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BOLESLAW WYSLOUCH: Good
morning, everybody.
00:00:25.980 --> 00:00:27.090
I'm Bolek Wyslouch.
00:00:27.090 --> 00:00:31.920
I'm a teacher substitute for
Professor Lee, who is now
00:00:31.920 --> 00:00:34.770
at some conference
in China, and he
00:00:34.770 --> 00:00:39.510
asked me to talk to you
about coupled oscillators.
00:00:39.510 --> 00:00:43.290
I understand that he introduced
the concept last time.
00:00:43.290 --> 00:00:45.430
You worked through
some examples.
00:00:45.430 --> 00:00:47.775
So what we are
going to do today is
00:00:47.775 --> 00:00:50.100
to basically go
through one or two
00:00:50.100 --> 00:00:55.500
examples of very straightforward
coupled oscillators, where
00:00:55.500 --> 00:00:59.790
I will introduce various kinds
of systematic calculational
00:00:59.790 --> 00:01:01.380
techniques, how
to set things up,
00:01:01.380 --> 00:01:05.129
how to prepare things
for calculations.
00:01:05.129 --> 00:01:09.150
And also, we may, depending
on how much time we have,
00:01:09.150 --> 00:01:14.560
start driving, have driven
coupled oscillators.
00:01:14.560 --> 00:01:20.990
And we will work on two,
again, simple physical systems,
00:01:20.990 --> 00:01:28.230
one that consists of two pendula
driven by forces of gravity,
00:01:28.230 --> 00:01:29.067
each of them.
00:01:29.067 --> 00:01:30.900
And then they are
connected with the spring.
00:01:30.900 --> 00:01:33.450
So each of those pendula,
each of those masses,
00:01:33.450 --> 00:01:36.805
will feel the effects of
gravity and effects of springs
00:01:36.805 --> 00:01:38.930
at the same time, and they
will talk to each other.
00:01:38.930 --> 00:01:40.720
There will be
coupling between them.
00:01:40.720 --> 00:01:44.520
So that's one physical
example which we'll consider.
00:01:44.520 --> 00:01:47.220
The other physical
example consists
00:01:47.220 --> 00:01:54.540
of two masses in the horizontal
frictionless track connected
00:01:54.540 --> 00:01:55.590
by a set of springs.
00:01:55.590 --> 00:01:58.260
So they are driven
by forces of spring.
00:01:58.260 --> 00:02:00.660
And those two systems are
very similar to each other,
00:02:00.660 --> 00:02:03.420
almost identical in
terms of calculations,
00:02:03.420 --> 00:02:05.610
and they exhibit
the same phenomena,
00:02:05.610 --> 00:02:07.350
and I will be able
to demonstrate
00:02:07.350 --> 00:02:09.340
several of the neat new things.
00:02:09.340 --> 00:02:11.580
And this particular
system is set up
00:02:11.580 --> 00:02:14.640
to introduce external
driving force, which will
00:02:14.640 --> 00:02:16.090
create a new set of phenomena.
00:02:16.090 --> 00:02:19.150
And we'll talk about it today.
00:02:19.150 --> 00:02:21.540
And what I would
like to stress today
00:02:21.540 --> 00:02:24.240
when we go through all
those calculations is,
00:02:24.240 --> 00:02:28.470
A, how do you convert
a given physical system
00:02:28.470 --> 00:02:32.310
with all the forces, et cetera,
into some sort of fixed form,
00:02:32.310 --> 00:02:36.690
fixed type of notation,
with which you can treat all
00:02:36.690 --> 00:02:39.150
possible coupled oscillators?
00:02:39.150 --> 00:02:41.370
And also we will discuss
various interesting--
00:02:41.370 --> 00:02:43.650
even though the system
is very simple, just
00:02:43.650 --> 00:02:47.980
two masses, a spring, a little
bit of gravity on top of that,
00:02:47.980 --> 00:02:53.340
the way they behave could
be extremely complex,
00:02:53.340 --> 00:02:55.500
but it can be
understood in terms
00:02:55.500 --> 00:02:58.710
of very simple systematic
way of looking things
00:02:58.710 --> 00:03:03.070
through normal modes
and normal frequencies,
00:03:03.070 --> 00:03:05.650
so the characteristic
frequencies of the system.
00:03:05.650 --> 00:03:08.070
So let's set things up.
00:03:08.070 --> 00:03:09.570
So we'll start.
00:03:09.570 --> 00:03:11.370
This will be our workhorse.
00:03:11.370 --> 00:03:15.300
And by the way, once
we understand two,
00:03:15.300 --> 00:03:18.930
we will then generalize to
infinite number of oscillators,
00:03:18.930 --> 00:03:22.770
which is actually-- so
this model, which consists
00:03:22.770 --> 00:03:26.460
of weights hanging under
the influence of gravity
00:03:26.460 --> 00:03:28.920
plus the springs
will be then used
00:03:28.920 --> 00:03:33.160
for many applications of the
concepts later in this course.
00:03:33.160 --> 00:03:39.430
So let's try to convert
this physical system
00:03:39.430 --> 00:03:41.590
into a set of equations.
00:03:41.590 --> 00:03:47.920
So we have a mass, m, hanging
from some sort of fixed
00:03:47.920 --> 00:03:52.360
support, another mass here,
same mass for simplicity.
00:03:52.360 --> 00:03:55.150
We connect them with
a string, and we know
00:03:55.150 --> 00:03:57.080
everything about this system.
00:03:57.080 --> 00:04:00.610
We know the length of each
of those pendula, which
00:04:00.610 --> 00:04:01.840
is the same.
00:04:01.840 --> 00:04:03.280
We know masses.
00:04:03.280 --> 00:04:07.330
We know spring constant of a
spring connecting those two
00:04:07.330 --> 00:04:08.860
things.
00:04:08.860 --> 00:04:11.695
The spring is initially
at its rest position
00:04:11.695 --> 00:04:16.269
such that when the two pendula
are hanging vertically,
00:04:16.269 --> 00:04:17.950
the spring is relaxed.
00:04:17.950 --> 00:04:20.209
But if you move it
away from verticality,
00:04:20.209 --> 00:04:24.790
the spring either
compresses or stretches.
00:04:24.790 --> 00:04:30.970
And everything is in Earth's
gravitational field, g.
00:04:30.970 --> 00:04:35.460
We assume that this is an
ideal system, highly idealized.
00:04:35.460 --> 00:04:40.180
We only consider motion with
small angle approximation, only
00:04:40.180 --> 00:04:42.290
small displacement.
00:04:42.290 --> 00:04:44.680
There's no drag force assumed.
00:04:44.680 --> 00:04:47.560
The spring is ideal,
et cetera, et cetera.
00:04:47.560 --> 00:04:51.670
Of course, this thing here
is very far from being ideal,
00:04:51.670 --> 00:04:54.940
but hopefully basic
behaviors are similar.
00:04:54.940 --> 00:04:57.890
It's approximately ideal.
00:04:57.890 --> 00:04:59.720
To study the motion
of this thing,
00:04:59.720 --> 00:05:02.480
to understand how it
works, let's try to--
00:05:02.480 --> 00:05:06.770
let's try to parameterize
it, and displace it
00:05:06.770 --> 00:05:08.780
from equilibrium, and
look at the forces,
00:05:08.780 --> 00:05:11.550
and try to calculate
equations of motion.
00:05:11.550 --> 00:05:13.520
So we will characterize
this system
00:05:13.520 --> 00:05:16.760
by two position coordinates.
00:05:16.760 --> 00:05:18.080
We will have x.
00:05:18.080 --> 00:05:19.580
We'll give this one number one.
00:05:19.580 --> 00:05:21.500
This will be number two.
00:05:21.500 --> 00:05:26.060
And we will have x subscript
1, which in general would
00:05:26.060 --> 00:05:26.960
depend on the time.
00:05:26.960 --> 00:05:29.600
This is the position of
this mass with respect
00:05:29.600 --> 00:05:31.320
to its equilibrium position.
00:05:31.320 --> 00:05:35.070
We will have x2 as
a function of t.
00:05:35.070 --> 00:05:37.100
Again, this tells us everything.
00:05:37.100 --> 00:05:39.770
And the full description
of the system
00:05:39.770 --> 00:05:42.920
is to know exactly what
happens to x1 and x2
00:05:42.920 --> 00:05:46.910
for all possible times.
00:05:46.910 --> 00:05:49.910
And we will impose some
initial conditions.
00:05:49.910 --> 00:05:52.650
We can come back to that later.
00:05:52.650 --> 00:05:55.550
So again, so the
coordinate system is this.
00:05:55.550 --> 00:05:58.790
When we start talking about
the system in principle
00:05:58.790 --> 00:06:01.910
in the case of
somewhat larger angles,
00:06:01.910 --> 00:06:04.425
you have to worry about
vertical positions as well.
00:06:04.425 --> 00:06:05.300
So we will introduce.
00:06:05.300 --> 00:06:09.500
So there is also a coordinate y,
which we will need temporarily
00:06:09.500 --> 00:06:12.040
to set things up.
00:06:12.040 --> 00:06:15.690
So x is, as I say, x is
measured from equilibrium.
00:06:15.690 --> 00:06:17.860
Y is positioned vertically.
00:06:17.860 --> 00:06:20.451
So to calculate the
equations of motion,
00:06:20.451 --> 00:06:21.700
we have to look at the forces.
00:06:21.700 --> 00:06:24.010
So let's look at what
are the forces acting,
00:06:24.010 --> 00:06:27.460
for example, on this
mass, the mass, which
00:06:27.460 --> 00:06:31.450
is-- if it's displaced
from a vertical position.
00:06:31.450 --> 00:06:37.420
Let's say this mass, mass 1,
has moved by some distance
00:06:37.420 --> 00:06:41.500
away from thing Temporarily,
let's introduce an angle here
00:06:41.500 --> 00:06:44.860
to characterize this
displacement from vertical.
00:06:44.860 --> 00:06:47.590
And let's write down all
the forces acting on this
00:06:47.590 --> 00:06:50.320
- force diagram
acting on this mass.
00:06:50.320 --> 00:06:55.570
So there is a tension in
the string or the rod.
00:06:55.570 --> 00:06:58.480
Let's call it T1.
00:06:58.480 --> 00:07:03.020
There is a force
of spring acting
00:07:03.020 --> 00:07:04.680
in a horizontal direction.
00:07:04.680 --> 00:07:06.350
This is a vector.
00:07:06.350 --> 00:07:11.840
And there is a force of
gravity acting on this
00:07:11.840 --> 00:07:14.130
in the vertical direction.
00:07:14.130 --> 00:07:16.610
We can write down those forces.
00:07:16.610 --> 00:07:18.050
We know a lot about them.
00:07:18.050 --> 00:07:23.810
This one is minus mg y-hat.
00:07:23.810 --> 00:07:33.660
This one is equal to k x2 minus
x1 in the x-hat direction.
00:07:33.660 --> 00:07:37.520
So this is the force which, when
the spring is displaced from
00:07:37.520 --> 00:07:40.730
equilibrium, there is a
spring force, Hooke force,
00:07:40.730 --> 00:07:42.480
in the direction of --
00:07:42.480 --> 00:07:43.940
in the usual direction.
00:07:43.940 --> 00:07:47.570
In this case, it's actually
in the opposite direction.
00:07:47.570 --> 00:07:49.710
And then there is a
tension the spring,
00:07:49.710 --> 00:07:53.370
which has to be calculated
such that we understand
00:07:53.370 --> 00:07:56.340
the acceleration of this object.
00:07:56.340 --> 00:07:59.210
So let's write
down the equations
00:07:59.210 --> 00:08:02.450
in the x-hat direction.
00:08:02.450 --> 00:08:07.990
This is m acceleration
of object number 1
00:08:07.990 --> 00:08:16.560
in x direction is equal to
minus T1 sine theta 1 plus k
00:08:16.560 --> 00:08:21.150
x2 minus x1.
00:08:21.150 --> 00:08:26.270
And in the y-hat
direction, we have
00:08:26.270 --> 00:08:39.340
m y1 direction is equal to
T cosine theta 1 minus mg.
00:08:39.340 --> 00:08:43.909
At the small angle for theta
1 much, much smaller than one,
00:08:43.909 --> 00:08:49.060
we can assume, that cos theta
1 is approximately equal to 1
00:08:49.060 --> 00:08:53.910
and sine theta 1
is equal to angle.
00:08:53.910 --> 00:08:55.710
We do the usual thing.
00:08:55.710 --> 00:08:59.250
So basically, in
this approximation,
00:08:59.250 --> 00:09:01.200
and also by looking
at the system,
00:09:01.200 --> 00:09:03.570
it's clear that the
system does not move,
00:09:03.570 --> 00:09:06.270
and the vertical
direction can be ignored.
00:09:06.270 --> 00:09:07.121
Yes?
00:09:07.121 --> 00:09:09.162
AUDIENCE: How do you know
which way [INAUDIBLE]??
00:09:09.162 --> 00:09:10.370
BOLESLAW WYSLOUCH: Excuse me?
00:09:10.370 --> 00:09:11.572
AUDIENCE: The [INAUDIBLE].
00:09:11.572 --> 00:09:13.323
How do you know which
way it [INAUDIBLE]??
00:09:13.323 --> 00:09:14.697
BOLESLAW WYSLOUCH:
How do I know?
00:09:14.697 --> 00:09:15.413
AUDIENCE: Yeah.
00:09:15.413 --> 00:09:16.395
[INAUDIBLE]
00:09:19.090 --> 00:09:21.920
BOLESLAW WYSLOUCH:
The spring force is--
00:09:21.920 --> 00:09:24.740
well, you have to
look at the mass 1.
