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PROFESSOR: Today we're going
to talk all about moments
00:00:23.680 --> 00:00:30.760
of inertia, and the last time
we did [? muddy ?] cards, there
00:00:30.760 --> 00:00:32.860
were a lot of questions.
00:00:32.860 --> 00:00:37.990
The most common question-- where
did all those terms come from?
00:00:37.990 --> 00:00:42.040
And I'm going to do
a brief kind of intro
00:00:42.040 --> 00:00:47.045
around that so as to facilitate
the rest of the discussions.
00:00:58.060 --> 00:00:59.940
Some basic assumptions
here today.
00:00:59.940 --> 00:01:13.190
We're going to
consider bodies that
00:01:13.190 --> 00:01:30.456
rotate about their center
of mass, or fixed points.
00:01:34.540 --> 00:01:38.210
That's the kind of
problems we're doing.
00:01:38.210 --> 00:01:42.610
And secondly-- all
through this lecture
00:01:42.610 --> 00:01:47.160
today, there's four or five
significant assumptions
00:01:47.160 --> 00:01:50.125
or conditions that
are really important
00:01:50.125 --> 00:01:51.125
to the whole discussion.
00:01:51.125 --> 00:01:52.630
I'm going to really
try to highlight them.
00:01:52.630 --> 00:01:53.754
And starting with two here.
00:01:53.754 --> 00:02:10.036
The other one is we're going to
use reference frames attached
00:02:10.036 --> 00:02:10.619
to the bodies.
00:02:13.850 --> 00:02:16.740
That's why we did all that
work in kinematics figuring out
00:02:16.740 --> 00:02:21.560
how to do velocities of
things that are in translating
00:02:21.560 --> 00:02:23.190
and rotating
references frame, it's
00:02:23.190 --> 00:02:25.220
so that we can do
these problems.
00:02:28.860 --> 00:02:38.500
So this said, here's
a inertial frame.
00:02:42.410 --> 00:02:46.000
Here's some rigid body.
00:02:46.000 --> 00:02:48.830
Here's a point on
it, that's my origin.
00:02:48.830 --> 00:02:56.400
And here is a set of coordinates
attached to the rigid body--
00:02:56.400 --> 00:03:00.340
an xyz system-- the body
rotates and translates,
00:03:00.340 --> 00:03:03.360
it goes with it.
00:03:03.360 --> 00:03:09.050
And if we consider a
little piece of this body,
00:03:09.050 --> 00:03:13.020
a little part-- I'll call
it little mass element
00:03:13.020 --> 00:03:23.110
mi-- from here to here is some
distance y with respect to A,
00:03:23.110 --> 00:03:30.060
then the angular momentum
of that little mass particle
00:03:30.060 --> 00:03:37.760
is by definition of angular
momentum Ri cross Pi.
00:03:40.660 --> 00:03:43.310
And the P is always
the linear momentum
00:03:43.310 --> 00:03:46.916
defined in an inertial frame.
00:03:46.916 --> 00:03:48.665
That's the definition
of angular momentum.
00:03:59.310 --> 00:04:04.050
So you have seen this
little demo before.
00:04:04.050 --> 00:04:08.630
So this is now my rigid
body that I'm talking about.
00:04:08.630 --> 00:04:14.330
In this case, the axis
of rotation is this one.
00:04:14.330 --> 00:04:17.410
And I'm just going to
define my rigid body
00:04:17.410 --> 00:04:21.360
as essentially rotating
about this fixed point,
00:04:21.360 --> 00:04:23.270
this is A now.
00:04:23.270 --> 00:04:32.830
And my coordinate system is
xyz, so the rotation is omega z.
00:04:32.830 --> 00:04:36.200
And this rigid body--
this is massless,
00:04:36.200 --> 00:04:39.470
consists of a single point mass.
00:04:39.470 --> 00:04:41.200
So we've done this
before, but we
00:04:41.200 --> 00:04:43.710
tended to do this
kind of problem
00:04:43.710 --> 00:04:44.910
using polar coordinates.
00:04:44.910 --> 00:04:47.790
And now I really wanted to
strictly think in terms of XYZ.
