WEBVTT
00:00:02.195 --> 00:00:04.520
The following content is
provided under a Creative
00:00:04.520 --> 00:00:05.910
Commons license.
00:00:05.910 --> 00:00:08.119
Your support will help
MIT OpenCourseWare
00:00:08.119 --> 00:00:12.210
continue to offer high-quality
educational resources for free.
00:00:12.210 --> 00:00:14.750
To make a donation or to
view additional materials
00:00:14.750 --> 00:00:18.710
from hundreds of MIT courses,
visit MIT OpenCourseWare
00:00:18.710 --> 00:00:19.580
at ocw.mit.edu.
00:00:23.270 --> 00:00:27.110
YEN-JIE LEE: OK, so
welcome back, everybody.
00:00:27.110 --> 00:00:29.980
Welcome back to 8.03.
00:00:29.980 --> 00:00:34.400
Today, we are going to
continue the discussion
00:00:34.400 --> 00:00:36.600
of the harmonic oscillators.
00:00:36.600 --> 00:00:40.550
And also, we will add
damping force into the game
00:00:40.550 --> 00:00:43.160
and see what will happen, OK?
00:00:43.160 --> 00:00:45.950
So this is actually what
we have learned last time
00:00:45.950 --> 00:00:47.900
from this slide.
00:00:47.900 --> 00:00:52.700
We have analyzed the physics
of a harmonic oscillator, which
00:00:52.700 --> 00:00:54.680
we actually
demonstrated last time.
00:00:54.680 --> 00:00:57.590
And you can see the
device still there.
00:00:57.590 --> 00:01:01.020
And Hooke's law,
actually the Hooke's law
00:01:01.020 --> 00:01:05.180
is actually far more general
than what we saw before.
00:01:05.180 --> 00:01:09.340
It works for all
small oscillations
00:01:09.340 --> 00:01:16.170
around about a point of
equilibrium position, OK?
00:01:16.170 --> 00:01:23.190
And that can be demonstrated
by multiple different kinds
00:01:23.190 --> 00:01:24.960
of physical systems.
00:01:24.960 --> 00:01:30.630
For example here, I have a
mass, which actually can only
00:01:30.630 --> 00:01:33.600
move along this track here.
00:01:33.600 --> 00:01:37.350
And if I put this mass
set free, then this thing
00:01:37.350 --> 00:01:43.160
is actually exercising
harmonic oscillation, OK?
00:01:43.160 --> 00:01:44.660
We can do this with
large amplitude.
00:01:44.660 --> 00:01:48.540
We can also do it
with small amplitude.
00:01:48.540 --> 00:01:52.680
And you see that,
huh, really, it works.
00:01:52.680 --> 00:01:54.080
Hooke's law actually works.
00:01:54.080 --> 00:01:56.940
And it predicts
exactly the same motion
00:01:56.940 --> 00:02:02.040
as to what you see
on the slide, OK?
00:02:02.040 --> 00:02:06.690
And we also have a little
bit more complicated system.
00:02:06.690 --> 00:02:11.009
For example, this
is some kind of rod.
00:02:11.009 --> 00:02:14.980
And you can actually fix one
point and make it oscillate.
00:02:14.980 --> 00:02:18.120
And you see that,
huh, it also does
00:02:18.120 --> 00:02:21.430
some kind of
harmonic oscillation.
00:02:21.430 --> 00:02:25.630
But now, what is actually
oscillating is the amplitude.
00:02:25.630 --> 00:02:29.110
The amplitude is actually
the angle with respect
00:02:29.110 --> 00:02:32.510
to the downward direction.
00:02:32.510 --> 00:02:36.340
And finally this is actually
the vertical version
00:02:36.340 --> 00:02:40.420
of this spring mass
system, which you will be
00:02:40.420 --> 00:02:43.080
analyzing that in your P-set.
00:02:43.080 --> 00:02:46.850
And you see that, huh, it
actually oscillates up and down
00:02:46.850 --> 00:02:48.410
harmonically.
00:02:48.410 --> 00:02:51.400
So that's all very nice.
00:02:51.400 --> 00:02:54.280
And we also have
learned one thing which
00:02:54.280 --> 00:02:55.930
is very, very interesting.
00:02:55.930 --> 00:03:02.020
It's that a complex exponential
is actually a pretty beautiful
00:03:02.020 --> 00:03:04.390
way to present the solution.
00:03:04.390 --> 00:03:09.070
And you will see it works
also when describing
00:03:09.070 --> 00:03:10.470
the damped oscillators.
00:03:10.470 --> 00:03:17.310
And we will see how it
works in the lecture today.
00:03:17.310 --> 00:03:21.240
I received several questions
during my office hour
00:03:21.240 --> 00:03:23.790
and through email or Piazza.
00:03:23.790 --> 00:03:29.640
There were some confusions about
doing the Taylor expansion, OK?
00:03:29.640 --> 00:03:34.450
So in lecture last time,
the equilibrium position
00:03:34.450 --> 00:03:36.630
is at x equal to 0.
00:03:36.630 --> 00:03:41.730
Therefore, I do Taylor
expansion around 0, OK?
00:03:41.730 --> 00:03:45.210
But in this case, if
the equilibrium position
00:03:45.210 --> 00:03:50.850
or the minima of the
potential is at x equal to L,
00:03:50.850 --> 00:03:55.050
then what you need to do
is to do a Taylor expansion
00:03:55.050 --> 00:03:58.700
around x equal to L,
just to make that really,
00:03:58.700 --> 00:04:00.460
really clear, OK?
00:04:00.460 --> 00:04:04.950
OK, I hope that will help
you with the P-set question.
00:04:04.950 --> 00:04:07.780
OK, so let's get
started immediately.
00:04:07.780 --> 00:04:13.200
So let's continue the discussion
of the equation of motion
00:04:13.200 --> 00:04:14.740
we arrived at last time.
00:04:14.740 --> 00:04:22.690
So we have M x double-dot and
this is equal to minus kx, OK?
00:04:22.690 --> 00:04:25.120
That is actually the
formula from last time.
00:04:25.120 --> 00:04:35.470
And we can actually
calculate the kinetic energy
00:04:35.470 --> 00:04:38.230
of this spring and mass system.
00:04:38.230 --> 00:04:45.820
And basically, this is
going to be equal to 1/2 M
00:04:45.820 --> 00:04:50.400
times x dot squared.
00:04:50.400 --> 00:04:53.740
OK, and we can also calculate
the potential energy
00:04:53.740 --> 00:04:55.920
of the spring.
00:04:55.920 --> 00:05:10.760
Potential energy, and that
is equal to 1/2 kx squared.
00:05:10.760 --> 00:05:13.940
We also know what would
be the total energy.
00:05:13.940 --> 00:05:19.520
The total energy would be
a sum of the kinetic energy
00:05:19.520 --> 00:05:21.360
and of the potential.
00:05:21.360 --> 00:05:25.280
Basically, you get this
formula, 1/2 M x dot
00:05:25.280 --> 00:05:33.840
squared plus 1/2 kx squared.
00:05:33.840 --> 00:05:37.420
One last time, we have solved
this equation of motion, right?
00:05:37.420 --> 00:05:48.890
So the solution we got is x
equal to A cosine omega 0 t
00:05:48.890 --> 00:05:51.470
plus phi.
00:05:51.470 --> 00:05:59.160
Well, omega 0 is equal to
a square root of k over M.
00:05:59.160 --> 00:06:01.460
Therefore, we can
actually calculate
00:06:01.460 --> 00:06:06.790
what would be the total energy
as a function of time, right?
00:06:06.790 --> 00:06:08.940
So if we calculate
that, we'll get
00:06:08.940 --> 00:06:19.410
E will be equal to 1/2 M A
squared omega 0 squared sine
00:06:19.410 --> 00:06:23.850
squared omega 0 t plus phi--
00:06:23.850 --> 00:06:26.940
so this is actually
the first term here--
00:06:26.940 --> 00:06:39.210
plus 1/2 kA squared cosine
squared omega 0 t plus phi, OK?
00:06:39.210 --> 00:06:44.570
Then, we also know that
this coefficient here
00:06:44.570 --> 00:06:47.250
is just kA squared, right?
00:06:47.250 --> 00:06:50.970
Because omega 0 is actually
equal to the square root of k
00:06:50.970 --> 00:06:58.470
over M. And if you replace this
omega 0 squared by k over M,
00:06:58.470 --> 00:07:02.160
then you actually arrive
at kA squared, OK?
00:07:02.160 --> 00:07:03.750
So that is actually very good.
00:07:03.750 --> 00:07:07.470
So that means I can
simplify the total energy.
00:07:07.470 --> 00:07:12.690
And what we are going to
get is 1/2 kA squared.
00:07:12.690 --> 00:07:15.180
I can take this factor out.
00:07:15.180 --> 00:07:19.680
And that will give me, inside
these brackets, I will get sine
00:07:19.680 --> 00:07:27.910
squared omega 0 t plus phi
plus cosine squared omega
00:07:27.910 --> 00:07:31.410
0 t plus phi.
00:07:31.410 --> 00:07:36.430
And this is actually
equal to 1, right?
00:07:36.430 --> 00:07:39.990
Just a reminder, sine squared
of theta plus cosine squared
00:07:39.990 --> 00:07:43.000
of theta is always equal to 1.
00:07:43.000 --> 00:07:45.360
So that gives me
this result. This
00:07:45.360 --> 00:07:52.750
is actually 1/2 kA squared, OK?
00:07:52.750 --> 00:07:56.740
So that is actually the
result. What does that mean?
00:07:56.740 --> 00:08:04.840
That means, if I actually
pull this mass harder,
00:08:04.840 --> 00:08:09.600
so that initially it
has larger amplitude,
00:08:09.600 --> 00:08:12.630
then the total energy
is actually proportioned
00:08:12.630 --> 00:08:15.360
to amplitude squared, OK?
00:08:15.360 --> 00:08:18.630
So I am storing more
and more energy.
00:08:18.630 --> 00:08:23.230
If I increase the
amplitude even more,
00:08:23.230 --> 00:08:26.430
then I am storing the
energy in this system.
00:08:26.430 --> 00:08:28.430
And it's proportional
to A squared.
00:08:28.430 --> 00:08:32.200
And also, if the spring
constant is larger,
00:08:32.200 --> 00:08:36.142
the same amplitude will
give you more energy.
00:08:36.142 --> 00:08:38.475
So that means that you can
store more energy if you have
00:08:38.475 --> 00:08:41.950
a larger string constant, OK?
00:08:41.950 --> 00:08:46.230
The most surprising thing
is that actually this
00:08:46.230 --> 00:08:50.020
is actually a constant, right?
00:08:50.020 --> 00:08:51.610
What does that mean?
00:08:51.610 --> 00:08:55.450
The total energy is
actually not variating
00:08:55.450 --> 00:08:56.873
as a function of time.
00:08:56.873 --> 00:08:57.740
You see?
00:08:57.740 --> 00:09:00.970
So total energy is constant, OK?
00:09:00.970 --> 00:09:06.970
So you can see from this slide
the total energy is actually
00:09:06.970 --> 00:09:11.522
showing us the sum,
which is the green curve.
00:09:11.522 --> 00:09:13.480
And the kinetic energy
and the potential energy
00:09:13.480 --> 00:09:17.500
are shown as red
and blue curves.
00:09:17.500 --> 00:09:23.050
You can see that the total
energy is actually constant.
