WEBVTT
00:00:00.050 --> 00:00:01.770
The following
content is provided
00:00:01.770 --> 00:00:04.010
under a Creative
Commons license.
00:00:04.010 --> 00:00:06.860
Your support will help MIT
OpenCourseWare continue
00:00:06.860 --> 00:00:10.720
to offer high quality
educational resources for free.
00:00:10.720 --> 00:00:13.330
To make a donation or
view additional materials
00:00:13.330 --> 00:00:17.209
from hundreds of MIT courses,
visit MIT OpenCourseWare
00:00:17.209 --> 00:00:17.834
at ocw.mit.edu.
00:00:21.430 --> 00:00:25.030
PROFESSOR: I'm Wit Busza,
professor of physics at MIT.
00:00:25.030 --> 00:00:29.120
I'm joining my colleague,
Professor Walter Lewin,
00:00:29.120 --> 00:00:31.980
to help you understand
the physics of waves
00:00:31.980 --> 00:00:33.200
and vibrations.
00:00:33.200 --> 00:00:37.360
Now you may well ask, why
spend so much effort on waves
00:00:37.360 --> 00:00:38.460
and vibrations.
00:00:38.460 --> 00:00:41.140
And the answer is very simple.
00:00:41.140 --> 00:00:48.100
If you take any system,
disturb it from equilibrium,
00:00:48.100 --> 00:00:52.410
from a stable equilibrium,
the resultant motion
00:00:52.410 --> 00:00:54.600
is waves and vibrations.
00:00:54.600 --> 00:00:57.100
So it's a very
common phenomenon.
00:00:57.100 --> 00:01:01.320
Not only is it that very
common, understanding waves
00:01:01.320 --> 00:01:04.220
and vibrations
have very important
00:01:04.220 --> 00:01:05.670
practical applications.
00:01:05.670 --> 00:01:09.060
And furthermore, the
fact that they exist,
00:01:09.060 --> 00:01:12.390
that this phenomenon exists,
has tremendous consequences
00:01:12.390 --> 00:01:14.210
on our world.
00:01:14.210 --> 00:01:18.680
If waves and vibrations were
different or didn't exist,
00:01:18.680 --> 00:01:23.450
you wouldn't recognize
our universe.
00:01:23.450 --> 00:01:27.780
What is the role I am
playing in this course?
00:01:27.780 --> 00:01:31.650
To answer that question,
I have to remind you
00:01:31.650 --> 00:01:34.200
what is the scientific method.
00:01:34.200 --> 00:01:38.950
In essence, the scientific
method has two components.
00:01:38.950 --> 00:01:43.230
The first, you
look around and you
00:01:43.230 --> 00:01:48.120
describe what you see in the
one and only language that
00:01:48.120 --> 00:01:50.770
can be used, or we
find that can be
00:01:50.770 --> 00:01:52.770
used for the
description of nature.
00:01:52.770 --> 00:01:55.920
That this mathematics, in terms
of mathematical equations.
00:01:59.130 --> 00:02:02.870
The second aspect is
since the universe
00:02:02.870 --> 00:02:05.930
is describable in terms
of mathematical equations,
00:02:05.930 --> 00:02:09.160
we can solve those equations.
00:02:09.160 --> 00:02:14.510
And that means predict
result of situations,
00:02:14.510 --> 00:02:19.080
of experiments, which
we've never seen before.
00:02:19.080 --> 00:02:23.200
Again, this is important
for two reasons.
00:02:23.200 --> 00:02:27.350
One, practical-- to be able
to predict what will happen.
00:02:27.350 --> 00:02:30.680
But the other far
more important is
00:02:30.680 --> 00:02:35.840
that it is the way we have,
the objective way we have
00:02:35.840 --> 00:02:40.290
of checking whether our
understanding of the universe
00:02:40.290 --> 00:02:42.160
is correct or not.
00:02:42.160 --> 00:02:49.000
If the predictions do not give
the right-- do not correspond
00:02:49.000 --> 00:02:51.900
to what one actually sees,
you know, your theory,
00:02:51.900 --> 00:02:55.250
your understanding is wrong.
00:02:55.250 --> 00:02:59.740
My role is related
to the second part.
00:02:59.740 --> 00:03:05.230
In other words, what I would
want to help you learn,
00:03:05.230 --> 00:03:10.730
take a given situation,
convert it into mathematics,
00:03:10.730 --> 00:03:15.200
solve it, and predict
what will happen.
00:03:15.200 --> 00:03:18.670
We call that problem solving.
00:03:18.670 --> 00:03:19.460
OK.
00:03:19.460 --> 00:03:23.255
Let me immediately start
with a concrete example.
00:03:30.010 --> 00:03:36.300
What I have here,
describing a situation which
00:03:36.300 --> 00:03:39.290
we would like to understand.
00:03:39.290 --> 00:03:42.450
Imagine you have
an ideal spring,
00:03:42.450 --> 00:03:46.190
a spring that obeys Hooke's law.
00:03:46.190 --> 00:03:50.380
As I've shown here is the
spring constant k, length,
00:03:50.380 --> 00:03:54.890
natural length l0, and your
suspend it from the ceiling.
00:03:54.890 --> 00:03:59.410
Imagine that you take
a mass, a small mass,
00:03:59.410 --> 00:04:03.990
m, and you attach
it to that spring.
00:04:03.990 --> 00:04:08.870
At some instant of time, and in
the proceeds of attaching it,
00:04:08.870 --> 00:04:10.170
you may stretch the string.
00:04:10.170 --> 00:04:13.200
So the spring may at this
instant not be stretched.
00:04:13.200 --> 00:04:15.680
But let's assume
while you're attaching
00:04:15.680 --> 00:04:17.680
and you've stretched
the spring a little bit,
00:04:17.680 --> 00:04:20.769
you're holding it, all right.
00:04:20.769 --> 00:04:24.820
At that instant, it's
velocity is 0, stationary.
00:04:24.820 --> 00:04:26.170
You let go.
00:04:26.170 --> 00:04:28.470
The question is,
what will happen?
00:04:28.470 --> 00:04:33.052
Can you predict what will be
the motion of that particle?
00:04:33.052 --> 00:04:34.800
You know, you've
seen this often.
00:04:34.800 --> 00:04:37.400
But a priori, it's not
obvious what will happen.
00:04:37.400 --> 00:04:39.560
The spring may pull the mass up.
00:04:39.560 --> 00:04:41.920
The mass may pull
the spring more down.
00:04:41.920 --> 00:04:43.610
It may oscillate.
00:04:43.610 --> 00:04:48.070
Everything, until you've
understood what's going on, you
00:04:48.070 --> 00:04:51.770
cannot predict the outcome.
00:04:51.770 --> 00:04:57.840
So let's assume that at
some instant of time,
00:04:57.840 --> 00:05:03.790
we call that time t, it's
as shown on the right.
00:05:03.790 --> 00:05:05.690
In order to be able
to describe this,
00:05:05.690 --> 00:05:09.940
I have to tell you
where these masses are
00:05:09.940 --> 00:05:11.170
at these various times.
00:05:11.170 --> 00:05:14.540
So I will define a
coordinate system.
00:05:14.540 --> 00:05:16.580
This is a one
dimensional situation.
00:05:16.580 --> 00:05:17.830
So I only need one coordinate.
00:05:17.830 --> 00:05:19.570
And I'll call it the y.
00:05:19.570 --> 00:05:21.920
My y will be up.
00:05:21.920 --> 00:05:23.000
All right.
00:05:23.000 --> 00:05:26.550
Now I also have to measure
things from some location.
00:05:26.550 --> 00:05:30.101
So I need to define what
I mean by y equals 0.
00:05:30.101 --> 00:05:34.450
And I will define y
equals 0, the position
00:05:34.450 --> 00:05:40.560
where if the mass is at that
location, the force of gravity
00:05:40.560 --> 00:05:44.130
pulling it down and the
spring force pulling it up
00:05:44.130 --> 00:05:49.610
cancel, so that there is no
net force on the particle.
00:05:49.610 --> 00:05:53.062
So y equals 0 is the
equilibrium position.
00:05:53.062 --> 00:05:55.130
And then the position
of the spring
00:05:55.130 --> 00:06:00.190
when it has no mass attached
is the distance y0 from that
00:06:00.190 --> 00:06:04.080
at t equals 0, the position
I will say is y initial.
00:06:04.080 --> 00:06:05.360
That's some number.
00:06:05.360 --> 00:06:07.190
So that's a known quantity.
00:06:07.190 --> 00:06:07.790
All right.
00:06:07.790 --> 00:06:12.890
And that any other instant time,
I defined it as y equals t.
00:06:12.890 --> 00:06:16.960
That is the physical situation
I wish to understand.
00:06:16.960 --> 00:06:20.340
I want to know what
happened with that spring.
00:06:20.340 --> 00:06:25.430
So now I will translate
that into mathematics.
00:06:25.430 --> 00:06:28.750
I will now try to give you
a mathematical description
00:06:28.750 --> 00:06:29.503
of that situation.
00:06:32.860 --> 00:06:40.040
So we know that we are dealing
with forces and masses.
00:06:40.040 --> 00:06:45.620
So to describe that, we
use Newtonian mechanics.
00:06:45.620 --> 00:06:49.910
So here is now my
mathematical description.
00:06:49.910 --> 00:06:56.470
The mass is a point m, of mass
m, on which two forces act.
00:06:56.470 --> 00:07:02.330
There is the force fs due to
the spring and the force fg
00:07:02.330 --> 00:07:06.680
due to the fact that this mass
is in the gravitational field,
00:07:06.680 --> 00:07:10.530
and therefore there is
a gravitational force
00:07:10.530 --> 00:07:11.860
on this mass, fg.
00:07:11.860 --> 00:07:13.350
OK.
00:07:13.350 --> 00:07:15.540
We call this a force diagram.
00:07:15.540 --> 00:07:17.930
Or some people call it
the free body diagram.
00:07:20.500 --> 00:07:26.700
Now this mass, because
of the force acting it,
00:07:26.700 --> 00:07:28.870
its motion will change.
00:07:28.870 --> 00:07:31.960
And it will have
an acceleration,
00:07:31.960 --> 00:07:34.230
which I will call a of t.
00:07:34.230 --> 00:07:39.430
And by the way, that of course
is the second derivative
00:07:39.430 --> 00:07:42.400
of y with respect to dt.
00:07:42.400 --> 00:07:42.990
It's a vector.
00:07:42.990 --> 00:07:46.090
It's in the y direction.
00:07:46.090 --> 00:07:49.030
And in order so I don't
have to write things over
00:07:49.030 --> 00:07:52.970
and too many things
in these equations,
00:07:52.970 --> 00:07:56.960
I will define the symbol
y with two dots on it
00:07:56.960 --> 00:08:01.310
as the second derivative
of y with respect to time.
