WEBVTT
00:00:03.500 --> 00:00:06.530
Now, we've been discussing
steady uniform precession,
00:00:06.530 --> 00:00:09.650
which is the simplest possible
case of a phenomenon that
00:00:09.650 --> 00:00:11.400
can be much more complicated.
00:00:11.400 --> 00:00:13.400
As an example and the
case we've been discussing
00:00:13.400 --> 00:00:17.510
so far where we release a
gyroscope from rest when it's
00:00:17.510 --> 00:00:20.060
horizontal, very
careful measurements
00:00:20.060 --> 00:00:22.250
would show that the
initial motion isn't
00:00:22.250 --> 00:00:25.390
just steady precession but
has some additional motion
00:00:25.390 --> 00:00:28.640
superimposed on it.
00:00:28.640 --> 00:00:31.250
The spin axis sort
of bounces or nods
00:00:31.250 --> 00:00:33.980
up and down as it precesses.
00:00:33.980 --> 00:00:35.510
However, friction
at the pivot point
00:00:35.510 --> 00:00:37.280
causes the amplitude
of the nodding
00:00:37.280 --> 00:00:41.330
to rapidly decay until it
settles into steady precession.
00:00:41.330 --> 00:00:44.090
This nodding motion
is called nutation.
00:00:44.090 --> 00:00:46.490
In our example, it
decays so quickly
00:00:46.490 --> 00:00:48.960
that we don't even notice it.
00:00:48.960 --> 00:00:52.040
But if you had a perfectly
frictionless pivot point,
00:00:52.040 --> 00:00:54.320
it would be a more
noticeable effect.
00:00:54.320 --> 00:00:57.770
And it would persist
for much longer.
00:00:57.770 --> 00:00:59.270
Now, there's an
interesting point
00:00:59.270 --> 00:01:03.460
about our horizontal gyroscope
that I'd like to point out.
00:01:07.600 --> 00:01:09.560
So here is my rod of length d.
00:01:09.560 --> 00:01:13.050
This is my point s.
00:01:13.050 --> 00:01:19.390
Here's my wheel that's rotating
with some angular velocity.
00:01:19.390 --> 00:01:22.830
And again, there's
the r hat direction.
00:01:22.830 --> 00:01:26.060
That's the direction that little
omega vector's pointing in.
00:01:26.060 --> 00:01:30.240
That's the k hat direction.
00:01:30.240 --> 00:01:32.680
And that's the
theta hat direction.
00:01:32.680 --> 00:01:36.509
Now, before I release it,
before I release the gyroscope
00:01:36.509 --> 00:01:39.150
from rest, the angular
momentum vector
00:01:39.150 --> 00:01:44.550
points in the r hat direction,
as drawn at this instant.
00:01:44.550 --> 00:01:46.830
When I release it
from rest, there
00:01:46.830 --> 00:01:49.140
is a torque acting
in the theta hat
00:01:49.140 --> 00:01:52.170
direction that causes
the angular momentum
00:01:52.170 --> 00:01:54.600
vector to rotate.
00:01:54.600 --> 00:01:56.082
And we've seen this
in some detail.
00:01:58.344 --> 00:01:59.759
But there's something
else to keep
00:01:59.759 --> 00:02:04.350
in mind, which is that now
when the system is precessing,
00:02:04.350 --> 00:02:08.639
that means that the center
of mass of the wheel
00:02:08.639 --> 00:02:13.050
is orbiting around the
z-axis, the vertical axis,
00:02:13.050 --> 00:02:16.470
through this pivot point.
00:02:16.470 --> 00:02:21.329
So you can think of that as a
point mass with the full mass
00:02:21.329 --> 00:02:28.650
of the gyroscope moving in
a circular path around point
00:02:28.650 --> 00:02:31.829
s with a radius d.
00:02:31.829 --> 00:02:36.840
That implies that there must be
a component of angular momentum
00:02:36.840 --> 00:02:39.960
pointing in the k hat
direction, pointing along
00:02:39.960 --> 00:02:44.550
the z-axis, corresponding
to the translational motion
00:02:44.550 --> 00:02:47.520
of the center of mass, the
circular translational motion
00:02:47.520 --> 00:02:53.950
of the center of mass, of the
gyroscope around the z-axis.
