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YEN-JIE LEE: OK, so
welcome back, everybody.
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00:00:27,110 --> 00:00:29,980
Welcome back to 8.03.
10
00:00:29,980 --> 00:00:34,400
Today, we are going to
continue the discussion
11
00:00:34,400 --> 00:00:36,600
of the harmonic oscillators.
12
00:00:36,600 --> 00:00:40,550
And also, we will add
damping force into the game
13
00:00:40,550 --> 00:00:43,160
and see what will happen, OK?
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00:00:43,160 --> 00:00:45,950
So this is actually what
we have learned last time
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00:00:45,950 --> 00:00:47,900
from this slide.
16
00:00:47,900 --> 00:00:52,700
We have analyzed the physics
of a harmonic oscillator, which
17
00:00:52,700 --> 00:00:54,680
we actually
demonstrated last time.
18
00:00:54,680 --> 00:00:57,590
And you can see the
device still there.
19
00:00:57,590 --> 00:01:01,020
And Hooke's law,
actually the Hooke's law
20
00:01:01,020 --> 00:01:05,180
is actually far more general
than what we saw before.
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00:01:05,180 --> 00:01:09,340
It works for all
small oscillations
22
00:01:09,340 --> 00:01:16,170
around about a point of
equilibrium position, OK?
23
00:01:16,170 --> 00:01:23,190
And that can be demonstrated
by multiple different kinds
24
00:01:23,190 --> 00:01:24,960
of physical systems.
25
00:01:24,960 --> 00:01:30,630
For example here, I have a
mass, which actually can only
26
00:01:30,630 --> 00:01:33,600
move along this track here.
27
00:01:33,600 --> 00:01:37,350
And if I put this mass
set free, then this thing
28
00:01:37,350 --> 00:01:43,160
is actually exercising
harmonic oscillation, OK?
29
00:01:43,160 --> 00:01:44,660
We can do this with
large amplitude.
30
00:01:44,660 --> 00:01:48,540
We can also do it
with small amplitude.
31
00:01:48,540 --> 00:01:52,680
And you see that,
huh, really, it works.
32
00:01:52,680 --> 00:01:54,080
Hooke's law actually works.
33
00:01:54,080 --> 00:01:56,940
And it predicts
exactly the same motion
34
00:01:56,940 --> 00:02:02,040
as to what you see
on the slide, OK?
35
00:02:02,040 --> 00:02:06,690
And we also have a little
bit more complicated system.
36
00:02:06,690 --> 00:02:11,009
For example, this
is some kind of rod.
37
00:02:11,009 --> 00:02:14,980
And you can actually fix one
point and make it oscillate.
38
00:02:14,980 --> 00:02:18,120
And you see that,
huh, it also does
39
00:02:18,120 --> 00:02:21,430
some kind of
harmonic oscillation.
40
00:02:21,430 --> 00:02:25,630
But now, what is actually
oscillating is the amplitude.
41
00:02:25,630 --> 00:02:29,110
The amplitude is actually
the angle with respect
42
00:02:29,110 --> 00:02:32,510
to the downward direction.
43
00:02:32,510 --> 00:02:36,340
And finally this is actually
the vertical version
44
00:02:36,340 --> 00:02:40,420
of this spring mass
system, which you will be
45
00:02:40,420 --> 00:02:43,080
analyzing that in your P-set.
46
00:02:43,080 --> 00:02:46,850
And you see that, huh, it
actually oscillates up and down
47
00:02:46,850 --> 00:02:48,410
harmonically.
48
00:02:48,410 --> 00:02:51,400
So that's all very nice.
49
00:02:51,400 --> 00:02:54,280
And we also have
learned one thing which
50
00:02:54,280 --> 00:02:55,930
is very, very interesting.
51
00:02:55,930 --> 00:03:02,020
It's that a complex exponential
is actually a pretty beautiful
52
00:03:02,020 --> 00:03:04,390
way to present the solution.
53
00:03:04,390 --> 00:03:09,070
And you will see it works
also when describing
54
00:03:09,070 --> 00:03:10,470
the damped oscillators.
55
00:03:10,470 --> 00:03:17,310
And we will see how it
works in the lecture today.
56
00:03:17,310 --> 00:03:21,240
I received several questions
during my office hour
57
00:03:21,240 --> 00:03:23,790
and through email or Piazza.
58
00:03:23,790 --> 00:03:29,640
There were some confusions about
doing the Taylor expansion, OK?
59
00:03:29,640 --> 00:03:34,450
So in lecture last time,
the equilibrium position
60
00:03:34,450 --> 00:03:36,630
is at x equal to 0.
61
00:03:36,630 --> 00:03:41,730
Therefore, I do Taylor
expansion around 0, OK?
62
00:03:41,730 --> 00:03:45,210
But in this case, if
the equilibrium position
63
00:03:45,210 --> 00:03:50,850
or the minima of the
potential is at x equal to L,
64
00:03:50,850 --> 00:03:55,050
then what you need to do
is to do a Taylor expansion
65
00:03:55,050 --> 00:03:58,700
around x equal to L,
just to make that really,
66
00:03:58,700 --> 00:04:00,460
really clear, OK?
67
00:04:00,460 --> 00:04:04,950
OK, I hope that will help
you with the P-set question.
68
00:04:04,950 --> 00:04:07,780
OK, so let's get
started immediately.
69
00:04:07,780 --> 00:04:13,200
So let's continue the discussion
of the equation of motion
70
00:04:13,200 --> 00:04:14,740
we arrived at last time.
71
00:04:14,740 --> 00:04:22,690
So we have M x double-dot and
this is equal to minus kx, OK?
72
00:04:22,690 --> 00:04:25,120
That is actually the
formula from last time.
73
00:04:25,120 --> 00:04:35,470
And we can actually
calculate the kinetic energy
74
00:04:35,470 --> 00:04:38,230
of this spring and mass system.
75
00:04:38,230 --> 00:04:45,820
And basically, this is
going to be equal to 1/2 M
76
00:04:45,820 --> 00:04:50,400
times x dot squared.
77
00:04:50,400 --> 00:04:53,740
OK, and we can also calculate
the potential energy
78
00:04:53,740 --> 00:04:55,920
of the spring.
79
00:04:55,920 --> 00:05:10,760
Potential energy, and that
is equal to 1/2 kx squared.
80
00:05:10,760 --> 00:05:13,940
We also know what would
be the total energy.
81
00:05:13,940 --> 00:05:19,520
The total energy would be
a sum of the kinetic energy
82
00:05:19,520 --> 00:05:21,360
and of the potential.
83
00:05:21,360 --> 00:05:25,280
Basically, you get this
formula, 1/2 M x dot
84
00:05:25,280 --> 00:05:33,840
squared plus 1/2 kx squared.
85
00:05:33,840 --> 00:05:37,420
One last time, we have solved
this equation of motion, right?
86
00:05:37,420 --> 00:05:48,890
So the solution we got is x
equal to A cosine omega 0 t
87
00:05:48,890 --> 00:05:51,470
plus phi.
88
00:05:51,470 --> 00:05:59,160
Well, omega 0 is equal to
a square root of k over M.
89
00:05:59,160 --> 00:06:01,460
Therefore, we can
actually calculate
90
00:06:01,460 --> 00:06:06,790
what would be the total energy
as a function of time, right?
91
00:06:06,790 --> 00:06:08,940
So if we calculate
that, we'll get
92
00:06:08,940 --> 00:06:19,410
E will be equal to 1/2 M A
squared omega 0 squared sine
93
00:06:19,410 --> 00:06:23,850
squared omega 0 t plus phi--
94
00:06:23,850 --> 00:06:26,940
so this is actually
the first term here--
95
00:06:26,940 --> 00:06:39,210
plus 1/2 kA squared cosine
squared omega 0 t plus phi, OK?
96
00:06:39,210 --> 00:06:44,570
Then, we also know that
this coefficient here
97
00:06:44,570 --> 00:06:47,250
is just kA squared, right?
98
00:06:47,250 --> 00:06:50,970
Because omega 0 is actually
equal to the square root of k
99
00:06:50,970 --> 00:06:58,470
over M. And if you replace this
omega 0 squared by k over M,
100
00:06:58,470 --> 00:07:02,160
then you actually arrive
at kA squared, OK?
101
00:07:02,160 --> 00:07:03,750
So that is actually very good.
102
00:07:03,750 --> 00:07:07,470
So that means I can
simplify the total energy.
103
00:07:07,470 --> 00:07:12,690
And what we are going to
get is 1/2 kA squared.
104
00:07:12,690 --> 00:07:15,180
I can take this factor out.
105
00:07:15,180 --> 00:07:19,680
And that will give me, inside
these brackets, I will get sine
106
00:07:19,680 --> 00:07:27,910
squared omega 0 t plus phi
plus cosine squared omega
107
00:07:27,910 --> 00:07:31,410
0 t plus phi.
108
00:07:31,410 --> 00:07:36,430
And this is actually
equal to 1, right?
109
00:07:36,430 --> 00:07:39,990
Just a reminder, sine squared
of theta plus cosine squared
110
00:07:39,990 --> 00:07:43,000
of theta is always equal to 1.
111
00:07:43,000 --> 00:07:45,360
So that gives me
this result. This
112
00:07:45,360 --> 00:07:52,750
is actually 1/2 kA squared, OK?
113
00:07:52,750 --> 00:07:56,740
So that is actually the
result. What does that mean?
114
00:07:56,740 --> 00:08:04,840
That means, if I actually
pull this mass harder,
115
00:08:04,840 --> 00:08:09,600
so that initially it
has larger amplitude,
116
00:08:09,600 --> 00:08:12,630
then the total energy
is actually proportioned
117
00:08:12,630 --> 00:08:15,360
to amplitude squared, OK?
118
00:08:15,360 --> 00:08:18,630
So I am storing more
and more energy.
119
00:08:18,630 --> 00:08:23,230
If I increase the
amplitude even more,
120
00:08:23,230 --> 00:08:26,430
then I am storing the
energy in this system.
121
00:08:26,430 --> 00:08:28,430
And it's proportional
to A squared.
122
00:08:28,430 --> 00:08:32,200
And also, if the spring
constant is larger,
123
00:08:32,200 --> 00:08:36,142
the same amplitude will
give you more energy.
124
00:08:36,142 --> 00:08:38,475
So that means that you can
store more energy if you have
125
00:08:38,475 --> 00:08:41,950
a larger string constant, OK?
126
00:08:41,950 --> 00:08:46,230
The most surprising thing
is that actually this
127
00:08:46,230 --> 00:08:50,020
is actually a constant, right?
128
00:08:50,020 --> 00:08:51,610
What does that mean?
129
00:08:51,610 --> 00:08:55,450
The total energy is
actually not variating
130
00:08:55,450 --> 00:08:56,873
as a function of time.
131
00:08:56,873 --> 00:08:57,740
You see?
132
00:08:57,740 --> 00:09:00,970
So total energy is constant, OK?
133
00:09:00,970 --> 00:09:06,970
So you can see from this slide
the total energy is actually
134
00:09:06,970 --> 00:09:11,522
showing us the sum,
which is the green curve.
135
00:09:11,522 --> 00:09:13,480
And the kinetic energy
and the potential energy
136
00:09:13,480 --> 00:09:17,500
are shown as red
and blue curves.
137
00:09:17,500 --> 00:09:23,050
You can see that the total
energy is actually constant.
138
00:09:23,050 --> 00:09:25,900
But this system
is very dynamical.
139
00:09:25,900 --> 00:09:26,410
You see?
140
00:09:26,410 --> 00:09:31,840
So that energy is actually
going back and forth
141
00:09:31,840 --> 00:09:38,840
between the spring and the mass
in the form of kinetic energy
142
00:09:38,840 --> 00:09:41,200
and in the form of
potential energy.
143
00:09:41,200 --> 00:09:45,820
But they are doing it so well,
such that the sum is actually
144
00:09:45,820 --> 00:09:47,125
a constant.
