1
00:00:26,000 --> 00:00:32,000
This is a brief,
so, the equation,
2
00:00:29,000 --> 00:00:35,000
and we got the characteristic
equation from the last time.
3
00:00:35,000 --> 00:00:41,000
The general topic for today is
going to be oscillations,
4
00:00:41,000 --> 00:00:47,000
which are extremely important
in the applications and in
5
00:00:47,000 --> 00:00:53,000
everyday life.
But, the oscillations,
6
00:00:50,000 --> 00:00:56,000
we know, are associated with a
complex root.
7
00:00:55,000 --> 00:01:01,000
So, they correspond to complex
roots of the characteristic
8
00:01:01,000 --> 00:01:07,000
equation.
r squared plus br plus k equals
9
00:01:06,000 --> 00:01:12,000
zero.
I'd like to begin.
10
00:01:09,000 --> 00:01:15,000
Most of the lecture will be
about discussing the relations
11
00:01:13,000 --> 00:01:19,000
between these numbers,
these constants,
12
00:01:16,000 --> 00:01:22,000
and the various properties that
the solutions,
13
00:01:19,000 --> 00:01:25,000
oscillatory solutions,
have.
14
00:01:21,000 --> 00:01:27,000
But, before that,
I'd like to begin by clearing
15
00:01:24,000 --> 00:01:30,000
up a couple of questions almost
everybody has at some point or
16
00:01:29,000 --> 00:01:35,000
other when they study the case
of complex roots.
17
00:01:34,000 --> 00:01:40,000
Complex roots are the case
which produce oscillations in
18
00:01:38,000 --> 00:01:44,000
the solutions.
That's the relation,
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00:01:40,000 --> 00:01:46,000
and that's why I'm talking
about this for the first few
20
00:01:44,000 --> 00:01:50,000
minutes.
Now, what is the problem?
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The complex roots,
of course, there will be two
22
00:01:50,000 --> 00:01:56,000
roots, and they occur at the
complex conjugates of each
23
00:01:54,000 --> 00:02:00,000
other.
So, they will be of the form a
24
00:01:57,000 --> 00:02:03,000
plus or minus bi.
Last time, I showed you,
25
00:02:02,000 --> 00:02:08,000
I took the root r equals a plus
bi,
26
00:02:08,000 --> 00:02:14,000
which leads to the solution.
The corresponding solution is a
27
00:02:14,000 --> 00:02:20,000
complex solution which is e to
the at, (a plus i b)t.
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And, what we did was the
29
00:02:24,000 --> 00:02:30,000
problem was to get real
solutions out of that.
30
00:02:29,000 --> 00:02:35,000
We needed two real solutions,
and the way I got them was by
31
00:02:36,000 --> 00:02:42,000
separating this into its real
part and its imaginary part.
32
00:02:44,000 --> 00:02:50,000
And, I proved a little theorem
for you that said both of those
33
00:02:47,000 --> 00:02:53,000
give solutions.
So, the real part was e to the
34
00:02:49,000 --> 00:02:55,000
a t times cosine b t,
and the imaginary
35
00:02:52,000 --> 00:02:58,000
part was e to the at sine b t.
36
00:02:55,000 --> 00:03:01,000
And, those were the two
solutions.
37
00:02:57,000 --> 00:03:03,000
So, here was y1.
And, the point was those,
38
00:02:59,000 --> 00:03:05,000
out of the complex solutions,
we got real solutions.
39
00:03:03,000 --> 00:03:09,000
We have to have real solutions
because we live in the real
40
00:03:07,000 --> 00:03:13,000
world.
The equation is real.
41
00:03:09,000 --> 00:03:15,000
Its coefficients are real.
They represent real quantities.
42
00:03:13,000 --> 00:03:19,000
That's the way the solutions,
therefore, have to be.
43
00:03:17,000 --> 00:03:23,000
So, these, the point is,
these are now real solutions,
44
00:03:21,000 --> 00:03:27,000
these two guys,
y1 and y2.
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00:03:23,000 --> 00:03:29,000
Now, the first question almost
everybody has,
46
00:03:26,000 --> 00:03:32,000
and I was pleased to see at the
end of the lecture,
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00:03:29,000 --> 00:03:35,000
a few people came up and asked
me, yeah, well,
48
00:03:33,000 --> 00:03:39,000
you took a plus bi,
but there was another root,
49
00:03:36,000 --> 00:03:42,000
a minus bi.
You didn't use that one.
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00:03:40,000 --> 00:03:46,000
That would give two more
solutions, right?
51
00:03:43,000 --> 00:03:49,000
Of course, they didn't say
that.
52
00:03:45,000 --> 00:03:51,000
They were too smart.
They just said,
53
00:03:47,000 --> 00:03:53,000
what about that other root?
Well, what about it?
54
00:03:50,000 --> 00:03:56,000
The reason I don't have to talk
about the other root is because
55
00:03:54,000 --> 00:04:00,000
although it does give to
solutions, it doesn't give two
56
00:03:58,000 --> 00:04:04,000
new ones.
Maybe I can indicate that most
57
00:04:02,000 --> 00:04:08,000
clearly here even though you
won't be able to take notes by
58
00:04:07,000 --> 00:04:13,000
just using colored chalk.
Suppose, instead of plus bi,
59
00:04:12,000 --> 00:04:18,000
I used a minus bi.
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00:04:16,000 --> 00:04:22,000
What would have changed?
Well, this would now become
61
00:04:20,000 --> 00:04:26,000
minus here.
Would this change?
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00:04:23,000 --> 00:04:29,000
No, because e to the minus ibt
is the cosine of
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00:04:28,000 --> 00:04:34,000
minus b, but that's the same as
the cosine of b.
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00:04:34,000 --> 00:04:40,000
How about here?
This would become the sine of
65
00:04:37,000 --> 00:04:43,000
minus bt.
But that's simply the negative
66
00:04:42,000 --> 00:04:48,000
of the sine of bt.
So, the only change would have
67
00:04:47,000 --> 00:04:53,000
been to put a minus sign there.
Now, I don't care if I get y2
68
00:04:52,000 --> 00:04:58,000
or negative y2 because what am I
going to do with it?
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00:04:57,000 --> 00:05:03,000
When I get it,
I'm going to write y,
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00:05:00,000 --> 00:05:06,000
the general solution,
as c1 y1 plus c2 y2.
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00:05:05,000 --> 00:05:11,000
So, if I get negative y2,
that just changes that
72
00:05:09,000 --> 00:05:15,000
arbitrary constant from c2 to
minus c2, which is just as
73
00:05:13,000 --> 00:05:19,000
arbitrary a constant.
So, in other words,
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00:05:17,000 --> 00:05:23,000
there's no reason to use the
other root because it doesn't
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00:05:22,000 --> 00:05:28,000
give anything new.
Now, there the story could
76
00:05:25,000 --> 00:05:31,000
stop.
And, I would like it to stop,
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00:05:28,000 --> 00:05:34,000
frankly, but I don't dare
because there's a second
78
00:05:33,000 --> 00:05:39,000
question.
And, I'm visiting recitations
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00:05:37,000 --> 00:05:43,000
not this semester,
but in previous semesters.
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00:05:40,000 --> 00:05:46,000
In 18.03, so many recitations
do this.
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00:05:42,000 --> 00:05:48,000
I have to partly inoculate you
against it, and partly tell you
82
00:05:46,000 --> 00:05:52,000
that some of the engineering
courses do do it,
83
00:05:50,000 --> 00:05:56,000
and therefore you probably
should learn it also.