00:04:47.790 --> 00:04:49.420
The answers come out
exactly the same.
00:04:49.420 --> 00:04:50.970
Polar coordinates also moved.
00:04:50.970 --> 00:04:54.470
You could do this in
rhat, thetahat, z.
00:04:54.470 --> 00:04:59.390
We're going to do it in
xyz attached to the body.
00:05:22.450 --> 00:05:26.150
And y is going into
the board, and we'll
00:05:26.150 --> 00:05:29.180
have no values in the y
direction for this problem.
00:05:29.180 --> 00:05:33.150
Here's our little
mass, we'll call it m1.
00:05:33.150 --> 00:05:38.480
It's at a position out here
x1, z1 are its coordinates,
00:05:38.480 --> 00:05:41.430
and y is zero.
00:05:41.430 --> 00:05:44.710
And we want to compute
its angular momentum,
00:05:44.710 --> 00:05:57.505
h1 with respect to A. That's
our R1A cross P1 and 0.
00:05:57.505 --> 00:06:02.800
And for this problem
that's x1 in the i hat
00:06:02.800 --> 00:06:09.080
plus z1 in the k hat,
that's the R-- cross
00:06:09.080 --> 00:06:15.030
P and P is mass
times velocity, m1.
00:06:15.030 --> 00:06:19.980
And the velocity is
just omega cross R.
00:06:19.980 --> 00:06:21.070
We've done this before.
00:06:21.070 --> 00:06:25.010
So we need a length
on this thing.
00:06:33.410 --> 00:06:35.510
We've done this enough
times before I'm assuming
00:06:35.510 --> 00:06:38.260
you can make this leap with me.
00:06:38.260 --> 00:06:46.510
The momentum is into the
board, in the y direction.
00:06:46.510 --> 00:06:53.620
The linear momentum
is R cross with that.
00:06:53.620 --> 00:07:02.260
And that will give us--
let me write it-- omega z.
00:07:02.260 --> 00:07:04.310
So that's the linear momentum.
00:07:04.310 --> 00:07:10.850
It's in the j direction,
its velocity is x1 omega z.
00:07:10.850 --> 00:07:14.120
The radius times the rotation
rate is the velocity,
00:07:14.120 --> 00:07:15.820
the direction is that direction.
00:07:15.820 --> 00:07:17.470
So this is your momentum.
00:07:17.470 --> 00:07:19.360
You multiply out
this cross product,
00:07:19.360 --> 00:07:34.410
you get two terms-- m1 x1
squared omega z k, minus m1 x1
00:07:34.410 --> 00:07:37.400
z1 omega z i.
00:07:40.070 --> 00:07:43.030
And these two terms
then we would identify.
00:07:43.030 --> 00:07:46.094
This is the angular momentum
in the k direction, angular
00:07:46.094 --> 00:07:47.260
momentum in the i direction.
00:07:47.260 --> 00:07:49.343
There doesn't happen to
be any in the z direction,
00:07:49.343 --> 00:07:50.760
in general there could be.
00:07:50.760 --> 00:07:55.035
But this is h for this
particle, particle one.
00:07:59.370 --> 00:08:01.760
I want to particularly
emphasize this one--
00:08:01.760 --> 00:08:04.380
oops, this is-- excuse me-- z.
00:08:04.380 --> 00:08:06.780
And this is the
piece we call hx.
00:08:10.050 --> 00:08:12.664
So there's a complement
of the angular momentum
00:08:12.664 --> 00:08:14.080
that's in the z
direction, there's
00:08:14.080 --> 00:08:15.520
a component in the x direction.
00:08:21.350 --> 00:08:24.040
We know that if
we compute-- just
00:08:24.040 --> 00:08:32.720
to tie it back to previous
work a little further-- d
00:08:32.720 --> 00:08:41.900
by dt of h1A gives us
the external torques
00:08:41.900 --> 00:08:44.370
in the system, a
vector of torques
00:08:44.370 --> 00:08:46.490
when we compute that out.
00:08:46.490 --> 00:08:49.270
I'm not going to calculate
it, I don't need it
00:08:49.270 --> 00:08:51.450
for the purposes of the
rest of the discussion.
