1
00:00:27,000 --> 00:00:29,984
So, last time,
we had driven damped
2
00:00:29,984 --> 00:00:33,143
oscillations,
and we did steady-state
3
00:00:33,143 --> 00:00:36,654
solutions.
And the steady-state solutions
4
00:00:36,654 --> 00:00:41,920
have no adjustable constants.
That is strange because at time
5
00:00:41,920 --> 00:00:45,343
T equals zero,
I can put the object in a
6
00:00:45,343 --> 00:00:48,853
certain position.
I can give it a certain
7
00:00:48,853 --> 00:00:52,364
velocity, and so I have two free
choices.
8
00:00:52,364 --> 00:00:57,805
And yet, those free choices did
not show up in our steady-state
9
00:00:57,805 --> 00:01:01,422
solution.
In other words,
10
00:01:01,422 --> 00:01:06,922
the system had lost its memory
over what happened at time T
11
00:01:06,922 --> 00:01:09,387
equals zero.
And therefore,
12
00:01:09,387 --> 00:01:14,318
I discussed with you that you
have to wait to go into
13
00:01:14,318 --> 00:01:18,491
steady-state.
And that is what I will address
14
00:01:18,491 --> 00:01:22,000
today.
There must be something missing
15
00:01:22,000 --> 00:01:27,784
what we did so that the time T
equals zero adjustables do show
16
00:01:27,784 --> 00:01:30,913
up.
So, let's return to our spring
17
00:01:30,913 --> 00:01:35,509
system.
So, we have here spring
18
00:01:35,509 --> 00:01:40,854
constant K mass M.
And, this object can be driven.
19
00:01:40,854 --> 00:01:47,290
You could do that with a force
directly on it here where you
20
00:01:47,290 --> 00:01:53,181
can shake on the left side,
omega zero squared equals K
21
00:01:53,181 --> 00:01:56,563
over M.
I will call it that way,
22
00:01:56,563 --> 00:02:03,000
and gamma equals B over M.
So, that's the damping.
23
00:02:03,000 --> 00:02:09,043
Now, let us assume that I don't
drive it, that I just give it a
24
00:02:09,043 --> 00:02:14,014
kick, time T equals zero.
I let it do its own thing.
25
00:02:14,014 --> 00:02:19,570
We discussed that two lectures
ago, so we get an un-driven
26
00:02:19,570 --> 00:02:23,956
situation, not driven.
And then, the solution,
27
00:02:23,956 --> 00:02:28,732
you should remember,
has this exponential decay in
28
00:02:28,732 --> 00:02:33,151
it.
[Is it?] X equals some value X
29
00:02:33,151 --> 00:02:39,455
that follows from the initial
conditions times E to the minus
30
00:02:39,455 --> 00:02:45,758
gamma over two times T times
cosine omega prime T plus alpha?
31
00:02:45,758 --> 00:02:51,957
And these, the X and the alpha
can be found if you know at T
32
00:02:51,957 --> 00:02:57,000
equals zero what X is,
and what X dot is.
33
00:02:57,000 --> 00:03:02,425
Omega prime is a frequency
which is the square root of
34
00:03:02,425 --> 00:03:07,338
omega zero squared minus gamma
squared over four,
35
00:03:07,338 --> 00:03:12,047
just a hair under omega zero,
but that depends,
36
00:03:12,047 --> 00:03:17,165
of course, on gamma.
So, this is the situation when
37
00:03:17,165 --> 00:03:22,283
we are not driving.
Let's now take a situation that
38
00:03:22,283 --> 00:03:24,842
we do drive.
So, we drive,
39
00:03:24,842 --> 00:03:28,834
for instance,
with a force that we apply
40
00:03:28,834 --> 00:03:36,000
directly to this object,
which is one way of driving it.
41
00:03:36,000 --> 00:03:40,081
And, that force is F zero times
cosine omega T.
42
00:03:40,081 --> 00:03:44,430
This omega is my omega.
I force that omega on that
43
00:03:44,430 --> 00:03:47,713
system.
I don't give a damn what omega
44
00:03:47,713 --> 00:03:51,262
prime is [or?] the term of what
omega is.
45
00:03:51,262 --> 00:03:54,634
And, that's tough luck for this
object.
46
00:03:54,634 --> 00:03:58,716
And, we know that in the
steady-state solution,
47
00:03:58,716 --> 00:04:04,037
only this omega survives.
That one will go.
48
00:04:04,037 --> 00:04:11,230
And, the solution that we have
seen that we derived last time
49
00:04:11,230 --> 00:04:16,984
is some amplitude,
A, times the cosine of omega T
50
00:04:16,984 --> 00:04:21,299
minus delta.
That is my omega that is
51
00:04:21,299 --> 00:04:25,974
nonnegotiable.
This is the steady-state.
52
00:04:25,974 --> 00:04:32,687
And, we derived that the
tangent of delta was omega gamma
53
00:04:32,687 --> 00:04:40,000
divided by omega zero squared
minus omega squared.
54
00:04:40,000 --> 00:04:45,923
And, we found for A this
monstrous solution.
55
00:04:45,923 --> 00:04:52,122
Remember, upstairs we had F
zero divided by M.
56
00:04:52,122 --> 00:04:59,974
And, downstairs we have the
square root omega zero squared
57
00:04:59,974 --> 00:05:07,000
minus omega squared plus omega
gamma squared.
58
00:05:07,000 --> 00:05:12,578
And we spent the entire lecture
on dealing with this A.
59
00:05:12,578 --> 00:05:17,330
We changed omega,
and we evaluated omega equals
60
00:05:17,330 --> 00:05:22,599
zero, omega at residents,
omega at very high values.
61
00:05:22,599 --> 00:05:27,454
There is no adjustable constant
in the solution.
62
00:05:27,454 --> 00:05:32,000
But, there was one in that
solution.
63
00:05:32,000 --> 00:05:36,566
Now, let us look at the
differential equations that we
64
00:05:36,566 --> 00:05:40,356
were solving.
And so, first we go through the
65
00:05:40,356 --> 00:05:44,147
un-driven system.
Un-driven means at T equals
66
00:05:44,147 --> 00:05:48,886
zero, you give it a kick,
and you just let it do its own
67
00:05:48,886 --> 00:05:51,987
thing.
The differential equation that
68
00:05:51,987 --> 00:05:57,156
we then had was X double lot
plus gamma X dot plus omega zero
69
00:05:57,156 --> 00:06:01,977
squared X equal zero.
That's the one we had.
70
00:06:01,977 --> 00:06:06,382
And we solved it and we found
that solution there.
71
00:06:06,382 --> 00:06:10,426
Then, we were driving it.
So, now we drive it.
72
00:06:10,426 --> 00:06:13,752
Now, what's the differential
equation?
73
00:06:13,752 --> 00:06:18,786
Well, we had an X double dot,
gamma X dot plus omega zero
74
00:06:18,786 --> 00:06:22,471
squared times X.
And now, we had here this
75
00:06:22,471 --> 00:06:26,157
driving term.
In the case that we directly
76
00:06:26,157 --> 00:06:31,460
put the force on the object,
then we had here F zero divided
77
00:06:31,460 --> 00:06:36,404
by M times cosine omega T
because the M comes in because
78
00:06:36,404 --> 00:06:40,000
you divide by M,
right?
79
00:06:40,000 --> 00:06:42,681
You had Newton's Second Law as
MA.
80
00:06:42,681 --> 00:06:47,558
But, in the case that you shake
the left side with your hand,
81
00:06:47,558 --> 00:06:52,353
with this [eta?] zero cosine
omega T, that is another way of
82
00:06:52,353 --> 00:06:56,416
effectively driving the object.
We worked that out.
83
00:06:56,416 --> 00:07:00,968
Then we have here eta zero
times omega zero squared times
84
00:07:00,968 --> 00:07:04,651
cosine omega T.
But in any case,
85
00:07:04,651 --> 00:07:10,327
you see here a driving term,
and we solve our equations.
86
00:07:10,327 --> 00:07:15,797
And, if you take this case,
then this is the solution.
87
00:07:15,797 --> 00:07:20,957
If you take this case,
then this takes the place of
88
00:07:20,957 --> 00:07:23,846
that.
Suppose now I take this
89
00:07:23,846 --> 00:07:29,935
solution and a substitute that
solution in this differential
90
00:07:29,935 --> 00:07:34,485
equation.
Then I get the result is zero
91
00:07:34,485 --> 00:07:36,714
because, look,
forget this.
92
00:07:36,714 --> 00:07:41,342
So, I put it in this part.
I get a zero because it fits
93
00:07:41,342 --> 00:07:46,142
this differential equation.
So, what is wrong with adding
94
00:07:46,142 --> 00:07:49,057
zero?
Therefore, if I add these two
95
00:07:49,057 --> 00:07:54,114
solutions, it must be solution
to this differential equation
96
00:07:54,114 --> 00:07:58,742
because you just add zero.
And, if you take a 18.03 and
97
00:07:58,742 --> 00:08:03,139
they use very nice terms.
They say, yes,
98
00:08:03,139 --> 00:08:07,699
of course, if you have the
special solution which is this
99
00:08:07,699 --> 00:08:11,282
one, you have to add the
homogenous solution.
100
00:08:11,282 --> 00:08:15,842
And this word homogeneous means
that you put the zero in.
101
00:08:15,842 --> 00:08:20,483
And so, the general solution is
really the sum of the two.
102
00:08:20,483 --> 00:08:24,066
By adding this one,
you effectively add zero.
103
00:08:24,066 --> 00:08:27,648
So, you add nothing,
but you get something in
104
00:08:27,648 --> 00:08:30,743
return.
What you get in return are your
105
00:08:30,743 --> 00:08:36,952
two adjustable constants.
So, now you can deal with the
106
00:08:36,952 --> 00:08:42,958
situation that at T equals zero,
you know exactly where that
107
00:08:42,958 --> 00:08:46,419
object is and what its velocity
is.
108
00:08:46,419 --> 00:08:51,305
And so, I'll write down out the
general solution,
109
00:08:51,305 --> 00:08:57,209
which is the one that governs
the major part of the lecture
110
00:08:57,209 --> 00:09:00,874
today.
So, X now, a function of time,
111
00:09:00,874 --> 00:09:08,000
is the steady-state solution,
A cosine omega T minus delta.
112
00:09:08,000 --> 00:09:11,855
This is my omega.
This is my will.
113
00:09:11,855 --> 00:09:18,163
That's Walter Lewin's omega
plus X times E to the minus
114
00:09:18,163 --> 00:09:25,172
gamma over two times T times
cosine omega prime T plus alpha.
115
00:09:25,172 --> 00:09:29,728
And, this is the will of the
oscillator.
116
00:09:29,728 --> 00:09:35,586
And this is my will.
They are two different omegas.
117
00:09:35,586 --> 00:09:38,344
And now you can see what
happens.
118
00:09:38,344 --> 00:09:41,879
You see that this term will
never die out.
119
00:09:41,879 --> 00:09:45,327
It will last forever,
and ever, and ever.
120
00:09:45,327 --> 00:09:50,500
But this one is going to die.
It's a one over EDK time of two
121
00:09:50,500 --> 00:09:53,603
over gamma.
And so, if two over gamma
122
00:09:53,603 --> 00:09:58,344
happens to be ten hours,
then you have to wait ten hours
123
00:09:58,344 --> 00:10:03,000
for this one to be down by a
factor of E.
124
00:10:03,000 --> 00:10:08,323
But, if T over gamma happens to
be 1 ms, then all you have to do
125
00:10:08,323 --> 00:10:12,887
is rate 1 ms for that term to go
down by a factor of E.
126
00:10:12,887 --> 00:10:16,014
So, this is the one that will
die out.
127
00:10:16,014 --> 00:10:20,492
That's why we have to wait.
And so, this is called the
128
00:10:20,492 --> 00:10:24,042
[trangent?].
