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I just recalling some of the
notation we are going to need

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for today, and a couple of the
facts that we're going to use,

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00:00:18,000 --> 00:00:24,000
plus trying to clear up a
couple of confusions that the

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00:00:23,000 --> 00:00:29,000
recitations report.
This can be thought of two

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ways.
It's a formal polynomial in D,

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in the letter D.
It just has the shape of the

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polynomial, D squared plus AD
plus B.

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A and B are constant
coefficients.

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But, it's also,
at the same time,

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if you think what it does,
it's a linear operator on

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functions.
It's a linear operator on

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functions like y of t.
You think of it both ways:

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formal polynomial because we
want to do things like factoring

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it, substituting two for D and
things like that.

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Those are things you do with
polynomials.

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You do them algebraically.
You can take the formal

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derivative of the polynomial
because it's just sums of

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powers.
On the other hand,

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as a linear operator,
it does something to functions.

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It differentiates them,
multiplies them by constants or

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something like that.
So it's, so to speak,

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has a dual aspect this way.
And, that's one of the things

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we are exploiting what we use
operator methods to solve

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differential equations.
Now, let me remind you of the

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key thing we were interested in.
f of t:

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not any old function,
we'll get to that next time,

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but f of t,
exponentials.

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So, it should be an exponential
or something like an

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exponential, or pretty close to
it, for example,

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something with sine t
and cosine t,

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or e to the,
that could be thought of as

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part of the real or imaginary
part of a complex exponential.

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And, maybe by the end of today,
we will have generalized that

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even little more.
But basically,

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I'm interested in exponentials.
Let's make it alpha complex.

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That will at least take care of
the cases, e to the ax times

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cosine bx, sin bx,

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which are the main cases.
Those are the main cases.

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Then, remember the little table
we made.

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I simply gave you the formula
for the particular solution.

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So, what we're looking for is
we already know how to solve the

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homogeneous equation.
What we want is that particular

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solution.
And then, the recipe for it I

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gave you, these things were
proved by the substitution rules

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and exponential shift rules.
The recipe was that if f of t

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was, let's make a
little table.

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f of t is, well,
it's always e to the a t.

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So, in other words,

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it's e to the a t.
The cases are,

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so yp, what is the yp?
Well, it is the normal case is

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yp equals e to that alpha t
divided by the

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00:03:17,000 --> 00:03:23,000
polynomial where you substitute,
you take that polynomial,

53
00:03:21,000 --> 00:03:27,000
and wherever you see a D,
you substitute the complex

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number, alpha.
There, I'm thinking of it as a

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formal polynomial.
I'm not thinking of it as an

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operator.
Now, this breaks down.

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So, that's the formula for the
particular solution.

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The only trouble is,
it breaks down if p of alpha is

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zero.
So, we have to assume that it's

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not.
Now, if p of alpha is zero,

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that means alpha is a root of
the polynomial,

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a zero of the polynomial is a
better word.

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So, in that case,
it will be e to the alpha t

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divided by p prime of alpha.

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Differentiate formally the
polynomials, --

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-- and you will get 2D plus A.

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And now, substitute in the
alpha.

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And, this will be okay provided
p prime of alpha

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is not zero.
That means that alpha is the

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simple root, simple zero of p.
And then, there's one more

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case, which, since I won't need
today, I won't write on the

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board.
But, you'll need it for

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homework.
So, make sure you know it.

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Another words,
if this is zero,

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then you've got a double root.
And, there is still a different

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formula.
And, this is wrong because I

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forgot the t.
Yes?

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I could tell on your faces.
That was before,

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and now we are up to today.
What we are interested in

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talking about today is what this
has to do with the phenomenon of

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resonance.
Everybody knows at least one

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case of resonance,
I hope.

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A little kid is on his swing,
right?

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Back and forth,
and they are very,

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very little,
so they want a push.

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Okay, well, everybody knows
that to make the swing go,

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a swing has a certain natural
frequency.

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It swings back and forth like
that.

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It's a simple pendulum.
It's actually damped,

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00:05:34,000 --> 00:05:40,000
but let's pretend that it
isn't.

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Everybody knows you want to
push a kid on a swing so that

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they go high.
You have to push with

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essentially the same frequency
that the natural frequency of

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the spring, of the swing is.
It's automatic,

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because when you come back
here, it gets to there,

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and that's where you push.
So, automatically,

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you time your pushes.
But if you want the kid to

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stop, you just do the opposite.
Push at the wrong time.

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So anyway, that's resonance.
Of course, there are more

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serious applications of it.
It's what made the Tacoma

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Bridge fall down,
and I think movies of that are

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now being shown not merely on
television, but in elementary

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school.
Resonance is what made,

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okay, more resonance stories
later.

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So, my aim is,
what is this physical

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phenomenon, that to get a big
amplitude you should have it

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match the frequency?
What does that have to do with

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a differential equation?
Well, the differential equation

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for that simple pendulum,
let's assume it's undamped,

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will be of the type y double
prime plus,

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I'm using t now since t is
time.

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That will be our new
independent variable,

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plus omega nought squared
is the natural

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frequency of the pendulum or of
the spring, or whatever it is

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that's doing the vibrating.
Yeah, any questions?

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What we're doing is driving
that with the cosine,

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with something of a different
frequency.

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So, this is the input,
or the driving term as it's

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often called,
or it's sometimes called the

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forcing term.
And, the point is I'm going to

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assume that the frequency is
different.

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The driving frequency is
different from the natural

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frequency.
So, this is the input

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frequency.
Okay, and now let's simply

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solve the equation and see what
we get.

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So, it's if I write it using
the operator,

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it's D squared plus omega
nought squared applied to y

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is equal to cosine.

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It's a good idea to do this
because the formulas are going

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to ask you to substitute into a
polynomial.

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So, it's good to have the
polynomial right in front of you

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to avoid the possibility of
error.

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Well, really what I want is the
particular solution.

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It's the particular solution
that's going to give me a pure

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oscillation.
And, the thing to do is,

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of course, since this cosine,
you want to make it complex.

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So, we are going to complexify
the equation in order to be able

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to solve it more easily,
and in order to be able to use

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those formulas.
So, the complex equation is

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going to be D squared plus omega
nought squared.

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Well, it's going to be a

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complex, particular solution.
So, I'll call it y tilde.

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And, on the right-hand side,
that's going to be e to the i

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00:09:04,000 --> 00:09:10,000
omega1 t.
Cosine is the real part of

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this.
So, when we get our answer,

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we want to be sure to take the
real part of the answer.