00:08:01.310 --> 00:08:04.120
y dot is the first
derivative-- in other words,
00:08:04.120 --> 00:08:07.120
the velocity, et cetera.
00:08:07.120 --> 00:08:11.450
And by the way, you may notice
I'm going very slowly here.
00:08:11.450 --> 00:08:13.870
I'm doing that intentionally.
00:08:13.870 --> 00:08:19.600
I'm going to go here in gory
detail every part, you know,
00:08:19.600 --> 00:08:23.320
because often I know
that when one goes
00:08:23.320 --> 00:08:26.900
to a lecture, or studies
in a book, et cetera,
00:08:26.900 --> 00:08:31.560
you look at some step
from one step to another.
00:08:31.560 --> 00:08:33.450
And you can't figure it out.
00:08:33.450 --> 00:08:37.289
The reason for it often is not
that you are not smart enough
00:08:37.289 --> 00:08:41.010
to do it, is but the
because the teacher
00:08:41.010 --> 00:08:43.120
or whoever wrote
the book, et cetera,
00:08:43.120 --> 00:08:45.760
is so familiar with
the material he will
00:08:45.760 --> 00:08:49.170
do several steps in
his head or her head,
00:08:49.170 --> 00:08:51.050
and you don't know about it.
00:08:51.050 --> 00:08:53.510
For this first
example, I will try
00:08:53.510 --> 00:08:55.790
to avoid anything of that kind.
00:08:55.790 --> 00:08:57.070
Later on, I'll go faster.
00:08:57.070 --> 00:08:59.130
And I'll do the same
as everybody else.
00:08:59.130 --> 00:09:02.900
But the moment, as I say,
I'm going in gory detail.
00:09:02.900 --> 00:09:03.590
OK.
00:09:03.590 --> 00:09:09.135
So this is the diagram,
this free body diagram,
00:09:09.135 --> 00:09:11.570
of the situation.
00:09:11.570 --> 00:09:15.750
And I know from
Newtonian mechanics
00:09:15.750 --> 00:09:20.360
that if there are forces
acting on that mass,
00:09:20.360 --> 00:09:24.010
that mass will have
an acceleration, which
00:09:24.010 --> 00:09:27.070
will be equal to the
net force acting on it,
00:09:27.070 --> 00:09:32.540
divided by the mass, the
inertia, of that system.
00:09:32.540 --> 00:09:35.100
So that is what a will be.
00:09:35.100 --> 00:09:38.570
I further know the
force is vector,
00:09:38.570 --> 00:09:41.830
so the net force
acting on this mass
00:09:41.830 --> 00:09:44.250
is the sum of those,
the vectorial sum,
00:09:44.250 --> 00:09:45.890
of those two forces.
00:09:45.890 --> 00:09:51.020
So f is the sum of the
force due to the spring
00:09:51.020 --> 00:09:54.280
and due to gravity.
00:09:54.280 --> 00:09:57.230
Next, we also know
something about the spring.
00:09:57.230 --> 00:09:59.050
I told you at the
beginning that I'm
00:09:59.050 --> 00:10:01.510
considering an ideal spring.
00:10:01.510 --> 00:10:03.750
So for the purpose
of this problem,
00:10:03.750 --> 00:10:07.370
I'm assuming I have this
fictitious thing, a spring
00:10:07.370 --> 00:10:10.070
which essentially has
no mass, massless,
00:10:10.070 --> 00:10:13.240
which obeys exactly Hooke's law.
00:10:13.240 --> 00:10:17.230
And here I can't help
digress and point out to you
00:10:17.230 --> 00:10:19.440
that that's a terrible misnomer.
00:10:19.440 --> 00:10:23.130
There is no Hooke's
law of nature.
00:10:23.130 --> 00:10:27.575
It is an empirical relation
which tells you the force
00:10:27.575 --> 00:10:31.490
that the spring exerts when you
stretch it a certain distance,
00:10:31.490 --> 00:10:32.620
all right.
00:10:32.620 --> 00:10:34.750
But anyway, it's
stuck historically.
00:10:34.750 --> 00:10:35.990
It's Hooke's law.
00:10:35.990 --> 00:10:38.810
So Hooke's law,
from Hooke's law,
00:10:38.810 --> 00:10:45.380
I know what will be the force,
fs, when the situation is as
00:10:45.380 --> 00:10:49.340
shown over there,
all right, at time t.
00:10:49.340 --> 00:10:53.930
So at this time, this
extension of this spring
00:10:53.930 --> 00:10:57.480
will be, of course, y0 minus yt.
00:10:57.480 --> 00:10:58.170
OK.
00:10:58.170 --> 00:11:01.910
And so I get that the
force due to the spring
00:11:01.910 --> 00:11:05.590
will be the spring constant
times its extension
00:11:05.590 --> 00:11:07.310
at that instant of time.
00:11:07.310 --> 00:11:08.900
It is a vector.
00:11:08.900 --> 00:11:18.320
And y0 is a bigger number than
yt, this is a positive number.
00:11:18.320 --> 00:11:22.540
Therefore, the stretched
spring will pull the mass up.
00:11:22.540 --> 00:11:24.080
So this is in the y direction.
00:11:24.080 --> 00:11:26.900
This is plus.
00:11:26.900 --> 00:11:28.890
How about the
gravitational force?
00:11:28.890 --> 00:11:34.630
Well, that is, of
course, the minus mg,
00:11:34.630 --> 00:11:37.390
the force of the
gravitational field on that.
00:11:37.390 --> 00:11:40.390
And it's minus in
the y direction
00:11:40.390 --> 00:11:45.110
here, because it's
pulling this mass down.
00:11:45.110 --> 00:11:45.980
OK.
00:11:45.980 --> 00:11:49.090
Now what else do we know?
00:11:49.090 --> 00:11:53.230
We know that we could get
everything done very carefully.
00:11:53.230 --> 00:11:56.030
We know that we
defined y equals 0
00:11:56.030 --> 00:11:58.250
to be the equilibrium position.
00:11:58.250 --> 00:12:05.935
Therefore, when y is 0, we know
that the second derivative of y
00:12:05.935 --> 00:12:06.760
is 0.
00:12:06.760 --> 00:12:08.350
It's not accelerating.
00:12:08.350 --> 00:12:10.840
So that's a condition
we must not forget.
00:12:10.840 --> 00:12:14.910
Another thing we know that
initially, in other words,
00:12:14.910 --> 00:12:20.770
at t equals 0, the position
of that mass is y initial.
00:12:20.770 --> 00:12:26.720
Finally, I told you that
the velocity of that mass
00:12:26.720 --> 00:12:30.230
was 0 at t equals 0, stationary.
00:12:30.230 --> 00:12:35.040
So this is the beginning of our
translating all the information
00:12:35.040 --> 00:12:38.940
we gathered here
into mathematics.
00:12:48.400 --> 00:12:52.150
Let me continue now
using this information
00:12:52.150 --> 00:12:57.210
and try to reduce it to the
minimum set of equations.
00:12:57.210 --> 00:13:03.190
From a equals fm, from this,
I get that the acceleration
00:13:03.190 --> 00:13:06.140
is the total force divided by m.
00:13:06.140 --> 00:13:10.380
I can now replace
these two forces
00:13:10.380 --> 00:13:12.790
from the information
I wrote over there.
00:13:12.790 --> 00:13:17.210
And so that is equal, 1
over-- this is f over m.
00:13:17.210 --> 00:13:19.590
It's 1 of m times
the net force, which
00:13:19.590 --> 00:13:22.430
is the force due to the
spring minus the force
00:13:22.430 --> 00:13:24.160
due to the gravity, OK.
00:13:24.160 --> 00:13:33.260
So from this, I can now actually
write an algebraic equation,
00:13:33.260 --> 00:13:34.730
rather than the vector one.
00:13:34.730 --> 00:13:37.185
Notice - ha ha.
00:13:37.185 --> 00:13:39.940
I have noticed myself
even something.
00:13:39.940 --> 00:13:45.230
Here this has to be
in the direction of y.
00:13:45.230 --> 00:13:47.070
OK.
00:13:47.070 --> 00:13:51.720
This is a vector equation,
but all the parts
00:13:51.720 --> 00:13:55.820
are in the same direction,
in the y direction.
00:13:55.820 --> 00:14:01.700
Therefore, I can rewrite this
just the equation for the one
00:14:01.700 --> 00:14:07.000
component and not bother to
write the y hats throughout.
00:14:07.000 --> 00:14:12.470
So this equation I've rewritten
now just removing y hat.
00:14:12.470 --> 00:14:17.860
So this is how the mass
will be accelerating.
00:14:17.860 --> 00:14:20.230
Unfortunately, it's
a single equation,
00:14:20.230 --> 00:14:22.260
but I have more than
one unknown in it.
00:14:22.260 --> 00:14:24.920
Because I don't know y0.
00:14:24.920 --> 00:14:26.600
And I don't know y of t.
00:14:26.600 --> 00:14:29.940
Clearly, I won't be able to
solve that equation, all right.
00:14:29.940 --> 00:14:33.280
But at that stage, I go
back to the information
00:14:33.280 --> 00:14:35.270
I told you at the beginning.
00:14:35.270 --> 00:14:40.420
We defined y equals 0
to be the place where
00:14:40.420 --> 00:14:44.500
a y double dot, the
second derivative, is 0.
00:14:44.500 --> 00:14:50.640
Therefore, I can write
that at position.
00:14:50.640 --> 00:14:54.510
When y of t is 0, this is 0.
00:14:54.510 --> 00:15:00.210
So 0 is equal to 1 over
m, into ky minus mg.
00:15:00.210 --> 00:15:04.540
And I immediately from
this get that ky0 is mg.
00:15:04.540 --> 00:15:09.800
Therefore, I have
found what y0 is.
00:15:09.800 --> 00:15:10.440
OK.
00:15:10.440 --> 00:15:11.220
Great.
00:15:11.220 --> 00:15:14.270
So there's only
one unknown here.
00:15:14.270 --> 00:15:18.390
So using this
information in here,
00:15:18.390 --> 00:15:22.530
I end up immediately
with this equation,
00:15:22.530 --> 00:15:24.880
that the second derivative
of y with respect
00:15:24.880 --> 00:15:27.500
to time, the
acceleration of the mass
00:15:27.500 --> 00:15:34.620
is equal to minus k
over m times y of t.
00:15:34.620 --> 00:15:41.630
At this stage, I will really
find this quantity, k over m.
00:15:41.630 --> 00:15:46.410
For the time being, you can look
at it as just for convenience,
00:15:46.410 --> 00:15:48.330
less to write on the board.
00:15:48.330 --> 00:15:54.110
But later, you see this
will help us understand
00:15:54.110 --> 00:15:56.310
how to deal with
different situations.