And it will die out faster the
129
00:10:24,042 --> 00:10:27,000
higher gamma is.
And this is called,
130
00:10:27,000 --> 00:10:31,056
then, the steady-state
solution, which ultimately
131
00:10:31,056 --> 00:10:35,451
survives.
Suppose I told you that at T
132
00:10:35,451 --> 00:10:40,180
equals zero, X equals zero,
and X dot also equals zero.
133
00:10:40,180 --> 00:10:45,084
And, I was so nasty to say,
oh, by the way when you solve
134
00:10:45,084 --> 00:10:49,288
for X and alpha would be a very
nice thing to do.
135
00:10:49,288 --> 00:10:53,228
But, it would take you 15
minutes of grinding;
136
00:10:53,228 --> 00:10:57,694
not so fast because,
remember, if you have an X dot,
137
00:10:57,694 --> 00:11:02,073
you have to take the time
derivative of this entire
138
00:11:02,073 --> 00:11:05,987
function.
You have a T here or you have a
139
00:11:05,987 --> 00:11:09,194
T there, and you have to
substitute in there time T
140
00:11:09,194 --> 00:11:11,951
equals zero.
And then, you have to make that
141
00:11:11,951 --> 00:11:14,517
equal zero.
And, it takes you 15 minutes.
142
00:11:14,517 --> 00:11:16,569
And, out pops a
[UNINTELLIGIBLE],
143
00:11:16,569 --> 00:11:18,877
a value for X,
and a value for alpha.
144
00:11:18,877 --> 00:11:21,891
You haven't learned much
physics when you do it,
145
00:11:21,891 --> 00:11:23,879
but you've learned some
algebra.
146
00:11:23,879 --> 00:11:27,214
And so, I decided not to spend
my time on doing that.
147
00:11:27,214 --> 00:11:30,356
But, in principle,
you must agree with me now that
148
00:11:30,356 --> 00:11:33,691
if I specify the initial
conditions, I don't have too
149
00:11:33,691 --> 00:11:37,948
call this zero.
I could call this X zero.
150
00:11:37,948 --> 00:11:42,314
I can do anything I want to.
I can give X dot any value I
151
00:11:42,314 --> 00:11:45,276
want to.
And they get unique values for
152
00:11:45,276 --> 00:11:48,395
X and for alpha.
And, that is ultimately,
153
00:11:48,395 --> 00:11:51,981
then, what the solution is.
I must add the two.
154
00:11:51,981 --> 00:11:55,333
The bottom line,
and that's really where the
155
00:11:55,333 --> 00:12:00,010
physics is, and that has to do
with problem two-five that you
156
00:12:00,010 --> 00:12:05,000
have this week on your plate is
then the following.
157
00:12:05,000 --> 00:12:09,102
If I make a plot of X as a
function of time,
158
00:12:09,102 --> 00:12:13,681
then the solution is really the
sum of these two.
159
00:12:13,681 --> 00:12:18,166
This is the steady-state
solution which has been
160
00:12:18,166 --> 00:12:22,363
amplitude, capital A,
which is nonnegotiable.
161
00:12:22,363 --> 00:12:26,943
It has nothing to do with the
initial conditions.
162
00:12:26,943 --> 00:12:32,000
It has its own omega,
Walter Lewin's omega.
163
00:12:32,000 --> 00:12:37,722
So, let's assume that this is
the period, T.
164
00:12:37,722 --> 00:12:43,577
And so, out of that pops a nice
cosinusoidal,
165
00:12:43,577 --> 00:12:48,768
sinusoidal curve.
Let me put it in here.
166
00:12:48,768 --> 00:12:54,756
And, this never changes.
This goes on forever,
167
00:12:54,756 --> 00:12:59,148
and ever, and ever.
And this here,
168
00:12:59,148 --> 00:13:04,205
this time, T,
is 2 pi divided by omega,
169
00:13:04,205 --> 00:13:10,528
my omega.
However, there is also this
170
00:13:10,528 --> 00:13:17,977
one, but this one dies out.
And so, I will now put in some
171
00:13:17,977 --> 00:13:24,380
kind of an exponential decay,
something like this.
172
00:13:24,380 --> 00:13:30,000
It has its own frequency,
omega prime.
173
00:13:30,000 --> 00:13:34,126
I can choose anything.
I can make omega larger,
174
00:13:34,126 --> 00:13:39,059
omega prime larger than omega.
I can make it smaller so,
175
00:13:39,059 --> 00:13:43,275
I just pick one,
and let's suppose that the zero
176
00:13:43,275 --> 00:13:46,684
crossings are here,
and here, and here,
177
00:13:46,684 --> 00:13:50,272
and here, and here.
And so, for instance,
178
00:13:50,272 --> 00:13:55,475
the curve than that has to be
added could be something like
179
00:13:55,475 --> 00:13:57,000
this.
180
00:13:57,000 --> 00:14:02,000
181
00:14:02,000 --> 00:14:07,463
And this time it's T prime is 2
pi divided by omega prime.
182
00:14:07,463 --> 00:14:13,501
And so, you see that the sum of
the two, which I will not try to
183
00:14:13,501 --> 00:14:18,581
sketch, is the solution.
But it is the pink one that's
184
00:14:18,581 --> 00:14:22,990
going to die out.
That's why sometimes you have
185
00:14:22,990 --> 00:14:27,111
to be patient,
except if it is 1 ms two over
186
00:14:27,111 --> 00:14:32,000
gamma, you don't have to be very
patient.
187
00:14:32,000 --> 00:14:34,614
Now, let us look at the
situation.
188
00:14:34,614 --> 00:14:38,814
And I can arrange that.
And I'm going to arrange that,
189
00:14:38,814 --> 00:14:42,379
that omega and omega prime are
close together.
190
00:14:42,379 --> 00:14:46,183
I can choose that.
The system cannot choose omega
191
00:14:46,183 --> 00:14:48,243
prime.
The system is stuck.
192
00:14:48,243 --> 00:14:51,808
This is omega prime.
The system has no choice.
193
00:14:51,808 --> 00:14:55,691
But, I have a choice.
I can make omega any value I
194
00:14:55,691 --> 00:14:58,781
want to.
So, I can make it very close to
195
00:14:58,781 --> 00:15:03,325
omega prime.
What do you think is going to
196
00:15:03,325 --> 00:15:06,372
happen now?
When I turn this system on,
197
00:15:06,372 --> 00:15:10,943
all of a sudden the driver,
and my own omega is very close
198
00:15:10,943 --> 00:15:14,712
to that omega prime.
It's true that the trangent
199
00:15:14,712 --> 00:15:18,160
will die out,
but let's say we take a system
200
00:15:18,160 --> 00:15:22,330
with a pretty high Q.
So, it doesn't die out so fast.
201
00:15:22,330 --> 00:15:25,377
What do you expect you're going
to see?
202
00:15:25,377 --> 00:15:28,183
Excellent.
You're going to see beats
203
00:15:28,183 --> 00:15:32,674
because now you have two
harmonic oscillations which have
204
00:15:32,674 --> 00:15:36,788
to be added.
But, the frequencies are a
205
00:15:36,788 --> 00:15:40,022
little different.
And, if this one survives long
206
00:15:40,022 --> 00:15:43,393
enough, there comes a time that
they are in phase.
207
00:15:43,393 --> 00:15:46,144
There comes a time they are out
of phase.
208
00:15:46,144 --> 00:15:50,066
You see very low amplitude.
And so, you are going to see a
209
00:15:50,066 --> 00:15:53,093
bit phenomenon.
And that is to do what I want
210
00:15:53,093 --> 00:15:55,638
you to see.
For that, we need a system
211
00:15:55,638 --> 00:15:59,560
preferably with a high IQ.
And then, the driving frequency
212
00:15:59,560 --> 00:16:03,000
has to be close to the omega
prime.
213
00:16:03,000 --> 00:16:07,025
And for that,
I have chosen this system here.
214
00:16:07,025 --> 00:16:11,507
This is an air track.
And, we can make the damping
215
00:16:11,507 --> 00:16:16,813
very low, unpleasantly low,
believe me, a very low value of
216
00:16:16,813 --> 00:16:22,028
gamma that when I turn on the
airflow, so here is a spring
217
00:16:22,028 --> 00:16:25,687
constant, K.
Here is one spring constant,
218
00:16:25,687 --> 00:16:30,078
K, and here is the mass.
And, there's very little
219
00:16:30,078 --> 00:16:34,095
friction.
And now I'm going to drive it
220
00:16:34,095 --> 00:16:37,997
just a little bit off resonance,
a little bit below the
221
00:16:37,997 --> 00:16:41,610
resonance frequency.
And, what you are going to see
222
00:16:41,610 --> 00:16:45,512
now is the sum of these two.
But since gamma is so low,
223
00:16:45,512 --> 00:16:49,342
it will take a long time for
this trangent to die out.
224
00:16:49,342 --> 00:16:52,377
And that is exactly what I want
you to see.
225
00:16:52,377 --> 00:16:55,123
In addition,
you're going to see beats.
226
00:16:55,123 --> 00:16:59,170
And as long as you see beating,
you know that you haven't
227
00:16:59,170 --> 00:17:03,000
reached the steady-state
solution yet.
228
00:17:03,000 --> 00:17:09,643
But, if you are patient and I'm
patient, we probably will see it
229
00:17:09,643 --> 00:17:14,283
go into steady-state,
but it may take several
230
00:17:14,283 --> 00:17:17,552
minutes.
So, you ready for that?
231
00:17:17,552 --> 00:17:23,458
I'm going to drive it here.
And, I start the driving out.
232
00:17:23,458 --> 00:17:28,730
So, relax and look at the
amplitude of this object.
233
00:17:28,730 --> 00:17:34,392
And see what happens.
Hey, the amplitude is going
234
00:17:34,392 --> 00:17:37,848
down, hey, hey,
hey, I call that a beat.
235
00:17:37,848 --> 00:17:41,835
Did you see that?
Did you see the amplitude go
236
00:17:41,835 --> 00:17:44,936
down?
The two frequencies were being
237
00:17:44,936 --> 00:17:48,924
against each other.
Now it's picking up again.
238
00:17:48,924 --> 00:17:53,177
It's nowhere near a steady
state: very low gamma,
239
00:17:53,177 --> 00:17:55,835
very high Q.
There we go again,
240
00:17:55,835 --> 00:18:00,000
amplitude way down picks up
again.
241
00:18:00,000 --> 00:18:03,626
Just be patient.
Let's be patient and see
242
00:18:03,626 --> 00:18:09,337
whether we have the privilege of
seeing it go into steady-state.
243
00:18:09,337 --> 00:18:13,597
It's a very high Q system.
Since I am just below
244
00:18:13,597 --> 00:18:19,399
resonance, the driver in the car
will be in phase when I am below
245
00:18:19,399 --> 00:18:22,481
resonance in steady-state
solution.
246
00:18:22,481 --> 00:18:26,198
So, this delta will be very
close to zero,
247
00:18:26,198 --> 00:18:29,099
below resonance,
above resonance,
248
00:18:29,099 --> 00:18:35,563
180° out of phase.
But, I am just below resonance.
249
00:18:35,563 --> 00:18:39,241
So, when we go into
steady-state,
250
00:18:39,241 --> 00:18:45,908
we also will see that we are
very close to a delta of zero.
251
00:18:45,908 --> 00:18:50,505
Now, let's see what the
amplitude is now,
252
00:18:50,505 --> 00:18:54,758
and whether the amplitude is
changing.
253
00:18:54,758 --> 00:19:02,000
Well, we are getting there.
It pays off to be patient.
254
00:19:02,000 --> 00:19:05,727
Later today,
I will do an experiment where
255
00:19:05,727 --> 00:19:08,818
two over gamma is two many
seconds.