145
00:09:47,125 --> 00:09:50,200
So the energy is
actually constant, OK?
146
00:09:50,200 --> 00:09:53,170
So that is actually
pretty beautiful.
147
00:09:53,170 --> 00:09:59,520
And it can be described very
well by these mathematics.
148
00:09:59,520 --> 00:10:00,550
Any questions from here?
149
00:10:04,380 --> 00:10:08,342
OK, so I would like to say
simple harmonic motion,
150
00:10:08,342 --> 00:10:09,800
actually, what you
are going to get
151
00:10:09,800 --> 00:10:12,780
is the energy is actually
conserved and independent
152
00:10:12,780 --> 00:10:14,300
of the time.
153
00:10:14,300 --> 00:10:18,130
And later, you will see
an example with damping.
154
00:10:18,130 --> 00:10:20,735
And you will see that
energy conservation
155
00:10:20,735 --> 00:10:24,320
is now no longer the case, OK?
156
00:10:24,320 --> 00:10:29,040
So let's immediately
jump to another example,
157
00:10:29,040 --> 00:10:33,270
which is actually involving
simple harmonic motion.
158
00:10:33,270 --> 00:10:41,100
So let's take this rod and
nail system as an example.
159
00:10:41,100 --> 00:10:44,660
If I actually slightly
move this rod,
160
00:10:44,660 --> 00:10:49,490
and then I release
that, then actually
161
00:10:49,490 --> 00:10:55,010
you will see simple harmonic
motion, also for this system.
162
00:10:55,010 --> 00:11:00,620
So let's actually do the
calculation as another example.
163
00:11:00,620 --> 00:11:02,720
So this is actually my system.
164
00:11:02,720 --> 00:11:05,920
I have this rod, OK?
165
00:11:05,920 --> 00:11:09,680
Now, I am assume that the
mass is actually uniformly
166
00:11:09,680 --> 00:11:15,370
distributed on this rod and
is nailed on the wall, OK?
167
00:11:15,370 --> 00:11:20,150
And the length of this
rod is actually l.
168
00:11:20,150 --> 00:11:25,430
So that means the center
of mass is actually at l/2
169
00:11:25,430 --> 00:11:29,600
with respect to the nail, OK?
170
00:11:29,600 --> 00:11:36,270
And also, this whole system
is set up on Earth, right?
171
00:11:36,270 --> 00:11:38,870
Therefore, there will
be gravitational force
172
00:11:38,870 --> 00:11:41,570
pointing downward, OK?
173
00:11:41,570 --> 00:11:46,600
So that means you have
gravitational force, Fg,
174
00:11:46,600 --> 00:11:49,670
pointing downward, OK?
175
00:11:49,670 --> 00:11:52,490
So this is actually
the system, which
176
00:11:52,490 --> 00:11:54,290
I would like to understand.
177
00:11:54,290 --> 00:11:58,490
And just a reminder, what are we
going to do afterwards in order
178
00:11:58,490 --> 00:12:02,420
to turn the whole system
into a language we
179
00:12:02,420 --> 00:12:04,640
know describes the nature?
180
00:12:04,640 --> 00:12:08,330
What are we going to do?
181
00:12:08,330 --> 00:12:09,040
Anybody?
182
00:12:13,360 --> 00:12:18,010
We are going to define
the coordinate system,
183
00:12:18,010 --> 00:12:20,980
so that I can
translate everything
184
00:12:20,980 --> 00:12:22,600
into mathematics, right?
185
00:12:22,600 --> 00:12:25,450
So that's actually what
we are always doing.
186
00:12:25,450 --> 00:12:26,950
And you will see
that we are always
187
00:12:26,950 --> 00:12:32,290
doing this in this class, OK?
188
00:12:32,290 --> 00:12:35,060
So what is actually
the coordinate system
189
00:12:35,060 --> 00:12:36,760
which I would like to use?
190
00:12:36,760 --> 00:12:42,100
Since this system is going to
be rotating back and forth,
191
00:12:42,100 --> 00:12:47,660
therefore, I would
like to define theta
192
00:12:47,660 --> 00:12:52,700
to be that angle with respect
to the axis, which essentially
193
00:12:52,700 --> 00:12:55,190
pointing downward, OK?
194
00:12:55,190 --> 00:13:02,510
So the origin of this coordinate
system uses theta equal to 0.
195
00:13:02,510 --> 00:13:14,100
This means that the rod is
actually pointing downward, OK?
196
00:13:14,100 --> 00:13:21,750
And also, I need to define what
is actually the positive value
197
00:13:21,750 --> 00:13:22,950
of the zeta, right?
198
00:13:22,950 --> 00:13:29,400
So I define anti-clockwise
direction to be positive, OK?
199
00:13:29,400 --> 00:13:34,370
So it is actually important
to actually first define that,
200
00:13:34,370 --> 00:13:38,170
then actually to translate
everything into mathematics,
201
00:13:38,170 --> 00:13:40,530
OK?
202
00:13:40,530 --> 00:13:42,880
So the initial condition
is the following.
203
00:13:42,880 --> 00:13:48,060
So I actually move this thing,
rotate this thing slightly.
204
00:13:48,060 --> 00:13:50,880
Then, I actually release
that really carefully
205
00:13:50,880 --> 00:13:55,140
without introducing any
initial velocity, OK?
206
00:13:55,140 --> 00:13:58,380
Therefore, I have two
initial conditions.
207
00:14:05,760 --> 00:14:11,750
OK, at t equal to 0, there
are two initial conditions.
208
00:14:11,750 --> 00:14:16,780
The first one is theta 0
is equal to theta initial.
209
00:14:21,130 --> 00:14:23,290
The second condition
is the same as what
210
00:14:23,290 --> 00:14:26,000
we have been doing last time.
211
00:14:26,000 --> 00:14:29,730
The initial velocity
or angular velocity
212
00:14:29,730 --> 00:14:30,980
is actually equal to 0.
213
00:14:30,980 --> 00:14:36,610
So that gives you theta
dot equal to 0, OK?
214
00:14:36,610 --> 00:14:43,030
Now, we have actually defined
the coordinate system.
215
00:14:43,030 --> 00:14:49,660
Now, we can actually
draw a force diagram,
216
00:14:49,660 --> 00:14:54,460
so that we can actually use
our knowledge about the physics
217
00:14:54,460 --> 00:14:57,020
to obtain the equation
of motion, right?
218
00:14:57,020 --> 00:14:59,680
So now, the force
diagram looks like this.
219
00:15:05,570 --> 00:15:10,420
So this is actually the
center of mass of this rod.
220
00:15:10,420 --> 00:15:14,300
And you have a force
pointing downward,
221
00:15:14,300 --> 00:15:17,540
which is due to the
gravitational force.
222
00:15:17,540 --> 00:15:21,660
Fg is equal to mg.
223
00:15:21,660 --> 00:15:23,620
It's pointing downward.
224
00:15:23,620 --> 00:15:28,960
The magnitude is
actually equal to mg.
225
00:15:28,960 --> 00:15:36,160
And also, we know the R vector.
226
00:15:36,160 --> 00:15:40,880
This vector has a length, l/2.
227
00:15:40,880 --> 00:15:49,228
It's pointing from the center
of mass of this rod to the nail,
228
00:15:49,228 --> 00:15:51,420
OK?
229
00:15:51,420 --> 00:15:57,720
And also, we know the angle
between these vectors,
230
00:15:57,720 --> 00:16:00,960
pointing from the center
of mass to the nail,
231
00:16:00,960 --> 00:16:04,920
and the vertical direction,
which we have already defined,
232
00:16:04,920 --> 00:16:07,180
which is actually called theta.
233
00:16:07,180 --> 00:16:10,260
Therefore, now, we
can actually calculate
234
00:16:10,260 --> 00:16:13,700
what would be the torque.
235
00:16:13,700 --> 00:16:19,970
Tau will be equal
to this R vector
236
00:16:19,970 --> 00:16:27,300
cross the force, total force
acting on the center mass.
237
00:16:27,300 --> 00:16:33,410
In this case, it's just Fg, OK?
238
00:16:33,410 --> 00:16:36,650
So now, we can actually
write this down explicitly.
239
00:16:36,650 --> 00:16:39,050
Since the whole
system is actually
240
00:16:39,050 --> 00:16:44,980
rotating on a single plane,
so there's only one plane
241
00:16:44,980 --> 00:16:46,590
this is sitting on.
242
00:16:46,590 --> 00:16:51,990
And it's actually going back and
forth only on this plane, OK?
243
00:16:51,990 --> 00:16:56,630
Therefore, actually, I
can drop all the arrows
244
00:16:56,630 --> 00:17:01,550
and write down the magnitude
of the tau directly.
245
00:17:01,550 --> 00:17:14,240
And this will be equal to
minus mg l/2 sine theta t, OK?
246
00:17:14,240 --> 00:17:15,319
Any questions so far?
247
00:17:20,079 --> 00:17:22,720
OK, so now, we have the torque.
248
00:17:22,720 --> 00:17:26,660
And we can make use of
the rotational version
249
00:17:26,660 --> 00:17:30,950
of Newton's Law to obtain the
equation of motion, right?
250
00:17:30,950 --> 00:17:35,180
So that should be
pretty straightforward.
251
00:17:35,180 --> 00:17:39,890
Tau will be equal to I, which
is the moment of inertia
252
00:17:39,890 --> 00:17:48,320
of the system,
times alpha t, OK?
253
00:17:48,320 --> 00:17:50,900
And just for your
information, I already
254
00:17:50,900 --> 00:17:56,490
calculated the I for you.
255
00:17:56,490 --> 00:18:02,970
I is equal to 1/3
ml squared, OK?
256
00:18:02,970 --> 00:18:06,060
So you can actually go
back home and actually do
257
00:18:06,060 --> 00:18:08,580
a check to see if I'm
telling the truth.
258
00:18:08,580 --> 00:18:11,890
And if you trust me,
then that's the answer,
259
00:18:11,890 --> 00:18:14,370
which is actually
1/3 ml squared,
260
00:18:14,370 --> 00:18:19,350
if the mass is actually
uniformly distributed
261
00:18:19,350 --> 00:18:22,050
on this rod, OK?
262
00:18:22,050 --> 00:18:37,410
So that would give me minus mgl
divided by 2 sine theta t, OK?
263
00:18:37,410 --> 00:18:43,140
So that is actually
coming from this side, OK?
264
00:18:43,140 --> 00:18:46,740
So now, I can actually
simplify this expression.
265
00:18:46,740 --> 00:18:50,550
I can now plug in the I
value into this equation.
266
00:18:50,550 --> 00:19:01,390
And I will get 1/3 ml squared
theta double-dot t, which
267
00:19:01,390 --> 00:19:02,880
is actually alpha, OK?
268
00:19:02,880 --> 00:19:05,250
Now, I write it as
theta double-dot.
269
00:19:05,250 --> 00:19:14,670
And that will be equal to minus
mgl over 2 sine theta, OK?
270
00:19:14,670 --> 00:19:17,470
I can move all the constants
to the right-hand side.
271
00:19:17,470 --> 00:19:20,970
Therefore, I get
theta double-dot t.
272
00:19:20,970 --> 00:19:23,610
This is equal to minus mgl.
273
00:19:27,380 --> 00:19:32,800
OK, actually, I can already
simplify this, right?
274
00:19:32,800 --> 00:19:34,400
These actually cancel.
275
00:19:34,400 --> 00:19:36,920
And the 1/l actually cancels.
276
00:19:36,920 --> 00:19:44,150
So therefore, I get minus
3g over 2 sine theta t.
277
00:19:48,050 --> 00:19:53,110
OK, as you know, we
actually defined omega
278
00:19:53,110 --> 00:19:57,380
to replace this constant
to make our life easier.
279
00:19:57,380 --> 00:20:05,050
So I can now define omega 0
equal to square root of 3g
280
00:20:05,050 --> 00:20:09,760
over 2l, OK?
281
00:20:09,760 --> 00:20:12,390
And that will give
you theta double-dot
282
00:20:12,390 --> 00:20:20,320
of t equal to minus omega
0 squared sin theta t.
