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PROFESSOR: Hi.
00:00:58.860 --> 00:01:04.610
Today we're going to
conclude our study of vectors
00:01:04.610 --> 00:01:07.650
as applied to
motion in the plane.
00:01:07.650 --> 00:01:09.880
Now recall that for
the last two units,
00:01:09.880 --> 00:01:12.120
we were discussing
polar coordinates.
00:01:12.120 --> 00:01:14.360
So today what we
would like to do
00:01:14.360 --> 00:01:20.400
is investigate what velocity
and acceleration vectors would
00:01:20.400 --> 00:01:25.360
have looked like, had we elected
to pick representative vectors
00:01:25.360 --> 00:01:27.750
in terms of polar coordinates.
00:01:27.750 --> 00:01:30.420
Now what do we mean by
representative vectors?
00:01:30.420 --> 00:01:33.910
We mean, of course,
analogs of i and j,
00:01:33.910 --> 00:01:37.920
just like T and N in
tangential-normal components
00:01:37.920 --> 00:01:41.400
were parallels to i and j.
00:01:41.400 --> 00:01:44.140
Let's take a look and
see what that means here.
00:01:44.140 --> 00:01:47.750
We call today's lecture
"Vectors in Polar Coordinates."
00:01:47.750 --> 00:01:50.250
The idea is that
we're given a curve
00:01:50.250 --> 00:01:53.430
C, which we have for
some reason or other
00:01:53.430 --> 00:01:57.330
elected to express in
terms of polar coordinates.
00:01:57.330 --> 00:02:02.440
The polar equation of the
curve C is r equals f of theta.
00:02:02.440 --> 00:02:05.250
A typical point on
the curve C would
00:02:05.250 --> 00:02:10.360
be called r comma theta say,
where theta was the angle made
00:02:10.360 --> 00:02:13.680
by the horizontal and
the radius vector,
00:02:13.680 --> 00:02:17.830
and r was the distance from
the origin to the point.
00:02:17.830 --> 00:02:20.120
Now what we're saying
is, that in terms
00:02:20.120 --> 00:02:23.630
of polar coordinates,
a very natural vector
00:02:23.630 --> 00:02:25.910
to pick-- especially
if we think later
00:02:25.910 --> 00:02:27.570
in terms of the
simple force fields
00:02:27.570 --> 00:02:30.160
that we've talked
about earlier, the idea
00:02:30.160 --> 00:02:34.370
that there may be a force
that has its line of action
00:02:34.370 --> 00:02:36.810
from the origin to the
point on the curve--
00:02:36.810 --> 00:02:40.060
a very natural vector
to choose is the vector
00:02:40.060 --> 00:02:42.920
that we elect to call u sub r.
00:02:42.920 --> 00:02:45.900
And what that vector
is, apparently,
00:02:45.900 --> 00:02:51.480
is the vector 1 unit
long having the direction
00:02:51.480 --> 00:02:56.640
of the radius vector R. So
here is u sub r over here.
00:02:56.640 --> 00:03:00.070
If we think of u sub r
as playing the role of i,
00:03:00.070 --> 00:03:02.550
then the vector which
plays the role of j
00:03:02.550 --> 00:03:07.700
should be a positive
90-degree rotation of u sub r.
00:03:07.700 --> 00:03:12.330
And we elect to call that vector
u sub theta, using the theta
00:03:12.330 --> 00:03:14.360
here more to indicate
the fact that we're
00:03:14.360 --> 00:03:17.550
using polar coordinates
than to indicate anything
00:03:17.550 --> 00:03:19.550
about the angle theta itself.
00:03:19.550 --> 00:03:22.600
In other words, notice that
u sub theta, by definition,
00:03:22.600 --> 00:03:26.350
is just a positive 90-degree
rotation of u sub r, where
00:03:26.350 --> 00:03:31.834
u sub r is a unit vector in the
direction of the radius vector.
00:03:31.834 --> 00:03:33.500
Now if we want to see
this in terms of i
00:03:33.500 --> 00:03:35.630
and j components,
what we're saying
00:03:35.630 --> 00:03:41.210
is that u sub r is a unit vector
whose i component is what?
00:03:41.210 --> 00:03:43.390
Since this angle
here is also theta,
00:03:43.390 --> 00:03:47.470
its i component is cosine
theta, and its j component
00:03:47.470 --> 00:03:48.700
is sine theta
00:03:48.700 --> 00:03:52.550
So u sub r is cosine
theta i plus sine theta j.
00:03:52.550 --> 00:03:55.980
u sub theta-- and I'm going
to do a little twist here
00:03:55.980 --> 00:03:58.470
that I didn't do in
the T and N components,
00:03:58.470 --> 00:04:00.379
just to show you
another approach.
00:04:00.379 --> 00:04:02.420
Rather than to start with
derivatives or anything
00:04:02.420 --> 00:04:05.750
like this, notice that what
we know about u sub theta
00:04:05.750 --> 00:04:08.260
is that it's
obtained from u sub r
00:04:08.260 --> 00:04:11.530
by a positive 90-degree
rotation of theta.
00:04:11.530 --> 00:04:16.910
So that means if I replace theta
in the expression for u sub r,
00:04:16.910 --> 00:04:21.200
by theta plus 90 degrees, that
should give me u sub theta.
00:04:21.200 --> 00:04:24.700
If I now remember my
trigonometric identities,
00:04:24.700 --> 00:04:28.210
that tells me that u sub
theta is minus sine theta
00:04:28.210 --> 00:04:31.220
i plus cosine theta j.