00:09:23.050 --> 00:09:25.900
But this system
is very dynamical.
00:09:25.900 --> 00:09:26.410
You see?
00:09:26.410 --> 00:09:31.840
So that energy is actually
going back and forth
00:09:31.840 --> 00:09:38.840
between the spring and the mass
in the form of kinetic energy
00:09:38.840 --> 00:09:41.200
and in the form of
potential energy.
00:09:41.200 --> 00:09:45.820
But they are doing it so well,
such that the sum is actually
00:09:45.820 --> 00:09:47.125
a constant.
00:09:47.125 --> 00:09:50.200
So the energy is
actually constant, OK?
00:09:50.200 --> 00:09:53.170
So that is actually
pretty beautiful.
00:09:53.170 --> 00:09:59.520
And it can be described very
well by these mathematics.
00:09:59.520 --> 00:10:00.550
Any questions from here?
00:10:04.380 --> 00:10:08.342
OK, so I would like to say
simple harmonic motion,
00:10:08.342 --> 00:10:09.800
actually, what you
are going to get
00:10:09.800 --> 00:10:12.780
is the energy is actually
conserved and independent
00:10:12.780 --> 00:10:14.300
of the time.
00:10:14.300 --> 00:10:18.130
And later, you will see
an example with damping.
00:10:18.130 --> 00:10:20.735
And you will see that
energy conservation
00:10:20.735 --> 00:10:24.320
is now no longer the case, OK?
00:10:24.320 --> 00:10:29.040
So let's immediately
jump to another example,
00:10:29.040 --> 00:10:33.270
which is actually involving
simple harmonic motion.
00:10:33.270 --> 00:10:41.100
So let's take this rod and
nail system as an example.
00:10:41.100 --> 00:10:44.660
If I actually slightly
move this rod,
00:10:44.660 --> 00:10:49.490
and then I release
that, then actually
00:10:49.490 --> 00:10:55.010
you will see simple harmonic
motion, also for this system.
00:10:55.010 --> 00:11:00.620
So let's actually do the
calculation as another example.
00:11:00.620 --> 00:11:02.720
So this is actually my system.
00:11:02.720 --> 00:11:05.920
I have this rod, OK?
00:11:05.920 --> 00:11:09.680
Now, I am assume that the
mass is actually uniformly
00:11:09.680 --> 00:11:15.370
distributed on this rod and
is nailed on the wall, OK?
00:11:15.370 --> 00:11:20.150
And the length of this
rod is actually l.
00:11:20.150 --> 00:11:25.430
So that means the center
of mass is actually at l/2
00:11:25.430 --> 00:11:29.600
with respect to the nail, OK?
00:11:29.600 --> 00:11:36.270
And also, this whole system
is set up on Earth, right?
00:11:36.270 --> 00:11:38.870
Therefore, there will
be gravitational force
00:11:38.870 --> 00:11:41.570
pointing downward, OK?
00:11:41.570 --> 00:11:46.600
So that means you have
gravitational force, Fg,
00:11:46.600 --> 00:11:49.670
pointing downward, OK?
00:11:49.670 --> 00:11:52.490
So this is actually
the system, which
00:11:52.490 --> 00:11:54.290
I would like to understand.
00:11:54.290 --> 00:11:58.490
And just a reminder, what are we
going to do afterwards in order
00:11:58.490 --> 00:12:02.420
to turn the whole system
into a language we
00:12:02.420 --> 00:12:04.640
know describes the nature?
00:12:04.640 --> 00:12:08.330
What are we going to do?
00:12:08.330 --> 00:12:09.040
Anybody?
00:12:13.360 --> 00:12:18.010
We are going to define
the coordinate system,
00:12:18.010 --> 00:12:20.980
so that I can
translate everything
00:12:20.980 --> 00:12:22.600
into mathematics, right?
00:12:22.600 --> 00:12:25.450
So that's actually what
we are always doing.
00:12:25.450 --> 00:12:26.950
And you will see
that we are always
00:12:26.950 --> 00:12:32.290
doing this in this class, OK?
00:12:32.290 --> 00:12:35.060
So what is actually
the coordinate system
00:12:35.060 --> 00:12:36.760
which I would like to use?
00:12:36.760 --> 00:12:42.100
Since this system is going to
be rotating back and forth,
00:12:42.100 --> 00:12:47.660
therefore, I would
like to define theta
00:12:47.660 --> 00:12:52.700
to be that angle with respect
to the axis, which essentially
00:12:52.700 --> 00:12:55.190
pointing downward, OK?
00:12:55.190 --> 00:13:02.510
So the origin of this coordinate
system uses theta equal to 0.
00:13:02.510 --> 00:13:14.100
This means that the rod is
actually pointing downward, OK?
00:13:14.100 --> 00:13:21.750
And also, I need to define what
is actually the positive value
00:13:21.750 --> 00:13:22.950
of the zeta, right?
00:13:22.950 --> 00:13:29.400
So I define anti-clockwise
direction to be positive, OK?
00:13:29.400 --> 00:13:34.370
So it is actually important
to actually first define that,
00:13:34.370 --> 00:13:38.170
then actually to translate
everything into mathematics,
00:13:38.170 --> 00:13:40.530
OK?
00:13:40.530 --> 00:13:42.880
So the initial condition
is the following.
00:13:42.880 --> 00:13:48.060
So I actually move this thing,
rotate this thing slightly.
00:13:48.060 --> 00:13:50.880
Then, I actually release
that really carefully
00:13:50.880 --> 00:13:55.140
without introducing any
initial velocity, OK?
00:13:55.140 --> 00:13:58.380
Therefore, I have two
initial conditions.
00:14:05.760 --> 00:14:11.750
OK, at t equal to 0, there
are two initial conditions.
00:14:11.750 --> 00:14:16.780
The first one is theta 0
is equal to theta initial.
00:14:21.130 --> 00:14:23.290
The second condition
is the same as what
00:14:23.290 --> 00:14:26.000
we have been doing last time.
00:14:26.000 --> 00:14:29.730
The initial velocity
or angular velocity
00:14:29.730 --> 00:14:30.980
is actually equal to 0.
00:14:30.980 --> 00:14:36.610
So that gives you theta
dot equal to 0, OK?
00:14:36.610 --> 00:14:43.030
Now, we have actually defined
the coordinate system.
00:14:43.030 --> 00:14:49.660
Now, we can actually
draw a force diagram,
00:14:49.660 --> 00:14:54.460
so that we can actually use
our knowledge about the physics
00:14:54.460 --> 00:14:57.020
to obtain the equation
of motion, right?
00:14:57.020 --> 00:14:59.680
So now, the force
diagram looks like this.
00:15:05.570 --> 00:15:10.420
So this is actually the
center of mass of this rod.
00:15:10.420 --> 00:15:14.300
And you have a force
pointing downward,
00:15:14.300 --> 00:15:17.540
which is due to the
gravitational force.
00:15:17.540 --> 00:15:21.660
Fg is equal to mg.
00:15:21.660 --> 00:15:23.620
It's pointing downward.
00:15:23.620 --> 00:15:28.960
The magnitude is
actually equal to mg.
00:15:28.960 --> 00:15:36.160
And also, we know the R vector.
00:15:36.160 --> 00:15:40.880
This vector has a length, l/2.
00:15:40.880 --> 00:15:49.228
It's pointing from the center
of mass of this rod to the nail,
00:15:49.228 --> 00:15:51.420
OK?
00:15:51.420 --> 00:15:57.720
And also, we know the angle
between these vectors,
00:15:57.720 --> 00:16:00.960
pointing from the center
of mass to the nail,
00:16:00.960 --> 00:16:04.920
and the vertical direction,
which we have already defined,
00:16:04.920 --> 00:16:07.180
which is actually called theta.
00:16:07.180 --> 00:16:10.260
Therefore, now, we
can actually calculate
00:16:10.260 --> 00:16:13.700
what would be the torque.
00:16:13.700 --> 00:16:19.970
Tau will be equal
to this R vector
00:16:19.970 --> 00:16:27.300
cross the force, total force
acting on the center mass.
00:16:27.300 --> 00:16:33.410
In this case, it's just Fg, OK?
00:16:33.410 --> 00:16:36.650
So now, we can actually
write this down explicitly.
00:16:36.650 --> 00:16:39.050
Since the whole
system is actually
00:16:39.050 --> 00:16:44.980
rotating on a single plane,
so there's only one plane
00:16:44.980 --> 00:16:46.590
this is sitting on.
00:16:46.590 --> 00:16:51.990
And it's actually going back and
forth only on this plane, OK?
00:16:51.990 --> 00:16:56.630
Therefore, actually, I
can drop all the arrows
00:16:56.630 --> 00:17:01.550
and write down the magnitude
of the tau directly.
00:17:01.550 --> 00:17:14.240
And this will be equal to
minus mg l/2 sine theta t, OK?
00:17:14.240 --> 00:17:15.319
Any questions so far?
00:17:20.079 --> 00:17:22.720
OK, so now, we have the torque.
00:17:22.720 --> 00:17:26.660
And we can make use of
the rotational version
00:17:26.660 --> 00:17:30.950
of Newton's Law to obtain the
equation of motion, right?
00:17:30.950 --> 00:17:35.180
So that should be
pretty straightforward.
00:17:35.180 --> 00:17:39.890
Tau will be equal to I, which
is the moment of inertia
00:17:39.890 --> 00:17:48.320
of the system,
times alpha t, OK?
00:17:48.320 --> 00:17:50.900
And just for your
information, I already
00:17:50.900 --> 00:17:56.490
calculated the I for you.
00:17:56.490 --> 00:18:02.970
I is equal to 1/3
ml squared, OK?
00:18:02.970 --> 00:18:06.060
So you can actually go
back home and actually do
00:18:06.060 --> 00:18:08.580
a check to see if I'm
telling the truth.
00:18:08.580 --> 00:18:11.890
And if you trust me,
then that's the answer,
00:18:11.890 --> 00:18:14.370
which is actually
1/3 ml squared,
00:18:14.370 --> 00:18:19.350
if the mass is actually
uniformly distributed
00:18:19.350 --> 00:18:22.050
on this rod, OK?
00:18:22.050 --> 00:18:37.410
So that would give me minus mgl
divided by 2 sine theta t, OK?
00:18:37.410 --> 00:18:43.140
So that is actually
coming from this side, OK?
00:18:43.140 --> 00:18:46.740
So now, I can actually
simplify this expression.
00:18:46.740 --> 00:18:50.550
I can now plug in the I
value into this equation.
00:18:50.550 --> 00:19:01.390
And I will get 1/3 ml squared
theta double-dot t, which
00:19:01.390 --> 00:19:02.880
is actually alpha, OK?
00:19:02.880 --> 00:19:05.250
Now, I write it as
theta double-dot.
00:19:05.250 --> 00:19:14.670
And that will be equal to minus
mgl over 2 sine theta, OK?
00:19:14.670 --> 00:19:17.470
I can move all the constants
to the right-hand side.
00:19:17.470 --> 00:19:20.970
Therefore, I get
theta double-dot t.
00:19:20.970 --> 00:19:23.610
This is equal to minus mgl.
00:19:27.380 --> 00:19:32.800
OK, actually, I can already
simplify this, right?
00:19:32.800 --> 00:19:34.400
These actually cancel.
00:19:34.400 --> 00:19:36.920
And the 1/l actually cancels.