283
00:20:24,610 --> 00:20:26,530
Any questions so far?
284
00:20:26,530 --> 00:20:28,820
A lot of calculations.
285
00:20:28,820 --> 00:20:31,480
But they should all be
pretty straightforward.
286
00:20:31,480 --> 00:20:35,900
And actually, we are done now.
287
00:20:35,900 --> 00:20:36,740
We are done.
288
00:20:36,740 --> 00:20:39,090
Because we have the
equation of motion.
289
00:20:39,090 --> 00:20:42,530
And the rest of the job
is to solve just it.
290
00:20:42,530 --> 00:20:46,040
So it is actually now the
problem of the math department.
291
00:20:46,040 --> 00:20:50,960
So can anybody actually tell me
the solution of the theta of t?
292
00:20:50,960 --> 00:20:54,025
Anybody?
293
00:20:54,025 --> 00:20:56,947
AUDIENCE: Unfortunately,
we'd have to approximate it.
294
00:20:56,947 --> 00:20:59,880
YEN-JIE LEE: That's
very unfortunate.
295
00:20:59,880 --> 00:21:04,600
So now, we are facing a
very difficult situation.
296
00:21:04,600 --> 00:21:08,970
We don't know how to solve
this equation in front of you.
297
00:21:08,970 --> 00:21:13,080
I don't know, OK?
298
00:21:13,080 --> 00:21:18,870
Of course, you can actually
solve it with a computer,
299
00:21:18,870 --> 00:21:23,020
or, if you want to go fancy,
solve it with your cellphone,
300
00:21:23,020 --> 00:21:24,020
if it doesn't explode.
301
00:21:28,740 --> 00:21:34,710
But it's not really nice
to do this in front of you.
302
00:21:34,710 --> 00:21:35,830
We don't learn too much.
303
00:21:35,830 --> 00:21:39,690
OK, so what are we going to do?
304
00:21:39,690 --> 00:21:45,720
So what we can do is actually
to consider a special case.
305
00:21:45,720 --> 00:21:50,930
So we know that this equation
of motion is exact, OK?
306
00:21:50,930 --> 00:21:53,780
So if you solve it,
it would describe
307
00:21:53,780 --> 00:21:57,350
the motion of this rod.
308
00:22:00,440 --> 00:22:04,920
Even with a large
angle, it works, OK?
309
00:22:04,920 --> 00:22:07,560
And now, in order
to actually show
310
00:22:07,560 --> 00:22:11,100
you the math in the
class, therefore
311
00:22:11,100 --> 00:22:14,090
actually I will do a
small approximation.
312
00:22:14,090 --> 00:22:17,130
So actually, I would
only work on the case
313
00:22:17,130 --> 00:22:21,390
that when the
amplitude is very small
314
00:22:21,390 --> 00:22:23,710
and see what is going to happen.
315
00:22:23,710 --> 00:22:27,150
So now, I'm considering
a special case.
316
00:22:27,150 --> 00:22:29,650
Up to now, everything is exact.
317
00:22:29,650 --> 00:22:34,340
And now, I am now going
to a special case.
318
00:22:34,340 --> 00:22:39,390
Theta t goes to 0, OK?
319
00:22:39,390 --> 00:22:41,390
Then, we can actually get this.
320
00:22:41,390 --> 00:22:48,020
Sine theta t is
roughly theta t, OK?
321
00:22:48,020 --> 00:22:50,000
Based on the Taylor
expansion, you
322
00:22:50,000 --> 00:22:53,870
can actually verify this, OK?
323
00:22:53,870 --> 00:23:02,120
So in this case, if we take
theta equal to 1 degree,
324
00:23:02,120 --> 00:23:06,626
then the ratio of the
sine theta and the theta
325
00:23:06,626 --> 00:23:12,710
is actually equal to
99.99%, which is very good.
326
00:23:12,710 --> 00:23:23,930
If I take it as 5 degrees,
then it's actually 99%.
327
00:23:23,930 --> 00:23:29,970
Even at 10 degrees,
it's actually 99.5%.
328
00:23:29,970 --> 00:23:34,700
Now, that shows you that sine
theta is so close to theta, OK?
329
00:23:34,700 --> 00:23:35,690
We are pretty safe.
330
00:23:35,690 --> 00:23:39,750
Because the difference
is smaller than 1%.
331
00:23:39,750 --> 00:23:42,890
OK, so that's very nice.
332
00:23:42,890 --> 00:23:50,800
After this approximation, I get
my final equation of motion.
333
00:23:50,800 --> 00:24:00,250
Theta double-dot t equal to
minus omega 0 squared theta t.
334
00:24:00,250 --> 00:24:04,375
Just a reminder, omega 0
is equal to square root
335
00:24:04,375 --> 00:24:11,400
of 3g over 2l, OK?
336
00:24:11,400 --> 00:24:17,870
We have solved this equation
last time, last lecture, right?
337
00:24:17,870 --> 00:24:19,200
It's exactly the same.
338
00:24:19,200 --> 00:24:21,790
OK, it happened to
be exactly the same.
339
00:24:21,790 --> 00:24:23,550
Therefore, I know
the solution will
340
00:24:23,550 --> 00:24:30,660
be theta of t equal to A
cosine omega 0 t plus phi.
341
00:24:34,000 --> 00:24:39,750
From the initial conditions,
which I have one and two,
342
00:24:39,750 --> 00:24:43,650
I am not going to go over
these calculation again.
343
00:24:43,650 --> 00:24:46,560
But again, we can
actually plug in 1 and 2
344
00:24:46,560 --> 00:24:50,850
to solve the unknown
A and the phi.
345
00:24:50,850 --> 00:24:53,400
If you do this
exercise, you will
346
00:24:53,400 --> 00:24:58,620
conclude that A is
equal to theta initial.
347
00:25:01,740 --> 00:25:08,900
And phi is equal to 0, OK?
348
00:25:08,900 --> 00:25:18,340
So the solution would be
theta of t equal to theta
349
00:25:18,340 --> 00:25:24,280
initial cosine omega 0 t.
350
00:25:27,400 --> 00:25:33,560
You can see that this actually
works for this system.
351
00:25:33,560 --> 00:25:35,940
Simple harmonic
oscillation actually
352
00:25:35,940 --> 00:25:41,540
described the motion of this
system as a function of time.
353
00:25:41,540 --> 00:25:45,470
You can also see a few
more examples shown here.
354
00:25:45,470 --> 00:25:49,270
Two of them you are going to
really work on in your P-set
355
00:25:49,270 --> 00:25:53,330
and also another one
involving circuits.
356
00:25:53,330 --> 00:25:57,290
If you have a capacitor
and you have an inductor,
357
00:25:57,290 --> 00:26:03,290
actually the size of
the current is also
358
00:26:03,290 --> 00:26:06,740
doing a simple
harmonic motion, OK?
359
00:26:06,740 --> 00:26:09,560
And as we actually
discussed before,
360
00:26:09,560 --> 00:26:12,570
the energy is always conserved.
361
00:26:12,570 --> 00:26:15,410
And that is actually stored
in different components
362
00:26:15,410 --> 00:26:16,890
of the system, OK?
363
00:26:20,520 --> 00:26:22,510
So we have done this.
364
00:26:22,510 --> 00:26:26,550
What is actually new today?
365
00:26:26,550 --> 00:26:30,100
What we are going to do
today is let's actually
366
00:26:30,100 --> 00:26:34,140
observe this phenomenon here.
367
00:26:34,140 --> 00:26:38,010
So this thing is actually
going to go back and forth.
368
00:26:38,010 --> 00:26:41,240
But it's actually not going
to do that forever, right?
369
00:26:41,240 --> 00:26:48,340
Something is happening, which
actually slows the motion down.
370
00:26:48,340 --> 00:26:52,060
I can also make use
of this system, OK?
371
00:26:52,060 --> 00:26:54,180
I start from here.
372
00:26:54,180 --> 00:26:56,320
And I'm not worried
that this actually
373
00:26:56,320 --> 00:26:58,330
goes out of this track.
374
00:26:58,330 --> 00:27:03,160
Because I know for sure
it will stop there.
375
00:27:03,160 --> 00:27:04,240
Why?
376
00:27:04,240 --> 00:27:09,780
Because the initial
amplitude is not going to--
377
00:27:09,780 --> 00:27:11,434
the amplitude is not
going to be larger
378
00:27:11,434 --> 00:27:12,850
than the initial
amplitude, right?
379
00:27:12,850 --> 00:27:15,480
So I'm not worried at all, OK?
380
00:27:15,480 --> 00:27:17,710
But you can see that
the amplitude is
381
00:27:17,710 --> 00:27:20,680
changing as a function of time.
382
00:27:20,680 --> 00:27:23,590
Apparently,
something is missing.
383
00:27:23,590 --> 00:27:28,050
And that is actually a
direct force, or friction,
384
00:27:28,050 --> 00:27:32,720
which is actually not
included in our calculation.
385
00:27:32,720 --> 00:27:40,860
So let's actually try to make
the calculation more realistic
386
00:27:40,860 --> 00:27:43,900
and see what is going to happen.
387
00:27:43,900 --> 00:27:51,500
So now, I will
introduce a drag force,
388
00:27:51,500 --> 00:27:58,300
which actually introduces
a torque tau drag, t,
389
00:27:58,300 --> 00:28:01,360
which is equal to minus b--
390
00:28:01,360 --> 00:28:03,670
b is actually some
kind of constant,
391
00:28:03,670 --> 00:28:06,970
which is given to you--
392
00:28:06,970 --> 00:28:14,680
theta dot t, which is actually
proportional to angular
393
00:28:14,680 --> 00:28:19,490
velocity of that rod, OK?
394
00:28:19,490 --> 00:28:24,350
And also of course, I keep
the original approximation.
395
00:28:24,350 --> 00:28:27,590
The theta is very
small, such that I
396
00:28:27,590 --> 00:28:30,810
don't have to deal with the
integration of sine theta, OK?
397
00:28:30,810 --> 00:28:33,880
So solving this, theta
double-dot equal to minus
398
00:28:33,880 --> 00:28:37,520
omega 0 squared sine theta
is a complicated function.
399
00:28:40,360 --> 00:28:46,030
You may ask, why do I actually
introduce a drag force
400
00:28:46,030 --> 00:28:49,770
proportional to the velocity?
401
00:28:49,770 --> 00:28:53,460
And why do I put a
minus sign there?
402
00:28:53,460 --> 00:28:58,380
That is actually because, if you
have a minus sign, that means,
403
00:28:58,380 --> 00:29:03,870
when this mass or that rod
is actually going downward,
404
00:29:03,870 --> 00:29:06,030
then the drag force
is really dragging it.
405
00:29:06,030 --> 00:29:09,750
Because it's actually in
the opposite direction
406
00:29:09,750 --> 00:29:14,460
of the velocity of the
mass or the angular
407
00:29:14,460 --> 00:29:16,470
velocity of the rod, OK?
408
00:29:16,470 --> 00:29:18,750
So I need a minus
sign there, OK?
409
00:29:18,750 --> 00:29:21,550
Otherwise, it's not
a drag force anymore.
410
00:29:21,550 --> 00:29:25,580
It's actually accelerating
the whole thing.
411
00:29:25,580 --> 00:29:33,878
Secondly, why do I choose that
to be proportional to theta dot
412
00:29:33,878 --> 00:29:37,750
or velocity?
413
00:29:37,750 --> 00:29:42,150
There's really no
much deeper reason.
414
00:29:42,150 --> 00:29:46,140
I choose this form
because I can actually
415
00:29:46,140 --> 00:29:48,460
solve it in front of you, OK?
416
00:29:48,460 --> 00:29:52,560
The reality is actually
between proportional
417
00:29:52,560 --> 00:29:57,210
to theta dot and theta dot
squared, for example, OK?
418
00:29:57,210 --> 00:29:59,940
This is actually a model
which I introduced here,
419
00:29:59,940 --> 00:30:04,440
which I can actually
solve it in front of you.