256
00:19:08,818 --> 00:19:12,000
So, all you have to do is wait
4 ms.
257
00:19:12,000 --> 00:19:17,090
So, I'll make up for the fact
that now we have to wait to
258
00:19:17,090 --> 00:19:20,272
long.
Let's take a look at this now.
259
00:19:20,272 --> 00:19:24,272
I think it looks terrific.
It looks terrific.
260
00:19:24,272 --> 00:19:28,090
In phase, I don't see much
beating anymore.
261
00:19:28,090 --> 00:19:33,000
It looks like the amplitude is
constant.
262
00:19:33,000 --> 00:19:38,522
I think we've killed this one,
and I think this one has
263
00:19:38,522 --> 00:19:42,306
survived.
If you increase the damping,
264
00:19:42,306 --> 00:19:47,215
this would happen,
of course, earlier that you go
265
00:19:47,215 --> 00:19:51,000
into the steady-state solution.
266
00:19:51,000 --> 00:20:00,000
267
00:20:00,000 --> 00:20:04,141
It looks great.
I don't see any change anymore
268
00:20:04,141 --> 00:20:08,466
in the amplitude.
So, that's the A that you have
269
00:20:08,466 --> 00:20:12,699
there, capital A,
and they are nicely in phase.
270
00:20:12,699 --> 00:20:17,852
It's a very high Q system,
so the change to go from delta
271
00:20:17,852 --> 00:20:23,098
zero to delta pi over two at
resonance takes place over an
272
00:20:23,098 --> 00:20:26,319
extremely narrow range of
frequency.
273
00:20:26,319 --> 00:20:32,867
So, they are still in phase.
All right, if we are driving
274
00:20:32,867 --> 00:20:38,295
this system with a force,
say, directly on the object,
275
00:20:38,295 --> 00:20:43,518
F zero cosine omega T,
then in steady state there is
276
00:20:43,518 --> 00:20:48,024
energy dissipation because there
is friction.
277
00:20:48,024 --> 00:20:52,427
And, where there is friction,
there is heat.
278
00:20:52,427 --> 00:20:57,753
And that means energy.
That means I had to do work to
279
00:20:57,753 --> 00:21:04,000
provide that energy to the
steady-state situation.
280
00:21:04,000 --> 00:21:09,760
So, as the thing is never
changing, it's A just going on
281
00:21:09,760 --> 00:21:13,530
forever and ever.
While that happens,
282
00:21:13,530 --> 00:21:18,871
I must put in energy,
which comes out in the form of
283
00:21:18,871 --> 00:21:22,432
heat.
So, let us return to the good
284
00:21:22,432 --> 00:21:27,354
old days of 8.01.
And, I want to remind you that
285
00:21:27,354 --> 00:21:33,429
work is the dot product between
a force and a displacement,
286
00:21:33,429 --> 00:21:36,621
DX.
It's a dot product.
287
00:21:36,621 --> 00:21:40,890
It's a scalar work.
A little bit of work is done by
288
00:21:40,890 --> 00:21:44,475
this force if it moves over a
distance, DX.
289
00:21:44,475 --> 00:21:49,426
If the two are perpendicular to
each other, then no work is
290
00:21:49,426 --> 00:21:52,329
done.
Satellite into circular orbit
291
00:21:52,329 --> 00:21:56,000
around the Earth:
no work is done.
292
00:21:56,000 --> 00:22:00,698
And so, now I can calculate
what the power is because the
293
00:22:00,698 --> 00:22:04,306
power is DW DT,
how many joules per second I
294
00:22:04,306 --> 00:22:09,088
had to put into the system.
And so, if I take a derivative
295
00:22:09,088 --> 00:22:13,031
times the derivative,
then I get F dotted with V
296
00:22:13,031 --> 00:22:16,052
because DX DT is simply the
velocity.
297
00:22:16,052 --> 00:22:19,324
Now, if I have a
one-dimensional system,
298
00:22:19,324 --> 00:22:23,770
and what I mean by that is the
force is either in this
299
00:22:23,770 --> 00:22:28,637
direction or in this direction
in the velocity is either in
300
00:22:28,637 --> 00:22:33,000
this direction or in this
direction.
301
00:22:33,000 --> 00:22:36,795
That's what I mean by a
one-dimensional system.
302
00:22:36,795 --> 00:22:40,837
Then I can delete the dot,
and then the signs will
303
00:22:40,837 --> 00:22:44,137
automatically take care of the
direction.
304
00:22:44,137 --> 00:22:47,437
The minus V is then this.
Plus V is that,
305
00:22:47,437 --> 00:22:50,984
and same for force.
So, I can kill the dots.
306
00:22:50,984 --> 00:22:54,615
So, now, I have to know what
the velocity is.
307
00:22:54,615 --> 00:22:58,492
In the steady-state solution.
Well, that's easy,
308
00:22:58,492 --> 00:23:02,700
because I go to the
steady-state solution here and I
309
00:23:02,700 --> 00:23:08,325
calculate what X dot is.
So I'll put that here,
310
00:23:08,325 --> 00:23:13,189
X dot, which is V.
So, the derivative of cosine
311
00:23:13,189 --> 00:23:17,101
omega T is minus omega times the
sine.
312
00:23:17,101 --> 00:23:23,127
So, I get minus omega times A
times the sine omega T minus
313
00:23:23,127 --> 00:23:25,876
delta.
That's the velocity.
314
00:23:25,876 --> 00:23:33,162
But, I know what the force is.
That is F zero cosine omega T.
315
00:23:33,162 --> 00:23:37,169
So, there we go,
F zero cosine omega T,
316
00:23:37,169 --> 00:23:41,175
but I'm going to put these in
also now.
317
00:23:41,175 --> 00:23:46,025
So, I get omega A,
and then I put in the cosine
318
00:23:46,025 --> 00:23:51,929
omega T, and now I'm going to
put in this one the sine of
319
00:23:51,929 --> 00:23:56,463
omega T minus delta.
So, take a deep breath.
320
00:23:56,463 --> 00:24:00,047
F zero cosine omega T is the
force.
321
00:24:00,047 --> 00:24:05,604
See that there?
And the velocity has been
322
00:24:05,604 --> 00:24:11,020
derived from the first
derivative of the steady-state
323
00:24:11,020 --> 00:24:17,270
solution, gives me a minus omega
A, sine omega T minus delta.
324
00:24:17,270 --> 00:24:23,416
Now, this, what we have here
between brackets can be written
325
00:24:23,416 --> 00:24:29,562
as the sine of omega T cosine
delta minus the cosine omega T
326
00:24:29,562 --> 00:24:35,485
times the sine of delta.
So, what you see here between
327
00:24:35,485 --> 00:24:39,085
brackets is the same as what you
see there.
328
00:24:39,085 --> 00:24:43,714
And I really interested in
knowing every moment in time
329
00:24:43,714 --> 00:24:46,885
what exactly the power is?
Not really.
330
00:24:46,885 --> 00:24:51,771
Most of the time I'm really
interested in knowing what the
331
00:24:51,771 --> 00:24:55,628
average power is,
that is required to keep the
332
00:24:55,628 --> 00:25:00,771
thing going, average over one
oscillation or over hundreds of
333
00:25:00,771 --> 00:25:05,188
oscillations.
That's really what I'm
334
00:25:05,188 --> 00:25:08,471
interested in,
not necessarily the
335
00:25:08,471 --> 00:25:11,952
instantaneous power.
In other words,
336
00:25:11,952 --> 00:25:17,324
most of the time my interest is
really in what this is,
337
00:25:17,324 --> 00:25:20,408
that is, the time average
value.
338
00:25:20,408 --> 00:25:25,780
And, think of it as being one
oscillation that is fine.
339
00:25:25,780 --> 00:25:29,361
But you can think of it also as
days.
340
00:25:29,361 --> 00:25:33,041
Now, I need some experts and
audience.
341
00:25:33,041 --> 00:25:39,109
I see here with cosine omega T,
and I see here with sine omega
342
00:25:39,109 --> 00:25:44,113
T.
What is the time average value
343
00:25:44,113 --> 00:25:48,990
of that product?
The time average value [between
344
00:25:48,990 --> 00:25:53,245
the?] sine omega T and the
cosine omega T,
345
00:25:53,245 --> 00:25:59,471
if I average it over one cycle
or two cycles or three cycles.
346
00:25:59,471 --> 00:26:03,000
Come on, high school,
yeah?
347
00:26:03,000 --> 00:26:08,003
What is the time average of
sine omega T times cosine omega
348
00:26:08,003 --> 00:26:11,194
T, time average over one period?
Zero.
349
00:26:11,194 --> 00:26:15,594
This one times this one times
average gives me zero.
350
00:26:15,594 --> 00:26:19,389
You don't believe it,
go back to high school.
351
00:26:19,389 --> 00:26:23,789
Ask your high school teacher.
He will agree with me.
352
00:26:23,789 --> 00:26:28,447
Here I see a cosine omega T,
and I see a cosine omega T
353
00:26:28,447 --> 00:26:32,314
there.
What is the time average of
354
00:26:32,314 --> 00:26:36,599
cosine squared omega T?
Ah, you guys are waking up.
355
00:26:36,599 --> 00:26:39,000
It's one half.
So, therefore,
356
00:26:39,000 --> 00:26:42,514
for this product,
sorry, for this product,
357
00:26:42,514 --> 00:26:45,857
I can write one half.
And so now we get,
358
00:26:45,857 --> 00:26:51,000
notice there is a minus here.
And so this minus picks up this
359
00:26:51,000 --> 00:26:53,400
minus.
So, it becomes a plus.
360
00:26:53,400 --> 00:26:57,257
So, I get F zero.
I get the half that is here.
361
00:26:57,257 --> 00:27:03,310
Then I get omega.
I get my A, and then I get the
362
00:27:03,310 --> 00:27:07,393
sine of delta.
Are we happy with that?
363
00:27:07,393 --> 00:27:11,696
So, you see,
it collapses into something
364
00:27:11,696 --> 00:27:16,662
that is relatively simple.
What is sine delta?
365
00:27:16,662 --> 00:27:20,855
Well, I remember what tangent
delta is.
366
00:27:20,855 --> 00:27:26,262
If I know tangent delta,
can I then calculate sine
367
00:27:26,262 --> 00:27:29,462
delta?
And, the answer is yes,
368
00:27:29,462 --> 00:27:33,936
of course.
If this is delta,
369
00:27:33,936 --> 00:27:40,430
and if this is omega gamma,
and this is omega zero squared
370
00:27:40,430 --> 00:27:46,012
minus omega squared,
then it only takes Pythagoras
371
00:27:46,012 --> 00:27:52,392
to calculate what this is.
And so I know that the sine of
372
00:27:52,392 --> 00:27:58,544
delta must be omega gamma
divided by the square root of
373
00:27:58,544 --> 00:28:03,898
omega zero squared minus omega
squared, squared,
374
00:28:03,898 --> 00:28:11,167
plus omega gamma squared.
So, yes, I do know what the
375
00:28:11,167 --> 00:28:16,681
sine of delta is.
So, now, I can come to a close
376
00:28:16,681 --> 00:28:22,430
by substituting in here what the
sine of delta is.
377
00:28:22,430 --> 00:28:27,005
And, I can substitute in here
what A is.
378
00:28:27,005 --> 00:28:31,759
Here is A.
You know what is nice?
379
00:28:31,759 --> 00:28:39,161
Look at the downstairs here.
It's the same as the downstairs
380
00:28:39,161 --> 00:28:42,799
there.
So, if I multiply them,
381
00:28:42,799 --> 00:28:49,448
the square root goes away.
So, if I can write down now
382
00:28:49,448 --> 00:28:54,842
what P averages,
go slowly, so we get F zero
383
00:28:54,842 --> 00:28:58,605
divided by two.
I get an omega.
384
00:28:58,605 --> 00:29:04,000
I will go to the A very
shortly.