00:08:51.450 --> 00:08:54.180
This one gives you
omega z dot term.
00:08:54.180 --> 00:08:55.911
This one gets two
terms because you have
00:08:55.911 --> 00:08:57.190
to take the derivative of i.
00:08:57.190 --> 00:09:00.430
So you get three
terms which are torque
00:09:00.430 --> 00:09:02.700
in the x, torque in the y,
torque in the z directions.
00:09:02.700 --> 00:09:04.410
Two of them are
static, because it's
00:09:04.410 --> 00:09:08.320
trying to bend this thing
back and out, and one of them,
00:09:08.320 --> 00:09:11.040
the z direction one, is the
one that makes it spin faster.
00:09:13.850 --> 00:09:16.910
So we've seen that,
that's a review.
00:09:16.910 --> 00:09:39.930
And to make the leap from
that to rigid bodies-- Capital
00:09:39.930 --> 00:09:43.570
H, now, a collection of
particles with respect
00:09:43.570 --> 00:09:51.870
to its the origin
attached to the rigid body
00:09:51.870 --> 00:09:54.082
is going to be the
summation over all
00:09:54.082 --> 00:09:55.165
the little mass particles.
00:10:00.740 --> 00:10:07.254
Their position vector crossed
with the linear momentum of all
00:10:07.254 --> 00:10:08.420
those little mass particles.
00:10:08.420 --> 00:10:10.490
Sum all that out.
00:10:10.490 --> 00:10:12.870
Because it's a rigid
body, this is always
00:10:12.870 --> 00:10:16.150
something of the form that
looks like omega cross R,
00:10:16.150 --> 00:10:20.290
m omega cross R. And
I can then write this
00:10:20.290 --> 00:10:29.310
as the summation of
[? oper ?] i of the mi,
00:10:29.310 --> 00:10:39.225
RiA cross omega with
respect to 0, cross RiA.
00:10:42.430 --> 00:10:49.140
That gives you the velocity, m
gives you the linear momentum,
00:10:49.140 --> 00:10:50.720
crossed with this
one again makes it
00:10:50.720 --> 00:10:52.690
into an angular momentum.
00:10:52.690 --> 00:10:59.780
So the angular momentum of every
particle-- the bits of this
00:10:59.780 --> 00:11:07.400
come out looking like
an R squared omega.
00:11:07.400 --> 00:11:13.410
Or at least has dimensions
of R squared omega.
00:11:13.410 --> 00:11:16.350
These can be like x1
squared, but these can also
00:11:16.350 --> 00:11:21.710
end up being terms like x1
z1, those cross product terms.
00:11:21.710 --> 00:11:27.240
But in general, these
things all add up.
00:11:27.240 --> 00:11:30.830
And this is a vector,
this is a vector,
00:11:30.830 --> 00:11:33.510
so this is a vector--
these are all vectors,
00:11:33.510 --> 00:11:34.540
the result's a vector.
00:11:39.040 --> 00:11:42.730
And therefore we would break
this down, this whole thing.
00:11:42.730 --> 00:11:45.750
Once you write all that
out and sum it all up,
00:11:45.750 --> 00:11:48.210
you're going to
get a piece of it
00:11:48.210 --> 00:11:52.440
that we would say that
is in the i hat direction
00:11:52.440 --> 00:11:54.580
and we call that Hx.
00:11:54.580 --> 00:11:58.460
Another piece that's in
the j hat direction, which
00:11:58.460 --> 00:12:03.540
we call Hy and Hz in the k hat.
00:12:03.540 --> 00:12:05.920
That's just how
that all shakes out.
00:12:05.920 --> 00:12:07.290
In general you get three parts.
00:12:11.750 --> 00:12:13.664
Yeah?
00:12:13.664 --> 00:12:17.648
AUDIENCE: Why do you have mass
multiplied by displacement
00:12:17.648 --> 00:12:19.650
and the momentum?
00:12:19.650 --> 00:12:24.170
PROFESSOR: Why do we have the
mass multiplied by a position
00:12:24.170 --> 00:12:27.210
vector times the momentum?