84
00:05:53,000 --> 00:05:59,000
So, there is another way of
proceeding, which is what you
85
00:05:57,000 --> 00:06:03,000
might have thought.
Hey, look, we got two complex
86
00:06:00,000 --> 00:06:06,000
roots.
That gives us two solutions,
87
00:06:04,000 --> 00:06:10,000
which are different.
Neither one is a constant
88
00:06:08,000 --> 00:06:14,000
multiple of the other.
So, the other approach is,
89
00:06:12,000 --> 00:06:18,000
use, as a general solution,
y equals, now,
90
00:06:16,000 --> 00:06:22,000
I'm going to put a capital C
here.
91
00:06:19,000 --> 00:06:25,000
You will see why in just a
second, times e to the (a plus b
92
00:06:24,000 --> 00:06:30,000
i) times t.
93
00:06:27,000 --> 00:06:33,000
And then, I will use the other
solution: C2 times e to the (a
94
00:06:32,000 --> 00:06:38,000
minus b i) t.
95
00:06:37,000 --> 00:06:43,000
These are two independent
solutions.
96
00:06:39,000 --> 00:06:45,000
And therefore,
can't I get the general
97
00:06:42,000 --> 00:06:48,000
solution in that form?
Now, in a sense,
98
00:06:45,000 --> 00:06:51,000
you can.
The whole problem is the
99
00:06:47,000 --> 00:06:53,000
following, of course,
that I'm only interested in
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00:06:51,000 --> 00:06:57,000
real solutions.
This is a complex function.
101
00:06:54,000 --> 00:07:00,000
This is another complex
function.
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00:06:56,000 --> 00:07:02,000
It's got an i in it,
in other words,
103
00:06:59,000 --> 00:07:05,000
when I write it out as u plus
iv.
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00:07:04,000 --> 00:07:10,000
If I expect to be able to get a
real solution out of that,
105
00:07:08,000 --> 00:07:14,000
that means I have to make,
allow these coefficients to be
106
00:07:13,000 --> 00:07:19,000
complex numbers,
and not real numbers.
107
00:07:16,000 --> 00:07:22,000
So, in other words,
what I'm saying is that an
108
00:07:20,000 --> 00:07:26,000
expression like this,
where the a plus bi and a minus
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00:07:24,000 --> 00:07:30,000
bi are complex roots of that
characteristic equation,
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00:07:28,000 --> 00:07:34,000
is formally a very general,
complex solution to the
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00:07:33,000 --> 00:07:39,000
equation.
And therefore,
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00:07:36,000 --> 00:07:42,000
the problem becomes,
how, from this expression,
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00:07:41,000 --> 00:07:47,000
do I get the real solutions?
So, the problem is,
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00:07:47,000 --> 00:07:53,000
I accept these as the complex
solutions.
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00:07:51,000 --> 00:07:57,000
My problem is,
to find among all these guys
116
00:07:56,000 --> 00:08:02,000
where C1 and C2 are allowed to
be complex, the problem is,
117
00:08:03,000 --> 00:08:09,000
which of the green solutions
are real?
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00:08:09,000 --> 00:08:15,000
Now, there are many ways of
getting the answer.
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00:08:12,000 --> 00:08:18,000
There is a super hack way.
The super hack way is to say,
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00:08:17,000 --> 00:08:23,000
well, this one is C1 plus i d1.
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00:08:20,000 --> 00:08:26,000
This is C2 plus i d2.
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00:08:23,000 --> 00:08:29,000
And, I'll write all this out in
terms of what it is,
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00:08:27,000 --> 00:08:33,000
you know, cosine plus i sine,
and don't forget the e to the
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00:08:31,000 --> 00:08:37,000
at.
And, I will write it all out,
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00:08:34,000 --> 00:08:40,000
and it will take an entire
board.
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00:08:36,000 --> 00:08:42,000
And then, I will just see what
the condition is.
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I'll write its real part,
and its imaginary part.
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00:08:42,000 --> 00:08:48,000
And then, I will say the
imaginary part has got to be
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00:08:46,000 --> 00:08:52,000
zero.
And, then I will see what it's
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00:08:48,000 --> 00:08:54,000
like.
That works fine.
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It just takes too much space.
And also, it doesn't teach you
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a few things that I think you
should know.
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So, I'm going to give another.
So, let's say we can answer
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this two ways:
by hack, in other words,
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multiply everything out.
Multiply all out,
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00:09:07,000 --> 00:09:13,000
make the imaginary part equal
zero.
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00:09:11,000 --> 00:09:17,000
Now, here's a better way,
in my opinion.
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What I'm trying to do is,
this is some complex function,
139
00:09:23,000 --> 00:09:29,000
u plus iv.
How do I know when a complex
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00:09:27,000 --> 00:09:33,000
function is real?
I want this to be real.
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00:09:34,000 --> 00:09:40,000
Well, the hack method
corresponds to,
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say, v must be equal to zero.
It's real if v is zero.
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So, expand it out,
and see why v is zero.
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00:09:49,000 --> 00:09:55,000
There's a slightly more subtle
method, which is to change i to
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minus i.
And, what?
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00:10:00,000 --> 00:10:06,000
And, see if it stays the same
because if I change i to minus i
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00:10:10,000 --> 00:10:16,000
and it turns out,
the expression doesn't change,
148
00:10:17,000 --> 00:10:23,000
then it must have been real,
if the expression doesn't
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00:10:26,000 --> 00:10:32,000
change when I change I to minus
I.
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00:10:33,000 --> 00:10:39,000
Well, sure.
But you will see it works.
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00:10:36,000 --> 00:10:42,000
Now, that's what I'm going to
apply to this.
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00:10:41,000 --> 00:10:47,000
If I want this to be real,
I phrase the question,
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00:10:46,000 --> 00:10:52,000
I rephrase the question for the
green solution as change,
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00:10:52,000 --> 00:10:58,000
so I'm going to change i to
minus i in the green thing,
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00:10:58,000 --> 00:11:04,000
and that's going to give me
what conditions,
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00:11:03,000 --> 00:11:09,000
and that will give conditions
on the C's.
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00:11:09,000 --> 00:11:15,000
Well, let's do it.
In fact, it's easier done than
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00:11:12,000 --> 00:11:18,000
talked about.
Let's change,
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00:11:14,000 --> 00:11:20,000
take the green solution,
and change.
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00:11:16,000 --> 00:11:22,000
Well, I better recopy it,
C1.
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00:11:18,000 --> 00:11:24,000
So, these are complex numbers.
That's why I wrote them as
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00:11:22,000 --> 00:11:28,000
capital letters because little
letters you tend to interpret as
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00:11:27,000 --> 00:11:33,000
real numbers.
So, C1 e to the (a plus b i)t,
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00:11:30,000 --> 00:11:36,000
I'll recopy it quickly,
plus C2 e to the (a minus b i).
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00:11:38,000 --> 00:11:44,000
Okay, we're going to change i
to negative i.
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00:11:41,000 --> 00:11:47,000
Now, here's a complex number.
What happens to it when you
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00:11:46,000 --> 00:11:52,000
change i to negative i?
You change it into its-- Class?
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00:11:51,000 --> 00:11:57,000
What do we change it to?
Its complex conjugate.
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00:11:55,000 --> 00:12:01,000
And, the notation for complex
conjugate is you put a bar over
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00:12:00,000 --> 00:12:06,000
it.
