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BOLESLAW WYSLOUCH: Good
morning, everybody.
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00:00:25,980 --> 00:00:27,090
I'm Bolek Wyslouch.
10
00:00:27,090 --> 00:00:31,920
I'm a teacher substitute for
Professor Lee, who is now
11
00:00:31,920 --> 00:00:34,770
at some conference
in China, and he
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00:00:34,770 --> 00:00:39,510
asked me to talk to you
about coupled oscillators.
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00:00:39,510 --> 00:00:43,290
I understand that he introduced
the concept last time.
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00:00:43,290 --> 00:00:45,430
You worked through
some examples.
15
00:00:45,430 --> 00:00:47,775
So what we are
going to do today is
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00:00:47,775 --> 00:00:50,100
to basically go
through one or two
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00:00:50,100 --> 00:00:55,500
examples of very straightforward
coupled oscillators, where
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00:00:55,500 --> 00:00:59,790
I will introduce various kinds
of systematic calculational
19
00:00:59,790 --> 00:01:01,380
techniques, how
to set things up,
20
00:01:01,380 --> 00:01:05,129
how to prepare things
for calculations.
21
00:01:05,129 --> 00:01:09,150
And also, we may, depending
on how much time we have,
22
00:01:09,150 --> 00:01:14,560
start driving, have driven
coupled oscillators.
23
00:01:14,560 --> 00:01:20,990
And we will work on two,
again, simple physical systems,
24
00:01:20,990 --> 00:01:28,230
one that consists of two pendula
driven by forces of gravity,
25
00:01:28,230 --> 00:01:29,067
each of them.
26
00:01:29,067 --> 00:01:30,900
And then they are
connected with the spring.
27
00:01:30,900 --> 00:01:33,450
So each of those pendula,
each of those masses,
28
00:01:33,450 --> 00:01:36,805
will feel the effects of
gravity and effects of springs
29
00:01:36,805 --> 00:01:38,930
at the same time, and they
will talk to each other.
30
00:01:38,930 --> 00:01:40,720
There will be
coupling between them.
31
00:01:40,720 --> 00:01:44,520
So that's one physical
example which we'll consider.
32
00:01:44,520 --> 00:01:47,220
The other physical
example consists
33
00:01:47,220 --> 00:01:54,540
of two masses in the horizontal
frictionless track connected
34
00:01:54,540 --> 00:01:55,590
by a set of springs.
35
00:01:55,590 --> 00:01:58,260
So they are driven
by forces of spring.
36
00:01:58,260 --> 00:02:00,660
And those two systems are
very similar to each other,
37
00:02:00,660 --> 00:02:03,420
almost identical in
terms of calculations,
38
00:02:03,420 --> 00:02:05,610
and they exhibit
the same phenomena,
39
00:02:05,610 --> 00:02:07,350
and I will be able
to demonstrate
40
00:02:07,350 --> 00:02:09,340
several of the neat new things.
41
00:02:09,340 --> 00:02:11,580
And this particular
system is set up
42
00:02:11,580 --> 00:02:14,640
to introduce external
driving force, which will
43
00:02:14,640 --> 00:02:16,090
create a new set of phenomena.
44
00:02:16,090 --> 00:02:19,150
And we'll talk about it today.
45
00:02:19,150 --> 00:02:21,540
And what I would
like to stress today
46
00:02:21,540 --> 00:02:24,240
when we go through all
those calculations is,
47
00:02:24,240 --> 00:02:28,470
A, how do you convert
a given physical system
48
00:02:28,470 --> 00:02:32,310
with all the forces, et cetera,
into some sort of fixed form,
49
00:02:32,310 --> 00:02:36,690
fixed type of notation,
with which you can treat all
50
00:02:36,690 --> 00:02:39,150
possible coupled oscillators?
51
00:02:39,150 --> 00:02:41,370
And also we will discuss
various interesting--
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00:02:41,370 --> 00:02:43,650
even though the system
is very simple, just
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00:02:43,650 --> 00:02:47,980
two masses, a spring, a little
bit of gravity on top of that,
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00:02:47,980 --> 00:02:53,340
the way they behave could
be extremely complex,
55
00:02:53,340 --> 00:02:55,500
but it can be
understood in terms
56
00:02:55,500 --> 00:02:58,710
of very simple systematic
way of looking things
57
00:02:58,710 --> 00:03:03,070
through normal modes
and normal frequencies,
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00:03:03,070 --> 00:03:05,650
so the characteristic
frequencies of the system.
59
00:03:05,650 --> 00:03:08,070
So let's set things up.
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00:03:08,070 --> 00:03:09,570
So we'll start.
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00:03:09,570 --> 00:03:11,370
This will be our workhorse.
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00:03:11,370 --> 00:03:15,300
And by the way, once
we understand two,
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00:03:15,300 --> 00:03:18,930
we will then generalize to
infinite number of oscillators,
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00:03:18,930 --> 00:03:22,770
which is actually-- so
this model, which consists
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00:03:22,770 --> 00:03:26,460
of weights hanging under
the influence of gravity
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00:03:26,460 --> 00:03:28,920
plus the springs
will be then used
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00:03:28,920 --> 00:03:33,160
for many applications of the
concepts later in this course.
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00:03:33,160 --> 00:03:39,430
So let's try to convert
this physical system
69
00:03:39,430 --> 00:03:41,590
into a set of equations.
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00:03:41,590 --> 00:03:47,920
So we have a mass, m, hanging
from some sort of fixed
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00:03:47,920 --> 00:03:52,360
support, another mass here,
same mass for simplicity.
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00:03:52,360 --> 00:03:55,150
We connect them with
a string, and we know
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00:03:55,150 --> 00:03:57,080
everything about this system.
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00:03:57,080 --> 00:04:00,610
We know the length of each
of those pendula, which
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00:04:00,610 --> 00:04:01,840
is the same.
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00:04:01,840 --> 00:04:03,280
We know masses.
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00:04:03,280 --> 00:04:07,330
We know spring constant of a
spring connecting those two
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00:04:07,330 --> 00:04:08,860
things.
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00:04:08,860 --> 00:04:11,695
The spring is initially
at its rest position
80
00:04:11,695 --> 00:04:16,269
such that when the two pendula
are hanging vertically,
81
00:04:16,269 --> 00:04:17,950
the spring is relaxed.
82
00:04:17,950 --> 00:04:20,209
But if you move it
away from verticality,
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00:04:20,209 --> 00:04:24,790
the spring either
compresses or stretches.
84
00:04:24,790 --> 00:04:30,970
And everything is in Earth's
gravitational field, g.
85
00:04:30,970 --> 00:04:35,460
We assume that this is an
ideal system, highly idealized.
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00:04:35,460 --> 00:04:40,180
We only consider motion with
small angle approximation, only
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00:04:40,180 --> 00:04:42,290
small displacement.
88
00:04:42,290 --> 00:04:44,680
There's no drag force assumed.
89
00:04:44,680 --> 00:04:47,560
The spring is ideal,
et cetera, et cetera.
90
00:04:47,560 --> 00:04:51,670
Of course, this thing here
is very far from being ideal,
91
00:04:51,670 --> 00:04:54,940
but hopefully basic
behaviors are similar.
92
00:04:54,940 --> 00:04:57,890
It's approximately ideal.
93
00:04:57,890 --> 00:04:59,720
To study the motion
of this thing,
94
00:04:59,720 --> 00:05:02,480
to understand how it
works, let's try to--
95
00:05:02,480 --> 00:05:06,770
let's try to parameterize
it, and displace it
96
00:05:06,770 --> 00:05:08,780
from equilibrium, and
look at the forces,
97
00:05:08,780 --> 00:05:11,550
and try to calculate
equations of motion.
98
00:05:11,550 --> 00:05:13,520
So we will characterize
this system
99
00:05:13,520 --> 00:05:16,760
by two position coordinates.
100
00:05:16,760 --> 00:05:18,080
We will have x.
101
00:05:18,080 --> 00:05:19,580
We'll give this one number one.
102
00:05:19,580 --> 00:05:21,500
This will be number two.
103
00:05:21,500 --> 00:05:26,060
And we will have x subscript
1, which in general would
104
00:05:26,060 --> 00:05:26,960
depend on the time.
105
00:05:26,960 --> 00:05:29,600
This is the position of
this mass with respect
106
00:05:29,600 --> 00:05:31,320
to its equilibrium position.
107
00:05:31,320 --> 00:05:35,070
We will have x2 as
a function of t.
108
00:05:35,070 --> 00:05:37,100
Again, this tells us everything.
109
00:05:37,100 --> 00:05:39,770
And the full description
of the system
110
00:05:39,770 --> 00:05:42,920
is to know exactly what
happens to x1 and x2
111
00:05:42,920 --> 00:05:46,910
for all possible times.
112
00:05:46,910 --> 00:05:49,910
And we will impose some
initial conditions.
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00:05:49,910 --> 00:05:52,650
We can come back to that later.
114
00:05:52,650 --> 00:05:55,550
So again, so the
coordinate system is this.
115
00:05:55,550 --> 00:05:58,790
When we start talking about
the system in principle
116
00:05:58,790 --> 00:06:01,910
in the case of
somewhat larger angles,
117
00:06:01,910 --> 00:06:04,425
you have to worry about
vertical positions as well.
118
00:06:04,425 --> 00:06:05,300
So we will introduce.
119
00:06:05,300 --> 00:06:09,500
So there is also a coordinate y,
which we will need temporarily
120
00:06:09,500 --> 00:06:12,040
to set things up.
121
00:06:12,040 --> 00:06:15,690
So x is, as I say, x is
measured from equilibrium.
122
00:06:15,690 --> 00:06:17,860
Y is positioned vertically.
123
00:06:17,860 --> 00:06:20,451
So to calculate the
equations of motion,
124
00:06:20,451 --> 00:06:21,700
we have to look at the forces.
125
00:06:21,700 --> 00:06:24,010
So let's look at what
are the forces acting,
126
00:06:24,010 --> 00:06:27,460
for example, on this
mass, the mass, which
127
00:06:27,460 --> 00:06:31,450
is-- if it's displaced
from a vertical position.
128
00:06:31,450 --> 00:06:37,420
Let's say this mass, mass 1,
has moved by some distance
129
00:06:37,420 --> 00:06:41,500
away from thing Temporarily,
let's introduce an angle here
130
00:06:41,500 --> 00:06:44,860
to characterize this
displacement from vertical.
131
00:06:44,860 --> 00:06:47,590
And let's write down all
the forces acting on this
132
00:06:47,590 --> 00:06:50,320
- force diagram
acting on this mass.
133
00:06:50,320 --> 00:06:55,570
So there is a tension in
the string or the rod.
134
00:06:55,570 --> 00:06:58,480
Let's call it T1.
135
00:06:58,480 --> 00:07:03,020
There is a force
of spring acting
136
00:07:03,020 --> 00:07:04,680
in a horizontal direction.
137
00:07:04,680 --> 00:07:06,350
This is a vector.
138
00:07:06,350 --> 00:07:11,840
And there is a force of
gravity acting on this
139
00:07:11,840 --> 00:07:14,130
in the vertical direction.
140
00:07:14,130 --> 00:07:16,610
We can write down those forces.
141
00:07:16,610 --> 00:07:18,050
We know a lot about them.
142
00:07:18,050 --> 00:07:23,810
This one is minus mg y-hat.
143
00:07:23,810 --> 00:07:33,660
This one is equal to k x2 minus
x1 in the x-hat direction.
144
00:07:33,660 --> 00:07:37,520
So this is the force which, when
the spring is displaced from
145
00:07:37,520 --> 00:07:40,730
equilibrium, there is a
spring force, Hooke force,
146
00:07:40,730 --> 00:07:42,480
in the direction of --
147
00:07:42,480 --> 00:07:43,940
in the usual direction.
148
00:07:43,940 --> 00:07:47,570
In this case, it's actually
in the opposite direction.
149
00:07:47,570 --> 00:07:49,710
And then there is a
tension the spring,
150
00:07:49,710 --> 00:07:53,370
which has to be calculated
such that we understand
151
00:07:53,370 --> 00:07:56,340
the acceleration of this object.
152
00:07:56,340 --> 00:07:59,210
So let's write
down the equations
153
00:07:59,210 --> 00:08:02,450
in the x-hat direction.
154
00:08:02,450 --> 00:08:07,990
This is m acceleration
of object number 1
155
00:08:07,990 --> 00:08:16,560
in x direction is equal to
minus T1 sine theta 1 plus k
156
00:08:16,560 --> 00:08:21,150
x2 minus x1.
157
00:08:21,150 --> 00:08:26,270
And in the y-hat
direction, we have
158
00:08:26,270 --> 00:08:39,340
m y1 direction is equal to
T cosine theta 1 minus mg.
159
00:08:39,340 --> 00:08:43,909
At the small angle for theta
1 much, much smaller than one,
160
00:08:43,909 --> 00:08:49,060
we can assume, that cos theta
1 is approximately equal to 1
161
00:08:49,060 --> 00:08:53,910
and sine theta 1
is equal to angle.
162
00:08:53,910 --> 00:08:55,710
We do the usual thing.
163
00:08:55,710 --> 00:08:59,250
So basically, in
this approximation,
164
00:08:59,250 --> 00:09:01,200
and also by looking
at the system,
165
00:09:01,200 --> 00:09:03,570
it's clear that the
system does not move,
166
00:09:03,570 --> 00:09:06,270
and the vertical
direction can be ignored.
167
00:09:06,270 --> 00:09:07,121
Yes?
168
00:09:07,121 --> 00:09:09,162
AUDIENCE: How do you know
which way [INAUDIBLE]??
169
00:09:09,162 --> 00:09:10,370
BOLESLAW WYSLOUCH: Excuse me?
170
00:09:10,370 --> 00:09:11,572
AUDIENCE: The [INAUDIBLE].
171
00:09:11,572 --> 00:09:13,323
How do you know which
way it [INAUDIBLE]??
172
00:09:13,323 --> 00:09:14,697
BOLESLAW WYSLOUCH:
How do I know?
173
00:09:14,697 --> 00:09:15,413
AUDIENCE: Yeah.
174
00:09:15,413 --> 00:09:16,395
[INAUDIBLE]
175
00:09:19,090 --> 00:09:21,920
BOLESLAW WYSLOUCH:
The spring force is--
176
00:09:21,920 --> 00:09:24,740
well, you have to
look at the mass 1.
177
00:09:24,740 --> 00:09:26,930
You are just looking at mass 1.
178
00:09:26,930 --> 00:09:29,570
So the spring is
connected to mass 1.