00:19:36.920 --> 00:19:44.150
So therefore, I get minus
3g over 2 sine theta t.
00:19:48.050 --> 00:19:53.110
OK, as you know, we
actually defined omega
00:19:53.110 --> 00:19:57.380
to replace this constant
to make our life easier.
00:19:57.380 --> 00:20:05.050
So I can now define omega 0
equal to square root of 3g
00:20:05.050 --> 00:20:09.760
over 2l, OK?
00:20:09.760 --> 00:20:12.390
And that will give
you theta double-dot
00:20:12.390 --> 00:20:20.320
of t equal to minus omega
0 squared sin theta t.
00:20:24.610 --> 00:20:26.530
Any questions so far?
00:20:26.530 --> 00:20:28.820
A lot of calculations.
00:20:28.820 --> 00:20:31.480
But they should all be
pretty straightforward.
00:20:31.480 --> 00:20:35.900
And actually, we are done now.
00:20:35.900 --> 00:20:36.740
We are done.
00:20:36.740 --> 00:20:39.090
Because we have the
equation of motion.
00:20:39.090 --> 00:20:42.530
And the rest of the job
is to solve just it.
00:20:42.530 --> 00:20:46.040
So it is actually now the
problem of the math department.
00:20:46.040 --> 00:20:50.960
So can anybody actually tell me
the solution of the theta of t?
00:20:50.960 --> 00:20:54.025
Anybody?
00:20:54.025 --> 00:20:56.947
AUDIENCE: Unfortunately,
we'd have to approximate it.
00:20:56.947 --> 00:20:59.880
YEN-JIE LEE: That's
very unfortunate.
00:20:59.880 --> 00:21:04.600
So now, we are facing a
very difficult situation.
00:21:04.600 --> 00:21:08.970
We don't know how to solve
this equation in front of you.
00:21:08.970 --> 00:21:13.080
I don't know, OK?
00:21:13.080 --> 00:21:18.870
Of course, you can actually
solve it with a computer,
00:21:18.870 --> 00:21:23.020
or, if you want to go fancy,
solve it with your cellphone,
00:21:23.020 --> 00:21:24.020
if it doesn't explode.
00:21:28.740 --> 00:21:34.710
But it's not really nice
to do this in front of you.
00:21:34.710 --> 00:21:35.830
We don't learn too much.
00:21:35.830 --> 00:21:39.690
OK, so what are we going to do?
00:21:39.690 --> 00:21:45.720
So what we can do is actually
to consider a special case.
00:21:45.720 --> 00:21:50.930
So we know that this equation
of motion is exact, OK?
00:21:50.930 --> 00:21:53.780
So if you solve it,
it would describe
00:21:53.780 --> 00:21:57.350
the motion of this rod.
00:22:00.440 --> 00:22:04.920
Even with a large
angle, it works, OK?
00:22:04.920 --> 00:22:07.560
And now, in order
to actually show
00:22:07.560 --> 00:22:11.100
you the math in the
class, therefore
00:22:11.100 --> 00:22:14.090
actually I will do a
small approximation.
00:22:14.090 --> 00:22:17.130
So actually, I would
only work on the case
00:22:17.130 --> 00:22:21.390
that when the
amplitude is very small
00:22:21.390 --> 00:22:23.710
and see what is going to happen.
00:22:23.710 --> 00:22:27.150
So now, I'm considering
a special case.
00:22:27.150 --> 00:22:29.650
Up to now, everything is exact.
00:22:29.650 --> 00:22:34.340
And now, I am now going
to a special case.
00:22:34.340 --> 00:22:39.390
Theta t goes to 0, OK?
00:22:39.390 --> 00:22:41.390
Then, we can actually get this.
00:22:41.390 --> 00:22:48.020
Sine theta t is
roughly theta t, OK?
00:22:48.020 --> 00:22:50.000
Based on the Taylor
expansion, you
00:22:50.000 --> 00:22:53.870
can actually verify this, OK?
00:22:53.870 --> 00:23:02.120
So in this case, if we take
theta equal to 1 degree,
00:23:02.120 --> 00:23:06.626
then the ratio of the
sine theta and the theta
00:23:06.626 --> 00:23:12.710
is actually equal to
99.99%, which is very good.
00:23:12.710 --> 00:23:23.930
If I take it as 5 degrees,
then it's actually 99%.
00:23:23.930 --> 00:23:29.970
Even at 10 degrees,
it's actually 99.5%.
00:23:29.970 --> 00:23:34.700
Now, that shows you that sine
theta is so close to theta, OK?
00:23:34.700 --> 00:23:35.690
We are pretty safe.
00:23:35.690 --> 00:23:39.750
Because the difference
is smaller than 1%.
00:23:39.750 --> 00:23:42.890
OK, so that's very nice.
00:23:42.890 --> 00:23:50.800
After this approximation, I get
my final equation of motion.
00:23:50.800 --> 00:24:00.250
Theta double-dot t equal to
minus omega 0 squared theta t.
00:24:00.250 --> 00:24:04.375
Just a reminder, omega 0
is equal to square root
00:24:04.375 --> 00:24:11.400
of 3g over 2l, OK?
00:24:11.400 --> 00:24:17.870
We have solved this equation
last time, last lecture, right?
00:24:17.870 --> 00:24:19.200
It's exactly the same.
00:24:19.200 --> 00:24:21.790
OK, it happened to
be exactly the same.
00:24:21.790 --> 00:24:23.550
Therefore, I know
the solution will
00:24:23.550 --> 00:24:30.660
be theta of t equal to A
cosine omega 0 t plus phi.
00:24:34.000 --> 00:24:39.750
From the initial conditions,
which I have one and two,
00:24:39.750 --> 00:24:43.650
I am not going to go over
these calculation again.
00:24:43.650 --> 00:24:46.560
But again, we can
actually plug in 1 and 2
00:24:46.560 --> 00:24:50.850
to solve the unknown
A and the phi.
00:24:50.850 --> 00:24:53.400
If you do this
exercise, you will
00:24:53.400 --> 00:24:58.620
conclude that A is
equal to theta initial.
00:25:01.740 --> 00:25:08.900
And phi is equal to 0, OK?
00:25:08.900 --> 00:25:18.340
So the solution would be
theta of t equal to theta
00:25:18.340 --> 00:25:24.280
initial cosine omega 0 t.
00:25:27.400 --> 00:25:33.560
You can see that this actually
works for this system.
00:25:33.560 --> 00:25:35.940
Simple harmonic
oscillation actually
00:25:35.940 --> 00:25:41.540
described the motion of this
system as a function of time.
00:25:41.540 --> 00:25:45.470
You can also see a few
more examples shown here.
00:25:45.470 --> 00:25:49.270
Two of them you are going to
really work on in your P-set
00:25:49.270 --> 00:25:53.330
and also another one
involving circuits.
00:25:53.330 --> 00:25:57.290
If you have a capacitor
and you have an inductor,
00:25:57.290 --> 00:26:03.290
actually the size of
the current is also
00:26:03.290 --> 00:26:06.740
doing a simple
harmonic motion, OK?
00:26:06.740 --> 00:26:09.560
And as we actually
discussed before,
00:26:09.560 --> 00:26:12.570
the energy is always conserved.
00:26:12.570 --> 00:26:15.410
And that is actually stored
in different components
00:26:15.410 --> 00:26:16.890
of the system, OK?
00:26:20.520 --> 00:26:22.510
So we have done this.
00:26:22.510 --> 00:26:26.550
What is actually new today?
00:26:26.550 --> 00:26:30.100
What we are going to do
today is let's actually
00:26:30.100 --> 00:26:34.140
observe this phenomenon here.
00:26:34.140 --> 00:26:38.010
So this thing is actually
going to go back and forth.
00:26:38.010 --> 00:26:41.240
But it's actually not going
to do that forever, right?
00:26:41.240 --> 00:26:48.340
Something is happening, which
actually slows the motion down.
00:26:48.340 --> 00:26:52.060
I can also make use
of this system, OK?
00:26:52.060 --> 00:26:54.180
I start from here.
00:26:54.180 --> 00:26:56.320
And I'm not worried
that this actually
00:26:56.320 --> 00:26:58.330
goes out of this track.
00:26:58.330 --> 00:27:03.160
Because I know for sure
it will stop there.
00:27:03.160 --> 00:27:04.240
Why?
00:27:04.240 --> 00:27:09.780
Because the initial
amplitude is not going to--
00:27:09.780 --> 00:27:11.434
the amplitude is not
going to be larger
00:27:11.434 --> 00:27:12.850
than the initial
amplitude, right?
00:27:12.850 --> 00:27:15.480
So I'm not worried at all, OK?
00:27:15.480 --> 00:27:17.710
But you can see that
the amplitude is
00:27:17.710 --> 00:27:20.680
changing as a function of time.
00:27:20.680 --> 00:27:23.590
Apparently,
something is missing.
00:27:23.590 --> 00:27:28.050
And that is actually a
direct force, or friction,
00:27:28.050 --> 00:27:32.720
which is actually not
included in our calculation.
00:27:32.720 --> 00:27:40.860
So let's actually try to make
the calculation more realistic
00:27:40.860 --> 00:27:43.900
and see what is going to happen.
00:27:43.900 --> 00:27:51.500
So now, I will
introduce a drag force,
00:27:51.500 --> 00:27:58.300
which actually introduces
a torque tau drag, t,
00:27:58.300 --> 00:28:01.360
which is equal to minus b--
00:28:01.360 --> 00:28:03.670
b is actually some
kind of constant,
00:28:03.670 --> 00:28:06.970
which is given to you--
00:28:06.970 --> 00:28:14.680
theta dot t, which is actually
proportional to angular
00:28:14.680 --> 00:28:19.490
velocity of that rod, OK?
00:28:19.490 --> 00:28:24.350
And also of course, I keep
the original approximation.
00:28:24.350 --> 00:28:27.590
The theta is very
small, such that I
00:28:27.590 --> 00:28:30.810
don't have to deal with the
integration of sine theta, OK?
00:28:30.810 --> 00:28:33.880
So solving this, theta
double-dot equal to minus
00:28:33.880 --> 00:28:37.520
omega 0 squared sine theta
is a complicated function.
00:28:40.360 --> 00:28:46.030
You may ask, why do I actually
introduce a drag force
00:28:46.030 --> 00:28:49.770
proportional to the velocity?
00:28:49.770 --> 00:28:53.460
And why do I put a
minus sign there?
00:28:53.460 --> 00:28:58.380
That is actually because, if you
have a minus sign, that means,
00:28:58.380 --> 00:29:03.870
when this mass or that rod
is actually going downward,
00:29:03.870 --> 00:29:06.030
then the drag force
is really dragging it.
00:29:06.030 --> 00:29:09.750
Because it's actually in
the opposite direction
00:29:09.750 --> 00:29:14.460
of the velocity of the
mass or the angular
00:29:14.460 --> 00:29:16.470
velocity of the rod, OK?
00:29:16.470 --> 00:29:18.750
So I need a minus
sign there, OK?
00:29:18.750 --> 00:29:21.550
Otherwise, it's not
a drag force anymore.
00:29:21.550 --> 00:29:25.580
It's actually accelerating
the whole thing.
00:29:25.580 --> 00:29:33.878
Secondly, why do I choose that
to be proportional to theta dot
00:29:33.878 --> 00:29:37.750
or velocity?