00:09:24.740 --> 00:09:26.930
You are just looking at mass 1.
00:09:26.930 --> 00:09:29.570
So the spring is
connected to mass 1.
00:09:29.570 --> 00:09:32.930
And the force of
the spring on mass 1
00:09:32.930 --> 00:09:39.390
is k times however the spring
is squashed or stretched,
00:09:39.390 --> 00:09:41.210
all right?
00:09:41.210 --> 00:09:44.960
So it knows about the existence
of mass 2, but only in a sense
00:09:44.960 --> 00:09:47.320
that you have to know
the position of mass 2.
00:09:47.320 --> 00:09:50.930
So we just assume
that x2 is something,
00:09:50.930 --> 00:09:53.820
and we just look
where the spring is.
00:09:53.820 --> 00:09:55.980
So that's why--
the force of spring
00:09:55.980 --> 00:09:59.350
depends on the difference
of position x1 minus x2.
00:10:02.560 --> 00:10:05.410
So this is written here.
00:10:05.410 --> 00:10:08.610
And in fact, interestingly,
the position of the mass 1
00:10:08.610 --> 00:10:11.380
itself is a negative sign here.
00:10:11.380 --> 00:10:14.080
So if you move mass
1, the spring force
00:10:14.080 --> 00:10:17.926
is in the right
direction, minus kx.
00:10:17.926 --> 00:10:19.360
All right?
00:10:19.360 --> 00:10:21.520
So there is no
motion x1, so we can
00:10:21.520 --> 00:10:25.750
conclude from here the T cosine1
is approximately equal to 1.
00:10:25.750 --> 00:10:28.960
So T is simply equal to mg.
00:10:28.960 --> 00:10:33.250
So the tension in the spring
can be assumed to be mg.
00:10:33.250 --> 00:10:35.000
We don't have to worry about it.
00:10:35.000 --> 00:10:37.390
And then we just plug in--
00:10:37.390 --> 00:10:40.780
also the angle can be
converted into position
00:10:40.780 --> 00:10:46.630
by realizing that the
distance times the angle
00:10:46.630 --> 00:10:49.180
is equal to displacement,
the usual geometry.
00:10:49.180 --> 00:10:52.120
The net result is that
by simplifying things,
00:10:52.120 --> 00:10:58.990
I can write down
equations for acceleration
00:10:58.990 --> 00:11:03.760
in the horizontal
direction for mass 1
00:11:03.760 --> 00:11:18.920
is equal to minus mg x1
over l plus k x2 minus x1.
00:11:18.920 --> 00:11:19.900
OK?
00:11:19.900 --> 00:11:26.730
So this is an equation
of motion for mass 1
00:11:26.730 --> 00:11:29.980
in our coupled system.
00:11:29.980 --> 00:11:33.750
And I could say
most of the terms
00:11:33.750 --> 00:11:37.860
have to do with a
motion of mass 1 itself.
00:11:37.860 --> 00:11:40.620
Mass 1 is its own pendulum.
00:11:40.620 --> 00:11:46.080
And mass 1 is feeling the
effect of the spring force.
00:11:46.080 --> 00:11:48.420
But because the
force of the spring
00:11:48.420 --> 00:11:50.670
depends on the difference
between positions,
00:11:50.670 --> 00:11:53.010
there is this coupling--
00:11:53.010 --> 00:11:58.200
so the motion of mass 1
knows of where mass 2 is.
00:11:58.200 --> 00:12:03.310
And motion of mass 2 influences
the motion of mass 1.
00:12:03.310 --> 00:12:05.970
That's how the
coupling shows up.
00:12:05.970 --> 00:12:08.760
So for most of those
problems, what you do is
00:12:08.760 --> 00:12:13.220
you simply focus on
the mass in question.
00:12:13.220 --> 00:12:15.690
You take all the forces,
you calculate them,
00:12:15.690 --> 00:12:17.610
and then this
coupling will somehow
00:12:17.610 --> 00:12:20.250
appear in the equations.
00:12:20.250 --> 00:12:26.080
So we can repeat exactly
the same calculation
00:12:26.080 --> 00:12:29.020
focusing on mass 2.
00:12:29.020 --> 00:12:33.210
And then the equation which you
will get will be very similar.
00:12:33.210 --> 00:12:36.820
Let me just slightly
rewrite this equation here
00:12:36.820 --> 00:12:40.390
to kind of combine
all the terms which
00:12:40.390 --> 00:12:42.420
depend on the position
of mass 1 with terms
00:12:42.420 --> 00:12:44.172
that depend on mass 2.
00:12:44.172 --> 00:12:53.910
So where m x-acceleration
is equal to minus k
00:12:53.910 --> 00:13:06.040
plus mg over l times
x1 plus k times x2.
00:13:06.040 --> 00:13:07.740
So this is the coupling term.
00:13:13.680 --> 00:13:17.710
This is what makes
those pendula coupled.
00:13:17.710 --> 00:13:18.370
All right?
00:13:18.370 --> 00:13:23.740
And then I can write almost
exactly the same equation
00:13:23.740 --> 00:13:31.420
of mass 2 with the proper
replacement of masses.
00:13:31.420 --> 00:13:37.080
So let me write this down
in the following way-- kx1
00:13:37.080 --> 00:13:45.420
minus k plus mg over l times x2.
00:13:50.440 --> 00:13:54.650
So the motion of
mass x1 depends on x1
00:13:54.650 --> 00:13:57.440
itself multiplied by
something with a spring
00:13:57.440 --> 00:13:59.710
term and gravitational
term and depends
00:13:59.710 --> 00:14:04.250
on the position of mass 2
only through the spring.
00:14:04.250 --> 00:14:10.970
Mass 2 also is mostly driven
by its own gravitational force
00:14:10.970 --> 00:14:15.950
of itself plus the spring
depends on the position of x2.
00:14:15.950 --> 00:14:18.410
But there is this
coupling term that
00:14:18.410 --> 00:14:20.330
depends on position of mass 1.
00:14:20.330 --> 00:14:23.480
So both of them feel the
neighbor on the other side,
00:14:23.480 --> 00:14:24.290
right?
00:14:24.290 --> 00:14:29.770
So if I keep this one
steady of x2 equals 0,
00:14:29.770 --> 00:14:31.580
then basically the
forces here is just
00:14:31.580 --> 00:14:34.310
the spring plus the gravity.
00:14:34.310 --> 00:14:37.910
If I move this one and keep
this one at 0, the force on this
00:14:37.910 --> 00:14:40.000
spring spring and gravity.
00:14:40.000 --> 00:14:44.010
But if this one is displaced,
and I move that guy,
00:14:44.010 --> 00:14:47.050
the forces on this one
are affected by the fact
00:14:47.050 --> 00:14:49.500
that number 2 changed.
00:14:49.500 --> 00:14:50.000
OK?
00:14:50.000 --> 00:14:52.250
Again, I was able to
determine those coupling
00:14:52.250 --> 00:14:56.842
terms by simply looking at
mass 1 itself, mass 2 itself.
00:14:56.842 --> 00:15:00.350
All right, so this is the
set of two coupled equations.
00:15:00.350 --> 00:15:03.440
I have accelerations
here for x1, x2,
00:15:03.440 --> 00:15:05.240
and I have positions here.
00:15:05.240 --> 00:15:08.420
It's like an oscillator
of position acceleration
00:15:08.420 --> 00:15:13.130
with a constant term except that
things here are a little mixed.
00:15:13.130 --> 00:15:16.970
And the trick in this
whole mathematics,
00:15:16.970 --> 00:15:19.670
and calculations,
and the way we do
00:15:19.670 --> 00:15:25.820
things is how do you solve
those coupled equations?
00:15:25.820 --> 00:15:26.860
OK?
00:15:26.860 --> 00:15:30.660
So what I would like to do is--
00:15:30.660 --> 00:15:32.950
and there is multiple
ways of doing that.
00:15:32.950 --> 00:15:34.840
So let me do everything.
00:15:34.840 --> 00:15:38.250
Let's write down everything
in the matrix form,
00:15:38.250 --> 00:15:40.170
because it turns out
that linear matrices are
00:15:40.170 --> 00:15:41.400
very useful for that.
00:15:41.400 --> 00:15:43.060
We will use them very, very--
00:15:43.060 --> 00:15:44.410
in a very simple way.
00:15:44.410 --> 00:15:48.210
So let's introduce to
them and show vector,
00:15:48.210 --> 00:15:53.790
which consists of x1 and x2.
00:15:53.790 --> 00:15:58.880
So basically, all the
position x1 and x2 are here.
00:15:58.880 --> 00:16:02.900
So we will be monitoring
the change of this x2
00:16:02.900 --> 00:16:04.460
as a function of time.
00:16:04.460 --> 00:16:09.900
We will introduce
a force matrix k,
00:16:09.900 --> 00:16:25.440
which is equal to k plus mg over
l minus k here, minus k here,
00:16:25.440 --> 00:16:29.600
k plus mg over l there.
00:16:29.600 --> 00:16:32.340
This is a two by two matrix.
00:16:32.340 --> 00:16:37.140
And then we need a third
matrix, mass matrix,
00:16:37.140 --> 00:16:45.070
which simply says that masses
are mass of first object is m
00:16:45.070 --> 00:16:46.610
and the other one
is also m, right?
00:16:49.130 --> 00:16:52.520
So these are three
matrices that basically
00:16:52.520 --> 00:16:56.640
contains exactly the same
information as out there.
00:16:56.640 --> 00:16:58.110
I probably need another matrix.
00:16:58.110 --> 00:17:01.440
I need an inverse matrix
for mass, which basically
00:17:01.440 --> 00:17:05.900
is 1 over m, 1 over m, 0 and 0.
00:17:05.900 --> 00:17:10.630
This is a inverted matrix.
00:17:10.630 --> 00:17:20.069
OK, and it turns out that after
I introduced these matrices,
00:17:20.069 --> 00:17:23.810
this set of equations
can be written simply
00:17:23.810 --> 00:17:30.700
as X, the second derivative
of the vector capital X,
00:17:30.700 --> 00:17:36.890
is equal to minus
m to the minus 1,
00:17:36.890 --> 00:17:43.480
this matrix, multiplying
matrix k and then multiplying
00:17:43.480 --> 00:17:45.953
vectors x again.
00:17:49.801 --> 00:17:52.210
All right?
00:17:52.210 --> 00:17:58.880
So this is exactly the
same as this, just written
00:17:58.880 --> 00:18:00.720
a different way.
00:18:00.720 --> 00:18:02.380
So it's only the
question of notation.
00:18:02.380 --> 00:18:06.190
So it turns out
it's very convenient
00:18:06.190 --> 00:18:11.240
to use matrix calculation
to do things faster.
00:18:11.240 --> 00:18:14.510
So instead of repeating writing,
all the x1s, x2, et cetera,
00:18:14.510 --> 00:18:19.480
instead I just stick them into
one or two element objects.
00:18:19.480 --> 00:18:21.800
I use matrices to
multiply things,
00:18:21.800 --> 00:18:23.540
and if I want to
know x1 and x2, I
00:18:23.540 --> 00:18:26.540
can always go, OK,
the top component
00:18:26.540 --> 00:18:30.450
of vector x, lower component of
vector x gives me the solution.
00:18:30.450 --> 00:18:31.010
Simple.
00:18:31.010 --> 00:18:31.510
Right?
00:18:34.690 --> 00:18:42.210
So let's try to use this
terminology to find solutions.
00:18:42.210 --> 00:18:46.550
So the question is how
do we find solutions
00:18:46.550 --> 00:18:47.880
to coupled oscillations.
00:18:47.880 --> 00:18:53.150
What is the most efficient way
of finding the most general
00:18:53.150 --> 00:18:56.060
motion of a coupled system?
00:18:56.060 --> 00:18:57.380
Anybody knows?
00:18:57.380 --> 00:18:59.690
What's the first thing?
00:18:59.690 --> 00:19:01.002
Yes?
00:19:01.002 --> 00:19:02.430
AUDIENCE: [INAUDIBLE].
00:19:02.430 --> 00:19:03.940
BOLESLAW WYSLOUCH:
Introduce what?
00:19:03.940 --> 00:19:06.031
AUDIENCE: [INAUDIBLE]
using complex notation.
00:19:06.031 --> 00:19:07.156
BOLESLAW WYSLOUCH: Coupled?
00:19:07.156 --> 00:19:07.930
AUDIENCE: Complex.
00:19:07.930 --> 00:19:09.990
BOLESLAW WYSLOUCH:
Complex oscillation.
00:19:09.990 --> 00:19:11.860
Yes, that's right.
00:19:11.860 --> 00:19:15.240
So all right, let's do it.
00:19:15.240 --> 00:19:17.310
But hold on.
00:19:17.310 --> 00:19:22.640
But what form of oscillation?
00:19:22.640 --> 00:19:28.100
OK, all kinds of complex numbers
can write, but any particular--
00:19:28.100 --> 00:19:28.975
AUDIENCE: [INAUDIBLE]
00:19:28.975 --> 00:19:30.475
BOLESLAW WYSLOUCH:
That's something.
00:19:30.475 --> 00:19:32.450
That's the physics
answer, all right?
00:19:32.450 --> 00:19:36.080
Complex notation is a
mathematical answer,
00:19:36.080 --> 00:19:38.220
how to solve a
mathematical equation.
00:19:38.220 --> 00:19:43.970
But the physics answer is to
find fixed frequency modes us
00:19:43.970 --> 00:19:46.340
such that the system,
the complete system,
00:19:46.340 --> 00:19:48.720
oscillates at one frequency.
00:19:48.720 --> 00:19:50.670
Everybody moves together.