00:15:56.310 --> 00:15:58.390
But for the time
being, you can just
00:15:58.390 --> 00:16:01.830
think of this as a
convenience, so I
00:16:01.830 --> 00:16:03.650
can write less on the board.
00:16:03.650 --> 00:16:08.640
And I end up finally
with one equation.
00:16:08.640 --> 00:16:11.590
The second derivative of
y with respect to time
00:16:11.590 --> 00:16:15.650
is equal to minus a constant,
that's k over m, right,
00:16:15.650 --> 00:16:18.210
times the value of y times t.
00:16:18.210 --> 00:16:22.960
This is the equation of
motion for this mass.
00:16:22.960 --> 00:16:28.420
It tells me in mathematical
form how the motion of that mass
00:16:28.420 --> 00:16:31.070
changes with time.
00:16:31.070 --> 00:16:35.320
I can now actually
predict what will
00:16:35.320 --> 00:16:39.840
happen in this
particular situation.
00:16:39.840 --> 00:16:46.570
Because I know what was
the motion of it at time 0.
00:16:46.570 --> 00:16:52.150
I know that at time 0, the
position was y initial.
00:16:52.150 --> 00:16:55.800
And the velocity was 0.
00:16:55.800 --> 00:17:03.120
These three lines are completely
equivalent from the point
00:17:03.120 --> 00:17:06.920
of view of understanding
the motion of the mass
00:17:06.920 --> 00:17:09.950
to our original description.
00:17:09.950 --> 00:17:14.010
This is a physical
description of the situation.
00:17:14.010 --> 00:17:17.920
This is a mathematical
description
00:17:17.920 --> 00:17:18.904
of the same situation.
00:17:22.450 --> 00:17:24.540
So we've achieved step one.
00:17:24.540 --> 00:17:27.730
We've translated a
physical situation
00:17:27.730 --> 00:17:30.310
into a mathematical one.
00:17:30.310 --> 00:17:33.260
Let me now try
from this, I should
00:17:33.260 --> 00:17:35.260
be able to predict
what this mass will do.
00:17:42.420 --> 00:17:42.920
OK.
00:17:47.660 --> 00:17:51.450
I'm now switching into
the world the mathematics.
00:17:51.450 --> 00:17:54.330
As I just am
repeating here, I've
00:17:54.330 --> 00:17:57.040
gone away from a
physical description
00:17:57.040 --> 00:17:59.460
to a mathematical description.
00:17:59.460 --> 00:18:02.820
This is pure mathematics.
00:18:02.820 --> 00:18:05.280
I have an equation, a
mathematical equation,
00:18:05.280 --> 00:18:07.470
for y of t.
00:18:07.470 --> 00:18:10.130
It's a second order
differential equation.
00:18:10.130 --> 00:18:15.310
I had the boundary conditions,
or initial conditions.
00:18:15.310 --> 00:18:19.490
I can solve that
using mathematics.
00:18:19.490 --> 00:18:20.090
OK.
00:18:20.090 --> 00:18:20.970
Let's do that.
00:18:20.970 --> 00:18:22.880
So I'm now doing
pure mathematics.
00:18:22.880 --> 00:18:25.490
I don't want to teach you math.
00:18:25.490 --> 00:18:28.540
That's the role of the
math department, all right.
00:18:28.540 --> 00:18:32.010
So how do I solve that equation?
00:18:32.010 --> 00:18:35.000
And let me tell
you how I solve it.
00:18:35.000 --> 00:18:39.680
I am make use of the
so-called uniqueness theorem.
00:18:39.680 --> 00:18:43.430
I know, or the
mathematicians have told me,
00:18:43.430 --> 00:18:58.360
that if I find a solution to
that equation which satisfies--
00:18:58.360 --> 00:19:03.390
if I find a solution which
satisfies that equation,
00:19:03.390 --> 00:19:09.080
and if it has the right
number of arbitrary constant,
00:19:09.080 --> 00:19:14.770
then I have found the one
and only general equation,
00:19:14.770 --> 00:19:18.600
which is a solution of that.
00:19:18.600 --> 00:19:19.490
Let me be concrete.
00:19:23.170 --> 00:19:39.780
y of t equals to a cosine omega
t plus phi, where A and phi are
00:19:39.780 --> 00:19:42.215
arbitrary, are arbitrary.
00:19:47.280 --> 00:19:49.310
They are some arbitrary numbers.
00:19:49.310 --> 00:19:50.450
But a number is there.
00:19:50.450 --> 00:19:53.470
This can be 7 and this
can be 21 degrees,
00:19:53.470 --> 00:19:55.998
or whatever, but any number.
00:19:55.998 --> 00:20:01.090
This equation satisfies
my differential equation.
00:20:01.090 --> 00:20:02.830
If you don't believe me, try it.
00:20:02.830 --> 00:20:06.960
Differentiate this
twice, all right,
00:20:06.960 --> 00:20:11.580
for any value of A and phi and
you'll satisfy that equation.
00:20:11.580 --> 00:20:16.490
So this is a solution which
satisfy that equation.
00:20:16.490 --> 00:20:19.390
It has the right number
of arbitrary constants,
00:20:19.390 --> 00:20:23.410
that two arbitrary
constants in here.
00:20:23.410 --> 00:20:25.830
And therefore, this
is the only solution
00:20:25.830 --> 00:20:29.130
in the universe
of that equation.
00:20:29.130 --> 00:20:30.300
OK.
00:20:30.300 --> 00:20:33.990
Now, so being a
physicist, I don't
00:20:33.990 --> 00:20:36.370
care how I got the solution.
00:20:36.370 --> 00:20:39.020
Once I had the
solution, if I know
00:20:39.020 --> 00:20:42.950
it's the only one
that exists, I'm home.
00:20:42.950 --> 00:20:46.037
Now you can say well, I
suppose I didn't guess it.
00:20:46.037 --> 00:20:47.120
Well, there are many ways.
00:20:47.120 --> 00:20:48.550
You go on the web and find it.
00:20:48.550 --> 00:20:49.940
You go in the book and find it.
00:20:49.940 --> 00:20:52.680
You ask your friends
what it is, all right.
00:20:52.680 --> 00:20:54.540
That's mathematics.
00:20:54.540 --> 00:20:58.540
And once you've found the
solution, we can go on.
00:20:58.540 --> 00:21:02.020
All right, so this is the
solution of that equation.
00:21:02.020 --> 00:21:05.430
Next, if that's
y, what is y dot?
00:21:05.430 --> 00:21:08.070
What is the rate of change of t?
00:21:08.070 --> 00:21:15.136
That's going to equal minus
omega 0 A sine omega 0 t
00:21:15.136 --> 00:21:17.424
plus phi, OK.
00:21:20.910 --> 00:21:23.585
Can I predict what will happen?
00:21:23.585 --> 00:21:24.590
All right.
00:21:24.590 --> 00:21:28.310
I still need, in order to
be able to predict what
00:21:28.310 --> 00:21:30.570
will happen, I need
to find out what
00:21:30.570 --> 00:21:36.090
are the values of
A and phi which
00:21:36.090 --> 00:21:40.250
satisfy the other
information right here.
00:21:40.250 --> 00:21:44.830
See, I told you that we
reduce that physical situation
00:21:44.830 --> 00:21:47.480
to a differential equation,
the equation of motion
00:21:47.480 --> 00:21:53.660
for this mass, including the
information about where it was
00:21:53.660 --> 00:21:57.780
at some instant of time, how
it was moving, et cetera.
00:21:57.780 --> 00:22:05.470
So I need to make sure that
this equation satisfies
00:22:05.470 --> 00:22:08.010
these boundary conditions.
00:22:08.010 --> 00:22:12.100
In other words, it its
thees boundary conditions
00:22:12.100 --> 00:22:17.130
which will determine
what are the A's and phi
00:22:17.130 --> 00:22:21.930
for the particular problem
that I had there, OK.
00:22:21.930 --> 00:22:26.560
And so what I do is-- let's,
for example, takes here.
00:22:26.560 --> 00:22:28.040
Because I see 0.
00:22:28.040 --> 00:22:36.000
Y dot t is 0, all
right, at t equals 0.
00:22:36.000 --> 00:22:45.100
Well, when t equals 0, this
is minus omega 0 A sine phi.
00:22:45.100 --> 00:22:46.440
OK.
00:22:46.440 --> 00:22:51.535
Therefore, I immediately
conclude that phi is 0.
00:22:55.020 --> 00:22:56.740
OK.
00:22:56.740 --> 00:23:00.840
Next, I know-- so now
that I know that phi is 0,
00:23:00.840 --> 00:23:03.850
I can go back to this equation.
00:23:03.850 --> 00:23:05.510
This is now 0.
00:23:05.510 --> 00:23:12.730
And we know that y at t
equals 0 is y initial.
00:23:12.730 --> 00:23:17.130
But that t equals
0 cosine of 0 is 1.
00:23:17.130 --> 00:23:21.010
Therefore, A is y initial.
00:23:21.010 --> 00:23:32.530
And so I get finitely y of t is
equal to y initial, all right,
00:23:32.530 --> 00:23:38.300
times cosine omega 0.
00:23:38.300 --> 00:23:41.340
Let me now replace
it with the 1-- well,
00:23:41.340 --> 00:23:45.570
let me leave it as
omega 0 t pluse 0.
00:23:45.570 --> 00:23:50.450
That, and I can rewrite
this, putting all the numbers
00:23:50.450 --> 00:23:58.280
that I have, y initial cosine.
00:23:58.280 --> 00:24:02.050
And I'll now even
replace omega 0 by k
00:24:02.050 --> 00:24:05.385
over n, square root
of k over n, times t.
00:24:10.420 --> 00:24:17.820
Notice there are no
unknown quantities in here.
00:24:17.820 --> 00:24:22.210
This tells me two things.
00:24:22.210 --> 00:24:28.590
At any instant of
time, I can calculate
00:24:28.590 --> 00:24:32.340
where this mass will be.
00:24:32.340 --> 00:24:35.360
It's given by this equation.
00:24:35.360 --> 00:24:39.230
Secondly, I can describe
the kind of motion it does.
00:24:39.230 --> 00:24:41.160
What is this equation?
00:24:41.160 --> 00:24:44.060
As a function of
time, this corresponds
00:24:44.060 --> 00:24:48.060
to an oscillating position y.
00:24:48.060 --> 00:24:51.160
So this mass, when I
let go, will oscillate.
00:24:55.140 --> 00:24:56.720
What will be the period?
00:24:56.720 --> 00:25:01.360
How long will it take before it
comes back to where it started?
00:25:01.360 --> 00:25:06.320
Well, the period t
will be how much time
00:25:06.320 --> 00:25:12.360
do I have to add to this
t, so that the angle here
00:25:12.360 --> 00:25:14.230
changes by 2 pi?