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I don't want the complex
answer, I want its real part.

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I want the real answer,
in other words,

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the really real answer,
the real real answer.

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So, now without further ado,
because of those beautiful,

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the problem has been solved
once and for all by using the

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substitution rule.
I did that for you on Monday.

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The answer is simply e to the i
omega1 t

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divided by what?
This polynomial with omega one

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substituted in for D.
So, sorry, i omega one,

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the complex
coefficient of t.

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So, it is substitute i omega
for D, I omega one for D,

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and you get (i omega one)
squared plus omega nought

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00:10:01,000 --> 00:10:07,000
squared.

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Well, let's make that look a
little bit better.

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This should be e to the (i
omega one t)

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divided by, now,
what's this?

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00:10:17,000 --> 00:10:23,000
This is simply omega nought
squared minus omega

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one squared.
But, I want the real part of

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it.
So, as one final,

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last step, the real part of
that is what we call just the

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real particular solution,
so, yp without the tilde

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00:10:37,000 --> 00:10:43,000
anymore.
And, the real part of this,

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well, this cosine plus i sine.
And, the denominator,

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luckily, turns out to be real.
So, it's simply going to be

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cosine omega one t.

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That's the top,
divided by this thing,

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omega nought squared minus
omega one squared.

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00:11:00,000 --> 00:11:06,000
In other words,

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that's the response.
This is the input,

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00:11:07,000 --> 00:11:13,000
and that's what came out.
Well, in other words,

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what one sees is,
regardless of what natural

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frequency this system wanted to
use for itself,

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00:11:20,000 --> 00:11:26,000
at least for this solution,
what it responds to is the

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driving frequency,
the input frequency.

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The only thing is that the
amplitude has changed,

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and in a rather dramatic way,
if omega1, depending on the

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00:11:39,000 --> 00:11:45,000
relative sizes of omega1 and
omega2.

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00:11:43,000 --> 00:11:49,000
Now, the interesting case is
when omega one is very close to

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00:11:48,000 --> 00:11:54,000
omega, the natural frequency.
When you push it with

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00:11:53,000 --> 00:11:59,000
approximately it's natural
frequency, then the solution is

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00:11:59,000 --> 00:12:05,000
big amplitude.
The amplitude is large.

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00:12:04,000 --> 00:12:10,000
So, the solution looks like the
frequency.

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00:12:07,000 --> 00:12:13,000
The input might have looked
like this.

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00:12:10,000 --> 00:12:16,000
Well, it's cosine,
so it ought to start up here.

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00:12:13,000 --> 00:12:19,000
The input might have looked
like this, but the response will

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00:12:18,000 --> 00:12:24,000
be a curve with the same
frequency and still a pure

193
00:12:22,000 --> 00:12:28,000
oscillation.
But, it will have much,

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00:12:24,000 --> 00:12:30,000
much bigger amplitude.
And, it's because the

195
00:12:28,000 --> 00:12:34,000
denominator, omega nought
squared minus omega

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00:12:32,000 --> 00:12:38,000
one squared,
is always zero.

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00:12:35,000 --> 00:12:41,000
So, the response will,
instead, look like this.

198
00:12:41,000 --> 00:12:47,000
Now, to all intents and
purposes, that's resonance.

199
00:12:44,000 --> 00:12:50,000
You are pushing something with
approximately the same

200
00:12:48,000 --> 00:12:54,000
frequency, something that wants
to oscillate.

201
00:12:52,000 --> 00:12:58,000
And, you are pushing it with
approximately the same frequency

202
00:12:57,000 --> 00:13:03,000
that it would like to oscillate
by itself.

203
00:13:00,000 --> 00:13:06,000
And, what that does is it
builds up the amplitude

204
00:13:05,000 --> 00:13:11,000
Well, what happens if omega one
is actually equal

205
00:13:10,000 --> 00:13:16,000
to omega zero?
So, that's the case I'd like to

206
00:13:14,000 --> 00:13:20,000
analyze for you now.
Suppose the two are equal,

207
00:13:18,000 --> 00:13:24,000
in other words.
Well, the problem is,

208
00:13:21,000 --> 00:13:27,000
of course, I can't use that
same solution.

209
00:13:24,000 --> 00:13:30,000
It isn't applicable.
But that's why I gave you,

210
00:13:28,000 --> 00:13:34,000
derived for you using the
exponential shift law last time,

211
00:13:33,000 --> 00:13:39,000
the second version,
when it is a root.

212
00:13:38,000 --> 00:13:44,000
So, if omega one equals omega
nought,

213
00:13:42,000 --> 00:13:48,000
so now our equation looks like
D squared plus omega nought

214
00:13:47,000 --> 00:13:53,000
squared, the natural frequency,
y.

215
00:13:51,000 --> 00:13:57,000
But this time,
the driving frequency,

216
00:13:54,000 --> 00:14:00,000
the input frequency,
is omega nought itself.

217
00:13:57,000 --> 00:14:03,000
Then, the same analysis,
a lot of it is,

218
00:14:00,000 --> 00:14:06,000
well, I'd better be careful.
I'd better be careful.

219
00:14:04,000 --> 00:14:10,000
Let's go through the analysis
again very rapidly.

220
00:14:10,000 --> 00:14:16,000
What we want to do is first
complexify it,

221
00:14:13,000 --> 00:14:19,000
and then solve.
So, the complex equation will

222
00:14:17,000 --> 00:14:23,000
be D squared plus omega nought
squared times y tilde equals e

223
00:14:22,000 --> 00:14:28,000
to the i omega nought t,
this time.

224
00:14:29,000 --> 00:14:35,000
But now, i omega is zero of
this polynomial.

225
00:14:32,000 --> 00:14:38,000
That's why I picked it,
right?

226
00:14:35,000 --> 00:14:41,000
If I plug in i omega zero,
I get i omega zero

227
00:14:40,000 --> 00:14:46,000
quantity squared plus omega
nought squared.

228
00:14:46,000 --> 00:14:52,000
That's zero.

229
00:14:48,000 --> 00:14:54,000
So, I'm in the second case.
So, i omega nought is a simple

230
00:14:53,000 --> 00:14:59,000
root, simple zero,
of D squared plus

231
00:14:58,000 --> 00:15:04,000
omega nought,
that polynomial squared.

232
00:15:04,000 --> 00:15:10,000
Therefore, the complex
particular solution is now t e

233
00:15:08,000 --> 00:15:14,000
to the i omega nought t
divided by p prime,

234
00:15:13,000 --> 00:15:19,000
where you plug in that root,
the i omega nought.