179
00:09:29,570 --> 00:09:32,930
And the force of
the spring on mass 1
180
00:09:32,930 --> 00:09:39,390
is k times however the spring
is squashed or stretched,
181
00:09:39,390 --> 00:09:41,210
all right?
182
00:09:41,210 --> 00:09:44,960
So it knows about the existence
of mass 2, but only in a sense
183
00:09:44,960 --> 00:09:47,320
that you have to know
the position of mass 2.
184
00:09:47,320 --> 00:09:50,930
So we just assume
that x2 is something,
185
00:09:50,930 --> 00:09:53,820
and we just look
where the spring is.
186
00:09:53,820 --> 00:09:55,980
So that's why--
the force of spring
187
00:09:55,980 --> 00:09:59,350
depends on the difference
of position x1 minus x2.
188
00:10:02,560 --> 00:10:05,410
So this is written here.
189
00:10:05,410 --> 00:10:08,610
And in fact, interestingly,
the position of the mass 1
190
00:10:08,610 --> 00:10:11,380
itself is a negative sign here.
191
00:10:11,380 --> 00:10:14,080
So if you move mass
1, the spring force
192
00:10:14,080 --> 00:10:17,926
is in the right
direction, minus kx.
193
00:10:17,926 --> 00:10:19,360
All right?
194
00:10:19,360 --> 00:10:21,520
So there is no
motion x1, so we can
195
00:10:21,520 --> 00:10:25,750
conclude from here the T cosine1
is approximately equal to 1.
196
00:10:25,750 --> 00:10:28,960
So T is simply equal to mg.
197
00:10:28,960 --> 00:10:33,250
So the tension in the spring
can be assumed to be mg.
198
00:10:33,250 --> 00:10:35,000
We don't have to worry about it.
199
00:10:35,000 --> 00:10:37,390
And then we just plug in--
200
00:10:37,390 --> 00:10:40,780
also the angle can be
converted into position
201
00:10:40,780 --> 00:10:46,630
by realizing that the
distance times the angle
202
00:10:46,630 --> 00:10:49,180
is equal to displacement,
the usual geometry.
203
00:10:49,180 --> 00:10:52,120
The net result is that
by simplifying things,
204
00:10:52,120 --> 00:10:58,990
I can write down
equations for acceleration
205
00:10:58,990 --> 00:11:03,760
in the horizontal
direction for mass 1
206
00:11:03,760 --> 00:11:18,920
is equal to minus mg x1
over l plus k x2 minus x1.
207
00:11:18,920 --> 00:11:19,900
OK?
208
00:11:19,900 --> 00:11:26,730
So this is an equation
of motion for mass 1
209
00:11:26,730 --> 00:11:29,980
in our coupled system.
210
00:11:29,980 --> 00:11:33,750
And I could say
most of the terms
211
00:11:33,750 --> 00:11:37,860
have to do with a
motion of mass 1 itself.
212
00:11:37,860 --> 00:11:40,620
Mass 1 is its own pendulum.
213
00:11:40,620 --> 00:11:46,080
And mass 1 is feeling the
effect of the spring force.
214
00:11:46,080 --> 00:11:48,420
But because the
force of the spring
215
00:11:48,420 --> 00:11:50,670
depends on the difference
between positions,
216
00:11:50,670 --> 00:11:53,010
there is this coupling--
217
00:11:53,010 --> 00:11:58,200
so the motion of mass 1
knows of where mass 2 is.
218
00:11:58,200 --> 00:12:03,310
And motion of mass 2 influences
the motion of mass 1.
219
00:12:03,310 --> 00:12:05,970
That's how the
coupling shows up.
220
00:12:05,970 --> 00:12:08,760
So for most of those
problems, what you do is
221
00:12:08,760 --> 00:12:13,220
you simply focus on
the mass in question.
222
00:12:13,220 --> 00:12:15,690
You take all the forces,
you calculate them,
223
00:12:15,690 --> 00:12:17,610
and then this
coupling will somehow
224
00:12:17,610 --> 00:12:20,250
appear in the equations.
225
00:12:20,250 --> 00:12:26,080
So we can repeat exactly
the same calculation
226
00:12:26,080 --> 00:12:29,020
focusing on mass 2.
227
00:12:29,020 --> 00:12:33,210
And then the equation which you
will get will be very similar.
228
00:12:33,210 --> 00:12:36,820
Let me just slightly
rewrite this equation here
229
00:12:36,820 --> 00:12:40,390
to kind of combine
all the terms which
230
00:12:40,390 --> 00:12:42,420
depend on the position
of mass 1 with terms
231
00:12:42,420 --> 00:12:44,172
that depend on mass 2.
232
00:12:44,172 --> 00:12:53,910
So where m x-acceleration
is equal to minus k
233
00:12:53,910 --> 00:13:06,040
plus mg over l times
x1 plus k times x2.
234
00:13:06,040 --> 00:13:07,740
So this is the coupling term.
235
00:13:13,680 --> 00:13:17,710
This is what makes
those pendula coupled.
236
00:13:17,710 --> 00:13:18,370
All right?
237
00:13:18,370 --> 00:13:23,740
And then I can write almost
exactly the same equation
238
00:13:23,740 --> 00:13:31,420
of mass 2 with the proper
replacement of masses.
239
00:13:31,420 --> 00:13:37,080
So let me write this down
in the following way-- kx1
240
00:13:37,080 --> 00:13:45,420
minus k plus mg over l times x2.
241
00:13:50,440 --> 00:13:54,650
So the motion of
mass x1 depends on x1
242
00:13:54,650 --> 00:13:57,440
itself multiplied by
something with a spring
243
00:13:57,440 --> 00:13:59,710
term and gravitational
term and depends
244
00:13:59,710 --> 00:14:04,250
on the position of mass 2
only through the spring.
245
00:14:04,250 --> 00:14:10,970
Mass 2 also is mostly driven
by its own gravitational force
246
00:14:10,970 --> 00:14:15,950
of itself plus the spring
depends on the position of x2.
247
00:14:15,950 --> 00:14:18,410
But there is this
coupling term that
248
00:14:18,410 --> 00:14:20,330
depends on position of mass 1.
249
00:14:20,330 --> 00:14:23,480
So both of them feel the
neighbor on the other side,
250
00:14:23,480 --> 00:14:24,290
right?
251
00:14:24,290 --> 00:14:29,770
So if I keep this one
steady of x2 equals 0,
252
00:14:29,770 --> 00:14:31,580
then basically the
forces here is just
253
00:14:31,580 --> 00:14:34,310
the spring plus the gravity.
254
00:14:34,310 --> 00:14:37,910
If I move this one and keep
this one at 0, the force on this
255
00:14:37,910 --> 00:14:40,000
spring spring and gravity.
256
00:14:40,000 --> 00:14:44,010
But if this one is displaced,
and I move that guy,
257
00:14:44,010 --> 00:14:47,050
the forces on this one
are affected by the fact
258
00:14:47,050 --> 00:14:49,500
that number 2 changed.
259
00:14:49,500 --> 00:14:50,000
OK?
260
00:14:50,000 --> 00:14:52,250
Again, I was able to
determine those coupling
261
00:14:52,250 --> 00:14:56,842
terms by simply looking at
mass 1 itself, mass 2 itself.
262
00:14:56,842 --> 00:15:00,350
All right, so this is the
set of two coupled equations.
263
00:15:00,350 --> 00:15:03,440
I have accelerations
here for x1, x2,
264
00:15:03,440 --> 00:15:05,240
and I have positions here.
265
00:15:05,240 --> 00:15:08,420
It's like an oscillator
of position acceleration
266
00:15:08,420 --> 00:15:13,130
with a constant term except that
things here are a little mixed.
267
00:15:13,130 --> 00:15:16,970
And the trick in this
whole mathematics,
268
00:15:16,970 --> 00:15:19,670
and calculations,
and the way we do
269
00:15:19,670 --> 00:15:25,820
things is how do you solve
those coupled equations?
270
00:15:25,820 --> 00:15:26,860
OK?
271
00:15:26,860 --> 00:15:30,660
So what I would like to do is--
272
00:15:30,660 --> 00:15:32,950
and there is multiple
ways of doing that.
273
00:15:32,950 --> 00:15:34,840
So let me do everything.
274
00:15:34,840 --> 00:15:38,250
Let's write down everything
in the matrix form,
275
00:15:38,250 --> 00:15:40,170
because it turns out
that linear matrices are
276
00:15:40,170 --> 00:15:41,400
very useful for that.
277
00:15:41,400 --> 00:15:43,060
We will use them very, very--
278
00:15:43,060 --> 00:15:44,410
in a very simple way.
279
00:15:44,410 --> 00:15:48,210
So let's introduce to
them and show vector,
280
00:15:48,210 --> 00:15:53,790
which consists of x1 and x2.
281
00:15:53,790 --> 00:15:58,880
So basically, all the
position x1 and x2 are here.
282
00:15:58,880 --> 00:16:02,900
So we will be monitoring
the change of this x2
283
00:16:02,900 --> 00:16:04,460
as a function of time.
284
00:16:04,460 --> 00:16:09,900
We will introduce
a force matrix k,
285
00:16:09,900 --> 00:16:25,440
which is equal to k plus mg over
l minus k here, minus k here,
286
00:16:25,440 --> 00:16:29,600
k plus mg over l there.
287
00:16:29,600 --> 00:16:32,340
This is a two by two matrix.
288
00:16:32,340 --> 00:16:37,140
And then we need a third
matrix, mass matrix,
289
00:16:37,140 --> 00:16:45,070
which simply says that masses
are mass of first object is m
290
00:16:45,070 --> 00:16:46,610
and the other one
is also m, right?
291
00:16:49,130 --> 00:16:52,520
So these are three
matrices that basically
292
00:16:52,520 --> 00:16:56,640
contains exactly the same
information as out there.
293
00:16:56,640 --> 00:16:58,110
I probably need another matrix.
294
00:16:58,110 --> 00:17:01,440
I need an inverse matrix
for mass, which basically
295
00:17:01,440 --> 00:17:05,900
is 1 over m, 1 over m, 0 and 0.
296
00:17:05,900 --> 00:17:10,630
This is a inverted matrix.
297
00:17:10,630 --> 00:17:20,069
OK, and it turns out that after
I introduced these matrices,
298
00:17:20,069 --> 00:17:23,810
this set of equations
can be written simply
299
00:17:23,810 --> 00:17:30,700
as X, the second derivative
of the vector capital X,
300
00:17:30,700 --> 00:17:36,890
is equal to minus
m to the minus 1,
301
00:17:36,890 --> 00:17:43,480
this matrix, multiplying
matrix k and then multiplying
302
00:17:43,480 --> 00:17:45,953
vectors x again.
303
00:17:49,801 --> 00:17:52,210
All right?
304
00:17:52,210 --> 00:17:58,880
So this is exactly the
same as this, just written
305
00:17:58,880 --> 00:18:00,720
a different way.
306
00:18:00,720 --> 00:18:02,380
So it's only the
question of notation.
307
00:18:02,380 --> 00:18:06,190
So it turns out
it's very convenient
308
00:18:06,190 --> 00:18:11,240
to use matrix calculation
to do things faster.
309
00:18:11,240 --> 00:18:14,510
So instead of repeating writing,
all the x1s, x2, et cetera,
310
00:18:14,510 --> 00:18:19,480
instead I just stick them into
one or two element objects.
311
00:18:19,480 --> 00:18:21,800
I use matrices to
multiply things,
312
00:18:21,800 --> 00:18:23,540
and if I want to
know x1 and x2, I
313
00:18:23,540 --> 00:18:26,540
can always go, OK,
the top component
314
00:18:26,540 --> 00:18:30,450
of vector x, lower component of
vector x gives me the solution.
315
00:18:30,450 --> 00:18:31,010
Simple.
316
00:18:31,010 --> 00:18:31,510
Right?
317
00:18:34,690 --> 00:18:42,210
So let's try to use this
terminology to find solutions.
318
00:18:42,210 --> 00:18:46,550
So the question is how
do we find solutions
319
00:18:46,550 --> 00:18:47,880
to coupled oscillations.
320
00:18:47,880 --> 00:18:53,150
What is the most efficient way
of finding the most general
321
00:18:53,150 --> 00:18:56,060
motion of a coupled system?
322
00:18:56,060 --> 00:18:57,380
Anybody knows?
323
00:18:57,380 --> 00:18:59,690
What's the first thing?
324
00:18:59,690 --> 00:19:01,002
Yes?
325
00:19:01,002 --> 00:19:02,430
AUDIENCE: [INAUDIBLE].
326
00:19:02,430 --> 00:19:03,940
BOLESLAW WYSLOUCH:
Introduce what?
327
00:19:03,940 --> 00:19:06,031
AUDIENCE: [INAUDIBLE]
using complex notation.
328
00:19:06,031 --> 00:19:07,156
BOLESLAW WYSLOUCH: Coupled?
329
00:19:07,156 --> 00:19:07,930
AUDIENCE: Complex.
330
00:19:07,930 --> 00:19:09,990
BOLESLAW WYSLOUCH:
Complex oscillation.
331
00:19:09,990 --> 00:19:11,860
Yes, that's right.
332
00:19:11,860 --> 00:19:15,240
So all right, let's do it.
333
00:19:15,240 --> 00:19:17,310
But hold on.
334
00:19:17,310 --> 00:19:22,640
But what form of oscillation?
335
00:19:22,640 --> 00:19:28,100
OK, all kinds of complex numbers
can write, but any particular--
336
00:19:28,100 --> 00:19:28,975
AUDIENCE: [INAUDIBLE]
337
00:19:28,975 --> 00:19:30,475
BOLESLAW WYSLOUCH:
That's something.
338
00:19:30,475 --> 00:19:32,450
That's the physics
answer, all right?
339
00:19:32,450 --> 00:19:36,080
Complex notation is a
mathematical answer,
340
00:19:36,080 --> 00:19:38,220
how to solve a
mathematical equation.
341
00:19:38,220 --> 00:19:43,970
But the physics answer is to
find fixed frequency modes us
342
00:19:43,970 --> 00:19:46,340
such that the system,
the complete system,
343
00:19:46,340 --> 00:19:48,720
oscillates at one frequency.
344
00:19:48,720 --> 00:19:50,670
Everybody moves together.
345
00:19:50,670 --> 00:19:53,050
This is so-called normal mode.