420
00:30:04,440 --> 00:30:07,110
On the other hand, you'll
see that it's actually not
421
00:30:07,110 --> 00:30:09,690
bad at all.
422
00:30:09,690 --> 00:30:12,900
It actually works and
describes the system,
423
00:30:12,900 --> 00:30:17,910
which will actually work to
perform the demo here, OK?
424
00:30:17,910 --> 00:30:24,390
And once we have introduced
this, the equation of motion
425
00:30:24,390 --> 00:30:26,810
will be modified.
426
00:30:26,810 --> 00:30:29,070
So let's come back to
the equation of motion.
427
00:30:29,070 --> 00:30:32,890
So you have to
theta double-dot t
428
00:30:32,890 --> 00:30:42,490
originally would be equal to
tau total t divided by I, OK?
429
00:30:42,490 --> 00:30:57,350
And now, this will become tau
t plus tau drag t divided by I.
430
00:30:57,350 --> 00:30:59,540
So there's an
additional time here.
431
00:30:59,540 --> 00:31:03,010
OK, if I simplify
this whole equation,
432
00:31:03,010 --> 00:31:12,320
then I get minus mgl
over 2 sine theta.
433
00:31:12,320 --> 00:31:16,960
And this is actually
roughly theta minus
434
00:31:16,960 --> 00:31:27,140
b theta dot divided by
1/3 of ml squared, OK?
435
00:31:27,140 --> 00:31:30,520
So you can see that I still
make this approximation sine
436
00:31:30,520 --> 00:31:34,940
theta roughly equal to theta.
437
00:31:34,940 --> 00:31:40,170
Then, I can actually write this
equation in the small angle
438
00:31:40,170 --> 00:31:40,670
case.
439
00:31:44,740 --> 00:31:53,000
OK, I get minus
3g over 2l theta t
440
00:31:53,000 --> 00:31:58,740
minus 3b over ml
squared theta dot t.
441
00:32:01,660 --> 00:32:10,490
OK, and now, as usual, I define
omega 0 squared equal to 3g
442
00:32:10,490 --> 00:32:11,960
over 2l.
443
00:32:11,960 --> 00:32:18,410
And I can also define
gamma is equal to 3b over
444
00:32:18,410 --> 00:32:25,050
ml squared, just to make
my life easier, right?
445
00:32:25,050 --> 00:32:32,740
Finally, we will arrive at this
expression, theta double-dot
446
00:32:32,740 --> 00:32:42,260
plus gamma theta dot plus
omega 0 squared theta.
447
00:32:42,260 --> 00:32:44,217
And that is equal to 0.
448
00:32:48,120 --> 00:32:56,340
So what you can see from here
is that we have actually derived
449
00:32:56,340 --> 00:32:58,950
the equation of motion, OK?
450
00:32:58,950 --> 00:33:01,140
We have derived the
equation of motion.
451
00:33:01,140 --> 00:33:05,660
And actually, part of the
work is actually really
452
00:33:05,660 --> 00:33:08,250
just solving this
equation of motion.
453
00:33:08,250 --> 00:33:09,840
And you don't really
have to solve it.
454
00:33:09,840 --> 00:33:13,650
Because you already get
the result from 18.03
455
00:33:13,650 --> 00:33:16,130
actually, if you remember.
456
00:33:16,130 --> 00:33:18,610
And we are going to
discuss the result.
457
00:33:18,610 --> 00:33:22,830
But before that, before I really
try to solve this equation,
458
00:33:22,830 --> 00:33:28,460
I would like to take a vote, OK?
459
00:33:28,460 --> 00:33:34,540
So here, I have two
different systems.
460
00:33:34,540 --> 00:33:37,980
They have equal amounts of mass.
461
00:33:37,980 --> 00:33:40,450
They are attached to a spring.
462
00:33:40,450 --> 00:33:44,440
If you do the same equation
of motion derivation,
463
00:33:44,440 --> 00:33:51,850
you will actually get exactly
the same equation of motion
464
00:33:51,850 --> 00:33:54,730
in that format, OK?
465
00:33:54,730 --> 00:33:56,380
So the form of the
equation of motion
466
00:33:56,380 --> 00:34:01,850
will be the same between this
system and that system, OK?
467
00:34:01,850 --> 00:34:06,380
I would like to ask you a
question about the oscillation
468
00:34:06,380 --> 00:34:07,950
frequency.
469
00:34:07,950 --> 00:34:12,230
So you can see that one of
them is actually a better mass.
470
00:34:12,230 --> 00:34:14,270
It's like a point-like particle.
471
00:34:14,270 --> 00:34:17,730
And the other one is
wearing a hat, OK?
472
00:34:17,730 --> 00:34:21,469
What is going to happen is
that this Mexican hat is
473
00:34:21,469 --> 00:34:26,270
going to be trying to
push the air away, right?
474
00:34:26,270 --> 00:34:30,710
Then, you may think,
OK, this Mexican thing
475
00:34:30,710 --> 00:34:33,290
is not really very important.
476
00:34:33,290 --> 00:34:35,150
Therefore, the
oscillation frequency
477
00:34:35,150 --> 00:34:37,120
may be the same, right?
478
00:34:37,120 --> 00:34:40,040
How many of you think the
oscillation frequency,
479
00:34:40,040 --> 00:34:42,889
if I actually tried to
perturb these two systems,
480
00:34:42,889 --> 00:34:44,740
would be the same?
481
00:34:44,740 --> 00:34:45,870
Raise your hands.
482
00:34:45,870 --> 00:34:49,730
1, 2, 3, 4, 5, 6, 7, 8--
483
00:34:49,730 --> 00:34:51,040
OK, we have 11.
484
00:34:53,639 --> 00:34:57,630
So the omega, the
predicted omega,
485
00:34:57,630 --> 00:35:00,903
will be equal to omega 0--
486
00:35:00,903 --> 00:35:02,670
11 of you.
487
00:35:02,670 --> 00:35:10,120
How many of you will think
that, because of this hat,
488
00:35:10,120 --> 00:35:16,050
this pushing this air away,
it's a lot of work to be done.
489
00:35:16,050 --> 00:35:21,810
Therefore, this is going to
slow down the oscillation.
490
00:35:21,810 --> 00:35:24,930
How many of you think
that is going to happen?
491
00:35:24,930 --> 00:35:26,040
1, 2, 3--
492
00:35:31,540 --> 00:35:32,490
OK, 17.
493
00:35:45,790 --> 00:35:48,190
It may happen to
you that you think
494
00:35:48,190 --> 00:35:54,100
this idea of wearing a
hat is really fashionable.
495
00:35:54,100 --> 00:35:55,950
Therefore, it got
really exciting
496
00:35:55,950 --> 00:35:58,950
and it oscillates faster.
497
00:35:58,950 --> 00:36:00,780
Can that happen?
498
00:36:00,780 --> 00:36:04,700
How many of you actually
think that is going to happen?
499
00:36:04,700 --> 00:36:09,150
OK, one-- you think so?
500
00:36:09,150 --> 00:36:10,990
Two.
501
00:36:10,990 --> 00:36:13,330
Very good, we have 2.
502
00:36:13,330 --> 00:36:14,080
What do you think?
503
00:36:14,080 --> 00:36:15,830
Where are the rest?
504
00:36:15,830 --> 00:36:21,700
Only 30 of you actually think
that is going to happen.
505
00:36:24,320 --> 00:36:27,600
OK, all the rest
think of the class
506
00:36:27,600 --> 00:36:29,960
think that this one
is going to-- pew!
507
00:36:29,960 --> 00:36:32,600
Disappear to the moon, OK?
508
00:36:32,600 --> 00:36:34,330
So that is actually the opinion.
509
00:36:34,330 --> 00:36:36,280
And we have completed the poll.
510
00:36:36,280 --> 00:36:40,130
And what we are going
to do is that we
511
00:36:40,130 --> 00:36:42,050
are going to solve
this system and see
512
00:36:42,050 --> 00:36:44,160
what is going to happen.
513
00:36:44,160 --> 00:36:48,410
And we will do that experiment
in front of you, OK?
514
00:36:48,410 --> 00:36:51,000
All right, so that's very nice.
515
00:36:51,000 --> 00:36:53,520
So now, we have this
question of motion.
516
00:36:53,520 --> 00:36:57,040
And now, I will pretend
that I'm from the math
517
00:36:57,040 --> 00:37:02,600
department for a bit and help
guide you through the solution.
518
00:37:02,600 --> 00:37:06,470
So now, I can use this trick.
519
00:37:06,470 --> 00:37:08,570
I can actually say
theta is actually
520
00:37:08,570 --> 00:37:14,730
the real part of the z,
which is a complex function.
521
00:37:14,730 --> 00:37:18,620
And as we learned
before, z of t,
522
00:37:18,620 --> 00:37:23,630
and I assume that to be
exponential I alpha t.
523
00:37:23,630 --> 00:37:26,130
So alpha is actually
some kind of constant,
524
00:37:26,130 --> 00:37:29,810
which I don't really know
what is the constant yet.
525
00:37:29,810 --> 00:37:36,590
OK, I can now actually
write the equation of motion
526
00:37:36,590 --> 00:37:38,390
in the form of z.
527
00:37:38,390 --> 00:37:47,127
Then basically, what I get is
z double-dot t plus gamma z dot
528
00:37:47,127 --> 00:37:54,290
t plus omega 0 squared z of t.
529
00:37:54,290 --> 00:37:58,250
And this is equal to 0, OK?
530
00:37:58,250 --> 00:38:02,060
So remember, exponential
function cannot be killed
531
00:38:02,060 --> 00:38:04,220
by differentiation, right?
532
00:38:04,220 --> 00:38:06,180
Therefore, it's
really convenient.
533
00:38:06,180 --> 00:38:07,410
You can see from here.
534
00:38:07,410 --> 00:38:10,640
Now, I can plug in
this expression--
535
00:38:10,640 --> 00:38:16,440
which I did this and guessed
to this equation of motion.
536
00:38:16,440 --> 00:38:20,910
Then what I am going to
get is minus alpha squared.
537
00:38:20,910 --> 00:38:24,130
Because you take I
alpha I alpha out
538
00:38:24,130 --> 00:38:28,640
of this exponential
function, right?
539
00:38:28,640 --> 00:38:32,630
Because you do double
differentiation.
540
00:38:32,630 --> 00:38:40,160
So you get minus alpha
squared plus i gamma alpha--
541
00:38:40,160 --> 00:38:43,680
because this is only
differentiated one time--
542
00:38:43,680 --> 00:38:47,120
plus omega 0 squared.
543
00:38:47,120 --> 00:38:51,570
And all those
things are actually
544
00:38:51,570 --> 00:38:56,480
multiplying this exponential
function, exponential i alpha t
545
00:38:56,480 --> 00:38:59,540
equal to 0, OK?
546
00:38:59,540 --> 00:39:01,455
So we will write
this expression.
547
00:39:01,455 --> 00:39:03,350
That is very nice.
548
00:39:03,350 --> 00:39:07,370
And we also know that,
this expression is
549
00:39:07,370 --> 00:39:09,680
going to be valid all the time.
550
00:39:09,680 --> 00:39:14,270
No matter what t you put in,
it should be valid, right?
551
00:39:14,270 --> 00:39:16,130
Because this is the
equation of motion.
552
00:39:16,130 --> 00:39:19,790
And we hope that this solution
will survive this test.
553
00:39:22,380 --> 00:39:28,110
So I can easily conclude
that this one is actually not
554
00:39:28,110 --> 00:39:29,960
equal to 0.
555
00:39:29,960 --> 00:39:32,640
It can be some value, not 0.
556
00:39:32,640 --> 00:39:35,400
So what is actually equal to 0?
557
00:39:35,400 --> 00:39:41,790
This first term is
actually equal to 0, OK?