00:25:14.230 --> 00:25:21.128
Well, that's obviously
2 pi root of m over k.
00:25:21.128 --> 00:25:23.510
OK.
00:25:23.510 --> 00:25:26.640
So I've achieved
what I wanted to do.
00:25:26.640 --> 00:25:29.910
I've taken a physical situation.
00:25:29.910 --> 00:25:35.160
And I have predicted if I
let go what will happen.
00:25:35.160 --> 00:25:40.710
This is the motion
it will experience.
00:25:40.710 --> 00:25:42.750
This is the period.
00:25:42.750 --> 00:25:44.830
I can predict the
time, et cetera.
00:25:44.830 --> 00:25:48.270
At this stage, let's
stop for a second
00:25:48.270 --> 00:25:52.390
and consider what we've done.
00:25:52.390 --> 00:25:56.050
Because it's the
essence of-- this
00:25:56.050 --> 00:25:59.010
is a good example of the essence
of the scientific method.
00:26:04.070 --> 00:26:05.855
We have taken a
physical situation.
00:26:08.870 --> 00:26:15.425
We've described it in
terms of mathematics.
00:26:20.040 --> 00:26:26.820
Then we made an act
of faith that if I
00:26:26.820 --> 00:26:33.410
take the mathematical
equations and I solve them,
00:26:33.410 --> 00:26:37.340
that the resultant
answer will actually
00:26:37.340 --> 00:26:41.890
correspond to what
nature will do.
00:26:41.890 --> 00:26:45.540
If you stop to think
about that, it's amazing.
00:26:45.540 --> 00:26:48.400
Nobody understands that fact.
00:26:48.400 --> 00:26:49.140
Why that's true.
00:26:49.140 --> 00:26:51.980
Why it happened.
00:26:51.980 --> 00:26:53.670
In other words,
nobody understands
00:26:53.670 --> 00:26:59.230
why nature can be described
in terms of mathematics.
00:26:59.230 --> 00:27:00.200
OK.
00:27:00.200 --> 00:27:06.990
But it is that fact which makes
the scientific method possible.
00:27:09.880 --> 00:27:15.325
Finally on this note, let me
give a quotation from Einstein
00:27:15.325 --> 00:27:20.110
which beautifully summarizes
what I've just said.
00:27:20.110 --> 00:27:24.850
And that is the following, "The
most incomprehensible thing
00:27:24.850 --> 00:27:29.780
about the universe is that
it is comprehensible."
00:27:29.780 --> 00:27:34.930
The fact that we can follow
this procedure is amazing.
00:27:38.000 --> 00:27:39.040
OK.
00:27:39.040 --> 00:27:44.180
Let me at this stage go and
take another example, all right.
00:27:54.820 --> 00:27:56.635
So let's take another example.
00:28:05.220 --> 00:28:08.340
Consider the
following situation.
00:28:08.340 --> 00:28:14.500
I take something like
a ruler, a uniform rod.
00:28:14.500 --> 00:28:19.165
And I put a nail through
it, some kind of a pivot.
00:28:19.165 --> 00:28:21.290
There is some pivot.
00:28:21.290 --> 00:28:24.340
I pivot the ruler on it.
00:28:24.340 --> 00:28:27.880
And it's hanging like this.
00:28:27.880 --> 00:28:28.760
OK.
00:28:28.760 --> 00:28:34.630
Let's assume the mass
is m of the ruler.
00:28:34.630 --> 00:28:35.785
The length is l.
00:28:41.830 --> 00:28:46.160
It's a uniform ruler,
a rod of some kind.
00:28:46.160 --> 00:28:54.630
And at t equals 0, I
give it an impulse.
00:28:54.630 --> 00:29:01.905
I give it a little impulse,
so we are now at t equals 0.
00:29:05.100 --> 00:29:07.200
We give it an impulse.
00:29:07.200 --> 00:29:10.665
At that instant, the ruler
is still hanging vertically.
00:29:15.350 --> 00:29:18.800
Let me, just so that when
you look on the board,
00:29:18.800 --> 00:29:21.720
you may be confused
in which plane I am.
00:29:21.720 --> 00:29:23.460
This is the vertical plane.
00:29:23.460 --> 00:29:24.600
So this is up.
00:29:29.900 --> 00:29:31.740
So I give it an impulse.
00:29:31.740 --> 00:29:38.480
So at that instant
of time, this ruler
00:29:38.480 --> 00:29:48.260
will have an angular velocity
which I will call theta dot.
00:29:53.740 --> 00:29:56.550
This is at time equals 0.
00:29:56.550 --> 00:29:58.910
And it has some
number as a result,
00:29:58.910 --> 00:30:01.670
depends how big an
impulse I gave it.
00:30:01.670 --> 00:30:04.460
And so that you remember
what I'm talking about,
00:30:04.460 --> 00:30:07.380
I like to give this,
instead of using a symbol,
00:30:07.380 --> 00:30:16.320
I'll call this angular
velocity at t equals 0.
00:30:16.320 --> 00:30:20.190
So this is some number, so
many radians per second.
00:30:20.190 --> 00:30:22.360
That's at t equals 0.
00:30:22.360 --> 00:30:25.260
And I'm now going to
follow this method again.
00:30:25.260 --> 00:30:29.430
I want to know what will
be the motion of this.
00:30:29.430 --> 00:30:31.240
What's going to
happen to this ruler.
00:30:31.240 --> 00:30:36.170
Is it going to start spinning
around this, like this forever?
00:30:36.170 --> 00:30:38.260
What will happen?
00:30:38.260 --> 00:30:43.740
So I will try to translate
this problem into mathematics.
00:30:47.750 --> 00:30:50.530
Because of the
mechanical constraint,
00:30:50.530 --> 00:30:55.520
at some instant of time,
the ruler may be doing this.
00:30:55.520 --> 00:30:57.950
Let's call this the time t.
00:30:57.950 --> 00:31:01.605
This is time t.
00:31:01.605 --> 00:31:04.310
And time t is like this.
00:31:04.310 --> 00:31:07.860
And I've got to define
some coordinate system.
00:31:07.860 --> 00:31:11.930
So I'll take this angle
from the vertical.
00:31:11.930 --> 00:31:14.621
And I call that theta at time t.
00:31:17.870 --> 00:31:19.540
That's why I call
this theta dot.
00:31:19.540 --> 00:31:22.220
This is the rate
of change of that.
00:31:22.220 --> 00:31:27.260
So at some instance of time,
it will be at this position,
00:31:27.260 --> 00:31:28.430
all right.
00:31:28.430 --> 00:31:31.320
At that instant of
time, it'll have
00:31:31.320 --> 00:31:35.110
a velocity in this direction.
00:31:35.110 --> 00:31:37.820
And we'll have an acceleration
in that direction.
00:31:37.820 --> 00:31:40.000
So for example, the
acceleration will
00:31:40.000 --> 00:31:43.400
be theta double dot at time t.
00:31:49.370 --> 00:31:54.680
And just so that at this stage,
I will still to remind you
00:31:54.680 --> 00:31:58.398
that's alpha, alpha time t.
00:32:01.266 --> 00:32:04.090
Because often alpha is
used as the acceleration.
00:32:04.090 --> 00:32:06.120
So at the moment,
I just want you
00:32:06.120 --> 00:32:09.220
so you can easy for you to
see what I'm talking about.
00:32:09.220 --> 00:32:14.880
So at some instance of time,
that is the physical situation.
00:32:14.880 --> 00:32:21.600
I would like to now convert
this into mathematics.
00:32:21.600 --> 00:32:23.350
Follow the same
procedure as before.
00:32:28.980 --> 00:32:33.920
I need to write the
equation of motion for this.
00:32:33.920 --> 00:32:39.180
And I need to write down
the initial conditions.
00:32:39.180 --> 00:32:42.820
So how do I do with that?
00:32:42.820 --> 00:32:47.250
So now I start off by
the free body diagram.
00:32:47.250 --> 00:32:50.420
Here is the pivot.
00:32:50.420 --> 00:32:51.930
That's the route.
00:32:55.800 --> 00:32:58.492
This angle is theta t.
00:33:05.120 --> 00:33:08.480
There will be a force acting.
00:33:08.480 --> 00:33:11.680
We're now dealing with
rigid body motion.
00:33:11.680 --> 00:33:18.460
So today we did Newtonian
mechanics for masses and forces
00:33:18.460 --> 00:33:20.610
through a single
point mass and forces.
00:33:20.610 --> 00:33:26.320
Now we are doing a Newtonian
dynamics for rigid body motion.
00:33:26.320 --> 00:33:30.675
You know that if a rigid body
is in the gravitational field,
00:33:30.675 --> 00:33:36.230
the gravity acts force fg.
00:33:36.230 --> 00:33:38.830
We can analyze it, as
if there was a force fg
00:33:38.830 --> 00:33:44.520
g acting through the center
of mass here of the body.
00:33:44.520 --> 00:33:46.705
So this length now is l over 2.
00:33:52.250 --> 00:33:55.060
So there will be
a force fg acting.
00:33:55.060 --> 00:34:03.620
And as a result, there will
be torques about this point.
00:34:03.620 --> 00:34:05.180
Now let me say the following.
00:34:05.180 --> 00:34:13.710
We are dealing here with motion,
rotations, in a single plane.
00:34:13.710 --> 00:34:17.040
And so we are dealing
about rotations,
00:34:17.040 --> 00:34:22.650
about an axis
through this point p.
00:34:22.650 --> 00:34:26.560
We're not dealing with
three dimensional rotations,
00:34:26.560 --> 00:34:30.710
but simple situation
where all the motion
00:34:30.710 --> 00:34:37.940
is about a single axis, which is
perpendicular to this point, p.
00:34:40.730 --> 00:34:43.580
There will be a
torque about p because
00:34:43.580 --> 00:34:46.550
of the gravitational force.
00:34:46.550 --> 00:34:50.389
And as a result, there's going
to be the acceleration, which
00:34:50.389 --> 00:34:55.636
as we've said over there,
is theta double dot of t.
00:35:00.180 --> 00:35:10.865
Now, we know that torques gives
rise to angular acceleration.
00:35:16.560 --> 00:35:23.710
Let me define that we will take
clockwise motion, clockwise
00:35:23.710 --> 00:35:34.860
motion, clockwise
rotations to be positive.
00:35:42.350 --> 00:35:48.600
So any rotation, this angle,
for example, I am sorry.
00:35:48.600 --> 00:35:50.160
I meant anti-clockwise.
00:35:53.630 --> 00:35:55.830
Anti-clockwise is positive.
00:35:55.830 --> 00:35:56.760
Look at this.