So, in other words,
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00:12:03,000 --> 00:12:09,000
when I do that,
the C1 changes to C1 bar,
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00:12:07,000 --> 00:12:13,000
complex conjugate,
the complex conjugate of C1.
173
00:12:11,000 --> 00:12:17,000
What happens to this guy?
This guy changes to e to the (a
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00:12:17,000 --> 00:12:23,000
minus b i) t.
This changes to the complex
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00:12:22,000 --> 00:12:28,000
conjugate of C2 now,
times e to the (a plus b i) t.
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00:12:27,000 --> 00:12:33,000
Well, I want these two to be
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00:12:32,000 --> 00:12:38,000
the same.
I want the two expressions the
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00:12:36,000 --> 00:12:42,000
same.
Why do I want them the same?
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00:12:40,000 --> 00:12:46,000
Because, if there's no change,
that will mean that it's real.
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00:12:46,000 --> 00:12:52,000
Now, when is that going to
happen?
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00:12:49,000 --> 00:12:55,000
That happens if,
well, here is this,
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00:12:53,000 --> 00:12:59,000
that.
If C2 should be equal to C1
183
00:12:56,000 --> 00:13:02,000
bar, that's only one condition.
There's another condition.
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00:13:02,000 --> 00:13:08,000
C2 bar should equal C1.
So, I get two conditions,
185
00:13:06,000 --> 00:13:12,000
but there's really only one
condition there because if this
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00:13:10,000 --> 00:13:16,000
is true, that's true too.
I simply put bars over both
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00:13:14,000 --> 00:13:20,000
things, and two bars cancel each
other out.
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00:13:18,000 --> 00:13:24,000
If you take the complex
conjugate and do it again,
189
00:13:21,000 --> 00:13:27,000
you get back where you started.
Change i to minus i,
190
00:13:25,000 --> 00:13:31,000
and then i to minus i again.
Well, never mind.
191
00:13:30,000 --> 00:13:36,000
Anyway, these are the same.
This equation doesn't say
192
00:13:36,000 --> 00:13:42,000
anything that the first one
didn't say already.
193
00:13:41,000 --> 00:13:47,000
So, this one is redundant.
And, our conclusion is that the
194
00:13:48,000 --> 00:13:54,000
real solutions to the equation
are, in their entirety,
195
00:13:55,000 --> 00:14:01,000
I now don't need both C2 and
C1.
196
00:14:00,000 --> 00:14:06,000
One of them will do,
and since I'm going to write it
197
00:14:03,000 --> 00:14:09,000
out as a complex number,
I will write it out in terms of
198
00:14:07,000 --> 00:14:13,000
its coefficient.
So, it's C1.
199
00:14:09,000 --> 00:14:15,000
Let's just simply write it.
C plus i times d,
200
00:14:14,000 --> 00:14:20,000
that's the coefficient.
That's what I called C1 before.
201
00:14:18,000 --> 00:14:24,000
And, that's times e to the (a
plus b i) t.
202
00:14:22,000 --> 00:14:28,000
There's no reason why I put bi
here and id there,
203
00:14:25,000 --> 00:14:31,000
in case you're wondering,
sheer caprice.
204
00:14:30,000 --> 00:14:36,000
And what's the other term?
Now, the other term is
205
00:14:34,000 --> 00:14:40,000
completely determined.
Its coefficient must be C minus
206
00:14:38,000 --> 00:14:44,000
i d times e to the
(a minus b i) t.
207
00:14:43,000 --> 00:14:49,000
In other words,
this thing is perfectly
208
00:14:46,000 --> 00:14:52,000
general.
Any complex number times that
209
00:14:49,000 --> 00:14:55,000
first root you use,
exponentiated,
210
00:14:52,000 --> 00:14:58,000
and the second term can be
described as the complex
211
00:14:56,000 --> 00:15:02,000
conjugate of the first.
The coefficient is the complex
212
00:15:03,000 --> 00:15:09,000
conjugate, and this part is the
complex conjugate of that.
213
00:15:10,000 --> 00:15:16,000
Now, it's in this form,
many engineers write the
214
00:15:15,000 --> 00:15:21,000
solution this way,
and physicists,
215
00:15:19,000 --> 00:15:25,000
too, so, scientists and
engineers we will include.
216
00:15:25,000 --> 00:15:31,000
Write the solution this way.
Write the real solutions this
217
00:15:32,000 --> 00:15:38,000
way in that complex form.
Well, why do they do something
218
00:15:35,000 --> 00:15:41,000
so perverse?
You will have to ask them.
219
00:15:38,000 --> 00:15:44,000
But, in fact,
when we studied Fourier series,
220
00:15:41,000 --> 00:15:47,000
we will probably have to do
something, have to do that at
221
00:15:45,000 --> 00:15:51,000
one point.
If you work a lot with complex
222
00:15:48,000 --> 00:15:54,000
numbers, it turns out to be,
in some ways,
223
00:15:51,000 --> 00:15:57,000
a more convenient
representation than the one I've
224
00:15:55,000 --> 00:16:01,000
given you in terms of sines and
cosines.
225
00:15:59,000 --> 00:16:05,000
Well, from this,
how would I get,
226
00:16:01,000 --> 00:16:07,000
suppose I insisted,
well, if someone gave it to me
227
00:16:05,000 --> 00:16:11,000
in that form,
I don't see how I would convert
228
00:16:08,000 --> 00:16:14,000
it back into sines and cosines.
And, I'd like to show you how
229
00:16:13,000 --> 00:16:19,000
to do that efficiently,
too, because,
230
00:16:16,000 --> 00:16:22,000
again, it's one of the
fundamental techniques that I
231
00:16:20,000 --> 00:16:26,000
think you should know.
And, I didn't get a chance to
232
00:16:24,000 --> 00:16:30,000
say it when we studied complex
numbers that first lecture.
233
00:16:28,000 --> 00:16:34,000
It's in the notes,
but it doesn't prove anything
234
00:16:32,000 --> 00:16:38,000
since I don't think it made you
use it in an example.
235
00:16:38,000 --> 00:16:44,000
So, the problem is,
now, by way of finishing this
236
00:16:43,000 --> 00:16:49,000
up, too, to change this to the
old form, I mean the form
237
00:16:50,000 --> 00:16:56,000
involving sines and cosines.
Now, again, there are two ways
238
00:16:56,000 --> 00:17:02,000
of doing it.
The hack way is you write it
239
00:17:01,000 --> 00:17:07,000
all out.
Well, e to the (a plus b i)t
240
00:17:03,000 --> 00:17:09,000
turns into e to the a t times
241
00:17:08,000 --> 00:17:14,000
cosine this plus i sine that.
And, the other term does,
242
00:17:12,000 --> 00:17:18,000
too.
And then you've got stuff out
243
00:17:14,000 --> 00:17:20,000
front.
And, the thing stretches over
244
00:17:17,000 --> 00:17:23,000
two boards.
But you group all the terms
245
00:17:20,000 --> 00:17:26,000
together.
You finally get it.
246
00:17:22,000 --> 00:17:28,000
By the way, when you do it,
you'll find that the imaginary
247
00:17:26,000 --> 00:17:32,000
part disappears completely.
It has to because that's the
248
00:17:31,000 --> 00:17:37,000
way we chose the coefficients.
So, here's the hack method.
249
00:17:37,000 --> 00:17:43,000
Write it all out:
blah, blah, blah,
250
00:17:40,000 --> 00:17:46,000
blah, blah, blah,
blah, and nicer.