346
00:19:53,050 --> 00:19:55,670
It turns out that
every of the system,
347
00:19:55,670 --> 00:19:57,280
depending on number
of dimensions,
348
00:19:57,280 --> 00:20:03,880
will have a certain number of
frequencies, normal modes, that
349
00:20:03,880 --> 00:20:05,190
would--
350
00:20:05,190 --> 00:20:08,050
the whole system oscillates
at the same frequency,
351
00:20:08,050 --> 00:20:12,850
both x1 and x2, undergoing
motion of the same frequency.
352
00:20:12,850 --> 00:20:14,670
We don't know what
the frequency is.
353
00:20:14,670 --> 00:20:17,150
We don't know it's
amplitude, et cetera.
354
00:20:17,150 --> 00:20:19,450
But it is the same.
355
00:20:19,450 --> 00:20:19,950
OK?
356
00:20:23,930 --> 00:20:28,850
So this means that I can
write that the whole vector
357
00:20:28,850 --> 00:20:35,630
x, both x1 and x2, are
undergoing the same oscillatory
358
00:20:35,630 --> 00:20:36,650
motion.
359
00:20:36,650 --> 00:20:40,151
So I propose that--
360
00:20:40,151 --> 00:20:46,010
so of course, we use the
usual trick that anytime
361
00:20:46,010 --> 00:20:50,960
we have a solution
in complex variables,
362
00:20:50,960 --> 00:20:54,870
we can always get back to real
things by taking a real part.
363
00:20:54,870 --> 00:20:58,430
So I understand you've
done this before.
364
00:20:58,430 --> 00:21:02,660
So let's introduce
variable z, just kind
365
00:21:02,660 --> 00:21:12,910
of a two-element vector, which
has a complex term, a fixed
366
00:21:12,910 --> 00:21:16,690
frequency, plus a
phase, a rhythm complex,
367
00:21:16,690 --> 00:21:24,700
multiplying vector A,
a fixed vector A. OK?
368
00:21:24,700 --> 00:21:36,730
And vector A is simply has two
components, A1, A2, or maybe
369
00:21:36,730 --> 00:21:40,290
I should write it differently.
370
00:21:40,290 --> 00:21:45,170
So vector A contains
information about some sort
371
00:21:45,170 --> 00:21:51,910
of initial conditions
for position x 1 2.
372
00:21:51,910 --> 00:21:55,040
Anyway, these are
two constant numbers.
373
00:21:55,040 --> 00:21:59,690
And also, we will, because
we have this phase here,
374
00:21:59,690 --> 00:22:02,480
because we keep phase
in this expression,
375
00:22:02,480 --> 00:22:06,570
we can assume and require
that is a real number.
376
00:22:06,570 --> 00:22:07,510
So A is real.
377
00:22:10,960 --> 00:22:13,980
It's a slightly different
way of doing things,
378
00:22:13,980 --> 00:22:17,330
but we can assume
this for now, right?
379
00:22:17,330 --> 00:22:20,830
So the solution which
is written here--
380
00:22:20,830 --> 00:22:25,710
it's some two numbers,
oscillatory term,
381
00:22:25,710 --> 00:22:30,710
with both x1 and x2 oscillating
with the same frequency,
382
00:22:30,710 --> 00:22:32,520
and this is our
postulated solution.
383
00:22:32,520 --> 00:22:36,560
So we plug it into the
equation, and we adjust things
384
00:22:36,560 --> 00:22:38,570
until it fits.
385
00:22:38,570 --> 00:22:44,060
So let's plug this into
our matrix calculation.
386
00:22:44,060 --> 00:22:46,169
And what you see here is that--
387
00:22:46,169 --> 00:22:46,960
so what do we have?
388
00:22:46,960 --> 00:22:50,870
So this is the term,
which is second time
389
00:22:50,870 --> 00:22:54,340
the derivative vector
X. And because vector--
390
00:22:54,340 --> 00:22:55,960
or vector Z really.
391
00:22:55,960 --> 00:22:57,880
So I have to do--
392
00:22:57,880 --> 00:22:58,770
so I plug this here.
393
00:22:58,770 --> 00:23:08,460
So Z double dot is simply equal
minus omega squared times Z.
394
00:23:08,460 --> 00:23:09,498
Right?
395
00:23:09,498 --> 00:23:10,740
Like this.
396
00:23:10,740 --> 00:23:13,050
So this is a simple thing.
397
00:23:13,050 --> 00:23:16,080
When I plug this in
here, my equation
398
00:23:16,080 --> 00:23:24,280
becomes an equation for
A. So I have minus omega
399
00:23:24,280 --> 00:23:30,060
squared z-hat, which maybe
I just write it immediately
400
00:23:30,060 --> 00:23:34,730
in terms of a complex term
by times the vector A.
401
00:23:34,730 --> 00:23:45,570
So I have e to i omega t plus
y times A is equal to minus M
402
00:23:45,570 --> 00:23:57,071
to minus 1 K times e to the i
omega t plus phi times vector
403
00:23:57,071 --> 00:24:00,840
A. OK?
404
00:24:00,840 --> 00:24:04,680
And this term is
a proportionality
405
00:24:04,680 --> 00:24:06,870
constant at any
given moment of time.
406
00:24:06,870 --> 00:24:10,400
So it goes through the
matrix multiplication.
407
00:24:10,400 --> 00:24:11,730
So you can just delete this.
408
00:24:11,730 --> 00:24:13,950
You can divide both sides.
409
00:24:13,950 --> 00:24:16,980
You have signs here.
410
00:24:16,980 --> 00:24:19,870
And then I have
an equation which
411
00:24:19,870 --> 00:24:23,430
is a linear matrix
equation, which
412
00:24:23,430 --> 00:24:30,570
is M minus 1 K times vector A.
413
00:24:30,570 --> 00:24:33,840
And I can rewrite it
a little bit again.
414
00:24:33,840 --> 00:24:40,320
So I can rewrite in this
minus 1 K minus omega squared
415
00:24:40,320 --> 00:24:47,860
times unity matrix times
vector A is equal to 0.
416
00:24:47,860 --> 00:24:53,110
So this is the
equation which we need
417
00:24:53,110 --> 00:25:01,879
to solve to obtain the solutions
to at least one normal mode,
418
00:25:01,879 --> 00:25:04,420
and we expect that there will
be two normal modes, because we
419
00:25:04,420 --> 00:25:05,086
have two masses.
420
00:25:09,100 --> 00:25:11,680
So now, this is--
421
00:25:16,250 --> 00:25:19,390
so this is some matrix,
two by two matrix,
422
00:25:19,390 --> 00:25:22,120
which we can know very
easily how to write.
423
00:25:22,120 --> 00:25:26,070
Multiplying a
vector gives you 0.
424
00:25:26,070 --> 00:25:32,790
It turns out that for this
to work, there are two--
425
00:25:32,790 --> 00:25:37,860
there is a criterion,
which has to be satisfied,
426
00:25:37,860 --> 00:25:40,270
namely the determinant
of the two by two matrix
427
00:25:40,270 --> 00:25:43,740
has to be equal to 0,
because if you take
428
00:25:43,740 --> 00:25:47,550
the determinant on both sides,
you have to have 0 on this side
429
00:25:47,550 --> 00:25:49,410
to be able to obtain
0 on the other side.
430
00:25:49,410 --> 00:25:53,250
So mathematically, the way
to find out the oscillating
431
00:25:53,250 --> 00:25:58,730
frequency is you take a
determinant of m minus 1 K
432
00:25:58,730 --> 00:26:04,590
minus i omega squared
must be equal to 0.
433
00:26:04,590 --> 00:26:09,580
So let's try to see how
to calculate things.
434
00:26:09,580 --> 00:26:19,970
So let's write down this matrix
explicitly using this and that.
435
00:26:19,970 --> 00:26:23,840
So let's write this down.
436
00:26:23,840 --> 00:26:28,590
So I take a big
object like this.
437
00:26:28,590 --> 00:26:34,490
And so in this
element here, I have
438
00:26:34,490 --> 00:26:36,680
to multiply this
matrix times that.
439
00:26:36,680 --> 00:26:40,640
If I multiply this
matrix, I simply divide
440
00:26:40,640 --> 00:26:43,430
all those effectively
multiplication of m
441
00:26:43,430 --> 00:26:47,500
minus 1 times this matrix
divides all the elements here
442
00:26:47,500 --> 00:26:49,200
by m.
443
00:26:49,200 --> 00:26:50,290
That's all there is to it.
444
00:26:50,290 --> 00:26:52,210
I just divide everything by m.
445
00:26:54,800 --> 00:27:03,970
So the first M minus 1 K
is k over m plus g over l.
446
00:27:03,970 --> 00:27:14,260
This is minus k over m minus k
over m k over m plus g over l.
447
00:27:17,338 --> 00:27:18,790
So this is multiplication.
448
00:27:18,790 --> 00:27:21,500
This is this term here.
449
00:27:21,500 --> 00:27:24,450
And then I have to do minus
unity matrix times omega
450
00:27:24,450 --> 00:27:25,360
squared.
451
00:27:25,360 --> 00:27:32,690
All this will do is it will
subtract omega squared here.
452
00:27:32,690 --> 00:27:33,880
I should write this.
453
00:27:38,300 --> 00:27:39,480
OK?
454
00:27:39,480 --> 00:27:40,510
So this is in this one.
455
00:27:40,510 --> 00:27:46,610
Maybe it would be more clear
if I move it over here.
456
00:27:46,610 --> 00:27:48,480
All right, so this
is the matrix that
457
00:27:48,480 --> 00:27:51,760
contains all the information
about our system,
458
00:27:51,760 --> 00:27:54,725
the mass, the gravitational
acceleration, the length,
459
00:27:54,725 --> 00:27:57,090
the spring strength, et cetera.
460
00:27:57,090 --> 00:28:03,180
And we assumed they oscillate
with a fixed frequency.
461
00:28:03,180 --> 00:28:09,540
So I have to find the
determinant of this matrix
462
00:28:09,540 --> 00:28:11,790
equal to 0.
463
00:28:11,790 --> 00:28:13,330
So how do I get that?
464
00:28:13,330 --> 00:28:18,440
And by the way,
you have a matrix,
465
00:28:18,440 --> 00:28:21,560
and you want to make sure
that its determinant is 0.
466
00:28:21,560 --> 00:28:23,910
It turns out the
only variable which
467
00:28:23,910 --> 00:28:27,870
we have to change
parameters of this matrix--
468
00:28:27,870 --> 00:28:30,440
you know, the spring constant
and the mass this affects
469
00:28:30,440 --> 00:28:31,470
is given.
470
00:28:31,470 --> 00:28:34,440
The system has been built.
It's hanging over there.
471
00:28:34,440 --> 00:28:36,400
I cannot change anything.
472
00:28:36,400 --> 00:28:40,590
So the only parameter here,
which I can change, or adjust,
473
00:28:40,590 --> 00:28:43,560
or find is omega square.
474
00:28:43,560 --> 00:28:46,980
So I will try all possible
matrices of this type
475
00:28:46,980 --> 00:28:52,740
until I find one or two that
have a determinant equal to 0.
476
00:28:52,740 --> 00:28:55,430
But if I find them,
this would correspond
477
00:28:55,430 --> 00:28:58,905
to the normal frequencies.
478
00:28:58,905 --> 00:29:01,230
OK?
479
00:29:01,230 --> 00:29:04,770
So how do I calculate
the determinant of a two
480
00:29:04,770 --> 00:29:05,980
by two matrix?
481
00:29:05,980 --> 00:29:10,280
I do this by this minus
this by that, right?
482
00:29:10,280 --> 00:29:15,590
So that of this matrix
is equal to k over
483
00:29:15,590 --> 00:29:24,110
m minus g plus g over
l minus omega squared.
484
00:29:24,110 --> 00:29:27,280
The two identical terms
so I can put the square
485
00:29:27,280 --> 00:29:32,960
and then minus this minus
k squared over m squared
486
00:29:32,960 --> 00:29:35,620
must be equal to 0.
487
00:29:35,620 --> 00:29:36,120
Right?
488
00:29:36,120 --> 00:29:40,710
So this is the equation
which we need to solve.
489
00:29:40,710 --> 00:29:45,960
We need to find which
parameter omega sets this to 0.
490
00:29:45,960 --> 00:29:53,440
And then this is a pretty
straightforward calculation,
491
00:29:53,440 --> 00:29:57,610
except if I don't have--
492
00:29:57,610 --> 00:29:59,220
I'll just use this one.
493
00:30:02,650 --> 00:30:05,080
OK, so let's rewrite
this a little bit.
494
00:30:05,080 --> 00:30:08,680
So this is basically equivalent
to the following equation
495
00:30:08,680 --> 00:30:16,120
g over l plus k over
m minus omega squared
496
00:30:16,120 --> 00:30:21,080
must be equal either to
plus or minus k over m.
497
00:30:21,080 --> 00:30:21,580
Right?
498
00:30:21,580 --> 00:30:23,960
I took a square
root of both sides.
499
00:30:23,960 --> 00:30:25,690
If you take a square
root, you have
500
00:30:25,690 --> 00:30:29,150
to worry about plus
and minus signs, right?
501
00:30:29,150 --> 00:30:32,200
So there are two solutions
which corresponds to plus here.
502
00:30:32,200 --> 00:30:34,660
The other one corresponds
to minus here.
503
00:30:34,660 --> 00:30:37,840
So solution number
1, which corresponds
504
00:30:37,840 --> 00:30:42,760
to plus sign right
here, it basically
505
00:30:42,760 --> 00:30:49,340
says that omega squared
is equal to g over l.
506
00:30:49,340 --> 00:30:49,840
Right?
507
00:30:49,840 --> 00:30:53,620
So there is one solution,
one oscillation,
508
00:30:53,620 --> 00:30:55,780
that does not depend
on the spring constant,
509
00:30:55,780 --> 00:30:58,390
because the spring
constant cancels.
510
00:30:58,390 --> 00:31:01,960
And there's a second
solution which
511
00:31:01,960 --> 00:31:09,340
corresponds to minus, where
omega squared is equal to g
512
00:31:09,340 --> 00:31:14,640
over l plus 2k over m.
513
00:31:17,630 --> 00:31:18,440
Right?
514
00:31:18,440 --> 00:31:21,740
Because there are two
possible solutions.
515
00:31:21,740 --> 00:31:23,340
And this is what we have.
516
00:31:23,340 --> 00:31:26,970
So we have a--
517
00:31:26,970 --> 00:31:33,420
so what this says is that if I
set my frequency to g over l,
518
00:31:33,420 --> 00:31:36,450
if I set the system to
oscillate to this frequency,
519
00:31:36,450 --> 00:31:40,260
then it will be--
520
00:31:40,260 --> 00:31:45,030
I will be able to set things up
such that it oscillates forever
521
00:31:45,030 --> 00:31:48,730
at this frequency, one
fixed frequency forever.