235
00:15:17,000 --> 00:15:23,000
Now, what's p prime?

236
00:15:19,000 --> 00:15:25,000
p prime is 2D, right?

237
00:15:22,000 --> 00:15:28,000
If I differentiate this
formally, as if D were a

238
00:15:26,000 --> 00:15:32,000
variable, the way you
differentiate polynomials,

239
00:15:29,000 --> 00:15:35,000
the derivative,
this is a constant,

240
00:15:32,000 --> 00:15:38,000
and the derivative is 2D.
So, the denominator should have

241
00:15:38,000 --> 00:15:44,000
two times for D.
You are going to plug in i

242
00:15:42,000 --> 00:15:48,000
omega zero.
So, it's 2 i omega zero.

243
00:15:47,000 --> 00:15:53,000
And now, I want the real part

244
00:15:51,000 --> 00:15:57,000
of that, which is what?
Well, think about it.

245
00:15:55,000 --> 00:16:01,000
The top is cosine plus i sine.
The real part is now going to

246
00:16:00,000 --> 00:16:06,000
come from the sine,
right, because it's cosine plus

247
00:16:05,000 --> 00:16:11,000
i sine.
But this i is going to divide

248
00:16:09,000 --> 00:16:15,000
out the i that goes with this
sine.

249
00:16:11,000 --> 00:16:17,000
And, therefore,
the real part is going to be t

250
00:16:15,000 --> 00:16:21,000
times the sine,
this time, of omega nought t.

251
00:16:18,000 --> 00:16:24,000
And, that's going to be divided

252
00:16:22,000 --> 00:16:28,000
by, well, the i canceled out the
i that was in front of the sine

253
00:16:27,000 --> 00:16:33,000
function.
And therefore,

254
00:16:28,000 --> 00:16:34,000
what's left is two omega nought
down below.

255
00:16:34,000 --> 00:16:40,000
So, that's our particular
solution now.

256
00:16:36,000 --> 00:16:42,000
Well, it looks different from
that guy.

257
00:16:39,000 --> 00:16:45,000
It doesn't look like that
anymore.

258
00:16:42,000 --> 00:16:48,000
What does it look like?
Well, it shows the way to plot

259
00:16:46,000 --> 00:16:52,000
such things is basically it's an
oscillation of frequency omega

260
00:16:50,000 --> 00:16:56,000
nought.
But, its amplitude is changing.

261
00:16:54,000 --> 00:17:00,000
So, the way to do it is,
as always, if you have a basic

262
00:16:58,000 --> 00:17:04,000
oscillation which is neither too
fast nor too slow,

263
00:17:02,000 --> 00:17:08,000
think of that as the thing,
and the other stuff multiplying

264
00:17:06,000 --> 00:17:12,000
it, think of it as changing the
amplitude of that oscillation

265
00:17:10,000 --> 00:17:16,000
with time.
So, the amplitude is that

266
00:17:14,000 --> 00:17:20,000
function, t divided by two omega
zero.

267
00:17:18,000 --> 00:17:24,000
So, just as we did when we
talked about damping,

268
00:17:21,000 --> 00:17:27,000
you plot that and it's negative
on the picture.

269
00:17:25,000 --> 00:17:31,000
So, this is the function whose
graph is t divided by two omega

270
00:17:29,000 --> 00:17:35,000
nought.
That's the changing amplitude,

271
00:17:34,000 --> 00:17:40,000
as it were.
And then, the function itself

272
00:17:37,000 --> 00:17:43,000
does what oscillation it can,
but it has to stay within those

273
00:17:41,000 --> 00:17:47,000
lines.
So, the thing that's

274
00:17:43,000 --> 00:17:49,000
oscillating is sine omega nought
t,

275
00:17:47,000 --> 00:17:53,000
which would like to be a pure
oscillation, but can't because

276
00:17:52,000 --> 00:17:58,000
its amplitude is being changed
by that thing.

277
00:17:55,000 --> 00:18:01,000
So, it's doing this,
and now the rest I have to

278
00:17:58,000 --> 00:18:04,000
leave to your imagination.
In other words,

279
00:18:03,000 --> 00:18:09,000
what happens when omega nought
is equal to, when the driving

280
00:18:07,000 --> 00:18:13,000
frequency is actually equal to
omega nought,

281
00:18:11,000 --> 00:18:17,000
mathematically this turns into
a different looking solution,

282
00:18:15,000 --> 00:18:21,000
one with steadily increasing
amplitude.

283
00:18:18,000 --> 00:18:24,000
The amplitude increases
linearly like the function t

284
00:18:22,000 --> 00:18:28,000
divided by two omega nought.

285
00:18:25,000 --> 00:18:31,000
Well, many people are upset by
this, slightly,

286
00:18:29,000 --> 00:18:35,000
in the sense that there is a
funny feeling.

287
00:18:32,000 --> 00:18:38,000
How is it that that solution
can turn into this one?

288
00:18:38,000 --> 00:18:44,000
If I simply let omega one go to
omega zero, what happens?

289
00:18:43,000 --> 00:18:49,000
Well, the pink curve just gets
taller and taller,

290
00:18:47,000 --> 00:18:53,000
and after a while all you see
of it is just a bunch of

291
00:18:52,000 --> 00:18:58,000
vertical lines which seem to be
spaced at whatever the right

292
00:18:57,000 --> 00:19:03,000
period is for that function.
It's sort of like being in a

293
00:19:03,000 --> 00:19:09,000
first story window and watching
a giraffe go by.

294
00:19:08,000 --> 00:19:14,000
All you see is that.
Okay.

295
00:19:20,000 --> 00:19:26,000
So, my concern is how does that
function turn into this one?

296
00:19:24,000 --> 00:19:30,000
I have something in mind to
remind you of,

297
00:19:27,000 --> 00:19:33,000
and that's why we'll go through
this little exercise.

298
00:19:31,000 --> 00:19:37,000
It's a simple exercise.
But the function of it is,

299
00:19:35,000 --> 00:19:41,000
of course that as omega one
goes to omega zero cannot

300
00:19:39,000 --> 00:19:45,000
possibly turn into this.
It's doing the wrong thing near

301
00:19:43,000 --> 00:19:49,000
zero.
It's already zooming up.