00:04:31.220 --> 00:04:34.580
By the way, if we now
look at this expression
00:04:34.580 --> 00:04:37.600
and compare it with the
expression for u sub r,
00:04:37.600 --> 00:04:42.750
we see at once that u sub theta
is the derivative of u sub
00:04:42.750 --> 00:04:44.710
r with respect to theta.
00:04:44.710 --> 00:04:47.970
And notice, as we said
before, that part of this
00:04:47.970 --> 00:04:50.550
should have been
known to us by now.
00:04:50.550 --> 00:04:53.930
Namely, since u sub
r varies with theta
00:04:53.930 --> 00:04:56.490
but it has a constant
magnitude, we
00:04:56.490 --> 00:05:00.030
know that the derivative of
u sub r with respect to theta
00:05:00.030 --> 00:05:03.980
has to be perpendicular
to u sub r.
00:05:03.980 --> 00:05:06.400
In other words, we knew
that u sub theta had
00:05:06.400 --> 00:05:09.275
to be either plus or minus
the derivative of u sub
00:05:09.275 --> 00:05:11.040
r with respect to
theta, but now we
00:05:11.040 --> 00:05:13.230
have a direct way
of showing this.
00:05:13.230 --> 00:05:16.090
And by the way, going
one step further,
00:05:16.090 --> 00:05:20.350
if we now differentiate u sub
theta with respect to theta,
00:05:20.350 --> 00:05:21.360
we get what?
00:05:21.360 --> 00:05:25.330
Minus cosine theta i
minus sine theta j,
00:05:25.330 --> 00:05:28.450
which is just minus u sub r.
00:05:28.450 --> 00:05:30.430
In other words, if
you differentiate
00:05:30.430 --> 00:05:33.010
u sub r with respect
to theta once,
00:05:33.010 --> 00:05:36.790
you get u sub theta,
just as we should.
00:05:36.790 --> 00:05:41.300
If you differentiate a second
time, you get minus u sub r.
00:05:41.300 --> 00:05:44.750
And therefore, it appears
that the operation
00:05:44.750 --> 00:05:48.160
of differentiating with
respect to theta rotates
00:05:48.160 --> 00:05:51.300
u sub r by 90 degrees.
00:05:51.300 --> 00:05:52.970
By the way, I should
mention that I
00:05:52.970 --> 00:05:54.880
could have made all
of these remarks
00:05:54.880 --> 00:05:58.537
when we were studying tangential
and normal components.
00:05:58.537 --> 00:06:00.370
In other words-- I just
wrote this out here,
00:06:00.370 --> 00:06:04.230
but this was true with T
and N. But the point was,
00:06:04.230 --> 00:06:07.200
we never had to
differentiate N with respect
00:06:07.200 --> 00:06:10.410
to t to find the
acceleration vector a.
00:06:10.410 --> 00:06:13.780
In other words, recall
that the key step in using
00:06:13.780 --> 00:06:16.760
tangential and normal
vectors-- and I'll mention this
00:06:16.760 --> 00:06:18.370
in a little bit
more detail later--
00:06:18.370 --> 00:06:21.430
was that the velocity
vector was simply
00:06:21.430 --> 00:06:26.790
ds/dt times the unit tangent
vector T. The coefficient of N
00:06:26.790 --> 00:06:28.670
was 0.
00:06:28.670 --> 00:06:32.780
In polar coordinates,
notice that v in general
00:06:32.780 --> 00:06:37.390
will have both a u sub r
and a u sub theta component.
00:06:37.390 --> 00:06:41.260
Therefore, to compute a, I have
to differentiate v with respect
00:06:41.260 --> 00:06:42.210
to t.
00:06:42.210 --> 00:06:44.030
That means, among
other things, I'm
00:06:44.030 --> 00:06:46.350
going to have to take
the derivative of u sub
00:06:46.350 --> 00:06:48.090
theta with respect to t.
00:06:48.090 --> 00:06:50.330
By the chain rule,
that's going to be
00:06:50.330 --> 00:06:52.150
the same as taking the
derivative of u sub
00:06:52.150 --> 00:06:55.820
theta with respect to
theta times d theta / dt.
00:06:55.820 --> 00:06:57.880
But the important point
is, is that someplace
00:06:57.880 --> 00:07:00.040
along the line in
studying kinematics
00:07:00.040 --> 00:07:02.150
and polar coordinates,
I am going
00:07:02.150 --> 00:07:06.070
to have to differentiate u sub
theta with respect to theta.
00:07:06.070 --> 00:07:07.950
And just to show you
again very, very quickly
00:07:07.950 --> 00:07:10.090
what I mean by
this, all I'm saying
00:07:10.090 --> 00:07:14.230
is we already know in kinematics
that the velocity vector is
00:07:14.230 --> 00:07:16.430
always tangential to the curve.
00:07:16.430 --> 00:07:18.040
Notice in this
particular diagram,
00:07:18.040 --> 00:07:22.090
for example, that if you look
at u sub r and u sub theta,
00:07:22.090 --> 00:07:24.760
if you think of a vector
whose direction is
00:07:24.760 --> 00:07:27.570
tangent to the curve at
this particular point,
00:07:27.570 --> 00:07:29.580
that vector will
have, in general,
00:07:29.580 --> 00:07:33.200
both a u sub theta and
a u sub r component.
00:07:33.200 --> 00:07:35.380
In fact, in this
particular diagram,
00:07:35.380 --> 00:07:39.595
I shouldn't say "in general,"
it will have u sub r and a u sub
00:07:39.595 --> 00:07:40.960
theta component.
00:07:40.960 --> 00:07:41.460
OK.