00:02:53.950 --> 00:02:56.650
But where did that k
component come from?
00:02:56.650 --> 00:02:59.500
There was no initial z component
of the angular momentum.
00:02:59.500 --> 00:03:02.530
We said that initially the
angular momentum just pointed
00:03:02.530 --> 00:03:04.720
in the r hat direction.
00:03:04.720 --> 00:03:07.120
And there's no torque
in the k hat direction.
00:03:07.120 --> 00:03:10.670
The only torque is in
the theta hat direction.
00:03:10.670 --> 00:03:14.522
So how do we end up with some
angular momentum pointing
00:03:14.522 --> 00:03:15.480
in the k hat direction?
00:03:15.480 --> 00:03:18.010
This seems to violate
the conservation
00:03:18.010 --> 00:03:21.079
of angular momentum.
00:03:21.079 --> 00:03:24.640
The solution to this puzzle is
that, in fact, the gyroscope
00:03:24.640 --> 00:03:29.050
doesn't remain precisely
horizontal when I release it.
00:03:29.050 --> 00:03:34.930
Instead, it dips by
a very small angle.
00:03:34.930 --> 00:03:36.160
So that's the horizontal.
00:03:36.160 --> 00:03:38.860
After I release
it, the gyroscope's
00:03:38.860 --> 00:03:41.770
actually dipped downward
by a small amount.
00:03:41.770 --> 00:03:44.165
I've exaggerated it
considerably in this drawing.
00:03:44.165 --> 00:03:47.161
We'll call that
angle delta theta.
00:03:49.810 --> 00:03:55.570
That small dip gives the
small negative component
00:03:55.570 --> 00:03:58.180
of the spin angular
momentum that's
00:03:58.180 --> 00:04:02.040
able to balance
the plus z angular
00:04:02.040 --> 00:04:04.840
momentum due to the
motion center of mass.
00:04:04.840 --> 00:04:07.960
Now, because the spin
angular momentum vector
00:04:07.960 --> 00:04:10.330
has such a large magnitude--
00:04:13.120 --> 00:04:14.910
that's my spin angular
momentum vector--
00:04:19.950 --> 00:04:23.550
only a small angle
is necessary in order
00:04:23.550 --> 00:04:26.790
to get enough of a
negative Lz component
00:04:26.790 --> 00:04:29.730
to balance out the
Lz corresponding
00:04:29.730 --> 00:04:32.490
to the center of mass motion.
00:04:32.490 --> 00:04:34.860
OK?
00:04:34.860 --> 00:04:37.560
But what we see is that
although gyroscopes
00:04:37.560 --> 00:04:41.850
seem like a remarkable
system, they're not magic.
00:04:41.850 --> 00:04:44.596
And, in fact, angular
momentum is conserved.
00:04:49.070 --> 00:04:52.240
The larger little
omega is, the faster
00:04:52.240 --> 00:04:53.750
the spin angular velocity is.
00:04:53.750 --> 00:04:56.100
And, therefore, the
larger the vector L
00:04:56.100 --> 00:05:00.720
is, the smaller that angle
is and the less of a dip
00:05:00.720 --> 00:05:02.640
that you get.
00:05:02.640 --> 00:05:05.910
So now we can write the
exact angular momentum
00:05:05.910 --> 00:05:07.020
for the gyroscope.
00:05:16.310 --> 00:05:17.990
This is my distance d.
00:05:17.990 --> 00:05:19.880
This is my pivot point s.
00:05:22.770 --> 00:05:26.010
My spin angular velocity
looks like that.
00:05:26.010 --> 00:05:32.159
Here is the r hat direction, the
k hat direction, and the theta
00:05:32.159 --> 00:05:36.360
hat direction into the screen.
00:05:36.360 --> 00:05:40.880
And we're orbiting--
the precession is around
00:05:40.880 --> 00:05:45.960
the z-axis with an angular
speed capital omega.
00:05:45.960 --> 00:05:50.430
Now, recall that
the total angular
00:05:50.430 --> 00:05:58.659
momentum with respect to point
s can be written in two parts.
00:05:58.659 --> 00:06:03.930
There's the angular momentum
due to the translational motion
00:06:03.930 --> 00:06:05.150
of the center of mass.