00:19:50.670 --> 00:19:53.050
This is so-called normal mode.
00:19:53.050 --> 00:19:55.670
It turns out that
every of the system,
00:19:55.670 --> 00:19:57.280
depending on number
of dimensions,
00:19:57.280 --> 00:20:03.880
will have a certain number of
frequencies, normal modes, that
00:20:03.880 --> 00:20:05.190
would--
00:20:05.190 --> 00:20:08.050
the whole system oscillates
at the same frequency,
00:20:08.050 --> 00:20:12.850
both x1 and x2, undergoing
motion of the same frequency.
00:20:12.850 --> 00:20:14.670
We don't know what
the frequency is.
00:20:14.670 --> 00:20:17.150
We don't know it's
amplitude, et cetera.
00:20:17.150 --> 00:20:19.450
But it is the same.
00:20:19.450 --> 00:20:19.950
OK?
00:20:23.930 --> 00:20:28.850
So this means that I can
write that the whole vector
00:20:28.850 --> 00:20:35.630
x, both x1 and x2, are
undergoing the same oscillatory
00:20:35.630 --> 00:20:36.650
motion.
00:20:36.650 --> 00:20:40.151
So I propose that--
00:20:40.151 --> 00:20:46.010
so of course, we use the
usual trick that anytime
00:20:46.010 --> 00:20:50.960
we have a solution
in complex variables,
00:20:50.960 --> 00:20:54.870
we can always get back to real
things by taking a real part.
00:20:54.870 --> 00:20:58.430
So I understand you've
done this before.
00:20:58.430 --> 00:21:02.660
So let's introduce
variable z, just kind
00:21:02.660 --> 00:21:12.910
of a two-element vector, which
has a complex term, a fixed
00:21:12.910 --> 00:21:16.690
frequency, plus a
phase, a rhythm complex,
00:21:16.690 --> 00:21:24.700
multiplying vector A,
a fixed vector A. OK?
00:21:24.700 --> 00:21:36.730
And vector A is simply has two
components, A1, A2, or maybe
00:21:36.730 --> 00:21:40.290
I should write it differently.
00:21:40.290 --> 00:21:45.170
So vector A contains
information about some sort
00:21:45.170 --> 00:21:51.910
of initial conditions
for position x 1 2.
00:21:51.910 --> 00:21:55.040
Anyway, these are
two constant numbers.
00:21:55.040 --> 00:21:59.690
And also, we will, because
we have this phase here,
00:21:59.690 --> 00:22:02.480
because we keep phase
in this expression,
00:22:02.480 --> 00:22:06.570
we can assume and require
that is a real number.
00:22:06.570 --> 00:22:07.510
So A is real.
00:22:10.960 --> 00:22:13.980
It's a slightly different
way of doing things,
00:22:13.980 --> 00:22:17.330
but we can assume
this for now, right?
00:22:17.330 --> 00:22:20.830
So the solution which
is written here--
00:22:20.830 --> 00:22:25.710
it's some two numbers,
oscillatory term,
00:22:25.710 --> 00:22:30.710
with both x1 and x2 oscillating
with the same frequency,
00:22:30.710 --> 00:22:32.520
and this is our
postulated solution.
00:22:32.520 --> 00:22:36.560
So we plug it into the
equation, and we adjust things
00:22:36.560 --> 00:22:38.570
until it fits.
00:22:38.570 --> 00:22:44.060
So let's plug this into
our matrix calculation.
00:22:44.060 --> 00:22:46.169
And what you see here is that--
00:22:46.169 --> 00:22:46.960
so what do we have?
00:22:46.960 --> 00:22:50.870
So this is the term,
which is second time
00:22:50.870 --> 00:22:54.340
the derivative vector
X. And because vector--
00:22:54.340 --> 00:22:55.960
or vector Z really.
00:22:55.960 --> 00:22:57.880
So I have to do--
00:22:57.880 --> 00:22:58.770
so I plug this here.
00:22:58.770 --> 00:23:08.460
So Z double dot is simply equal
minus omega squared times Z.
00:23:08.460 --> 00:23:09.498
Right?
00:23:09.498 --> 00:23:10.740
Like this.
00:23:10.740 --> 00:23:13.050
So this is a simple thing.
00:23:13.050 --> 00:23:16.080
When I plug this in
here, my equation
00:23:16.080 --> 00:23:24.280
becomes an equation for
A. So I have minus omega
00:23:24.280 --> 00:23:30.060
squared z-hat, which maybe
I just write it immediately
00:23:30.060 --> 00:23:34.730
in terms of a complex term
by times the vector A.
00:23:34.730 --> 00:23:45.570
So I have e to i omega t plus
y times A is equal to minus M
00:23:45.570 --> 00:23:57.071
to minus 1 K times e to the i
omega t plus phi times vector
00:23:57.071 --> 00:24:00.840
A. OK?
00:24:00.840 --> 00:24:04.680
And this term is
a proportionality
00:24:04.680 --> 00:24:06.870
constant at any
given moment of time.
00:24:06.870 --> 00:24:10.400
So it goes through the
matrix multiplication.
00:24:10.400 --> 00:24:11.730
So you can just delete this.
00:24:11.730 --> 00:24:13.950
You can divide both sides.
00:24:13.950 --> 00:24:16.980
You have signs here.
00:24:16.980 --> 00:24:19.870
And then I have
an equation which
00:24:19.870 --> 00:24:23.430
is a linear matrix
equation, which
00:24:23.430 --> 00:24:30.570
is M minus 1 K times vector A.
00:24:30.570 --> 00:24:33.840
And I can rewrite it
a little bit again.
00:24:33.840 --> 00:24:40.320
So I can rewrite in this
minus 1 K minus omega squared
00:24:40.320 --> 00:24:47.860
times unity matrix times
vector A is equal to 0.
00:24:47.860 --> 00:24:53.110
So this is the
equation which we need
00:24:53.110 --> 00:25:01.879
to solve to obtain the solutions
to at least one normal mode,
00:25:01.879 --> 00:25:04.420
and we expect that there will
be two normal modes, because we
00:25:04.420 --> 00:25:05.086
have two masses.
00:25:09.100 --> 00:25:11.680
So now, this is--
00:25:16.250 --> 00:25:19.390
so this is some matrix,
two by two matrix,
00:25:19.390 --> 00:25:22.120
which we can know very
easily how to write.
00:25:22.120 --> 00:25:26.070
Multiplying a
vector gives you 0.
00:25:26.070 --> 00:25:32.790
It turns out that for this
to work, there are two--
00:25:32.790 --> 00:25:37.860
there is a criterion,
which has to be satisfied,
00:25:37.860 --> 00:25:40.270
namely the determinant
of the two by two matrix
00:25:40.270 --> 00:25:43.740
has to be equal to 0,
because if you take
00:25:43.740 --> 00:25:47.550
the determinant on both sides,
you have to have 0 on this side
00:25:47.550 --> 00:25:49.410
to be able to obtain
0 on the other side.
00:25:49.410 --> 00:25:53.250
So mathematically, the way
to find out the oscillating
00:25:53.250 --> 00:25:58.730
frequency is you take a
determinant of m minus 1 K
00:25:58.730 --> 00:26:04.590
minus i omega squared
must be equal to 0.
00:26:04.590 --> 00:26:09.580
So let's try to see how
to calculate things.
00:26:09.580 --> 00:26:19.970
So let's write down this matrix
explicitly using this and that.
00:26:19.970 --> 00:26:23.840
So let's write this down.
00:26:23.840 --> 00:26:28.590
So I take a big
object like this.
00:26:28.590 --> 00:26:34.490
And so in this
element here, I have
00:26:34.490 --> 00:26:36.680
to multiply this
matrix times that.
00:26:36.680 --> 00:26:40.640
If I multiply this
matrix, I simply divide
00:26:40.640 --> 00:26:43.430
all those effectively
multiplication of m
00:26:43.430 --> 00:26:47.500
minus 1 times this matrix
divides all the elements here
00:26:47.500 --> 00:26:49.200
by m.
00:26:49.200 --> 00:26:50.290
That's all there is to it.
00:26:50.290 --> 00:26:52.210
I just divide everything by m.
00:26:54.800 --> 00:27:03.970
So the first M minus 1 K
is k over m plus g over l.
00:27:03.970 --> 00:27:14.260
This is minus k over m minus k
over m k over m plus g over l.
00:27:17.338 --> 00:27:18.790
So this is multiplication.
00:27:18.790 --> 00:27:21.500
This is this term here.
00:27:21.500 --> 00:27:24.450
And then I have to do minus
unity matrix times omega
00:27:24.450 --> 00:27:25.360
squared.
00:27:25.360 --> 00:27:32.690
All this will do is it will
subtract omega squared here.
00:27:32.690 --> 00:27:33.880
I should write this.
00:27:38.300 --> 00:27:39.480
OK?
00:27:39.480 --> 00:27:40.510
So this is in this one.
00:27:40.510 --> 00:27:46.610
Maybe it would be more clear
if I move it over here.
00:27:46.610 --> 00:27:48.480
All right, so this
is the matrix that
00:27:48.480 --> 00:27:51.760
contains all the information
about our system,
00:27:51.760 --> 00:27:54.725
the mass, the gravitational
acceleration, the length,
00:27:54.725 --> 00:27:57.090
the spring strength, et cetera.
00:27:57.090 --> 00:28:03.180
And we assumed they oscillate
with a fixed frequency.
00:28:03.180 --> 00:28:09.540
So I have to find the
determinant of this matrix
00:28:09.540 --> 00:28:11.790
equal to 0.
00:28:11.790 --> 00:28:13.330
So how do I get that?
00:28:13.330 --> 00:28:18.440
And by the way,
you have a matrix,
00:28:18.440 --> 00:28:21.560
and you want to make sure
that its determinant is 0.
00:28:21.560 --> 00:28:23.910
It turns out the
only variable which
00:28:23.910 --> 00:28:27.870
we have to change
parameters of this matrix--
00:28:27.870 --> 00:28:30.440
you know, the spring constant
and the mass this affects
00:28:30.440 --> 00:28:31.470
is given.
00:28:31.470 --> 00:28:34.440
The system has been built.
It's hanging over there.
00:28:34.440 --> 00:28:36.400
I cannot change anything.
00:28:36.400 --> 00:28:40.590
So the only parameter here,
which I can change, or adjust,
00:28:40.590 --> 00:28:43.560
or find is omega square.
00:28:43.560 --> 00:28:46.980
So I will try all possible
matrices of this type
00:28:46.980 --> 00:28:52.740
until I find one or two that
have a determinant equal to 0.
00:28:52.740 --> 00:28:55.430
But if I find them,
this would correspond
00:28:55.430 --> 00:28:58.905
to the normal frequencies.
00:28:58.905 --> 00:29:01.230
OK?
00:29:01.230 --> 00:29:04.770
So how do I calculate
the determinant of a two
00:29:04.770 --> 00:29:05.980
by two matrix?
00:29:05.980 --> 00:29:10.280
I do this by this minus
this by that, right?
00:29:10.280 --> 00:29:15.590
So that of this matrix
is equal to k over
00:29:15.590 --> 00:29:24.110
m minus g plus g over
l minus omega squared.
00:29:24.110 --> 00:29:27.280
The two identical terms
so I can put the square
00:29:27.280 --> 00:29:32.960
and then minus this minus
k squared over m squared
00:29:32.960 --> 00:29:35.620
must be equal to 0.
00:29:35.620 --> 00:29:36.120
Right?
00:29:36.120 --> 00:29:40.710
So this is the equation
which we need to solve.
00:29:40.710 --> 00:29:45.960
We need to find which
parameter omega sets this to 0.
00:29:45.960 --> 00:29:53.440
And then this is a pretty
straightforward calculation,
00:29:53.440 --> 00:29:57.610
except if I don't have--
00:29:57.610 --> 00:29:59.220
I'll just use this one.
00:30:02.650 --> 00:30:05.080
OK, so let's rewrite
this a little bit.
00:30:05.080 --> 00:30:08.680
So this is basically equivalent
to the following equation
00:30:08.680 --> 00:30:16.120
g over l plus k over
m minus omega squared
00:30:16.120 --> 00:30:21.080
must be equal either to
plus or minus k over m.
00:30:21.080 --> 00:30:21.580
Right?
00:30:21.580 --> 00:30:23.960
I took a square
root of both sides.
00:30:23.960 --> 00:30:25.690
If you take a square
root, you have
00:30:25.690 --> 00:30:29.150
to worry about plus
and minus signs, right?
00:30:29.150 --> 00:30:32.200
So there are two solutions
which corresponds to plus here.
00:30:32.200 --> 00:30:34.660
The other one corresponds
to minus here.
00:30:34.660 --> 00:30:37.840
So solution number
1, which corresponds
00:30:37.840 --> 00:30:42.760
to plus sign right
here, it basically
00:30:42.760 --> 00:30:49.340
says that omega squared
is equal to g over l.
00:30:49.340 --> 00:30:49.840
Right?
00:30:49.840 --> 00:30:53.620
So there is one solution,
one oscillation,
00:30:53.620 --> 00:30:55.780
that does not depend
on the spring constant,
00:30:55.780 --> 00:30:58.390
because the spring
constant cancels.
00:30:58.390 --> 00:31:01.960
And there's a second
solution which
00:31:01.960 --> 00:31:09.340
corresponds to minus, where
omega squared is equal to g
00:31:09.340 --> 00:31:14.640
over l plus 2k over m.
00:31:17.630 --> 00:31:18.440
Right?
00:31:18.440 --> 00:31:21.740
Because there are two
possible solutions.
00:31:21.740 --> 00:31:23.340
And this is what we have.