385
00:29:04,000 --> 00:29:09,968
I will first pick omega gamma
here, which is my sine delta.
386
00:29:09,968 --> 00:29:14,907
So, that makes this a square.
And, I get a gamma.
387
00:29:14,907 --> 00:29:20,155
And, now I turn to the A,
which is an F zero over M.
388
00:29:20,155 --> 00:29:24,786
So, I get a square here.
And, I get an M here.
389
00:29:24,786 --> 00:29:30,137
And now, I get this downstairs
times this downstairs.
390
00:29:30,137 --> 00:29:36,105
The square root disappears,
and so I get omega zero squared
391
00:29:36,105 --> 00:29:40,736
minus omega squared,
squared, plus omega gamma
392
00:29:40,736 --> 00:29:44,422
squared.
Almost end of story.
393
00:29:44,422 --> 00:29:46,698
I'm going to rewrite it a
little.
394
00:29:46,698 --> 00:29:48,690
And Tony French,
in his book,
395
00:29:48,690 --> 00:29:51,393
rewrites it in,
again, a different way.
396
00:29:51,393 --> 00:29:55,020
He loves to work with Q's.
He puts the Q's in there.
397
00:29:55,020 --> 00:30:00,000
I'm going to divide upstairs
and downstairs by omega squared.
398
00:30:00,000 --> 00:30:05,768
And, when I do that,
I got F zero squared.
399
00:30:05,768 --> 00:30:11,256
I've got a gamma here.
I get two M here.
400
00:30:11,256 --> 00:30:19,839
And then, I get here omega zero
squared divided by omega minus
401
00:30:19,839 --> 00:30:24,341
omega squared plus gamma
squared.
402
00:30:24,341 --> 00:30:30,367
That's what I get.
That's not the only way you can
403
00:30:30,367 --> 00:30:33,804
write it, but that's one way you
can write it.
404
00:30:33,804 --> 00:30:37,241
Let me check that.
I have five zeroes squared.
405
00:30:37,241 --> 00:30:40,372
I have gamma,
two M here, and we have this
406
00:30:40,372 --> 00:30:43,350
downstairs.
So, the time has come now to
407
00:30:43,350 --> 00:30:45,794
try to see through this
equation.
408
00:30:45,794 --> 00:30:50,071
You remember last lecture,
we spent the whole lecture not
409
00:30:50,071 --> 00:30:54,042
to look at this dumb equation,
but to see through it.
410
00:30:54,042 --> 00:31:00,000
And we were able to see through
it, see all its idiosyncrasies.
411
00:31:00,000 --> 00:31:05,114
Let's look at the
idiosyncrasies of this average
412
00:31:05,114 --> 00:31:10,229
power over one cycle,
over a multiple of cycles.
413
00:31:10,229 --> 00:31:15,235
Well, let us first make gamma
infinitely large.
414
00:31:15,235 --> 00:31:18,608
Don't look even at the
equation.
415
00:31:18,608 --> 00:31:24,049
That means there is an infinite
amount of friction.
416
00:31:24,049 --> 00:31:30,035
The system never gets going.
There's no way it will ever
417
00:31:30,035 --> 00:31:33,735
move.
So, clearly the average power
418
00:31:33,735 --> 00:31:37,985
must go to zero.
And, indeed,
419
00:31:37,985 --> 00:31:41,022
you see that.
The gamma is downstairs.
420
00:31:41,022 --> 00:31:44,634
If gamma goes to infinity,
the power is zero.
421
00:31:44,634 --> 00:31:47,589
Let's make the mass infinitely
large.
422
00:31:47,589 --> 00:31:52,268
If the mass in infinitely
large, you have an infinity high
423
00:31:52,268 --> 00:31:55,223
inertia.
Nothing will ever get going.
424
00:31:55,223 --> 00:31:58,261
No force will ever get the mass
going.
425
00:31:58,261 --> 00:32:02,776
Well, that means you expect
that the power goes to zero.
426
00:32:02,776 --> 00:32:04,828
And, indeed,
you see here,
427
00:32:04,828 --> 00:32:10,000
if M goes to infinity,
the power goes to zero.
428
00:32:10,000 --> 00:32:12,193
Let's say the force goes to
zero.
429
00:32:12,193 --> 00:32:15,688
We are not even driving it.
Well, if we are not even
430
00:32:15,688 --> 00:32:19,663
driving it, I hope you will
agree with me you don't have to
431
00:32:19,663 --> 00:32:21,171
put in any work,
right?
432
00:32:21,171 --> 00:32:24,049
Nothing gets going.
So, clearly you expect,
433
00:32:24,049 --> 00:32:26,517
then, that the power will go to
zero.
434
00:32:26,517 --> 00:32:28,915
And, indeed,
if F zero goes to zero,
435
00:32:28,915 --> 00:32:34,390
you see that the power is zero.
Suppose you make omega zero.
436
00:32:34,390 --> 00:32:38,317
So that means there will never
be any velocity.
437
00:32:38,317 --> 00:32:43,097
You never pick up any velocity.
It takes infinitely long.
438
00:32:43,097 --> 00:32:45,317
Omega is zero.
So, clearly,
439
00:32:45,317 --> 00:32:49,329
if nothing ever moves the power
will go to zero.
440
00:32:49,329 --> 00:32:51,890
Now, look.
If omega makes zero,
441
00:32:51,890 --> 00:32:55,560
this is zero.
But this one goes to infinity.
442
00:32:55,560 --> 00:33:00,000
And, therefore,
the power goes to zero.
443
00:33:00,000 --> 00:33:05,076
So, you need no equations for
that, just common sense to
444
00:33:05,076 --> 00:33:09,599
immediately conclude that this
has to be the case.
445
00:33:09,599 --> 00:33:13,015
Suppose you go to infinity with
omega.
446
00:33:13,015 --> 00:33:16,892
Very, very fast.
Well, if you go infinitely
447
00:33:16,892 --> 00:33:22,615
fast, because of the inertia of
the system, it can never react,
448
00:33:22,615 --> 00:33:26,861
can never get going.
So I will predict that the
449
00:33:26,861 --> 00:33:32,064
power must go to zero.
And, if you put omega infinity
450
00:33:32,064 --> 00:33:35,436
in here, this goes to zero.
This goes to infinity,
451
00:33:35,436 --> 00:33:39,634
and so the power goes to zero.
So, all this is complete common
452
00:33:39,634 --> 00:33:41,836
sense.
All of this you could have
453
00:33:41,836 --> 00:33:45,896
predicted without that equation.
But, isn't it nice that the
454
00:33:45,896 --> 00:33:49,888
equation supports my intuition?
So, now comes the question,
455
00:33:49,888 --> 00:33:53,741
if omega goes to omega zero,
and that is the reason why I
456
00:33:53,741 --> 00:33:57,320
wrote it in this way.
Notice when omega goes to zero,
457
00:33:57,320 --> 00:34:02,000
when omega goes to omega zero,
this one goes to zero.
458
00:34:02,000 --> 00:34:06,162
And therefore,
that is the frequency at which
459
00:34:06,162 --> 00:34:12,216
the average power is the maximum
[ever?], can never be any higher
460
00:34:12,216 --> 00:34:16,000
because it's independent of
gamma, right,
461
00:34:16,000 --> 00:34:19,310
my omega?
So, I have an omega equals
462
00:34:19,310 --> 00:34:23,189
omega zero.
We reach the maximum value for
463
00:34:23,189 --> 00:34:26,310
power.
We can never go any higher.
464
00:34:26,310 --> 00:34:31,920
It's exactly at omega zero.
And, this is zero.
465
00:34:31,920 --> 00:34:36,613
You lose one gamma.
And so you find that this
466
00:34:36,613 --> 00:34:42,480
value, then, becomes F zero
squared divided by 2M gamma.
467
00:34:42,480 --> 00:34:47,813
And, you can write that,
rewrite have a little bit.
468
00:34:47,813 --> 00:34:51,973
I do that because Tony French
likes Q's.
469
00:34:51,973 --> 00:34:55,813
And so, I write it with a Q in
there.
470
00:34:55,813 --> 00:35:02,000
Remember that Q is omega zero
divided by gamma.
471
00:35:02,000 --> 00:35:09,514
So, I can rewrite this as Q
times F zero squared upstairs.
472
00:35:09,514 --> 00:35:15,578
So, I get the Q in there,
which is always nice,
473
00:35:15,578 --> 00:35:22,434
divided by 2M omega zero.
So, this is the same thing.
474
00:35:22,434 --> 00:35:28,102
So, if I now plot,
make a curve for you of P
475
00:35:28,102 --> 00:35:35,090
average, not P maximum,
but P average as a function of
476
00:35:35,090 --> 00:35:43,000
frequency, so here is omega and
here is P average.
477
00:35:43,000 --> 00:35:47,576
Here is omega zero.
It goes through a maximum
478
00:35:47,576 --> 00:35:51,735
exactly at omega zero.
It starts at zero,
479
00:35:51,735 --> 00:35:57,976
you see, and it ends at zero.
And, it sweeps up to a maximum,
480
00:35:57,976 --> 00:36:03,903
and then it goes down again.
And, for reasonable values of
481
00:36:03,903 --> 00:36:09,000
Q, these curves look extremely
symmetric.
482
00:36:09,000 --> 00:36:17,460
And so, this value here is then
the average max,
483
00:36:17,460 --> 00:36:23,039
which is that value,
this value.
484
00:36:23,039 --> 00:36:33,840
If we look at the width of this
curve, at half maximum of the
485
00:36:33,840 --> 00:36:38,542
power,
so this is one half times P
486
00:36:38,542 --> 00:36:43,534
maximum, then you can show,
and it is not so difficult
487
00:36:43,534 --> 00:36:47,301
algebraically,
but I will not attempt it,
488
00:36:47,301 --> 00:36:52,858
you can show that the width at
half maximum is very close to
489
00:36:52,858 --> 00:36:56,249
gamma.
Remember, gamma and omega have
490
00:36:56,249 --> 00:37:00,110
the same unit.
It's one divided by second.
491
00:37:00,110 --> 00:37:03,972
In other words,
if you go to half maximum,
492
00:37:03,972 --> 00:37:10,000
this point here is omega zero
minus gamma over two.
493
00:37:10,000 --> 00:37:15,447
And, this point here is omega
zero plus gamma over two.
494
00:37:15,447 --> 00:37:21,096
And so, you see immediately,
which of course makes sense,
495
00:37:21,096 --> 00:37:27,048
that if gamma is very small,
that the peak gets very narrow.
496
00:37:27,048 --> 00:37:33,000
And, if gamma is very high,
the peak gets very broad.
497
00:37:33,000 --> 00:37:36,300
That's intuitively quite
pleasing.
498
00:37:36,300 --> 00:37:39,599
High Q systems,
very narrow peaks.
499
00:37:39,599 --> 00:37:45,500
And, that's the way that Tony
French likes to plot his data.
500
00:37:45,500 --> 00:37:49,599
I will show you that on the
overhead here,
501
00:37:49,599 --> 00:37:53,300
this is just a picture from
your book.
502
00:37:53,300 --> 00:37:57,699
What Tony does here,
he plots not omega here,
503
00:37:57,699 --> 00:38:01,699
but he plots omega divided by
omega zero.
504
00:38:01,699 --> 00:38:07,000
So, that means the resonance is
at one.
505
00:38:07,000 --> 00:38:10,837
And, he doesn't plot the
average value for P here,
506
00:38:10,837 --> 00:38:15,066
but he plots it into strange
units into the unit F zero
507
00:38:15,066 --> 00:38:17,572
squared divided by 2M omega
zero.
508
00:38:17,572 --> 00:38:22,036
So, now, he effectively can
compare the vertical axis with
509
00:38:22,036 --> 00:38:26,891
the Q value because he likes the
fact that it is Q times higher
510
00:38:26,891 --> 00:38:30,180
than something.