00:12:27.210 --> 00:12:30.205
Because that is the definition
of angular momentum.
00:12:34.450 --> 00:12:36.763
It's R cross P, all right?
00:12:36.763 --> 00:12:37.596
[INTERPOSING VOICES]
00:12:46.310 --> 00:12:48.120
PROFESSOR: Ah, I
see what I've done.
00:12:48.120 --> 00:12:48.950
Yeah you're right.
00:12:48.950 --> 00:12:52.540
This m doesn't belong--
I got ahead of myself--
00:12:52.540 --> 00:12:54.470
it doesn't belong here.
00:12:54.470 --> 00:12:57.680
It pops out of this to here.
00:12:57.680 --> 00:12:59.752
Thanks for catching that.
00:12:59.752 --> 00:13:00.460
Absolutely right.
00:13:15.420 --> 00:13:22.530
So every term here is
made up of things that
00:13:22.530 --> 00:13:30.310
look like x1, z1, m1 omega x.
00:13:30.310 --> 00:13:34.150
Or omega y, or omega z,
because this is a vector,
00:13:34.150 --> 00:13:37.450
and it can have three
components-- a piece in the i,
00:13:37.450 --> 00:13:39.960
a piece in the j, a
piece in the k direction.
00:13:39.960 --> 00:13:45.190
So there's a lot
of possible terms,
00:13:45.190 --> 00:13:49.950
and we're in the practice
of writing this out.
00:13:53.570 --> 00:14:00.250
This H vector can
be written then
00:14:00.250 --> 00:14:05.330
as a-- I'm going to
run out of room here.
00:14:19.220 --> 00:14:22.220
So the Hx term-- the
piece that comes out
00:14:22.220 --> 00:14:26.410
with all little bits
in the i direction--
00:14:26.410 --> 00:14:35.532
when you just break it apart,
we would write it Ixx omega
00:14:35.532 --> 00:14:46.725
x, plus Ixy omega
y, plus Ixz omega z.
00:14:49.330 --> 00:14:54.610
All the i terms that
float out of this,
00:14:54.610 --> 00:14:58.060
we just collect them
together-- all the ones that
00:14:58.060 --> 00:15:02.150
are multiplied by omega x
ends up being some terms that
00:15:02.150 --> 00:15:11.570
look like sums of mi--
x, mi z squared terms,
00:15:11.570 --> 00:15:13.726
and there's some
y squared terms.
00:15:13.726 --> 00:15:15.350
But we collect them
all together and we
00:15:15.350 --> 00:15:18.730
call this constant
in front of it Ixx.
00:15:18.730 --> 00:15:21.410
Just what floats
out of this stuff.
00:15:21.410 --> 00:15:25.030
We break it into three pieces,
the part of the angular
00:15:25.030 --> 00:15:26.865
momentum due to the
rotation in the x,
00:15:26.865 --> 00:15:28.780
the part due to the
rotation in the y,
00:15:28.780 --> 00:15:31.400
the part due to the
rotation in the z.
00:15:31.400 --> 00:15:34.436
And you do the
same thing for Hy,
00:15:34.436 --> 00:15:43.930
and you get in Iyz omega
x plus an Iyy omega
00:15:43.930 --> 00:15:53.660
y plus an Iyz omega z, and
you're finally get and Hz term.
00:15:53.660 --> 00:16:04.935
Izx omega x, Izy
omega y, Izz omega z.
00:16:07.560 --> 00:16:10.775
That's everything that falls out
of just doing this calculation.
00:16:19.890 --> 00:16:24.430
And where in the habit of
writing that in a matrix
00:16:24.430 --> 00:16:32.870
notation as the
product of this thing
00:16:32.870 --> 00:16:33.985
called the inertia matrix.
00:16:41.120 --> 00:16:43.240
And so forth, with
this bottom term
00:16:43.240 --> 00:16:52.130
being Izz multiplied by
omega x, omega y, omega z.
00:16:52.130 --> 00:16:57.310
You multiply that times the top
row, you get Hx, middle row,
00:16:57.310 --> 00:17:01.240
you get Hy, bottom
row, you get the Hz.