00:31:23.340 --> 00:31:26.970
So we have a--
00:31:26.970 --> 00:31:33.420
so what this says is that if I
set my frequency to g over l,
00:31:33.420 --> 00:31:36.450
if I set the system to
oscillate to this frequency,
00:31:36.450 --> 00:31:40.260
then it will be--
00:31:40.260 --> 00:31:45.030
I will be able to set things up
such that it oscillates forever
00:31:45.030 --> 00:31:48.730
at this frequency, one
fixed frequency forever.
00:31:48.730 --> 00:31:49.730
And this is interesting.
00:31:49.730 --> 00:31:50.420
This is a frequency.
00:31:50.420 --> 00:31:52.530
It does not depend on the
strength of the spring.
00:31:52.530 --> 00:31:54.500
How is it possible?
00:31:54.500 --> 00:31:57.380
Somehow spring is
irrelevant for this motion.
00:31:57.380 --> 00:32:00.060
And it turns out that there
is a very simple oscillation,
00:32:00.060 --> 00:32:02.570
easy to see, if
basically that this
00:32:02.570 --> 00:32:05.160
is a frequency of
a single pendulum.
00:32:05.160 --> 00:32:06.680
So basically, you
got both pendula
00:32:06.680 --> 00:32:10.520
going together,
each of them happily
00:32:10.520 --> 00:32:13.160
oscillating by themselves.
00:32:13.160 --> 00:32:16.020
And the spring is completely
irrelevant for this motion.
00:32:16.020 --> 00:32:18.700
If I cut it off, the
motion will not change.
00:32:18.700 --> 00:32:22.030
It just happens that two
identical pendula are going
00:32:22.030 --> 00:32:24.570
at their own natural frequency.
00:32:24.570 --> 00:32:26.600
So the force of
spring is irrelevant.
00:32:26.600 --> 00:32:27.360
Nothing happens.
00:32:27.360 --> 00:32:28.730
This is a normal mode.
00:32:28.730 --> 00:32:32.770
And it can go forever at
this particular frequency.
00:32:32.770 --> 00:32:33.690
OK?
00:32:33.690 --> 00:32:38.120
The other option is
usually symmetrically.
00:32:38.120 --> 00:32:41.280
I move them away
from each other.
00:32:41.280 --> 00:32:45.200
And this is the motion where,
again, it's not exactly
00:32:45.200 --> 00:32:50.710
ideal small angle oscillation,
but let me try again,
00:32:50.710 --> 00:32:52.730
I guess with less.
00:32:52.730 --> 00:32:54.830
So this is the situation
where the spring really
00:32:54.830 --> 00:32:56.920
comes in at full force.
00:32:56.920 --> 00:33:00.144
It's being stretched
maximally, because they go away
00:33:00.144 --> 00:33:00.810
from each other.
00:33:00.810 --> 00:33:03.360
So very quickly, the
spring is stretched.
00:33:03.360 --> 00:33:07.060
And they go together so it's
stretch from both sides.
00:33:07.060 --> 00:33:10.940
And the whole system oscillates
at the same frequency,
00:33:10.940 --> 00:33:15.760
and because of this
additional force of spring,
00:33:15.760 --> 00:33:19.670
the frequency is actually
higher, it's larger.
00:33:19.670 --> 00:33:23.260
It oscillates faster.
00:33:23.260 --> 00:33:25.210
All right, so that's
the first step
00:33:25.210 --> 00:33:26.590
in understanding the system.
00:33:26.590 --> 00:33:30.370
We now know that there are
two oscillations and two
00:33:30.370 --> 00:33:31.880
normal frequencies.
00:33:31.880 --> 00:33:34.600
And the next step to
finish our understanding
00:33:34.600 --> 00:33:37.560
of the system in a mathematical
way, to describe it fully,
00:33:37.560 --> 00:33:40.810
I have to know what is
the shape of oscillations.
00:33:40.810 --> 00:33:44.140
I simply showed you here
so you know what to expect.
00:33:44.140 --> 00:33:48.910
But I have to be able to dig
it out from the equations.
00:33:48.910 --> 00:33:53.800
And the way to dig it out
is to find vector A. See,
00:33:53.800 --> 00:34:01.360
our real equation of
motion is up here.
00:34:01.360 --> 00:34:02.950
This is an equation of motion.
00:34:02.950 --> 00:34:08.710
This is, I have to now
find the vector A, which
00:34:08.710 --> 00:34:11.620
when you plug it in, it works--
00:34:11.620 --> 00:34:13.810
it satisfies this equation.
00:34:13.810 --> 00:34:16.929
So I already know what are
the two possible omegas--
00:34:16.929 --> 00:34:21.219
they can do it, but still
I have to find vector A.
00:34:21.219 --> 00:34:23.650
So I have to solve two
separate independent problems.
00:34:23.650 --> 00:34:26.600
One is finding vector
A for this situation
00:34:26.600 --> 00:34:28.900
and then find the vector
A for that situation
00:34:28.900 --> 00:34:30.070
and see if it works.
00:34:30.070 --> 00:34:31.969
So I had to plug in the whole.
00:34:31.969 --> 00:34:34.980
I had to plug it into
the whole equation.
00:34:34.980 --> 00:34:38.440
And you can show
that if you set--
00:34:38.440 --> 00:34:45.730
if you set omega squared to g
over l, and you plug it into--
00:34:45.730 --> 00:34:47.860
if you plug it into
this equation, what
00:34:47.860 --> 00:34:50.230
you get is a matrix
equation which
00:34:50.230 --> 00:34:56.600
looks like this-- k
over m minus k over m
00:34:56.600 --> 00:35:01.050
minus k over m k over m.
00:35:01.050 --> 00:35:03.320
And this is because--
00:35:03.320 --> 00:35:08.760
[I try to-- so if you plug omega
squared here equal to g over
00:35:08.760 --> 00:35:14.120
l, then this cancels out,
and this cancels out.
00:35:14.120 --> 00:35:16.420
So you plug it in
here, and you get
00:35:16.420 --> 00:35:22.660
this very simple, very simple
matrix that has k over m terms.
00:35:22.660 --> 00:35:25.210
So the question is
what sort of thing
00:35:25.210 --> 00:35:29.660
can you put here to get 0.
00:35:29.660 --> 00:35:34.750
What kind of vector you can
plug into those two places
00:35:34.750 --> 00:35:40.450
such that the matrix times
vector will end up with 0?
00:35:40.450 --> 00:35:45.250
One example is that basically
amplitude is the same.
00:35:45.250 --> 00:35:47.070
Both of them move together.
00:35:47.070 --> 00:35:51.131
So you plug 1 here and 1 here.
00:35:51.131 --> 00:35:51.630
Right?
00:35:51.630 --> 00:35:54.890
So this is a good solution.
00:35:54.890 --> 00:35:58.880
And every other solution
is a linear multiplication
00:35:58.880 --> 00:36:01.220
of this one for this
frequency, right?
00:36:01.220 --> 00:36:04.790
There is k over m times 1
minus k over m gives you 0.
00:36:04.790 --> 00:36:07.280
So this is a good solution for--
00:36:07.280 --> 00:36:10.250
so this is solution number 1.
00:36:10.250 --> 00:36:12.350
What about this thing here?
00:36:12.350 --> 00:36:23.650
If I plug this omega
squared into this matrix,
00:36:23.650 --> 00:36:26.300
it's g over l plus 2k over m.
00:36:26.300 --> 00:36:32.330
If I plug it in here, then this
matrix is way more complicated.
00:36:32.330 --> 00:36:34.470
It will actually
look very similar,
00:36:34.470 --> 00:36:37.780
but with important differences.
00:36:37.780 --> 00:36:42.030
So this one will
look minus k over m
00:36:42.030 --> 00:36:49.346
minus k over m minus k
over m minus k over m.
00:36:49.346 --> 00:36:52.240
OK?
00:36:52.240 --> 00:36:58.430
And then again, for this second
possible normal frequency,
00:36:58.430 --> 00:37:01.310
I have to find the vector
A, which corresponds
00:37:01.310 --> 00:37:02.510
to that frequency motion.
00:37:02.510 --> 00:37:07.260
And it turns out that they are
the same, but the sign changes.
00:37:07.260 --> 00:37:13.620
So one possible solution
is 1 and minus 1.
00:37:13.620 --> 00:37:18.450
If I plug in 1 minus 1, then
this matrix times the vector
00:37:18.450 --> 00:37:20.620
gives you automatically 0.
00:37:20.620 --> 00:37:23.680
So this is the second
possible normal mode.
00:37:23.680 --> 00:37:24.640
All right?
00:37:24.640 --> 00:37:28.390
So this is a systematic
way to solve equations.
00:37:28.390 --> 00:37:31.090
You plug in all
the information you
00:37:31.090 --> 00:37:34.240
know about the system
into a two by two matrix.
00:37:34.240 --> 00:37:38.110
And then you calculate
the normal mode.
00:37:38.110 --> 00:37:41.284
And then you calculate a
shape of a normal mode.
00:37:43.950 --> 00:37:47.770
Is that clear?
00:37:47.770 --> 00:37:51.570
Any questions at this time?
00:37:51.570 --> 00:37:52.710
Right?
00:37:52.710 --> 00:37:55.240
So in principle, we know,
now, at the end of the day,
00:37:55.240 --> 00:37:58.824
I still want to know how much
1 moves, how much 2 moves.
00:37:58.824 --> 00:38:00.240
So we have to put
it all together.
00:38:00.240 --> 00:38:04.560
We have identified the frequency
and the kind of, in the matrix
00:38:04.560 --> 00:38:06.420
notation, shape of the node.
00:38:06.420 --> 00:38:08.130
But of course,
the final solution
00:38:08.130 --> 00:38:11.790
is a linear superposition
of all possible normal modes
00:38:11.790 --> 00:38:14.560
with described
position of mass 1,
00:38:14.560 --> 00:38:15.810
position of mass 2, et cetera.
00:38:15.810 --> 00:38:19.590
So let's do a little bit of--
00:38:19.590 --> 00:38:23.700
so maybe graphically I can
write down that this is the--
00:38:23.700 --> 00:38:27.300
this is the oscillation
that corresponds
00:38:27.300 --> 00:38:32.130
to this type of mode, to those
two masses move together.
00:38:32.130 --> 00:38:34.920
And this is oscillation
that corresponds to the mode
00:38:34.920 --> 00:38:40.240
where masses move in
opposite directions.
00:38:40.240 --> 00:38:45.120
At any moment of time,
in this normal mode,
00:38:45.120 --> 00:38:49.360
at any moment of time,
wherever mass 1 is,
00:38:49.360 --> 00:38:53.830
mass 2 is minus the distance
away from its own equilibrium.
00:38:53.830 --> 00:38:56.080
So if this one is plus
1 centimeter here,
00:38:56.080 --> 00:38:57.580
the other one is
minus 1 centimeter.
00:38:57.580 --> 00:38:58.770
This one is minus 5.
00:38:58.770 --> 00:39:01.250
This one is plus 5 and so on.
00:39:01.250 --> 00:39:05.396
Whereas in this mode, both
of them move together.
00:39:05.396 --> 00:39:06.140
All right?
00:39:06.140 --> 00:39:11.730
So let's try to go back to the--
00:39:11.730 --> 00:39:13.510
you can get rid of this one.
00:39:13.510 --> 00:39:16.740
Let's try to go back, and
now with this knowledge,
00:39:16.740 --> 00:39:24.154
let's write down x1 and x2 for
positions of the two masses.
00:39:33.270 --> 00:39:45.306
So x1-- so basically,
the x will have to be--
00:39:45.306 --> 00:39:46.630
I used z there.
00:39:46.630 --> 00:39:52.510
So x will be real of vector z.
00:39:52.510 --> 00:39:55.850
So I take my complex numbers
and take a real of them.
00:39:55.850 --> 00:39:59.470
So from an exponent, I
will end up with a cosine
00:39:59.470 --> 00:40:00.640
appropriately and so on.
00:40:00.640 --> 00:40:03.980
And then I will use
the [INAUDIBLE]..
00:40:03.980 --> 00:40:10.030
So this is real part
of e to the i omega
00:40:10.030 --> 00:40:14.530
plus phi where omega is one
of the two possibilities.
00:40:14.530 --> 00:40:21.980
Omega t plus phi times vector
A, which we've identified here,
00:40:21.980 --> 00:40:26.200
and times some additional--
00:40:26.200 --> 00:40:30.700
these are those vectors A
this is one possible amplitude
00:40:30.700 --> 00:40:31.830
of notation.
00:40:31.830 --> 00:40:34.090
But in general, it
can be anything.
00:40:34.090 --> 00:40:34.810
You can multiply.
00:40:34.810 --> 00:40:38.210
You can have small oscillations,
large oscillations.
00:40:38.210 --> 00:40:40.000
So there is some
overall amplitude.
00:40:40.000 --> 00:40:41.660
But the shape always
has to be simple.
00:40:41.660 --> 00:40:45.430
They either go together,
or they go opposite.
00:40:45.430 --> 00:40:46.880
So to make it more
general, I have
00:40:46.880 --> 00:40:48.680
to give some multiplicative
factor there.
00:40:51.670 --> 00:40:56.560
So if I do everything, I
end up with x, the mode 1
00:40:56.560 --> 00:41:00.610
will in general have some
sort of overall constant C,
00:41:00.610 --> 00:41:08.900
cosine omega 1 t plus phi
1 times the vector 1, 1.
00:41:08.900 --> 00:41:10.220
This will be for x1.
00:41:10.220 --> 00:41:12.020
This will be for x2.