00:35:56.760 --> 00:36:01.715
If this rotates like
that to this angle, this
00:36:01.715 --> 00:36:03.900
I take to be a
positive number, it's
00:36:03.900 --> 00:36:06.200
an anti-clockwise rotation.
00:36:06.200 --> 00:36:09.710
Similarly, if this acceleration
is a positive number,
00:36:09.710 --> 00:36:13.140
it's accelerating
in this direction.
00:36:13.140 --> 00:36:18.430
Since we are dealing with
rotations about a single axis,
00:36:18.430 --> 00:36:22.800
we don't have to go to
the vector formulation.
00:36:22.800 --> 00:36:27.500
We can consider it
just the magnitude.
00:36:27.500 --> 00:36:36.380
And we know that
the acceleration
00:36:36.380 --> 00:36:43.660
is equal to the torque divided
by the moment of inertia.
00:36:43.660 --> 00:36:45.845
Or you may have seen
it the other way.
00:36:45.845 --> 00:36:48.580
Torque Equals I alpha.
00:36:56.620 --> 00:36:58.090
I prefer it this way.
00:36:58.090 --> 00:36:59.640
For me, it's more logical.
00:36:59.640 --> 00:37:03.040
The angular acceleration is
a consequence of the torque.
00:37:03.040 --> 00:37:05.940
So I write it like that.
00:37:05.940 --> 00:37:12.460
So this is the dynamic
equation, which tells you
00:37:12.460 --> 00:37:17.530
how the motion of this
mass changes with time.
00:37:17.530 --> 00:37:18.860
All right.
00:37:18.860 --> 00:37:26.460
So alpha is theta
double dot of t.
00:37:26.460 --> 00:37:28.240
OK.
00:37:28.240 --> 00:37:32.120
What is the torque at
that instant of time?
00:37:32.120 --> 00:37:42.860
Well, you know general torque
is r cross f, all right.
00:37:42.860 --> 00:37:44.730
That's true in three dimension.
00:37:44.730 --> 00:37:46.590
So it will apply here.
00:37:46.590 --> 00:37:58.960
So the torque is going to be
this force times this distance.
00:37:58.960 --> 00:38:00.000
OK.
00:38:00.000 --> 00:38:01.870
So it's going to
be-- let's write it.
00:38:01.870 --> 00:38:15.030
The force is mg right,
times l over 2 sine theta,
00:38:15.030 --> 00:38:16.160
theta of time t.
00:38:16.160 --> 00:38:18.800
OK.
00:38:18.800 --> 00:38:27.300
That's the torque about this
axis p on this rod, all right.
00:38:27.300 --> 00:38:35.010
And it's divided by I, where
I is the moment of inertia
00:38:35.010 --> 00:38:39.100
of this rod about
an axis through p
00:38:39.100 --> 00:38:41.670
perpendicular to the board.
00:38:41.670 --> 00:38:43.020
OK.
00:38:43.020 --> 00:38:45.990
Now we need to
calculate the moment,
00:38:45.990 --> 00:38:49.650
in order to continue further,
we need to calculate I.
00:38:49.650 --> 00:38:52.500
Since we know this
mass of the rod.
00:38:52.500 --> 00:38:54.560
And we know it's a uniform rod.
00:38:54.560 --> 00:38:59.280
And we know it's length
l, we can calculate it.
00:38:59.280 --> 00:39:00.490
You know how to do it.
00:39:00.490 --> 00:39:04.440
If you don't, you can look it
up in the book on mechanics,
00:39:04.440 --> 00:39:05.150
all right.
00:39:05.150 --> 00:39:07.950
Or just look up the
moments of inertia.
00:39:07.950 --> 00:39:16.060
And you will find that
the moment of inertia,
00:39:16.060 --> 00:39:22.600
you will find that the moment
of inertia I for a rod like that
00:39:22.600 --> 00:39:29.700
is 1/3 the mass times
the length squared.
00:39:29.700 --> 00:39:30.930
OK.
00:39:30.930 --> 00:39:32.330
So now I have to continue.
00:39:32.330 --> 00:39:35.190
But I've run out of board space.
00:39:35.190 --> 00:39:38.860
So I'm going to erase
the board at the far end.
00:39:38.860 --> 00:39:42.730
And we'll continue from there.
00:39:42.730 --> 00:39:45.440
So I erased the board.
00:39:45.440 --> 00:39:48.930
And then so that you don't have
to look backwards and forwards,
00:39:48.930 --> 00:39:53.470
I've started rewriting it and
I realized that I actually
00:39:53.470 --> 00:39:55.047
missed the negative sign.
00:39:55.047 --> 00:39:56.380
So I'm going to correct it here.
00:39:56.380 --> 00:39:58.710
So that's why it's
completely written out.
00:39:58.710 --> 00:40:01.650
So let me just remind you.
00:40:01.650 --> 00:40:05.040
The situation we
have is this rod,
00:40:05.040 --> 00:40:07.930
which at time t, we
define this angle
00:40:07.930 --> 00:40:10.410
to be theta t, the
rotation of the rod.
00:40:10.410 --> 00:40:13.230
It has an acceleration,
theta double dot t.
00:40:13.230 --> 00:40:18.090
And we are considering rotations
about an axis perpendicular
00:40:18.090 --> 00:40:20.540
to the board through
this point here.
00:40:20.540 --> 00:40:21.450
OK.
00:40:21.450 --> 00:40:24.540
We know, that was the
last thing we did,
00:40:24.540 --> 00:40:29.780
that the acceleration is
given by the torque divided
00:40:29.780 --> 00:40:31.430
by the moment of inertia.
00:40:31.430 --> 00:40:33.330
All right.
00:40:33.330 --> 00:40:38.450
The torque is mg, l over
2 sine theta, I derived it
00:40:38.450 --> 00:40:40.790
for you before, divided by I.
00:40:40.790 --> 00:40:44.430
But what I neglected
to put a negative sign.
00:40:44.430 --> 00:40:46.760
And that you could do
in your head, right.
00:40:46.760 --> 00:40:53.040
Consider we've taken all
the rotations to be positive
00:40:53.040 --> 00:40:55.050
if they're anti-clockwise.
00:40:55.050 --> 00:40:58.210
So this angle is a
positive rotation.
00:40:58.210 --> 00:41:02.420
This would be, this direction
would be a positive rotation.
00:41:02.420 --> 00:41:05.980
But the torque if
you look at this,
00:41:05.980 --> 00:41:08.270
there is a force
acting down on this.
00:41:08.270 --> 00:41:11.970
So about this point,
it's trying to rotate
00:41:11.970 --> 00:41:14.380
this in the clockwise direction.
00:41:14.380 --> 00:41:15.580
And so it's minus.
00:41:15.580 --> 00:41:19.020
And I didn't-- it would have
naturally come out if I did
00:41:19.020 --> 00:41:25.912
the full vector calculation,
the torque is R times F.
00:41:25.912 --> 00:41:28.120
It would have come out, the
sign would have come out.
00:41:28.120 --> 00:41:30.450
So that's where this
minus sign comes in.
00:41:30.450 --> 00:41:33.480
OK, so this is where we got
on the board over there.
00:41:33.480 --> 00:41:35.720
And now let's continue.
00:41:35.720 --> 00:41:38.810
We can replace I from here.
00:41:38.810 --> 00:41:45.530
And we get that
theta double dot of t
00:41:45.530 --> 00:42:00.380
is equal to minus, all right,
3 halves, 3 l, 3 halves, not l.
00:42:00.380 --> 00:42:01.700
It's divided by 2.
00:42:01.700 --> 00:42:02.800
l is at the bottom.
00:42:02.800 --> 00:42:04.060
Sorry.
00:42:04.060 --> 00:42:13.936
3 halves g over l times
sine theta, theta of t.
00:42:13.936 --> 00:42:14.436
OK.
00:42:17.540 --> 00:42:20.770
I'm sorry.
00:42:20.770 --> 00:42:25.770
Sine theta of t.
00:42:25.770 --> 00:42:33.160
OK, as before, to simplify it,
I will write omega, I'll define.
00:42:33.160 --> 00:42:47.880
Let's define omega 0 squared to
be equal to 3 halves g over l.
00:42:47.880 --> 00:42:53.760
With this definition,
we get that theta double
00:42:53.760 --> 00:43:05.750
dot t is equal to minus omega
0 squared sine theta of t.
00:43:05.750 --> 00:43:06.770
OK.
00:43:06.770 --> 00:43:13.680
So this is our equation of
motion for this problem.
00:43:13.680 --> 00:43:16.350
That's the equation of motion.
00:43:16.350 --> 00:43:20.140
And these are the
boundary conditions.
00:43:20.140 --> 00:43:25.670
So these three equations
are a translation
00:43:25.670 --> 00:43:28.770
of this problem in the
language of mathematics.
00:43:33.270 --> 00:43:37.910
If we now want to
predict what will happen
00:43:37.910 --> 00:43:41.250
to this rod at
some other time, we
00:43:41.250 --> 00:43:43.380
have to solve these equations.
00:43:43.380 --> 00:43:47.710
And admit now, I have a problem.
00:43:47.710 --> 00:43:52.140
If you remember, when we
did it for the spring,
00:43:52.140 --> 00:43:58.310
the equation of motion was one
where I guessed the answer.
00:43:58.310 --> 00:44:04.800
I don't know what the
answer is of this.
00:44:04.800 --> 00:44:07.520
If you go into
books, you will find
00:44:07.520 --> 00:44:11.590
that this is not one of
the differential equations
00:44:11.590 --> 00:44:13.800
which you can
analytically solve.
00:44:13.800 --> 00:44:16.650
It's, in fact, a second
order differential equations
00:44:16.650 --> 00:44:19.700
with transcendental
functions in it.
00:44:19.700 --> 00:44:23.870
So this is not something
we know the answer to.
00:44:23.870 --> 00:44:28.530
So the only thing if I want to
now predict what will happen,
00:44:28.530 --> 00:44:31.660
I have to numerically
solve this.
00:44:31.660 --> 00:44:35.160
And then I can-- I have
enough information.
00:44:35.160 --> 00:44:39.390
I can numerically solve this
equation with these boundary
00:44:39.390 --> 00:44:41.980
conditions and predict
what will happen.
00:44:41.980 --> 00:44:44.840
That's not very instructive
for the purpose of course
00:44:44.840 --> 00:44:46.120
at the moment.
00:44:46.120 --> 00:44:48.300
So let me do something else.
00:44:48.300 --> 00:44:50.930
OK, let me modify the problem.
00:44:50.930 --> 00:44:55.280
Rather than take the
problem we took, let me say,
00:44:55.280 --> 00:44:59.420
how about if I took
this rod and gave it
00:44:59.420 --> 00:45:01.720
only a very tiny impulse.
00:45:01.720 --> 00:45:04.460
So this angle is small.