251
00:17:42,000 --> 00:17:48,000
Nicer, and teach you something
you're supposed to know.
252
00:17:47,000 --> 00:17:53,000
Write it this way.
First of all,
253
00:17:50,000 --> 00:17:56,000
you notice that both terms have
an e to the a t
254
00:17:55,000 --> 00:18:01,000
factor.
Let's get rid of that right
255
00:17:58,000 --> 00:18:04,000
away.
I'm pulling it out front
256
00:18:02,000 --> 00:18:08,000
because that's automatically
real, and therefore,
257
00:18:06,000 --> 00:18:12,000
isn't going to affect the rest
of the answer at all.
258
00:18:10,000 --> 00:18:16,000
So, let's pull out that,
and what's left?
259
00:18:14,000 --> 00:18:20,000
Well, what's left,
you see, involves just the two
260
00:18:18,000 --> 00:18:24,000
parameters, C and d,
so I'm going to have a C term.
261
00:18:22,000 --> 00:18:28,000
And, I'm going to have a d
term.
262
00:18:25,000 --> 00:18:31,000
What multiplies the arbitrary
constant, C?
263
00:18:30,000 --> 00:18:36,000
Answer: after I remove the e to
the a t,
264
00:18:34,000 --> 00:18:40,000
what multiplies it is,
e to the b i t plus e to the --
265
00:18:39,000 --> 00:18:45,000
e to the b i t.
Let's write it i b t.
266
00:18:43,000 --> 00:18:49,000
And, the other term is plus e
to the negative i b t.
267
00:18:48,000 --> 00:18:54,000
See how I got that,
268
00:18:51,000 --> 00:18:57,000
pulled it out?
And, how about the d?
269
00:18:54,000 --> 00:19:00,000
What goes with d?
d goes with,
270
00:18:57,000 --> 00:19:03,000
well, first of all,
there's an I in front that i
271
00:19:01,000 --> 00:19:07,000
better not forget.
And then, the rest of it is i.
272
00:19:07,000 --> 00:19:13,000
So, it's i d times,
it's e to the b i t,
273
00:19:11,000 --> 00:19:17,000
e to the i b t minus,
now, e to the minus i b t.
274
00:19:16,000 --> 00:19:22,000
So, that's the way the solution
275
00:19:22,000 --> 00:19:28,000
looks.
It doesn't look a lot better,
276
00:19:25,000 --> 00:19:31,000
but now you must use the magic
formulas, which,
277
00:19:30,000 --> 00:19:36,000
I want you to know as well as
you know Euler's formula,
278
00:19:35,000 --> 00:19:41,000
even better than you know
Euler's formula.
279
00:19:41,000 --> 00:19:47,000
They're a consequence of
Euler's formula.
280
00:19:43,000 --> 00:19:49,000
They're Euler's formula read
backwards.
281
00:19:45,000 --> 00:19:51,000
Euler's formula says you've got
a complex exponential here.
282
00:19:49,000 --> 00:19:55,000
Here's how to write it in terms
of sines and cosines.
283
00:19:52,000 --> 00:19:58,000
The backwards thing says you've
got a sine or a cosine.
284
00:19:55,000 --> 00:20:01,000
Here is the way to write it in
terms of complex exponentials.
285
00:20:00,000 --> 00:20:06,000
And, remember,
the way to do it is,
286
00:20:04,000 --> 00:20:10,000
cosine a is equal to e to the i
a t, i a, plus e to the negative
287
00:20:11,000 --> 00:20:17,000
i a divided by two.
288
00:20:17,000 --> 00:20:23,000
And, sine of a is almost the
same thing, except you use a
289
00:20:24,000 --> 00:20:30,000
minus sign.
And, what everybody forgets,
290
00:20:28,000 --> 00:20:34,000
you have to divide by i.
So, this is a backwards version
291
00:20:35,000 --> 00:20:41,000
of Euler's formula.
The two of them taken together
292
00:20:38,000 --> 00:20:44,000
are equivalent to Euler's
formula.
293
00:20:40,000 --> 00:20:46,000
If I took cosine a,
multiply this through by i,
294
00:20:44,000 --> 00:20:50,000
and added them up,
on the right-hand side I'd get
295
00:20:47,000 --> 00:20:53,000
exactly e to the ia.
I'd get Euler's formula,
296
00:20:51,000 --> 00:20:57,000
in other words.
All right, so,
297
00:20:53,000 --> 00:20:59,000
what does this come out to be,
finally?
298
00:20:55,000 --> 00:21:01,000
This particular sum of
exponentials,
299
00:20:58,000 --> 00:21:04,000
you should always recognize as
real.
300
00:21:02,000 --> 00:21:08,000
You know it's real because when
you change i to minus i,
301
00:21:07,000 --> 00:21:13,000
the two terms switch.
And therefore,
302
00:21:10,000 --> 00:21:16,000
the expression doesn't change.
What is it?
303
00:21:14,000 --> 00:21:20,000
This part is twice the cosine
of bt.
304
00:21:18,000 --> 00:21:24,000
What's this part?
This part is 2 i times the sine
305
00:21:22,000 --> 00:21:28,000
of bt.
And so, what does the whole
306
00:21:27,000 --> 00:21:33,000
thing come to be?
It is e to the a t times 2C
307
00:21:33,000 --> 00:21:39,000
cosine bt plus i times,
did I lose possibly a,
308
00:21:38,000 --> 00:21:44,000
no it's okay,
minus i times i is minus,
309
00:21:43,000 --> 00:21:49,000
so, minus 2d times the sine of
bt.
310
00:21:50,000 --> 00:21:56,000
Shall I write that out?
311
00:21:55,000 --> 00:22:01,000
So, in other words,
it's e to the a t times 2C
312
00:21:58,000 --> 00:22:04,000
cosine b t minus 2d times the
sine of b t,
313
00:22:03,000 --> 00:22:09,000
which is, since 2C and negative
314
00:22:07,000 --> 00:22:13,000
2d are just arbitrary constants,
just as arbitrary as the
315
00:22:12,000 --> 00:22:18,000
constants of C and d themselves
are.
316
00:22:15,000 --> 00:22:21,000
This is our old form of writing
the real solution.
317
00:22:19,000 --> 00:22:25,000
Here's the way using science
and cosines, and there's the way
318
00:22:24,000 --> 00:22:30,000
that uses complex numbers and
complex functions throughout.
319
00:22:30,000 --> 00:22:36,000
Notice they both have two
arbitrary constants in them,
320
00:22:33,000 --> 00:22:39,000
C and d, two arbitrary
constants.
321
00:22:36,000 --> 00:22:42,000
That, you expect.
But that has two arbitrary
322
00:22:39,000 --> 00:22:45,000
constants in it,
too, just the real and
323
00:22:42,000 --> 00:22:48,000
imaginary parts of that complex
coefficient, C plus i d.
324
00:22:46,000 --> 00:22:52,000
Well, that took half the
325
00:22:48,000 --> 00:22:54,000
period, and it was a long,
I don't consider it a
326
00:22:52,000 --> 00:22:58,000
digression because learning
those ways of dealing with
327
00:22:56,000 --> 00:23:02,000
complex numbers of complex
functions is a fairly important
328
00:23:00,000 --> 00:23:06,000
goal in this course,
actually.
329
00:23:04,000 --> 00:23:10,000
But let's get back now to
studying what the oscillations
330
00:23:07,000 --> 00:23:13,000
actually look like.