00:07:41.460 --> 00:07:47.210
So far so good, but now I want
to make one little caution,
00:07:47.210 --> 00:07:50.540
a caution which is not at all
self-evident, at least to me,
00:07:50.540 --> 00:07:52.850
and which gave me
great difficulty myself
00:07:52.850 --> 00:07:54.560
when I was a student.
00:07:54.560 --> 00:07:59.130
And that is, my feeling was
that u sub r was simply the unit
00:07:59.130 --> 00:08:00.785
vector in the direction of r.
00:08:00.785 --> 00:08:02.630
In fact, I said that
earlier, that u sub
00:08:02.630 --> 00:08:06.410
r was the unit vector in the
direction of the radius vector
00:08:06.410 --> 00:08:09.990
R. And this is one of the
reasons why even though
00:08:09.990 --> 00:08:12.240
Professor Thomas in the
textbook doesn't make such
00:08:12.240 --> 00:08:18.120
an issue over this, why I am
such a bug on using the phrase
00:08:18.120 --> 00:08:20.610
"sense" as well as direction.
00:08:20.610 --> 00:08:25.210
And that is, my claim
is that the unit vector
00:08:25.210 --> 00:08:30.140
u sub r need not be the
radius vector R divided
00:08:30.140 --> 00:08:33.980
by the magnitude of R. It'll
have the same direction,
00:08:33.980 --> 00:08:36.220
but watch what
happens with sense.
00:08:36.220 --> 00:08:38.409
Instead of talking about
this thing abstractly,
00:08:38.409 --> 00:08:40.460
let me give you a
concrete example.
00:08:40.460 --> 00:08:43.309
Let's take the curve
which in polar coordinates
00:08:43.309 --> 00:08:46.400
has the equation r
equals cosine theta.
00:08:46.400 --> 00:08:47.540
OK?
00:08:47.540 --> 00:08:50.490
As you recall, this would be
this particular circle here.
00:08:50.490 --> 00:08:53.490
Now let's take theta
to be 120 degrees.
00:08:53.490 --> 00:08:57.350
If I use our definition for u
sub r, which is cosine theta
00:08:57.350 --> 00:09:00.700
i plus sine theta
j, and replace theta
00:09:00.700 --> 00:09:04.650
by 120 degrees, what I
get is, is that u sub r
00:09:04.650 --> 00:09:09.180
is cos 120 degrees i
plus sine 120 degrees j.
00:09:09.180 --> 00:09:12.080
Remember that the cosine
of 120 is minus 1/2
00:09:12.080 --> 00:09:16.610
and the sine of 120 is plus
1/2 the square root of 3.
00:09:16.610 --> 00:09:21.800
u sub r turns out to
be minus 1/2 i plus 1/2
00:09:21.800 --> 00:09:23.890
the square root of 3 j.
00:09:23.890 --> 00:09:28.490
On the other hand, my claim is
that when theta is 120 degrees,
00:09:28.490 --> 00:09:30.900
what point are we at the curve?
00:09:30.900 --> 00:09:32.840
You see, if I take
theta to be 120
00:09:32.840 --> 00:09:39.190
degrees-- notice that
when theta is 120 degrees,
00:09:39.190 --> 00:09:40.790
r is negative 1/2.
00:09:43.550 --> 00:09:47.330
And that therefore I'm
at the point P_0 here.
00:09:47.330 --> 00:09:51.440
Recall that by definition,
R, the radius vector,
00:09:51.440 --> 00:09:53.980
is measured from the
origin to the point.
00:09:53.980 --> 00:09:57.650
In other words, according
to our previous definition,
00:09:57.650 --> 00:10:00.090
it's this vector,
which would be called
00:10:00.090 --> 00:10:03.660
R. But our definition
says that it's
00:10:03.660 --> 00:10:07.320
this vector, which is u sub r.
00:10:07.320 --> 00:10:09.890
And in fact, if you
just check the figures
00:10:09.890 --> 00:10:11.940
that we've obtained
over here, notice
00:10:11.940 --> 00:10:16.230
that u sub r has its i
component equal to minus 1/2,
00:10:16.230 --> 00:10:20.710
its j component being plus
1/2 the square root of 3.
00:10:20.710 --> 00:10:22.980
Therefore x is negative.
00:10:22.980 --> 00:10:24.480
y is positive.
00:10:24.480 --> 00:10:26.960
But if x is negative
and y is positive,
00:10:26.960 --> 00:10:30.790
you're in the second quadrant,
not the fourth quadrant.
00:10:30.790 --> 00:10:31.290
You see?
00:10:31.290 --> 00:10:34.970
In other words, u sub r is
almost the radius vector.
00:10:34.970 --> 00:10:39.380
In fact, it would have been,
if in polar coordinates,
00:10:39.380 --> 00:10:41.360
little r happened
to be positive.
00:10:41.360 --> 00:10:44.760
In fact, let me summarize
that in a different way.
00:10:44.760 --> 00:10:47.980
Let's assume that we have a
curve C whose polar equation
00:10:47.980 --> 00:10:49.970
is r equals f of theta.
00:10:49.970 --> 00:10:51.540
Then the idea is this.
00:10:51.540 --> 00:10:55.280
If f of theta happens to
be at least as big as 0,
00:10:55.280 --> 00:10:59.090
then u sub r is equal
to the radius vector
00:10:59.090 --> 00:11:01.400
R divided by its magnitude.
00:11:01.400 --> 00:11:04.740
In other words, u sub r will
be in the same direction
00:11:04.740 --> 00:11:06.800
as the radius vector.
00:11:06.800 --> 00:11:09.860
And by the way, recall this is
just another way of saying r.