00:06:05.150 --> 00:06:07.350
And we know the
center of mass is just
00:06:07.350 --> 00:06:10.290
orbiting around the z-axis.
00:06:10.290 --> 00:06:17.260
So I'll call that, in fact,
the orbital angular momentum.
00:06:17.260 --> 00:06:23.160
So this is due to
the translation
00:06:23.160 --> 00:06:27.660
of the center of mass
with respect to point s.
00:06:27.660 --> 00:06:34.440
And then the second term is
due to the rotational angular
00:06:34.440 --> 00:06:39.540
momentum, or spin
angular momentum,
00:06:39.540 --> 00:06:41.220
relative to the center of mass.
00:06:41.220 --> 00:06:49.780
So this is due to rotation
about the center of mass.
00:06:49.780 --> 00:06:53.409
So again, the general angular
momentum of a rigid body
00:06:53.409 --> 00:06:59.300
is equal to the center of
mass translational angular
00:06:59.300 --> 00:07:02.350
momentum plus the angular
momentum due to rotation,
00:07:02.350 --> 00:07:05.140
a pure rotation, about
the center of mass.
00:07:05.140 --> 00:07:08.690
So let's write each
of these terms.
00:07:08.690 --> 00:07:13.540
The orbital angular
momentum, that
00:07:13.540 --> 00:07:16.390
is the translational angular
momentum of the center of mass,
00:07:16.390 --> 00:07:20.800
is just due to the motion
of the center of mass
00:07:20.800 --> 00:07:25.360
of the gyroscope, which is
at radius d with respect
00:07:25.360 --> 00:07:30.370
to point s, moving in a circle
with angular speed capital
00:07:30.370 --> 00:07:32.900
omega.
00:07:32.900 --> 00:07:38.409
So that's equal to the
mass times the center
00:07:38.409 --> 00:07:44.140
of mass velocity times
the radius of the circle.
00:07:44.140 --> 00:07:49.940
And that angular momentum
is in the k hat direction.
00:07:49.940 --> 00:07:52.210
But the center of
mass velocity is
00:07:52.210 --> 00:07:54.340
just equal to-- since
it's a circular motion--
00:07:54.340 --> 00:07:56.560
is just equal to the
radius of the circle,
00:07:56.560 --> 00:08:00.100
d, times the angular speed
of the circular motion
00:08:00.100 --> 00:08:02.890
of the center of mass,
which is capital omega.
00:08:02.890 --> 00:08:11.620
So I can write this
as m capital omega d
00:08:11.620 --> 00:08:14.930
squared in the k hat direction.
00:08:14.930 --> 00:08:19.930
So that's the angular momentum
due to the translational motion
00:08:19.930 --> 00:08:22.690
of the center of
mass around point s,
00:08:22.690 --> 00:08:26.450
what I'm calling the
orbital angular momentum.
00:08:26.450 --> 00:08:32.169
Now, the spin angular
momentum, we've
00:08:32.169 --> 00:08:35.380
been talking about
the rapid spin
00:08:35.380 --> 00:08:40.539
of the wheel around its axis.
00:08:40.539 --> 00:08:45.490
So that's given by
the moment of inertia
00:08:45.490 --> 00:08:51.380
about that axis times the
spin angular velocity.
00:08:51.380 --> 00:08:56.890
And that's pointing in
the r hat direction.
00:08:56.890 --> 00:08:58.780
But there's a subtlety here.
00:08:58.780 --> 00:09:02.140
It turns out that is
not the only rotation
00:09:02.140 --> 00:09:06.520
about the center of mass that
this wheel is undergoing.
00:09:06.520 --> 00:09:10.480
And actually because of that,
I'm going to call this I1.
00:09:10.480 --> 00:09:13.000
And let me just
draw a picture here.
00:09:13.000 --> 00:09:24.580
So for my disk rotating
around this axis,
00:09:24.580 --> 00:09:29.250
the relevant moment of
inertia is what I'll call I1.
00:09:29.250 --> 00:09:34.773
And for a disk, we know that
would be 1/2 m r squared.
00:09:37.960 --> 00:09:42.190
The other rotation
is a subtle one.
00:09:42.190 --> 00:09:44.200
Notice in this
drawing, suppose I
00:09:44.200 --> 00:09:50.110
were to draw a dot on the
outside face of the wheel.