331
00:23:27,000 --> 00:23:33,000
Okay, well, I'd like to save a
little time, but very quickly,
332
00:23:34,000 --> 00:23:40,000
you don't have to reproduce
this sketch.
333
00:23:39,000 --> 00:23:45,000
I remember very well from
Friday to Monday,
334
00:23:45,000 --> 00:23:51,000
but I can't expect you to for a
variety of reasons.
335
00:23:51,000 --> 00:23:57,000
I mean, I have to think about
this stuff all weekend.
336
00:23:58,000 --> 00:24:04,000
And you, God forbid.
So, here's the picture,
337
00:24:03,000 --> 00:24:09,000
and I won't explain anymore
what's in it,
338
00:24:06,000 --> 00:24:12,000
except there's the mass.
Here is the spring constant,
339
00:24:09,000 --> 00:24:15,000
the spring with its constant
here.
340
00:24:11,000 --> 00:24:17,000
Here's the dashpot with its
constant.
341
00:24:13,000 --> 00:24:19,000
The equation is from Newton's
law: m x double,
342
00:24:16,000 --> 00:24:22,000
so this will be x,
and here's, let's say,
343
00:24:19,000 --> 00:24:25,000
the equilibrium point is over
here.
344
00:24:21,000 --> 00:24:27,000
It looks like m x double prime;
we derived this last time,
345
00:24:25,000 --> 00:24:31,000
plus c x prime plus k x equals
zero.
346
00:24:30,000 --> 00:24:36,000
And now, if I put that in
standard form,
347
00:24:33,000 --> 00:24:39,000
it's going to look like x
double prime plus c over m x
348
00:24:39,000 --> 00:24:45,000
prime plus k over m times x
equals zero.
349
00:24:45,000 --> 00:24:51,000
And, finally,
350
00:24:47,000 --> 00:24:53,000
the standard form in which your
book writes it,
351
00:24:52,000 --> 00:24:58,000
which is good,
it's a standard form in general
352
00:24:56,000 --> 00:25:02,000
that is used in the science and
engineering courses.
353
00:25:02,000 --> 00:25:08,000
One writes this as,
just to be perverse,
354
00:25:05,000 --> 00:25:11,000
I'm going to change x back to
y, okay, mostly just to be
355
00:25:11,000 --> 00:25:17,000
eclectic, to get you used to
every conceivable notation.
356
00:25:19,000 --> 00:25:25,000
So, I'm going to write this to
change x to y.
357
00:25:22,000 --> 00:25:28,000
So, that's going to become y
double prime.
358
00:25:26,000 --> 00:25:32,000
And now, this is given a new
name, p, except to get rid of
359
00:25:30,000 --> 00:25:36,000
lots of twos,
which would really screw up the
360
00:25:33,000 --> 00:25:39,000
formulas, make it 2p.
You will see why in a minute.
361
00:25:38,000 --> 00:25:44,000
So, there's 2p times y prime,
and this thing we
362
00:25:42,000 --> 00:25:48,000
are going to call omega nought
squared.
363
00:25:46,000 --> 00:25:52,000
Now, that's okay.
It's a positive number.
364
00:25:49,000 --> 00:25:55,000
Any positive number is the
square of some other positive
365
00:25:53,000 --> 00:25:59,000
number.
Take a square root.
366
00:25:55,000 --> 00:26:01,000
You will see why,
it makes the formulas much
367
00:25:59,000 --> 00:26:05,000
pretty to call it that.
And, it makes also a lot of
368
00:26:04,000 --> 00:26:10,000
things much easier to remember.
So, all I'm doing is changing
369
00:26:08,000 --> 00:26:14,000
the names of the constants that
way in order to get better
370
00:26:13,000 --> 00:26:19,000
formulas, easier to remember
formulas at the end.
371
00:26:16,000 --> 00:26:22,000
Now, we are interested in the
case where there is
372
00:26:20,000 --> 00:26:26,000
oscillations.
In other words,
373
00:26:22,000 --> 00:26:28,000
I only care about the case in
which this has complex roots,
374
00:26:27,000 --> 00:26:33,000
because if it has just real
roots, that's the over-damped
375
00:26:31,000 --> 00:26:37,000
case.
I don't get any oscillations.
376
00:26:35,000 --> 00:26:41,000
By far, oscillations are by far
the more important of the cases,
377
00:26:40,000 --> 00:26:46,000
I mean, just because,
I don't know,
378
00:26:43,000 --> 00:26:49,000
I could go on for five minutes
listing things that oscillate,
379
00:26:48,000 --> 00:26:54,000
oscillations,
you know, like this.
380
00:26:51,000 --> 00:26:57,000
So they can oscillate by going
to sleep, and waking up,
381
00:26:56,000 --> 00:27:02,000
and going to sleep,
and waking up.
382
00:26:59,000 --> 00:27:05,000
They could oscillate.
So, that means we're going to
383
00:27:03,000 --> 00:27:09,000
get complex roots.
The characteristic equation is
384
00:27:07,000 --> 00:27:13,000
going to be r squared plus 2p.
So, p is a constant,
385
00:27:10,000 --> 00:27:16,000
now, right?
Often, p I use in this position
386
00:27:13,000 --> 00:27:19,000
to indicate a function of t.
But here, p is a constant.
387
00:27:16,000 --> 00:27:22,000
So, r squared plus 2p times r
plus omega nought squared is
388
00:27:20,000 --> 00:27:26,000
equal to zero.
389
00:27:23,000 --> 00:27:29,000
Now, what are its roots?
Well, you see right away the
390
00:27:27,000 --> 00:27:33,000
first advantage in putting in
the two there.
391
00:27:31,000 --> 00:27:37,000
When I use the quadratic
formula, it's negative 2p over
392
00:27:34,000 --> 00:27:40,000
two.
Remember that two in the
393
00:27:37,000 --> 00:27:43,000
denominator.
So, that's simply negative p.
394
00:27:40,000 --> 00:27:46,000
And, how about the rest?
Plus or minus the square root
395
00:27:44,000 --> 00:27:50,000
of, now do it in your head.
4p squared minus 4 omega nought
396
00:27:48,000 --> 00:27:54,000
squared.
So, there's a four in both of
397
00:27:53,000 --> 00:27:59,000
those terms.
When I pull it outside becomes
398
00:27:56,000 --> 00:28:02,000
two.
And, the two in the denominator
399
00:27:59,000 --> 00:28:05,000
is lurking, waiting to
annihilate it.
400
00:28:03,000 --> 00:28:09,000
So, that two disappears
entirely, and it will we are
401
00:28:06,000 --> 00:28:12,000
left with is,
simply, p squared minus omega
402
00:28:09,000 --> 00:28:15,000
nought squared.
403
00:28:11,000 --> 00:28:17,000
Now, whenever people write
quadratic equations,
404
00:28:14,000 --> 00:28:20,000
and arbitrarily put a two in
there, it's because they were
405
00:28:18,000 --> 00:28:24,000
going to want to solve the
quadratic equation using the
406
00:28:21,000 --> 00:28:27,000
quadratic formula,
and they don't want all those
407
00:28:24,000 --> 00:28:30,000
twos and fours to be cluttering
up the formula.
408
00:28:29,000 --> 00:28:35,000
That's what we are doing here.
Okay, now, the first case is
409
00:28:33,000 --> 00:28:39,000
where p is equal to zero.
This is going to explain
410
00:28:37,000 --> 00:28:43,000
immediately why I wrote that
omega nought squared,
411
00:28:41,000 --> 00:28:47,000
as you probably already know
from physics.