00:45:04.460 --> 00:45:11.200
Let me make the angles
sufficiently small, such
00:45:11.200 --> 00:45:18.850
that sine theta of t
is always approximately
00:45:18.850 --> 00:45:24.200
equal to theta of t.
00:45:24.200 --> 00:45:27.690
Depends how well you
want to approximate this.
00:45:27.690 --> 00:45:31.580
But typically, if you use
your calculator or computer,
00:45:31.580 --> 00:45:36.000
up to about 10 degrees, that
approximation is pretty good.
00:45:36.000 --> 00:45:40.490
So I will now change my problem.
00:45:40.490 --> 00:45:44.400
And I said OK, let's see whether
we can predict analytically
00:45:44.400 --> 00:45:47.990
the motion of the rod where
I give the impulse, which
00:45:47.990 --> 00:45:52.280
is sufficiently small, that
this angle is always small.
00:45:52.280 --> 00:45:58.830
Under those conditions, note
that my equation of motion
00:45:58.830 --> 00:46:09.430
becomes theta double dot of t
is equal to minus omega squared
00:46:09.430 --> 00:46:12.970
times theta at t.
00:46:12.970 --> 00:46:17.875
Because sine theta t is always
approximately equal to theta t
00:46:17.875 --> 00:46:20.070
if I take the
angle small enough.
00:46:26.800 --> 00:46:30.180
And eureka, I can
solve that one.
00:46:30.180 --> 00:46:32.705
Because that's exactly the
same equation we solved before.
00:46:43.880 --> 00:46:46.910
OK, so we get the
solution to that equation,
00:46:46.910 --> 00:46:58.320
is theta of t is some constant
cosine omega 0 t plus phi.
00:47:02.530 --> 00:47:07.580
As before, A and phi are
some arbitrary constants.
00:47:07.580 --> 00:47:11.360
And clearly, if it worked over
there for that same equation,
00:47:11.360 --> 00:47:12.040
it works here.
00:47:12.040 --> 00:47:16.760
The only difference here is we
have theta of t instead y of t.
00:47:16.760 --> 00:47:19.270
That's just different
symbols, but the solution
00:47:19.270 --> 00:47:21.960
is exactly the same.
00:47:21.960 --> 00:47:25.070
So we know that's the
solution of this equation.
00:47:25.070 --> 00:47:27.159
We know the boundary conditions.
00:47:27.159 --> 00:47:28.950
Therefore, we can
predict what will happen.
00:47:28.950 --> 00:47:30.700
Let's continue and do that.
00:47:30.700 --> 00:47:38.630
So from here, you get theta
dot of t is equal minus omega 0
00:47:38.630 --> 00:47:46.890
A sine omega 0 t plus phase.
00:47:46.890 --> 00:47:48.500
OK.
00:47:48.500 --> 00:47:52.610
And we have to put in
the boundary conditions.
00:47:52.610 --> 00:47:54.280
OK.
00:47:54.280 --> 00:48:06.790
Now at t equals 0, OK, we get
that this is at t equals 0.
00:48:06.790 --> 00:48:13.000
So at t equals 0,
we get theta of t 0
00:48:13.000 --> 00:48:22.130
is equal to A cosine phi.
00:48:22.130 --> 00:48:22.860
OK.
00:48:22.860 --> 00:48:30.810
Therefore, phi is pi over 2.
00:48:30.810 --> 00:48:32.460
That's a possible value of phi.
00:48:35.160 --> 00:48:43.640
Now that gives me that
if I is pi over 2 here,
00:48:43.640 --> 00:48:55.050
we get to that theta dot of t,
which is equal to the angular,
00:48:55.050 --> 00:49:04.400
angular velocity at t
equal to 0, all right,
00:49:04.400 --> 00:49:23.500
will be equal to minus omega 0
A sine omega ) t plus pi over 2.
00:49:26.990 --> 00:49:31.810
OK, from which I
can get that A is
00:49:31.810 --> 00:49:33.860
angular velocity over t plus 0.
00:49:33.860 --> 00:49:39.400
And so my final solution
is that theta of t
00:49:39.400 --> 00:49:52.210
is equal to angular
velocity at t equal 0
00:49:52.210 --> 00:50:00.230
divided omega 0
sine, sine omega O t.
00:50:03.080 --> 00:50:03.580
OK.
00:50:03.580 --> 00:50:06.300
And I want to make sure I'm not
making a sign mistake again.
00:50:06.300 --> 00:50:07.510
I'm not.
00:50:07.510 --> 00:50:08.380
All right.
00:50:08.380 --> 00:50:12.990
And so, and omega 0
we know, and so the
00:50:12.990 --> 00:50:16.640
in terms of knowing
quantities, the answer
00:50:16.640 --> 00:50:25.340
is angular velocity
of t equals 0 over.
00:50:25.340 --> 00:50:30.210
And omega 0, we have
found defined to be that,
00:50:30.210 --> 00:50:39.590
so the square root
of 3 g over 2 l times
00:50:39.590 --> 00:50:50.090
sine square root
3g over 2l times t.
00:50:50.090 --> 00:50:54.150
Now this is theta t.
00:50:54.150 --> 00:50:56.780
So we have completely
solved the problem.
00:50:56.780 --> 00:50:58.550
And we have
predicted the motion.
00:50:58.550 --> 00:51:04.040
So as before,
following this process
00:51:04.040 --> 00:51:07.300
of taking the
physical situation,
00:51:07.300 --> 00:51:10.010
describing it in
terms of mathematics,
00:51:10.010 --> 00:51:12.600
solving the
mathematical equations,
00:51:12.600 --> 00:51:14.380
including all the
information we have
00:51:14.380 --> 00:51:16.470
about the problem,
the boundary--
00:51:16.470 --> 00:51:19.590
initial conditions or
boundary conditions,
00:51:19.590 --> 00:51:24.500
we can predict what will
happen to this angle
00:51:24.500 --> 00:51:28.670
as a function of time, and
also the kind of motion
00:51:28.670 --> 00:51:31.150
this is an accelerating motion.
00:51:31.150 --> 00:51:35.140
I can also predict, as before,
that the period of this
00:51:35.140 --> 00:51:43.570
will be 2 pi, 2 pi square
root of 2l over 3g, et cetera.
00:51:46.720 --> 00:51:49.350
OK.
00:51:49.350 --> 00:52:01.750
Now, one of the things you'll
notice, that in some ways,
00:52:01.750 --> 00:52:03.650
it seems I'm repeating myself.
00:52:06.760 --> 00:52:09.520
We took completely
different situations,
00:52:09.520 --> 00:52:13.490
and yet the result, the
equations of motion,
00:52:13.490 --> 00:52:17.520
and the results, have
good very similar form.
00:52:17.520 --> 00:52:22.470
Now this is part of the beauty
of the scientific method.
00:52:22.470 --> 00:52:29.080
Because it turns
out that very many
00:52:29.080 --> 00:52:32.940
different physical
situations can
00:52:32.940 --> 00:52:37.180
be described by the same
mathematical equations.
00:52:37.180 --> 00:52:43.470
So once you've solved
the problem for one
00:52:43.470 --> 00:52:48.620
physical situation, you have
automatically have solved it
00:52:48.620 --> 00:52:53.340
for an almost infinite number
of other situations which
00:52:53.340 --> 00:52:57.510
are described by this
same mathematics.
00:52:57.510 --> 00:53:01.790
Finally, let me do just more
as a question of practice,
00:53:01.790 --> 00:53:06.050
one more problem of this
kind that apparently
00:53:06.050 --> 00:53:07.630
seems to be
completely different.
00:53:07.630 --> 00:53:10.485
I'll take a problem from
electricity and magnetism.
00:53:14.960 --> 00:53:17.170
Let me consider the
following situation.
00:53:25.120 --> 00:53:29.620
So now we're going to
a different problem.
00:53:29.620 --> 00:53:34.540
The physical situation is
suppose I have two plates, two
00:53:34.540 --> 00:53:38.830
metal plates, and I
connect them with a wire.
00:53:38.830 --> 00:53:46.660
Schematically, it
consists of a capacitor C
00:53:46.660 --> 00:53:49.507
connected to an inductor.
00:53:53.620 --> 00:53:56.320
This is a schematic
representation
00:53:56.320 --> 00:54:00.090
of two parallel plates
connected, short
00:54:00.090 --> 00:54:03.860
circuited by a wire.
00:54:03.860 --> 00:54:10.150
I will assume for simplicity
here that these wires have
00:54:10.150 --> 00:54:14.830
no resistance,
superconducting, all right.
00:54:14.830 --> 00:54:21.370
Any loop like has
an inductant L.
00:54:21.370 --> 00:54:24.870
And the capacity
between these is C.
00:54:24.870 --> 00:54:27.600
So this is an L C circuit.
00:54:27.600 --> 00:54:32.110
And I'm going to assume that
at time equal 0, so this is now
00:54:32.110 --> 00:54:41.520
time equals 0, I have a
charge here, minus Q0 plus Q0
00:54:41.520 --> 00:54:44.580
here, OK.
00:54:44.580 --> 00:54:47.970
And let's assume that at time
there's even a current flowing,
00:54:47.970 --> 00:54:50.500
so I is 0 here.
00:54:54.420 --> 00:54:59.780
So this is a system which is
disturbed from equilibrium.
00:54:59.780 --> 00:55:02.907
And what will happen
is a function of time.
00:55:05.590 --> 00:55:10.990
I will do the same and
almost boring you to tears,
00:55:10.990 --> 00:55:13.230
I'm going to, you'll
see I'm essentially
00:55:13.230 --> 00:55:15.790
doing the same problem again.
00:55:15.790 --> 00:55:23.520
I will now consider this circuit
at some arbitrary time t,
00:55:23.520 --> 00:55:29.420
derive the equation of motion
for the charges in the current,
00:55:29.420 --> 00:55:32.640
therefore translate
this physical situation,
00:55:32.640 --> 00:55:34.430
or describe this
physical situation
00:55:34.430 --> 00:55:37.560
in terms of
mathematics, deriving
00:55:37.560 --> 00:55:40.960
mathematical
equations, solve them,
00:55:40.960 --> 00:55:42.680
and predict what will happen.
00:55:42.680 --> 00:55:46.370
So I just follow what I
just did a second ago.
00:55:46.370 --> 00:55:55.590
So at some instant of
time, that same circuit L
00:55:55.590 --> 00:56:00.530
will have some current
I of t to the charge
00:56:00.530 --> 00:56:06.840
minus Q of t plus
Q of t, all right.
00:56:06.840 --> 00:56:07.960
This at time t.
00:56:12.830 --> 00:56:19.170
So from this, I can derive
the equation of motion.
00:56:22.500 --> 00:56:32.340
Let me remind you
about Faraday's law.
