761
00:55:02,240 --> 00:55:06,960
However, life is
hard because there
762
00:55:06,960 --> 00:55:13,010
are many things which actually,
we ignored in this example.
763
00:55:13,010 --> 00:55:16,620
One apparent thing,
which we actually ignore,
764
00:55:16,620 --> 00:55:18,840
is the direct force.
765
00:55:18,840 --> 00:55:24,420
So you can see that before I was
actually making this pendulum
766
00:55:24,420 --> 00:55:27,480
oscillate back and forth.
767
00:55:27,480 --> 00:55:29,430
What is happening now?
768
00:55:29,430 --> 00:55:31,930
There are not
oscillating anymore.
769
00:55:31,930 --> 00:55:33,660
Why?
770
00:55:33,660 --> 00:55:37,020
Well, they stopped being.
771
00:55:37,020 --> 00:55:40,440
Apparently,
something is missing.
772
00:55:40,440 --> 00:55:46,930
When I actually
moved this system,
773
00:55:46,930 --> 00:55:51,490
if I turn off the air so
that there's friction,
774
00:55:51,490 --> 00:55:53,590
then it doesn't really move.
775
00:55:53,590 --> 00:55:58,820
If I increase a bit, the
air so that the slide have
776
00:55:58,820 --> 00:56:02,330
some slight freedom,
then actually, you
777
00:56:02,330 --> 00:56:05,990
can see that you move
a bit then you stop.
778
00:56:05,990 --> 00:56:12,625
If I increase this
some more, you
779
00:56:12,625 --> 00:56:18,660
can see that the amplitude
becomes smaller and smaller.
780
00:56:18,660 --> 00:56:22,400
So in the following lecture,
what we are going to do
781
00:56:22,400 --> 00:56:27,110
is to study how to actually
include a direct force into it
782
00:56:27,110 --> 00:56:31,210
again and of course, using the
same machinery which we have
783
00:56:31,210 --> 00:56:34,160
learned from here and
see if we can actually
784
00:56:34,160 --> 00:56:35,890
solve this problem.
785
00:56:35,890 --> 00:56:36,950
Thank you very much.
786
00:56:36,950 --> 00:56:39,360
We actually end
up earlier today.
787
00:56:39,360 --> 00:56:40,520
Sorry for that.
788
00:56:40,520 --> 00:56:43,280
And maybe I will make the
lecture longer next time.
789
00:56:45,890 --> 00:56:48,340
And if you have any
questions about what
790
00:56:48,340 --> 00:56:55,480
we have covered today, I'm
here available to help you.