412
00:28:44,000 --> 00:28:50,000
If p is equal to zero,
the mass isn't zero.
413
00:28:48,000 --> 00:28:54,000
Otherwise, nothing good would
be happening here.
414
00:28:52,000 --> 00:28:58,000
It must be that the damping is
zero.
415
00:28:55,000 --> 00:29:01,000
So, p is equal to zero
corresponds to undamped.
416
00:29:00,000 --> 00:29:06,000
There is no dashpot.
The oscillations are undamped.
417
00:29:03,000 --> 00:29:09,000
And, the equation,
then, becomes the solutions,
418
00:29:06,000 --> 00:29:12,000
then, are, well,
the equation becomes the
419
00:29:09,000 --> 00:29:15,000
equation of simple harmonic
motion, which,
420
00:29:12,000 --> 00:29:18,000
I think you already are used to
writing in this form.
421
00:29:15,000 --> 00:29:21,000
And, the reason you're writing
in this form because you know
422
00:29:19,000 --> 00:29:25,000
when you do that,
this becomes the circular
423
00:29:22,000 --> 00:29:28,000
frequency of the oscillations.
The solutions are pure
424
00:29:26,000 --> 00:29:32,000
oscillations,
and omega nought is
425
00:29:29,000 --> 00:29:35,000
the circular frequency.
So, right away from the
426
00:29:33,000 --> 00:29:39,000
equation itself,
if you write it in this form,
427
00:29:37,000 --> 00:29:43,000
you can read off what the
frequency of the solutions is
428
00:29:41,000 --> 00:29:47,000
going to be, the circular
frequency of the solutions.
429
00:29:45,000 --> 00:29:51,000
Now, the solutions themselves,
of course, look like,
430
00:29:49,000 --> 00:29:55,000
the general solutions look like
y equal, in this particular
431
00:29:54,000 --> 00:30:00,000
case, the p part is zero.
This is zero.
432
00:29:57,000 --> 00:30:03,000
It's simply,
so, in this case,
433
00:29:59,000 --> 00:30:05,000
r is equal to omega nought i
times omega naught plus or
434
00:30:03,000 --> 00:30:09,000
minus, but as before we don't
bother with the minus sign since
435
00:30:08,000 --> 00:30:14,000
one of those roots is good
enough.
436
00:30:13,000 --> 00:30:19,000
And then, the solutions are
simply c1 cosine omega nought t
437
00:30:16,000 --> 00:30:22,000
plus c2 sine omega nought t.
438
00:30:20,000 --> 00:30:26,000
That's if you write it out in
439
00:30:23,000 --> 00:30:29,000
the sign, and if you write it
using the trigonometric
440
00:30:26,000 --> 00:30:32,000
identity, then the other way of
writing it is a times the cosine
441
00:30:30,000 --> 00:30:36,000
of omega nought t.
442
00:30:34,000 --> 00:30:40,000
But now you will have to put it
a phase lag.
443
00:30:37,000 --> 00:30:43,000
So, you have those two forms of
writing it.
444
00:30:41,000 --> 00:30:47,000
And, I assume you remember
writing the little triangle,
445
00:30:45,000 --> 00:30:51,000
which converts one into the
other.
446
00:30:48,000 --> 00:30:54,000
Okay, so this justifies calling
this omega nought squared
447
00:30:53,000 --> 00:30:59,000
rather than k over m.
448
00:30:56,000 --> 00:31:02,000
And now, the question is what
does the damp case look like?
449
00:31:01,000 --> 00:31:07,000
It requires a somewhat closer
analysis, and it requires a
450
00:31:06,000 --> 00:31:12,000
certain amount of thinking.
So, let's begin with an epsilon
451
00:31:13,000 --> 00:31:19,000
bit of thinking.
So, here's my question.
452
00:31:18,000 --> 00:31:24,000
So, in the damped case,
I want to be sure that I'm
453
00:31:24,000 --> 00:31:30,000
getting oscillations.
When do I get oscillations if,
454
00:31:30,000 --> 00:31:36,000
well, we get oscillations if
those roots are really complex,
455
00:31:37,000 --> 00:31:43,000
and not masquerading.
Now, when are the roots going
456
00:31:43,000 --> 00:31:49,000
to be really complex?
This has to be,
457
00:31:46,000 --> 00:31:52,000
the inside has to be negative.
p squared minus omega squared
458
00:31:52,000 --> 00:31:58,000
must be negative.
459
00:31:56,000 --> 00:32:02,000
p squared minus omega nought
squared must be less than zero
460
00:32:01,000 --> 00:32:07,000
so that we are taking a square
root of negative number,
461
00:32:06,000 --> 00:32:12,000
and we are getting a real
complex roots,
462
00:32:09,000 --> 00:32:15,000
really complex roots.
In other words,
463
00:32:14,000 --> 00:32:20,000
now, this says,
remember these numbers are all
464
00:32:17,000 --> 00:32:23,000
positive, p and omega nought are
positive.
465
00:32:21,000 --> 00:32:27,000
So, the condition is that p
should be
466
00:32:25,000 --> 00:32:31,000
less than omega nought.
In other words,
467
00:32:28,000 --> 00:32:34,000
the damping should be less than
the circular frequency,
468
00:32:32,000 --> 00:32:38,000
except p is not the damping.
It's half the damping,
469
00:32:38,000 --> 00:32:44,000
and it's not really the damping
either because it involved the
470
00:32:43,000 --> 00:32:49,000
m, too.
You'd better just call it p.
471
00:32:47,000 --> 00:32:53,000
Naturally, I could write the
condition out in terms of c,
472
00:32:52,000 --> 00:32:58,000
m, and k.
So, your book does that,
473
00:32:55,000 --> 00:33:01,000
but I'm not going to.
It gives it in terms of c,
474
00:32:59,000 --> 00:33:05,000
m, and k, which somebody might
want to know.
475
00:33:03,000 --> 00:33:09,000
But, you know,
we don't have to do everything
476
00:33:08,000 --> 00:33:14,000
here.
Okay, so let's assume that this
477
00:33:12,000 --> 00:33:18,000
is true.
What is the solution look like?
478
00:33:15,000 --> 00:33:21,000
Well, we already experimented
with that last time.
479
00:33:19,000 --> 00:33:25,000
Remember, there was some
guiding thing which was an
480
00:33:23,000 --> 00:33:29,000
exponential.
And then, down here,
481
00:33:26,000 --> 00:33:32,000
we wrote the negative.
So, this was an exponential.
482
00:33:31,000 --> 00:33:37,000
In fact, it was the
exponential, e to the negative
483
00:33:35,000 --> 00:33:41,000
pt.
And, in between that,
484
00:33:38,000 --> 00:33:44,000
the curve tried to do its
thing.
485
00:33:41,000 --> 00:33:47,000
So, the solution looks sort of
like this.
486
00:33:45,000 --> 00:33:51,000
It oscillated,
but it had to use that
487
00:33:48,000 --> 00:33:54,000
exponential function as its
guidelines, as its amplitude,
488
00:33:53,000 --> 00:33:59,000
in other words.
Now, this is a truly terrible
489
00:33:57,000 --> 00:34:03,000
picture.
It's so terrible,
490
00:34:01,000 --> 00:34:07,000
it's unusable.
Okay, this picture never
491
00:34:05,000 --> 00:34:11,000
happened.
Unfortunately,
492
00:34:07,000 --> 00:34:13,000
this is not my forte along with
a lot of other things.