00:29:37.750 --> 00:29:42.150
There's really no
much deeper reason.
00:29:42.150 --> 00:29:46.140
I choose this form
because I can actually
00:29:46.140 --> 00:29:48.460
solve it in front of you, OK?
00:29:48.460 --> 00:29:52.560
The reality is actually
between proportional
00:29:52.560 --> 00:29:57.210
to theta dot and theta dot
squared, for example, OK?
00:29:57.210 --> 00:29:59.940
This is actually a model
which I introduced here,
00:29:59.940 --> 00:30:04.440
which I can actually
solve it in front of you.
00:30:04.440 --> 00:30:07.110
On the other hand, you'll
see that it's actually not
00:30:07.110 --> 00:30:09.690
bad at all.
00:30:09.690 --> 00:30:12.900
It actually works and
describes the system,
00:30:12.900 --> 00:30:17.910
which will actually work to
perform the demo here, OK?
00:30:17.910 --> 00:30:24.390
And once we have introduced
this, the equation of motion
00:30:24.390 --> 00:30:26.810
will be modified.
00:30:26.810 --> 00:30:29.070
So let's come back to
the equation of motion.
00:30:29.070 --> 00:30:32.890
So you have to
theta double-dot t
00:30:32.890 --> 00:30:42.490
originally would be equal to
tau total t divided by I, OK?
00:30:42.490 --> 00:30:57.350
And now, this will become tau
t plus tau drag t divided by I.
00:30:57.350 --> 00:30:59.540
So there's an
additional time here.
00:30:59.540 --> 00:31:03.010
OK, if I simplify
this whole equation,
00:31:03.010 --> 00:31:12.320
then I get minus mgl
over 2 sine theta.
00:31:12.320 --> 00:31:16.960
And this is actually
roughly theta minus
00:31:16.960 --> 00:31:27.140
b theta dot divided by
1/3 of ml squared, OK?
00:31:27.140 --> 00:31:30.520
So you can see that I still
make this approximation sine
00:31:30.520 --> 00:31:34.940
theta roughly equal to theta.
00:31:34.940 --> 00:31:40.170
Then, I can actually write this
equation in the small angle
00:31:40.170 --> 00:31:40.670
case.
00:31:44.740 --> 00:31:53.000
OK, I get minus
3g over 2l theta t
00:31:53.000 --> 00:31:58.740
minus 3b over ml
squared theta dot t.
00:32:01.660 --> 00:32:10.490
OK, and now, as usual, I define
omega 0 squared equal to 3g
00:32:10.490 --> 00:32:11.960
over 2l.
00:32:11.960 --> 00:32:18.410
And I can also define
gamma is equal to 3b over
00:32:18.410 --> 00:32:25.050
ml squared, just to make
my life easier, right?
00:32:25.050 --> 00:32:32.740
Finally, we will arrive at this
expression, theta double-dot
00:32:32.740 --> 00:32:42.260
plus gamma theta dot plus
omega 0 squared theta.
00:32:42.260 --> 00:32:44.217
And that is equal to 0.
00:32:48.120 --> 00:32:56.340
So what you can see from here
is that we have actually derived
00:32:56.340 --> 00:32:58.950
the equation of motion, OK?
00:32:58.950 --> 00:33:01.140
We have derived the
equation of motion.
00:33:01.140 --> 00:33:05.660
And actually, part of the
work is actually really
00:33:05.660 --> 00:33:08.250
just solving this
equation of motion.
00:33:08.250 --> 00:33:09.840
And you don't really
have to solve it.
00:33:09.840 --> 00:33:13.650
Because you already get
the result from 18.03
00:33:13.650 --> 00:33:16.130
actually, if you remember.
00:33:16.130 --> 00:33:18.610
And we are going to
discuss the result.
00:33:18.610 --> 00:33:22.830
But before that, before I really
try to solve this equation,
00:33:22.830 --> 00:33:28.460
I would like to take a vote, OK?
00:33:28.460 --> 00:33:34.540
So here, I have two
different systems.
00:33:34.540 --> 00:33:37.980
They have equal amounts of mass.
00:33:37.980 --> 00:33:40.450
They are attached to a spring.
00:33:40.450 --> 00:33:44.440
If you do the same equation
of motion derivation,
00:33:44.440 --> 00:33:51.850
you will actually get exactly
the same equation of motion
00:33:51.850 --> 00:33:54.730
in that format, OK?
00:33:54.730 --> 00:33:56.380
So the form of the
equation of motion
00:33:56.380 --> 00:34:01.850
will be the same between this
system and that system, OK?
00:34:01.850 --> 00:34:06.380
I would like to ask you a
question about the oscillation
00:34:06.380 --> 00:34:07.950
frequency.
00:34:07.950 --> 00:34:12.230
So you can see that one of
them is actually a better mass.
00:34:12.230 --> 00:34:14.270
It's like a point-like particle.
00:34:14.270 --> 00:34:17.730
And the other one is
wearing a hat, OK?
00:34:17.730 --> 00:34:21.469
What is going to happen is
that this Mexican hat is
00:34:21.469 --> 00:34:26.270
going to be trying to
push the air away, right?
00:34:26.270 --> 00:34:30.710
Then, you may think,
OK, this Mexican thing
00:34:30.710 --> 00:34:33.290
is not really very important.
00:34:33.290 --> 00:34:35.150
Therefore, the
oscillation frequency
00:34:35.150 --> 00:34:37.120
may be the same, right?
00:34:37.120 --> 00:34:40.040
How many of you think the
oscillation frequency,
00:34:40.040 --> 00:34:42.889
if I actually tried to
perturb these two systems,
00:34:42.889 --> 00:34:44.740
would be the same?
00:34:44.740 --> 00:34:45.870
Raise your hands.
00:34:45.870 --> 00:34:49.730
1, 2, 3, 4, 5, 6, 7, 8--
00:34:49.730 --> 00:34:51.040
OK, we have 11.
00:34:53.639 --> 00:34:57.630
So the omega, the
predicted omega,
00:34:57.630 --> 00:35:00.903
will be equal to omega 0--
00:35:00.903 --> 00:35:02.670
11 of you.
00:35:02.670 --> 00:35:10.120
How many of you will think
that, because of this hat,
00:35:10.120 --> 00:35:16.050
this pushing this air away,
it's a lot of work to be done.
00:35:16.050 --> 00:35:21.810
Therefore, this is going to
slow down the oscillation.
00:35:21.810 --> 00:35:24.930
How many of you think
that is going to happen?
00:35:24.930 --> 00:35:26.040
1, 2, 3--
00:35:31.540 --> 00:35:32.490
OK, 17.
00:35:45.790 --> 00:35:48.190
It may happen to
you that you think
00:35:48.190 --> 00:35:54.100
this idea of wearing a
hat is really fashionable.
00:35:54.100 --> 00:35:55.950
Therefore, it got
really exciting
00:35:55.950 --> 00:35:58.950
and it oscillates faster.
00:35:58.950 --> 00:36:00.780
Can that happen?
00:36:00.780 --> 00:36:04.700
How many of you actually
think that is going to happen?
00:36:04.700 --> 00:36:09.150
OK, one-- you think so?
00:36:09.150 --> 00:36:10.990
Two.
00:36:10.990 --> 00:36:13.330
Very good, we have 2.
00:36:13.330 --> 00:36:14.080
What do you think?
00:36:14.080 --> 00:36:15.830
Where are the rest?
00:36:15.830 --> 00:36:21.700
Only 30 of you actually think
that is going to happen.
00:36:24.320 --> 00:36:27.600
OK, all the rest
think of the class
00:36:27.600 --> 00:36:29.960
think that this one
is going to-- pew!
00:36:29.960 --> 00:36:32.600
Disappear to the moon, OK?
00:36:32.600 --> 00:36:34.330
So that is actually the opinion.
00:36:34.330 --> 00:36:36.280
And we have completed the poll.
00:36:36.280 --> 00:36:40.130
And what we are going
to do is that we
00:36:40.130 --> 00:36:42.050
are going to solve
this system and see
00:36:42.050 --> 00:36:44.160
what is going to happen.
00:36:44.160 --> 00:36:48.410
And we will do that experiment
in front of you, OK?
00:36:48.410 --> 00:36:51.000
All right, so that's very nice.
00:36:51.000 --> 00:36:53.520
So now, we have this
question of motion.
00:36:53.520 --> 00:36:57.040
And now, I will pretend
that I'm from the math
00:36:57.040 --> 00:37:02.600
department for a bit and help
guide you through the solution.
00:37:02.600 --> 00:37:06.470
So now, I can use this trick.
00:37:06.470 --> 00:37:08.570
I can actually say
theta is actually
00:37:08.570 --> 00:37:14.730
the real part of the z,
which is a complex function.
00:37:14.730 --> 00:37:18.620
And as we learned
before, z of t,
00:37:18.620 --> 00:37:23.630
and I assume that to be
exponential I alpha t.
00:37:23.630 --> 00:37:26.130
So alpha is actually
some kind of constant,
00:37:26.130 --> 00:37:29.810
which I don't really know
what is the constant yet.
00:37:29.810 --> 00:37:36.590
OK, I can now actually
write the equation of motion
00:37:36.590 --> 00:37:38.390
in the form of z.
00:37:38.390 --> 00:37:47.127
Then basically, what I get is
z double-dot t plus gamma z dot
00:37:47.127 --> 00:37:54.290
t plus omega 0 squared z of t.
00:37:54.290 --> 00:37:58.250
And this is equal to 0, OK?
00:37:58.250 --> 00:38:02.060
So remember, exponential
function cannot be killed
00:38:02.060 --> 00:38:04.220
by differentiation, right?
00:38:04.220 --> 00:38:06.180
Therefore, it's
really convenient.
00:38:06.180 --> 00:38:07.410
You can see from here.
00:38:07.410 --> 00:38:10.640
Now, I can plug in
this expression--
00:38:10.640 --> 00:38:16.440
which I did this and guessed
to this equation of motion.
00:38:16.440 --> 00:38:20.910
Then what I am going to
get is minus alpha squared.
00:38:20.910 --> 00:38:24.130
Because you take I
alpha I alpha out
00:38:24.130 --> 00:38:28.640
of this exponential
function, right?
00:38:28.640 --> 00:38:32.630
Because you do double
differentiation.
00:38:32.630 --> 00:38:40.160
So you get minus alpha
squared plus i gamma alpha--
00:38:40.160 --> 00:38:43.680
because this is only
differentiated one time--
00:38:43.680 --> 00:38:47.120
plus omega 0 squared.
00:38:47.120 --> 00:38:51.570
And all those
things are actually
00:38:51.570 --> 00:38:56.480
multiplying this exponential
function, exponential i alpha t
00:38:56.480 --> 00:38:59.540
equal to 0, OK?
00:38:59.540 --> 00:39:01.455
So we will write
this expression.
00:39:01.455 --> 00:39:03.350
That is very nice.
00:39:03.350 --> 00:39:07.370
And we also know that,
this expression is
00:39:07.370 --> 00:39:09.680
going to be valid all the time.
00:39:09.680 --> 00:39:14.270
No matter what t you put in,
it should be valid, right?
00:39:14.270 --> 00:39:16.130
Because this is the
equation of motion.
00:39:16.130 --> 00:39:19.790
And we hope that this solution
will survive this test.