00:11:09.860 --> 00:11:12.260
What we're saying is
again, if we had never
00:11:12.260 --> 00:11:15.210
let little r in polar
coordinates be negative,
00:11:15.210 --> 00:11:17.400
no problem would have occurred.
00:11:17.400 --> 00:11:19.180
But we do let little
r be negative.
00:11:19.180 --> 00:11:21.740
So we have to be a mite careful.
00:11:21.740 --> 00:11:23.770
The careful part comes in where?
00:11:23.770 --> 00:11:25.770
If r happens to be negative.
00:11:25.770 --> 00:11:29.800
In which case, u sub r
has the opposite sense
00:11:29.800 --> 00:11:32.320
of the radius vector
capital R, which
00:11:32.320 --> 00:11:34.050
we saw in the previous example.
00:11:34.050 --> 00:11:36.260
In other words, in
this case, u sub r
00:11:36.260 --> 00:11:40.100
is the negative of
the radius vector
00:11:40.100 --> 00:11:42.220
divided by its magnitude.
00:11:42.220 --> 00:11:43.400
In what case is that?
00:11:43.400 --> 00:11:45.550
If r happens to be negative.
00:11:45.550 --> 00:11:50.150
The important point to notice,
however, that in either case,
00:11:50.150 --> 00:11:57.220
the radius vector R is equal to
the polar coordinate r times u
00:11:57.220 --> 00:11:58.000
sub r.
00:11:58.000 --> 00:12:00.590
In other words, if r
happens to be positive,
00:12:00.590 --> 00:12:02.810
these two vectors
have the same sense.
00:12:02.810 --> 00:12:06.030
If r happens to be
negative, these two vectors
00:12:06.030 --> 00:12:07.600
have the opposite sense.
00:12:07.600 --> 00:12:10.120
In other words, in either
case, this expression
00:12:10.120 --> 00:12:11.350
is always correct.
00:12:11.350 --> 00:12:13.320
But the important
thing to notice
00:12:13.320 --> 00:12:18.180
is that the sense of u sub r
is determined by theta, not
00:12:18.180 --> 00:12:21.150
by r, not by f of theta.
00:12:21.150 --> 00:12:22.530
OK?
00:12:22.530 --> 00:12:26.120
At any rate, once we have this
particular recipe established,
00:12:26.120 --> 00:12:30.230
we can now go ahead and
study motion in the plane.
00:12:30.230 --> 00:12:32.910
Namely, notice that
our radius vector
00:12:32.910 --> 00:12:38.020
R is now given by the polar
coordinate r times u sub r.
00:12:38.020 --> 00:12:40.200
Or I guess I should
say here that I'm
00:12:40.200 --> 00:12:42.720
assuming that the
equation of motion
00:12:42.720 --> 00:12:46.160
is given by r is some
function of theta.
00:12:46.160 --> 00:12:48.960
That's the r that
I'm using in here.
00:12:48.960 --> 00:12:51.760
At any rate, what is
the velocity vector?
00:12:51.760 --> 00:12:55.470
By definition, it's just the
derivative of the radius vector
00:12:55.470 --> 00:12:57.120
with respect to time.
00:12:57.120 --> 00:13:01.060
That's just d/dt
of r times u sub r.
00:13:01.060 --> 00:13:04.210
Now keep in mind,
that r and u sub r
00:13:04.210 --> 00:13:06.470
are both functions
of time-- namely,
00:13:06.470 --> 00:13:09.240
the distance of the
particle from the origin
00:13:09.240 --> 00:13:11.760
as well as the direction
of the line of action
00:13:11.760 --> 00:13:14.450
that joins the
particle to the origin
00:13:14.450 --> 00:13:16.550
will, in general,
depend on time.
00:13:16.550 --> 00:13:19.280
Consequently, I must use
the product rule here.
00:13:19.280 --> 00:13:22.220
I already know that I can use
the product rule for vector
00:13:22.220 --> 00:13:25.860
and/or scalar functions and
any combination thereof.
00:13:25.860 --> 00:13:29.170
So I just differentiate this
thing with respect to time.
00:13:29.170 --> 00:13:30.060
I get what?
00:13:30.060 --> 00:13:32.730
This is dr/dt-- in other
words, the derivative
00:13:32.730 --> 00:13:36.680
of the first times the
second, which is u sub r, OK?
00:13:36.680 --> 00:13:39.680
Plus the first
times the derivative
00:13:39.680 --> 00:13:41.980
of the second, which
is the derivative of u
00:13:41.980 --> 00:13:44.540
sub r with respect to time.
00:13:44.540 --> 00:13:46.570
Now keep in mind,
again, there is
00:13:46.570 --> 00:13:48.550
nothing wrong with this recipe.
00:13:48.550 --> 00:13:52.280
But what I would like is to have
the velocity expressed in u sub
00:13:52.280 --> 00:13:54.490
r and u sub theta components.
00:13:54.490 --> 00:13:58.380
So far, I have it expressed
in terms of u sub r component
00:13:58.380 --> 00:14:01.280
and a d(u sub r)/dt component.
00:14:01.280 --> 00:14:04.970
But now, here again is why the
chain rule is so important.
00:14:04.970 --> 00:14:10.380
Keep in mind that I already know
that if this expression here
00:14:10.380 --> 00:14:13.840
had been the derivative
of u sub r with respect
00:14:13.840 --> 00:14:18.120
to theta instead of
with respect to t,
00:14:18.120 --> 00:14:21.120
what would this
expression have been?
00:14:21.120 --> 00:14:22.990
It would have been u sub theta.