558
00:39:41,790 --> 00:39:46,560
Therefore, I can now
solve this equation;
559
00:39:46,560 --> 00:39:50,370
minus alpha squared plus
i gamma alpha plus omega 0
560
00:39:50,370 --> 00:39:52,530
squared equal to 0.
561
00:39:52,530 --> 00:39:54,780
I can solve it, OK?
562
00:39:54,780 --> 00:39:59,650
If I do that, then
I would get alpha
563
00:39:59,650 --> 00:40:08,310
is equal to i gamma plus/minus
square root of 4 omega 0
564
00:40:08,310 --> 00:40:13,840
squared minus gamma
squared divided by 2.
565
00:40:13,840 --> 00:40:17,940
This is actually the
second order polynomial.
566
00:40:17,940 --> 00:40:21,110
And that is actually equal to 0.
567
00:40:21,110 --> 00:40:23,800
Therefore, you can
actually solve it easily.
568
00:40:23,800 --> 00:40:26,490
And this is actually
the solution.
569
00:40:26,490 --> 00:40:30,810
And I can write it down in
a slightly different form.
570
00:40:30,810 --> 00:40:36,780
i gamma over 2 plus/minus
square root of omega 0
571
00:40:36,780 --> 00:40:43,180
squared minus gamma
squared over 4, OK?
572
00:40:43,180 --> 00:40:44,160
Any questions so far?
573
00:40:44,160 --> 00:40:45,060
Am I going too fast?
574
00:40:48,450 --> 00:40:49,290
Everything's OK?
575
00:40:53,484 --> 00:41:00,005
OK, So you can see that alpha
is equal to this expression.
576
00:41:00,005 --> 00:41:03,880
And I would like to
consider a situation
577
00:41:03,880 --> 00:41:13,020
where omega 0 is much,
much larger than gamma, OK?
578
00:41:13,020 --> 00:41:15,310
Just a reminder of
what is gamma, OK?
579
00:41:15,310 --> 00:41:18,010
Maybe you've got
already a bit confused.
580
00:41:18,010 --> 00:41:19,480
What is gamma?
581
00:41:19,480 --> 00:41:24,630
Gamma is related to the strength
of the direct force, right?
582
00:41:24,630 --> 00:41:27,760
It is actually 3b
over ml squared, OK?
583
00:41:27,760 --> 00:41:37,090
b is actually determining the
size of the direct force, OK?
584
00:41:37,090 --> 00:41:40,816
So I would like to
consider a situation.
585
00:41:40,816 --> 00:41:48,790
The first situation is if
omega 0 squared is larger
586
00:41:48,790 --> 00:41:53,320
than gamma squared over 4.
587
00:41:53,320 --> 00:42:02,540
So in that case, the
drag force is small.
588
00:42:02,540 --> 00:42:04,150
It is not huge.
589
00:42:04,150 --> 00:42:06,540
It's small, OK?
590
00:42:06,540 --> 00:42:12,140
If that is the case, this
is actually real, right?
591
00:42:12,140 --> 00:42:14,420
Because omega 0
squared is larger
592
00:42:14,420 --> 00:42:16,370
than gamma squared over 4.
593
00:42:16,370 --> 00:42:18,950
Therefore, this is real, OK?
594
00:42:18,950 --> 00:42:23,870
So now, I can actually
define omega squared,
595
00:42:23,870 --> 00:42:31,210
define that as omega 0 squared
minus gamma squared over 4, OK?
596
00:42:33,820 --> 00:42:49,090
And this will become i gamma
over 2 plus/minus omega, OK?
597
00:42:49,090 --> 00:42:53,110
So that means I would
have two solutions coming
598
00:42:53,110 --> 00:42:57,040
from this exercise.
599
00:42:57,040 --> 00:43:04,345
Z plus of t is equal to
exponential minus gamma over 2
600
00:43:04,345 --> 00:43:11,010
t exponential i omega t, OK?
601
00:43:11,010 --> 00:43:19,740
And the second solution, if
I take one of the plus sign
602
00:43:19,740 --> 00:43:22,080
and one of the minus
sign solutions,
603
00:43:22,080 --> 00:43:24,570
then the second solution
would be exponential
604
00:43:24,570 --> 00:43:34,150
minus i gamma over 2 t
exponential minus i omega t,
605
00:43:34,150 --> 00:43:35,240
OK?
606
00:43:35,240 --> 00:43:36,124
Any questions so far?
607
00:43:40,000 --> 00:43:46,160
OK, so we would like to
go back to theta, right?
608
00:43:46,160 --> 00:43:47,670
So what would be the theta?
609
00:43:50,210 --> 00:43:55,040
So that means I would have a
theta 1 of t, which is actually
610
00:43:55,040 --> 00:43:56,630
taking the real part.
611
00:43:56,630 --> 00:44:02,630
So it's theta plus maybe,
taking the real part of z plus.
612
00:44:02,630 --> 00:44:06,890
And that will give you
exponential minus gamma
613
00:44:06,890 --> 00:44:16,610
over 2 t cosine omega t, OK?
614
00:44:16,610 --> 00:44:20,640
I'm just plugging in the
solution to this equation, OK?
615
00:44:25,240 --> 00:44:29,790
Theta minus t would be
equal to exponential,
616
00:44:29,790 --> 00:44:38,050
and this gamma over
2 t sine omega t, OK?
617
00:44:38,050 --> 00:44:43,480
Finally, the full
solution of theta of t
618
00:44:43,480 --> 00:44:49,200
would be a linear combination
of these two solution, right?
619
00:44:49,200 --> 00:44:54,180
Therefore, you will get theta
of t equal to exponential
620
00:44:54,180 --> 00:45:01,080
minus gamma over 2 t a
(is some kind of constant)
621
00:45:01,080 --> 00:45:08,280
times cosine omega t
plus b sine omega t.
622
00:45:12,401 --> 00:45:16,450
And of course, from the
last time, as you will know,
623
00:45:16,450 --> 00:45:28,030
this can also be written as A
cosine omega t plus phi, OK?
624
00:45:28,030 --> 00:45:32,440
Any questions so far?
625
00:45:32,440 --> 00:45:36,410
OK, very good.
626
00:45:36,410 --> 00:45:40,500
So we have actually already
solved this equation.
627
00:45:40,500 --> 00:45:44,400
And of course, we can
actually plug this back
628
00:45:44,400 --> 00:45:49,710
into this equation of motion.
629
00:45:49,710 --> 00:45:51,716
And you will see
that it really works.
630
00:45:51,716 --> 00:45:53,090
And I'm not going
to do that now.
631
00:45:53,090 --> 00:45:55,610
But you can actually
go back home and check.
632
00:45:55,610 --> 00:45:59,250
And if you believe me, it works.
633
00:45:59,250 --> 00:46:04,650
And also at the same time, it
got two undetermined constants,
634
00:46:04,650 --> 00:46:08,880
since this is a second
order differential equation.
635
00:46:08,880 --> 00:46:11,835
Therefore, huh, this
thing actually works.
636
00:46:11,835 --> 00:46:14,220
It has two arbitrary constants.
637
00:46:14,220 --> 00:46:16,530
Therefore, that is
actually the one and only
638
00:46:16,530 --> 00:46:19,680
one solution in
the universe which
639
00:46:19,680 --> 00:46:25,220
satisfies the equation of motion
or satisfies that differential
640
00:46:25,220 --> 00:46:27,020
question, OK?
641
00:46:27,020 --> 00:46:33,090
So this thing actually
has dramatic consequences.
642
00:46:33,090 --> 00:46:34,500
The first thing
which we learn is
643
00:46:34,500 --> 00:46:39,300
that, as a function of time,
what is going to happen?
644
00:46:39,300 --> 00:46:46,390
The amplitude is now becoming
exponential minus gamma
645
00:46:46,390 --> 00:46:52,140
over 2 t times A. This is
actually the amplitude.
646
00:46:52,140 --> 00:46:57,840
The amplitude is
decreasing exponentially.
647
00:46:57,840 --> 00:47:00,570
So that is actually
the first prediction
648
00:47:00,570 --> 00:47:03,150
coming from this exercise, OK?
649
00:47:03,150 --> 00:47:07,480
The second prediction is
that this thing is still
650
00:47:07,480 --> 00:47:08,260
oscillating.
651
00:47:08,260 --> 00:47:12,400
Because you've got the cosine
omega t plus phi there,
652
00:47:12,400 --> 00:47:13,760
you see?
653
00:47:13,760 --> 00:47:16,720
So the damping
motion is going to be
654
00:47:16,720 --> 00:47:21,550
like going up and down, up
and down, and get tired.
655
00:47:21,550 --> 00:47:25,030
Therefore, the amplitude
becomes smaller, and smaller,
656
00:47:25,030 --> 00:47:26,320
and smaller.
657
00:47:26,320 --> 00:47:29,960
But it's never 0, right?
658
00:47:29,960 --> 00:47:31,190
It's never 0, OK?
659
00:47:31,190 --> 00:47:34,160
It's actually going to be
oscillating down, down, down,
660
00:47:34,160 --> 00:47:35,550
so small I couldn't see it.
661
00:47:35,550 --> 00:47:39,590
But it's still oscillating, OK?
662
00:47:39,590 --> 00:47:44,610
Finally, we actually
have also the answer
663
00:47:44,610 --> 00:47:49,050
to the original
question we posed, OK?
664
00:47:49,050 --> 00:47:54,200
So now, you can see that the
oscillation frequency is omega,
665
00:47:54,200 --> 00:47:55,080
OK?
666
00:47:55,080 --> 00:48:01,060
Originally, before we
introduced the drag force,
667
00:48:01,060 --> 00:48:05,570
omega 0, which is the
oscillation frequency,
668
00:48:05,570 --> 00:48:06,985
is actually an
angular frequency.
669
00:48:06,985 --> 00:48:10,930
It's actually the square
root of 3g over 2l.
670
00:48:10,930 --> 00:48:17,380
And you can see that the
new omega, the oscillation
671
00:48:17,380 --> 00:48:24,650
frequency with drag force,
is the square root of this,
672
00:48:24,650 --> 00:48:29,000
omega 0 squared minus
gamma squared over 4.
673
00:48:29,000 --> 00:48:32,230
So what this
actually tells us is
674
00:48:32,230 --> 00:48:38,400
that this is going
to be smaller,
675
00:48:38,400 --> 00:48:40,740
because of the drag force, OK?
676
00:48:40,740 --> 00:48:43,980
So that's a prediction.
677
00:48:43,980 --> 00:48:47,080
Let's do the experiment and
see what is going to happen.
678
00:48:47,080 --> 00:48:49,740
So let's take a look
at these two systems.
679
00:48:49,740 --> 00:48:54,870
They have the identical mass,
which our technical instructor
680
00:48:54,870 --> 00:48:57,330
actually carefully prepared.
681
00:48:57,330 --> 00:49:00,600
They have the same mass,
even though one actually
682
00:49:00,600 --> 00:49:01,980
looks a bit funny.
683
00:49:01,980 --> 00:49:04,740
The other one looks normal, OK?
684
00:49:04,740 --> 00:49:10,590
Now, what I'm going to do
is to really try and see
685
00:49:10,590 --> 00:49:13,110
which one is actually
oscillating faster, OK?
686
00:49:13,110 --> 00:49:14,802
So let's see.
687
00:49:14,802 --> 00:49:17,380
I release them at the same time.
688
00:49:17,380 --> 00:49:19,470
And you can see
that originally they
689
00:49:19,470 --> 00:49:22,120
seem to be oscillating
at the same frequency.
690
00:49:22,120 --> 00:49:27,570
But you can see very clearly
that the one with the hat
691
00:49:27,570 --> 00:49:31,230
is actually
oscillating slower, OK?
692
00:49:31,230 --> 00:49:33,810
So you can see
that, OK, 17 of you
693
00:49:33,810 --> 00:49:37,440
actually got the correct answer.
694
00:49:37,440 --> 00:49:40,380
And the most important
thing is that you
695
00:49:40,380 --> 00:49:42,780
can see that this
simple mass actually
696
00:49:42,780 --> 00:49:46,460
describes and predicts
what is going to happen
697
00:49:46,460 --> 00:49:48,930
in my little experiment.