00:39:22.380 --> 00:39:28.110
So I can easily conclude
that this one is actually not
00:39:28.110 --> 00:39:29.960
equal to 0.
00:39:29.960 --> 00:39:32.640
It can be some value, not 0.
00:39:32.640 --> 00:39:35.400
So what is actually equal to 0?
00:39:35.400 --> 00:39:41.790
This first term is
actually equal to 0, OK?
00:39:41.790 --> 00:39:46.560
Therefore, I can now
solve this equation;
00:39:46.560 --> 00:39:50.370
minus alpha squared plus
i gamma alpha plus omega 0
00:39:50.370 --> 00:39:52.530
squared equal to 0.
00:39:52.530 --> 00:39:54.780
I can solve it, OK?
00:39:54.780 --> 00:39:59.650
If I do that, then
I would get alpha
00:39:59.650 --> 00:40:08.310
is equal to i gamma plus/minus
square root of 4 omega 0
00:40:08.310 --> 00:40:13.840
squared minus gamma
squared divided by 2.
00:40:13.840 --> 00:40:17.940
This is actually the
second order polynomial.
00:40:17.940 --> 00:40:21.110
And that is actually equal to 0.
00:40:21.110 --> 00:40:23.800
Therefore, you can
actually solve it easily.
00:40:23.800 --> 00:40:26.490
And this is actually
the solution.
00:40:26.490 --> 00:40:30.810
And I can write it down in
a slightly different form.
00:40:30.810 --> 00:40:36.780
i gamma over 2 plus/minus
square root of omega 0
00:40:36.780 --> 00:40:43.180
squared minus gamma
squared over 4, OK?
00:40:43.180 --> 00:40:44.160
Any questions so far?
00:40:44.160 --> 00:40:45.060
Am I going too fast?
00:40:48.450 --> 00:40:49.290
Everything's OK?
00:40:53.484 --> 00:41:00.005
OK, So you can see that alpha
is equal to this expression.
00:41:00.005 --> 00:41:03.880
And I would like to
consider a situation
00:41:03.880 --> 00:41:13.020
where omega 0 is much,
much larger than gamma, OK?
00:41:13.020 --> 00:41:15.310
Just a reminder of
what is gamma, OK?
00:41:15.310 --> 00:41:18.010
Maybe you've got
already a bit confused.
00:41:18.010 --> 00:41:19.480
What is gamma?
00:41:19.480 --> 00:41:24.630
Gamma is related to the strength
of the direct force, right?
00:41:24.630 --> 00:41:27.760
It is actually 3b
over ml squared, OK?
00:41:27.760 --> 00:41:37.090
b is actually determining the
size of the direct force, OK?
00:41:37.090 --> 00:41:40.816
So I would like to
consider a situation.
00:41:40.816 --> 00:41:48.790
The first situation is if
omega 0 squared is larger
00:41:48.790 --> 00:41:53.320
than gamma squared over 4.
00:41:53.320 --> 00:42:02.540
So in that case, the
drag force is small.
00:42:02.540 --> 00:42:04.150
It is not huge.
00:42:04.150 --> 00:42:06.540
It's small, OK?
00:42:06.540 --> 00:42:12.140
If that is the case, this
is actually real, right?
00:42:12.140 --> 00:42:14.420
Because omega 0
squared is larger
00:42:14.420 --> 00:42:16.370
than gamma squared over 4.
00:42:16.370 --> 00:42:18.950
Therefore, this is real, OK?
00:42:18.950 --> 00:42:23.870
So now, I can actually
define omega squared,
00:42:23.870 --> 00:42:31.210
define that as omega 0 squared
minus gamma squared over 4, OK?
00:42:33.820 --> 00:42:49.090
And this will become i gamma
over 2 plus/minus omega, OK?
00:42:49.090 --> 00:42:53.110
So that means I would
have two solutions coming
00:42:53.110 --> 00:42:57.040
from this exercise.
00:42:57.040 --> 00:43:04.345
Z plus of t is equal to
exponential minus gamma over 2
00:43:04.345 --> 00:43:11.010
t exponential i omega t, OK?
00:43:11.010 --> 00:43:19.740
And the second solution, if
I take one of the plus sign
00:43:19.740 --> 00:43:22.080
and one of the minus
sign solutions,
00:43:22.080 --> 00:43:24.570
then the second solution
would be exponential
00:43:24.570 --> 00:43:34.150
minus i gamma over 2 t
exponential minus i omega t,
00:43:34.150 --> 00:43:35.240
OK?
00:43:35.240 --> 00:43:36.124
Any questions so far?
00:43:40.000 --> 00:43:46.160
OK, so we would like to
go back to theta, right?
00:43:46.160 --> 00:43:47.670
So what would be the theta?
00:43:50.210 --> 00:43:55.040
So that means I would have a
theta 1 of t, which is actually
00:43:55.040 --> 00:43:56.630
taking the real part.
00:43:56.630 --> 00:44:02.630
So it's theta plus maybe,
taking the real part of z plus.
00:44:02.630 --> 00:44:06.890
And that will give you
exponential minus gamma
00:44:06.890 --> 00:44:16.610
over 2 t cosine omega t, OK?
00:44:16.610 --> 00:44:20.640
I'm just plugging in the
solution to this equation, OK?
00:44:25.240 --> 00:44:29.790
Theta minus t would be
equal to exponential,
00:44:29.790 --> 00:44:38.050
and this gamma over
2 t sine omega t, OK?
00:44:38.050 --> 00:44:43.480
Finally, the full
solution of theta of t
00:44:43.480 --> 00:44:49.200
would be a linear combination
of these two solution, right?
00:44:49.200 --> 00:44:54.180
Therefore, you will get theta
of t equal to exponential
00:44:54.180 --> 00:45:01.080
minus gamma over 2 t a
(is some kind of constant)
00:45:01.080 --> 00:45:08.280
times cosine omega t
plus b sine omega t.
00:45:12.401 --> 00:45:16.450
And of course, from the
last time, as you will know,
00:45:16.450 --> 00:45:28.030
this can also be written as A
cosine omega t plus phi, OK?
00:45:28.030 --> 00:45:32.440
Any questions so far?
00:45:32.440 --> 00:45:36.410
OK, very good.
00:45:36.410 --> 00:45:40.500
So we have actually already
solved this equation.
00:45:40.500 --> 00:45:44.400
And of course, we can
actually plug this back
00:45:44.400 --> 00:45:49.710
into this equation of motion.
00:45:49.710 --> 00:45:51.716
And you will see
that it really works.
00:45:51.716 --> 00:45:53.090
And I'm not going
to do that now.
00:45:53.090 --> 00:45:55.610
But you can actually
go back home and check.
00:45:55.610 --> 00:45:59.250
And if you believe me, it works.
00:45:59.250 --> 00:46:04.650
And also at the same time, it
got two undetermined constants,
00:46:04.650 --> 00:46:08.880
since this is a second
order differential equation.
00:46:08.880 --> 00:46:11.835
Therefore, huh, this
thing actually works.
00:46:11.835 --> 00:46:14.220
It has two arbitrary constants.
00:46:14.220 --> 00:46:16.530
Therefore, that is
actually the one and only
00:46:16.530 --> 00:46:19.680
one solution in
the universe which
00:46:19.680 --> 00:46:25.220
satisfies the equation of motion
or satisfies that differential
00:46:25.220 --> 00:46:27.020
question, OK?
00:46:27.020 --> 00:46:33.090
So this thing actually
has dramatic consequences.
00:46:33.090 --> 00:46:34.500
The first thing
which we learn is
00:46:34.500 --> 00:46:39.300
that, as a function of time,
what is going to happen?
00:46:39.300 --> 00:46:46.390
The amplitude is now becoming
exponential minus gamma
00:46:46.390 --> 00:46:52.140
over 2 t times A. This is
actually the amplitude.
00:46:52.140 --> 00:46:57.840
The amplitude is
decreasing exponentially.
00:46:57.840 --> 00:47:00.570
So that is actually
the first prediction
00:47:00.570 --> 00:47:03.150
coming from this exercise, OK?
00:47:03.150 --> 00:47:07.480
The second prediction is
that this thing is still
00:47:07.480 --> 00:47:08.260
oscillating.
00:47:08.260 --> 00:47:12.400
Because you've got the cosine
omega t plus phi there,
00:47:12.400 --> 00:47:13.760
you see?
00:47:13.760 --> 00:47:16.720
So the damping
motion is going to be
00:47:16.720 --> 00:47:21.550
like going up and down, up
and down, and get tired.
00:47:21.550 --> 00:47:25.030
Therefore, the amplitude
becomes smaller, and smaller,
00:47:25.030 --> 00:47:26.320
and smaller.
00:47:26.320 --> 00:47:29.960
But it's never 0, right?
00:47:29.960 --> 00:47:31.190
It's never 0, OK?
00:47:31.190 --> 00:47:34.160
It's actually going to be
oscillating down, down, down,
00:47:34.160 --> 00:47:35.550
so small I couldn't see it.
00:47:35.550 --> 00:47:39.590
But it's still oscillating, OK?
00:47:39.590 --> 00:47:44.610
Finally, we actually
have also the answer
00:47:44.610 --> 00:47:49.050
to the original
question we posed, OK?
00:47:49.050 --> 00:47:54.200
So now, you can see that the
oscillation frequency is omega,
00:47:54.200 --> 00:47:55.080
OK?
00:47:55.080 --> 00:48:01.060
Originally, before we
introduced the drag force,
00:48:01.060 --> 00:48:05.570
omega 0, which is the
oscillation frequency,
00:48:05.570 --> 00:48:06.985
is actually an
angular frequency.
00:48:06.985 --> 00:48:10.930
It's actually the square
root of 3g over 2l.
00:48:10.930 --> 00:48:17.380
And you can see that the
new omega, the oscillation
00:48:17.380 --> 00:48:24.650
frequency with drag force,
is the square root of this,
00:48:24.650 --> 00:48:29.000
omega 0 squared minus
gamma squared over 4.
00:48:29.000 --> 00:48:32.230
So what this
actually tells us is
00:48:32.230 --> 00:48:38.400
that this is going
to be smaller,
00:48:38.400 --> 00:48:40.740
because of the drag force, OK?
00:48:40.740 --> 00:48:43.980
So that's a prediction.
00:48:43.980 --> 00:48:47.080
Let's do the experiment and
see what is going to happen.
00:48:47.080 --> 00:48:49.740
So let's take a look
at these two systems.
00:48:49.740 --> 00:48:54.870
They have the identical mass,
which our technical instructor
00:48:54.870 --> 00:48:57.330
actually carefully prepared.
00:48:57.330 --> 00:49:00.600
They have the same mass,
even though one actually
00:49:00.600 --> 00:49:01.980
looks a bit funny.
00:49:01.980 --> 00:49:04.740
The other one looks normal, OK?
00:49:04.740 --> 00:49:10.590
Now, what I'm going to do
is to really try and see
00:49:10.590 --> 00:49:13.110
which one is actually
oscillating faster, OK?
00:49:13.110 --> 00:49:14.802
So let's see.
00:49:14.802 --> 00:49:17.380
I release them at the same time.
00:49:17.380 --> 00:49:19.470
And you can see
that originally they
00:49:19.470 --> 00:49:22.120
seem to be oscillating
at the same frequency.
00:49:22.120 --> 00:49:27.570
But you can see very clearly
that the one with the hat
00:49:27.570 --> 00:49:31.230
is actually
oscillating slower, OK?