And, he has plotted this in
511
00:38:30,180 --> 00:38:35,410
terms of that something.
And so, if you take a curve for
512
00:38:35,410 --> 00:38:38,442
Q equals ten,
which has the peak here in
513
00:38:38,442 --> 00:38:42,018
power, you see,
indeed, that he finds that very
514
00:38:42,018 --> 00:38:45,050
close on his scale to ten.
Notice, also,
515
00:38:45,050 --> 00:38:48,705
the nice symmetry.
And, you see for lower values
516
00:38:48,705 --> 00:38:51,815
of Q, which are curves here
that, indeed,
517
00:38:51,815 --> 00:38:55,780
the peak gets broader.
The width of this peak is one
518
00:38:55,780 --> 00:38:59,123
over Q because,
remember, this axis is omega
519
00:38:59,123 --> 00:39:03,981
divided by omega zero.
So, if the width is gamma on
520
00:39:03,981 --> 00:39:07,872
that peak, it is now gamma
divided by omega zero in this
521
00:39:07,872 --> 00:39:10,136
plot.
And, gamma divided by omega
522
00:39:10,136 --> 00:39:13,391
zero is one over Q.
So, here, the width in this
523
00:39:13,391 --> 00:39:17,070
presentation is directly
inversely proportional to Q.
524
00:39:17,070 --> 00:39:20,113
So, if Q is ten,
then the width there is one
525
00:39:20,113 --> 00:39:22,660
tenth.
He then shows you another plot
526
00:39:22,660 --> 00:39:24,924
whereby he does what I did
there.
527
00:39:24,924 --> 00:39:28,816
He plots it as a function of
omega, not as a function of
528
00:39:28,816 --> 00:39:34,739
omega divided by omega zero.
And then, he emphasizes the
529
00:39:34,739 --> 00:39:41,028
fact that the width here is that
gamma that I mentioned at half
530
00:39:41,028 --> 00:39:46,811
the maximum power you get here
[the width/with?] of gamma.
531
00:39:46,811 --> 00:39:52,797
And, this is a picture that I
chose verbatim from your book.
532
00:39:52,797 --> 00:39:56,550
This is the best moment for the
break.
533
00:39:56,550 --> 00:40:02,148
That means the mini quiz.
I realize it's a bit early;
534
00:40:02,148 --> 00:40:05,251
we are only 40 minutes into the
lecture.
535
00:40:05,251 --> 00:40:09,071
But it's a natural point.
You will see what comes
536
00:40:09,071 --> 00:40:12,175
afterwards.
It's better that we make the
537
00:40:12,175 --> 00:40:14,084
break now.
So, therefore,
538
00:40:14,084 --> 00:40:18,779
I need some help from people
who are willing to hand out the
539
00:40:18,779 --> 00:40:21,962
mini quiz.
It would be nice if I can find
540
00:40:21,962 --> 00:40:24,668
the mini quizzes.
I have them here,
541
00:40:24,668 --> 00:40:30,000
but someone took them.
Oh, no, they are still there.
542
00:40:30,000 --> 00:40:35,578
I have a nice conspiracy.
Afterwards, after the break,
543
00:40:35,578 --> 00:40:41,684
we will collect them this time
in some boxes so that it's a
544
00:40:41,684 --> 00:40:47,684
little bit more organized.
And so, I'm returning to an RLC
545
00:40:47,684 --> 00:40:53,578
circuit, which we discussed
earlier, the good old days of
546
00:40:53,578 --> 00:40:56,842
8.02.
I'm going to drive it now,
547
00:40:56,842 --> 00:41:04,000
not with the battery but with
an alternating power supply.
548
00:41:04,000 --> 00:41:11,000
549
00:41:11,000 --> 00:41:14,000
V zero cosine omega T.
550
00:41:14,000 --> 00:41:18,000
551
00:41:18,000 --> 00:41:23,579
Yeah, put it in here,
thank you.
552
00:41:23,579 --> 00:41:33,119
So, here's the circuit,
resistor R, self inductance L,
553
00:41:33,119 --> 00:41:39,157
capacitor C.
And, I have to write down,
554
00:41:39,157 --> 00:41:46,688
now, the differential equation.
I will adopt a positive current
555
00:41:46,688 --> 00:41:51,789
in this direction.
That will be my positive
556
00:41:51,789 --> 00:41:56,283
current.
The charge here on this right
557
00:41:56,283 --> 00:42:00,534
plate I will call Q.
And, therefore,
558
00:42:00,534 --> 00:42:06,000
by that definition,
I is then DQ DT.
559
00:42:06,000 --> 00:42:10,680
Sign sensitive.
I call the potential difference
560
00:42:10,680 --> 00:42:15,870
over this capacitor,
in going from the right side to
561
00:42:15,870 --> 00:42:19,228
the left side,
I call that V of C.
562
00:42:19,228 --> 00:42:25,333
That, then, is Q divided by C.
All of that is sign sensitive.
563
00:42:25,333 --> 00:42:31,336
I go a round this circuit and I
want to calculate the closed
564
00:42:31,336 --> 00:42:38,000
loop integral of E dot DL.
And that closed loop integral
565
00:42:38,000 --> 00:42:43,172
of E dot DL is not zero,
which many books tell you,
566
00:42:43,172 --> 00:42:47,827
even many professors tell you.
It is not zero,
567
00:42:47,827 --> 00:42:53,413
but it is minus D phi DT.
This is [UNINTELLIGIBLE] Law,
568
00:42:53,413 --> 00:42:59,000
and this runs our economy.
Because of the magnetic flux
569
00:42:59,000 --> 00:43:04,586
change in closed loops,
we can generate induced [EMS?],
570
00:43:04,586 --> 00:43:09,980
which run our economy.
Look at the lights.
571
00:43:09,980 --> 00:43:15,818
Luckily, this is not zero.
This [fires?] in magnetic flux
572
00:43:15,818 --> 00:43:21,969
that goes through a surface,
any surface that you can attach
573
00:43:21,969 --> 00:43:26,764
to this closed loop.
So, I've done this before.
574
00:43:26,764 --> 00:43:29,996
So, I can do it a little
faster.
575
00:43:29,996 --> 00:43:35,000
I go from here to here.
So, that is IR.
576
00:43:35,000 --> 00:43:39,653
There is no electric field
inside this ideal self inductor
577
00:43:39,653 --> 00:43:44,061
because a superconducting wire
cannot be an [E?] field.
578
00:43:44,061 --> 00:43:48,877
So, that is zero going from
here to there when I go over the
579
00:43:48,877 --> 00:43:50,673
capacitor.
I get my VC,
580
00:43:50,673 --> 00:43:55,571
and here, depending upon the
phase, if I assume this plus and
581
00:43:55,571 --> 00:43:58,510
this minus, but you can reverse
that.
582
00:43:58,510 --> 00:44:01,693
Then, I would get,
when I walk into this
583
00:44:01,693 --> 00:44:07,000
direction, I would get minus V
zero cosine omega T.
584
00:44:07,000 --> 00:44:11,284
But, if you feel like reversing
it, I have no problem with that.
585
00:44:11,284 --> 00:44:13,596
That's just a matter of 180°
phase.
586
00:44:13,596 --> 00:44:15,705
[It's known here from?]
physics.
587
00:44:15,705 --> 00:44:18,085
And, this now equals minus D
phi DT.
588
00:44:18,085 --> 00:44:21,962
The only thing where you apply
[UNINTELLIGIBLE] law is you
589
00:44:21,962 --> 00:44:25,906
should always integrate in the
direction that you have your
590
00:44:25,906 --> 00:44:28,695
current assumed,
that it is minus L DI DT.
591
00:44:28,695 --> 00:44:32,707
If you do it in the opposite
direction, then it is plus L DI
592
00:44:32,707 --> 00:44:36,047
DT.
I have learned a certain
593
00:44:36,047 --> 00:44:39,802
discipline in my life.
It took me many years.
594
00:44:39,802 --> 00:44:44,239
So, you have a long way to go.
And I always go in the
595
00:44:44,239 --> 00:44:47,994
direction of I,
so I never have to think that
596
00:44:47,994 --> 00:44:52,517
this is minus L DI DT.
This now covers the minus D phi
597
00:44:52,517 --> 00:44:54,650
DT.
So, now, what do I do?
598
00:44:54,650 --> 00:44:58,576
I bring the L in.
And, I take one more time the
599
00:44:58,576 --> 00:45:02,074
derivative.
And so, I get L times I double
600
00:45:02,074 --> 00:45:08,786
dot plus R times I dot plus VC.
But I take the time derivative.
601
00:45:08,786 --> 00:45:13,191
So, the Q dot becomes I.
So, I get I divided by C.
602
00:45:13,191 --> 00:45:18,404
And that now becomes the timed
derivative of this function.
603
00:45:18,404 --> 00:45:23,707
But, it goes to the right side,
which makes the minus sign a
604
00:45:23,707 --> 00:45:26,943
plus.
But, when I take the derivative
605
00:45:26,943 --> 00:45:31,078
of the cosine omega T,
I get a minus omega out.
606
00:45:31,078 --> 00:45:36,112
So, I get here minus P zero
times omega times the sign of
607
00:45:36,112 --> 00:45:41,000
omega T.
This is the differential
608
00:45:41,000 --> 00:45:48,625
equation that has to be solved.
And, I will divide this out by
609
00:45:48,625 --> 00:45:52,625
L.
I will divide everything by L.
610
00:45:52,625 --> 00:45:59,250
I will put the C a little
higher, and so with R over L
611
00:45:59,250 --> 00:46:05,875
gamma, and with omega zero
squared equals one over LC,
612
00:46:05,875 --> 00:46:11,125
this becomes,
then, I double dot plus gamma
613
00:46:11,125 --> 00:46:18,000
times I dot plus omega zero
squared times I.
614
00:46:18,000 --> 00:46:25,483
That now equals minus V zero
divided by L times omega times
615
00:46:25,483 --> 00:46:30,645
the sign of omega T.
But, here, you see a
616
00:46:30,645 --> 00:46:36,571
differential equation.
And that differential equation
617
00:46:36,571 --> 00:46:39,571
looks amazingly similar to this
one.
618
00:46:39,571 --> 00:46:43,000
And so, you should be able to
solve that.
619
00:46:43,000 --> 00:46:46,685
In fact, you wouldn't even want
to solve it.
620
00:46:46,685 --> 00:46:50,199
You can write down immediately
the answer.
621
00:46:50,199 --> 00:46:53,971
You're going to get an I,
which is an I zero,
622
00:46:53,971 --> 00:46:57,142
which takes the place of that A
there.
623
00:46:57,142 --> 00:47:01,000
This is a steady state
solution.
624
00:47:01,000 --> 00:47:06,767
I only go for steady state
solution times the sign of omega
625
00:47:06,767 --> 00:47:10,546
T minus delta:
no adjustable constants.
626
00:47:10,546 --> 00:47:15,917
It's a steady-state solution
that I have, steady state.
627
00:47:15,917 --> 00:47:21,883
And, I will leave you to find
me I zero, and you can work out
628
00:47:21,883 --> 00:47:26,060
what delta is.
That is part of your problem
629
00:47:26,060 --> 00:47:29,939
set anyhow.
But, with the knowledge that
630
00:47:29,939 --> 00:47:34,116
you have here,
you could write it down in a
631
00:47:34,116 --> 00:47:39,480
manner seconds.
So, without my telling you what
632
00:47:39,480 --> 00:47:43,320
I zero is, at least working it
out algebraically,
633
00:47:43,320 --> 00:47:47,159
we can talk 8.02.
And then, we can make all kinds
634
00:47:47,159 --> 00:47:51,239
of predictions without even
looking at the equation.
635
00:47:51,239 --> 00:47:55,400
So, that's interesting.