00:17:01.240 --> 00:17:04.700
So where this stuff
comes from is just
00:17:04.700 --> 00:17:10.770
from carrying out the summation
over all the mass bits.
00:17:10.770 --> 00:17:13.504
So it basically starts with the
definition of angular momentum.
00:17:13.504 --> 00:17:15.420
And this is just a
convenient way to write it.
00:17:19.910 --> 00:17:33.540
So for example, the Ixz term--
the one in the upper right
00:17:33.540 --> 00:17:40.090
there that's part of
Hx, the Ixz term--
00:17:40.090 --> 00:17:45.850
is minus the summation of
all the little mass bits
00:17:45.850 --> 00:17:50.170
times their location xi zi.
00:17:50.170 --> 00:17:55.331
And this, the xz,
always matches xz.
00:17:55.331 --> 00:17:57.580
And when you want to do this
over a continuous object,
00:17:57.580 --> 00:17:59.820
you integrate that.
00:17:59.820 --> 00:18:10.410
And for Izz, this is
the summation over i,
00:18:10.410 --> 00:18:21.920
all the bits of mi xi
squared plus yi squared.
00:18:21.920 --> 00:18:24.610
So in general this,
by summations, that's
00:18:24.610 --> 00:18:26.880
where these terms come from.
00:18:26.880 --> 00:18:27.530
They just come.
00:18:27.530 --> 00:18:29.670
They've started with
this calculation.
00:18:34.030 --> 00:18:35.905
And in general, from
that calculation.
00:18:39.040 --> 00:18:43.990
So for our one particle
system, for this thing.
00:18:43.990 --> 00:18:49.700
So where this is x and z,
it's one single particle.
00:18:49.700 --> 00:18:51.425
So for our one particle system.
00:19:19.310 --> 00:19:22.700
And there's no y1
in here because it's
00:19:22.700 --> 00:19:24.870
a zero in this problem.
00:19:24.870 --> 00:19:30.410
The coordinate of that little
mass particle is x1, 0, y1.
00:19:30.410 --> 00:19:33.380
So in general, this is x1
squared plus yi squared,
00:19:33.380 --> 00:19:34.030
but that's 0.
00:19:34.030 --> 00:19:36.130
So it's just mx1 squared.
00:19:47.220 --> 00:19:49.640
So that's where all
these things come from.
00:19:49.640 --> 00:19:56.900
Now let's kind of look at--
the units of these things
00:19:56.900 --> 00:20:01.040
are always mass
times length squared.
00:20:01.040 --> 00:20:08.500
The diagonal terms the Ixx,
Iyy, Izz terms will all
00:20:08.500 --> 00:20:12.190
be of the form of
something squared.
00:20:12.190 --> 00:20:17.080
And what it is is it's
always just the distance
00:20:17.080 --> 00:20:24.360
of the mass particles from the
axis of rotation, call that r.
00:20:24.360 --> 00:20:27.300
It's just a summation of
this perpendicular distance
00:20:27.300 --> 00:20:29.020
from the axis of rotation.
00:20:29.020 --> 00:20:32.990
For Izz, the axis
of rotation is z.
00:20:32.990 --> 00:20:36.190
And this piece here
is always the distance
00:20:36.190 --> 00:20:41.720
squared that the mass particle
is from the axis of rotation.
00:20:41.720 --> 00:20:47.050
Now that's the set up for today.
00:20:47.050 --> 00:20:49.880
And when we want to get onto
the real meat of the discussion,
00:20:49.880 --> 00:20:56.040
talking about things like
what principal axes are
00:20:56.040 --> 00:20:56.980
and so forth.
00:21:10.540 --> 00:21:13.060
Now we're going to move on to
the part of this conversation
00:21:13.060 --> 00:21:14.425
about principal axes.
00:21:29.820 --> 00:21:32.900
So a really important point.
00:21:36.280 --> 00:21:43.840
For every rigid body,
even weird ones.
00:21:43.840 --> 00:21:48.320
For every rigid body, there
is a coordinate system
00:21:48.320 --> 00:21:53.030
that you can attach to this
body, an xyz set of orthogonal
00:21:53.030 --> 00:21:56.210
coordinates that you can
fix in this body such
00:21:56.210 --> 00:22:02.455
that you can make this
inertia matrix be diagonal.