302
00:19:45,000 --> 00:19:51,000
But, the point is,
this is not the only particular

303
00:19:49,000 --> 00:19:55,000
solution on the block.
Any solution whatsoever of the

304
00:19:52,000 --> 00:19:58,000
differential equation,
the inhomogeneous equation,

305
00:19:56,000 --> 00:20:02,000
is a particular solution.
It's like Fred Rogers:

306
00:20:01,000 --> 00:20:07,000
everybody is special.
Okay, so all solutions are

307
00:20:05,000 --> 00:20:11,000
special.
We don't have to use that one.

308
00:20:08,000 --> 00:20:14,000
So, I will use,
where are all the other

309
00:20:11,000 --> 00:20:17,000
solutions?
So, I'm going back to the

310
00:20:14,000 --> 00:20:20,000
equation D squared plus omega
zero squared,

311
00:20:18,000 --> 00:20:24,000
applied to y,

312
00:20:20,000 --> 00:20:26,000
is equal to cosine
omega one t.

313
00:20:24,000 --> 00:20:30,000
Now, the particular solution we
found was that one,

314
00:20:27,000 --> 00:20:33,000
cosine omega one t divided by
that omega nought squared minus

315
00:20:32,000 --> 00:20:38,000
omega one squared.

316
00:20:39,000 --> 00:20:45,000
What do the other particular
solutions look like?

317
00:20:44,000 --> 00:20:50,000
Well, in general,
any particular solution will

318
00:20:49,000 --> 00:20:55,000
look like that one we found,
what is it, omega nought

319
00:20:54,000 --> 00:21:00,000
squared minus omega
one squared,

320
00:21:01,000 --> 00:21:07,000
plus I'm allowed to add to it
any piece of the complementary

321
00:21:07,000 --> 00:21:13,000
solution.
Equally particular,

322
00:21:11,000 --> 00:21:17,000
and equally good,
as a particular solution is

323
00:21:14,000 --> 00:21:20,000
this plus anything which solved
the homogeneous equation.

324
00:21:18,000 --> 00:21:24,000
Now, all I'm going to do is
pick out one good function which

325
00:21:23,000 --> 00:21:29,000
solves the homogeneous equation,
and here it is.

326
00:21:26,000 --> 00:21:32,000
It's the function minus cosine.
In fact, what does solve the

327
00:21:32,000 --> 00:21:38,000
homogeneous equation?
Well, it's solved by sine omega

328
00:21:36,000 --> 00:21:42,000
nought t,
cosine omega nought t,

329
00:21:40,000 --> 00:21:46,000
and any linear combination of

330
00:21:44,000 --> 00:21:50,000
those.
So, out of all those functions,

331
00:21:47,000 --> 00:21:53,000
the one I'm going to pick is
cosine omega nought t.

332
00:21:51,000 --> 00:21:57,000
And, I'm going to divide it by
this same guy.

333
00:21:55,000 --> 00:22:01,000
So, this is part of the
complementary solution.

334
00:22:00,000 --> 00:22:06,000
That's what we call the
complementary solution,

335
00:22:02,000 --> 00:22:08,000
the solution to the associated
homogeneous equation,

336
00:22:06,000 --> 00:22:12,000
to the reduced equation.
Call it what you like.

337
00:22:08,000 --> 00:22:14,000
So, this is one of the guys in
there, and it's still a

338
00:22:12,000 --> 00:22:18,000
particular solution to take the
one I first found,

339
00:22:15,000 --> 00:22:21,000
and add to it anything which
solves the homogeneous equation.

340
00:22:19,000 --> 00:22:25,000
I showed you that when we first
set out to solve the

341
00:22:22,000 --> 00:22:28,000
inhomogeneous equation in
general.

342
00:22:24,000 --> 00:22:30,000
Now, why do I pick that?
Well, I'm going to now

343
00:22:27,000 --> 00:22:33,000
calculate, what's the limit?
So, these guys are also good

344
00:22:31,000 --> 00:22:37,000
solutions to that.
This is a good solution to that

345
00:22:35,000 --> 00:22:41,000
equation, this equation.
All I'm going to do now is

346
00:22:38,000 --> 00:22:44,000
calculate the limit as omega one
approaches omega zero of this

347
00:22:42,000 --> 00:22:48,000
function.
Well, what is that?

348
00:22:46,000 --> 00:22:52,000
It's cosine omega one t minus
cosine omega zero t divided by

349
00:22:50,000 --> 00:22:56,000
omega nought squared minus omega
one squared.

350
00:22:57,000 --> 00:23:03,000
Now, you see why I did that.
If I let just this guy,

351
00:23:02,000 --> 00:23:08,000
omega one approaches
omega zero,

352
00:23:07,000 --> 00:23:13,000
I get infinity.
I don't get anything.

353
00:23:10,000 --> 00:23:16,000
But, this is different here
because I fixed it up,

354
00:23:14,000 --> 00:23:20,000
now.
The denominator becomes zero,

355
00:23:17,000 --> 00:23:23,000
but so does the numerator.
In other words,

356
00:23:21,000 --> 00:23:27,000
I've put myself in position to
use L'Hopital rule.

357
00:23:26,000 --> 00:23:32,000
So, let's L'Hopital it.
It's the limit.

358
00:23:29,000 --> 00:23:35,000
As omega one approaches omega
zero, and what do you do?

359
00:23:34,000 --> 00:23:40,000
You differentiate the top and
the bottom with respect to what?

360
00:23:42,000 --> 00:23:48,000
Right, with respect to omega
one.

361
00:23:44,000 --> 00:23:50,000
Omega one is the variable.
That's what's changing.

362
00:23:47,000 --> 00:23:53,000
The t that I'm thinking of is,
I'm thinking,

363
00:23:50,000 --> 00:23:56,000
for the temporary fixed.
This has a fixed value.

364
00:23:53,000 --> 00:23:59,000
Omega nought is fixed.
All that's changing in this

365
00:23:57,000 --> 00:24:03,000
limit operation is omega one.
And therefore,

366
00:24:01,000 --> 00:24:07,000
it's with respect to omega one
that I differentiate it.

367
00:24:05,000 --> 00:24:11,000
You got that?
Well, you are in no position to

368
00:24:08,000 --> 00:24:14,000
say yes or no,
so I shouldn't even ask the

369
00:24:11,000 --> 00:24:17,000
question, but okay,
rhetorical question.

370
00:24:14,000 --> 00:24:20,000
All right, let's differentiate
this expression,

371
00:24:17,000 --> 00:24:23,000
the top and bottom with respect
to omega one.

372
00:24:20,000 --> 00:24:26,000
So, the derivative of the top
with respect to omega one is

373
00:24:24,000 --> 00:24:30,000
negative sine omega one t.