So, everything that I'm going
636
00:47:55,400 --> 00:47:59,480
to tell you now I do without
knowing what I zero is.
637
00:47:59,480 --> 00:48:03,000
And, it better work out that
way.
638
00:48:03,000 --> 00:48:06,753
Suppose I make omega go to
zero.
639
00:48:06,753 --> 00:48:11,717
Remember 8.02?
Remember the word reactants
640
00:48:11,717 --> 00:48:18,618
that a capacitor has a certain
reactant, which is one over
641
00:48:18,618 --> 00:48:22,493
omega C, which has units of
ohms?
642
00:48:22,493 --> 00:48:30,000
If omega goes to zero,
these reactants go to infinity.
643
00:48:30,000 --> 00:48:36,547
No current can ever flow.
So, I predict that I zero will
644
00:48:36,547 --> 00:48:41,428
go to zero.
Suppose my omega goes to omega
645
00:48:41,428 --> 00:48:45,476
zero.
Now, your memory may fail you
646
00:48:45,476 --> 00:48:49,880
here on 8.02.
But, we have a wonderful
647
00:48:49,880 --> 00:48:56,666
demonstration in 8.02 that at
resonance, one over omega C,
648
00:48:56,666 --> 00:49:02,857
which is the reactant of the
capacitor minus omega L,
649
00:49:02,857 --> 00:49:10,000
which is the reactance of the
inductor, is zero.
650
00:49:10,000 --> 00:49:13,450
That determines,
actually, the resonance.
651
00:49:13,450 --> 00:49:18,282
And, when this is the case,
perhaps you remember that the
652
00:49:18,282 --> 00:49:22,164
system doesn't even know there
is a capacitor,
653
00:49:22,164 --> 00:49:26,392
and it doesn't even know there
is a self inductor.
654
00:49:26,392 --> 00:49:32,000
The two at all moments in time
exactly cancel each other.
655
00:49:32,000 --> 00:49:35,346
And therefore,
Ohm's law holds.
656
00:49:35,346 --> 00:49:38,469
There is no L.
There is no C.
657
00:49:38,469 --> 00:49:43,711
There is only the power supply
and the resistor.
658
00:49:43,711 --> 00:49:50,403
And so, since Ohm's law says V
equals IR, you must get I zero
659
00:49:50,403 --> 00:49:56,538
equals V zero divided by R.
That is what you must get at
660
00:49:56,538 --> 00:50:01,000
resonance.
It's nonnegotiable.
661
00:50:01,000 --> 00:50:05,253
Now, if omega goes to infinity,
omega L says,
662
00:50:05,253 --> 00:50:08,250
ha, ha, yeah,
over my dead body.
663
00:50:08,250 --> 00:50:12,601
No current ever,
imagine, a very fast changing
664
00:50:12,601 --> 00:50:15,791
signal.
That's what the whole self
665
00:50:15,791 --> 00:50:20,432
inductance is about.
It doesn't want any changes.
666
00:50:20,432 --> 00:50:23,622
It's conservative like you and
me.
667
00:50:23,622 --> 00:50:28,939
And so, the self inductance
says, sorry, the reactant is
668
00:50:28,939 --> 00:50:33,000
infinitely high,
no current.
669
00:50:33,000 --> 00:50:37,214
And so, I zero goes to zero.
And so, I make these
670
00:50:37,214 --> 00:50:40,990
predictions.
And that's always nice that you
671
00:50:40,990 --> 00:50:46,082
can use the knowledge to make
predictions without even ever
672
00:50:46,082 --> 00:50:49,858
having looked at this
differential equation.
673
00:50:49,858 --> 00:50:55,302
Any of these predictions I made
to not come out of my knowledge
674
00:50:55,302 --> 00:51:00,395
of that differential equation.
So, if we make now a plot of
675
00:51:00,395 --> 00:51:04,170
the current I zero,
that is not the current,
676
00:51:04,170 --> 00:51:09,000
but that is the maximum
possible current.
677
00:51:09,000 --> 00:51:15,327
It is that I zero that you see
here without having solved it.
678
00:51:15,327 --> 00:51:21,654
I can look now what it's going
to do exactly at I omega zero.
679
00:51:21,654 --> 00:51:26,505
It will go to a maximum.
It will start at zero.
680
00:51:26,505 --> 00:51:31,883
I will go to a maximum,
and then it will fall off at
681
00:51:31,883 --> 00:51:35,574
zero.
And, this value here is V zero
682
00:51:35,574 --> 00:51:39,931
divided by R.
This is not power.
683
00:51:39,931 --> 00:51:45,080
I have now plotted current.
The power, of course,
684
00:51:45,080 --> 00:51:50,337
would go as I squared R.
That is the heat that you
685
00:51:50,337 --> 00:51:56,344
dissipate in the resistor.
So, that will go as I squared.
686
00:51:56,344 --> 00:52:01,065
I have plotted here I as a
function of not T,
687
00:52:01,065 --> 00:52:06,000
but this is I as a function of
omega.
688
00:52:06,000 --> 00:52:10,503
You know when you catch an
error that I make,
689
00:52:10,503 --> 00:52:14,393
you get partial credit for this
course.
690
00:52:14,393 --> 00:52:19,409
So please, when you see me make
a mistake, scream.
691
00:52:19,409 --> 00:52:22,582
So, this is omega.
Here is zero.
692
00:52:22,582 --> 00:52:27,086
And, this is what I want to
demonstrate, now.
693
00:52:27,086 --> 00:52:32,000
I'm not going to show you I
zero only.
694
00:52:32,000 --> 00:52:36,325
But, what I'm going to do is
the following.
695
00:52:36,325 --> 00:52:41,887
I'm going to show you what I is
as a function of omega,
696
00:52:41,887 --> 00:52:47,139
here being the resonance.
Let us suppose I pick this
697
00:52:47,139 --> 00:52:50,023
omega.
Well, then, this is my
698
00:52:50,023 --> 00:52:53,730
solution.
So, yes, the amplitude is I
699
00:52:53,730 --> 00:52:57,335
zero.
Here is the sine omega T minus
700
00:52:57,335 --> 00:53:00,116
delta.
So, all you will see,
701
00:53:00,116 --> 00:53:03,000
then, is this.
702
00:53:03,000 --> 00:53:07,000
703
00:53:07,000 --> 00:53:10,507
Plus zero, minus plus zero,
if I do it here,
704
00:53:10,507 --> 00:53:15,156
then it goes [MAKES NOISE].
And, the demonstration that we
705
00:53:15,156 --> 00:53:20,049
have prepared for you is one
whereby we will sweep omega from
706
00:53:20,049 --> 00:53:25,269
zero to a value which I remember
in terms of Hertz is about 2,000
707
00:53:25,269 --> 00:53:30,000
hertz, and we'll do that in one
sixth of a second.
708
00:53:30,000 --> 00:53:35,940
And so, what you see is you see
this as an envelope,
709
00:53:35,940 --> 00:53:41,997
which is the I zero envelope.
But, you will see this.
710
00:53:41,997 --> 00:53:48,403
And then, [it speeds?] back.
And so, you see two things.
711
00:53:48,403 --> 00:53:52,130
You'll see the sine omega T
term.
712
00:53:52,130 --> 00:53:57,721
But, as omega changes,
you will see it go through
713
00:53:57,721 --> 00:54:02,380
resonance.
And then you'll see it go over
714
00:54:02,380 --> 00:54:07,039
resonance.
So, I will give you the values
715
00:54:07,039 --> 00:54:13,950
that we have chosen.
R equals 50 ohms.
716
00:54:13,950 --> 00:54:22,107
L equals 50 millihenry.
And, we choose C 0.5 µF,
717
00:54:22,107 --> 00:54:32,000
which is substantially higher
than what we did before.
718
00:54:32,000 --> 00:54:38,043
And, we choose C so high
because we want a low Q system.
719
00:54:38,043 --> 00:54:42,879
Omega zero is now 6.3 times ten
to the third.
720
00:54:42,879 --> 00:54:49,142
It is in radians per second.
So, F zero is about 1,000 Hz.
721
00:54:49,142 --> 00:54:54,527
And, we're going to sweep it
from zero over 1,000,
722
00:54:54,527 --> 00:54:58,043
which is resonant to about
2,000.
723
00:54:58,043 --> 00:55:04,382
And then we sweep it back.
And so, the Q of this system,
724
00:55:04,382 --> 00:55:08,490
which is omega zero divided by
gamma, is about 6.3.
725
00:55:08,490 --> 00:55:13,337
And, what the V zero is of that
circuit is not so important.
726
00:55:13,337 --> 00:55:17,526
But it is [four fold?],
but that's not so important.
727
00:55:17,526 --> 00:55:21,716
And so, what we're going to
show you, we measure the
728
00:55:21,716 --> 00:55:25,577
potential difference over a very
small resistor,
729
00:55:25,577 --> 00:55:31,000
which is somewhere in that
circuit, 1.7 ohms I believe.
730
00:55:31,000 --> 00:55:36,219
And so, that potential
difference over that resistor is
731
00:55:36,219 --> 00:55:38,539
IR.
And, R is a constant,
732
00:55:38,539 --> 00:55:42,309
and that's what we're going to
show you.
733
00:55:42,309 --> 00:55:47,433
So, we're going to show you
something that is directly
734
00:55:47,433 --> 00:55:51,493
linear proportional with I.
It's not power.
735
00:55:51,493 --> 00:55:54,296
It's I.
We want to know power.
736
00:55:54,296 --> 00:56:00,000
You have to square it.
I square R is the power.
737
00:56:00,000 --> 00:56:05,851
And, we're going to sweep it
1/6 of a second this way,
738
00:56:05,851 --> 00:56:11,261
and 1/6 of a second back.
Why did I only take into
739
00:56:11,261 --> 00:56:14,905
account the steady state
solution?
740
00:56:14,905 --> 00:56:20,867
Why don't I have to also
include the trangent solution,
741
00:56:20,867 --> 00:56:27,160
which with this experiment took
us five minutes to finally
742
00:56:27,160 --> 00:56:31,025
arrive at the steady state
solution?
743
00:56:31,025 --> 00:56:36,647
Why am I leaving it out?
I can't hear you.
744
00:56:36,647 --> 00:56:40,038
Where is the sound coming from?
Yes.
745
00:56:40,038 --> 00:56:44,593
What is two over gamma,
which is the decay time?
746
00:56:44,593 --> 00:56:49,244
One over EDK time?
Well, there is two over gamma.
747
00:56:49,244 --> 00:56:52,441
I didn't write down what gamma
is.
748
00:56:52,441 --> 00:56:55,639
Gamma is 1,000.
Gamma is R over L,
749
00:56:55,639 --> 00:57:00,000
right?
Gamma is R over L is 1,000.
750
00:57:00,000 --> 00:57:05,863
So, two over gamma,
which is the one over EDK time,
751
00:57:05,863 --> 00:57:10,201
is 2 ms.
Or, another way of putting it
752
00:57:10,201 --> 00:57:14,071
is that in about two
oscillations,
753
00:57:14,071 --> 00:57:16,768
I'm down by a factor,
E.
754
00:57:16,768 --> 00:57:22,280
Remember, Q over pi
oscillations will reduce the
755
00:57:22,280 --> 00:57:28,260
amplitude by a factor of E.
So, in two oscillations,
756
00:57:28,260 --> 00:57:36,000
already the trangent phenomenon
is effectively killed.
757
00:57:36,000 --> 00:57:39,373
And so, I don't have to take it
into account.
758
00:57:39,373 --> 00:57:43,054
You won't even notice it.
So, I told you earlier,
759
00:57:43,054 --> 00:57:45,968
when the decay time here was
very long.
760
00:57:45,968 --> 00:57:49,955
I'll show you another
experiment where the decay time
761
00:57:49,955 --> 00:57:52,638
is extremely short.