00:22:05.150 --> 00:22:07.410
Any weird body at
all, there is a set
00:22:07.410 --> 00:22:10.810
of coordinates that
if you use that set,
00:22:10.810 --> 00:22:13.570
this matrix turns
out to be diagonal.
00:22:13.570 --> 00:22:20.640
And what that means in
dynamics terms is if you then
00:22:20.640 --> 00:22:26.790
rotate the object about
one of those axes,
00:22:26.790 --> 00:22:29.150
it will be dynamically balanced.
00:22:31.860 --> 00:22:39.200
And so when you rotate a shaft--
this is an object for which I
00:22:39.200 --> 00:22:43.960
know-- it's a
circular uniform disk,
00:22:43.960 --> 00:22:47.140
one of the axes through the
center is a principal axis.
00:22:47.140 --> 00:22:51.500
And if I rotate the object about
that-- this one the hole's kind
00:22:51.500 --> 00:22:53.690
of out around so
it wobbles a bit
00:22:53.690 --> 00:22:58.450
but if I rotate it
about that it will
00:22:58.450 --> 00:23:02.600
make no torques about this
axis that are because it's
00:23:02.600 --> 00:23:05.120
unbalanced.
00:23:05.120 --> 00:23:08.590
It's essentially the dynamic
meaning of principal axis--
00:23:08.590 --> 00:23:11.500
an axis about which you can
rotate the thing and it'll just
00:23:11.500 --> 00:23:18.030
be smooth, no
away-from-the-axis torques.
00:23:18.030 --> 00:23:19.790
So that's so those
are pretty important.
00:23:19.790 --> 00:23:22.450
We want to know what
those are for rigid bodies
00:23:22.450 --> 00:23:24.160
and how to find them.
00:23:24.160 --> 00:23:26.590
So there's a mathematical
way to find them
00:23:26.590 --> 00:23:30.630
and you can just read
about in the textbook.
00:23:30.630 --> 00:23:37.770
It's sort of a-- I'm trying to
think of the mathematical term.
00:23:37.770 --> 00:23:40.690
I forgot it.
00:23:40.690 --> 00:23:42.220
But what I want
to teach you today
00:23:42.220 --> 00:23:44.890
is for many, many
objects, if you just
00:23:44.890 --> 00:23:48.000
look at their symmetries, you
can figure out by common sense
00:23:48.000 --> 00:23:49.452
where their principal axes are.
00:23:49.452 --> 00:23:51.035
So that's what we're
going to do next.
00:24:01.410 --> 00:24:04.660
We started off saying that
we're talking about bodies that
00:24:04.660 --> 00:24:07.520
are either rotating about
their centers of mass
00:24:07.520 --> 00:24:11.030
or about some other fixed point.
00:24:11.030 --> 00:24:18.470
So we're going to define
our principal axes
00:24:18.470 --> 00:24:23.990
assuming we are rotating
about the center of mass.
00:24:23.990 --> 00:24:28.120
There is an easy method known
as the parallel axis theorem
00:24:28.120 --> 00:24:29.920
to get to any other point.
00:24:29.920 --> 00:24:33.075
So why do we care about doing
it about the center of mass?
00:24:37.950 --> 00:24:40.200
Why is that useful to
know about the properties
00:24:40.200 --> 00:24:41.930
around the center for
dynamics purposes?
00:24:41.930 --> 00:24:43.560
What kind of dynamics
problems do you
00:24:43.560 --> 00:24:46.720
care about the center of mass?
00:24:46.720 --> 00:24:48.327
Rotation about the
center of mass.
00:24:48.327 --> 00:24:49.243
AUDIENCE: [INAUDIBLE].
00:24:52.820 --> 00:24:55.070
PROFESSOR: So when I throw
this thing in the air, it's
00:24:55.070 --> 00:24:57.320
rotating, it's translating,
and when it rotates,
00:24:57.320 --> 00:24:59.260
what's it's rotating about?
00:24:59.260 --> 00:25:00.302
AUDIENCE: Center of mass.