00:49:31.230 --> 00:49:33.810
So you can see
that, OK, 17 of you
00:49:33.810 --> 00:49:37.440
actually got the correct answer.
00:49:37.440 --> 00:49:40.380
And the most important
thing is that you
00:49:40.380 --> 00:49:42.780
can see that this
simple mass actually
00:49:42.780 --> 00:49:46.460
describes and predicts
what is going to happen
00:49:46.460 --> 00:49:48.930
in my little experiment.
00:49:48.930 --> 00:49:51.340
So that is actually really cool.
00:49:51.340 --> 00:49:54.890
And I think it's time
to take a little break.
00:49:54.890 --> 00:49:59.250
And then, we will come back
and look at other solutions.
00:49:59.250 --> 00:50:01.380
And of course, you
are welcome to come
00:50:01.380 --> 00:50:04.836
to the front to play with
those demonstrations.
00:50:11.180 --> 00:50:13.510
So there are two
small issues which
00:50:13.510 --> 00:50:16.970
were raised during the break.
00:50:16.970 --> 00:50:22.730
So the first one is that, if you
actually calculate the torque
00:50:22.730 --> 00:50:25.520
from this equation--
00:50:25.520 --> 00:50:27.440
so I made a mistake.
00:50:27.440 --> 00:50:31.490
The R vector should be
actually pointing from the nail
00:50:31.490 --> 00:50:33.140
to the center of mass, OK?
00:50:33.140 --> 00:50:35.660
So I think that's
a trivial mistake.
00:50:35.660 --> 00:50:38.630
So if you do this,
then you can actually
00:50:38.630 --> 00:50:43.010
calculate the tau
equal to R cross F.
00:50:43.010 --> 00:50:46.200
Then, you actually get
this minus sign, OK?
00:50:46.200 --> 00:50:51.580
So if I make a mistake in
pointing towards the nail,
00:50:51.580 --> 00:50:54.650
then you will get
no minus sign, then
00:50:54.650 --> 00:50:56.750
that didn't really work, OK?
00:50:56.750 --> 00:51:00.500
So very good, I'm very
happy that you are actually
00:51:00.500 --> 00:51:04.490
paying very much attention
to capture those.
00:51:04.490 --> 00:51:07.040
The second issue is that--
00:51:07.040 --> 00:51:09.590
so now, I'm saying
that, OK, now I
00:51:09.590 --> 00:51:13.180
have the solution in
the complex format.
00:51:13.180 --> 00:51:17.450
So I have a Z plus and
I have a Z minus, OK?
00:51:17.450 --> 00:51:21.170
And then I would like to go
to the real world, right?
00:51:21.170 --> 00:51:23.720
Because the imaginary
thing is actually
00:51:23.720 --> 00:51:27.740
hidden in some kind of motion
in the actual dimension,
00:51:27.740 --> 00:51:31.680
et cetera, I would like
to go back to reality, OK?
00:51:31.680 --> 00:51:36.790
And what I said in the class
is that I take the real part
00:51:36.790 --> 00:51:38.310
of one of the solutions.
00:51:38.310 --> 00:51:41.230
And I can also take
a real part of i
00:51:41.230 --> 00:51:43.440
times one of the solutions.
00:51:43.440 --> 00:51:46.400
But of course, you
can also do this
00:51:46.400 --> 00:51:50.440
by doing a linear combination
of the solutions, right?
00:51:50.440 --> 00:51:53.100
As we actually
discussed last time,
00:51:53.100 --> 00:51:55.100
the linear combination
of the solutions
00:51:55.100 --> 00:51:59.990
is also a solution to the
same equation of motion,
00:51:59.990 --> 00:52:02.010
since this one is
actually linear.
00:52:02.010 --> 00:52:06.960
Therefore, what I
actually do is actually
00:52:06.960 --> 00:52:11.850
to sum the two solutions, Z plus
and Z minus and divide it by 2.
00:52:11.850 --> 00:52:20.320
Or actually, I can actually do
a minus i/2 times Z plus minus Z
00:52:20.320 --> 00:52:21.260
minus, OK?
00:52:21.260 --> 00:52:24.740
And then I can also extract
this sign term here, OK?
00:52:24.740 --> 00:52:28.610
So that should be the correct
explanation of the two
00:52:28.610 --> 00:52:33.620
solutions in the real axis, OK?
00:52:33.620 --> 00:52:36.300
Any questions so far?
00:52:36.300 --> 00:52:40.200
Thank you very much
for capturing those.
00:52:40.200 --> 00:52:41.910
Ok, so now, you can
see that we have
00:52:41.910 --> 00:52:46.530
been discussing the equation of
motion of this functional form.
00:52:46.530 --> 00:52:49.780
And the one thing which is
really, really interesting
00:52:49.780 --> 00:52:57.170
is that the solution, when we
take a small drag force limit,
00:52:57.170 --> 00:53:01.560
actually we arrive at a
beautiful solution that
00:53:01.560 --> 00:53:06.560
looks like this, A
exponential minus gamma over 2
00:53:06.560 --> 00:53:09.370
t cosine omega t plus phi.
00:53:09.370 --> 00:53:13.080
That actually predicts
the oscillation, OK?
00:53:13.080 --> 00:53:17.310
At the same time, it also
says that the amplitude
00:53:17.310 --> 00:53:23.060
is actually going to drop
exponentially, but never 0, OK?
00:53:23.060 --> 00:53:28.650
Finally, we also know that
this solution actually tells us
00:53:28.650 --> 00:53:33.840
that, if we have a spring mass
system oscillating up and down,
00:53:33.840 --> 00:53:40.770
if we have a rod like what
we actually solve in a class,
00:53:40.770 --> 00:53:46.010
this object is going to pass
through 0, the equilibrium
00:53:46.010 --> 00:53:48.450
position, an infinite
number of times, right?
00:53:48.450 --> 00:53:51.630
Because the cosine
is always there.
00:53:51.630 --> 00:53:53.587
Therefore, although
the amplitude
00:53:53.587 --> 00:53:55.170
will become very
small, but it's still
00:53:55.170 --> 00:53:59.910
oscillating forever until
the end of the universe, OK?
00:53:59.910 --> 00:54:03.560
All right, so that's actually
what we have learned.
00:54:03.560 --> 00:54:06.660
And also, one thing which
we learned last time
00:54:06.660 --> 00:54:12.010
is that simple harmonic
motion, like this one, which
00:54:12.010 --> 00:54:15.310
we were just showing
here, or this one,
00:54:15.310 --> 00:54:18.330
which is actually a mass
oscillating back and forth
00:54:18.330 --> 00:54:24.830
on the track, is actually just a
projection of a circular motion
00:54:24.830 --> 00:54:26.760
in a complex plane, OK?
00:54:26.760 --> 00:54:30.770
And what we are really
seeing here in front of you
00:54:30.770 --> 00:54:34.950
is actually a projection
to the real axis, OK?
00:54:34.950 --> 00:54:37.850
So that's actually a
really remarkable result
00:54:37.850 --> 00:54:40.530
and a beautiful picture.
00:54:40.530 --> 00:54:44.670
And of course, we can actually
also plug in the solution
00:54:44.670 --> 00:54:47.530
with damping.
00:54:47.530 --> 00:54:50.900
So what is actually the
picture in this language,
00:54:50.900 --> 00:54:54.130
in this exact same language?
00:54:54.130 --> 00:54:58.270
If we actually follow
the locus, then basically
00:54:58.270 --> 00:55:00.980
what you are going to see
is that this thing actually
00:55:00.980 --> 00:55:02.530
spirals.
00:55:02.530 --> 00:55:06.400
And the amplitude is actually
getting smaller and smaller
00:55:06.400 --> 00:55:10.820
and is sucked into this
black hole in the 0, 0, OK?
00:55:10.820 --> 00:55:14.440
So you can see that
now the picture
00:55:14.440 --> 00:55:18.170
looks as if there is
something really rotating
00:55:18.170 --> 00:55:19.660
in the complex plane.
00:55:19.660 --> 00:55:21.790
And it's actually approaching 0.
00:55:21.790 --> 00:55:23.770
Because the
amplitude is actually
00:55:23.770 --> 00:55:25.670
getting smaller and smaller.
00:55:25.670 --> 00:55:29.960
But this whole thing
is still rotating, OK?
00:55:29.960 --> 00:55:33.370
OK, that's really nice.
00:55:33.370 --> 00:55:39.360
All right, so now, this is
actually a special case.
00:55:39.360 --> 00:55:45.000
When we actually assume that
gamma is actually pretty small.
00:55:45.000 --> 00:55:49.800
So you have very
small drag force, OK?
00:55:49.800 --> 00:55:54.330
So let's actually check
what would happen.
00:55:54.330 --> 00:55:58.200
If I now start to
increase the drag force,
00:55:58.200 --> 00:56:02.700
make this hat larger,
larger, and larger,
00:56:02.700 --> 00:56:08.660
introducing more and more
drag, what is going to happen?
00:56:08.660 --> 00:56:13.790
OK, so now, I consider
the second situation,
00:56:13.790 --> 00:56:22.190
omega 0 squared equal
to gamma squared over 4.
00:56:22.190 --> 00:56:25.520
OK, so when the
gamma is very small,
00:56:25.520 --> 00:56:29.150
what we see is that this is
actually underdamped, right?
00:56:29.150 --> 00:56:32.150
So the damping is really small.
00:56:32.150 --> 00:56:38.050
But if I increase the
gamma to a critical value,
00:56:38.050 --> 00:56:41.840
now omega 0 squared happens
to be equal to gamma
00:56:41.840 --> 00:56:43.960
squared over 4, OK?
00:56:43.960 --> 00:56:56.660
I call this a critically
damped oscillator, OK?
00:56:56.660 --> 00:56:58.770
So what does that mean?
00:56:58.770 --> 00:57:07.070
That means omega is
equal to 0, you see?
00:57:07.070 --> 00:57:10.190
This is our definition
of omega, right?
00:57:10.190 --> 00:57:15.080
If omega 0 squared is equal
to gamma squared of over 4,
00:57:15.080 --> 00:57:19.670
then omega is equal to 0.
00:57:19.670 --> 00:57:22.970
That is actually
the critical moment
00:57:22.970 --> 00:57:28.720
the system stops
oscillating, OK?
00:57:28.720 --> 00:57:33.050
So it is not
oscillating anymore.
00:57:33.050 --> 00:57:38.220
So now, I can actually
start from the solution
00:57:38.220 --> 00:57:41.150
I obtained from 1, OK?
00:57:41.150 --> 00:57:43.910
Then, I can actually
now make use
00:57:43.910 --> 00:57:48.950
of these two solutions, the
theta plus and the theta minus.
00:57:53.730 --> 00:58:00.250
Theta plus t would be equal
to exponential minus gamma
00:58:00.250 --> 00:58:05.940
over 2 t cosine omega t.
00:58:05.940 --> 00:58:09.240
When omega goes to
0, what is going
00:58:09.240 --> 00:58:13.640
to happen is that this is
actually becoming, which value?
00:58:13.640 --> 00:58:15.600
Anybody know?
00:58:15.600 --> 00:58:19.010
If omega is 0, what
is going to happen?
00:58:19.010 --> 00:58:20.180
1, yeah.
00:58:20.180 --> 00:58:21.650
OK, 1, right?