374
00:24:28,000 --> 00:24:34,000
But, I have to use the chain
rule.

375
00:24:32,000 --> 00:24:38,000
That's differentiating with
respect to this argument,

376
00:24:35,000 --> 00:24:41,000
this variable.
But now, I must take times the

377
00:24:38,000 --> 00:24:44,000
derivative of this thing with
respect to omega one.

378
00:24:43,000 --> 00:24:49,000
And that is t is the constant,
so times t.

379
00:24:46,000 --> 00:24:52,000
And, how about the bottom?
The derivative of the bottom

380
00:24:49,000 --> 00:24:55,000
with respect to omega one is,
well, that's a constant.

381
00:24:53,000 --> 00:24:59,000
So, it becomes zero.
And, this becomes negative two

382
00:24:57,000 --> 00:25:03,000
omega one.
So, it's the limit of this

383
00:25:01,000 --> 00:25:07,000
expression as omega one
approaches omega zero.

384
00:25:05,000 --> 00:25:11,000
And now it's not indeterminate

385
00:25:09,000 --> 00:25:15,000
anymore.
The answer is,

386
00:25:10,000 --> 00:25:16,000
the negative signs cancel.
It's simply t sine omega nought

387
00:25:15,000 --> 00:25:21,000
t divided by two omega nought.

388
00:25:19,000 --> 00:25:25,000
So, that's how we get that

389
00:25:21,000 --> 00:25:27,000
solution.
It is a limit as omega one,

390
00:25:24,000 --> 00:25:30,000
but not of the
particular solution we found

391
00:25:28,000 --> 00:25:34,000
first, but of this other one.
Now, it's still too much

392
00:25:34,000 --> 00:25:40,000
algebra.
I mean, what's going on here?

393
00:25:37,000 --> 00:25:43,000
Well, that's something else you
should know.

394
00:25:41,000 --> 00:25:47,000
Okay, so my question is,
therefore, what does this mean?

395
00:25:46,000 --> 00:25:52,000
What's the geometric meaning of
all this?

396
00:25:50,000 --> 00:25:56,000
In other words,
what does that function look

397
00:25:54,000 --> 00:26:00,000
like?
Well, that's another

398
00:25:56,000 --> 00:26:02,000
trigonometric identity,
which in your book is just

399
00:26:01,000 --> 00:26:07,000
buried as half of one line sort
of casual as if everybody knows

400
00:26:07,000 --> 00:26:13,000
it, and I know that virtually no
one knows it.

401
00:26:13,000 --> 00:26:19,000
But, here's your chance.
So, the cosine of B minus the

402
00:26:17,000 --> 00:26:23,000
cosine of A can be expressed
as a product of

403
00:26:22,000 --> 00:26:28,000
signs.
It's the sine of (A minus B)

404
00:26:25,000 --> 00:26:31,000
over two times the sine of (A
plus B) over two,

405
00:26:29,000 --> 00:26:35,000
I believe.

406
00:26:33,000 --> 00:26:39,000
My only uncertainty:
is there a two in front of

407
00:26:37,000 --> 00:26:43,000
that?
I think there has to be.

408
00:26:40,000 --> 00:26:46,000
Let me check.
Sorry.

409
00:26:42,000 --> 00:26:48,000
Is there a two?
I wouldn't trust my memory

410
00:26:46,000 --> 00:26:52,000
anyway.
I'd look it up.

411
00:26:49,000 --> 00:26:55,000
I did look it up,
two, yes.

412
00:26:51,000 --> 00:26:57,000
If you had to prove that,
you could use the sine formula

413
00:26:57,000 --> 00:27:03,000
to expand this out.
That would be a bad way to do

414
00:27:03,000 --> 00:27:09,000
it.
The best way is to use complex

415
00:27:05,000 --> 00:27:11,000
numbers.
Express the sign in terms of

416
00:27:08,000 --> 00:27:14,000
complex numbers,
exponentials,

417
00:27:10,000 --> 00:27:16,000
you know, the backwards Euler
formula.

418
00:27:13,000 --> 00:27:19,000
Then do it here,
and then just multiply those

419
00:27:17,000 --> 00:27:23,000
two expressions involving
exponentials together,

420
00:27:20,000 --> 00:27:26,000
and cancel, cancel,
cancel, cancel,

421
00:27:23,000 --> 00:27:29,000
cancel, and this is what you
will end up with.

422
00:27:26,000 --> 00:27:32,000
You see why I did this.
It's because this has that

423
00:27:31,000 --> 00:27:37,000
form.
So, let's apply that formula to

424
00:27:33,000 --> 00:27:39,000
it.
So, what's the left-hand side?

425
00:27:36,000 --> 00:27:42,000
B is omega one t, and A is
omega nought t.

426
00:27:39,000 --> 00:27:45,000
So, this is omega one t,

427
00:27:42,000 --> 00:27:48,000
and this is omega nought t

428
00:27:44,000 --> 00:27:50,000
All right, so what we get is

429
00:27:47,000 --> 00:27:53,000
that the cosine of omega one t
minus the cosine of omega nought

430
00:27:51,000 --> 00:27:57,000
t, which is exactly the

431
00:27:54,000 --> 00:28:00,000
numerator of this function that
I'm trying to get a handle on.

432
00:28:00,000 --> 00:28:06,000
Then we will divide it by its
amplitude.

433
00:28:02,000 --> 00:28:08,000
So, that's this constant factor
that's real.

434
00:28:05,000 --> 00:28:11,000
It's a small number because I'm
thinking of omega one

435
00:28:10,000 --> 00:28:16,000
as being rather close to omega
zero,

436
00:28:13,000 --> 00:28:19,000
and getting closer and closer.
What does this tell us about

437
00:28:17,000 --> 00:28:23,000
the right-hand side?
Well, the right-hand side is

438
00:28:21,000 --> 00:28:27,000
twice the sine of A minus B.

439
00:28:24,000 --> 00:28:30,000
Now, that's good because these
guys sort of resemble each

440
00:28:28,000 --> 00:28:34,000
other.
So, that's (omega nought minus

441
00:28:32,000 --> 00:28:38,000
omega one) times t.

442
00:28:35,000 --> 00:28:41,000
That's A minus B,
and I'm supposed to divide that

443
00:28:39,000 --> 00:28:45,000
by two.
And then, the other one will be

444
00:28:42,000 --> 00:28:48,000
the same thing with plus:
sine omega nought plus omega

445
00:28:46,000 --> 00:28:52,000
one over two times t.