00:25:00.302 --> 00:25:01.385
PROFESSOR: Center of mass.
00:25:01.385 --> 00:25:03.185
So there's just lots
and lots of problem
00:25:03.185 --> 00:25:08.239
in which in fact the rotation
is going to occur around
00:25:08.239 --> 00:25:09.030
the center of mass.
00:25:09.030 --> 00:25:12.830
Any time the thing is off
there and there's nothing
00:25:12.830 --> 00:25:14.940
constraining its
rotational motion,
00:25:14.940 --> 00:25:17.980
the rotation will be
about the center of mass.
00:25:17.980 --> 00:25:24.250
So that's a good enough reason,
and a very practical reason
00:25:24.250 --> 00:25:26.470
then for computing
these things or knowing
00:25:26.470 --> 00:25:30.090
how to find these things
about the center of mass.
00:25:30.090 --> 00:25:34.690
All right so we're now
going to do principal axes,
00:25:34.690 --> 00:25:40.270
and we're going to teach
you some symmetry rules.
00:25:45.160 --> 00:25:48.510
Maybe first a
little common sense.
00:25:48.510 --> 00:25:52.880
This thing-- is this
a principal axis?
00:25:52.880 --> 00:25:53.980
No way, right?
00:25:53.980 --> 00:25:57.210
And we know for a fact that
it has off-diagonal terms
00:25:57.210 --> 00:25:59.700
because I've defined
my coordinates as being
00:25:59.700 --> 00:26:03.840
an x, y, z, like this, and this
is often at some strange angle.
00:26:03.840 --> 00:26:08.970
So as a practical
matter, how could I
00:26:08.970 --> 00:26:12.880
alter this object so that
this is a principal axis?
00:26:12.880 --> 00:26:14.689
Essentially, how would
you balance this?
00:26:14.689 --> 00:26:15.605
AUDIENCE: [INAUDIBLE].
00:26:19.480 --> 00:26:22.200
PROFESSOR: Take the
mass off, fix it.
00:26:22.200 --> 00:26:23.385
[LAUGHTER]
00:26:23.385 --> 00:26:24.766
Or?
00:26:24.766 --> 00:26:27.730
All right.
00:26:27.730 --> 00:26:29.390
Where do I put it?
00:26:29.390 --> 00:26:31.692
Down here?
00:26:31.692 --> 00:26:32.775
Ah, you want it like this.
00:26:37.117 --> 00:26:40.380
This is steel going
into aluminum,
00:26:40.380 --> 00:26:42.755
I have to be careful that
I don't strip the threads.
00:26:50.170 --> 00:26:50.670
There we go.
00:26:54.610 --> 00:26:59.070
And now test one, this
is axis of rotation,
00:26:59.070 --> 00:27:00.840
is it dynamically in balance?
00:27:00.840 --> 00:27:03.830
Smooth as silk.
00:27:03.830 --> 00:27:07.210
This now is a principal axis.
00:27:07.210 --> 00:27:09.110
Its mass distribution
is symmetric.
00:27:12.790 --> 00:27:16.070
Where is there a plane of
symmetry for this problem,
00:27:16.070 --> 00:27:17.537
for this object?
00:27:17.537 --> 00:27:20.078
That's what we're going to talk
about now, planes of symmetry
00:27:20.078 --> 00:27:23.430
and axis of symmetry.
00:27:23.430 --> 00:27:26.292
Ah, you're saying one like back?
00:27:26.292 --> 00:27:28.000
You're right, that's
a plane of symmetry.
00:27:28.000 --> 00:27:29.416
Is there another
one for this one?
00:27:32.150 --> 00:27:32.930
Like this.
00:27:32.930 --> 00:27:35.650
So there's two
planes of symmetry.
00:27:35.650 --> 00:27:40.250
So let's say come up
with some symmetry rules.
00:27:44.380 --> 00:28:16.290
So this first one
is there exists--
00:28:16.290 --> 00:28:19.450
by this I mean diagonal,
it's a diagonal matrix.
00:28:19.450 --> 00:28:21.980
You can find a set
of orthogonal axes
00:28:21.980 --> 00:28:24.690
such that this
matrix is diagonal.