00:58:21.650 --> 00:58:27.155
So that will give me exponential
minus gamma over 2 t.
00:58:30.010 --> 00:58:32.730
Theta minus t--
00:58:32.730 --> 00:58:35.880
OK, I can do the same trick
and see what will happen.
00:58:35.880 --> 00:58:39.550
So I take theta minus
t, which is actually
00:58:39.550 --> 00:58:42.110
obtained from the
exercise number one
00:58:42.110 --> 00:58:44.450
when we discussed the
underdamped system.
00:58:44.450 --> 00:58:47.250
Then, you actually get
exponential minus gamma
00:58:47.250 --> 00:58:52.010
over 2 t sine omega t.
00:58:52.010 --> 00:59:00.140
When omega goes to 0, actually
then I get 0 this time, OK?
00:59:00.140 --> 00:59:01.920
So that doesn't
really work, right?
00:59:01.920 --> 00:59:05.080
Because if I have a
solution which is 0, then
00:59:05.080 --> 00:59:06.800
it's not describing
anything, right?
00:59:06.800 --> 00:59:09.410
I can always add
0 to the solution.
00:59:09.410 --> 00:59:10.700
But that doesn't help you.
00:59:10.700 --> 00:59:17.470
OK, so instead of taking
the limit of this function,
00:59:17.470 --> 00:59:21.170
actually we choose
to actually do
00:59:21.170 --> 00:59:25.960
theta minus t divided by omega.
00:59:25.960 --> 00:59:31.310
And then, we actually make
this omega approaching 0.
00:59:31.310 --> 00:59:34.310
Then basically, I get
exponential minus gamma
00:59:34.310 --> 00:59:42.280
over 2 t sine omega t
divided by omega, OK?
00:59:42.280 --> 00:59:48.210
If I have this omega
approaching to 0,
00:59:48.210 --> 00:59:53.130
then this is actually roughly
just exponential minus gamma
00:59:53.130 --> 00:59:57.930
over 2 t omega t over omega.
00:59:57.930 --> 01:00:05.010
And this is actually giving
you t times exponential minus
01:00:05.010 --> 01:00:06.220
gamma over 2 t.
01:00:09.580 --> 01:00:10.904
Any questions so far?
01:00:13.870 --> 01:00:15.556
Yes.
01:00:15.556 --> 01:00:18.448
AUDIENCE: Completely
unrelated, but is
01:00:18.448 --> 01:00:22.600
that a negative sign in front
of the theta minus negative 1/2?
01:00:22.600 --> 01:00:23.750
YEN-JIE LEE: This one?
01:00:23.750 --> 01:00:24.690
AUDIENCE: Yeah.
01:00:24.690 --> 01:00:25.440
YEN-JIE LEE: Yeah.
01:00:25.440 --> 01:00:28.465
So actually, OK, yeah.
01:00:28.465 --> 01:00:30.840
AUDIENCE: In front of the
1/2 is that a negative sign?
01:00:30.840 --> 01:00:32.590
YEN-JIE LEE: Yes, this
is a negative sign.
01:00:35.090 --> 01:00:39.990
OK, any other questions?
01:00:39.990 --> 01:00:45.840
OK, so you can see that now
I arrive at two solutions.
01:00:45.840 --> 01:00:50.320
One is actually proportional
to exponential minus gamma
01:00:50.320 --> 01:00:51.980
over 2 t.
01:00:51.980 --> 01:00:54.550
The other one is actually
proportional to t
01:00:54.550 --> 01:00:59.150
times exponential minus
gamma over 2 t, OK?
01:00:59.150 --> 01:01:03.680
So you can see that the cosine
or sine term disappeared,
01:01:03.680 --> 01:01:04.700
right?
01:01:04.700 --> 01:01:10.000
So that means you are
never oscillating, OK?
01:01:10.000 --> 01:01:13.210
So this is actually what
we see in this slide,
01:01:13.210 --> 01:01:15.730
this so-called
critically damped, OK?
01:01:15.730 --> 01:01:18.730
When actually,
omega 0 squared is
01:01:18.730 --> 01:01:23.830
equal to gamma squared over 4.
01:01:23.830 --> 01:01:26.620
And you can see that
what is going to happen
01:01:26.620 --> 01:01:34.540
is that this mass or this
rod is going to pass 0
01:01:34.540 --> 01:01:39.070
only one time at most, OK?
01:01:39.070 --> 01:01:41.980
And it could actually
never passed 0,
01:01:41.980 --> 01:01:45.210
if you actually set up the
initial condition correctly,
01:01:45.210 --> 01:01:45.710
OK?
01:01:45.710 --> 01:01:50.380
So one thing which I can do is
I really shoot this mass really,
01:01:50.380 --> 01:01:53.230
really, very forcefully,
so that I have
01:01:53.230 --> 01:01:56.040
a very large initial velocity.
01:01:56.040 --> 01:01:58.690
And what it actually
is going to do,
01:01:58.690 --> 01:02:00.630
like the right-hand
side diagram,
01:02:00.630 --> 01:02:03.550
is that, oh, you
overshoot the 0 a bit.
01:02:03.550 --> 01:02:09.040
Then, it goes back
almost exponentially, OK?
01:02:09.040 --> 01:02:13.830
So at most, you can
only pass through 0 one
01:02:13.830 --> 01:02:18.830
time, if you do this kind
of initial condition, OK?
01:02:18.830 --> 01:02:22.520
So that is actually
pretty interesting.
01:02:22.520 --> 01:02:29.370
And there are practical
applications of this solution,
01:02:29.370 --> 01:02:30.030
actually.
01:02:30.030 --> 01:02:32.870
So for example, we
have the door closed.
01:02:32.870 --> 01:02:34.490
So it's also here, right?
01:02:34.490 --> 01:02:38.030
The door closed, you would
like to have the door go back
01:02:38.030 --> 01:02:42.290
to the original closed
mold, the position
01:02:42.290 --> 01:02:45.740
of equilibrium position
actually really fast, OK?
01:02:45.740 --> 01:02:50.870
So what you can do is really
design this door close
01:02:50.870 --> 01:02:54.700
so that it actually matches
with the critical dampness
01:02:54.700 --> 01:02:58.460
situation, of your condition, so
that actually you would go back
01:02:58.460 --> 01:03:02.478
to 0 really quick, OK?
01:03:02.478 --> 01:03:06.380
Any questions?
01:03:06.380 --> 01:03:11.760
OK, so now, what we
could do is that, instead
01:03:11.760 --> 01:03:21.810
of having a very small drag
force, or we'll a slightly
01:03:21.810 --> 01:03:23.610
larger drag force,
so that actually
01:03:23.610 --> 01:03:28.320
reach the critically damped
situation, what we could do
01:03:28.320 --> 01:03:33.820
is that we put the whole
system into water, right?
01:03:33.820 --> 01:03:37.450
Then, the drag force
will be very big, OK?
01:03:37.450 --> 01:03:42.160
And we would like to see
what is going to happen, OK?
01:03:42.160 --> 01:03:46.690
So in this case is
the third situation.
01:03:46.690 --> 01:03:50.680
The third situation is that
omega 0 squared is actually
01:03:50.680 --> 01:03:55.960
smaller than gamma
squared over 4.
01:03:55.960 --> 01:04:03.220
So you have huge drag force, OK?
01:04:06.050 --> 01:04:09.770
So that would give
you a situation
01:04:09.770 --> 01:04:17.037
which is called
overdamped oscillator.
01:04:21.870 --> 01:04:27.140
Now, I have, again,
alpha is equal to i gamma
01:04:27.140 --> 01:04:32.600
over 2 plus/minus
square root of omega 0
01:04:32.600 --> 01:04:37.180
squared minus gamma
squared over 4, right?
01:04:37.180 --> 01:04:39.820
I'm just copying from here, OK?
01:04:39.820 --> 01:04:42.960
And that will be equal to i--
01:04:42.960 --> 01:04:45.850
I can take out the i, OK?--
01:04:45.850 --> 01:04:50.800
gamma over 2
plus/minus square root
01:04:50.800 --> 01:04:56.777
of gamma squared over 4
minus omega 0 squared.
01:05:02.140 --> 01:05:10.250
Now, I can actually
define gamma plus/minus
01:05:10.250 --> 01:05:18.270
equal to gamma over 2
plus/minus square root of gamma
01:05:18.270 --> 01:05:25.580
squared over 4 minus
omega 0 squared, OK?
01:05:25.580 --> 01:05:29.240
Then basically, the solution--
01:05:29.240 --> 01:05:32.340
actually now, I already
have the solution.
01:05:32.340 --> 01:05:36.720
So basically, the two solutions
would be looking like this.
01:05:36.720 --> 01:05:41.670
Theta of t would be
equal to A plus some kind
01:05:41.670 --> 01:05:48.990
of constant exponential
minus gamma plus t plus A
01:05:48.990 --> 01:05:56.070
minus exponential minus
gamma minus t, OK?
01:05:56.070 --> 01:05:59.070
Because this is actually
becoming already--
01:05:59.070 --> 01:06:04.730
OK, so alpha is actually
i times gamma plus/minus.
01:06:04.730 --> 01:06:08.360
Therefore, if you put
it back into this,
01:06:08.360 --> 01:06:09.990
then basically what
you are getting
01:06:09.990 --> 01:06:18.450
is exponential minus gamma
plus t or exponential minus
01:06:18.450 --> 01:06:20.700
gamma minus t, OK?
01:06:20.700 --> 01:06:22.920
So that's already
a real function.
01:06:22.920 --> 01:06:26.610
And the linear combination
of these two solutions
01:06:26.610 --> 01:06:33.030
is our final, full solution
to the equation of motion.
01:06:33.030 --> 01:06:36.670
OK, again, what we
are going to see
01:06:36.670 --> 01:06:41.700
is that actually the
drag force is huge.
01:06:41.700 --> 01:06:45.090
I just throw the whole
system into water.
01:06:45.090 --> 01:06:49.840
And the water is really trying
to stop the oscillation, really
01:06:49.840 --> 01:06:50.760
very much.
01:06:50.760 --> 01:06:54.030
Therefore, you can
see that, huh, again,
01:06:54.030 --> 01:06:57.510
I don't have any
oscillation, OK?
01:06:57.510 --> 01:07:00.180
If I am very, very
strong, I really
01:07:00.180 --> 01:07:06.450
start the initial velocity
or initial angular
01:07:06.450 --> 01:07:12.300
velocity really high, I actually
give a huge amount of energy
01:07:12.300 --> 01:07:15.960
into the system,
then, at most again, I
01:07:15.960 --> 01:07:20.310
can actually have the system
to pass through the equilibrium
01:07:20.310 --> 01:07:22.530
position only one time.
01:07:22.530 --> 01:07:28.740
Then, this whole system
will slowly recover,
01:07:28.740 --> 01:07:36.390
because exponential
function we see here.
01:07:36.390 --> 01:07:42.571
The amplitude is going to be
decaying exponentially, OK?
01:07:42.571 --> 01:07:45.560
Any questions?
01:07:45.560 --> 01:07:50.180
So let's actually do a quick
demonstration here, OK?
01:07:50.180 --> 01:07:56.540
So here, this is actually the
original little ball here,
01:07:56.540 --> 01:07:59.060
a metal one, which
actually you can
01:07:59.060 --> 01:08:04.790
see that this is really going
to go back and forth really
01:08:04.790 --> 01:08:05.720
nicely.