00:14:22.990 --> 00:14:26.060
We solved that earlier
in the lecture.
00:14:26.060 --> 00:14:27.640
Well, here's what we do.
00:14:27.640 --> 00:14:29.380
We say, OK, these
aren't the same.
00:14:29.380 --> 00:14:32.040
But let's cross this out.
00:14:32.040 --> 00:14:33.594
Let's use the chain rule.
00:14:33.594 --> 00:14:35.135
And by the chain
rule, the derivative
00:14:35.135 --> 00:14:38.970
of u sub r with respect to t
is the same as the derivative
00:14:38.970 --> 00:14:44.310
of u sub r with respect to
theta-- times d theta / dt.
00:14:44.310 --> 00:14:47.640
And rewriting this so
that it becomes legible,
00:14:47.640 --> 00:14:53.560
we have that the velocity vector
is dr/dt times u sub r plus r d
00:14:53.560 --> 00:14:56.690
theta / dt times u sub theta.
00:14:56.690 --> 00:15:02.610
And again notice that as
far as u sub r and u sub
00:15:02.610 --> 00:15:06.080
theta are concerned, even if
I did not notice my subtlety--
00:15:06.080 --> 00:15:09.120
and by the way, I'm going to
leave this for the exercises--
00:15:09.120 --> 00:15:13.560
but even if I didn't notice the
subtlety that u sub r need not
00:15:13.560 --> 00:15:16.960
have the same sense as the
radius vector r-- notice
00:15:16.960 --> 00:15:20.060
that if I did not try to
draw this thing to scale,
00:15:20.060 --> 00:15:23.770
I can still get the-- I
shouldn't have said the scale--
00:15:23.770 --> 00:15:27.500
but if I didn't try to
graph the answer here given
00:15:27.500 --> 00:15:31.090
r as a function of theta and
theta is a function of t,
00:15:31.090 --> 00:15:34.400
notice that dr/dt
and r d theta / dt
00:15:34.400 --> 00:15:38.570
are well-defined arithmetically
with no possible chance
00:15:38.570 --> 00:15:40.440
of making a geometrical mistake.
00:15:40.440 --> 00:15:42.680
The place that you can
make the biggest mistake
00:15:42.680 --> 00:15:45.210
is if you automatically
think that u sub r must
00:15:45.210 --> 00:15:48.410
have the same sense as
capital R. But as I say,
00:15:48.410 --> 00:15:50.380
we'll leave any
additional discussion
00:15:50.380 --> 00:15:52.720
of that for the exercises.
00:15:52.720 --> 00:15:55.100
I should also point
out that when I first
00:15:55.100 --> 00:15:57.570
learned this recipe
myself, it turned out
00:15:57.570 --> 00:16:00.580
that we were ahead of--
the physics class was ahead
00:16:00.580 --> 00:16:01.780
of the math class.
00:16:01.780 --> 00:16:03.890
And we learned this thing
in the physics class
00:16:03.890 --> 00:16:05.460
almost intuitively.
00:16:05.460 --> 00:16:07.900
In other words, as
a geometric aside,
00:16:07.900 --> 00:16:10.550
notice that if I'm
given the curve
00:16:10.550 --> 00:16:13.410
and say s indicates the
direction of increasing arc
00:16:13.410 --> 00:16:20.800
length here, what I could do is
think of a little differential
00:16:20.800 --> 00:16:21.850
region here.
00:16:21.850 --> 00:16:24.480
Namely, here's my
radius vector r,
00:16:24.480 --> 00:16:28.550
and here's my velocity vector
in the direction of u sub r.
00:16:28.550 --> 00:16:31.080
Then I take a little
increment of angle d theta,
00:16:31.080 --> 00:16:35.210
and I now think of v sub
theta, which is at right angles
00:16:35.210 --> 00:16:40.560
to v sub r, as being tangent
to the circle I would have
00:16:40.560 --> 00:16:44.290
obtained if I had imagined
that this particular point--
00:16:44.290 --> 00:16:48.140
the particle was being viewed
with respect to the circle
00:16:48.140 --> 00:16:49.800
rather than to the curve itself.
00:16:49.800 --> 00:16:52.260
To make a long story
short, what I'm driving at
00:16:52.260 --> 00:16:54.810
is that physically,
it's very easy
00:16:54.810 --> 00:16:58.305
to justify that the
magnitude of the u sub r
00:16:58.305 --> 00:17:02.220
component of the velocity
is the magnitude of dr/dt--
00:17:02.220 --> 00:17:05.440
how fast the radius vector
is changing instantaneously.
00:17:05.440 --> 00:17:09.510
On the other hand, notice that
for the u sub theta component,
00:17:09.510 --> 00:17:14.460
this arc length is given in
differential form by r d theta.
00:17:14.460 --> 00:17:17.940
If I divide the arc length
by the time, which is dt,
00:17:17.940 --> 00:17:23.380
I get r d theta divided by dt,
which leads to r d theta / dt,
00:17:23.380 --> 00:17:27.470
which is the same expression
that we got analytically.
00:17:27.470 --> 00:17:29.620
But the point that I
want to bring out here
00:17:29.620 --> 00:17:33.720
is that our derivation required
no geometrical physical
00:17:33.720 --> 00:17:34.640
insight.
00:17:34.640 --> 00:17:36.560
And the reason that I
want to bring this out
00:17:36.560 --> 00:17:39.950
is I followed this argument
fine in my elementary physics
00:17:39.950 --> 00:17:40.970
course.