446
00:28:50,000 --> 00:28:56,000
Now, how big is this,
approximately?

447
00:28:52,000 --> 00:28:58,000
Remember, think of omega one
as close to omega zero.

448
00:28:56,000 --> 00:29:02,000
Then, this is approximately

449
00:28:59,000 --> 00:29:05,000
omega zero.
So this part is approximately

450
00:29:03,000 --> 00:29:09,000
sine of omega zero t.

451
00:29:06,000 --> 00:29:12,000
This part, on the other hand,
that's a very small thing.

452
00:29:09,000 --> 00:29:15,000
Okay, now what I want to know
is what does this function look

453
00:29:13,000 --> 00:29:19,000
like?
The interest in knowing what

454
00:29:15,000 --> 00:29:21,000
the function looks like it is
because we want to be able to

455
00:29:19,000 --> 00:29:25,000
see that it's limited is that
thing.

456
00:29:21,000 --> 00:29:27,000
You can't tell what's what its
limit is, geometrically,

457
00:29:25,000 --> 00:29:31,000
unless you know it looks like.
So, what does it look like?

458
00:29:30,000 --> 00:29:36,000
Well, again,
the way to analyze it is the

459
00:29:35,000 --> 00:29:41,000
thing, that thing.
What you think of is,

460
00:29:41,000 --> 00:29:47,000
yeah, of course you cannot
divide one side of equality

461
00:29:49,000 --> 00:29:55,000
without dividing the equation by
the other side.

462
00:29:56,000 --> 00:30:02,000
So, that's got to be there,
too.

463
00:30:02,000 --> 00:30:08,000
Now, what does that look like?
Well, the way to think of it

464
00:30:06,000 --> 00:30:12,000
is, here is something with a
normal sort of frequency,

465
00:30:10,000 --> 00:30:16,000
omega nought.
It's doing its thing.

466
00:30:14,000 --> 00:30:20,000
It's a sine curve.
It's doing that.

467
00:30:16,000 --> 00:30:22,000
What's this?
Think of all this part as

468
00:30:19,000 --> 00:30:25,000
varying amplitude.
It's just another example of

469
00:30:23,000 --> 00:30:29,000
what I gave you before.
Here is a basic,

470
00:30:26,000 --> 00:30:32,000
pure oscillation,
and now, think of everything

471
00:30:29,000 --> 00:30:35,000
else that's multiplying it as
varying its amplitude.

472
00:30:35,000 --> 00:30:41,000
All right, so what does that
thing look like?

473
00:30:38,000 --> 00:30:44,000
Well, first what we want to do
is plot the amplitude lines.

474
00:30:44,000 --> 00:30:50,000
Now, what will they be?
This is sine of an extremely

475
00:30:48,000 --> 00:30:54,000
small number times t.
The frequency is small.

476
00:30:52,000 --> 00:30:58,000
How does the sine curve look if
its frequency is very low,

477
00:30:57,000 --> 00:31:03,000
very close to zero?
Well, that must mean its period

478
00:31:02,000 --> 00:31:08,000
is very large.
Here's something with a big

479
00:31:05,000 --> 00:31:11,000
frequency.
Here's something with a very,

480
00:31:08,000 --> 00:31:14,000
very low frequency.
Now, with a low frequency,

481
00:31:11,000 --> 00:31:17,000
it would hardly get off the
ground and get up to one here,

482
00:31:15,000 --> 00:31:21,000
and it would do that.
But, it's made to look a little

483
00:31:19,000 --> 00:31:25,000
more presentable because of this
coefficient in front,

484
00:31:23,000 --> 00:31:29,000
which is rather large.
And so, what this thing looks

485
00:31:27,000 --> 00:31:33,000
like, I won't pause to analyze
it more exactly.

486
00:31:32,000 --> 00:31:38,000
It's something which goes up at
a reasonable rate for quite a

487
00:31:36,000 --> 00:31:42,000
while, and let's say that's
quite awhile.

488
00:31:39,000 --> 00:31:45,000
And then it comes down,
and then it goes,

489
00:31:42,000 --> 00:31:48,000
and so on.
Of course, in figuring out its

490
00:31:45,000 --> 00:31:51,000
amplitude, we have to be willing
to draw its negative,

491
00:31:49,000 --> 00:31:55,000
too.
And since I didn't figure

492
00:31:51,000 --> 00:31:57,000
things out right,
I can at least make it cross,

493
00:31:54,000 --> 00:32:00,000
right?
Okay.

494
00:31:55,000 --> 00:32:01,000
So, this is a picture of this
slowly varying amplitude.

495
00:32:01,000 --> 00:32:07,000
And in between,
this is the function which is

496
00:32:05,000 --> 00:32:11,000
doing the oscillation,
as well as it can.

497
00:32:08,000 --> 00:32:14,000
But, it has to stay within that
amplitude.

498
00:32:12,000 --> 00:32:18,000
So, it's doing this.
Now, what happens?

499
00:32:15,000 --> 00:32:21,000
As omega one approaches omega
zero,

500
00:32:21,000 --> 00:32:27,000
this frequency gets closer and
closer to zero,

501
00:32:25,000 --> 00:32:31,000
which means the period of that
dotted line gets further and

502
00:32:30,000 --> 00:32:36,000
further out, goes to infinity,
and you never do ultimately get

503
00:32:35,000 --> 00:32:41,000
a chance to come down again.
All you can see is the initial

504
00:32:42,000 --> 00:32:48,000
part, where it's rising and
rising.

505
00:32:45,000 --> 00:32:51,000
And, that's how this curve
turns into that one.

506
00:32:49,000 --> 00:32:55,000
Now, of course,
this curve is enormously

507
00:32:52,000 --> 00:32:58,000
interesting.
You must have had this

508
00:32:55,000 --> 00:33:01,000
somewhere.
That's the phenomenon of what

509
00:32:59,000 --> 00:33:05,000
are called beats.
Too frequencies--

510
00:33:03,000 --> 00:33:09,000
Your book has half a page
explaining this.

511
00:33:05,000 --> 00:33:11,000
That's the half a page where he
gives you this identity,

512
00:33:09,000 --> 00:33:15,000
except it gives it in a wrong
form, so that it's hard to

513
00:33:13,000 --> 00:33:19,000
figure out.
But anyway, the beats are two

514
00:33:16,000 --> 00:33:22,000
frequencies when you combine
them, the two frequencies being

515
00:33:20,000 --> 00:33:26,000
two combined pure oscillations
where the frequencies are very

516
00:33:24,000 --> 00:33:30,000
close to each other.
What you get is a curve which

517
00:33:27,000 --> 00:33:33,000
looks like that.
And, of course,

518
00:33:31,000 --> 00:33:37,000
what you hear is the envelope
of the curve.