While I'm at it,
762
00:57:52,638 --> 00:57:56,626
the experiment is set up there.
You will see it here.
763
00:57:56,626 --> 00:58:00,000
I'm going to make this 100
ohms.
764
00:58:00,000 --> 00:58:05,740
I'm going to double it to show
you that this point will exactly
765
00:58:05,740 --> 00:58:09,814
go down by a factor of two
because, remember,
766
00:58:09,814 --> 00:58:12,685
the peak is V zero divided by
R.
767
00:58:12,685 --> 00:58:17,592
And, I'm not changing P zero.
And so, when I double R,
768
00:58:17,592 --> 00:58:22,870
you will see it come down to
here, and you will see it get
769
00:58:22,870 --> 00:58:26,018
broader.
And then, I will go to 150
770
00:58:26,018 --> 00:58:30,000
ohms.
That's easy to do for us.
771
00:58:30,000 --> 00:58:34,324
So, we go to 150 ohms.
So, at 100 ohms,
772
00:58:34,324 --> 00:58:37,851
the Q is 3.15.
And, at 150 ohms,
773
00:58:37,851 --> 00:58:42,858
the Q is three times smaller.
It's about 2.1.
774
00:58:42,858 --> 00:58:47,979
So, at 150 ohms,
it would be one third of this
775
00:58:47,979 --> 00:58:53,896
height, somewhere here.
And the peak will be broader.
776
00:58:53,896 --> 00:59:00,382
And, all of that you get today
very easily by changing the
777
00:59:00,382 --> 00:59:06,229
resistance.
Again, this is not power.
778
00:59:06,229 --> 00:59:13,205
Power is I squared times R.
This is simply the current.
779
00:59:13,205 --> 00:59:21,215
All right, I will give you the
correct light setting because we
780
00:59:21,215 --> 00:59:27,028
have to make it a little darker.
And, then we,
781
00:59:27,028 --> 00:59:32,181
and there it is.
So, noticed the vertical
782
00:59:32,181 --> 00:59:37,272
oscillations go so fast that
your eyes cannot even follow
783
00:59:37,272 --> 00:59:40,363
them.
But, that's the sine omega T.
784
00:59:40,363 --> 00:59:45,090
And then, when it sweeps over
resonance, you see very
785
00:59:45,090 --> 00:59:49,818
dramatically this value which is
V zero divided by R.
786
00:59:49,818 --> 00:59:55,181
Now, notice on the scale here
that Marcos has set the I zero
787
00:59:55,181 --> 00:59:58,181
at resonance at three scale
units.
788
00:59:58,181 --> 1:00:03,454
So, it's nice when we go from
our 50 ohms to 100 ohms to go
789
1:00:03,454 --> 1:00:09,013
down by a factor of two.
And, you can check that.
790
1:00:09,013 --> 1:00:13,627
And when we go to 150 ohms,
it should go down from three
791
1:00:13,627 --> 1:00:17,402
units to one unit.
So, it could quantitatively
792
1:00:17,402 --> 1:00:20,842
check this.
And, you'll see that the curve
793
1:00:20,842 --> 1:00:23,778
gets broader because it has a
low Q.
794
1:00:23,778 --> 1:00:26,379
So, now I will make it 100
ohms.
795
1:00:26,379 --> 1:00:32,000
You see, it's down by a factor
of two from here to here.
796
1:00:32,000 --> 1:00:36,446
And, you may have noticed that
it also gets broader.
797
1:00:36,446 --> 1:00:40,196
And now, I will go to 150 ohms.
And you see,
798
1:00:40,196 --> 1:00:45,079
it's down by a factor of three.
And again, it is broader.
799
1:00:45,079 --> 1:00:50,136
So, this is an amazing way how
with RLC circuits you can do
800
1:00:50,136 --> 1:00:55,193
wonderful things because you can
manipulate omega zero very
801
1:00:55,193 --> 1:00:59,901
easily by changing L and C.
And, you can manipulate the
802
1:00:59,901 --> 1:01:04,000
driving frequency also very
easily.
803
1:01:04,000 --> 1:01:09,682
So, it is clear that systems
respond strongly when they're
804
1:01:09,682 --> 1:01:13,271
exposed to their resonance
frequency.
805
1:01:13,271 --> 1:01:19,052
We've seen that for pendulums.
We've seen that for springs.
806
1:01:19,052 --> 1:01:24,436
We've seen at 401 glass.
And we've seen this now for an
807
1:01:24,436 --> 1:01:28,623
RLC circuit.
So, these systems at resonance
808
1:01:28,623 --> 1:01:36,000
absorb a large amount of energy
per unit time out of the driver.
809
1:01:36,000 --> 1:01:43,533
I have here two tuning forks,
which have an extremely high Q.
810
1:01:43,533 --> 1:01:50,690
My attempts to measure it,
I conclude it's way larger than
811
1:01:50,690 --> 1:01:55,838
even 1,000.
And they have exactly the same
812
1:01:55,838 --> 1:02:01,488
frequency: both 256 Hz,
this one and this one,
813
1:02:01,488 --> 1:02:06,374
both 256 Hz.
That's' the way they are
814
1:02:06,374 --> 1:02:11,717
designed to a high degree of
accuracy, to better than a
815
1:02:11,717 --> 1:02:16,268
fraction of 1 Hz.
But, the Q's are so high that
816
1:02:16,268 --> 1:02:21,314
if you were to plot,
if you drive these tuning forks
817
1:02:21,314 --> 1:02:27,151
and you are to plot here this
average power as a function of
818
1:02:27,151 --> 1:02:32,000
omega, then you get something
like this.
819
1:02:32,000 --> 1:02:36,468
It means you have to drive it
exactly at the right frequency.
820
1:02:36,468 --> 1:02:39,446
Otherwise, it will not go into
resonance.
821
1:02:39,446 --> 1:02:42,053
Well, we know how to get this
going.
822
1:02:42,053 --> 1:02:45,478
You just bang it.
That means you dump the whole
823
1:02:45,478 --> 1:02:48,904
spectrum on it.
It picks out the frequency that
824
1:02:48,904 --> 1:02:53,148
likes, and now I'm going to show
you something remarkable.
825
1:02:53,148 --> 1:02:56,276
When this one generates 256
pressure waves,
826
1:02:56,276 --> 1:03:00,000
this one feels those pressure
waves.
827
1:03:00,000 --> 1:03:05,914
And it loves it because it's
just at the right frequency.
828
1:03:05,914 --> 1:03:11,933
And so, it starts oscillate.
And so, when I stop this one,
829
1:03:11,933 --> 1:03:17,003
you'll hear this one.
And that's called resonance
830
1:03:17,003 --> 1:03:20,277
absorption.
Let's do that first.
831
1:03:20,277 --> 1:03:26,613
Now, you must understand that
the sound waves go from here to
832
1:03:26,613 --> 1:03:30,099
there.
Not much power reaches that
833
1:03:30,099 --> 1:03:33,785
point.
So, when I stop this one,
834
1:03:33,785 --> 1:03:36,642
you hear sound but it's not
overwhelming.
835
1:03:36,642 --> 1:03:39,000
So, you have to be very quiet.
836
1:03:39,000 --> 1:03:48,000
837
1:03:48,000 --> 1:03:49,000
Hear it?
838
1:03:49,000 --> 1:03:53,000
839
1:03:53,000 --> 1:03:55,928
And I can do the same by
hitting this one,
840
1:03:55,928 --> 1:03:59,000
and then this one will start to
resonate.
841
1:03:59,000 --> 1:04:06,000
842
1:04:06,000 --> 1:04:12,059
Now, if the driving frequency
is off by a fraction of the
843
1:04:12,059 --> 1:04:17,578
hertz, 1 Hz is enough,
1 Hz difference because the Q
844
1:04:17,578 --> 1:04:23,962
is so high that this system will
not be able to get this one
845
1:04:23,962 --> 1:04:27,858
going.
And, I can make this frequency
846
1:04:27,858 --> 1:04:35,000
a little lower than 256 Hz by
putting this weight on here.
847
1:04:35,000 --> 1:04:39,607
And, we have measured the
frequency at this loaded way.
848
1:04:39,607 --> 1:04:43,617
It's roughly 255 Hz.
And, you are somewhere here
849
1:04:43,617 --> 1:04:48,054
because it's so narrow.
The resonance absorption peak
850
1:04:48,054 --> 1:04:51,382
is very sharp.
By the way, this is power
851
1:04:51,382 --> 1:04:55,307
because what reaches here is
joules per second.
852
1:04:55,307 --> 1:05:00,000
That's what gets it going,
energy per second.
853
1:05:00,000 --> 1:05:03,467
So it's really a power
transfer.
854
1:05:03,467 --> 1:05:08,723
So now, I change the frequency,
and there we go.
855
1:05:08,723 --> 1:05:12,750
Nothing.
But just change this by 1 Hz
856
1:05:12,750 --> 1:05:15,657
and you hear nothing.
Dead.
857
1:05:15,657 --> 1:05:20,690
So, now you see,
you get some respect for high
858
1:05:20,690 --> 1:05:24,493
Q's.
If you want to get a resonance
859
1:05:24,493 --> 1:05:31,315
absorption in the high Q system.
You've got to be dead on that
860
1:05:31,315 --> 1:05:36,536
frequency.
So, if you, for instance,
861
1:05:36,536 --> 1:05:42,014
banged all the keys on the
piano, and this one would be
862
1:05:42,014 --> 1:05:48,304
nearby, it would only start to
resonate if one of those strings
863
1:05:48,304 --> 1:05:51,550
would produce exactly the 256
Hz.
864
1:05:51,550 --> 1:05:56,217
Otherwise, it would not.
It ignores everything.
865
1:05:56,217 --> 1:06:02,000
It's only sensitive to that
resonance frequency.
866
1:06:02,000 --> 1:06:07,289
You probably in high school
have learned a little bit about
867
1:06:07,289 --> 1:06:11,120
atomic physics.
And, you probably know that
868
1:06:11,120 --> 1:06:16,592
electrons have discrete energy
levels and discrete orbits and
869
1:06:16,592 --> 1:06:19,785
atoms.
And, you can excite the atom.
870
1:06:19,785 --> 1:06:23,706
You can bring an electron in a
higher orbit,
871
1:06:23,706 --> 1:06:30,000
discrete orbits which cost you
a discrete amount of energy.
872
1:06:30,000 --> 1:06:34,308
And, when the atom recombines,
when the electron falls back,
873
1:06:34,308 --> 1:06:38,178
you get that energy back,
exactly the same amount that
874
1:06:38,178 --> 1:06:41,610
you had to put in.
And, that energy that you get
875
1:06:41,610 --> 1:06:45,845
back comes out most of the time
in the form of what we call
876
1:06:45,845 --> 1:06:49,643
electromagnetic radiation.
I know that in 8.03 we are
877
1:06:49,643 --> 1:06:53,294
going to deal with
electromagnetic radiation in the
878
1:06:53,294 --> 1:06:55,922
future.
But, it's enough for now that
879
1:06:55,922 --> 1:06:58,332
you know that light,
infrared, UV,
880
1:06:58,332 --> 1:07:01,691
gamma rays, x-rays,
all of that radio emission,
881
1:07:01,691 --> 1:07:06,000
all of that is electromagnetic
radiation.
882
1:07:06,000 --> 1:07:12,139
And so, here I have the energy
level, energy increasing,
883
1:07:12,139 --> 1:07:18,390
here with an electron in orbit.
That's the energy of that
884
1:07:18,390 --> 1:07:22,855
electron.
A higher energy state is when I
885
1:07:22,855 --> 1:07:28,213
bring this electron here.
I cannot do anything in
886
1:07:28,213 --> 1:07:34,078
between.
Quantum mechanics says it's one
887
1:07:34,078 --> 1:07:39,210
or the other.