00:17:40.970 --> 00:17:43.930
The place I got hung up is
that the instructor then went
00:17:43.930 --> 00:17:47.290
into a fantastic hand-waving
type of demonstration
00:17:47.290 --> 00:17:49.530
and showed us how the
acceleration looked
00:17:49.530 --> 00:17:52.620
in terms of u sub r and
u sub theta components.
00:17:52.620 --> 00:17:54.720
And actually, that
was a blessing for me,
00:17:54.720 --> 00:17:56.730
because it was that
day that I decided
00:17:56.730 --> 00:17:58.960
to become a math major
rather than a physics
00:17:58.960 --> 00:18:02.089
major, which was a blessing both
for me and society, I guess.
00:18:02.089 --> 00:18:03.630
But the thing that
I want to show you
00:18:03.630 --> 00:18:06.810
is that the beauty of
our mathematical approach
00:18:06.810 --> 00:18:10.460
is that we can now obtain
a, the acceleration vector,
00:18:10.460 --> 00:18:13.370
from the velocity vector
without having to know
00:18:13.370 --> 00:18:14.950
any great physical insight.
00:18:14.950 --> 00:18:18.790
In fact, we have to know no
physical insight to do this.
00:18:18.790 --> 00:18:23.520
Namely, by definition, a is
the derivative of the velocity
00:18:23.520 --> 00:18:25.330
vector with respect to time.
00:18:25.330 --> 00:18:27.170
We also have seen
that the velocity
00:18:27.170 --> 00:18:31.050
vector is the expression
that I have here in brackets.
00:18:31.050 --> 00:18:33.910
So I have to differentiate
that with respect to time.
00:18:33.910 --> 00:18:38.190
Notice that this is the sum
of two terms, one of which
00:18:38.190 --> 00:18:41.100
is a product of two factors,
and the other of which
00:18:41.100 --> 00:18:43.170
is a product of three factors.
00:18:43.170 --> 00:18:46.900
And by the way, among other
things to review here,
00:18:46.900 --> 00:18:49.380
this is the first
time in this course
00:18:49.380 --> 00:18:53.040
that we have actually
had to use the product
00:18:53.040 --> 00:18:59.630
rule for a function consisting
of three variable factors.
00:18:59.630 --> 00:19:01.810
Even though we discussed
this in part one,
00:19:01.810 --> 00:19:04.650
here is a case where in
a real-life situation,
00:19:04.650 --> 00:19:09.550
what we need is the derivative
rule for a product of three
00:19:09.550 --> 00:19:10.110
functions.
00:19:10.110 --> 00:19:14.370
At any rate, this is done
in great detail in the text.
00:19:14.370 --> 00:19:16.200
I do it more in the notes.
00:19:16.200 --> 00:19:18.360
So I'm just going to
hit the highlights here.
00:19:18.360 --> 00:19:22.840
The point is I now differentiate
this sum term by term.
00:19:22.840 --> 00:19:25.530
Namely, to differentiate
this, I take
00:19:25.530 --> 00:19:30.210
the derivative of the
first term times the second
00:19:30.210 --> 00:19:35.030
plus the first term times
the derivative of the second.
00:19:35.030 --> 00:19:37.410
Now to differentiate
this term, I
00:19:37.410 --> 00:19:40.940
have to differentiate a
product of three factors.
00:19:40.940 --> 00:19:43.780
And recall-- and by the way,
as I told you in part one,
00:19:43.780 --> 00:19:47.280
whenever I say "recall," that
means if you don't recall,
00:19:47.280 --> 00:19:50.180
it's my polite way of
saying, "look it up."
00:19:50.180 --> 00:19:53.870
But to differentiate a
product of three functions,
00:19:53.870 --> 00:19:56.710
we write the product
down three times,
00:19:56.710 --> 00:19:59.910
and each time differentiate
a different factor.
00:19:59.910 --> 00:20:01.880
For example, the
first time we'll
00:20:01.880 --> 00:20:05.600
differentiate r with respect
to t, which is dr/dt.
00:20:05.600 --> 00:20:07.270
The second time,
we'll differentiate
00:20:07.270 --> 00:20:09.750
d theta / dt with
respect to t, which
00:20:09.750 --> 00:20:12.380
is the second derivative
of theta with respect to t.
00:20:12.380 --> 00:20:14.080
And the third time,
we'll differentiate
00:20:14.080 --> 00:20:17.850
u sub theta with respect to t,
which is the derivative of u
00:20:17.850 --> 00:20:19.540
sub theta with respect to t.
00:20:19.540 --> 00:20:22.260
And summarizing that,
what do I have here?
00:20:22.260 --> 00:20:28.180
I have dr/dt d theta / dt times
u sub theta plus r d^2 theta /
00:20:28.180 --> 00:20:32.900
dt squared times u sub theta
plus r d theta / dt times
00:20:32.900 --> 00:20:36.570
the derivative of u sub
theta with respect to t.
00:20:36.570 --> 00:20:40.010
Now the point is that if I
look at these five terms,
00:20:40.010 --> 00:20:43.610
some of them are in nice form.
00:20:43.610 --> 00:20:45.720
Namely, here's a u sub r term.
00:20:45.720 --> 00:20:47.470
Here's a u sub theta term.
00:20:47.470 --> 00:20:49.170
Here's a u sub theta term.
00:20:49.170 --> 00:20:52.090
But these terms are
sort of mongrelized.
00:20:52.090 --> 00:20:56.960
Namely, what I have to do here
is utilize the chain rule.
00:20:56.960 --> 00:20:58.800
And remember that the
derivative of u sub
00:20:58.800 --> 00:21:02.650
r with respect to theta
would have been u sub theta.