00:41:12.020 --> 00:41:21.660
And the mode number 2 will
be C2 cosine omega 2 times
00:41:21.660 --> 00:41:27.170
t plus phi 2 times 1 minus 1.
00:41:27.170 --> 00:41:28.970
All right?
00:41:28.970 --> 00:41:32.150
So let's see what
things are adjustable
00:41:32.150 --> 00:41:34.110
and what things are fixed.
00:41:34.110 --> 00:41:37.370
So the omega 1 and
omega 2 are fixed
00:41:37.370 --> 00:41:41.900
given by the construction of
the two coupled oscillators.
00:41:41.900 --> 00:41:44.060
This shape, 1 and
1, and 1 minus 1
00:41:44.060 --> 00:41:47.060
is fixed, because these are the
shape of normal modes, which
00:41:47.060 --> 00:41:48.930
corresponds to
those frequencies.
00:41:48.930 --> 00:41:52.670
So we have only four
constants-- overall amplitude c1
00:41:52.670 --> 00:41:54.040
for normal mode 1.
00:41:54.040 --> 00:41:56.930
Overall amplitude c2 for
normal mode 2 and then
00:41:56.930 --> 00:42:01.010
the relative phase of
those two normal modes.
00:42:01.010 --> 00:42:03.260
And the superposition
of x1 plus x2
00:42:03.260 --> 00:42:08.790
gives you the most general
combination of possible motion.
00:42:08.790 --> 00:42:11.090
So if I write this
down now in terms
00:42:11.090 --> 00:42:14.540
of position of number
1 and number 2,
00:42:14.540 --> 00:42:18.332
so I have a position of
x1 as a function of time.
00:42:18.332 --> 00:42:19.790
In general, it will
look like this.
00:42:19.790 --> 00:42:24.690
It will be some sort of
constant alpha, cosine omega 1 t
00:42:24.690 --> 00:42:32.720
plus phi plus constant
beta cosine omega 2 times
00:42:32.720 --> 00:42:36.610
t plus phi 2 plus phi 1.
00:42:36.610 --> 00:42:40.670
So mass number 1, this
is position of mass 1,
00:42:40.670 --> 00:42:45.400
will in general be a
superposition of the two
00:42:45.400 --> 00:42:47.740
possible oscillations.
00:42:47.740 --> 00:42:54.000
The position of mass 2
will be very similar,
00:42:54.000 --> 00:42:55.970
but there will be a very
important difference
00:42:55.970 --> 00:43:00.820
between the alpha
cosine omega 1 t
00:43:00.820 --> 00:43:08.340
plus phi 1 minus beta
cosine omega 2t plus phi 2.
00:43:10.870 --> 00:43:13.900
This is very important
to understand exactly how
00:43:13.900 --> 00:43:16.350
this equation came about.
00:43:16.350 --> 00:43:19.470
You see, this is the influence
of the symmetric mode,
00:43:19.470 --> 00:43:21.550
where the two
things are together.
00:43:21.550 --> 00:43:23.880
So they are multiplied
by alpha, some sort
00:43:23.880 --> 00:43:30.550
of arbitrary constant, but
with exactly the same sign.
00:43:30.550 --> 00:43:32.290
And this is the part
which corresponds
00:43:32.290 --> 00:43:37.040
to a second mode, which is
with different frequencies.
00:43:37.040 --> 00:43:39.600
And there is an opposite
sign between this amplitude
00:43:39.600 --> 00:43:41.250
and that amplitude.
00:43:41.250 --> 00:43:47.100
So you have only
four coefficients--
00:43:47.100 --> 00:43:53.970
alpha, beta, phi 1, and phi
2, which are determined,
00:43:53.970 --> 00:43:56.028
which need initial conditions.
00:44:03.210 --> 00:44:06.770
So any arbitrary mode-- this
is the most general motion
00:44:06.770 --> 00:44:10.550
of the two coupled
oscillator systems.
00:44:10.550 --> 00:44:14.100
And to describe it in
specifically-- defined
00:44:14.100 --> 00:44:16.250
for a specific
configuration, you
00:44:16.250 --> 00:44:20.540
will have to determine the
values of alphas and phis.
00:44:20.540 --> 00:44:22.560
OK?
00:44:22.560 --> 00:44:28.940
So what I want to do is I want
to write down a specific motion
00:44:28.940 --> 00:44:31.790
for the following situation.
00:44:31.790 --> 00:44:37.040
So I keep position of x1 at 0.
00:44:37.040 --> 00:44:41.410
It's not moving, so
the velocity is 0.
00:44:41.410 --> 00:44:45.760
I displaced this one by
a small positive amount.
00:44:45.760 --> 00:44:51.270
So the position of number 2 at
t equals 0 is different than 0--
00:44:51.270 --> 00:44:53.970
some displacement
x0 or something.
00:44:53.970 --> 00:44:55.645
And its velocity is 0.
00:44:55.645 --> 00:44:56.520
And then I let it go.
00:44:59.230 --> 00:45:03.690
Again, this is not the ideal
decoupled oscillator, right?
00:45:03.690 --> 00:45:08.220
OK, and then you see
the things start moving.
00:45:08.220 --> 00:45:11.980
Let me try to show it again,
because it's not exactly here,
00:45:11.980 --> 00:45:13.520
so this one will be going on.
00:45:13.520 --> 00:45:15.580
So let's say this
one is running,
00:45:15.580 --> 00:45:18.550
and then I let this one go.
00:45:18.550 --> 00:45:23.390
And what you see here is
that this one is moving,
00:45:23.390 --> 00:45:24.740
and then that starts to move.
00:45:24.740 --> 00:45:26.170
This one stops.
00:45:26.170 --> 00:45:27.630
That starts moving.
00:45:27.630 --> 00:45:29.460
It starts being
complicated, right?
00:45:29.460 --> 00:45:31.860
It's kind of complicated motion.
00:45:31.860 --> 00:45:35.070
But whatever this
motion is, we know
00:45:35.070 --> 00:45:38.880
that it's simply those cosines
which are kind of adding up
00:45:38.880 --> 00:45:42.330
to give you this impression of
rather a complicated motion,
00:45:42.330 --> 00:45:43.360
right?
00:45:43.360 --> 00:45:46.515
So again, I let this one out.
00:45:46.515 --> 00:45:47.710
I let it go.
00:45:47.710 --> 00:45:50.870
This might be 0.
00:45:50.870 --> 00:45:52.160
So this one slows down.
00:45:52.160 --> 00:45:55.630
This starts going.
00:45:55.630 --> 00:45:57.280
And this one then slows down.
00:45:57.280 --> 00:45:59.290
The other one starts going.
00:45:59.290 --> 00:46:02.600
They kind of talk to each other.
00:46:02.600 --> 00:46:05.670
And it's this
combination of cosines.
00:46:05.670 --> 00:46:08.120
All right, so let's try
to write to simplify
00:46:08.120 --> 00:46:11.530
this for a specific case of
specific initial conditions.
00:46:26.910 --> 00:46:34.000
So I said x1 equals 0, to
equal 0 x1 velocity at 0
00:46:34.000 --> 00:46:35.480
is equal to 0.
00:46:35.480 --> 00:46:37.790
So those ones are not moving.
00:46:37.790 --> 00:46:46.470
X2 at 0 is equal to some sort
of x0 and x2 velocity at 0
00:46:46.470 --> 00:46:47.780
is equal to 0.
00:46:47.780 --> 00:46:49.110
So this one is displaced.
00:46:49.110 --> 00:46:50.510
They are all stationary.
00:46:50.510 --> 00:46:51.670
This one is at position 0.
00:46:51.670 --> 00:46:56.540
If I plug this in, it turns
out without lots of details
00:46:56.540 --> 00:47:02.060
that what you will get
to is that alpha will
00:47:02.060 --> 00:47:06.690
be equal to x0 divided by 2.
00:47:06.690 --> 00:47:10.850
Beta will be equal to
minus x0 divided by 2.
00:47:10.850 --> 00:47:14.840
And phi 1 will be equal
to phi 2 equal to 0.
00:47:14.840 --> 00:47:16.100
You can check.
00:47:16.100 --> 00:47:18.380
If you plug it into
those equations,
00:47:18.380 --> 00:47:23.120
if you plug t equals 0, phi
is equal to 0, et cetera,
00:47:23.120 --> 00:47:24.650
you will see that it works.
00:47:24.650 --> 00:47:28.850
So you can write down the
specific case of x1 of t
00:47:28.850 --> 00:47:41.726
to be x0 over 2 cosine omega1
t minus cosine omega2 t.
00:47:41.726 --> 00:47:44.640
It's because beta
has a negative sign.
00:47:44.640 --> 00:47:57.770
And x2 of t will be equal
to x0 over 2 cosine omega1 t
00:47:57.770 --> 00:48:01.860
plus cosine omega2 t.
00:48:01.860 --> 00:48:03.640
OK?
00:48:03.640 --> 00:48:06.580
So each of those objects
effectively feels
00:48:06.580 --> 00:48:09.460
the effects of
omega 1 and omega 2,
00:48:09.460 --> 00:48:11.860
but in a slightly different way.
00:48:11.860 --> 00:48:14.470
That's why their relative
motions are different.
00:48:14.470 --> 00:48:19.600
So what I will do now is I
will show you an animation.
00:48:19.600 --> 00:48:22.180
Hopefully, it works.
00:48:22.180 --> 00:48:24.970
And we will have time--
00:48:24.970 --> 00:48:28.850
since on the computer, you
can make things perfect.
00:48:28.850 --> 00:48:29.380
Let's do it.
00:48:29.380 --> 00:48:33.640
So I'll have-- running
a MatLab simulation.
00:48:33.640 --> 00:48:36.210
Let's see how it goes.
00:48:36.210 --> 00:48:38.560
Large.
00:48:38.560 --> 00:48:40.680
So what is going on
here is the following.
00:48:40.680 --> 00:48:42.050
I took some initial conditions.
00:48:42.050 --> 00:48:43.850
I'm not sure if it's
exactly the same.
00:48:43.850 --> 00:48:47.050
This was for the course
that I taught some time ago.
00:48:47.050 --> 00:48:49.840
What you see here
is the following--
00:48:49.840 --> 00:48:56.660
you have the green is the
normal mode, number 1.
00:48:59.570 --> 00:49:03.810
The magenta is
normal mode number 2.
00:49:03.810 --> 00:49:07.080
And blue and the red
are the actual pendula.
00:49:09.900 --> 00:49:10.740
All right?
00:49:10.740 --> 00:49:13.970
And the motion of
blue and red is simply
00:49:13.970 --> 00:49:16.920
a linear sum of the two.
00:49:16.920 --> 00:49:18.510
And what you see here is--
00:49:18.510 --> 00:49:21.590
and then I plot the
position of the blue and red
00:49:21.590 --> 00:49:24.390
in color, the function of time.
00:49:24.390 --> 00:49:25.190
So you see this--
00:49:25.190 --> 00:49:28.750
the fact that let's
say red is now stopped,
00:49:28.750 --> 00:49:30.640
and the blue is at maximum.
00:49:30.640 --> 00:49:34.890
And now, the red is picking up.
00:49:34.890 --> 00:49:38.430
And now the blue
stopped, and the red
00:49:38.430 --> 00:49:40.890
is going full swing, et cetera.
00:49:40.890 --> 00:49:43.070
And this is exactly what--
00:49:43.070 --> 00:49:46.269
this is the computer simulation
that shows you that one of them
00:49:46.269 --> 00:49:48.060
is going up, the other
one down, et cetera.
00:49:48.060 --> 00:49:49.874
And this is for the
certain combination
00:49:49.874 --> 00:49:50.790
of initial conditions.
00:49:50.790 --> 00:49:54.210
I could go change initial
conditions in my program
00:49:54.210 --> 00:49:55.830
and have a different behavior.
00:49:55.830 --> 00:49:59.850
But whatever happens,
I would be able to--
00:49:59.850 --> 00:50:05.430
it will always be a
combination of the two motions.
00:50:05.430 --> 00:50:14.150
Now, is there a way to disable
one of the normal modes?
00:50:14.150 --> 00:50:16.204
How would you disable
one of the normal modes?
00:50:19.452 --> 00:50:21.400
Is there a quick
way to set things
00:50:21.400 --> 00:50:26.700
up such that the second normal
mode, whichever you choose,
00:50:26.700 --> 00:50:28.450
doesn't show up in
their equations at all?
00:50:34.294 --> 00:50:36.729
AUDIENCE: You said [INAUDIBLE].
00:50:36.729 --> 00:50:37.703
BOLESLAW WYSLOUCH: Hmm?
00:50:37.703 --> 00:50:40.138
AUDIENCE: [INAUDIBLE]
00:50:44.934 --> 00:50:46.600
BOLESLAW WYSLOUCH:
Yeah, so what you do,
00:50:46.600 --> 00:50:48.500
is you just change the
initial conditions.
00:50:48.500 --> 00:50:50.170
So you set it up
at T equal to 0.
00:50:50.170 --> 00:50:55.690
I have initial conditions
that basically favor or demand
00:50:55.690 --> 00:50:59.320
that only in this
general equation
00:50:59.320 --> 00:51:02.080
either alpha or
beta is equal to 0.
00:51:02.080 --> 00:51:08.740
So for example, one possibility
is I move both of them
00:51:08.740 --> 00:51:11.680
at the same distance, and I
just let them go like this such
00:51:11.680 --> 00:51:15.010
that the spring is
irrelevant, right?
00:51:15.010 --> 00:51:17.679
How would I do it in my program?
00:51:17.679 --> 00:51:18.220
I don't know.