522
00:31:48,730 --> 00:31:49,730
And this is interesting.
523
00:31:49,730 --> 00:31:50,420
This is a frequency.
524
00:31:50,420 --> 00:31:52,530
It does not depend on the
strength of the spring.
525
00:31:52,530 --> 00:31:54,500
How is it possible?
526
00:31:54,500 --> 00:31:57,380
Somehow spring is
irrelevant for this motion.
527
00:31:57,380 --> 00:32:00,060
And it turns out that there
is a very simple oscillation,
528
00:32:00,060 --> 00:32:02,570
easy to see, if
basically that this
529
00:32:02,570 --> 00:32:05,160
is a frequency of
a single pendulum.
530
00:32:05,160 --> 00:32:06,680
So basically, you
got both pendula
531
00:32:06,680 --> 00:32:10,520
going together,
each of them happily
532
00:32:10,520 --> 00:32:13,160
oscillating by themselves.
533
00:32:13,160 --> 00:32:16,020
And the spring is completely
irrelevant for this motion.
534
00:32:16,020 --> 00:32:18,700
If I cut it off, the
motion will not change.
535
00:32:18,700 --> 00:32:22,030
It just happens that two
identical pendula are going
536
00:32:22,030 --> 00:32:24,570
at their own natural frequency.
537
00:32:24,570 --> 00:32:26,600
So the force of
spring is irrelevant.
538
00:32:26,600 --> 00:32:27,360
Nothing happens.
539
00:32:27,360 --> 00:32:28,730
This is a normal mode.
540
00:32:28,730 --> 00:32:32,770
And it can go forever at
this particular frequency.
541
00:32:32,770 --> 00:32:33,690
OK?
542
00:32:33,690 --> 00:32:38,120
The other option is
usually symmetrically.
543
00:32:38,120 --> 00:32:41,280
I move them away
from each other.
544
00:32:41,280 --> 00:32:45,200
And this is the motion where,
again, it's not exactly
545
00:32:45,200 --> 00:32:50,710
ideal small angle oscillation,
but let me try again,
546
00:32:50,710 --> 00:32:52,730
I guess with less.
547
00:32:52,730 --> 00:32:54,830
So this is the situation
where the spring really
548
00:32:54,830 --> 00:32:56,920
comes in at full force.
549
00:32:56,920 --> 00:33:00,144
It's being stretched
maximally, because they go away
550
00:33:00,144 --> 00:33:00,810
from each other.
551
00:33:00,810 --> 00:33:03,360
So very quickly, the
spring is stretched.
552
00:33:03,360 --> 00:33:07,060
And they go together so it's
stretch from both sides.
553
00:33:07,060 --> 00:33:10,940
And the whole system oscillates
at the same frequency,
554
00:33:10,940 --> 00:33:15,760
and because of this
additional force of spring,
555
00:33:15,760 --> 00:33:19,670
the frequency is actually
higher, it's larger.
556
00:33:19,670 --> 00:33:23,260
It oscillates faster.
557
00:33:23,260 --> 00:33:25,210
All right, so that's
the first step
558
00:33:25,210 --> 00:33:26,590
in understanding the system.
559
00:33:26,590 --> 00:33:30,370
We now know that there are
two oscillations and two
560
00:33:30,370 --> 00:33:31,880
normal frequencies.
561
00:33:31,880 --> 00:33:34,600
And the next step to
finish our understanding
562
00:33:34,600 --> 00:33:37,560
of the system in a mathematical
way, to describe it fully,
563
00:33:37,560 --> 00:33:40,810
I have to know what is
the shape of oscillations.
564
00:33:40,810 --> 00:33:44,140
I simply showed you here
so you know what to expect.
565
00:33:44,140 --> 00:33:48,910
But I have to be able to dig
it out from the equations.
566
00:33:48,910 --> 00:33:53,800
And the way to dig it out
is to find vector A. See,
567
00:33:53,800 --> 00:34:01,360
our real equation of
motion is up here.
568
00:34:01,360 --> 00:34:02,950
This is an equation of motion.
569
00:34:02,950 --> 00:34:08,710
This is, I have to now
find the vector A, which
570
00:34:08,710 --> 00:34:11,620
when you plug it in, it works--
571
00:34:11,620 --> 00:34:13,810
it satisfies this equation.
572
00:34:13,810 --> 00:34:16,929
So I already know what are
the two possible omegas--
573
00:34:16,929 --> 00:34:21,219
they can do it, but still
I have to find vector A.
574
00:34:21,219 --> 00:34:23,650
So I have to solve two
separate independent problems.
575
00:34:23,650 --> 00:34:26,600
One is finding vector
A for this situation
576
00:34:26,600 --> 00:34:28,900
and then find the vector
A for that situation
577
00:34:28,900 --> 00:34:30,070
and see if it works.
578
00:34:30,070 --> 00:34:31,969
So I had to plug in the whole.
579
00:34:31,969 --> 00:34:34,980
I had to plug it into
the whole equation.
580
00:34:34,980 --> 00:34:38,440
And you can show
that if you set--
581
00:34:38,440 --> 00:34:45,730
if you set omega squared to g
over l, and you plug it into--
582
00:34:45,730 --> 00:34:47,860
if you plug it into
this equation, what
583
00:34:47,860 --> 00:34:50,230
you get is a matrix
equation which
584
00:34:50,230 --> 00:34:56,600
looks like this-- k
over m minus k over m
585
00:34:56,600 --> 00:35:01,050
minus k over m k over m.
586
00:35:01,050 --> 00:35:03,320
And this is because--
587
00:35:03,320 --> 00:35:08,760
[I try to-- so if you plug omega
squared here equal to g over
588
00:35:08,760 --> 00:35:14,120
l, then this cancels out,
and this cancels out.
589
00:35:14,120 --> 00:35:16,420
So you plug it in
here, and you get
590
00:35:16,420 --> 00:35:22,660
this very simple, very simple
matrix that has k over m terms.
591
00:35:22,660 --> 00:35:25,210
So the question is
what sort of thing
592
00:35:25,210 --> 00:35:29,660
can you put here to get 0.
593
00:35:29,660 --> 00:35:34,750
What kind of vector you can
plug into those two places
594
00:35:34,750 --> 00:35:40,450
such that the matrix times
vector will end up with 0?
595
00:35:40,450 --> 00:35:45,250
One example is that basically
amplitude is the same.
596
00:35:45,250 --> 00:35:47,070
Both of them move together.
597
00:35:47,070 --> 00:35:51,131
So you plug 1 here and 1 here.
598
00:35:51,131 --> 00:35:51,630
Right?
599
00:35:51,630 --> 00:35:54,890
So this is a good solution.
600
00:35:54,890 --> 00:35:58,880
And every other solution
is a linear multiplication
601
00:35:58,880 --> 00:36:01,220
of this one for this
frequency, right?
602
00:36:01,220 --> 00:36:04,790
There is k over m times 1
minus k over m gives you 0.
603
00:36:04,790 --> 00:36:07,280
So this is a good solution for--
604
00:36:07,280 --> 00:36:10,250
so this is solution number 1.
605
00:36:10,250 --> 00:36:12,350
What about this thing here?
606
00:36:12,350 --> 00:36:23,650
If I plug this omega
squared into this matrix,
607
00:36:23,650 --> 00:36:26,300
it's g over l plus 2k over m.
608
00:36:26,300 --> 00:36:32,330
If I plug it in here, then this
matrix is way more complicated.
609
00:36:32,330 --> 00:36:34,470
It will actually
look very similar,
610
00:36:34,470 --> 00:36:37,780
but with important differences.
611
00:36:37,780 --> 00:36:42,030
So this one will
look minus k over m
612
00:36:42,030 --> 00:36:49,346
minus k over m minus k
over m minus k over m.
613
00:36:49,346 --> 00:36:52,240
OK?
614
00:36:52,240 --> 00:36:58,430
And then again, for this second
possible normal frequency,
615
00:36:58,430 --> 00:37:01,310
I have to find the vector
A, which corresponds
616
00:37:01,310 --> 00:37:02,510
to that frequency motion.
617
00:37:02,510 --> 00:37:07,260
And it turns out that they are
the same, but the sign changes.
618
00:37:07,260 --> 00:37:13,620
So one possible solution
is 1 and minus 1.
619
00:37:13,620 --> 00:37:18,450
If I plug in 1 minus 1, then
this matrix times the vector
620
00:37:18,450 --> 00:37:20,620
gives you automatically 0.
621
00:37:20,620 --> 00:37:23,680
So this is the second
possible normal mode.
622
00:37:23,680 --> 00:37:24,640
All right?
623
00:37:24,640 --> 00:37:28,390
So this is a systematic
way to solve equations.
624
00:37:28,390 --> 00:37:31,090
You plug in all
the information you
625
00:37:31,090 --> 00:37:34,240
know about the system
into a two by two matrix.
626
00:37:34,240 --> 00:37:38,110
And then you calculate
the normal mode.
627
00:37:38,110 --> 00:37:41,284
And then you calculate a
shape of a normal mode.
628
00:37:43,950 --> 00:37:47,770
Is that clear?
629
00:37:47,770 --> 00:37:51,570
Any questions at this time?
630
00:37:51,570 --> 00:37:52,710
Right?
631
00:37:52,710 --> 00:37:55,240
So in principle, we know,
now, at the end of the day,
632
00:37:55,240 --> 00:37:58,824
I still want to know how much
1 moves, how much 2 moves.
633
00:37:58,824 --> 00:38:00,240
So we have to put
it all together.
634
00:38:00,240 --> 00:38:04,560
We have identified the frequency
and the kind of, in the matrix
635
00:38:04,560 --> 00:38:06,420
notation, shape of the node.
636
00:38:06,420 --> 00:38:08,130
But of course,
the final solution
637
00:38:08,130 --> 00:38:11,790
is a linear superposition
of all possible normal modes
638
00:38:11,790 --> 00:38:14,560
with described
position of mass 1,
639
00:38:14,560 --> 00:38:15,810
position of mass 2, et cetera.
640
00:38:15,810 --> 00:38:19,590
So let's do a little bit of--
641
00:38:19,590 --> 00:38:23,700
so maybe graphically I can
write down that this is the--
642
00:38:23,700 --> 00:38:27,300
this is the oscillation
that corresponds
643
00:38:27,300 --> 00:38:32,130
to this type of mode, to those
two masses move together.
644
00:38:32,130 --> 00:38:34,920
And this is oscillation
that corresponds to the mode
645
00:38:34,920 --> 00:38:40,240
where masses move in
opposite directions.
646
00:38:40,240 --> 00:38:45,120
At any moment of time,
in this normal mode,
647
00:38:45,120 --> 00:38:49,360
at any moment of time,
wherever mass 1 is,
648
00:38:49,360 --> 00:38:53,830
mass 2 is minus the distance
away from its own equilibrium.
649
00:38:53,830 --> 00:38:56,080
So if this one is plus
1 centimeter here,
650
00:38:56,080 --> 00:38:57,580
the other one is
minus 1 centimeter.
651
00:38:57,580 --> 00:38:58,770
This one is minus 5.
652
00:38:58,770 --> 00:39:01,250
This one is plus 5 and so on.
653
00:39:01,250 --> 00:39:05,396
Whereas in this mode, both
of them move together.
654
00:39:05,396 --> 00:39:06,140
All right?
655
00:39:06,140 --> 00:39:11,730
So let's try to go back to the--
656
00:39:11,730 --> 00:39:13,510
you can get rid of this one.
657
00:39:13,510 --> 00:39:16,740
Let's try to go back, and
now with this knowledge,
658
00:39:16,740 --> 00:39:24,154
let's write down x1 and x2 for
positions of the two masses.
659
00:39:33,270 --> 00:39:45,306
So x1-- so basically,
the x will have to be--
660
00:39:45,306 --> 00:39:46,630
I used z there.
661
00:39:46,630 --> 00:39:52,510
So x will be real of vector z.
662
00:39:52,510 --> 00:39:55,850
So I take my complex numbers
and take a real of them.
663
00:39:55,850 --> 00:39:59,470
So from an exponent, I
will end up with a cosine
664
00:39:59,470 --> 00:40:00,640
appropriately and so on.
665
00:40:00,640 --> 00:40:03,980
And then I will use
the [INAUDIBLE]..
666
00:40:03,980 --> 00:40:10,030
So this is real part
of e to the i omega
667
00:40:10,030 --> 00:40:14,530
plus phi where omega is one
of the two possibilities.
668
00:40:14,530 --> 00:40:21,980
Omega t plus phi times vector
A, which we've identified here,
669
00:40:21,980 --> 00:40:26,200
and times some additional--
670
00:40:26,200 --> 00:40:30,700
these are those vectors A
this is one possible amplitude
671
00:40:30,700 --> 00:40:31,830
of notation.
672
00:40:31,830 --> 00:40:34,090
But in general, it
can be anything.
673
00:40:34,090 --> 00:40:34,810
You can multiply.
674
00:40:34,810 --> 00:40:38,210
You can have small oscillations,
large oscillations.
675
00:40:38,210 --> 00:40:40,000
So there is some
overall amplitude.
676
00:40:40,000 --> 00:40:41,660
But the shape always
has to be simple.
677
00:40:41,660 --> 00:40:45,430
They either go together,
or they go opposite.
678
00:40:45,430 --> 00:40:46,880
So to make it more
general, I have
679
00:40:46,880 --> 00:40:48,680
to give some multiplicative
factor there.
680
00:40:51,670 --> 00:40:56,560
So if I do everything, I
end up with x, the mode 1
681
00:40:56,560 --> 00:41:00,610
will in general have some
sort of overall constant C,
682
00:41:00,610 --> 00:41:08,900
cosine omega 1 t plus phi
1 times the vector 1, 1.
683
00:41:08,900 --> 00:41:10,220
This will be for x1.
684
00:41:10,220 --> 00:41:12,020
This will be for x2.
685
00:41:12,020 --> 00:41:21,660
And the mode number 2 will
be C2 cosine omega 2 times
686
00:41:21,660 --> 00:41:27,170
t plus phi 2 times 1 minus 1.
687
00:41:27,170 --> 00:41:28,970
All right?
688
00:41:28,970 --> 00:41:32,150
So let's see what
things are adjustable
689
00:41:32,150 --> 00:41:34,110
and what things are fixed.
690
00:41:34,110 --> 00:41:37,370
So the omega 1 and
omega 2 are fixed
691
00:41:37,370 --> 00:41:41,900
given by the construction of
the two coupled oscillators.