00:21:02.650 --> 00:21:06.310
The derivative of u sub
theta with respect to theta
00:21:06.310 --> 00:21:08.240
would have been minus u sub r.
00:21:08.240 --> 00:21:09.860
So by the chain
rule-- you see what
00:21:09.860 --> 00:21:13.400
I'm going to do is, I'll
replace each of these terms
00:21:13.400 --> 00:21:16.170
by their chain rule expression.
00:21:16.170 --> 00:21:17.864
Then I'll collect terms.
00:21:17.864 --> 00:21:19.280
And the reason I'm
going over this
00:21:19.280 --> 00:21:24.060
fairly rapidly is that it is
a problem of sheer mechanics.
00:21:24.060 --> 00:21:27.560
But the punch line is that
if I now collect my terms,
00:21:27.560 --> 00:21:32.080
the acceleration vector has as
its u sub r component d^2 r /
00:21:32.080 --> 00:21:36.310
dt squared minus r d
theta / dt squared.
00:21:36.310 --> 00:21:41.975
And the u sub theta component is
r d^2 theta / dt squared plus 2
00:21:41.975 --> 00:21:44.329
dr/dt d theta / dt.
00:21:44.329 --> 00:21:45.870
And the beautiful
part, from my point
00:21:45.870 --> 00:21:47.600
of view about all
of this, is if I
00:21:47.600 --> 00:21:50.240
don't understand
any physics at all,
00:21:50.240 --> 00:21:52.470
this particular result is valid.
00:21:52.470 --> 00:21:54.720
It's mathematically
self-contained.
00:21:54.720 --> 00:21:57.420
Now certainly there
is no harm in a man
00:21:57.420 --> 00:22:00.700
who understands physics well
enough to say, look at it.
00:22:00.700 --> 00:22:04.740
This is the acceleration in
the radius direction alone.
00:22:04.740 --> 00:22:08.180
And this is some kind of a
correction factor proportional
00:22:08.180 --> 00:22:11.070
to the square of the
angular velocity, see,
00:22:11.070 --> 00:22:13.990
d theta / dt being angular
velocity and what have you.
00:22:13.990 --> 00:22:16.170
And go through this
particular thing.
00:22:16.170 --> 00:22:17.910
I'm saying fine,
if you can do that.
00:22:17.910 --> 00:22:19.120
But notice the beauty.
00:22:19.120 --> 00:22:22.310
This complicated
expression gives us
00:22:22.310 --> 00:22:25.560
the acceleration vector in
terms of u sub r and u sub
00:22:25.560 --> 00:22:27.570
theta with no hand-waving.
00:22:27.570 --> 00:22:29.950
It's mathematically
self-contained.
00:22:29.950 --> 00:22:32.230
And by the way, keep
in mind that one
00:22:32.230 --> 00:22:35.880
of the reasons that we study
polar coordinate motion
00:22:35.880 --> 00:22:37.780
is the fact that,
in many cases, we
00:22:37.780 --> 00:22:42.200
are going to be dealing
with a central force field.
00:22:42.200 --> 00:22:50.910
And the interesting thing
is that in a central--
00:22:50.910 --> 00:22:56.350
I'll just abbreviate this--
in a central force situation,
00:22:56.350 --> 00:22:57.620
this expression is 0.
00:22:57.620 --> 00:22:59.360
See, central force means what?
00:22:59.360 --> 00:23:01.810
That the force is in
the radial direction.
00:23:01.810 --> 00:23:03.340
That means all of
the acceleration--
00:23:03.340 --> 00:23:05.100
if you're using
Newtonian physics,
00:23:05.100 --> 00:23:08.090
F equals ma-- all
the acceleration is
00:23:08.090 --> 00:23:09.790
in the direction of u sub r.
00:23:09.790 --> 00:23:13.150
Therefore, the component in
the direction of u sub theta
00:23:13.150 --> 00:23:14.300
must be 0.
00:23:14.300 --> 00:23:16.820
So this fairly
complicated expression--
00:23:16.820 --> 00:23:22.310
r d^2 theta / dt squared plus
2 dr/dt d theta / dt equals 0
00:23:22.310 --> 00:23:27.060
becomes the fundamental equation
for central force field motion.
00:23:27.060 --> 00:23:30.400
But again, we'll talk about
that more in the exercises.
00:23:30.400 --> 00:23:32.850
What I wanted to do now
was to make what I think
00:23:32.850 --> 00:23:35.130
is a very important summary.
00:23:35.130 --> 00:23:38.180
And that is that when we're
studying the position vector
00:23:38.180 --> 00:23:44.950
R, and the velocity vector v,
and the acceleration vector a,
00:23:44.950 --> 00:23:48.890
that none of these depend
on the coordinate system.
00:23:48.890 --> 00:23:51.610
It's only their
components that do.
00:23:51.610 --> 00:23:54.150
In other words, at
the expense of having
00:23:54.150 --> 00:23:58.140
a fairly jumbled figure which I
rationalize here-- it is small,
00:23:58.140 --> 00:24:00.290
but I think it is
clear from context.
00:24:00.290 --> 00:24:03.260
What I'm saying is, let's
suppose I have a curve C,
00:24:03.260 --> 00:24:07.410
and some point P_0
on this curve C.
00:24:07.410 --> 00:24:10.530
I can draw in the pair
of orthogonal vectors
00:24:10.530 --> 00:24:12.200
i and j in the plane.
00:24:12.200 --> 00:24:15.490
I can draw in the pair
of orthogonal vectors
00:24:15.490 --> 00:24:17.510
u sub r-- "orthogonal"
means perpendicular,
00:24:17.510 --> 00:24:21.090
if we haven't said that before--
u sub r and u sub theta.