00:51:18.220 --> 00:51:21.440
I can, for example--
00:51:21.440 --> 00:51:28.510
I can, for example, set one
of the initial conditions to--
00:51:31.730 --> 00:51:32.670
this is still running.
00:51:32.670 --> 00:51:35.230
The old one is still running.
00:51:35.230 --> 00:51:40.151
So this is the moment.
00:51:40.151 --> 00:51:42.400
So what I did is I just
changed the initial condition.
00:51:44.960 --> 00:51:49.370
And you see, this is the type
of motion where one of the modes
00:51:49.370 --> 00:51:52.415
has stopped, just
you switched it off,
00:51:52.415 --> 00:51:54.540
and the other one is going
on, and then, of course,
00:51:54.540 --> 00:51:57.030
the total motion
is equal to that.
00:51:57.030 --> 00:51:59.720
And both of them happily go
with a constant amplitude.
00:51:59.720 --> 00:52:05.340
There is no shifting of
energy from one to another.
00:52:05.340 --> 00:52:10.130
So you can have all kinds of
motions by simply adjusting
00:52:10.130 --> 00:52:11.060
initial conditions.
00:52:11.060 --> 00:52:16.640
And those motions can be
done a very different way.
00:52:16.640 --> 00:52:19.810
So do you know--
00:52:19.810 --> 00:52:23.780
so this is how we can have
different shape of motion,
00:52:23.780 --> 00:52:26.300
depending on the
initial condition.
00:52:26.300 --> 00:52:30.830
Is there another
way for me to change
00:52:30.830 --> 00:52:34.551
the way this system behaves?
00:52:34.551 --> 00:52:35.300
Let's say I take--
00:52:35.300 --> 00:52:37.740
I have exactly
this system, and I
00:52:37.740 --> 00:52:41.220
want to change, for example,
the frequency of oscillations.
00:52:41.220 --> 00:52:44.200
How will I do it?
00:52:44.200 --> 00:52:46.838
It could be a very
expensive proposition, yes?
00:52:46.838 --> 00:52:47.786
AUDIENCE: Drive it?
00:52:47.786 --> 00:52:51.290
BOLESLAW WYSLOUCH: Yes, but
I don't want to drive it yet.
00:52:51.290 --> 00:52:54.100
I just want to have
it free oscillation.
00:52:54.100 --> 00:52:55.014
Yes?
00:52:55.014 --> 00:52:57.309
AUDIENCE: [INAUDIBLE]
00:52:57.309 --> 00:52:58.850
BOLESLAW WYSLOUCH:
Yeah, I could come
00:52:58.850 --> 00:53:00.930
and scratch it
away a little bit.
00:53:00.930 --> 00:53:03.580
And yes, the equations
depends on the mass.
00:53:03.580 --> 00:53:04.720
But I don't want to touch.
00:53:04.720 --> 00:53:06.494
I want to just have this thing.
00:53:06.494 --> 00:53:08.410
I don't want to make any
physical modification
00:53:08.410 --> 00:53:09.610
to the system.
00:53:09.610 --> 00:53:12.302
However, I can move it
into different places,
00:53:12.302 --> 00:53:14.260
any place you can think
of where I could really
00:53:14.260 --> 00:53:16.440
modify the solution.
00:53:16.440 --> 00:53:17.083
Yeah?
00:53:17.083 --> 00:53:18.030
AUDIENCE: To the moon.
00:53:18.030 --> 00:53:19.696
BOLESLAW WYSLOUCH:
To the moon, exactly.
00:53:19.696 --> 00:53:22.250
I could put it with me some
spaceship, and go to a place
00:53:22.250 --> 00:53:25.160
where the gravity
is different, right?
00:53:25.160 --> 00:53:26.690
Why not?
00:53:26.690 --> 00:53:28.480
So what would happen?
00:53:28.480 --> 00:53:35.440
So if gravity changes, then
basically what will happen
00:53:35.440 --> 00:53:38.044
is both this term and
that term will change.
00:53:38.044 --> 00:53:39.460
The spring will
remained the same.
00:53:39.460 --> 00:53:40.960
The mass will remain the same.
00:53:40.960 --> 00:53:44.560
So the relative magnitude
of omega 1 and omega 2
00:53:44.560 --> 00:53:46.900
will change.
00:53:46.900 --> 00:53:48.130
OK?
00:53:48.130 --> 00:53:51.910
So let's say, in fact, do
I have it in this one here?
00:53:51.910 --> 00:53:52.430
Yes.
00:53:52.430 --> 00:53:56.060
So let's say I do again.
00:53:56.060 --> 00:53:59.840
So this is what I
had before, right?
00:53:59.840 --> 00:54:03.770
So this is the one here which
is operating here on earth,
00:54:03.770 --> 00:54:05.690
and I let it go.
00:54:05.690 --> 00:54:07.610
I displaced it by
a certain distance.
00:54:07.610 --> 00:54:10.310
Let's say 1 millimeter,
and that's how it's gone.
00:54:10.310 --> 00:54:12.985
So now, let's take it
to, for example, Jupiter.
00:54:15.560 --> 00:54:17.860
So what do you think will
happen when we go to Jupiter.
00:54:20.510 --> 00:54:24.710
Jupiter, g, is much larger.
00:54:24.710 --> 00:54:26.560
OK?
00:54:26.560 --> 00:54:27.810
So what would happen to those?
00:54:32.350 --> 00:54:35.320
So the frequency
would be larger.
00:54:35.320 --> 00:54:37.860
Things will be faster, right?
00:54:37.860 --> 00:54:39.760
That's the higher frequency.
00:54:39.760 --> 00:54:43.474
But also the difference
between two frequencies
00:54:43.474 --> 00:54:44.140
will be smaller.
00:54:46.419 --> 00:54:48.460
And what happens when the
difference in frequency
00:54:48.460 --> 00:54:50.459
is smaller?
00:54:50.459 --> 00:54:52.500
You saw that there's the
fact that the energy was
00:54:52.500 --> 00:54:55.140
moving from one to the other.
00:54:55.140 --> 00:54:56.940
The thing would take--
00:54:56.940 --> 00:54:59.760
so one of them was oscillating,
the other one is stationary,
00:54:59.760 --> 00:55:02.251
then the other one would
pick up, et cetera.
00:55:02.251 --> 00:55:03.750
Do you think this
transfer of energy
00:55:03.750 --> 00:55:04.800
will be faster or slower?
00:55:08.650 --> 00:55:13.030
Two omegas closer to each other.
00:55:13.030 --> 00:55:15.540
Any guesses?
00:55:15.540 --> 00:55:16.290
AUDIENCE: Smaller.
00:55:16.290 --> 00:55:17.915
BOLESLAW WYSLOUCH:
Take kind of longer.
00:55:17.915 --> 00:55:19.560
Let's see what happens, right?
00:55:19.560 --> 00:55:24.489
So we go on the rocket,
and nowadays, you
00:55:24.489 --> 00:55:25.780
don't have to go to the rocket.
00:55:25.780 --> 00:55:26.779
Just remove one comment.
00:55:29.400 --> 00:55:32.220
And I went from about 10
meters per square second
00:55:32.220 --> 00:55:37.539
to 25 meters per square second,
and this is what is happening.
00:55:37.539 --> 00:55:38.080
Look at this.
00:55:38.080 --> 00:55:41.992
So first of all, this
identical system-- everything
00:55:41.992 --> 00:55:42.700
at the same time.
00:55:42.700 --> 00:55:43.283
It's the same.
00:55:43.283 --> 00:55:47.420
And so you see that
oscillations are much faster.
00:55:47.420 --> 00:55:51.940
So a number of amplitude
changes per second is larger.
00:55:51.940 --> 00:55:55.860
But it takes much
longer for the energy.
00:55:55.860 --> 00:55:57.340
So the red one is now stopping.
00:55:57.340 --> 00:56:01.480
It's now slowly coming up.
00:56:01.480 --> 00:56:04.470
So because the two frequencies
are closer to each other,
00:56:04.470 --> 00:56:07.560
they stay--
00:56:07.560 --> 00:56:12.890
it takes longer for them to
shift from one to the other.
00:56:12.890 --> 00:56:14.030
OK?
00:56:14.030 --> 00:56:16.670
So we are done at Jupiter.
00:56:16.670 --> 00:56:21.350
Let's now go to the Moon,
which has much lower
00:56:21.350 --> 00:56:23.772
gravitational acceleration.
00:56:23.772 --> 00:56:24.730
Let's see what happens.
00:56:27.990 --> 00:56:33.530
Again by logical
argument-- if something--
00:56:33.530 --> 00:56:36.110
so the smaller
gravitation accelerations
00:56:36.110 --> 00:56:39.830
means that the
frequency is now lower.
00:56:39.830 --> 00:56:43.370
So the pendula will move slower.
00:56:43.370 --> 00:56:45.779
However, the difference
between frequency
00:56:45.779 --> 00:56:47.570
will be larger, because
the spring is still
00:56:47.570 --> 00:56:48.981
the same strength.
00:56:48.981 --> 00:56:51.230
So it turns out that even
though everything is slower,
00:56:51.230 --> 00:56:54.840
but the energy transfer
will actually be faster.
00:56:54.840 --> 00:56:59.130
So let's try to see what
happens on the Moon.
00:56:59.130 --> 00:57:00.425
It's OK.
00:57:03.964 --> 00:57:09.040
It's a little bit not completely
clear what's going on,
00:57:09.040 --> 00:57:14.067
but you see, actually the motion
is kind of a little strange.
00:57:14.067 --> 00:57:14.900
Look at the red one.
00:57:14.900 --> 00:57:16.170
The red one is stopping.
00:57:16.170 --> 00:57:19.885
Then it's going halfway out.
00:57:19.885 --> 00:57:22.450
It looks kind of
messy, doesn't it?
00:57:22.450 --> 00:57:24.610
And so it doesn't show
up here very well,
00:57:24.610 --> 00:57:28.530
because the parameters have
changed so much that I have--
00:57:28.530 --> 00:57:39.990
I have those fixed
pictures which are--
00:57:39.990 --> 00:57:40.830
just a second.
00:57:40.830 --> 00:57:41.530
I'll show you.
00:57:45.790 --> 00:57:47.540
So this is the picture on the--
00:57:47.540 --> 00:57:51.530
some sort of stationary
picture on the Earth.
00:57:51.530 --> 00:57:53.730
I saw one of them
up, the other one--
00:57:53.730 --> 00:57:56.489
you see them shift
from one to the other.
00:57:56.489 --> 00:57:58.780
And you can see kind of the
frequency of how the energy
00:57:58.780 --> 00:58:00.400
shifts from one to the other.
00:58:00.400 --> 00:58:03.790
And also you can see the
frequency going up and down
00:58:03.790 --> 00:58:06.430
for the same exact conditions.
00:58:06.430 --> 00:58:10.060
This is now, just a
moment, this is a Jupiter.
00:58:10.060 --> 00:58:12.580
So Jupiter, you see that
the frequency itself it's
00:58:12.580 --> 00:58:13.970
much higher.
00:58:13.970 --> 00:58:20.190
And the energy transfer between
the two things takes longer.
00:58:20.190 --> 00:58:23.520
And on the Moon however,
the oscillations
00:58:23.520 --> 00:58:25.170
actually look really weird.
00:58:25.170 --> 00:58:27.180
This is an example
of one of them.
00:58:27.180 --> 00:58:32.370
It's kind of, you know, the two
frequencies are so far away,
00:58:32.370 --> 00:58:36.960
and it's really not even
a nice oscillatory motion.
00:58:36.960 --> 00:58:39.471
It's some sort of--
00:58:39.471 --> 00:58:43.530
it's much less
obvious that this is
00:58:43.530 --> 00:58:47.250
a superposition of two cosines,
because they kind of are
00:58:47.250 --> 00:58:48.920
exactly out of phase.
00:58:48.920 --> 00:58:51.270
So the motion is
kind of complete.
00:58:51.270 --> 00:58:52.620
Anyway, so this is--
00:58:52.620 --> 00:58:56.190
actually, so the lesson
is that the exact shape,
00:58:56.190 --> 00:58:59.810
the exact motion, we
know that can always be
00:58:59.810 --> 00:59:01.950
decomposed into simple motions.
00:59:01.950 --> 00:59:03.520
If you put them
together, things may
00:59:03.520 --> 00:59:05.340
get really interesting
and complicated,
00:59:05.340 --> 00:59:08.970
depending on what sort of
frequencies we are running
00:59:08.970 --> 00:59:10.620
and what sort of--
00:59:10.620 --> 00:59:14.670
what sort of initial
conditions we have.
00:59:14.670 --> 00:59:15.340
All right?
00:59:15.340 --> 00:59:15.970
Yes?
00:59:15.970 --> 00:59:18.490
Any questions?
00:59:18.490 --> 00:59:19.940
Yes?
00:59:19.940 --> 00:59:23.093
AUDIENCE: It's talking about
the center mass of the system
00:59:23.093 --> 00:59:24.920
or just one of the two --?
00:59:24.920 --> 00:59:26.461
BOLESLAW WYSLOUCH:
This one, I think,
00:59:26.461 --> 00:59:28.650
this one is just one of them.
00:59:28.650 --> 00:59:30.990
Actually, the one--
on the difference-- it
00:59:30.990 --> 00:59:32.890
normally doesn't matter.
00:59:32.890 --> 00:59:35.790
What matters this
is the frequency
00:59:35.790 --> 00:59:39.840
and how these move to the other.
00:59:39.840 --> 00:59:42.780
OK?
00:59:42.780 --> 00:59:45.891
Let's just forget about it.
00:59:45.891 --> 00:59:48.500
Just keep it.