519
00:33:34,000 --> 00:33:40,000
You hear the dotted lines.
Well, you hear this.

520
00:33:37,000 --> 00:33:43,000
You hear that,
too.

521
00:33:39,000 --> 00:33:45,000
But, what you hear is-- And,
that's how good violinists and

522
00:33:43,000 --> 00:33:49,000
cellists, and so on,
tune their instruments.

523
00:33:46,000 --> 00:33:52,000
They get one string right,
and then the other strings are

524
00:33:51,000 --> 00:33:57,000
tuned by listening.
They don't actually listen for

525
00:33:54,000 --> 00:34:00,000
the sound of the note.
They listened just for the

526
00:33:58,000 --> 00:34:04,000
beats, wah, wah,
wah, wah, and they turn the peg

527
00:34:02,000 --> 00:34:08,000
and it goes wah,
wah, wah, wah,

528
00:34:04,000 --> 00:34:10,000
and then finally as soon as the
wahs disappear,

529
00:34:07,000 --> 00:34:13,000
they know that the two strings
are in tune.

530
00:34:13,000 --> 00:34:19,000
A piano tuner does the same
thing.

531
00:34:16,000 --> 00:34:22,000
Of course, I,
being a very bad cellist,

532
00:34:19,000 --> 00:34:25,000
use a tuner.
That's another solution,

533
00:34:22,000 --> 00:34:28,000
a more modern solution.
Okay.

534
00:34:49,000 --> 00:34:55,000
Oh well.
Let's give it a try.

535
00:34:51,000 --> 00:34:57,000
The bad news is that problem
six in your problem set,

536
00:34:55,000 --> 00:35:01,000
I didn't ask you about the
undamped case.

537
00:34:58,000 --> 00:35:04,000
I thought, since you are mature
citizens, you could be asked

538
00:35:03,000 --> 00:35:09,000
about the damped case.

539
00:35:21,000 --> 00:35:27,000
I warn you, first of all you
have to get the notation.

540
00:35:26,000 --> 00:35:32,000
This is probably the most
important thing I'll do with

541
00:35:31,000 --> 00:35:37,000
this.
Your book uses this, resonance.

542
00:36:02,000 --> 00:36:08,000
I'm optimistic.
[LAUGHTER] Let's say zero or f

543
00:36:06,000 --> 00:36:12,000
of t.
It doesn't matter.

544
00:36:09,000 --> 00:36:15,000
In other words,
the constants,

545
00:36:12,000 --> 00:36:18,000
the book uses two sets of
constants to describe these

546
00:36:17,000 --> 00:36:23,000
equations.
If it's a spring,

547
00:36:20,000 --> 00:36:26,000
and not even talking about RLC
circuits, the spring mass,

548
00:36:26,000 --> 00:36:32,000
damping, k, spring constant.
Then you divide out by m and

549
00:36:33,000 --> 00:36:39,000
you get this.
You're familiar with that.

550
00:36:36,000 --> 00:36:42,000
And, it's only after you
divided out by the m that you're

551
00:36:41,000 --> 00:36:47,000
allowed to call this the square
of the natural frequency.

552
00:36:46,000 --> 00:36:52,000
So, omega naught is the natural
frequency, the natural undamped

553
00:36:51,000 --> 00:36:57,000
frequency.
If this term were not there,

554
00:36:54,000 --> 00:37:00,000
that omega nought
would give the frequency with

555
00:36:59,000 --> 00:37:05,000
which the system,
the little spring would like to

556
00:37:03,000 --> 00:37:09,000
vibrate by itself.
Now, further complication is

557
00:37:08,000 --> 00:37:14,000
that the visual uses neither of
these.

558
00:37:11,000 --> 00:37:17,000
The visual uses x double dot
plus b times x prime,

559
00:37:16,000 --> 00:37:22,000
I think we will have to fix
this in the future,

560
00:37:20,000 --> 00:37:26,000
but for now,
just live with it,

561
00:37:22,000 --> 00:37:28,000
plus kx,
and that's some function,

562
00:37:27,000 --> 00:37:33,000
again, a function.
So, in other words,

563
00:37:30,000 --> 00:37:36,000
the problem is that b is okay,
can't be confused with c.

564
00:37:37,000 --> 00:37:43,000
On the other hand,
this is not the same k as that.

565
00:37:42,000 --> 00:37:48,000
What I'm trying to say is,
don't automatically go to a

566
00:37:48,000 --> 00:37:54,000
formula one place,
and assume it's the same

567
00:37:53,000 --> 00:37:59,000
formula in another place.
You have to use these

568
00:37:58,000 --> 00:38:04,000
equivalences.
You have to look and see how

569
00:38:03,000 --> 00:38:09,000
the basic equation was written,
and then figure out what the

570
00:38:08,000 --> 00:38:14,000
constant should be.
Now, there was something

571
00:38:12,000 --> 00:38:18,000
called, when we analyzed this
before, and this has happened in

572
00:38:17,000 --> 00:38:23,000
recitation, there was the
natural, damped frequency.

573
00:38:22,000 --> 00:38:28,000
I'll call it the natural,
damped frequency.

574
00:38:25,000 --> 00:38:31,000
The book calls it the
pseudo-frequency.

575
00:38:30,000 --> 00:38:36,000
It's called pseudo-frequency
because the function,

576
00:38:33,000 --> 00:38:39,000
if you have zero on the right
hand side, but have damping,

577
00:38:38,000 --> 00:38:44,000
the function isn't periodic.
It decays.

578
00:38:41,000 --> 00:38:47,000
It does this.
Nonetheless,

579
00:38:43,000 --> 00:38:49,000
it still crosses the t-axis at
regular intervals,

580
00:38:47,000 --> 00:38:53,000
and therefore,
almost everybody just casually

581
00:38:50,000 --> 00:38:56,000
refers to it as the frequency,
and understands it's the

582
00:38:54,000 --> 00:39:00,000
natural damped frequency.
Now, the relation between them

583
00:38:59,000 --> 00:39:05,000
is given by the little picture I
drew you once.

584
00:39:02,000 --> 00:39:08,000
But, I didn't emphasize it
enough.

585
00:39:07,000 --> 00:39:13,000
Here is omega nought.