01:08:05.720 --> 01:08:09.860
And you can see that,
because of the friction,
01:08:09.860 --> 01:08:12.710
actually the
amplitude is becoming
01:08:12.710 --> 01:08:15.200
smaller and smaller, OK?
01:08:15.200 --> 01:08:19.189
So that actually matches
with this situation, right?
01:08:19.189 --> 01:08:22.850
So it's actually an
underdamped situation.
01:08:22.850 --> 01:08:26.529
This ball, in an
idealized situation,
01:08:26.529 --> 01:08:32.359
is going to go through 0
infinite number of times, OK?
01:08:32.359 --> 01:08:35.279
So now, what I am
going to do is now
01:08:35.279 --> 01:08:39.500
I change this ball to something
which is different, OK?
01:08:39.500 --> 01:08:43.569
This is actually
made of magnets, OK?
01:08:43.569 --> 01:08:45.770
And let's see what
is going to happen.
01:08:45.770 --> 01:08:47.950
So now, you can see that,
because this is actually
01:08:47.950 --> 01:08:51.460
made of magnets, therefore, the
drag force will be colossal,
01:08:51.460 --> 01:08:53.396
will be very, very big.
01:08:53.396 --> 01:08:56.378
And let's see what will happen.
01:08:59.859 --> 01:09:03.460
You see that the
drag force is huge.
01:09:03.460 --> 01:09:06.899
Therefore, you see I
put it here so that it
01:09:06.899 --> 01:09:08.699
has big initial velocity.
01:09:12.611 --> 01:09:16.800
It only passes
through 0 once, right?
01:09:16.800 --> 01:09:19.750
Of course, it now is actually
approaching the zero really,
01:09:19.750 --> 01:09:21.319
really slowly, exponentially.
01:09:21.319 --> 01:09:23.460
But it is not 0, OK?
01:09:23.460 --> 01:09:27.870
So it only passes through the
0 if you believe the math,
01:09:27.870 --> 01:09:30.580
only once, OK?
01:09:30.580 --> 01:09:35.470
Just to show that
this is a real deal--
01:09:35.470 --> 01:09:36.859
OK, now, whoa, right?
01:09:42.770 --> 01:09:48.189
Oh, I'm not trying to
destroy the classroom, OK?
01:09:48.189 --> 01:09:50.229
So you can actually
play with this
01:09:50.229 --> 01:09:52.670
after we finish
your lecture, OK?
01:09:55.200 --> 01:09:58.170
I would like to
ask you a question.
01:09:58.170 --> 01:10:01.930
After we learned this
from this lecture,
01:10:01.930 --> 01:10:08.920
there are three situations,
underdamped, critically damped,
01:10:08.920 --> 01:10:11.230
and overdamped, OK?
01:10:11.230 --> 01:10:13.550
I would like to ask
you two questions.
01:10:13.550 --> 01:10:18.130
The first one is through
this demonstration, OK?
01:10:18.130 --> 01:10:25.190
So, now I have a system
which is nicely constructed.
01:10:25.190 --> 01:10:27.990
I hope you can see it, OK?
01:10:27.990 --> 01:10:29.560
You can see it.
01:10:29.560 --> 01:10:36.870
And this system is made
of a torsional spring.
01:10:36.870 --> 01:10:39.930
And also, there's
a pad here, which
01:10:39.930 --> 01:10:42.170
is connected to the spring, OK?
01:10:42.170 --> 01:10:44.790
If I actually
perturb this thing,
01:10:44.790 --> 01:10:49.770
it's going to be oscillating
back and forth before I turn
01:10:49.770 --> 01:10:56.070
on the power, so that the
lower part is actually
01:10:56.070 --> 01:10:58.930
you have a magnet, OK?
01:10:58.930 --> 01:11:01.140
It's not turned on yet, OK?
01:11:01.140 --> 01:11:04.260
And this magnet is
going to provide
01:11:04.260 --> 01:11:10.590
a drag force to actually change
the behavior of the system, OK?
01:11:10.590 --> 01:11:15.480
So you can see that, before
I turn on the magnetic field,
01:11:15.480 --> 01:11:18.120
the whole system is actually
oscillating back and forth
01:11:18.120 --> 01:11:19.170
really nicely.
01:11:19.170 --> 01:11:23.450
As we predicted, small
amplitude vibration
01:11:23.450 --> 01:11:26.190
is harmonic oscillation, OK?
01:11:26.190 --> 01:11:27.760
So that's very nice.
01:11:27.760 --> 01:11:32.700
So now, what am I going to
do is to turn on the power
01:11:32.700 --> 01:11:35.190
and see what is going to happen.
01:11:35.190 --> 01:11:42.450
After I turn on the power,
there's an electric field, OK?
01:11:42.450 --> 01:11:45.730
And this is actually
going to be--
01:11:45.730 --> 01:11:49.870
OK, so the magnetic field
is actually turned down.
01:11:49.870 --> 01:11:53.370
Therefore, it is actually
acting like a drag force
01:11:53.370 --> 01:11:55.060
to this system, OK?
01:11:55.060 --> 01:11:57.910
So let's actually see
what is going to happen.
01:11:57.910 --> 01:12:01.540
Now, I release this.
01:12:01.540 --> 01:12:05.060
The behavior of the
system looks like this.
01:12:05.060 --> 01:12:09.730
It first oscillates,
and then it stops.
01:12:09.730 --> 01:12:14.940
So the question is, is this a
critically damped, underdamped,
01:12:14.940 --> 01:12:16.530
or overdamped system?
01:12:16.530 --> 01:12:19.360
Anybody knows?
01:12:19.360 --> 01:12:19.860
Yeah?
01:12:19.860 --> 01:12:21.330
AUDIENCE: Underdamped.
01:12:21.330 --> 01:12:23.120
YEN-JIE LEE: Yes,
this is underdamped.
01:12:23.120 --> 01:12:24.670
How do I see that?
01:12:24.670 --> 01:12:28.360
That is because, when
I do this experiment,
01:12:28.360 --> 01:12:33.870
you would pass through
0s multiple times.
01:12:33.870 --> 01:12:37.280
Therefore, there are
oscillations coming into play.
01:12:37.280 --> 01:12:40.700
Therefore, I can conclude
that the drag force is not
01:12:40.700 --> 01:12:41.750
large enough.
01:12:41.750 --> 01:12:44.900
So that is actually an
underdamped situation, OK?
01:12:44.900 --> 01:12:47.990
And the next time, we are
going to drag this system.
01:12:47.990 --> 01:12:51.050
I have a second
question for you.
01:12:51.050 --> 01:12:55.850
So now, your friends
know that you took 8.03.
01:12:55.850 --> 01:12:58.190
Therefore, they will
wonder if you can actually
01:12:58.190 --> 01:13:04.880
design a car suspension system,
to see if you can actually
01:13:04.880 --> 01:13:06.860
make this design for them.
01:13:06.860 --> 01:13:12.500
When you design this
car, which condition
01:13:12.500 --> 01:13:16.610
will you consider
to set up the car?
01:13:16.610 --> 01:13:22.700
Will you set it up as
underdamped, critically damped,
01:13:22.700 --> 01:13:24.710
or overdamped?
01:13:24.710 --> 01:13:27.998
How many of you actually think
it should be underdamped?
01:13:31.910 --> 01:13:33.650
No, nobody?
01:13:33.650 --> 01:13:39.390
How many of you actually
think it should be overdamped?
01:13:39.390 --> 01:13:45.110
1, 2, 3, 4, OK.
01:13:45.110 --> 01:13:49.250
How many of you actually think
it should be critically damped?
01:13:49.250 --> 01:13:51.360
OK, the majority of
you think that should
01:13:51.360 --> 01:13:52.930
be the correct design.
01:13:52.930 --> 01:13:59.680
So if you have the car designed
as an underdamped situation,
01:13:59.680 --> 01:14:01.470
then, when you
drive the car, you
01:14:01.470 --> 01:14:03.280
are going to have
very funny style.
01:14:03.280 --> 01:14:05.760
You are going to have this.
01:14:05.760 --> 01:14:07.470
This is the style.
01:14:07.470 --> 01:14:11.580
So the car is going to be
oscillating all the time, OK?
01:14:11.580 --> 01:14:14.290
Because it's going to be there.
01:14:14.290 --> 01:14:18.270
And it's really damping
really slowly, OK?
01:14:18.270 --> 01:14:22.860
If you design it
to be overdamped,
01:14:22.860 --> 01:14:24.930
it would become
very bumpy, right?
01:14:24.930 --> 01:14:29.040
So let's take a limit
of infinitely large drag
01:14:29.040 --> 01:14:30.600
force constant, OK?
01:14:30.600 --> 01:14:34.590
Then, it's like, when you
hit some bump, you go woo!
01:14:34.590 --> 01:14:36.360
Wow!
01:14:36.360 --> 01:14:40.710
It doesn't really help you
to reduce the amplitude, OK?
01:14:40.710 --> 01:14:44.440
So the correct
answer is you would
01:14:44.440 --> 01:14:50.010
give the advice that you would
do it critically damped, OK?
01:14:50.010 --> 01:14:54.420
So before we end
the section today,
01:14:54.420 --> 01:14:57.820
I would like to pose
a question to you.
01:14:57.820 --> 01:15:02.500
The thing which we have learned
from simple harmonic motion
01:15:02.500 --> 01:15:07.090
is that the energy is conserved
in a simple harmonic motion,
01:15:07.090 --> 01:15:07.590
OK?
01:15:07.590 --> 01:15:13.500
I have the Fs, the spring force,
proportional to minus k times
01:15:13.500 --> 01:15:15.000
x.
01:15:15.000 --> 01:15:18.360
And the energy is conserved, OK?
01:15:18.360 --> 01:15:23.430
But if I add a drag force in
the form or minus b times v,
01:15:23.430 --> 01:15:25.170
energy is not conserved, right?
01:15:25.170 --> 01:15:28.070
So you can see that it
was actually oscillating.
01:15:28.070 --> 01:15:31.090
Now, it's not
oscillating, right?
01:15:31.090 --> 01:15:34.200
This thing has stopped
oscillating, OK?
01:15:34.200 --> 01:15:38.400
Why is that the
case mathematically?
01:15:38.400 --> 01:15:42.600
OK, we know what is
happening physically
01:15:42.600 --> 01:15:43.980
in this physical system.
01:15:43.980 --> 01:15:50.250
Because OK, this Mexican hat
is trying to push the air away.
01:15:50.250 --> 01:15:53.280
So what is going to happen
is that it's transferring
01:15:53.280 --> 01:15:59.040
the energy from this system to
the molecules of the air, OK?
01:15:59.040 --> 01:16:00.780
So it's accelerating the air.
01:16:00.780 --> 01:16:02.700
So the energy goes away.
01:16:02.700 --> 01:16:06.660
But why the mathematical
form looks so similar
01:16:06.660 --> 01:16:09.120
and it does different things?
01:16:09.120 --> 01:16:10.820
And think about it.
01:16:10.820 --> 01:16:14.890
And I'm not going to talk
about the answer today.
01:16:14.890 --> 01:16:17.910
And thank you very much.
01:16:17.910 --> 01:16:20.890
And we will continue
next time to see
01:16:20.890 --> 01:16:24.770
what we can learn if I start
to drive the oscillator.
01:16:24.770 --> 01:16:26.490
Bye-bye.