And, if this difference is
888
1:07:39,210 --> 1:07:45,394
delta E in energy,
this E stands now for energy.
889
1:07:45,394 --> 1:07:52,236
Then, when the electron falls
back from here to here,
890
1:07:52,236 --> 1:08:00,000
it emits electromagnetic
radiation with this energy.
891
1:08:00,000 --> 1:08:06,781
But, if I radiate onto this
atom, electromagnetic radiation
892
1:08:06,781 --> 1:08:13,212
with exactly that energy,
then this electron can go from
893
1:08:13,212 --> 1:08:18,123
here to there.
And that is called resonance
894
1:08:18,123 --> 1:08:21,513
absorption.
Now, let us stick,
895
1:08:21,513 --> 1:08:28,178
for now, to visible light.
The higher the energy the bluer
896
1:08:28,178 --> 1:08:33,089
the light is,
or as modern physicists would
897
1:08:33,089 --> 1:08:40,777
say, the higher the frequency.
And the lower the energy,
898
1:08:40,777 --> 1:08:45,777
the redder the light,
the lower the frequency.
899
1:08:45,777 --> 1:08:52,333
So, our visible light that we
can see with our eyes goes all
900
1:08:52,333 --> 1:08:57,555
the way from the red,
low energy, to the violet,
901
1:08:57,555 --> 1:09:02,000
high-energy.
Let's go to the sun.
902
1:09:02,000 --> 1:09:07,561
The sun radiates in the visible
spectrum all the way from the
903
1:09:07,561 --> 1:09:11,918
red to the violet.
But, in the solar atmosphere,
904
1:09:11,918 --> 1:09:15,533
R elements.
And, when these elements see
905
1:09:15,533 --> 1:09:20,817
just the right energy from that
spectrum to which they are
906
1:09:20,817 --> 1:09:26,471
exposed, they love to take out
of that spectrum just the right
907
1:09:26,471 --> 1:09:30,457
energy that gets them into an
excited state,
908
1:09:30,457 --> 1:09:35,000
which is called resonance
absorption.
909
1:09:35,000 --> 1:09:39,651
And so, that energy is removed
from the spectrum.
910
1:09:39,651 --> 1:09:45,658
So, when you look at the solar
spectrum, there are bands in the
911
1:09:45,658 --> 1:09:49,147
spectrum, but the colors are
missing.
912
1:09:49,147 --> 1:09:54,379
Absorption in the spectrum,
dark bends in the spectrum.
913
1:09:54,379 --> 1:10:00,000
They were discovered in 1802 by
William Wollaston.
914
1:10:00,000 --> 1:10:04,300
And in 1814,
Fraunhofer had cataloged 475 of
915
1:10:04,300 --> 1:10:08,399
these lines.
And, they are now referred to
916
1:10:08,399 --> 1:10:13,899
as Fraunhofer absorption lines.
Even though they did not
917
1:10:13,899 --> 1:10:18,899
understand the physics,
this is a quantum mechanics
918
1:10:18,899 --> 1:10:23,600
picture that came from Niels
Bohr, 20th Century,
919
1:10:23,600 --> 1:10:28,899
even though they did not
understand what happened they
920
1:10:28,899 --> 1:10:34,399
had noticed that these black
lines in the solar spectrum
921
1:10:34,399 --> 1:10:40,199
coincided with emission lines in
the spectrum that they can
922
1:10:40,199 --> 1:10:48,000
generate in the laboratory by
heating up the various elements.
923
1:10:48,000 --> 1:10:52,765
And so, without understanding
why, they were able to say,
924
1:10:52,765 --> 1:10:57,617
ah, I see magnesium in the sun.
I see aluminum in the sun,
925
1:10:57,617 --> 1:11:02,991
in the solar atmosphere.
And so, that opened a whole new
926
1:11:02,991 --> 1:11:08,975
industry of spectroscopy which
allowed astronomers to determine
927
1:11:08,975 --> 1:11:13,994
the chemical composition of the
atmospheres of stars.
928
1:11:13,994 --> 1:11:19,978
And, it was in 1868 that Joseph
Lockyer found at least one dark
929
1:11:19,978 --> 1:11:25,383
line which did not coincide with
any emission line in the
930
1:11:25,383 --> 1:11:29,147
laboratory.
There is no element on Earth
931
1:11:29,147 --> 1:11:33,780
that he could say,
that must be the cause of that
932
1:11:33,780 --> 1:11:38,710
dark line.
And so, he calls it helium
933
1:11:38,710 --> 1:11:43,296
because the Greek word for the
sun is helios.
934
1:11:43,296 --> 1:11:49,446
So, it helium is an element
that was first discovered on the
935
1:11:49,446 --> 1:11:53,407
sun before it was later found on
Earth.
936
1:11:53,407 --> 1:11:59,348
I want to show you resonance
absorption on the scale of an
937
1:11:59,348 --> 1:12:02,788
atom.
And, the way I'm going to do
938
1:12:02,788 --> 1:12:08,000
that, the setup is here,
is as follows.
939
1:12:08,000 --> 1:12:13,230
We have a carbon [arc?].
Think of that as being the sun,
940
1:12:13,230 --> 1:12:17,794
which produces a spectrum,
a beautiful continuous
941
1:12:17,794 --> 1:12:21,407
spectrum.
I will show you that spectrum
942
1:12:21,407 --> 1:12:25,021
all the way from the red to the
violet.
943
1:12:25,021 --> 1:12:30,631
And then we have here a burner.
And we're going to put table
944
1:12:30,631 --> 1:12:35,291
salt here on the grid,
which dissociates the table
945
1:12:35,291 --> 1:12:41,943
salts, gives me sodium gas.
That's what I want because
946
1:12:41,943 --> 1:12:48,283
sodium, when you heat it,
can produce an emission line in
947
1:12:48,283 --> 1:12:52,132
yellow.
But if you can produce that
948
1:12:52,132 --> 1:12:58,358
emission line when the electron
goes from here to there,
949
1:12:58,358 --> 1:13:03,000
it's the 11th electron by the
way.
950
1:13:03,000 --> 1:13:07,135
It's the most outer electron of
sodium, 11 protons,
951
1:13:07,135 --> 1:13:10,195
11 electrons.
So, if it can produce an
952
1:13:10,195 --> 1:13:13,834
emission line when it goes from
here to here,
953
1:13:13,834 --> 1:13:16,563
it can also,
resonance absorption,
954
1:13:16,563 --> 1:13:20,864
namely when it sees that yellow
line, the energy that
955
1:13:20,864 --> 1:13:25,330
corresponds with the yellow
line, it sucks it up and it
956
1:13:25,330 --> 1:13:28,804
produces, then,
a dark line because when it
957
1:13:28,804 --> 1:13:31,864
absorbs out of here,
this yellow line,
958
1:13:31,864 --> 1:13:36,000
it re-emits it almost
immediately.
959
1:13:36,000 --> 1:13:39,333
But the re-emission will be in
all directions.
960
1:13:39,333 --> 1:13:43,703
And so, what's left over here
is very little of that yellow.
961
1:13:43,703 --> 1:13:46,888
And so, a dark line appears in
the spectrum.
962
1:13:46,888 --> 1:13:49,259
That then is the absorption
line.
963
1:13:49,259 --> 1:13:53,037
Its power, what I'm going to
show you, because light
964
1:13:53,037 --> 1:13:57,777
intensity, which I will show you
there on that screen is how many
965
1:13:57,777 --> 1:14:02,988
joules per second?
So, it is resonance absorption
966
1:14:02,988 --> 1:14:06,077
of power.
Now, there is a catch.
967
1:14:06,077 --> 1:14:12,254
And the catch is that probably
only you up here will be able to
968
1:14:12,254 --> 1:14:15,741
see it.
And, others could come down.
969
1:14:15,741 --> 1:14:21,121
You're going to see the
spectrum here first of the sun,
970
1:14:21,121 --> 1:14:27,000
which is my carbon arc.
Then I will put in sodium.
971
1:14:27,000 --> 1:14:30,594
And you will see an
unbelievable,
972
1:14:30,594 --> 1:14:35,311
unimaginable,
beautiful, sharp light like a
973
1:14:35,311 --> 1:14:40,590
razor in the yellow.
But you've got to be close.
974
1:14:40,590 --> 1:14:45,644
So, let's first,
Marcos, if you manage to open
975
1:14:45,644 --> 1:14:51,148
the gas, he knows exactly where
that gas valve is,
976
1:14:51,148 --> 1:14:58,000
then I will ignite it.
We won't put it yet in beam.
977
1:14:58,000 --> 1:15:04,000
978
1:15:04,000 --> 1:15:08,684
OK, so we're going to make it
completely dark.
979
1:15:08,684 --> 1:15:12,118
So, here are the [sold?]
crystals.
980
1:15:12,118 --> 1:15:17,011
And, we're going to show you
the spectrum there.
981
1:15:17,011 --> 1:15:21,070
We try to actually make you see
it here.
982
1:15:21,070 --> 1:15:25,442
[UNINTELLIGIBLE] that didn't
work out well.
983
1:15:25,442 --> 1:15:29,085
OK, so I'm going to turn on the
sun.
984
1:15:29,085 --> 1:15:34,199
There's the sun.
OK, now we make it completely
985
1:15:34,199 --> 1:15:38,448
dark, and I will give you a
minute or so for your eyes to
986
1:15:38,448 --> 1:15:41,103
adjust.
So, you see a spectrum here;
987
1:15:41,103 --> 1:15:45,275
you see a spectrum there.
How we do that is our problem.
988
1:15:45,275 --> 1:15:49,675
And, you will know how we do
that in a month or so when you
989
1:15:49,675 --> 1:15:53,393
will learn about gradings.
You will get a grading,
990
1:15:53,393 --> 1:15:56,579
actually, from us.
There is a grading here,
991
1:15:56,579 --> 1:16:00,979
a wonderful piece of physics,
which decomposes the light in
992
1:16:00,979 --> 1:16:05,000
colors, works way better than a
prism.
993
1:16:05,000 --> 1:16:08,585
And, you get one on the right
side.
994
1:16:08,585 --> 1:16:13,225
And, you get a mirror image on
the left side.
995
1:16:13,225 --> 1:16:16,810
Look here and let your eyes
adjust.
996
1:16:16,810 --> 1:16:20,396
And then comes the moment of
truth.
997
1:16:20,396 --> 1:16:24,509
I'm going to put in here now
the sodium.
998
1:16:24,509 --> 1:16:27,778
Unbelievable.
I see a line here,
999
1:16:27,778 --> 1:16:31,785
sharp as a razor blade.
Can you see it,
1000
1:16:31,785 --> 1:16:36,000
Nicole?
Isn't it incredible?
1001
1:16:36,000 --> 1:16:39,600
Come closer.
All of you, come closer;
1002
1:16:39,600 --> 1:16:43,899
look at that line.
Come on; come out of your
1003
1:16:43,899 --> 1:16:46,300
seats.
Look at that line,
1004
1:16:46,300 --> 1:16:52,000
and I will move the sodium out.
Now I'm at the sodium out.
1005
1:16:52,000 --> 1:16:56,300
And now I move it in again.
And there it is.
1006
1:16:56,300 --> 1:17:00,000
There it is.
You see that?
1007
1:17:00,000 --> 1:17:03,843
Isn't that amazing?
And now I move it out.
1008
1:17:03,843 --> 1:17:06,000
And now I move it in.
1009
1:17:06,000 --> 1:17:12,000
1010
1:17:12,000 --> 1:17:47,550
Isn't that a fantastic line?
Look at that line.
1011
1:17:47,550 --> 1:18:21,555
Look at that line.
Resonance absorption of an
1012
1:18:21,555 --> 1:18:54,015
extremely high Q system on an
atomic scale.
1013
1:18:54,015 --> 1:19:16,427
I hope you can sleep tonight.
[LAUGHTER] See you Thursday.