493
00:34:12,000 --> 00:34:18,000
All right, let's try it better.
Here's our better picture.
494
00:34:18,000 --> 00:34:24,000
Okay, there's the exponential.
At this point,
495
00:34:22,000 --> 00:34:28,000
I'm supposed to have a lecture
demonstration.
496
00:34:26,000 --> 00:34:32,000
It's supposed to go up on the
thing, so you can all see it.
497
00:34:34,000 --> 00:34:40,000
But then, you wouldn't be able
to copy it.
498
00:34:37,000 --> 00:34:43,000
So, at least we are on even
terms now.
499
00:34:40,000 --> 00:34:46,000
Okay, how does the actual curve
look?
500
00:34:43,000 --> 00:34:49,000
Well, I'm just trying to be
fair.
501
00:34:45,000 --> 00:34:51,000
That's all.
Okay, after a while,
502
00:34:48,000 --> 00:34:54,000
the point is,
just so we have something to
503
00:34:51,000 --> 00:34:57,000
aim at, let's say,
okay, here we are going to go,
504
00:34:55,000 --> 00:35:01,000
we're going to get down through
there.
505
00:35:00,000 --> 00:35:06,000
Okay then, this is our better
curve.
506
00:35:03,000 --> 00:35:09,000
Okay, so I am a solution,
a particular solution
507
00:35:08,000 --> 00:35:14,000
satisfying this initial
condition.
508
00:35:12,000 --> 00:35:18,000
I started here,
and that was my initial
509
00:35:16,000 --> 00:35:22,000
velocity.
The slope of that thing gave me
510
00:35:20,000 --> 00:35:26,000
the initial velocity.
Now, the interesting question
511
00:35:26,000 --> 00:35:32,000
is, the first,
in some ways,
512
00:35:28,000 --> 00:35:34,000
the most interesting question,
though there will be others,
513
00:35:35,000 --> 00:35:41,000
too, is what is this spacing?
Well, that's a period.
514
00:35:42,000 --> 00:35:48,000
And now, it's half a period.
I clearly ought to think of
515
00:35:47,000 --> 00:35:53,000
this as the whole period.
So, let's call that,
516
00:35:51,000 --> 00:35:57,000
I'm going to call this pi over,
so this spacing here,
517
00:35:56,000 --> 00:36:02,000
from there to there,
I will call that pi divided by
518
00:36:01,000 --> 00:36:07,000
omega one because this,
from here to here,
519
00:36:05,000 --> 00:36:11,000
should be, I hope,
twice that, two pi over omega
520
00:36:10,000 --> 00:36:16,000
one.
Now, my question is,
521
00:36:14,000 --> 00:36:20,000
so this, for a solution,
it's, in fact,
522
00:36:18,000 --> 00:36:24,000
is going to cross the axis
regularly in that way.
523
00:36:24,000 --> 00:36:30,000
My question is,
how does this period,
524
00:36:28,000 --> 00:36:34,000
so this is going to be its half
period.
525
00:36:34,000 --> 00:36:40,000
I will put period in quotation
marks because this isn't really
526
00:36:39,000 --> 00:36:45,000
a periodic function because it's
decreasing all the time in
527
00:36:43,000 --> 00:36:49,000
amplitude.
But, it's trying to be
528
00:36:46,000 --> 00:36:52,000
periodic.
At lease it's doing something
529
00:36:49,000 --> 00:36:55,000
periodically.
It's crossing the axis
530
00:36:52,000 --> 00:36:58,000
periodically.
So, this is the half period.
531
00:36:55,000 --> 00:37:01,000
Two pi over omega one
would be its full
532
00:37:00,000 --> 00:37:06,000
period.
What I want to know is,
533
00:37:02,000 --> 00:37:08,000
how does that half period,
or how does-- omega one is
534
00:37:07,000 --> 00:37:13,000
called its pseudo-frequency.
This should really be called
535
00:37:13,000 --> 00:37:19,000
its pseudo-period.
Everything is pseudo.
536
00:37:16,000 --> 00:37:22,000
Everything is fake here.
Like, the amoeba has its fake
537
00:37:21,000 --> 00:37:27,000
foot and stuff like that.
Okay, so this is its
538
00:37:24,000 --> 00:37:30,000
pseudo-period,
pseudo-frequency,
539
00:37:27,000 --> 00:37:33,000
pseudo-circular frequency,
but that's hopeless.
540
00:37:31,000 --> 00:37:37,000
I guess it should be circular
pseudo-frequency,
541
00:37:35,000 --> 00:37:41,000
or I don't know how you say
that.
542
00:37:39,000 --> 00:37:45,000
I don't think pseudo is a word
all by itself,
543
00:37:46,000 --> 00:37:52,000
not even in 18.03,
circular.
544
00:37:50,000 --> 00:37:56,000
Okay, here's my question.
If the damping goes up,
545
00:37:58,000 --> 00:38:04,000
this is the damping term.
If the damping goes up,
546
00:38:06,000 --> 00:38:12,000
what happens to the
pseudo-frequency?
547
00:38:11,000 --> 00:38:17,000
The frequency is how often the
curve, this is high-frequency,
548
00:38:19,000 --> 00:38:25,000
and this is low-frequency,
okay?
549
00:38:23,000 --> 00:38:29,000
So, my question is,
which way does the frequency
550
00:38:29,000 --> 00:38:35,000
go?
If the damping goes up,
551
00:38:33,000 --> 00:38:39,000
does the frequency go up or
down?
552
00:38:38,000 --> 00:38:44,000
Down.
I mean, I'm just asking you to
553
00:38:42,000 --> 00:38:48,000
answer intuitively on the basis
of your intuition about how this
554
00:38:50,000 --> 00:38:56,000
thing explains,
how this thing goes,
555
00:38:55,000 --> 00:39:01,000
and now let's get the formula.
What, in fact,
556
00:39:01,000 --> 00:39:07,000
is omega one?
What is omega one?
557
00:39:05,000 --> 00:39:11,000
The answer is when I solve the
equation, so,
558
00:39:09,000 --> 00:39:15,000
r is now, so in other words,
if omega one is,
559
00:39:13,000 --> 00:39:19,000
sorry, if I have p,
if p is no longer zero as it
560
00:39:18,000 --> 00:39:24,000
was in the undamped case,
what is the root,
561
00:39:22,000 --> 00:39:28,000
now?
Okay, well, the root is minus p
562
00:39:25,000 --> 00:39:31,000
plus or minus the square root of
p squared, --
563
00:39:31,000 --> 00:39:37,000
-- now I'm going to write it
this way, minus,
564
00:39:34,000 --> 00:39:40,000
to indicate that it's really a
negative number,
565
00:39:38,000 --> 00:39:44,000
omega squared minus p squared.
566
00:39:42,000 --> 00:39:48,000
Now, I'm going to call this,
because you see when I change
567
00:39:47,000 --> 00:39:53,000
this to sines and cosines,
the square root of this number
568
00:39:52,000 --> 00:39:58,000
is what's going to become that
new frequency.
569
00:39:55,000 --> 00:40:01,000
I'm going to call that minus p
plus or minus the square root of
570
00:40:00,000 --> 00:40:06,000
minus omega one squared.
That's going to be the new
571
00:40:06,000 --> 00:40:12,000
frequency.
And therefore,
572
00:40:08,000 --> 00:40:14,000
the root is going to change so
that the corresponding solution
573
00:40:13,000 --> 00:40:19,000
is going to look,
how?