698
00:49:48,930 --> 00:49:51,340
So that is actually really cool.
699
00:49:51,340 --> 00:49:54,890
And I think it's time
to take a little break.
700
00:49:54,890 --> 00:49:59,250
And then, we will come back
and look at other solutions.
701
00:49:59,250 --> 00:50:01,380
And of course, you
are welcome to come
702
00:50:01,380 --> 00:50:04,836
to the front to play with
those demonstrations.
703
00:50:11,180 --> 00:50:13,510
So there are two
small issues which
704
00:50:13,510 --> 00:50:16,970
were raised during the break.
705
00:50:16,970 --> 00:50:22,730
So the first one is that, if you
actually calculate the torque
706
00:50:22,730 --> 00:50:25,520
from this equation--
707
00:50:25,520 --> 00:50:27,440
so I made a mistake.
708
00:50:27,440 --> 00:50:31,490
The R vector should be
actually pointing from the nail
709
00:50:31,490 --> 00:50:33,140
to the center of mass, OK?
710
00:50:33,140 --> 00:50:35,660
So I think that's
a trivial mistake.
711
00:50:35,660 --> 00:50:38,630
So if you do this,
then you can actually
712
00:50:38,630 --> 00:50:43,010
calculate the tau
equal to R cross F.
713
00:50:43,010 --> 00:50:46,200
Then, you actually get
this minus sign, OK?
714
00:50:46,200 --> 00:50:51,580
So if I make a mistake in
pointing towards the nail,
715
00:50:51,580 --> 00:50:54,650
then you will get
no minus sign, then
716
00:50:54,650 --> 00:50:56,750
that didn't really work, OK?
717
00:50:56,750 --> 00:51:00,500
So very good, I'm very
happy that you are actually
718
00:51:00,500 --> 00:51:04,490
paying very much attention
to capture those.
719
00:51:04,490 --> 00:51:07,040
The second issue is that--
720
00:51:07,040 --> 00:51:09,590
so now, I'm saying
that, OK, now I
721
00:51:09,590 --> 00:51:13,180
have the solution in
the complex format.
722
00:51:13,180 --> 00:51:17,450
So I have a Z plus and
I have a Z minus, OK?
723
00:51:17,450 --> 00:51:21,170
And then I would like to go
to the real world, right?
724
00:51:21,170 --> 00:51:23,720
Because the imaginary
thing is actually
725
00:51:23,720 --> 00:51:27,740
hidden in some kind of motion
in the actual dimension,
726
00:51:27,740 --> 00:51:31,680
et cetera, I would like
to go back to reality, OK?
727
00:51:31,680 --> 00:51:36,790
And what I said in the class
is that I take the real part
728
00:51:36,790 --> 00:51:38,310
of one of the solutions.
729
00:51:38,310 --> 00:51:41,230
And I can also take
a real part of i
730
00:51:41,230 --> 00:51:43,440
times one of the solutions.
731
00:51:43,440 --> 00:51:46,400
But of course, you
can also do this
732
00:51:46,400 --> 00:51:50,440
by doing a linear combination
of the solutions, right?
733
00:51:50,440 --> 00:51:53,100
As we actually
discussed last time,
734
00:51:53,100 --> 00:51:55,100
the linear combination
of the solutions
735
00:51:55,100 --> 00:51:59,990
is also a solution to the
same equation of motion,
736
00:51:59,990 --> 00:52:02,010
since this one is
actually linear.
737
00:52:02,010 --> 00:52:06,960
Therefore, what I
actually do is actually
738
00:52:06,960 --> 00:52:11,850
to sum the two solutions, Z plus
and Z minus and divide it by 2.
739
00:52:11,850 --> 00:52:20,320
Or actually, I can actually do
a minus i/2 times Z plus minus Z
740
00:52:20,320 --> 00:52:21,260
minus, OK?
741
00:52:21,260 --> 00:52:24,740
And then I can also extract
this sign term here, OK?
742
00:52:24,740 --> 00:52:28,610
So that should be the correct
explanation of the two
743
00:52:28,610 --> 00:52:33,620
solutions in the real axis, OK?
744
00:52:33,620 --> 00:52:36,300
Any questions so far?
745
00:52:36,300 --> 00:52:40,200
Thank you very much
for capturing those.
746
00:52:40,200 --> 00:52:41,910
Ok, so now, you can
see that we have
747
00:52:41,910 --> 00:52:46,530
been discussing the equation of
motion of this functional form.
748
00:52:46,530 --> 00:52:49,780
And the one thing which is
really, really interesting
749
00:52:49,780 --> 00:52:57,170
is that the solution, when we
take a small drag force limit,
750
00:52:57,170 --> 00:53:01,560
actually we arrive at a
beautiful solution that
751
00:53:01,560 --> 00:53:06,560
looks like this, A
exponential minus gamma over 2
752
00:53:06,560 --> 00:53:09,370
t cosine omega t plus phi.
753
00:53:09,370 --> 00:53:13,080
That actually predicts
the oscillation, OK?
754
00:53:13,080 --> 00:53:17,310
At the same time, it also
says that the amplitude
755
00:53:17,310 --> 00:53:23,060
is actually going to drop
exponentially, but never 0, OK?
756
00:53:23,060 --> 00:53:28,650
Finally, we also know that
this solution actually tells us
757
00:53:28,650 --> 00:53:33,840
that, if we have a spring mass
system oscillating up and down,
758
00:53:33,840 --> 00:53:40,770
if we have a rod like what
we actually solve in a class,
759
00:53:40,770 --> 00:53:46,010
this object is going to pass
through 0, the equilibrium
760
00:53:46,010 --> 00:53:48,450
position, an infinite
number of times, right?
761
00:53:48,450 --> 00:53:51,630
Because the cosine
is always there.
762
00:53:51,630 --> 00:53:53,587
Therefore, although
the amplitude
763
00:53:53,587 --> 00:53:55,170
will become very
small, but it's still
764
00:53:55,170 --> 00:53:59,910
oscillating forever until
the end of the universe, OK?
765
00:53:59,910 --> 00:54:03,560
All right, so that's actually
what we have learned.
766
00:54:03,560 --> 00:54:06,660
And also, one thing which
we learned last time
767
00:54:06,660 --> 00:54:12,010
is that simple harmonic
motion, like this one, which
768
00:54:12,010 --> 00:54:15,310
we were just showing
here, or this one,
769
00:54:15,310 --> 00:54:18,330
which is actually a mass
oscillating back and forth
770
00:54:18,330 --> 00:54:24,830
on the track, is actually just a
projection of a circular motion
771
00:54:24,830 --> 00:54:26,760
in a complex plane, OK?
772
00:54:26,760 --> 00:54:30,770
And what we are really
seeing here in front of you
773
00:54:30,770 --> 00:54:34,950
is actually a projection
to the real axis, OK?
774
00:54:34,950 --> 00:54:37,850
So that's actually a
really remarkable result
775
00:54:37,850 --> 00:54:40,530
and a beautiful picture.
776
00:54:40,530 --> 00:54:44,670
And of course, we can actually
also plug in the solution
777
00:54:44,670 --> 00:54:47,530
with damping.
778
00:54:47,530 --> 00:54:50,900
So what is actually the
picture in this language,
779
00:54:50,900 --> 00:54:54,130
in this exact same language?
780
00:54:54,130 --> 00:54:58,270
If we actually follow
the locus, then basically
781
00:54:58,270 --> 00:55:00,980
what you are going to see
is that this thing actually
782
00:55:00,980 --> 00:55:02,530
spirals.
783
00:55:02,530 --> 00:55:06,400
And the amplitude is actually
getting smaller and smaller
784
00:55:06,400 --> 00:55:10,820
and is sucked into this
black hole in the 0, 0, OK?
785
00:55:10,820 --> 00:55:14,440
So you can see that
now the picture
786
00:55:14,440 --> 00:55:18,170
looks as if there is
something really rotating
787
00:55:18,170 --> 00:55:19,660
in the complex plane.
788
00:55:19,660 --> 00:55:21,790
And it's actually approaching 0.
789
00:55:21,790 --> 00:55:23,770
Because the
amplitude is actually
790
00:55:23,770 --> 00:55:25,670
getting smaller and smaller.
791
00:55:25,670 --> 00:55:29,960
But this whole thing
is still rotating, OK?
792
00:55:29,960 --> 00:55:33,370
OK, that's really nice.
793
00:55:33,370 --> 00:55:39,360
All right, so now, this is
actually a special case.
794
00:55:39,360 --> 00:55:45,000
When we actually assume that
gamma is actually pretty small.
795
00:55:45,000 --> 00:55:49,800
So you have very
small drag force, OK?
796
00:55:49,800 --> 00:55:54,330
So let's actually check
what would happen.
797
00:55:54,330 --> 00:55:58,200
If I now start to
increase the drag force,
798
00:55:58,200 --> 00:56:02,700
make this hat larger,
larger, and larger,
799
00:56:02,700 --> 00:56:08,660
introducing more and more
drag, what is going to happen?
800
00:56:08,660 --> 00:56:13,790
OK, so now, I consider
the second situation,
801
00:56:13,790 --> 00:56:22,190
omega 0 squared equal
to gamma squared over 4.
802
00:56:22,190 --> 00:56:25,520
OK, so when the
gamma is very small,
803
00:56:25,520 --> 00:56:29,150
what we see is that this is
actually underdamped, right?
804
00:56:29,150 --> 00:56:32,150
So the damping is really small.
805
00:56:32,150 --> 00:56:38,050
But if I increase the
gamma to a critical value,
806
00:56:38,050 --> 00:56:41,840
now omega 0 squared happens
to be equal to gamma
807
00:56:41,840 --> 00:56:43,960
squared over 4, OK?
808
00:56:43,960 --> 00:56:56,660
I call this a critically
damped oscillator, OK?
809
00:56:56,660 --> 00:56:58,770
So what does that mean?
810
00:56:58,770 --> 00:57:07,070
That means omega is
equal to 0, you see?
811
00:57:07,070 --> 00:57:10,190
This is our definition
of omega, right?
812
00:57:10,190 --> 00:57:15,080
If omega 0 squared is equal
to gamma squared of over 4,
813
00:57:15,080 --> 00:57:19,670
then omega is equal to 0.
814
00:57:19,670 --> 00:57:22,970
That is actually
the critical moment
815
00:57:22,970 --> 00:57:28,720
the system stops
oscillating, OK?
816
00:57:28,720 --> 00:57:33,050
So it is not
oscillating anymore.
817
00:57:33,050 --> 00:57:38,220
So now, I can actually
start from the solution
818
00:57:38,220 --> 00:57:41,150
I obtained from 1, OK?
819
00:57:41,150 --> 00:57:43,910
Then, I can actually
now make use
820
00:57:43,910 --> 00:57:48,950
of these two solutions, the
theta plus and the theta minus.
821
00:57:53,730 --> 00:58:00,250
Theta plus t would be equal
to exponential minus gamma
822
00:58:00,250 --> 00:58:05,940
over 2 t cosine omega t.
823
00:58:05,940 --> 00:58:09,240
When omega goes to
0, what is going
824
00:58:09,240 --> 00:58:13,640
to happen is that this is
actually becoming, which value?
825
00:58:13,640 --> 00:58:15,600
Anybody know?
826
00:58:15,600 --> 00:58:19,010
If omega is 0, what
is going to happen?
827
00:58:19,010 --> 00:58:20,180
1, yeah.
828
00:58:20,180 --> 00:58:21,650
OK, 1, right?
829
00:58:21,650 --> 00:58:27,155
So that will give me exponential
minus gamma over 2 t.
830
00:58:30,010 --> 00:58:32,730
Theta minus t--
831
00:58:32,730 --> 00:58:35,880
OK, I can do the same trick
and see what will happen.
832
00:58:35,880 --> 00:58:39,550
So I take theta minus
t, which is actually
833
00:58:39,550 --> 00:58:42,110
obtained from the
exercise number one
834
00:58:42,110 --> 00:58:44,450
when we discussed the
underdamped system.