00:56:32.340 --> 00:56:41.290
You know that if you have
current coil in the loop,
00:56:41.290 --> 00:56:48.780
it produces magnetic
flux in that loop.
00:56:48.780 --> 00:56:54.000
The changing flux gives rise
to an EMF around that loop.
00:56:54.000 --> 00:56:57.130
To be specific, Faraday's
law I can write.
00:56:57.130 --> 00:57:06.050
If I take this circuit of
the wire, the integral of E
00:57:06.050 --> 00:57:16.310
dot dl around a closed loop,
that is equal to minus du phi.
00:57:16.310 --> 00:57:17.540
5.
00:57:17.540 --> 00:57:18.980
Now watch out.
00:57:18.980 --> 00:57:21.830
The Greek alphabet has a
limited number of letters
00:57:21.830 --> 00:57:27.080
so you'll find one constant
is reusing the letters.
00:57:27.080 --> 00:57:29.020
But at the moment
not to confuse you,
00:57:29.020 --> 00:57:32.730
I'm going to put
here magnetic flux,
00:57:32.730 --> 00:57:36.425
total magnetic, total magnetic.
00:57:41.050 --> 00:57:44.970
So phi is the
total magnetic flux
00:57:44.970 --> 00:57:49.416
linking this circuit,
all right, dt.
00:58:00.750 --> 00:58:05.285
So Faraday's law tells us
that the integral of ED
00:58:05.285 --> 00:58:09.480
all around this loop
will be equal two
00:58:09.480 --> 00:58:12.320
minus the rate of
change magnetic flux.
00:58:12.320 --> 00:58:17.100
This is the dynamic
equation which
00:58:17.100 --> 00:58:18.930
tells you how this behaves.
00:58:18.930 --> 00:58:24.130
It is the analogous to
Newton's law f equals ma
00:58:24.130 --> 00:58:30.910
in the case of our
mass, or 2 torque is I
00:58:30.910 --> 00:58:33.290
alpha in the case of
rotations, et cetera.
00:58:33.290 --> 00:58:35.570
This is the non
dynamic equations.
00:58:35.570 --> 00:58:38.220
So let me calculate this.
00:58:38.220 --> 00:58:41.940
And now I'm going through
the-- around this.
00:58:41.940 --> 00:58:46.800
And you find since this wire
I'm assuming is superconducting,
00:58:46.800 --> 00:58:49.600
there can be no electric
field inside it.
00:58:49.600 --> 00:58:52.880
So the contribution
to this line integral
00:58:52.880 --> 00:58:56.040
is 0 when I go through the wire.
00:58:56.040 --> 00:58:59.780
So the only place where this
line integral is non-zero
00:58:59.780 --> 00:59:02.390
is between the
plates, all right.
00:59:02.390 --> 00:59:04.910
And that is simply the
potential difference
00:59:04.910 --> 00:59:09.560
between those,
which is q over c.
00:59:09.560 --> 00:59:10.170
OK.
00:59:10.170 --> 00:59:17.700
Q at time t over c, where
c is the capacitance is.
00:59:17.700 --> 00:59:22.360
That is the integral of
EDL around that loop.
00:59:22.360 --> 00:59:39.090
And that's going to be equal
to minus D dI at time t dt.
00:59:39.090 --> 00:59:39.960
All right.
00:59:39.960 --> 00:59:44.950
Because the magnetic
flux, this is
00:59:44.950 --> 00:59:50.510
by definition of the
inductance or first inductance
00:59:50.510 --> 00:59:55.230
is that the total flux
linking the circuit when
00:59:55.230 --> 00:59:59.660
the current flowing in it
is I of I, the total flux
00:59:59.660 --> 01:00:04.870
is L times I. Here, I have the
rate of change of that flux,
01:00:04.870 --> 01:00:07.140
so it's equal to this.
01:00:07.140 --> 01:00:08.310
OK.
01:00:08.310 --> 01:00:13.330
Now we know by
charge conservation,
01:00:13.330 --> 01:00:15.538
that the current I of t.
01:00:19.130 --> 01:00:20.040
What is the current?
01:00:20.040 --> 01:00:22.130
It's the charge is
flowing per second
01:00:22.130 --> 01:00:28.010
will be equal to the number of
charges per second that arrive
01:00:28.010 --> 01:00:39.160
at this plate here or the part
from there is equal to dQ dt.
01:00:39.160 --> 01:00:39.800
OK.
01:00:39.800 --> 01:00:42.480
Or in other words, Q dot.
01:00:46.290 --> 01:00:47.415
OK, let me continue.
01:00:57.120 --> 01:01:04.390
So from these two equations,
right, the dIdt therefore,
01:01:04.390 --> 01:01:07.970
this is the second thing,
so I end up from there
01:01:07.970 --> 01:01:13.770
that Q double dot,
second derivative of t,
01:01:13.770 --> 01:01:21.895
is equal to minus right
1 over LC times Q of t.
01:01:24.750 --> 01:01:25.900
Eureka.
01:01:25.900 --> 01:01:28.510
We have once again
the same equation.
01:01:28.510 --> 01:01:32.410
This I can define as
before, omega O squared.
01:01:32.410 --> 01:01:37.900
If I define that as
one over LC, all right,
01:01:37.900 --> 01:01:45.730
then what I have here is Q of
t double dot is equal to minus
01:01:45.730 --> 01:01:48.720
omega ) squared Q of t.
01:01:52.060 --> 01:01:55.830
Again, we have come
to the same equation.
01:01:55.830 --> 01:01:59.440
This is the same
equation of motion
01:01:59.440 --> 01:02:02.570
as we came in the
other two situations.
01:02:02.570 --> 01:02:05.360
So the answer will be the same.
01:02:05.360 --> 01:02:07.930
The variables will
be different here.
01:02:07.930 --> 01:02:12.770
It'll be the charge that
will be changing with time,
01:02:12.770 --> 01:02:15.250
while there in the one
case was the angle.
01:02:15.250 --> 01:02:21.000
In the other case was
the position of the mass.
01:02:21.000 --> 01:02:24.250
OK, and the solution
to this problem,
01:02:24.250 --> 01:02:29.100
I can now write
immediately, is Q of t
01:02:29.100 --> 01:02:36.040
is A cosine omega 0 t plus fe.
01:02:36.040 --> 01:02:40.110
Note that this is the fe is
nothing to do with that phi.
01:02:40.110 --> 01:02:40.880
OK.
01:02:40.880 --> 01:02:50.710
And Q dot of t, which
by the way is I of t,
01:02:50.710 --> 01:03:02.330
is equal to minus omega o
A sine omega o t plus phi.
01:03:02.330 --> 01:03:03.550
OK.
01:03:03.550 --> 01:03:09.220
Now as before, what
actually happens
01:03:09.220 --> 01:03:14.180
depends on the
initial conditions.
01:03:14.180 --> 01:03:19.640
And we look at that
picture on the top board.
01:03:19.640 --> 01:03:23.980
We know that initially Q is Q 0.
01:03:23.980 --> 01:03:28.570
And we know that
initially Q dot t is I0.
01:03:28.570 --> 01:03:30.800
To save time, I'll
just immediately write.
01:03:30.800 --> 01:03:36.040
You can do that in your head.
01:03:36.040 --> 01:03:41.150
And if you the
write that out, you
01:03:41.150 --> 01:03:48.120
find that if you've used
those two conditions,
01:03:48.120 --> 01:03:59.660
you find that time phi this
time is equal to minus IO over
01:03:59.660 --> 01:04:08.660
Q0 omega 0 and A is equal
to Q0 over cosine phi.
01:04:11.470 --> 01:04:15.090
I saved time without just
solving algebraic equations.
01:04:15.090 --> 01:04:17.680
Take these two equations.
01:04:17.680 --> 01:04:22.590
Consider t equals 0, the
values of those quantities,
01:04:22.590 --> 01:04:25.810
and just solve for the two
unknowns and you get this.
01:04:25.810 --> 01:04:30.790
And so once again, we have
predicted what will happen.
01:04:30.790 --> 01:04:35.420
And what I would like
to just at this stage
01:04:35.420 --> 01:04:40.360
emphasize that although
we have taken three
01:04:40.360 --> 01:04:45.430
different physical
situations, in each case,
01:04:45.430 --> 01:04:52.590
we took the system, displaced
it from equilibrium, let go
01:04:52.590 --> 01:04:56.110
and we wanted to see
what will happen.
01:04:56.110 --> 01:04:59.770
In all the cases,
it turned out that
01:04:59.770 --> 01:05:03.450
the mathematical description,
the mathematical equations
01:05:03.450 --> 01:05:05.730
are identical in form.
01:05:05.730 --> 01:05:08.890
And so they gave,
not surprisingly,
01:05:08.890 --> 01:05:12.040
the same kind of motion.
01:05:12.040 --> 01:05:15.550
This motion that we
see in all those cases,
01:05:15.550 --> 01:05:17.980
we called simple
harmonic motion.
01:05:21.260 --> 01:05:24.230
It has the characteristic that
if you displace the system
01:05:24.230 --> 01:05:29.830
from equilibrium, it oscillates
with harmonic motion,
01:05:29.830 --> 01:05:34.180
meaning it oscillates
as sine or a cosine
01:05:34.180 --> 01:05:37.110
of different phases, et cetera.
01:05:37.110 --> 01:05:40.700
If you tell me any
one of these systems
01:05:40.700 --> 01:05:43.060
where it was at any
instant of time,
01:05:43.060 --> 01:05:45.880
I can predict it
forever in the future.
01:05:48.910 --> 01:05:51.545
Now finally, the
last few minutes.
01:05:55.810 --> 01:06:00.020
Some of you may have noticed
that in each of these--
01:06:00.020 --> 01:06:01.940
or I told you at
the beginning that I
01:06:01.940 --> 01:06:08.530
can take a physical
situation and describe it
01:06:08.530 --> 01:06:13.730
in terms of mathematics, and
thereby predict the future.
01:06:13.730 --> 01:06:18.630
But in each case, I in
some ways almost cheated.
01:06:18.630 --> 01:06:21.880
I said let's consider
an ideal spring.
01:06:21.880 --> 01:06:25.680
We'll assume it has no
mass, that it exactly
01:06:25.680 --> 01:06:27.880
obeys Hooke's law.
01:06:27.880 --> 01:06:31.810
Or when it came to
that rod, I assumed
01:06:31.810 --> 01:06:36.800
that it's only displaced
by small amounts,
01:06:36.800 --> 01:06:44.930
so that's sine theta equals--
I can approximate with theta.
01:06:44.930 --> 01:06:48.650
In the case of the
electrical circuit,
01:06:48.650 --> 01:06:51.820
it maybe not so
obvious what I assumed,
01:06:51.820 --> 01:06:56.610
but I certainly made assumptions
about that the wire is
01:06:56.610 --> 01:06:58.930
perfectly conducting, et cetera.