692
00:41:41,900 --> 00:41:44,060
This shape, 1 and
1, and 1 minus 1
693
00:41:44,060 --> 00:41:47,060
is fixed, because these are the
shape of normal modes, which
694
00:41:47,060 --> 00:41:48,930
corresponds to
those frequencies.
695
00:41:48,930 --> 00:41:52,670
So we have only four
constants-- overall amplitude c1
696
00:41:52,670 --> 00:41:54,040
for normal mode 1.
697
00:41:54,040 --> 00:41:56,930
Overall amplitude c2 for
normal mode 2 and then
698
00:41:56,930 --> 00:42:01,010
the relative phase of
those two normal modes.
699
00:42:01,010 --> 00:42:03,260
And the superposition
of x1 plus x2
700
00:42:03,260 --> 00:42:08,790
gives you the most general
combination of possible motion.
701
00:42:08,790 --> 00:42:11,090
So if I write this
down now in terms
702
00:42:11,090 --> 00:42:14,540
of position of number
1 and number 2,
703
00:42:14,540 --> 00:42:18,332
so I have a position of
x1 as a function of time.
704
00:42:18,332 --> 00:42:19,790
In general, it will
look like this.
705
00:42:19,790 --> 00:42:24,690
It will be some sort of
constant alpha, cosine omega 1 t
706
00:42:24,690 --> 00:42:32,720
plus phi plus constant
beta cosine omega 2 times
707
00:42:32,720 --> 00:42:36,610
t plus phi 2 plus phi 1.
708
00:42:36,610 --> 00:42:40,670
So mass number 1, this
is position of mass 1,
709
00:42:40,670 --> 00:42:45,400
will in general be a
superposition of the two
710
00:42:45,400 --> 00:42:47,740
possible oscillations.
711
00:42:47,740 --> 00:42:54,000
The position of mass 2
will be very similar,
712
00:42:54,000 --> 00:42:55,970
but there will be a very
important difference
713
00:42:55,970 --> 00:43:00,820
between the alpha
cosine omega 1 t
714
00:43:00,820 --> 00:43:08,340
plus phi 1 minus beta
cosine omega 2t plus phi 2.
715
00:43:10,870 --> 00:43:13,900
This is very important
to understand exactly how
716
00:43:13,900 --> 00:43:16,350
this equation came about.
717
00:43:16,350 --> 00:43:19,470
You see, this is the influence
of the symmetric mode,
718
00:43:19,470 --> 00:43:21,550
where the two
things are together.
719
00:43:21,550 --> 00:43:23,880
So they are multiplied
by alpha, some sort
720
00:43:23,880 --> 00:43:30,550
of arbitrary constant, but
with exactly the same sign.
721
00:43:30,550 --> 00:43:32,290
And this is the part
which corresponds
722
00:43:32,290 --> 00:43:37,040
to a second mode, which is
with different frequencies.
723
00:43:37,040 --> 00:43:39,600
And there is an opposite
sign between this amplitude
724
00:43:39,600 --> 00:43:41,250
and that amplitude.
725
00:43:41,250 --> 00:43:47,100
So you have only
four coefficients--
726
00:43:47,100 --> 00:43:53,970
alpha, beta, phi 1, and phi
2, which are determined,
727
00:43:53,970 --> 00:43:56,028
which need initial conditions.
728
00:44:03,210 --> 00:44:06,770
So any arbitrary mode-- this
is the most general motion
729
00:44:06,770 --> 00:44:10,550
of the two coupled
oscillator systems.
730
00:44:10,550 --> 00:44:14,100
And to describe it in
specifically-- defined
731
00:44:14,100 --> 00:44:16,250
for a specific
configuration, you
732
00:44:16,250 --> 00:44:20,540
will have to determine the
values of alphas and phis.
733
00:44:20,540 --> 00:44:22,560
OK?
734
00:44:22,560 --> 00:44:28,940
So what I want to do is I want
to write down a specific motion
735
00:44:28,940 --> 00:44:31,790
for the following situation.
736
00:44:31,790 --> 00:44:37,040
So I keep position of x1 at 0.
737
00:44:37,040 --> 00:44:41,410
It's not moving, so
the velocity is 0.
738
00:44:41,410 --> 00:44:45,760
I displaced this one by
a small positive amount.
739
00:44:45,760 --> 00:44:51,270
So the position of number 2 at
t equals 0 is different than 0--
740
00:44:51,270 --> 00:44:53,970
some displacement
x0 or something.
741
00:44:53,970 --> 00:44:55,645
And its velocity is 0.
742
00:44:55,645 --> 00:44:56,520
And then I let it go.
743
00:44:59,230 --> 00:45:03,690
Again, this is not the ideal
decoupled oscillator, right?
744
00:45:03,690 --> 00:45:08,220
OK, and then you see
the things start moving.
745
00:45:08,220 --> 00:45:11,980
Let me try to show it again,
because it's not exactly here,
746
00:45:11,980 --> 00:45:13,520
so this one will be going on.
747
00:45:13,520 --> 00:45:15,580
So let's say this
one is running,
748
00:45:15,580 --> 00:45:18,550
and then I let this one go.
749
00:45:18,550 --> 00:45:23,390
And what you see here is
that this one is moving,
750
00:45:23,390 --> 00:45:24,740
and then that starts to move.
751
00:45:24,740 --> 00:45:26,170
This one stops.
752
00:45:26,170 --> 00:45:27,630
That starts moving.
753
00:45:27,630 --> 00:45:29,460
It starts being
complicated, right?
754
00:45:29,460 --> 00:45:31,860
It's kind of complicated motion.
755
00:45:31,860 --> 00:45:35,070
But whatever this
motion is, we know
756
00:45:35,070 --> 00:45:38,880
that it's simply those cosines
which are kind of adding up
757
00:45:38,880 --> 00:45:42,330
to give you this impression of
rather a complicated motion,
758
00:45:42,330 --> 00:45:43,360
right?
759
00:45:43,360 --> 00:45:46,515
So again, I let this one out.
760
00:45:46,515 --> 00:45:47,710
I let it go.
761
00:45:47,710 --> 00:45:50,870
This might be 0.
762
00:45:50,870 --> 00:45:52,160
So this one slows down.
763
00:45:52,160 --> 00:45:55,630
This starts going.
764
00:45:55,630 --> 00:45:57,280
And this one then slows down.
765
00:45:57,280 --> 00:45:59,290
The other one starts going.
766
00:45:59,290 --> 00:46:02,600
They kind of talk to each other.
767
00:46:02,600 --> 00:46:05,670
And it's this
combination of cosines.
768
00:46:05,670 --> 00:46:08,120
All right, so let's try
to write to simplify
769
00:46:08,120 --> 00:46:11,530
this for a specific case of
specific initial conditions.
770
00:46:26,910 --> 00:46:34,000
So I said x1 equals 0, to
equal 0 x1 velocity at 0
771
00:46:34,000 --> 00:46:35,480
is equal to 0.
772
00:46:35,480 --> 00:46:37,790
So those ones are not moving.
773
00:46:37,790 --> 00:46:46,470
X2 at 0 is equal to some sort
of x0 and x2 velocity at 0
774
00:46:46,470 --> 00:46:47,780
is equal to 0.
775
00:46:47,780 --> 00:46:49,110
So this one is displaced.
776
00:46:49,110 --> 00:46:50,510
They are all stationary.
777
00:46:50,510 --> 00:46:51,670
This one is at position 0.
778
00:46:51,670 --> 00:46:56,540
If I plug this in, it turns
out without lots of details
779
00:46:56,540 --> 00:47:02,060
that what you will get
to is that alpha will
780
00:47:02,060 --> 00:47:06,690
be equal to x0 divided by 2.
781
00:47:06,690 --> 00:47:10,850
Beta will be equal to
minus x0 divided by 2.
782
00:47:10,850 --> 00:47:14,840
And phi 1 will be equal
to phi 2 equal to 0.
783
00:47:14,840 --> 00:47:16,100
You can check.
784
00:47:16,100 --> 00:47:18,380
If you plug it into
those equations,
785
00:47:18,380 --> 00:47:23,120
if you plug t equals 0, phi
is equal to 0, et cetera,
786
00:47:23,120 --> 00:47:24,650
you will see that it works.
787
00:47:24,650 --> 00:47:28,850
So you can write down the
specific case of x1 of t
788
00:47:28,850 --> 00:47:41,726
to be x0 over 2 cosine omega1
t minus cosine omega2 t.
789
00:47:41,726 --> 00:47:44,640
It's because beta
has a negative sign.
790
00:47:44,640 --> 00:47:57,770
And x2 of t will be equal
to x0 over 2 cosine omega1 t
791
00:47:57,770 --> 00:48:01,860
plus cosine omega2 t.
792
00:48:01,860 --> 00:48:03,640
OK?
793
00:48:03,640 --> 00:48:06,580
So each of those objects
effectively feels
794
00:48:06,580 --> 00:48:09,460
the effects of
omega 1 and omega 2,
795
00:48:09,460 --> 00:48:11,860
but in a slightly different way.
796
00:48:11,860 --> 00:48:14,470
That's why their relative
motions are different.
797
00:48:14,470 --> 00:48:19,600
So what I will do now is I
will show you an animation.
798
00:48:19,600 --> 00:48:22,180
Hopefully, it works.
799
00:48:22,180 --> 00:48:24,970
And we will have time--
800
00:48:24,970 --> 00:48:28,850
since on the computer, you
can make things perfect.
801
00:48:28,850 --> 00:48:29,380
Let's do it.
802
00:48:29,380 --> 00:48:33,640
So I'll have-- running
a MatLab simulation.
803
00:48:33,640 --> 00:48:36,210
Let's see how it goes.
804
00:48:36,210 --> 00:48:38,560
Large.
805
00:48:38,560 --> 00:48:40,680
So what is going on
here is the following.
806
00:48:40,680 --> 00:48:42,050
I took some initial conditions.
807
00:48:42,050 --> 00:48:43,850
I'm not sure if it's
exactly the same.
808
00:48:43,850 --> 00:48:47,050
This was for the course
that I taught some time ago.
809
00:48:47,050 --> 00:48:49,840
What you see here
is the following--
810
00:48:49,840 --> 00:48:56,660
you have the green is the
normal mode, number 1.
811
00:48:59,570 --> 00:49:03,810
The magenta is
normal mode number 2.
812
00:49:03,810 --> 00:49:07,080
And blue and the red
are the actual pendula.
813
00:49:09,900 --> 00:49:10,740
All right?
814
00:49:10,740 --> 00:49:13,970
And the motion of
blue and red is simply
815
00:49:13,970 --> 00:49:16,920
a linear sum of the two.
816
00:49:16,920 --> 00:49:18,510
And what you see here is--
817
00:49:18,510 --> 00:49:21,590
and then I plot the
position of the blue and red
818
00:49:21,590 --> 00:49:24,390
in color, the function of time.
819
00:49:24,390 --> 00:49:25,190
So you see this--
820
00:49:25,190 --> 00:49:28,750
the fact that let's
say red is now stopped,
821
00:49:28,750 --> 00:49:30,640
and the blue is at maximum.
822
00:49:30,640 --> 00:49:34,890
And now, the red is picking up.
823
00:49:34,890 --> 00:49:38,430
And now the blue
stopped, and the red
824
00:49:38,430 --> 00:49:40,890
is going full swing, et cetera.
825
00:49:40,890 --> 00:49:43,070
And this is exactly what--
826
00:49:43,070 --> 00:49:46,269
this is the computer simulation
that shows you that one of them
827
00:49:46,269 --> 00:49:48,060
is going up, the other
one down, et cetera.
828
00:49:48,060 --> 00:49:49,874
And this is for the
certain combination
829
00:49:49,874 --> 00:49:50,790
of initial conditions.
830
00:49:50,790 --> 00:49:54,210
I could go change initial
conditions in my program
831
00:49:54,210 --> 00:49:55,830
and have a different behavior.
832
00:49:55,830 --> 00:49:59,850
But whatever happens,
I would be able to--
833
00:49:59,850 --> 00:50:05,430
it will always be a
combination of the two motions.
834
00:50:05,430 --> 00:50:14,150
Now, is there a way to disable
one of the normal modes?
835
00:50:14,150 --> 00:50:16,204
How would you disable
one of the normal modes?
836
00:50:19,452 --> 00:50:21,400
Is there a quick
way to set things
837
00:50:21,400 --> 00:50:26,700
up such that the second normal
mode, whichever you choose,
838
00:50:26,700 --> 00:50:28,450
doesn't show up in
their equations at all?
839
00:50:34,294 --> 00:50:36,729
AUDIENCE: You said [INAUDIBLE].
840
00:50:36,729 --> 00:50:37,703
BOLESLAW WYSLOUCH: Hmm?
841
00:50:37,703 --> 00:50:40,138
AUDIENCE: [INAUDIBLE]
842
00:50:44,934 --> 00:50:46,600
BOLESLAW WYSLOUCH:
Yeah, so what you do,
843
00:50:46,600 --> 00:50:48,500
is you just change the
initial conditions.
844
00:50:48,500 --> 00:50:50,170
So you set it up
at T equal to 0.
845
00:50:50,170 --> 00:50:55,690
I have initial conditions
that basically favor or demand
846
00:50:55,690 --> 00:50:59,320
that only in this
general equation
847
00:50:59,320 --> 00:51:02,080
either alpha or
beta is equal to 0.
848
00:51:02,080 --> 00:51:08,740
So for example, one possibility
is I move both of them
849
00:51:08,740 --> 00:51:11,680
at the same distance, and I
just let them go like this such
850
00:51:11,680 --> 00:51:15,010
that the spring is
irrelevant, right?
851
00:51:15,010 --> 00:51:17,679
How would I do it in my program?
852
00:51:17,679 --> 00:51:18,220
I don't know.
853
00:51:18,220 --> 00:51:21,440
I can, for example--
854
00:51:21,440 --> 00:51:28,510
I can, for example, set one
of the initial conditions to--
855
00:51:31,730 --> 00:51:32,670
this is still running.
856
00:51:32,670 --> 00:51:35,230
The old one is still running.
857
00:51:35,230 --> 00:51:40,151
So this is the moment.
858
00:51:40,151 --> 00:51:42,400
So what I did is I just
changed the initial condition.
859
00:51:44,960 --> 00:51:49,370
And you see, this is the type
of motion where one of the modes
860
00:51:49,370 --> 00:51:52,415
has stopped, just
you switched it off,
861
00:51:52,415 --> 00:51:54,540
and the other one is going
on, and then, of course,
862
00:51:54,540 --> 00:51:57,030
the total motion
is equal to that.