835
00:58:44,450 --> 00:58:47,250
Then, you actually get
exponential minus gamma
836
00:58:47,250 --> 00:58:52,010
over 2 t sine omega t.
837
00:58:52,010 --> 00:59:00,140
When omega goes to 0, actually
then I get 0 this time, OK?
838
00:59:00,140 --> 00:59:01,920
So that doesn't
really work, right?
839
00:59:01,920 --> 00:59:05,080
Because if I have a
solution which is 0, then
840
00:59:05,080 --> 00:59:06,800
it's not describing
anything, right?
841
00:59:06,800 --> 00:59:09,410
I can always add
0 to the solution.
842
00:59:09,410 --> 00:59:10,700
But that doesn't help you.
843
00:59:10,700 --> 00:59:17,470
OK, so instead of taking
the limit of this function,
844
00:59:17,470 --> 00:59:21,170
actually we choose
to actually do
845
00:59:21,170 --> 00:59:25,960
theta minus t divided by omega.
846
00:59:25,960 --> 00:59:31,310
And then, we actually make
this omega approaching 0.
847
00:59:31,310 --> 00:59:34,310
Then basically, I get
exponential minus gamma
848
00:59:34,310 --> 00:59:42,280
over 2 t sine omega t
divided by omega, OK?
849
00:59:42,280 --> 00:59:48,210
If I have this omega
approaching to 0,
850
00:59:48,210 --> 00:59:53,130
then this is actually roughly
just exponential minus gamma
851
00:59:53,130 --> 00:59:57,930
over 2 t omega t over omega.
852
00:59:57,930 --> 01:00:05,010
And this is actually giving
you t times exponential minus
853
01:00:05,010 --> 01:00:06,220
gamma over 2 t.
854
01:00:09,580 --> 01:00:10,904
Any questions so far?
855
01:00:13,870 --> 01:00:15,556
Yes.
856
01:00:15,556 --> 01:00:18,448
AUDIENCE: Completely
unrelated, but is
857
01:00:18,448 --> 01:00:22,600
that a negative sign in front
of the theta minus negative 1/2?
858
01:00:22,600 --> 01:00:23,750
YEN-JIE LEE: This one?
859
01:00:23,750 --> 01:00:24,690
AUDIENCE: Yeah.
860
01:00:24,690 --> 01:00:25,440
YEN-JIE LEE: Yeah.
861
01:00:25,440 --> 01:00:28,465
So actually, OK, yeah.
862
01:00:28,465 --> 01:00:30,840
AUDIENCE: In front of the
1/2 is that a negative sign?
863
01:00:30,840 --> 01:00:32,590
YEN-JIE LEE: Yes, this
is a negative sign.
864
01:00:35,090 --> 01:00:39,990
OK, any other questions?
865
01:00:39,990 --> 01:00:45,840
OK, so you can see that now
I arrive at two solutions.
866
01:00:45,840 --> 01:00:50,320
One is actually proportional
to exponential minus gamma
867
01:00:50,320 --> 01:00:51,980
over 2 t.
868
01:00:51,980 --> 01:00:54,550
The other one is actually
proportional to t
869
01:00:54,550 --> 01:00:59,150
times exponential minus
gamma over 2 t, OK?
870
01:00:59,150 --> 01:01:03,680
So you can see that the cosine
or sine term disappeared,
871
01:01:03,680 --> 01:01:04,700
right?
872
01:01:04,700 --> 01:01:10,000
So that means you are
never oscillating, OK?
873
01:01:10,000 --> 01:01:13,210
So this is actually what
we see in this slide,
874
01:01:13,210 --> 01:01:15,730
this so-called
critically damped, OK?
875
01:01:15,730 --> 01:01:18,730
When actually,
omega 0 squared is
876
01:01:18,730 --> 01:01:23,830
equal to gamma squared over 4.
877
01:01:23,830 --> 01:01:26,620
And you can see that
what is going to happen
878
01:01:26,620 --> 01:01:34,540
is that this mass or this
rod is going to pass 0
879
01:01:34,540 --> 01:01:39,070
only one time at most, OK?
880
01:01:39,070 --> 01:01:41,980
And it could actually
never passed 0,
881
01:01:41,980 --> 01:01:45,210
if you actually set up the
initial condition correctly,
882
01:01:45,210 --> 01:01:45,710
OK?
883
01:01:45,710 --> 01:01:50,380
So one thing which I can do is
I really shoot this mass really,
884
01:01:50,380 --> 01:01:53,230
really, very forcefully,
so that I have
885
01:01:53,230 --> 01:01:56,040
a very large initial velocity.
886
01:01:56,040 --> 01:01:58,690
And what it actually
is going to do,
887
01:01:58,690 --> 01:02:00,630
like the right-hand
side diagram,
888
01:02:00,630 --> 01:02:03,550
is that, oh, you
overshoot the 0 a bit.
889
01:02:03,550 --> 01:02:09,040
Then, it goes back
almost exponentially, OK?
890
01:02:09,040 --> 01:02:13,830
So at most, you can
only pass through 0 one
891
01:02:13,830 --> 01:02:18,830
time, if you do this kind
of initial condition, OK?
892
01:02:18,830 --> 01:02:22,520
So that is actually
pretty interesting.
893
01:02:22,520 --> 01:02:29,370
And there are practical
applications of this solution,
894
01:02:29,370 --> 01:02:30,030
actually.
895
01:02:30,030 --> 01:02:32,870
So for example, we
have the door closed.
896
01:02:32,870 --> 01:02:34,490
So it's also here, right?
897
01:02:34,490 --> 01:02:38,030
The door closed, you would
like to have the door go back
898
01:02:38,030 --> 01:02:42,290
to the original closed
mold, the position
899
01:02:42,290 --> 01:02:45,740
of equilibrium position
actually really fast, OK?
900
01:02:45,740 --> 01:02:50,870
So what you can do is really
design this door close
901
01:02:50,870 --> 01:02:54,700
so that it actually matches
with the critical dampness
902
01:02:54,700 --> 01:02:58,460
situation, of your condition, so
that actually you would go back
903
01:02:58,460 --> 01:03:02,478
to 0 really quick, OK?
904
01:03:02,478 --> 01:03:06,380
Any questions?
905
01:03:06,380 --> 01:03:11,760
OK, so now, what we
could do is that, instead
906
01:03:11,760 --> 01:03:21,810
of having a very small drag
force, or we'll a slightly
907
01:03:21,810 --> 01:03:23,610
larger drag force,
so that actually
908
01:03:23,610 --> 01:03:28,320
reach the critically damped
situation, what we could do
909
01:03:28,320 --> 01:03:33,820
is that we put the whole
system into water, right?
910
01:03:33,820 --> 01:03:37,450
Then, the drag force
will be very big, OK?
911
01:03:37,450 --> 01:03:42,160
And we would like to see
what is going to happen, OK?
912
01:03:42,160 --> 01:03:46,690
So in this case is
the third situation.
913
01:03:46,690 --> 01:03:50,680
The third situation is that
omega 0 squared is actually
914
01:03:50,680 --> 01:03:55,960
smaller than gamma
squared over 4.
915
01:03:55,960 --> 01:04:03,220
So you have huge drag force, OK?
916
01:04:06,050 --> 01:04:09,770
So that would give
you a situation
917
01:04:09,770 --> 01:04:17,037
which is called
overdamped oscillator.
918
01:04:21,870 --> 01:04:27,140
Now, I have, again,
alpha is equal to i gamma
919
01:04:27,140 --> 01:04:32,600
over 2 plus/minus
square root of omega 0
920
01:04:32,600 --> 01:04:37,180
squared minus gamma
squared over 4, right?
921
01:04:37,180 --> 01:04:39,820
I'm just copying from here, OK?
922
01:04:39,820 --> 01:04:42,960
And that will be equal to i--
923
01:04:42,960 --> 01:04:45,850
I can take out the i, OK?--
924
01:04:45,850 --> 01:04:50,800
gamma over 2
plus/minus square root
925
01:04:50,800 --> 01:04:56,777
of gamma squared over 4
minus omega 0 squared.
926
01:05:02,140 --> 01:05:10,250
Now, I can actually
define gamma plus/minus
927
01:05:10,250 --> 01:05:18,270
equal to gamma over 2
plus/minus square root of gamma
928
01:05:18,270 --> 01:05:25,580
squared over 4 minus
omega 0 squared, OK?
929
01:05:25,580 --> 01:05:29,240
Then basically, the solution--
930
01:05:29,240 --> 01:05:32,340
actually now, I already
have the solution.
931
01:05:32,340 --> 01:05:36,720
So basically, the two solutions
would be looking like this.
932
01:05:36,720 --> 01:05:41,670
Theta of t would be
equal to A plus some kind
933
01:05:41,670 --> 01:05:48,990
of constant exponential
minus gamma plus t plus A
934
01:05:48,990 --> 01:05:56,070
minus exponential minus
gamma minus t, OK?
935
01:05:56,070 --> 01:05:59,070
Because this is actually
becoming already--
936
01:05:59,070 --> 01:06:04,730
OK, so alpha is actually
i times gamma plus/minus.
937
01:06:04,730 --> 01:06:08,360
Therefore, if you put
it back into this,
938
01:06:08,360 --> 01:06:09,990
then basically what
you are getting
939
01:06:09,990 --> 01:06:18,450
is exponential minus gamma
plus t or exponential minus
940
01:06:18,450 --> 01:06:20,700
gamma minus t, OK?
941
01:06:20,700 --> 01:06:22,920
So that's already
a real function.
942
01:06:22,920 --> 01:06:26,610
And the linear combination
of these two solutions
943
01:06:26,610 --> 01:06:33,030
is our final, full solution
to the equation of motion.
944
01:06:33,030 --> 01:06:36,670
OK, again, what we
are going to see
945
01:06:36,670 --> 01:06:41,700
is that actually the
drag force is huge.
946
01:06:41,700 --> 01:06:45,090
I just throw the whole
system into water.
947
01:06:45,090 --> 01:06:49,840
And the water is really trying
to stop the oscillation, really
948
01:06:49,840 --> 01:06:50,760
very much.
949
01:06:50,760 --> 01:06:54,030
Therefore, you can
see that, huh, again,
950
01:06:54,030 --> 01:06:57,510
I don't have any
oscillation, OK?
951
01:06:57,510 --> 01:07:00,180
If I am very, very
strong, I really
952
01:07:00,180 --> 01:07:06,450
start the initial velocity
or initial angular
953
01:07:06,450 --> 01:07:12,300
velocity really high, I actually
give a huge amount of energy
954
01:07:12,300 --> 01:07:15,960
into the system,
then, at most again, I
955
01:07:15,960 --> 01:07:20,310
can actually have the system
to pass through the equilibrium
956
01:07:20,310 --> 01:07:22,530
position only one time.
957
01:07:22,530 --> 01:07:28,740
Then, this whole system
will slowly recover,
958
01:07:28,740 --> 01:07:36,390
because exponential
function we see here.
959
01:07:36,390 --> 01:07:42,571
The amplitude is going to be
decaying exponentially, OK?
960
01:07:42,571 --> 01:07:45,560
Any questions?
961
01:07:45,560 --> 01:07:50,180
So let's actually do a quick
demonstration here, OK?
962
01:07:50,180 --> 01:07:56,540
So here, this is actually the
original little ball here,
963
01:07:56,540 --> 01:07:59,060
a metal one, which
actually you can
964
01:07:59,060 --> 01:08:04,790
see that this is really going
to go back and forth really
965
01:08:04,790 --> 01:08:05,720
nicely.
966
01:08:05,720 --> 01:08:09,860
And you can see that,
because of the friction,
967
01:08:09,860 --> 01:08:12,710
actually the
amplitude is becoming
968
01:08:12,710 --> 01:08:15,200
smaller and smaller, OK?
969
01:08:15,200 --> 01:08:19,189
So that actually matches
with this situation, right?