586
00:39:09,000 --> 00:39:15,000
Here is the right angle.
The side is omega one,

587
00:39:12,000 --> 00:39:18,000
and this side is the
damping.

588
00:39:15,000 --> 00:39:21,000
So, in other words,
this is fixed because it's

589
00:39:19,000 --> 00:39:25,000
fixed by the spring.
That's the natural frequency of

590
00:39:23,000 --> 00:39:29,000
the spring, by itself.
If you are damping near the

591
00:39:27,000 --> 00:39:33,000
motion, then the more you damped
it, the bigger this side gets,

592
00:39:31,000 --> 00:39:37,000
and therefore the smaller omega
one is, the bigger the damping,

593
00:39:36,000 --> 00:39:42,000
then the smaller the frequency
with which the damped thing

594
00:39:40,000 --> 00:39:46,000
vibrates.
That sort of intuitive,

595
00:39:44,000 --> 00:39:50,000
and vice versa.
If you decrease the damping to

596
00:39:47,000 --> 00:39:53,000
almost zero, well,
then you'll make omega one

597
00:39:49,000 --> 00:39:55,000
almost the same size as omega
zero.

598
00:39:51,000 --> 00:39:57,000
This must be a right angle,
and therefore,

599
00:39:54,000 --> 00:40:00,000
if there's very little damping,
the natural damped frequency

600
00:39:57,000 --> 00:40:03,000
will be almost the same as the
original frequency,

601
00:40:00,000 --> 00:40:06,000
the natural frequency.
So, the relation between them

602
00:40:05,000 --> 00:40:11,000
is that omega one squared is
equal to omega nought squared

603
00:40:11,000 --> 00:40:17,000
minus p squared,

604
00:40:15,000 --> 00:40:21,000
and this comes from the
characteristic roots from the

605
00:40:20,000 --> 00:40:26,000
characteristic roots of the
damped equation.

606
00:40:26,000 --> 00:40:32,000
So, we did that before.
I'm just reminding you of it.

607
00:40:31,000 --> 00:40:37,000
Now, the third frequency which
now enters, and that I'm asking

608
00:40:37,000 --> 00:40:43,000
you about on the problem set is
if you've got a damped spring,

609
00:40:43,000 --> 00:40:49,000
okay, what happens when you
impose a motion on it with yet a

610
00:40:49,000 --> 00:40:55,000
third frequency?
In other words,

611
00:40:53,000 --> 00:40:59,000
drive the damped spring.
I don't care.

612
00:40:56,000 --> 00:41:02,000
I switched to y,
since I'm in y mode.

613
00:41:02,000 --> 00:41:08,000
So, our equation looks like
this, just as it did before,

614
00:41:06,000 --> 00:41:12,000
except now going to drive that
with an undetermined frequency,

615
00:41:10,000 --> 00:41:16,000
cosine omega t.

616
00:41:13,000 --> 00:41:19,000
And, my question,
now, is, see,

617
00:41:15,000 --> 00:41:21,000
it's not going to be able to
resonate in the correct-- you

618
00:41:19,000 --> 00:41:25,000
really only get true resonance
when you don't have damping.

619
00:41:24,000 --> 00:41:30,000
That's the only time where the
amplitude can build up

620
00:41:27,000 --> 00:41:33,000
indefinitely.
But nonetheless,

621
00:41:31,000 --> 00:41:37,000
for all practical purposes,
and there's always some damping

622
00:41:37,000 --> 00:41:43,000
unless you are a perfect vacuum
or something,

623
00:41:42,000 --> 00:41:48,000
there's almost always some
damping.

624
00:41:46,000 --> 00:41:52,000
So, p isn't zero,
can't be exactly zero.

625
00:41:50,000 --> 00:41:56,000
So, the problem is,
which omega gives,

626
00:41:54,000 --> 00:42:00,000
which frequency in the input,
which input frequency gives the

627
00:42:00,000 --> 00:42:06,000
maximal amplitude for the
response?

628
00:42:20,000 --> 00:42:26,000
We solved that problem when it
was undamped,

629
00:42:22,000 --> 00:42:28,000
and the answer was easy.
Omega should equal omega zero.

630
00:42:26,000 --> 00:42:32,000
But, when it's damped,
the answer is different.

631
00:42:30,000 --> 00:42:36,000
And, I'm not asking you to do
it in general.

632
00:42:33,000 --> 00:42:39,000
I'm giving you some numbers.
But nonetheless,

633
00:42:37,000 --> 00:42:43,000
it still must be the case.
So, I'm giving you,

634
00:42:40,000 --> 00:42:46,000
I give you specific values of p
and omega zero.

635
00:42:45,000 --> 00:42:51,000
That's on the problem set.
Of course, one of them is tied

636
00:42:50,000 --> 00:42:56,000
to your recitation.
But, the answer is,

637
00:42:53,000 --> 00:42:59,000
I'm going to give you the
general formula for the answer

638
00:42:58,000 --> 00:43:04,000
to make sure that you don't get
wildly astray.

639
00:43:03,000 --> 00:43:09,000
Let's call that omega r,

640
00:43:05,000 --> 00:43:11,000
the resonant omega.
This isn't true resonance.

641
00:43:09,000 --> 00:43:15,000
Your book calls it practical
resonance.

642
00:43:11,000 --> 00:43:17,000
Again, most people just call it
resonance.

643
00:43:15,000 --> 00:43:21,000
So, you know what I mean,
type of thing.

644
00:43:18,000 --> 00:43:24,000
It is omega r is very much like
that.

645
00:43:20,000 --> 00:43:26,000
Maybe I should have written
this one down in the same form.

646
00:43:25,000 --> 00:43:31,000
Omega one is the square root of
omega nought squared minus p

647
00:43:29,000 --> 00:43:35,000
squared.

648
00:43:34,000 --> 00:43:40,000
What would you expect?
Well, what I would expect is

649
00:43:37,000 --> 00:43:43,000
that omega r should be omega
one.

650
00:43:40,000 --> 00:43:46,000
The damped system has a natural
frequency.

651
00:43:43,000 --> 00:43:49,000
The resonant frequency should
be the same as that natural

652
00:43:47,000 --> 00:43:53,000
frequency with which the damped
system wants to do its thing.

653
00:43:52,000 --> 00:43:58,000
And the answer is,
that's not right.

654
00:43:54,000 --> 00:44:00,000
It is the square root.
It's a little lower.

655
00:43:57,000 --> 00:44:03,000
It's a little lower.
It is omega nought squared

656
00:44:01,000 --> 00:44:07,000
minus two p squared.