00:24:21.090 --> 00:24:23.690
I can draw those in.
00:24:23.690 --> 00:24:28.650
And I can draw in
T and N. Now all
00:24:28.650 --> 00:24:32.130
I know is that if I have
the velocity vector v,
00:24:32.130 --> 00:24:34.560
it must be tangential
to the curve.
00:24:34.560 --> 00:24:37.390
Hopefully by this time, we
realize that the acceleration
00:24:37.390 --> 00:24:39.930
vector has no such restriction.
00:24:39.930 --> 00:24:43.710
Let's just draw in a v and an a,
call these the velocity vectors
00:24:43.710 --> 00:24:45.690
and the acceleration vectors.
00:24:45.690 --> 00:24:50.190
The point is that v and a are
determined by the motion--
00:24:50.190 --> 00:24:52.180
not by the coordinate system.
00:24:52.180 --> 00:24:54.660
In other words, when we're
talking about the velocity
00:24:54.660 --> 00:24:58.850
of this particle at the point
P:0, its velocity is the same,
00:24:58.850 --> 00:25:01.230
no matter what coordinate
system we're using.
00:25:01.230 --> 00:25:03.480
It just happens that
if we're dealing
00:25:03.480 --> 00:25:06.870
with Cartesian coordinates,
the velocity vector
00:25:06.870 --> 00:25:10.600
is dx/dt i plus dy/dt j.
00:25:10.600 --> 00:25:13.060
In other words, it's this
particular combination
00:25:13.060 --> 00:25:14.310
of i and j.
00:25:14.310 --> 00:25:16.920
If we're using T
and N components,
00:25:16.920 --> 00:25:21.080
the particular combination
of T and N is what?
00:25:21.080 --> 00:25:28.020
ds/dt times the unit
tangent vector plus 0 N.
00:25:28.020 --> 00:25:31.620
And if we happen to be
using polar coordinates,
00:25:31.620 --> 00:25:35.586
the expression is dr/dt * u
sub r plus r * d theta / dt
00:25:35.586 --> 00:25:37.140
* u sub theta.
00:25:37.140 --> 00:25:39.095
But let me circle
these, because it's
00:25:39.095 --> 00:25:41.070
the same velocity in each case.
00:25:41.070 --> 00:25:44.860
We use this when horizontal and
vertical motion are important.
00:25:44.860 --> 00:25:48.960
We use this when we're
interested in motion
00:25:48.960 --> 00:25:50.250
along the curve.
00:25:50.250 --> 00:25:52.960
And we use this primarily
in central force fields.
00:25:52.960 --> 00:25:54.200
But it makes no difference.
00:25:54.200 --> 00:25:55.860
It's the same velocity vector.
00:25:55.860 --> 00:26:00.580
And in a similar way, it's also
the same acceleration vector,
00:26:00.580 --> 00:26:02.940
whichever system
you happen to use.
00:26:02.940 --> 00:26:05.880
Namely, if we use
Cartesian coordinates,
00:26:05.880 --> 00:26:08.260
the acceleration vector
is the second derivative
00:26:08.260 --> 00:26:15.280
of x with respect to t times i
plus the second derivative of y
00:26:15.280 --> 00:26:18.570
with respect to t times j.
00:26:18.570 --> 00:26:22.550
That same vector, if we express
it in T and N components,
00:26:22.550 --> 00:26:26.200
is d^2 s / dt squared
times T, plus kappa,
00:26:26.200 --> 00:26:32.500
the curvature number, times
ds/dt squared times N.
00:26:32.500 --> 00:26:35.770
And if we express it in
terms of polar coordinates,
00:26:35.770 --> 00:26:38.430
as we just saw earlier
in our lecture,
00:26:38.430 --> 00:26:41.330
this is the expression
that we get.
00:26:41.330 --> 00:26:45.880
In other words then,
this summarizes our study
00:26:45.880 --> 00:26:52.460
of motion in the plane
using either Cartesian
00:26:52.460 --> 00:26:58.260
or polar or tangential
and normal components.
00:26:58.260 --> 00:27:01.250
You see, the point is
that we pick whichever
00:27:01.250 --> 00:27:06.250
coordinate system happens to be
of the greatest interest to us,
00:27:06.250 --> 00:27:08.160
the greatest value to us.
00:27:08.160 --> 00:27:10.540
We make the coordinate
system our slave,
00:27:10.540 --> 00:27:12.230
rather than the
other way around,
00:27:12.230 --> 00:27:15.370
and tackle the problem from
that particular point of view.
00:27:15.370 --> 00:27:20.270
At any rate, that ends this
phase of our particular course.
00:27:20.270 --> 00:27:22.510
And in the next
phase of our course,
00:27:22.510 --> 00:27:26.150
we get to probably what is
the most fundamental building
00:27:26.150 --> 00:27:28.410
block of the entire course.
00:27:28.410 --> 00:27:30.430
We get to that
particular topic which,
00:27:30.430 --> 00:27:34.400
by and large, most courses in
functions of several variables
00:27:34.400 --> 00:27:35.450
begin with.
00:27:35.450 --> 00:27:39.930
But we'll talk about that
more the next time we meet.
00:27:39.930 --> 00:27:43.750
And until that time, goodbye.
00:27:43.750 --> 00:27:46.130
Funding for the
publication of this video
00:27:46.130 --> 00:27:51.000
was provided by the Gabriella
and Paul Rosenbaum Foundation.
00:27:51.000 --> 00:27:55.180
Help OCW continue to provide
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00:27:55.180 --> 00:27:59.575
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