00:59:48.500 --> 00:59:53.120
So let me now talk
about this thing, which
00:59:53.120 --> 00:59:57.650
is called beat phenomenon,
because when you look
00:59:57.650 --> 01:00:01.660
at the motion of one
of those objects,
01:00:01.660 --> 01:00:04.430
or the difference between
them or whatever, there's
01:00:04.430 --> 01:00:08.907
something kind of interesting
which can be extracted
01:00:08.907 --> 01:00:09.740
for those equations.
01:00:09.740 --> 01:00:11.400
Let's look at these
equations here.
01:00:11.400 --> 01:00:12.380
Let's look at mass 1.
01:00:16.960 --> 01:00:19.610
This is mass 1 and mass 2.
01:00:19.610 --> 01:00:23.143
So I can rewrite those solutions
a little bit different.
01:00:30.390 --> 01:00:34.030
And so what I want
to do is I want to--
01:00:34.030 --> 01:00:38.660
you see, this is a
difference of two cosines.
01:00:38.660 --> 01:00:40.700
This is a sum of two cosines.
01:00:40.700 --> 01:00:43.602
There are lots of neat
trigonometrical identities
01:00:43.602 --> 01:00:44.310
which we can use.
01:00:44.310 --> 01:00:46.520
So we just-- we do
zero physics here.
01:00:46.520 --> 01:00:50.790
We just rewrite the
trigonometrical formulas.
01:00:50.790 --> 01:00:54.540
So I do exactly this,
but I rewrite it.
01:00:54.540 --> 01:00:59.650
I use, for example, some of--
you have cosine alpha plus
01:00:59.650 --> 01:01:04.005
cosine beta is equal to--
01:01:04.005 --> 01:01:13.330
two cosine-- is equal to two
cosine alpha plus beta divided
01:01:13.330 --> 01:01:20.000
by 2 multiplied by cosine
alpha minus beta divided by 2.
01:01:20.000 --> 01:01:20.500
Right?
01:01:20.500 --> 01:01:23.950
That's the
trigonometric identity.
01:01:23.950 --> 01:01:24.940
Right?
01:01:24.940 --> 01:01:27.520
So let's just use this to write
this down and what you get
01:01:27.520 --> 01:01:29.060
is x1--
01:01:29.060 --> 01:01:40.560
x1 of t is equal to
minus x0 sine of omega 1
01:01:40.560 --> 01:01:50.430
plus omega 2 divided by 2 times
sine omega 1 minus omega 2
01:01:50.430 --> 01:01:54.240
divided by 2 times t.
01:01:54.240 --> 01:02:03.970
And x2 t is equal to x0,
some amplitude cosine omega 1
01:02:03.970 --> 01:02:11.410
plus omega 2 divided by 2
cosine omega 1 minus omega 2
01:02:11.410 --> 01:02:13.660
divided by t.
01:02:13.660 --> 01:02:17.230
So again, we did
zero physics here.
01:02:17.230 --> 01:02:20.252
We just rewrote the simple
trigonometric equations.
01:02:20.252 --> 01:02:22.210
But what you see is
something interesting here.
01:02:22.210 --> 01:02:27.580
So there is-- we have
those two frequencies which
01:02:27.580 --> 01:02:29.170
are playing a role.
01:02:29.170 --> 01:02:31.847
And for example, at Jupiter,
those two frequencies
01:02:31.847 --> 01:02:33.430
are actually very
close to each other,
01:02:33.430 --> 01:02:35.780
because everything is
dominated by the gravity,
01:02:35.780 --> 01:02:37.960
and we have a very weak spring.
01:02:37.960 --> 01:02:39.850
So the omega 1 and
omega 2 actually
01:02:39.850 --> 01:02:42.800
are very close to each other.
01:02:42.800 --> 01:02:45.650
So this thing, this
term here, kind of
01:02:45.650 --> 01:02:47.840
goes omega 1 plus
omega 2 divided by 2
01:02:47.840 --> 01:02:50.860
is like omega, right?
01:02:50.860 --> 01:02:55.520
100 plus 105 divided
by 2 is about 100.
01:02:55.520 --> 01:02:57.400
Whereas this one here
carries information
01:02:57.400 --> 01:03:00.420
about the difference
of frequencies--
01:03:00.420 --> 01:03:04.260
100, 102, the difference
is 2, which is very small.
01:03:04.260 --> 01:03:07.260
So how would this look like?
01:03:07.260 --> 01:03:10.910
So if you make a plot
under some conditions,
01:03:10.910 --> 01:03:17.750
you can, let's say,
so the two frequencies
01:03:17.750 --> 01:03:19.250
are close to each other.
01:03:24.810 --> 01:03:29.900
So if omega 1 is
close to omega 2--
01:03:29.900 --> 01:03:37.350
for example, omega 1 is
0.9 times omega 2, right?
01:03:37.350 --> 01:03:41.420
This is roughly what we have
on Earth in case of our system
01:03:41.420 --> 01:03:43.630
here.
01:03:43.630 --> 01:03:47.770
Then omega 1 plus
omega 2 divided 2
01:03:47.770 --> 01:03:55.400
would be about 0.95 omega
1, omega 2, I think,
01:03:55.400 --> 01:03:57.660
which is approximately
equal to omega 2
01:03:57.660 --> 01:04:02.880
or omega 1 and omega 1
minus omega 2 divided by 2
01:04:02.880 --> 01:04:07.340
will be about minus
0.05 times omega 2--
01:04:07.340 --> 01:04:12.100
much, much smaller than that.
01:04:12.100 --> 01:04:16.420
So we have-- so this term here--
01:04:16.420 --> 01:04:21.790
it basically oscillates
at the frequency of omega,
01:04:21.790 --> 01:04:24.900
of the frequency of the
individual pendulum.
01:04:24.900 --> 01:04:28.060
And the other term is
much, much smaller.
01:04:28.060 --> 01:04:29.010
How does this look?
01:04:29.010 --> 01:04:35.300
Well, it turns out that if
you make a sketch of this,
01:04:35.300 --> 01:04:38.972
if you do signs, for
example, it looks like this.
01:04:44.268 --> 01:04:44.768
OK?
01:04:47.666 --> 01:04:50.622
So there are in fact two--
01:04:50.622 --> 01:04:53.080
when you look at this picture,
you can see two frequencies.
01:04:53.080 --> 01:04:56.470
One which is clear the
oscillation of the--
01:04:56.470 --> 01:05:00.700
high-frequency oscillation
of things moving up and down.
01:05:00.700 --> 01:05:05.830
But there's also this kind
of overarching frequency
01:05:05.830 --> 01:05:08.800
of much smaller
frequency, and this
01:05:08.800 --> 01:05:12.290
is what corresponds to a
difference of two things.
01:05:12.290 --> 01:05:16.990
So in a sense, if you
look at this formula here,
01:05:16.990 --> 01:05:18.970
you have oscillation,
which is happening
01:05:18.970 --> 01:05:22.930
very quickly with a typical
oscillation of the system.
01:05:22.930 --> 01:05:26.780
But this is like a
modulation of the amplitude.
01:05:26.780 --> 01:05:29.440
So the amplitude of
the signal is changing.
01:05:29.440 --> 01:05:31.240
And this is what you see here.
01:05:31.240 --> 01:05:34.410
This is exactly the
picture out there.
01:05:34.410 --> 01:05:35.810
So the system oscillates.
01:05:35.810 --> 01:05:41.980
So one of those pendula,
either of them, is moving fast.
01:05:41.980 --> 01:05:43.240
But it's going faster.
01:05:43.240 --> 01:05:46.030
It's amplitude is larger,
and after some time,
01:05:46.030 --> 01:05:47.230
it slows down to 0.
01:05:47.230 --> 01:05:50.590
It goes higher and
slows down to 0.
01:05:50.590 --> 01:05:51.550
And you've seen this.
01:05:51.550 --> 01:05:57.700
We can do it again
here that both of them
01:05:57.700 --> 01:05:59.950
oscillate at roughly
the same frequency,
01:05:59.950 --> 01:06:03.430
but their individual
amplitudes are changing.
01:06:03.430 --> 01:06:06.020
And this transmission of--
01:06:06.020 --> 01:06:09.415
you know, one of them moving
full blast, the other one
01:06:09.415 --> 01:06:10.660
moving full blast.
01:06:10.660 --> 01:06:14.500
There's this kind of
frequency of energy
01:06:14.500 --> 01:06:19.720
moving from one to the other,
which is something called beat.
01:06:19.720 --> 01:06:23.470
This a beat system,
beat phenomenon somehow
01:06:23.470 --> 01:06:27.850
that energy is moving from
one place to another one.
01:06:27.850 --> 01:06:32.230
And we can have
some demonstration
01:06:32.230 --> 01:06:33.680
of how this happens.
01:06:33.680 --> 01:06:34.960
So we see this here.
01:06:34.960 --> 01:06:37.660
We see it on the pendula.
01:06:37.660 --> 01:06:41.864
We saw it on the
computer simulation.
01:06:41.864 --> 01:06:43.280
But now what we
are going to do is
01:06:43.280 --> 01:06:45.950
we're going to try
to hear it, right?
01:06:45.950 --> 01:06:50.030
So this is a demonstration
which maybe it works, maybe not.
01:06:50.030 --> 01:06:52.820
So let me-- it will work, OK?
01:06:52.820 --> 01:06:54.770
So let me explain what we have.
01:06:54.770 --> 01:06:58.860
So we have two speakers.
01:06:58.860 --> 01:07:06.675
And they basically go on very,
very similar frequencies,
01:07:06.675 --> 01:07:07.470
all right?
01:07:07.470 --> 01:07:11.340
So they both work at
similar frequencies.
01:07:14.436 --> 01:07:17.130
And so when I switched
on, you should hear--
01:07:17.130 --> 01:07:18.628
hear the sound.
01:07:18.628 --> 01:07:19.562
[HUM SOUND]
01:07:20.062 --> 01:07:20.969
OK?
01:07:20.969 --> 01:07:22.010
So this is the frequency.
01:07:22.010 --> 01:07:26.170
I believe it's just one of
them is working, and you know,
01:07:26.170 --> 01:07:28.680
this is just one
pendulum that is going
01:07:28.680 --> 01:07:31.350
on that given frequency, right?
01:07:31.350 --> 01:07:33.420
Then I will switch a
second loudspeaker.
01:07:36.390 --> 01:07:37.875
[HUM SOUND]
01:08:05.595 --> 01:08:08.070
Can you hear this kind of--
01:08:08.070 --> 01:08:09.060
wiggle?
01:08:09.060 --> 01:08:11.535
We'll change the
frequency a little.
01:08:15.495 --> 01:08:18.960
This is another frequency
of the original sound.
01:08:18.960 --> 01:08:22.490
And it's kind of the loudness of
the sound overall is changing.
01:08:26.482 --> 01:08:29.476
All right?
01:08:29.476 --> 01:08:31.971
This is faster.
01:08:31.971 --> 01:08:35.960
This is kind of extra,
extra sound which
01:08:35.960 --> 01:08:40.137
you hear is the difference of
mainly the frequency is not
01:08:40.137 --> 01:08:42.089
stable here, so I'll change it.
01:08:46.490 --> 01:08:48.319
Right?
01:08:48.319 --> 01:08:52.180
So this is, again, this is a
single one, perfectly constant
01:08:52.180 --> 01:08:54.939
frequency, no
change in amplitude,
01:08:54.939 --> 01:08:55.870
no change in loudness.
01:08:58.560 --> 01:09:01.892
Put them together, right?
01:09:01.892 --> 01:09:02.850
That's what they do.
01:09:02.850 --> 01:09:06.140
So if you have two, and I
can adjust the frequency,
01:09:06.140 --> 01:09:10.670
and the frequency is close,
then this frequency of changing
01:09:10.670 --> 01:09:12.000
is very slow.
01:09:12.000 --> 01:09:14.590
So you can actually hear it.
01:09:14.590 --> 01:09:16.819
Let me switch it off.
01:09:16.819 --> 01:09:18.500
So this is the effect of beats.
01:09:18.500 --> 01:09:26.359
I can maybe show you another
simulation of this works.
01:09:26.359 --> 01:09:29.200
Let's See.
01:09:29.200 --> 01:09:36.120
This one is oops, just a second.
01:09:36.120 --> 01:09:37.779
Let's see what it is.
01:09:42.170 --> 01:09:47.649
OK, so this is just
a single frequency.
01:09:47.649 --> 01:09:50.220
OK, again, I plot some pendulum.
01:09:50.220 --> 01:09:53.069
Then I can plot--
01:09:53.069 --> 01:09:56.618
sorry, no this one is this.
01:09:56.618 --> 01:10:03.923
I can-- this one.
01:10:03.923 --> 01:10:05.675
OK, we'll just plot it here.
01:10:09.680 --> 01:10:10.770
Maybe we can see.
01:10:10.770 --> 01:10:13.820
So there's a red one,
and there's a blue one.
01:10:13.820 --> 01:10:15.860
And I plot two
plots independently
01:10:15.860 --> 01:10:16.740
on top of each other.
01:10:16.740 --> 01:10:18.480
So they have an amplitude of 1.
01:10:18.480 --> 01:10:21.360
And clearly, you see that they
have a different frequency.
01:10:21.360 --> 01:10:23.810
So the red one is going
with some frequency.
01:10:23.810 --> 01:10:25.880
The blue one is going
with some other frequency.
01:10:25.880 --> 01:10:27.980
Sometimes they agree.
01:10:27.980 --> 01:10:30.170
Sometimes they do
not agree, right?
01:10:30.170 --> 01:10:33.250
And the places where they meet--
01:10:33.250 --> 01:10:34.680
they are on top of each other.