01:06:58.930 --> 01:07:02.750
And I didn't discuss
in detail what
01:07:02.750 --> 01:07:04.800
happens in between the
plates or the capacitor
01:07:04.800 --> 01:07:06.204
where the fields are, et cetera.
01:07:06.204 --> 01:07:07.370
One is doing approximations.
01:07:10.190 --> 01:07:14.510
In reality, if you look
at any physical situation
01:07:14.510 --> 01:07:19.150
in the world, it's always
incredibly complicated.
01:07:23.340 --> 01:07:27.990
It's never that you have an
idealized situation like this.
01:07:27.990 --> 01:07:32.570
So to what extent does
what we have just done
01:07:32.570 --> 01:07:35.660
correspond-- is
it useful at all.
01:07:35.660 --> 01:07:40.930
And the way I'll answer
it is by another example.
01:07:40.930 --> 01:07:45.040
This is the last thing I'll do
on the top of a simple harmonic
01:07:45.040 --> 01:07:47.520
motion, last problem.
01:07:47.520 --> 01:07:51.290
Suppose I'm looking
out of the window.
01:07:51.290 --> 01:08:00.110
And I see there is a tree and a
branch and a bird lands on it.
01:08:00.110 --> 01:08:04.740
Do I understand
what will happen?
01:08:04.740 --> 01:08:07.576
It's clearly
extremely complicated.
01:08:07.576 --> 01:08:10.800
The mechanics of the
branch is complicated.
01:08:10.800 --> 01:08:13.240
There is air friction.
01:08:13.240 --> 01:08:15.460
Nothing is simple about it.
01:08:15.460 --> 01:08:19.685
And yet you and I can
predict what will happen.
01:08:19.685 --> 01:08:21.850
You know what will happen.
01:08:21.850 --> 01:08:27.520
As the bird lands, it'll
start oscillating and finally
01:08:27.520 --> 01:08:34.830
come to rest, very much
like harmonic motion.
01:08:34.830 --> 01:08:41.120
I claim I can use the word,
I understand what's going on.
01:08:41.120 --> 01:08:44.149
And the reason why I claim
that is the following.
01:08:44.149 --> 01:08:49.439
That to understand
something, all
01:08:49.439 --> 01:08:54.160
I would like to understand
the general features of what's
01:08:54.160 --> 01:08:54.680
going on.
01:08:54.680 --> 01:08:58.350
I don't need to know what
every atom in the branch
01:08:58.350 --> 01:09:02.020
is going on in the
process of trying
01:09:02.020 --> 01:09:04.410
to understand what
the bird is doing.
01:09:04.410 --> 01:09:06.990
If I want to understand
what atoms are doing,
01:09:06.990 --> 01:09:09.979
that's a different story.
01:09:09.979 --> 01:09:20.330
And so one of the important
abilities we have to develop
01:09:20.330 --> 01:09:26.890
is to be able to, when
you see some situation,
01:09:26.890 --> 01:09:33.120
model it in terms of the
most important aspects
01:09:33.120 --> 01:09:35.779
of the situation.
01:09:35.779 --> 01:09:38.000
And let me be concrete.
01:09:38.000 --> 01:09:45.080
In this case, I can say look,
I can model this approximately
01:09:45.080 --> 01:09:49.460
as the branch I'll
treat as a spring,
01:09:49.460 --> 01:09:51.500
of some spring constant k.
01:09:51.500 --> 01:09:54.430
The bird I'm going
to treat as a mass m.
01:09:54.430 --> 01:09:57.940
And I'm going to
consider this situation
01:09:57.940 --> 01:10:07.210
to be modeled by a mass being
placed on a spring and let go.
01:10:07.210 --> 01:10:09.580
Now is that going
to be exactly this?
01:10:09.580 --> 01:10:11.330
No.
01:10:11.330 --> 01:10:15.200
But from the point of
view of understanding
01:10:15.200 --> 01:10:18.470
the general features
of this, it will
01:10:18.470 --> 01:10:23.310
be a reasonable approximation.
01:10:23.310 --> 01:10:28.080
Now how can I check, this
is the scientific method,
01:10:28.080 --> 01:10:33.350
that this is a good
approximation is the following.
01:10:33.350 --> 01:10:35.950
Make a prediction.
01:10:35.950 --> 01:10:41.160
Suppose when I see
the bird landing,
01:10:41.160 --> 01:10:52.760
it makes five oscillations, five
oscillations in ten seconds.
01:10:55.380 --> 01:10:57.650
OK.
01:10:57.650 --> 01:11:03.190
I can predict approximately
once the oscillations have
01:11:03.190 --> 01:11:10.030
died out how much the bird
has compressed, distorted,
01:11:10.030 --> 01:11:11.230
this branch.
01:11:11.230 --> 01:11:15.230
In other words, from
the moment it landed,
01:11:15.230 --> 01:11:19.550
what distance will the
branch, its position,
01:11:19.550 --> 01:11:23.682
be lowered when
it comes to rest.
01:11:26.990 --> 01:11:28.590
So let's model it.
01:11:28.590 --> 01:11:32.830
I'll model it first
of all, as I did here,
01:11:32.830 --> 01:11:38.260
as a mass and a spring
problem using-- this is now
01:11:38.260 --> 01:11:40.930
the same problem we did at
the beginning that is still
01:11:40.930 --> 01:11:42.630
on the board here.
01:11:42.630 --> 01:11:46.890
I can calculate that for
this idealized situation,
01:11:46.890 --> 01:11:50.240
the period will be equal
to 2 pi square root m
01:11:50.240 --> 01:11:54.110
over k, for this
idealized model.
01:11:54.110 --> 01:11:57.760
In reality, there's friction.
01:11:57.760 --> 01:12:02.850
So this oscillation
will be dumped out.
01:12:02.850 --> 01:12:06.580
And you would have learned
from Professor Walter Lewin
01:12:06.580 --> 01:12:12.120
that if you have a
damped oscillator,
01:12:12.120 --> 01:12:20.420
the frequency of oscillations
does not depend significantly
01:12:20.420 --> 01:12:24.830
on how much damping it is,
provided it is weak damping.
01:12:24.830 --> 01:12:30.200
So I would make the
assumption that the period
01:12:30.200 --> 01:12:33.200
will be given by that.
01:12:33.200 --> 01:12:38.750
I also know that at the time
when the motion has been damped
01:12:38.750 --> 01:12:44.620
out and the bird has come
to rest, at that instant,
01:12:44.620 --> 01:12:46.810
there's no net
force on the bird.
01:12:46.810 --> 01:12:49.290
And so the force
of gravity on it
01:12:49.290 --> 01:12:54.560
will be equal to the restoring
force due to the spring.
01:12:54.560 --> 01:12:57.400
So mg is equal to kl.
01:12:57.400 --> 01:13:03.170
From this, I get that
m over k is l over g.
01:13:03.170 --> 01:13:04.680
All right.
01:13:04.680 --> 01:13:07.860
But we know what the period is.
01:13:07.860 --> 01:13:10.580
We said five oscillations
in 10 seconds.
01:13:10.580 --> 01:13:14.250
So the period is two seconds.
01:13:14.250 --> 01:13:17.830
So two seconds will
equal 2 pi divided
01:13:17.830 --> 01:13:25.820
by but this, which is l over
g, the square root of l over g.
01:13:25.820 --> 01:13:29.540
Square this and calculate
the one unknown l
01:13:29.540 --> 01:13:33.210
and you get one meter.
01:13:33.210 --> 01:13:36.050
So my prediction
is that this bird
01:13:36.050 --> 01:13:44.514
will, after it's settled down,
roughly be lower by one meter.
01:13:44.514 --> 01:13:46.430
It's certainly not going
to be one millimeter.
01:13:46.430 --> 01:13:48.410
It's not going to
be one centimeter.
01:13:48.410 --> 01:13:50.740
It's not going be 10 meters.
01:13:50.740 --> 01:13:53.620
And if you go and
you measure it,
01:13:53.620 --> 01:13:56.920
you find this is
approximately correct.
01:13:56.920 --> 01:14:01.860
The fact that I can
predict it is for me
01:14:01.860 --> 01:14:05.550
the same as saying I
understand what's going on.
01:14:05.550 --> 01:14:08.270
I realize it's not exact.
01:14:08.270 --> 01:14:13.070
But with the approximations
that I've made,
01:14:13.070 --> 01:14:17.030
I get an answer which
is consistent with what
01:14:17.030 --> 01:14:19.250
is observed.
01:14:19.250 --> 01:14:25.690
Today I have tried to tell you
what my role in this course
01:14:25.690 --> 01:14:32.210
is, what I'm trying
to help you learn.
01:14:32.210 --> 01:14:35.910
I intentionally
went very slowly.
01:14:35.910 --> 01:14:39.270
I used the word gory detail.
01:14:39.270 --> 01:14:42.200
I tried, in particularly
in the first problem,
01:14:42.200 --> 01:14:46.140
not to miss any steps.
01:14:46.140 --> 01:14:51.870
And what we covered
today is the phenomenon
01:14:51.870 --> 01:14:54.330
of simple harmonic motion.
01:14:54.330 --> 01:15:01.870
It occurs whenever you
have any system which
01:15:01.870 --> 01:15:07.460
is displaced from equilibrium
where the restoring force is
01:15:07.460 --> 01:15:11.060
proportional to
the displacement.
01:15:11.060 --> 01:15:13.460
And it illustrated
that you could
01:15:13.460 --> 01:15:18.010
have very, very different
physical situations which,
01:15:18.010 --> 01:15:20.990
when translated
into mathematics,
01:15:20.990 --> 01:15:24.840
give essentially
the same problem.
01:15:24.840 --> 01:15:32.760
So it's a beautiful example
of the scientific method where
01:15:32.760 --> 01:15:40.310
we utilize this same--
well, once we've learned it
01:15:40.310 --> 01:15:46.780
for one system, we can apply
the results to another system.
01:15:46.780 --> 01:15:50.160
So as I said, today I did
simple harmonic motion.
01:15:50.160 --> 01:15:53.730
Next time, we would be
considering problems
01:15:53.730 --> 01:15:56.130
to do with simple
harmonic motion,
01:15:56.130 --> 01:15:59.560
but which includes
friction, damping.
01:15:59.560 --> 01:16:04.700
We'll then go on to talk about
harmonic oscillators which
01:16:04.700 --> 01:16:05.630
are driven.
01:16:05.630 --> 01:16:07.690
So we have driven
harmonic motion.
01:16:07.690 --> 01:16:09.460
And gradually in
the course, we'll
01:16:09.460 --> 01:16:13.900
go to more and more decrease
of freedom, waves, et cetera.