574
00:40:15,000 --> 00:40:21,000
Well, it's going to be e to the
negative pt times,
575
00:40:20,000 --> 00:40:26,000
let's write it out first in
terms of sines and cosines,
576
00:40:25,000 --> 00:40:31,000
times the cosine of,
well, the square root of omega
577
00:40:29,000 --> 00:40:35,000
one squared is omega one.
578
00:40:35,000 --> 00:40:41,000
But, there's an i out front
because of the negative sign in
579
00:40:39,000 --> 00:40:45,000
front of that.
So, it's going to be the cosine
580
00:40:42,000 --> 00:40:48,000
of omega one t
plus c2 times the sine of omega
581
00:40:47,000 --> 00:40:53,000
one t.
Or, if you prefer to write it
582
00:40:51,000 --> 00:40:57,000
out in the other form,
it's e to the minus p t times
583
00:40:54,000 --> 00:41:00,000
some amplitude,
which depends on c1 and c2,
584
00:40:57,000 --> 00:41:03,000
times the cosine of omega one t
minus the phase lag.
585
00:41:02,000 --> 00:41:08,000
Now, when I do that,
586
00:41:07,000 --> 00:41:13,000
you see omega one is
this pseudo-frequency.
587
00:41:13,000 --> 00:41:19,000
In other words,
this number omega one is the
588
00:41:17,000 --> 00:41:23,000
same one that I identified here.
And, why is that?
589
00:41:23,000 --> 00:41:29,000
Well, because,
what are two successive times?
590
00:41:27,000 --> 00:41:33,000
Suppose it crosses,
suppose the solution crosses
591
00:41:33,000 --> 00:41:39,000
the x-axis, sorry,
y-- the t-axis.
592
00:41:38,000 --> 00:41:44,000
For the first time,
at the point t1,
593
00:41:42,000 --> 00:41:48,000
what's the next time it crosses
t2?
594
00:41:46,000 --> 00:41:52,000
Let's jump to the two times
across it.
595
00:41:51,000 --> 00:41:57,000
So, I want this to be a whole
period, not a half period.
596
00:41:57,000 --> 00:42:03,000
What's t2?
Well, I say that t2 is nothing
597
00:42:02,000 --> 00:42:08,000
but 2 pi divided by omega one.
598
00:42:07,000 --> 00:42:13,000
And, you can see that because
when I plug in,
599
00:42:10,000 --> 00:42:16,000
if it's zero,
if I have a point where it's
600
00:42:14,000 --> 00:42:20,000
zero, so, omega one t minus phi,
601
00:42:18,000 --> 00:42:24,000
when will it be zero for the
first time?
602
00:42:22,000 --> 00:42:28,000
Well, that will be when the
cosine has to be zero.
603
00:42:26,000 --> 00:42:32,000
So, it will be some multiple
of, it will be,
604
00:42:30,000 --> 00:42:36,000
say, pi over two.
Then, the next time this
605
00:42:35,000 --> 00:42:41,000
happens will be,
if that happens at t1,
606
00:42:39,000 --> 00:42:45,000
then the next time it happens
will be at t1 plus 2 pi divided
607
00:42:45,000 --> 00:42:51,000
by omega one.
608
00:42:49,000 --> 00:42:55,000
That will also be pi over two
plus how much?
609
00:42:54,000 --> 00:43:00,000
Plus 2 pi, which is the next
time the cosine gets around and
610
00:43:00,000 --> 00:43:06,000
is doing its thing,
becoming zero as it goes down,
611
00:43:05,000 --> 00:43:11,000
not as it's coming up again.
In other words,
612
00:43:11,000 --> 00:43:17,000
this is what you should add to
the first time to get this
613
00:43:17,000 --> 00:43:23,000
second time that the cosine
becomes zero coming in the
614
00:43:23,000 --> 00:43:29,000
direction from top to the
bottom.
615
00:43:26,000 --> 00:43:32,000
So, this is,
in fact, the frequency with
616
00:43:30,000 --> 00:43:36,000
which it's crossing the axis.
Now, notice,
617
00:43:36,000 --> 00:43:42,000
I'm running out of boards.
What a disaster!
618
00:43:41,000 --> 00:43:47,000
In that expression,
take a look at it.
619
00:43:46,000 --> 00:43:52,000
I want to know what depends on
what.
620
00:43:50,000 --> 00:43:56,000
So, p, in that,
we got constants.
621
00:43:54,000 --> 00:44:00,000
We got p.
We got phi.
622
00:43:57,000 --> 00:44:03,000
We got A.
What else we got?
623
00:44:00,000 --> 00:44:06,000
Omega one.
What do these things depend
624
00:44:07,000 --> 00:44:13,000
upon?
You've got to keep it firmly in
625
00:44:11,000 --> 00:44:17,000
mind.
This depends only on the ODE.
626
00:44:14,000 --> 00:44:20,000
It's basically the damping.
It depends on c and m.
627
00:44:19,000 --> 00:44:25,000
Essentially, it's c over 2m
628
00:44:23,000 --> 00:44:29,000
actually.
How about phi?
629
00:44:25,000 --> 00:44:31,000
Well, phi, what else depends
only on the ODE?
630
00:44:30,000 --> 00:44:36,000
Omega one depends
only on the ODE.
631
00:44:36,000 --> 00:44:42,000
What's the formula for omega
one?
632
00:44:38,000 --> 00:44:44,000
Omega one squared.
633
00:44:40,000 --> 00:44:46,000
Where do we have it?
Omega one squared,
634
00:44:43,000 --> 00:44:49,000
I never wrote the formula for
you.
635
00:44:46,000 --> 00:44:52,000
So, we have omega nought
squared minus p squared equals
636
00:44:50,000 --> 00:44:56,000
omega one squared.
637
00:44:53,000 --> 00:44:59,000
What's the relation between
them?
638
00:44:56,000 --> 00:45:02,000
That's the Pythagorean theorem.
If this is omega nought,
639
00:45:00,000 --> 00:45:06,000
then this omega one,
this is p.
640
00:45:04,000 --> 00:45:10,000
They make a little,
right triangle in other words.
641
00:45:09,000 --> 00:45:15,000
The omega one depends on the
spring.
642
00:45:13,000 --> 00:45:19,000
So, it's equal to,
well, it's equal to that thing.
643
00:45:19,000 --> 00:45:25,000
So, it depends on the damping.
It depends upon the damping,
644
00:45:25,000 --> 00:45:31,000
and it depends on the spring
constant.
645
00:45:30,000 --> 00:45:36,000
How about the phi and the A?
What do they depend on?
646
00:45:36,000 --> 00:45:42,000
They depend upon the initial
conditions.
647
00:45:42,000 --> 00:45:48,000
So, the mass of constants,
they have different functions.
648
00:45:47,000 --> 00:45:53,000
What's making this complicated
is that our answer needs four
649
00:45:53,000 --> 00:45:59,000
parameters to describe it.
This tells you how fast it's
650
00:45:59,000 --> 00:46:05,000
coming down.
This tells you the phase lag.
651
00:46:03,000 --> 00:46:09,000
This amplitude modifies,
it tells you whether the
652
00:46:08,000 --> 00:46:14,000
exponential curve starts going,
is like that or goes like this.
653
00:46:15,000 --> 00:46:21,000
And, finally,
the omega one is this
654
00:46:18,000 --> 00:46:24,000
pseudo-frequency,
which tells you how it's
655
00:46:22,000 --> 00:46:28,000
bobbing up and down.