863
00:51:57,030 --> 00:51:59,720
And both of them happily go
with a constant amplitude.
864
00:51:59,720 --> 00:52:05,340
There is no shifting of
energy from one to another.
865
00:52:05,340 --> 00:52:10,130
So you can have all kinds of
motions by simply adjusting
866
00:52:10,130 --> 00:52:11,060
initial conditions.
867
00:52:11,060 --> 00:52:16,640
And those motions can be
done a very different way.
868
00:52:16,640 --> 00:52:19,810
So do you know--
869
00:52:19,810 --> 00:52:23,780
so this is how we can have
different shape of motion,
870
00:52:23,780 --> 00:52:26,300
depending on the
initial condition.
871
00:52:26,300 --> 00:52:30,830
Is there another
way for me to change
872
00:52:30,830 --> 00:52:34,551
the way this system behaves?
873
00:52:34,551 --> 00:52:35,300
Let's say I take--
874
00:52:35,300 --> 00:52:37,740
I have exactly
this system, and I
875
00:52:37,740 --> 00:52:41,220
want to change, for example,
the frequency of oscillations.
876
00:52:41,220 --> 00:52:44,200
How will I do it?
877
00:52:44,200 --> 00:52:46,838
It could be a very
expensive proposition, yes?
878
00:52:46,838 --> 00:52:47,786
AUDIENCE: Drive it?
879
00:52:47,786 --> 00:52:51,290
BOLESLAW WYSLOUCH: Yes, but
I don't want to drive it yet.
880
00:52:51,290 --> 00:52:54,100
I just want to have
it free oscillation.
881
00:52:54,100 --> 00:52:55,014
Yes?
882
00:52:55,014 --> 00:52:57,309
AUDIENCE: [INAUDIBLE]
883
00:52:57,309 --> 00:52:58,850
BOLESLAW WYSLOUCH:
Yeah, I could come
884
00:52:58,850 --> 00:53:00,930
and scratch it
away a little bit.
885
00:53:00,930 --> 00:53:03,580
And yes, the equations
depends on the mass.
886
00:53:03,580 --> 00:53:04,720
But I don't want to touch.
887
00:53:04,720 --> 00:53:06,494
I want to just have this thing.
888
00:53:06,494 --> 00:53:08,410
I don't want to make any
physical modification
889
00:53:08,410 --> 00:53:09,610
to the system.
890
00:53:09,610 --> 00:53:12,302
However, I can move it
into different places,
891
00:53:12,302 --> 00:53:14,260
any place you can think
of where I could really
892
00:53:14,260 --> 00:53:16,440
modify the solution.
893
00:53:16,440 --> 00:53:17,083
Yeah?
894
00:53:17,083 --> 00:53:18,030
AUDIENCE: To the moon.
895
00:53:18,030 --> 00:53:19,696
BOLESLAW WYSLOUCH:
To the moon, exactly.
896
00:53:19,696 --> 00:53:22,250
I could put it with me some
spaceship, and go to a place
897
00:53:22,250 --> 00:53:25,160
where the gravity
is different, right?
898
00:53:25,160 --> 00:53:26,690
Why not?
899
00:53:26,690 --> 00:53:28,480
So what would happen?
900
00:53:28,480 --> 00:53:35,440
So if gravity changes, then
basically what will happen
901
00:53:35,440 --> 00:53:38,044
is both this term and
that term will change.
902
00:53:38,044 --> 00:53:39,460
The spring will
remained the same.
903
00:53:39,460 --> 00:53:40,960
The mass will remain the same.
904
00:53:40,960 --> 00:53:44,560
So the relative magnitude
of omega 1 and omega 2
905
00:53:44,560 --> 00:53:46,900
will change.
906
00:53:46,900 --> 00:53:48,130
OK?
907
00:53:48,130 --> 00:53:51,910
So let's say, in fact, do
I have it in this one here?
908
00:53:51,910 --> 00:53:52,430
Yes.
909
00:53:52,430 --> 00:53:56,060
So let's say I do again.
910
00:53:56,060 --> 00:53:59,840
So this is what I
had before, right?
911
00:53:59,840 --> 00:54:03,770
So this is the one here which
is operating here on earth,
912
00:54:03,770 --> 00:54:05,690
and I let it go.
913
00:54:05,690 --> 00:54:07,610
I displaced it by
a certain distance.
914
00:54:07,610 --> 00:54:10,310
Let's say 1 millimeter,
and that's how it's gone.
915
00:54:10,310 --> 00:54:12,985
So now, let's take it
to, for example, Jupiter.
916
00:54:15,560 --> 00:54:17,860
So what do you think will
happen when we go to Jupiter.
917
00:54:20,510 --> 00:54:24,710
Jupiter, g, is much larger.
918
00:54:24,710 --> 00:54:26,560
OK?
919
00:54:26,560 --> 00:54:27,810
So what would happen to those?
920
00:54:32,350 --> 00:54:35,320
So the frequency
would be larger.
921
00:54:35,320 --> 00:54:37,860
Things will be faster, right?
922
00:54:37,860 --> 00:54:39,760
That's the higher frequency.
923
00:54:39,760 --> 00:54:43,474
But also the difference
between two frequencies
924
00:54:43,474 --> 00:54:44,140
will be smaller.
925
00:54:46,419 --> 00:54:48,460
And what happens when the
difference in frequency
926
00:54:48,460 --> 00:54:50,459
is smaller?
927
00:54:50,459 --> 00:54:52,500
You saw that there's the
fact that the energy was
928
00:54:52,500 --> 00:54:55,140
moving from one to the other.
929
00:54:55,140 --> 00:54:56,940
The thing would take--
930
00:54:56,940 --> 00:54:59,760
so one of them was oscillating,
the other one is stationary,
931
00:54:59,760 --> 00:55:02,251
then the other one would
pick up, et cetera.
932
00:55:02,251 --> 00:55:03,750
Do you think this
transfer of energy
933
00:55:03,750 --> 00:55:04,800
will be faster or slower?
934
00:55:08,650 --> 00:55:13,030
Two omegas closer to each other.
935
00:55:13,030 --> 00:55:15,540
Any guesses?
936
00:55:15,540 --> 00:55:16,290
AUDIENCE: Smaller.
937
00:55:16,290 --> 00:55:17,915
BOLESLAW WYSLOUCH:
Take kind of longer.
938
00:55:17,915 --> 00:55:19,560
Let's see what happens, right?
939
00:55:19,560 --> 00:55:24,489
So we go on the rocket,
and nowadays, you
940
00:55:24,489 --> 00:55:25,780
don't have to go to the rocket.
941
00:55:25,780 --> 00:55:26,779
Just remove one comment.
942
00:55:29,400 --> 00:55:32,220
And I went from about 10
meters per square second
943
00:55:32,220 --> 00:55:37,539
to 25 meters per square second,
and this is what is happening.
944
00:55:37,539 --> 00:55:38,080
Look at this.
945
00:55:38,080 --> 00:55:41,992
So first of all, this
identical system-- everything
946
00:55:41,992 --> 00:55:42,700
at the same time.
947
00:55:42,700 --> 00:55:43,283
It's the same.
948
00:55:43,283 --> 00:55:47,420
And so you see that
oscillations are much faster.
949
00:55:47,420 --> 00:55:51,940
So a number of amplitude
changes per second is larger.
950
00:55:51,940 --> 00:55:55,860
But it takes much
longer for the energy.
951
00:55:55,860 --> 00:55:57,340
So the red one is now stopping.
952
00:55:57,340 --> 00:56:01,480
It's now slowly coming up.
953
00:56:01,480 --> 00:56:04,470
So because the two frequencies
are closer to each other,
954
00:56:04,470 --> 00:56:07,560
they stay--
955
00:56:07,560 --> 00:56:12,890
it takes longer for them to
shift from one to the other.
956
00:56:12,890 --> 00:56:14,030
OK?
957
00:56:14,030 --> 00:56:16,670
So we are done at Jupiter.
958
00:56:16,670 --> 00:56:21,350
Let's now go to the Moon,
which has much lower
959
00:56:21,350 --> 00:56:23,772
gravitational acceleration.
960
00:56:23,772 --> 00:56:24,730
Let's see what happens.
961
00:56:27,990 --> 00:56:33,530
Again by logical
argument-- if something--
962
00:56:33,530 --> 00:56:36,110
so the smaller
gravitation accelerations
963
00:56:36,110 --> 00:56:39,830
means that the
frequency is now lower.
964
00:56:39,830 --> 00:56:43,370
So the pendula will move slower.
965
00:56:43,370 --> 00:56:45,779
However, the difference
between frequency
966
00:56:45,779 --> 00:56:47,570
will be larger, because
the spring is still
967
00:56:47,570 --> 00:56:48,981
the same strength.
968
00:56:48,981 --> 00:56:51,230
So it turns out that even
though everything is slower,
969
00:56:51,230 --> 00:56:54,840
but the energy transfer
will actually be faster.
970
00:56:54,840 --> 00:56:59,130
So let's try to see what
happens on the Moon.
971
00:56:59,130 --> 00:57:00,425
It's OK.
972
00:57:03,964 --> 00:57:09,040
It's a little bit not completely
clear what's going on,
973
00:57:09,040 --> 00:57:14,067
but you see, actually the motion
is kind of a little strange.
974
00:57:14,067 --> 00:57:14,900
Look at the red one.
975
00:57:14,900 --> 00:57:16,170
The red one is stopping.
976
00:57:16,170 --> 00:57:19,885
Then it's going halfway out.
977
00:57:19,885 --> 00:57:22,450
It looks kind of
messy, doesn't it?
978
00:57:22,450 --> 00:57:24,610
And so it doesn't show
up here very well,
979
00:57:24,610 --> 00:57:28,530
because the parameters have
changed so much that I have--
980
00:57:28,530 --> 00:57:39,990
I have those fixed
pictures which are--
981
00:57:39,990 --> 00:57:40,830
just a second.
982
00:57:40,830 --> 00:57:41,530
I'll show you.
983
00:57:45,790 --> 00:57:47,540
So this is the picture on the--
984
00:57:47,540 --> 00:57:51,530
some sort of stationary
picture on the Earth.
985
00:57:51,530 --> 00:57:53,730
I saw one of them
up, the other one--
986
00:57:53,730 --> 00:57:56,489
you see them shift
from one to the other.
987
00:57:56,489 --> 00:57:58,780
And you can see kind of the
frequency of how the energy
988
00:57:58,780 --> 00:58:00,400
shifts from one to the other.
989
00:58:00,400 --> 00:58:03,790
And also you can see the
frequency going up and down
990
00:58:03,790 --> 00:58:06,430
for the same exact conditions.
991
00:58:06,430 --> 00:58:10,060
This is now, just a
moment, this is a Jupiter.
992
00:58:10,060 --> 00:58:12,580
So Jupiter, you see that
the frequency itself it's
993
00:58:12,580 --> 00:58:13,970
much higher.
994
00:58:13,970 --> 00:58:20,190
And the energy transfer between
the two things takes longer.
995
00:58:20,190 --> 00:58:23,520
And on the Moon however,
the oscillations
996
00:58:23,520 --> 00:58:25,170
actually look really weird.
997
00:58:25,170 --> 00:58:27,180
This is an example
of one of them.
998
00:58:27,180 --> 00:58:32,370
It's kind of, you know, the two
frequencies are so far away,
999
00:58:32,370 --> 00:58:36,960
and it's really not even
a nice oscillatory motion.
1000
00:58:36,960 --> 00:58:39,471
It's some sort of--
1001
00:58:39,471 --> 00:58:43,530
it's much less
obvious that this is
1002
00:58:43,530 --> 00:58:47,250
a superposition of two cosines,
because they kind of are
1003
00:58:47,250 --> 00:58:48,920
exactly out of phase.
1004
00:58:48,920 --> 00:58:51,270
So the motion is
kind of complete.
1005
00:58:51,270 --> 00:58:52,620
Anyway, so this is--
1006
00:58:52,620 --> 00:58:56,190
actually, so the lesson
is that the exact shape,
1007
00:58:56,190 --> 00:58:59,810
the exact motion, we
know that can always be
1008
00:58:59,810 --> 00:59:01,950
decomposed into simple motions.
1009
00:59:01,950 --> 00:59:03,520
If you put them
together, things may
1010
00:59:03,520 --> 00:59:05,340
get really interesting
and complicated,
1011
00:59:05,340 --> 00:59:08,970
depending on what sort of
frequencies we are running
1012
00:59:08,970 --> 00:59:10,620
and what sort of--
1013
00:59:10,620 --> 00:59:14,670
what sort of initial
conditions we have.
1014
00:59:14,670 --> 00:59:15,340
All right?
1015
00:59:15,340 --> 00:59:15,970
Yes?
1016
00:59:15,970 --> 00:59:18,490
Any questions?
1017
00:59:18,490 --> 00:59:19,940
Yes?
1018
00:59:19,940 --> 00:59:23,093
AUDIENCE: It's talking about
the center mass of the system
1019
00:59:23,093 --> 00:59:24,920
or just one of the two --?
1020
00:59:24,920 --> 00:59:26,461
BOLESLAW WYSLOUCH:
This one, I think,
1021
00:59:26,461 --> 00:59:28,650
this one is just one of them.
1022
00:59:28,650 --> 00:59:30,990
Actually, the one--
on the difference-- it
1023
00:59:30,990 --> 00:59:32,890
normally doesn't matter.
1024
00:59:32,890 --> 00:59:35,790
What matters this
is the frequency
1025
00:59:35,790 --> 00:59:39,840
and how these move to the other.
1026
00:59:39,840 --> 00:59:42,780
OK?
1027
00:59:42,780 --> 00:59:45,891
Let's just forget about it.
1028
00:59:45,891 --> 00:59:48,500
Just keep it.
1029
00:59:48,500 --> 00:59:53,120
So let me now talk
about this thing, which
1030
00:59:53,120 --> 00:59:57,650
is called beat phenomenon,
because when you look
1031
00:59:57,650 --> 01:00:01,660
at the motion of one
of those objects,
1032
01:00:01,660 --> 01:00:04,430
or the difference between
them or whatever, there's
1033
01:00:04,430 --> 01:00:08,907
something kind of interesting
which can be extracted
1034
01:00:08,907 --> 01:00:09,740
for those equations.
1035
01:00:09,740 --> 01:00:11,400
Let's look at these
equations here.
1036
01:00:11,400 --> 01:00:12,380
Let's look at mass 1.