970
01:08:19,189 --> 01:08:22,850
So it's actually an
underdamped situation.
971
01:08:22,850 --> 01:08:26,529
This ball, in an
idealized situation,
972
01:08:26,529 --> 01:08:32,359
is going to go through 0
infinite number of times, OK?
973
01:08:32,359 --> 01:08:35,279
So now, what I am
going to do is now
974
01:08:35,279 --> 01:08:39,500
I change this ball to something
which is different, OK?
975
01:08:39,500 --> 01:08:43,569
This is actually
made of magnets, OK?
976
01:08:43,569 --> 01:08:45,770
And let's see what
is going to happen.
977
01:08:45,770 --> 01:08:47,950
So now, you can see that,
because this is actually
978
01:08:47,950 --> 01:08:51,460
made of magnets, therefore, the
drag force will be colossal,
979
01:08:51,460 --> 01:08:53,396
will be very, very big.
980
01:08:53,396 --> 01:08:56,378
And let's see what will happen.
981
01:08:59,859 --> 01:09:03,460
You see that the
drag force is huge.
982
01:09:03,460 --> 01:09:06,899
Therefore, you see I
put it here so that it
983
01:09:06,899 --> 01:09:08,699
has big initial velocity.
984
01:09:12,611 --> 01:09:16,800
It only passes
through 0 once, right?
985
01:09:16,800 --> 01:09:19,750
Of course, it now is actually
approaching the zero really,
986
01:09:19,750 --> 01:09:21,319
really slowly, exponentially.
987
01:09:21,319 --> 01:09:23,460
But it is not 0, OK?
988
01:09:23,460 --> 01:09:27,870
So it only passes through the
0 if you believe the math,
989
01:09:27,870 --> 01:09:30,580
only once, OK?
990
01:09:30,580 --> 01:09:35,470
Just to show that
this is a real deal--
991
01:09:35,470 --> 01:09:36,859
OK, now, whoa, right?
992
01:09:42,770 --> 01:09:48,189
Oh, I'm not trying to
destroy the classroom, OK?
993
01:09:48,189 --> 01:09:50,229
So you can actually
play with this
994
01:09:50,229 --> 01:09:52,670
after we finish
your lecture, OK?
995
01:09:55,200 --> 01:09:58,170
I would like to
ask you a question.
996
01:09:58,170 --> 01:10:01,930
After we learned this
from this lecture,
997
01:10:01,930 --> 01:10:08,920
there are three situations,
underdamped, critically damped,
998
01:10:08,920 --> 01:10:11,230
and overdamped, OK?
999
01:10:11,230 --> 01:10:13,550
I would like to ask
you two questions.
1000
01:10:13,550 --> 01:10:18,130
The first one is through
this demonstration, OK?
1001
01:10:18,130 --> 01:10:25,190
So, now I have a system
which is nicely constructed.
1002
01:10:25,190 --> 01:10:27,990
I hope you can see it, OK?
1003
01:10:27,990 --> 01:10:29,560
You can see it.
1004
01:10:29,560 --> 01:10:36,870
And this system is made
of a torsional spring.
1005
01:10:36,870 --> 01:10:39,930
And also, there's
a pad here, which
1006
01:10:39,930 --> 01:10:42,170
is connected to the spring, OK?
1007
01:10:42,170 --> 01:10:44,790
If I actually
perturb this thing,
1008
01:10:44,790 --> 01:10:49,770
it's going to be oscillating
back and forth before I turn
1009
01:10:49,770 --> 01:10:56,070
on the power, so that the
lower part is actually
1010
01:10:56,070 --> 01:10:58,930
you have a magnet, OK?
1011
01:10:58,930 --> 01:11:01,140
It's not turned on yet, OK?
1012
01:11:01,140 --> 01:11:04,260
And this magnet is
going to provide
1013
01:11:04,260 --> 01:11:10,590
a drag force to actually change
the behavior of the system, OK?
1014
01:11:10,590 --> 01:11:15,480
So you can see that, before
I turn on the magnetic field,
1015
01:11:15,480 --> 01:11:18,120
the whole system is actually
oscillating back and forth
1016
01:11:18,120 --> 01:11:19,170
really nicely.
1017
01:11:19,170 --> 01:11:23,450
As we predicted, small
amplitude vibration
1018
01:11:23,450 --> 01:11:26,190
is harmonic oscillation, OK?
1019
01:11:26,190 --> 01:11:27,760
So that's very nice.
1020
01:11:27,760 --> 01:11:32,700
So now, what am I going to
do is to turn on the power
1021
01:11:32,700 --> 01:11:35,190
and see what is going to happen.
1022
01:11:35,190 --> 01:11:42,450
After I turn on the power,
there's an electric field, OK?
1023
01:11:42,450 --> 01:11:45,730
And this is actually
going to be--
1024
01:11:45,730 --> 01:11:49,870
OK, so the magnetic field
is actually turned down.
1025
01:11:49,870 --> 01:11:53,370
Therefore, it is actually
acting like a drag force
1026
01:11:53,370 --> 01:11:55,060
to this system, OK?
1027
01:11:55,060 --> 01:11:57,910
So let's actually see
what is going to happen.
1028
01:11:57,910 --> 01:12:01,540
Now, I release this.
1029
01:12:01,540 --> 01:12:05,060
The behavior of the
system looks like this.
1030
01:12:05,060 --> 01:12:09,730
It first oscillates,
and then it stops.
1031
01:12:09,730 --> 01:12:14,940
So the question is, is this a
critically damped, underdamped,
1032
01:12:14,940 --> 01:12:16,530
or overdamped system?
1033
01:12:16,530 --> 01:12:19,360
Anybody knows?
1034
01:12:19,360 --> 01:12:19,860
Yeah?
1035
01:12:19,860 --> 01:12:21,330
AUDIENCE: Underdamped.
1036
01:12:21,330 --> 01:12:23,120
YEN-JIE LEE: Yes,
this is underdamped.
1037
01:12:23,120 --> 01:12:24,670
How do I see that?
1038
01:12:24,670 --> 01:12:28,360
That is because, when
I do this experiment,
1039
01:12:28,360 --> 01:12:33,870
you would pass through
0s multiple times.
1040
01:12:33,870 --> 01:12:37,280
Therefore, there are
oscillations coming into play.
1041
01:12:37,280 --> 01:12:40,700
Therefore, I can conclude
that the drag force is not
1042
01:12:40,700 --> 01:12:41,750
large enough.
1043
01:12:41,750 --> 01:12:44,900
So that is actually an
underdamped situation, OK?
1044
01:12:44,900 --> 01:12:47,990
And the next time, we are
going to drag this system.
1045
01:12:47,990 --> 01:12:51,050
I have a second
question for you.
1046
01:12:51,050 --> 01:12:55,850
So now, your friends
know that you took 8.03.
1047
01:12:55,850 --> 01:12:58,190
Therefore, they will
wonder if you can actually
1048
01:12:58,190 --> 01:13:04,880
design a car suspension system,
to see if you can actually
1049
01:13:04,880 --> 01:13:06,860
make this design for them.
1050
01:13:06,860 --> 01:13:12,500
When you design this
car, which condition
1051
01:13:12,500 --> 01:13:16,610
will you consider
to set up the car?
1052
01:13:16,610 --> 01:13:22,700
Will you set it up as
underdamped, critically damped,
1053
01:13:22,700 --> 01:13:24,710
or overdamped?
1054
01:13:24,710 --> 01:13:27,998
How many of you actually think
it should be underdamped?
1055
01:13:31,910 --> 01:13:33,650
No, nobody?
1056
01:13:33,650 --> 01:13:39,390
How many of you actually
think it should be overdamped?
1057
01:13:39,390 --> 01:13:45,110
1, 2, 3, 4, OK.
1058
01:13:45,110 --> 01:13:49,250
How many of you actually think
it should be critically damped?
1059
01:13:49,250 --> 01:13:51,360
OK, the majority of
you think that should
1060
01:13:51,360 --> 01:13:52,930
be the correct design.
1061
01:13:52,930 --> 01:13:59,680
So if you have the car designed
as an underdamped situation,
1062
01:13:59,680 --> 01:14:01,470
then, when you
drive the car, you
1063
01:14:01,470 --> 01:14:03,280
are going to have
very funny style.
1064
01:14:03,280 --> 01:14:05,760
You are going to have this.
1065
01:14:05,760 --> 01:14:07,470
This is the style.
1066
01:14:07,470 --> 01:14:11,580
So the car is going to be
oscillating all the time, OK?
1067
01:14:11,580 --> 01:14:14,290
Because it's going to be there.
1068
01:14:14,290 --> 01:14:18,270
And it's really damping
really slowly, OK?
1069
01:14:18,270 --> 01:14:22,860
If you design it
to be overdamped,
1070
01:14:22,860 --> 01:14:24,930
it would become
very bumpy, right?
1071
01:14:24,930 --> 01:14:29,040
So let's take a limit
of infinitely large drag
1072
01:14:29,040 --> 01:14:30,600
force constant, OK?
1073
01:14:30,600 --> 01:14:34,590
Then, it's like, when you
hit some bump, you go woo!
1074
01:14:34,590 --> 01:14:36,360
Wow!
1075
01:14:36,360 --> 01:14:40,710
It doesn't really help you
to reduce the amplitude, OK?
1076
01:14:40,710 --> 01:14:44,440
So the correct
answer is you would
1077
01:14:44,440 --> 01:14:50,010
give the advice that you would
do it critically damped, OK?
1078
01:14:50,010 --> 01:14:54,420
So before we end
the section today,
1079
01:14:54,420 --> 01:14:57,820
I would like to pose
a question to you.
1080
01:14:57,820 --> 01:15:02,500
The thing which we have learned
from simple harmonic motion
1081
01:15:02,500 --> 01:15:07,090
is that the energy is conserved
in a simple harmonic motion,
1082
01:15:07,090 --> 01:15:07,590
OK?
1083
01:15:07,590 --> 01:15:13,500
I have the Fs, the spring force,
proportional to minus k times
1084
01:15:13,500 --> 01:15:15,000
x.
1085
01:15:15,000 --> 01:15:18,360
And the energy is conserved, OK?
1086
01:15:18,360 --> 01:15:23,430
But if I add a drag force in
the form or minus b times v,
1087
01:15:23,430 --> 01:15:25,170
energy is not conserved, right?
1088
01:15:25,170 --> 01:15:28,070
So you can see that it
was actually oscillating.
1089
01:15:28,070 --> 01:15:31,090
Now, it's not
oscillating, right?
1090
01:15:31,090 --> 01:15:34,200
This thing has stopped
oscillating, OK?
1091
01:15:34,200 --> 01:15:38,400
Why is that the
case mathematically?
1092
01:15:38,400 --> 01:15:42,600
OK, we know what is
happening physically
1093
01:15:42,600 --> 01:15:43,980
in this physical system.
1094
01:15:43,980 --> 01:15:50,250
Because OK, this Mexican hat
is trying to push the air away.
1095
01:15:50,250 --> 01:15:53,280
So what is going to happen
is that it's transferring
1096
01:15:53,280 --> 01:15:59,040
the energy from this system to
the molecules of the air, OK?
1097
01:15:59,040 --> 01:16:00,780
So it's accelerating the air.
1098
01:16:00,780 --> 01:16:02,700
So the energy goes away.
1099
01:16:02,700 --> 01:16:06,660
But why the mathematical
form looks so similar
1100
01:16:06,660 --> 01:16:09,120
and it does different things?
1101
01:16:09,120 --> 01:16:10,820
And think about it.
1102
01:16:10,820 --> 01:16:14,890
And I'm not going to talk
about the answer today.
1103
01:16:14,890 --> 01:16:17,910
And thank you very much.
1104
01:16:17,910 --> 01:16:20,890
And we will continue
next time to see
1105
01:16:20,890 --> 01:16:24,770
what we can learn if I start
to drive the oscillator.
1106
01:16:24,770 --> 01:16:26,490
Bye-bye.