00:09:52.870 --> 00:09:55.600
When this wheel precessed
around 180 degrees
00:09:55.600 --> 00:10:00.280
to the other side,
that dot on the outside
00:10:00.280 --> 00:10:03.520
would be facing in the
minus r hat direction now
00:10:03.520 --> 00:10:08.980
or would be pointing to the
left rather than to the right.
00:10:08.980 --> 00:10:12.310
And what that means is
that this disk has actually
00:10:12.310 --> 00:10:14.560
rotated about its diameter.
00:10:14.560 --> 00:10:22.050
So this disk has rotated
around a diameter like this.
00:10:22.050 --> 00:10:26.375
And that rotation is at the
slower angular speed capital
00:10:26.375 --> 00:10:26.875
omega.
00:10:30.040 --> 00:10:34.210
And it takes one full orbit for
it to rotate entirely around.
00:10:34.210 --> 00:10:38.879
If that rotation
weren't happening--
00:10:38.879 --> 00:10:40.670
this wouldn't make
sense physically the way
00:10:40.670 --> 00:10:41.720
I have this set up.
00:10:41.720 --> 00:10:44.540
But as an object, if that
rotation weren't happening,
00:10:44.540 --> 00:10:48.710
what that would mean is that
this face with a dot on it
00:10:48.710 --> 00:10:53.600
would always be pointing to the
right as the disk moved around.
00:10:53.600 --> 00:10:56.707
If it had an independent pivot
point right at the center,
00:10:56.707 --> 00:10:58.790
it would be physically
possible for it to do that.
00:10:58.790 --> 00:11:00.498
That's not what's
happening in this case.
00:11:00.498 --> 00:11:03.740
In this case, this face is
always pointing outward,
00:11:03.740 --> 00:11:06.260
radially away from
the pivot point.
00:11:06.260 --> 00:11:10.460
And that results in a
rotation around this diameter.
00:11:10.460 --> 00:11:14.732
Now, it turns out that moment
of inertia, which I'll call I2,
00:11:14.732 --> 00:11:19.620
happens to be half the moment
of inertia for this axis.
00:11:19.620 --> 00:11:26.350
So in this case, it's
one 1/4 m r squared.
00:11:26.350 --> 00:11:29.860
So that is another
kind of rotation
00:11:29.860 --> 00:11:31.630
that's happening about
the center of mass.
00:11:31.630 --> 00:11:35.950
And so there's an additional
angular momentum term
00:11:35.950 --> 00:11:38.780
arising from that rotation.
00:11:38.780 --> 00:11:43.510
And that's equal to
I2 times the angular
00:11:43.510 --> 00:11:48.310
velocity of that rotation,
which is capital omega.
00:11:48.310 --> 00:11:52.360
And because the axis
there is the z-axis,
00:11:52.360 --> 00:11:58.270
this is pointing in
the k hat direction.
00:11:58.270 --> 00:12:08.530
So the total angular
momentum of the gyroscope
00:12:08.530 --> 00:12:18.040
is I1 times omega in the r
hat direction plus I2 times
00:12:18.040 --> 00:12:22.270
capital omega in
the k hat direction
00:12:22.270 --> 00:12:33.320
plus m capital omega d squared
in the k hat direction.
00:12:33.320 --> 00:12:36.620
The first term, the
r hat component,
00:12:36.620 --> 00:12:40.650
is the only part
that's rotating.
00:12:40.650 --> 00:12:46.870
So this is a rotating vector.
00:12:46.870 --> 00:12:49.515
The k hat terms are constant.
00:12:53.140 --> 00:12:56.530
This is the exact expression
for the angular momentum
00:12:56.530 --> 00:12:57.700
of a gyroscope.
00:12:57.700 --> 00:13:03.430
And now we see the gyroscopic
approximation more precisely
00:13:03.430 --> 00:13:06.790
is saying that this
rotating term dominates
00:13:06.790 --> 00:13:08.440
over the other two terms.
00:13:08.440 --> 00:13:11.440
So it's actually that this
term is very large compared
00:13:11.440 --> 00:13:14.470
to either of these two
terms, which, as we can see,
00:13:14.470 --> 00:13:17.410
is roughly equivalent to saying
that little omega is very
00:13:17.410 --> 00:13:20.560
large compared to big omega.