1037
01:00:16,960 --> 01:00:19,610
This is mass 1 and mass 2.
1038
01:00:19,610 --> 01:00:23,143
So I can rewrite those solutions
a little bit different.
1039
01:00:30,390 --> 01:00:34,030
And so what I want
to do is I want to--
1040
01:00:34,030 --> 01:00:38,660
you see, this is a
difference of two cosines.
1041
01:00:38,660 --> 01:00:40,700
This is a sum of two cosines.
1042
01:00:40,700 --> 01:00:43,602
There are lots of neat
trigonometrical identities
1043
01:00:43,602 --> 01:00:44,310
which we can use.
1044
01:00:44,310 --> 01:00:46,520
So we just-- we do
zero physics here.
1045
01:00:46,520 --> 01:00:50,790
We just rewrite the
trigonometrical formulas.
1046
01:00:50,790 --> 01:00:54,540
So I do exactly this,
but I rewrite it.
1047
01:00:54,540 --> 01:00:59,650
I use, for example, some of--
you have cosine alpha plus
1048
01:00:59,650 --> 01:01:04,005
cosine beta is equal to--
1049
01:01:04,005 --> 01:01:13,330
two cosine-- is equal to two
cosine alpha plus beta divided
1050
01:01:13,330 --> 01:01:20,000
by 2 multiplied by cosine
alpha minus beta divided by 2.
1051
01:01:20,000 --> 01:01:20,500
Right?
1052
01:01:20,500 --> 01:01:23,950
That's the
trigonometric identity.
1053
01:01:23,950 --> 01:01:24,940
Right?
1054
01:01:24,940 --> 01:01:27,520
So let's just use this to write
this down and what you get
1055
01:01:27,520 --> 01:01:29,060
is x1--
1056
01:01:29,060 --> 01:01:40,560
x1 of t is equal to
minus x0 sine of omega 1
1057
01:01:40,560 --> 01:01:50,430
plus omega 2 divided by 2 times
sine omega 1 minus omega 2
1058
01:01:50,430 --> 01:01:54,240
divided by 2 times t.
1059
01:01:54,240 --> 01:02:03,970
And x2 t is equal to x0,
some amplitude cosine omega 1
1060
01:02:03,970 --> 01:02:11,410
plus omega 2 divided by 2
cosine omega 1 minus omega 2
1061
01:02:11,410 --> 01:02:13,660
divided by t.
1062
01:02:13,660 --> 01:02:17,230
So again, we did
zero physics here.
1063
01:02:17,230 --> 01:02:20,252
We just rewrote the simple
trigonometric equations.
1064
01:02:20,252 --> 01:02:22,210
But what you see is
something interesting here.
1065
01:02:22,210 --> 01:02:27,580
So there is-- we have
those two frequencies which
1066
01:02:27,580 --> 01:02:29,170
are playing a role.
1067
01:02:29,170 --> 01:02:31,847
And for example, at Jupiter,
those two frequencies
1068
01:02:31,847 --> 01:02:33,430
are actually very
close to each other,
1069
01:02:33,430 --> 01:02:35,780
because everything is
dominated by the gravity,
1070
01:02:35,780 --> 01:02:37,960
and we have a very weak spring.
1071
01:02:37,960 --> 01:02:39,850
So the omega 1 and
omega 2 actually
1072
01:02:39,850 --> 01:02:42,800
are very close to each other.
1073
01:02:42,800 --> 01:02:45,650
So this thing, this
term here, kind of
1074
01:02:45,650 --> 01:02:47,840
goes omega 1 plus
omega 2 divided by 2
1075
01:02:47,840 --> 01:02:50,860
is like omega, right?
1076
01:02:50,860 --> 01:02:55,520
100 plus 105 divided
by 2 is about 100.
1077
01:02:55,520 --> 01:02:57,400
Whereas this one here
carries information
1078
01:02:57,400 --> 01:03:00,420
about the difference
of frequencies--
1079
01:03:00,420 --> 01:03:04,260
100, 102, the difference
is 2, which is very small.
1080
01:03:04,260 --> 01:03:07,260
So how would this look like?
1081
01:03:07,260 --> 01:03:10,910
So if you make a plot
under some conditions,
1082
01:03:10,910 --> 01:03:17,750
you can, let's say,
so the two frequencies
1083
01:03:17,750 --> 01:03:19,250
are close to each other.
1084
01:03:24,810 --> 01:03:29,900
So if omega 1 is
close to omega 2--
1085
01:03:29,900 --> 01:03:37,350
for example, omega 1 is
0.9 times omega 2, right?
1086
01:03:37,350 --> 01:03:41,420
This is roughly what we have
on Earth in case of our system
1087
01:03:41,420 --> 01:03:43,630
here.
1088
01:03:43,630 --> 01:03:47,770
Then omega 1 plus
omega 2 divided 2
1089
01:03:47,770 --> 01:03:55,400
would be about 0.95 omega
1, omega 2, I think,
1090
01:03:55,400 --> 01:03:57,660
which is approximately
equal to omega 2
1091
01:03:57,660 --> 01:04:02,880
or omega 1 and omega 1
minus omega 2 divided by 2
1092
01:04:02,880 --> 01:04:07,340
will be about minus
0.05 times omega 2--
1093
01:04:07,340 --> 01:04:12,100
much, much smaller than that.
1094
01:04:12,100 --> 01:04:16,420
So we have-- so this term here--
1095
01:04:16,420 --> 01:04:21,790
it basically oscillates
at the frequency of omega,
1096
01:04:21,790 --> 01:04:24,900
of the frequency of the
individual pendulum.
1097
01:04:24,900 --> 01:04:28,060
And the other term is
much, much smaller.
1098
01:04:28,060 --> 01:04:29,010
How does this look?
1099
01:04:29,010 --> 01:04:35,300
Well, it turns out that if
you make a sketch of this,
1100
01:04:35,300 --> 01:04:38,972
if you do signs, for
example, it looks like this.
1101
01:04:44,268 --> 01:04:44,768
OK?
1102
01:04:47,666 --> 01:04:50,622
So there are in fact two--
1103
01:04:50,622 --> 01:04:53,080
when you look at this picture,
you can see two frequencies.
1104
01:04:53,080 --> 01:04:56,470
One which is clear the
oscillation of the--
1105
01:04:56,470 --> 01:05:00,700
high-frequency oscillation
of things moving up and down.
1106
01:05:00,700 --> 01:05:05,830
But there's also this kind
of overarching frequency
1107
01:05:05,830 --> 01:05:08,800
of much smaller
frequency, and this
1108
01:05:08,800 --> 01:05:12,290
is what corresponds to a
difference of two things.
1109
01:05:12,290 --> 01:05:16,990
So in a sense, if you
look at this formula here,
1110
01:05:16,990 --> 01:05:18,970
you have oscillation,
which is happening
1111
01:05:18,970 --> 01:05:22,930
very quickly with a typical
oscillation of the system.
1112
01:05:22,930 --> 01:05:26,780
But this is like a
modulation of the amplitude.
1113
01:05:26,780 --> 01:05:29,440
So the amplitude of
the signal is changing.
1114
01:05:29,440 --> 01:05:31,240
And this is what you see here.
1115
01:05:31,240 --> 01:05:34,410
This is exactly the
picture out there.
1116
01:05:34,410 --> 01:05:35,810
So the system oscillates.
1117
01:05:35,810 --> 01:05:41,980
So one of those pendula,
either of them, is moving fast.
1118
01:05:41,980 --> 01:05:43,240
But it's going faster.
1119
01:05:43,240 --> 01:05:46,030
It's amplitude is larger,
and after some time,
1120
01:05:46,030 --> 01:05:47,230
it slows down to 0.
1121
01:05:47,230 --> 01:05:50,590
It goes higher and
slows down to 0.
1122
01:05:50,590 --> 01:05:51,550
And you've seen this.
1123
01:05:51,550 --> 01:05:57,700
We can do it again
here that both of them
1124
01:05:57,700 --> 01:05:59,950
oscillate at roughly
the same frequency,
1125
01:05:59,950 --> 01:06:03,430
but their individual
amplitudes are changing.
1126
01:06:03,430 --> 01:06:06,020
And this transmission of--
1127
01:06:06,020 --> 01:06:09,415
you know, one of them moving
full blast, the other one
1128
01:06:09,415 --> 01:06:10,660
moving full blast.
1129
01:06:10,660 --> 01:06:14,500
There's this kind of
frequency of energy
1130
01:06:14,500 --> 01:06:19,720
moving from one to the other,
which is something called beat.
1131
01:06:19,720 --> 01:06:23,470
This a beat system,
beat phenomenon somehow
1132
01:06:23,470 --> 01:06:27,850
that energy is moving from
one place to another one.
1133
01:06:27,850 --> 01:06:32,230
And we can have
some demonstration
1134
01:06:32,230 --> 01:06:33,680
of how this happens.
1135
01:06:33,680 --> 01:06:34,960
So we see this here.
1136
01:06:34,960 --> 01:06:37,660
We see it on the pendula.
1137
01:06:37,660 --> 01:06:41,864
We saw it on the
computer simulation.
1138
01:06:41,864 --> 01:06:43,280
But now what we
are going to do is
1139
01:06:43,280 --> 01:06:45,950
we're going to try
to hear it, right?
1140
01:06:45,950 --> 01:06:50,030
So this is a demonstration
which maybe it works, maybe not.
1141
01:06:50,030 --> 01:06:52,820
So let me-- it will work, OK?
1142
01:06:52,820 --> 01:06:54,770
So let me explain what we have.
1143
01:06:54,770 --> 01:06:58,860
So we have two speakers.
1144
01:06:58,860 --> 01:07:06,675
And they basically go on very,
very similar frequencies,
1145
01:07:06,675 --> 01:07:07,470
all right?
1146
01:07:07,470 --> 01:07:11,340
So they both work at
similar frequencies.
1147
01:07:14,436 --> 01:07:17,130
And so when I switched
on, you should hear--
1148
01:07:17,130 --> 01:07:18,628
hear the sound.
1149
01:07:18,628 --> 01:07:19,562
[HUM SOUND]
1150
01:07:20,062 --> 01:07:20,969
OK?
1151
01:07:20,969 --> 01:07:22,010
So this is the frequency.
1152
01:07:22,010 --> 01:07:26,170
I believe it's just one of
them is working, and you know,
1153
01:07:26,170 --> 01:07:28,680
this is just one
pendulum that is going
1154
01:07:28,680 --> 01:07:31,350
on that given frequency, right?
1155
01:07:31,350 --> 01:07:33,420
Then I will switch a
second loudspeaker.
1156
01:07:36,390 --> 01:07:37,875
[HUM SOUND]
1157
01:08:05,595 --> 01:08:08,070
Can you hear this kind of--
1158
01:08:08,070 --> 01:08:09,060
wiggle?
1159
01:08:09,060 --> 01:08:11,535
We'll change the
frequency a little.
1160
01:08:15,495 --> 01:08:18,960
This is another frequency
of the original sound.
1161
01:08:18,960 --> 01:08:22,490
And it's kind of the loudness of
the sound overall is changing.
1162
01:08:26,482 --> 01:08:29,476
All right?
1163
01:08:29,476 --> 01:08:31,971
This is faster.
1164
01:08:31,971 --> 01:08:35,960
This is kind of extra,
extra sound which
1165
01:08:35,960 --> 01:08:40,137
you hear is the difference of
mainly the frequency is not
1166
01:08:40,137 --> 01:08:42,089
stable here, so I'll change it.
1167
01:08:46,490 --> 01:08:48,319
Right?
1168
01:08:48,319 --> 01:08:52,180
So this is, again, this is a
single one, perfectly constant
1169
01:08:52,180 --> 01:08:54,939
frequency, no
change in amplitude,
1170
01:08:54,939 --> 01:08:55,870
no change in loudness.
1171
01:08:58,560 --> 01:09:01,892
Put them together, right?
1172
01:09:01,892 --> 01:09:02,850
That's what they do.
1173
01:09:02,850 --> 01:09:06,140
So if you have two, and I
can adjust the frequency,
1174
01:09:06,140 --> 01:09:10,670
and the frequency is close,
then this frequency of changing
1175
01:09:10,670 --> 01:09:12,000
is very slow.
1176
01:09:12,000 --> 01:09:14,590
So you can actually hear it.
1177
01:09:14,590 --> 01:09:16,819
Let me switch it off.
1178
01:09:16,819 --> 01:09:18,500
So this is the effect of beats.
1179
01:09:18,500 --> 01:09:26,359
I can maybe show you another
simulation of this works.
1180
01:09:26,359 --> 01:09:29,200
Let's See.
1181
01:09:29,200 --> 01:09:36,120
This one is oops, just a second.
1182
01:09:36,120 --> 01:09:37,779
Let's see what it is.
1183
01:09:42,170 --> 01:09:47,649
OK, so this is just
a single frequency.
1184
01:09:47,649 --> 01:09:50,220
OK, again, I plot some pendulum.
1185
01:09:50,220 --> 01:09:53,069
Then I can plot--
1186
01:09:53,069 --> 01:09:56,618
sorry, no this one is this.
1187
01:09:56,618 --> 01:10:03,923
I can-- this one.
1188
01:10:03,923 --> 01:10:05,675
OK, we'll just plot it here.
1189
01:10:09,680 --> 01:10:10,770
Maybe we can see.
1190
01:10:10,770 --> 01:10:13,820
So there's a red one,
and there's a blue one.
1191
01:10:13,820 --> 01:10:15,860
And I plot two
plots independently
1192
01:10:15,860 --> 01:10:16,740
on top of each other.
1193
01:10:16,740 --> 01:10:18,480
So they have an amplitude of 1.
1194
01:10:18,480 --> 01:10:21,360
And clearly, you see that they
have a different frequency.
1195
01:10:21,360 --> 01:10:23,810
So the red one is going
with some frequency.
1196
01:10:23,810 --> 01:10:25,880
The blue one is going
with some other frequency.
1197
01:10:25,880 --> 01:10:27,980
Sometimes they agree.
1198
01:10:27,980 --> 01:10:30,170
Sometimes they do
not agree, right?
1199
01:10:30,170 --> 01:10:33,250
And the places where they meet--
1200
01:10:33,250 --> 01:10:34,680
they are on top of each other.
1201
01:10:34,680 --> 01:10:38,400
This is where when you
add them up together,
1202
01:10:38,400 --> 01:10:40,430
this is where they
will be large.
1203
01:10:40,430 --> 01:10:42,710
In the places where
they're out of phase,
1204
01:10:42,710 --> 01:10:44,560
they will cancel each other.