01:10:34.680 --> 01:10:38.400
This is where when you
add them up together,
01:10:38.400 --> 01:10:40.430
this is where they
will be large.
01:10:40.430 --> 01:10:42.710
In the places where
they're out of phase,
01:10:42.710 --> 01:10:44.560
they will cancel each other.
01:10:44.560 --> 01:10:47.210
So if you take two
of those together,
01:10:47.210 --> 01:10:50.680
same amplitude, just
slightly different frequency,
01:10:50.680 --> 01:10:52.056
and you simply make a linear--
01:10:57.020 --> 01:11:01.780
superposition of the two, you
will get exactly the beating
01:11:01.780 --> 01:11:02.280
effect.
01:11:02.280 --> 01:11:04.400
So I just took two
of those pictures
01:11:04.400 --> 01:11:07.040
before I added them together
and got exactly that.
01:11:07.040 --> 01:11:09.080
You have a maximum,
minima, et cetera.
01:11:09.080 --> 01:11:13.760
And you see this overall beat
frequency, and the carrier,
01:11:13.760 --> 01:11:16.100
it's called carrier frequency.
01:11:16.100 --> 01:11:20.390
And this is something that,
again, happens very often.
01:11:20.390 --> 01:11:23.700
There's another
demonstration here.
01:11:23.700 --> 01:11:27.470
I have two tuning
forks, and they
01:11:27.470 --> 01:11:29.390
are very similar frequency.
01:11:29.390 --> 01:11:33.900
So first, I will show you
that they are coupled.
01:11:33.900 --> 01:11:38.420
They are coupled
because I gave this guy
01:11:38.420 --> 01:11:40.810
some initial condition.
01:11:40.810 --> 01:11:41.640
It's going.
01:11:41.640 --> 01:11:43.440
Then I stop it.
01:11:43.440 --> 01:11:46.140
But there's still sound,
because the second one picked up
01:11:46.140 --> 01:11:48.350
some energy, and it took off.
01:11:48.350 --> 01:11:50.100
Of course, you don't see them.
01:11:50.100 --> 01:11:53.640
So basically, what I'm saying is
that I [TONE] give this energy.
01:11:53.640 --> 01:11:55.610
This one is
completely stationary.
01:11:55.610 --> 01:11:58.926
Now energy is slowly
moving to the other one.
01:11:58.926 --> 01:12:00.800
I stop this guy, and
this guy is still going.
01:12:03.160 --> 01:12:05.320
So the energy is
being transferred
01:12:05.320 --> 01:12:07.720
by this air oscillating here.
01:12:07.720 --> 01:12:12.070
The coupling goes through the
air to the sound here, right?
01:12:12.070 --> 01:12:13.960
And they have very
similar frequency.
01:12:13.960 --> 01:12:17.260
So they are nicely coupled.
01:12:17.260 --> 01:12:19.999
But what we can also do--
01:12:19.999 --> 01:12:21.478
we can [TONE].
01:12:25.261 --> 01:12:25.760
Right?
01:12:25.760 --> 01:12:27.620
So they're both going.
01:12:27.620 --> 01:12:28.702
Do you hear the beats?
01:12:31.594 --> 01:12:32.558
[TONE]
01:12:34.968 --> 01:12:36.900
Not really.
01:12:36.900 --> 01:12:41.040
In fact, if they would have
exactly identical frequency,
01:12:41.040 --> 01:12:41.540
right?
01:12:41.540 --> 01:12:43.790
If they will be
perfectly the same,
01:12:43.790 --> 01:12:46.290
then the difference
would be 0, and there
01:12:46.290 --> 01:12:47.400
will be no beats at all.
01:12:47.400 --> 01:12:49.290
The period of beats
will be infinitely long,
01:12:49.290 --> 01:12:52.186
so it will take forever
for us to hear anything.
01:12:52.186 --> 01:12:54.060
So what we can do-- we
can break one of them.
01:12:54.060 --> 01:12:58.750
We can add some sort of weight.
01:12:58.750 --> 01:12:59.800
Some are here.
01:12:59.800 --> 01:13:02.450
There's some magic place
where it works best.
01:13:02.450 --> 01:13:04.370
So what I would do is
I will break this one.
01:13:04.370 --> 01:13:06.590
I will modify its frequency.
01:13:06.590 --> 01:13:07.910
That's another way to modify.
01:13:07.910 --> 01:13:09.701
I don't have to go to
Jupiter to modify it,
01:13:09.701 --> 01:13:13.980
because this one is just
a little mass here, right?
01:13:13.980 --> 01:13:14.900
[TONE]
01:13:18.770 --> 01:13:19.440
Ah, cool.
01:13:19.440 --> 01:13:23.160
AUDIENCE: Is that [INAUDIBLE]?
01:13:23.160 --> 01:13:27.278
BOLESLAW WYSLOUCH: Really,
this is actually a huge effect.
01:13:27.278 --> 01:13:28.262
[TONE]
01:13:30.230 --> 01:13:32.930
You can clearly see that they
are going up and down, up
01:13:32.930 --> 01:13:36.560
and down, because the frequency
is slightly different.
01:13:36.560 --> 01:13:39.260
So now, this thing is probably--
01:13:39.260 --> 01:13:42.070
I know it's a period, a
fraction of a second, right?
01:13:42.070 --> 01:13:42.752
Yes?
01:13:42.752 --> 01:13:45.112
AUDIENCE: Should both of
those sine and cosines
01:13:45.112 --> 01:13:47.460
have Ts in their arguments?
01:13:47.460 --> 01:13:51.005
BOLESLAW WYSLOUCH:
Of course always.
01:13:51.005 --> 01:13:54.280
They are both time
dependent, yeah.
01:13:54.280 --> 01:13:57.080
This is the fast
thing, and this is
01:13:57.080 --> 01:14:02.100
this time-dependent
modulation, yeah.
01:14:02.100 --> 01:14:07.287
All right, so
where are my notes?
01:14:07.287 --> 01:14:11.110
So this is the--
01:14:11.110 --> 01:14:13.790
this is how the--
01:14:13.790 --> 01:14:15.910
so we were able to
set up the system,
01:14:15.910 --> 01:14:17.770
put in some of the
matrix equation,
01:14:17.770 --> 01:14:20.310
kind of solved it, found
two frequencies, et cetera.
01:14:20.310 --> 01:14:21.450
There is one more--
01:14:21.450 --> 01:14:23.500
one additional
trick, which you can
01:14:23.500 --> 01:14:29.140
do to describe the motion
of a coupled pendula.
01:14:29.140 --> 01:14:34.940
And that is, in a sense,
force mathematically,
01:14:34.940 --> 01:14:39.340
force the normal modes
from sort of early on, to
01:14:39.340 --> 01:14:42.720
instead of, so far, when
we talked about pendula,
01:14:42.720 --> 01:14:48.300
we describe their motion in
terms of motion of number
01:14:48.300 --> 01:14:50.503
1, motion of number 2.
01:14:50.503 --> 01:14:52.970
It turns out we can
rewrite the equation
01:14:52.970 --> 01:14:56.370
into some sort of
new variables, where,
01:14:56.370 --> 01:15:00.980
so-called normal coordinates,
where you'll simultaneously
01:15:00.980 --> 01:15:05.120
describe both of them
and then kind of mix
01:15:05.120 --> 01:15:08.470
them together to
have a new formula,
01:15:08.470 --> 01:15:11.090
just rewrite the equation
in terms of new variables.
01:15:11.090 --> 01:15:13.565
So you do change of variables.
01:15:13.565 --> 01:15:19.520
So instead of keeping track
of x1 and x2 independently,
01:15:19.520 --> 01:15:22.550
you define something
which I called
01:15:22.550 --> 01:15:30.300
u1, which is simply x1
plus x2, and I define
01:15:30.300 --> 01:15:36.020
u2, which is x1 minus x2.
01:15:36.020 --> 01:15:38.670
So instead of talking about
x1 and x2 independently,
01:15:38.670 --> 01:15:41.340
I have a sum of
them and difference.
01:15:41.340 --> 01:15:42.190
Why not?
01:15:42.190 --> 01:15:42.800
Right?
01:15:42.800 --> 01:15:43.960
Two variables.
01:15:43.960 --> 01:15:47.360
I can always go back and
get x1 and x2 if I want to.
01:15:47.360 --> 01:15:51.620
So if one tells me that u1
is 1 centimeter and u2 2
01:15:51.620 --> 01:15:54.230
centimeters, I can always
go and get x1 and x2
01:15:54.230 --> 01:15:55.700
if I want to, right?
01:15:55.700 --> 01:15:57.060
So I can do it.
01:15:57.060 --> 01:16:01.865
And it turns out that if
I plot those variables
01:16:01.865 --> 01:16:05.150
in, in other words, I take
the original equations, which
01:16:05.150 --> 01:16:09.660
I conveniently erased and
make a sum or difference,
01:16:09.660 --> 01:16:12.470
it turns out that this
coupling kind of separates.
01:16:12.470 --> 01:16:17.450
So I will end up having
two separate equations
01:16:17.450 --> 01:16:18.290
for this one.
01:16:18.290 --> 01:16:20.990
So in general, the
equation of motion
01:16:20.990 --> 01:16:23.870
would be-- would look
like, so let's say
01:16:23.870 --> 01:16:33.620
I can write down m x1 plus
x2 is equal to minus m g
01:16:33.620 --> 01:16:44.110
over l times x1 plus x2.
01:16:44.110 --> 01:16:47.780
OK, this is when I
add two equations.
01:16:47.780 --> 01:16:52.370
And the other equation
when I subtract them--
01:16:52.370 --> 01:17:06.676
minus x2 is equal to minus mg
over l plus 2k x1 minus x2.
01:17:06.676 --> 01:17:09.120
I think that's
what is coming out.
01:17:09.120 --> 01:17:15.520
So if I add and subtract the two
original equations of motion,
01:17:15.520 --> 01:17:17.530
which I don't know if
I have them somewhere,
01:17:17.530 --> 01:17:19.680
and you can look
back, then you end up
01:17:19.680 --> 01:17:25.170
having those crossed
terms drop out.
01:17:25.170 --> 01:17:28.110
And you have one, which has only
this coefficient, the other one
01:17:28.110 --> 01:17:30.930
which has that coefficient.
01:17:30.930 --> 01:17:33.480
And this immediately--
and it looks--
01:17:33.480 --> 01:17:37.050
if I now write it in terms
of normal coordinates,
01:17:37.050 --> 01:17:44.190
then I have that m u1 double
dot is equal to simply minus mg
01:17:44.190 --> 01:17:53.910
over l, u1, and m u2 double
dot is equal to minus mg over
01:17:53.910 --> 01:17:58.610
l plus 2k times u2.
01:17:58.610 --> 01:18:03.050
And if you look at
those two equations,
01:18:03.050 --> 01:18:04.990
it turns out that
they are not coupled.
01:18:04.990 --> 01:18:13.380
Each of them is a question
of a one-dimensional harmonic
01:18:13.380 --> 01:18:15.150
oscillator.
01:18:15.150 --> 01:18:17.120
The first part one
only depends on u1.
01:18:17.120 --> 01:18:20.940
The second one
only depends on u2.
01:18:20.940 --> 01:18:26.230
And you can see the oscillating
frequency with your own eyes.
01:18:26.230 --> 01:18:29.860
So no, the determinants needed
no matrices, no nothing.
01:18:29.860 --> 01:18:32.710
We just added and subtracted
the two equations,
01:18:32.710 --> 01:18:35.790
and things magically separated.
01:18:35.790 --> 01:18:37.060
All right?
01:18:37.060 --> 01:18:41.350
So sometimes, especially in case
of very simple and symmetric
01:18:41.350 --> 01:18:43.420
systems, if you
introduce new variables,
01:18:43.420 --> 01:18:45.200
you can simplify your
life tremendously,
01:18:45.200 --> 01:18:51.418
and these are called normal
variables, normal coordinates.
01:18:58.390 --> 01:19:01.380
And it turns out that
you can always do that.
01:19:01.380 --> 01:19:03.450
So you can always have
a linear combination
01:19:03.450 --> 01:19:07.470
of parameters for arbitrary
size coupled oscillators system
01:19:07.470 --> 01:19:11.510
where you combine
different coordinates,
01:19:11.510 --> 01:19:15.600
and you basically force
the system to behave
01:19:15.600 --> 01:19:21.630
in a way in which it induces
the single oscillation, single
01:19:21.630 --> 01:19:22.560
frequency.
01:19:22.560 --> 01:19:25.800
So this is, again, a
very powerful trick,
01:19:25.800 --> 01:19:28.470
but usually for most
cases, you can do that
01:19:28.470 --> 01:19:31.620
only after you have solved
it, after you've found out
01:19:31.620 --> 01:19:32.760
normal modes, et cetera.
01:19:32.760 --> 01:19:35.410
So after you know your normal
mode, then you can say, ha, ha,
01:19:35.410 --> 01:19:38.100
I can I can introduce
normal variables
01:19:38.100 --> 01:19:39.480
and make things simpler.
01:19:39.480 --> 01:19:42.660
But at the end of the day
for complicated systems
01:19:42.660 --> 01:19:44.520
that work is the same.
01:19:44.520 --> 01:19:47.910
But for simple systems like
this one where there is
01:19:47.910 --> 01:19:51.040
a good symmetry, you can do it.
01:19:51.040 --> 01:19:54.870
Anyway, so I think we
are done for today.
01:19:54.870 --> 01:19:59.370
And on Tuesday, we'll continue
with forced oscillators.
01:19:59.370 --> 01:19:59.870
All right?
01:19:59.870 --> 01:20:01.720
Thank you.