1205
01:10:44,560 --> 01:10:47,210
So if you take two
of those together,
1206
01:10:47,210 --> 01:10:50,680
same amplitude, just
slightly different frequency,
1207
01:10:50,680 --> 01:10:52,056
and you simply make a linear--
1208
01:10:57,020 --> 01:11:01,780
superposition of the two, you
will get exactly the beating
1209
01:11:01,780 --> 01:11:02,280
effect.
1210
01:11:02,280 --> 01:11:04,400
So I just took two
of those pictures
1211
01:11:04,400 --> 01:11:07,040
before I added them together
and got exactly that.
1212
01:11:07,040 --> 01:11:09,080
You have a maximum,
minima, et cetera.
1213
01:11:09,080 --> 01:11:13,760
And you see this overall beat
frequency, and the carrier,
1214
01:11:13,760 --> 01:11:16,100
it's called carrier frequency.
1215
01:11:16,100 --> 01:11:20,390
And this is something that,
again, happens very often.
1216
01:11:20,390 --> 01:11:23,700
There's another
demonstration here.
1217
01:11:23,700 --> 01:11:27,470
I have two tuning
forks, and they
1218
01:11:27,470 --> 01:11:29,390
are very similar frequency.
1219
01:11:29,390 --> 01:11:33,900
So first, I will show you
that they are coupled.
1220
01:11:33,900 --> 01:11:38,420
They are coupled
because I gave this guy
1221
01:11:38,420 --> 01:11:40,810
some initial condition.
1222
01:11:40,810 --> 01:11:41,640
It's going.
1223
01:11:41,640 --> 01:11:43,440
Then I stop it.
1224
01:11:43,440 --> 01:11:46,140
But there's still sound,
because the second one picked up
1225
01:11:46,140 --> 01:11:48,350
some energy, and it took off.
1226
01:11:48,350 --> 01:11:50,100
Of course, you don't see them.
1227
01:11:50,100 --> 01:11:53,640
So basically, what I'm saying is
that I [TONE] give this energy.
1228
01:11:53,640 --> 01:11:55,610
This one is
completely stationary.
1229
01:11:55,610 --> 01:11:58,926
Now energy is slowly
moving to the other one.
1230
01:11:58,926 --> 01:12:00,800
I stop this guy, and
this guy is still going.
1231
01:12:03,160 --> 01:12:05,320
So the energy is
being transferred
1232
01:12:05,320 --> 01:12:07,720
by this air oscillating here.
1233
01:12:07,720 --> 01:12:12,070
The coupling goes through the
air to the sound here, right?
1234
01:12:12,070 --> 01:12:13,960
And they have very
similar frequency.
1235
01:12:13,960 --> 01:12:17,260
So they are nicely coupled.
1236
01:12:17,260 --> 01:12:19,999
But what we can also do--
1237
01:12:19,999 --> 01:12:21,478
we can [TONE].
1238
01:12:25,261 --> 01:12:25,760
Right?
1239
01:12:25,760 --> 01:12:27,620
So they're both going.
1240
01:12:27,620 --> 01:12:28,702
Do you hear the beats?
1241
01:12:31,594 --> 01:12:32,558
[TONE]
1242
01:12:34,968 --> 01:12:36,900
Not really.
1243
01:12:36,900 --> 01:12:41,040
In fact, if they would have
exactly identical frequency,
1244
01:12:41,040 --> 01:12:41,540
right?
1245
01:12:41,540 --> 01:12:43,790
If they will be
perfectly the same,
1246
01:12:43,790 --> 01:12:46,290
then the difference
would be 0, and there
1247
01:12:46,290 --> 01:12:47,400
will be no beats at all.
1248
01:12:47,400 --> 01:12:49,290
The period of beats
will be infinitely long,
1249
01:12:49,290 --> 01:12:52,186
so it will take forever
for us to hear anything.
1250
01:12:52,186 --> 01:12:54,060
So what we can do-- we
can break one of them.
1251
01:12:54,060 --> 01:12:58,750
We can add some sort of weight.
1252
01:12:58,750 --> 01:12:59,800
Some are here.
1253
01:12:59,800 --> 01:13:02,450
There's some magic place
where it works best.
1254
01:13:02,450 --> 01:13:04,370
So what I would do is
I will break this one.
1255
01:13:04,370 --> 01:13:06,590
I will modify its frequency.
1256
01:13:06,590 --> 01:13:07,910
That's another way to modify.
1257
01:13:07,910 --> 01:13:09,701
I don't have to go to
Jupiter to modify it,
1258
01:13:09,701 --> 01:13:13,980
because this one is just
a little mass here, right?
1259
01:13:13,980 --> 01:13:14,900
[TONE]
1260
01:13:18,770 --> 01:13:19,440
Ah, cool.
1261
01:13:19,440 --> 01:13:23,160
AUDIENCE: Is that [INAUDIBLE]?
1262
01:13:23,160 --> 01:13:27,278
BOLESLAW WYSLOUCH: Really,
this is actually a huge effect.
1263
01:13:27,278 --> 01:13:28,262
[TONE]
1264
01:13:30,230 --> 01:13:32,930
You can clearly see that they
are going up and down, up
1265
01:13:32,930 --> 01:13:36,560
and down, because the frequency
is slightly different.
1266
01:13:36,560 --> 01:13:39,260
So now, this thing is probably--
1267
01:13:39,260 --> 01:13:42,070
I know it's a period, a
fraction of a second, right?
1268
01:13:42,070 --> 01:13:42,752
Yes?
1269
01:13:42,752 --> 01:13:45,112
AUDIENCE: Should both of
those sine and cosines
1270
01:13:45,112 --> 01:13:47,460
have Ts in their arguments?
1271
01:13:47,460 --> 01:13:51,005
BOLESLAW WYSLOUCH:
Of course always.
1272
01:13:51,005 --> 01:13:54,280
They are both time
dependent, yeah.
1273
01:13:54,280 --> 01:13:57,080
This is the fast
thing, and this is
1274
01:13:57,080 --> 01:14:02,100
this time-dependent
modulation, yeah.
1275
01:14:02,100 --> 01:14:07,287
All right, so
where are my notes?
1276
01:14:07,287 --> 01:14:11,110
So this is the--
1277
01:14:11,110 --> 01:14:13,790
this is how the--
1278
01:14:13,790 --> 01:14:15,910
so we were able to
set up the system,
1279
01:14:15,910 --> 01:14:17,770
put in some of the
matrix equation,
1280
01:14:17,770 --> 01:14:20,310
kind of solved it, found
two frequencies, et cetera.
1281
01:14:20,310 --> 01:14:21,450
There is one more--
1282
01:14:21,450 --> 01:14:23,500
one additional
trick, which you can
1283
01:14:23,500 --> 01:14:29,140
do to describe the motion
of a coupled pendula.
1284
01:14:29,140 --> 01:14:34,940
And that is, in a sense,
force mathematically,
1285
01:14:34,940 --> 01:14:39,340
force the normal modes
from sort of early on, to
1286
01:14:39,340 --> 01:14:42,720
instead of, so far, when
we talked about pendula,
1287
01:14:42,720 --> 01:14:48,300
we describe their motion in
terms of motion of number
1288
01:14:48,300 --> 01:14:50,503
1, motion of number 2.
1289
01:14:50,503 --> 01:14:52,970
It turns out we can
rewrite the equation
1290
01:14:52,970 --> 01:14:56,370
into some sort of
new variables, where,
1291
01:14:56,370 --> 01:15:00,980
so-called normal coordinates,
where you'll simultaneously
1292
01:15:00,980 --> 01:15:05,120
describe both of them
and then kind of mix
1293
01:15:05,120 --> 01:15:08,470
them together to
have a new formula,
1294
01:15:08,470 --> 01:15:11,090
just rewrite the equation
in terms of new variables.
1295
01:15:11,090 --> 01:15:13,565
So you do change of variables.
1296
01:15:13,565 --> 01:15:19,520
So instead of keeping track
of x1 and x2 independently,
1297
01:15:19,520 --> 01:15:22,550
you define something
which I called
1298
01:15:22,550 --> 01:15:30,300
u1, which is simply x1
plus x2, and I define
1299
01:15:30,300 --> 01:15:36,020
u2, which is x1 minus x2.
1300
01:15:36,020 --> 01:15:38,670
So instead of talking about
x1 and x2 independently,
1301
01:15:38,670 --> 01:15:41,340
I have a sum of
them and difference.
1302
01:15:41,340 --> 01:15:42,190
Why not?
1303
01:15:42,190 --> 01:15:42,800
Right?
1304
01:15:42,800 --> 01:15:43,960
Two variables.
1305
01:15:43,960 --> 01:15:47,360
I can always go back and
get x1 and x2 if I want to.
1306
01:15:47,360 --> 01:15:51,620
So if one tells me that u1
is 1 centimeter and u2 2
1307
01:15:51,620 --> 01:15:54,230
centimeters, I can always
go and get x1 and x2
1308
01:15:54,230 --> 01:15:55,700
if I want to, right?
1309
01:15:55,700 --> 01:15:57,060
So I can do it.
1310
01:15:57,060 --> 01:16:01,865
And it turns out that if
I plot those variables
1311
01:16:01,865 --> 01:16:05,150
in, in other words, I take
the original equations, which
1312
01:16:05,150 --> 01:16:09,660
I conveniently erased and
make a sum or difference,
1313
01:16:09,660 --> 01:16:12,470
it turns out that this
coupling kind of separates.
1314
01:16:12,470 --> 01:16:17,450
So I will end up having
two separate equations
1315
01:16:17,450 --> 01:16:18,290
for this one.
1316
01:16:18,290 --> 01:16:20,990
So in general, the
equation of motion
1317
01:16:20,990 --> 01:16:23,870
would be-- would look
like, so let's say
1318
01:16:23,870 --> 01:16:33,620
I can write down m x1 plus
x2 is equal to minus m g
1319
01:16:33,620 --> 01:16:44,110
over l times x1 plus x2.
1320
01:16:44,110 --> 01:16:47,780
OK, this is when I
add two equations.
1321
01:16:47,780 --> 01:16:52,370
And the other equation
when I subtract them--
1322
01:16:52,370 --> 01:17:06,676
minus x2 is equal to minus mg
over l plus 2k x1 minus x2.
1323
01:17:06,676 --> 01:17:09,120
I think that's
what is coming out.
1324
01:17:09,120 --> 01:17:15,520
So if I add and subtract the two
original equations of motion,
1325
01:17:15,520 --> 01:17:17,530
which I don't know if
I have them somewhere,
1326
01:17:17,530 --> 01:17:19,680
and you can look
back, then you end up
1327
01:17:19,680 --> 01:17:25,170
having those crossed
terms drop out.
1328
01:17:25,170 --> 01:17:28,110
And you have one, which has only
this coefficient, the other one
1329
01:17:28,110 --> 01:17:30,930
which has that coefficient.
1330
01:17:30,930 --> 01:17:33,480
And this immediately--
and it looks--
1331
01:17:33,480 --> 01:17:37,050
if I now write it in terms
of normal coordinates,
1332
01:17:37,050 --> 01:17:44,190
then I have that m u1 double
dot is equal to simply minus mg
1333
01:17:44,190 --> 01:17:53,910
over l, u1, and m u2 double
dot is equal to minus mg over
1334
01:17:53,910 --> 01:17:58,610
l plus 2k times u2.
1335
01:17:58,610 --> 01:18:03,050
And if you look at
those two equations,
1336
01:18:03,050 --> 01:18:04,990
it turns out that
they are not coupled.
1337
01:18:04,990 --> 01:18:13,380
Each of them is a question
of a one-dimensional harmonic
1338
01:18:13,380 --> 01:18:15,150
oscillator.
1339
01:18:15,150 --> 01:18:17,120
The first part one
only depends on u1.
1340
01:18:17,120 --> 01:18:20,940
The second one
only depends on u2.
1341
01:18:20,940 --> 01:18:26,230
And you can see the oscillating
frequency with your own eyes.
1342
01:18:26,230 --> 01:18:29,860
So no, the determinants needed
no matrices, no nothing.
1343
01:18:29,860 --> 01:18:32,710
We just added and subtracted
the two equations,
1344
01:18:32,710 --> 01:18:35,790
and things magically separated.
1345
01:18:35,790 --> 01:18:37,060
All right?
1346
01:18:37,060 --> 01:18:41,350
So sometimes, especially in case
of very simple and symmetric
1347
01:18:41,350 --> 01:18:43,420
systems, if you
introduce new variables,
1348
01:18:43,420 --> 01:18:45,200
you can simplify your
life tremendously,
1349
01:18:45,200 --> 01:18:51,418
and these are called normal
variables, normal coordinates.
1350
01:18:58,390 --> 01:19:01,380
And it turns out that
you can always do that.
1351
01:19:01,380 --> 01:19:03,450
So you can always have
a linear combination
1352
01:19:03,450 --> 01:19:07,470
of parameters for arbitrary
size coupled oscillators system
1353
01:19:07,470 --> 01:19:11,510
where you combine
different coordinates,
1354
01:19:11,510 --> 01:19:15,600
and you basically force
the system to behave
1355
01:19:15,600 --> 01:19:21,630
in a way in which it induces
the single oscillation, single
1356
01:19:21,630 --> 01:19:22,560
frequency.
1357
01:19:22,560 --> 01:19:25,800
So this is, again, a
very powerful trick,
1358
01:19:25,800 --> 01:19:28,470
but usually for most
cases, you can do that
1359
01:19:28,470 --> 01:19:31,620
only after you have solved
it, after you've found out
1360
01:19:31,620 --> 01:19:32,760
normal modes, et cetera.
1361
01:19:32,760 --> 01:19:35,410
So after you know your normal
mode, then you can say, ha, ha,
1362
01:19:35,410 --> 01:19:38,100
I can I can introduce
normal variables
1363
01:19:38,100 --> 01:19:39,480
and make things simpler.
1364
01:19:39,480 --> 01:19:42,660
But at the end of the day
for complicated systems
1365
01:19:42,660 --> 01:19:44,520
that work is the same.
1366
01:19:44,520 --> 01:19:47,910
But for simple systems like
this one where there is
1367
01:19:47,910 --> 01:19:51,040
a good symmetry, you can do it.
1368
01:19:51,040 --> 01:19:54,870
Anyway, so I think we
are done for today.
1369
01:19:54,870 --> 01:19:59,370
And on Tuesday, we'll continue
with forced oscillators.
1370
01:19:59,370 --> 01:19:59,870
All right?
1371
01:19:59,870 --> 01:20:01,720
Thank you.