00:28:24.690 --> 00:28:27.545
That's what we mean when we
go to find principal axes.
00:28:30.990 --> 00:28:32.615
And pick a board here-- rules.
00:28:50.560 --> 00:28:52.186
And we're going to
have three of them,
00:28:52.186 --> 00:28:53.560
so just leave room
on your paper.
00:28:53.560 --> 00:28:54.400
We're going to build these up.
00:28:54.400 --> 00:28:56.590
And I'm going to talk about it,
and then you add another one
00:28:56.590 --> 00:28:57.190
and so forth.
00:29:12.540 --> 00:29:13.320
So rule one.
00:29:21.375 --> 00:29:24.826
AUDIENCE: [INAUDIBLE] the
inertia matrix [INAUDIBLE].
00:29:24.826 --> 00:29:26.798
PROFESSOR: What is
the inertia matrix--
00:29:26.798 --> 00:29:28.780
AUDIENCE: No, I
mean the second one.
00:29:28.780 --> 00:29:30.520
PROFESSOR: This thing
with the diagonals?
00:29:30.520 --> 00:29:33.500
This is going to have all the
off-diagonal terms are zeroes.
00:29:37.310 --> 00:29:38.470
That's what I mean.
00:29:38.470 --> 00:29:43.070
It's a diagonal matrix, the mass
moments of inertia and products
00:29:43.070 --> 00:29:48.160
of inertia-- you only have
Ixx, Iyy, Izz if you properly
00:29:48.160 --> 00:29:51.620
pick this set of orthogonal
coordinates attached
00:29:51.620 --> 00:29:54.060
to the body-- if
you pick them right.
00:29:54.060 --> 00:29:56.510
What it means if then you
spin that object about one
00:29:56.510 --> 00:30:00.560
of those axes, it's in balance.
00:30:00.560 --> 00:30:41.080
So rule one, if there it is
an axis of symmetry-- Remember
00:30:41.080 --> 00:30:44.385
the caveat here is we're talking
about uniform density objects
00:30:44.385 --> 00:30:48.156
or at least objects in which
the density is symmetrically
00:30:48.156 --> 00:30:48.655
distributed.
00:30:51.200 --> 00:30:53.860
So the geometric symmetry
and mass symmetry
00:30:53.860 --> 00:30:56.690
mean the same thing.
00:30:56.690 --> 00:30:57.630
What does that mean?
00:30:57.630 --> 00:30:58.840
So axis of symmetry.
00:31:03.440 --> 00:31:06.455
Does this have an
axis of symmetry?
00:31:09.680 --> 00:31:10.689
Where?
00:31:10.689 --> 00:31:14.601
AUDIENCE: It has multiple
ones down the middle
00:31:14.601 --> 00:31:16.527
and it's also
symmetric [INAUDIBLE].
00:31:16.527 --> 00:31:18.110
PROFESSOR: That's a
plane of symmetry,
00:31:18.110 --> 00:31:20.625
the [? other ones you think, ?]
axis of symmetry.
00:31:20.625 --> 00:31:22.725
AUDIENCE: Well I mean the
line right through it.
00:31:22.725 --> 00:31:23.600
PROFESSOR: Which way?
00:31:23.600 --> 00:31:25.001
AUDIENCE: Well any.
00:31:25.001 --> 00:31:25.935
PROFESSOR: Nope.
00:31:25.935 --> 00:31:27.650
AUDIENCE: Perpendicular to it.
00:31:27.650 --> 00:31:29.530
PROFESSOR: Axis
of symmetry means
00:31:29.530 --> 00:31:33.800
that there is a mirror
image from that point--
00:31:33.800 --> 00:31:36.540
that point mirror image over
here is always identical.
00:31:36.540 --> 00:31:40.670
But you think any from angle, it
doesn't matter where you start,
00:31:40.670 --> 00:31:42.250
it's reflected through the axis.
00:31:42.250 --> 00:31:44.590
So things that are
circularly shaped
00:31:44.590 --> 00:31:47.760
tend to have axes of symmetry.
00:31:47.760 --> 00:31:51.680
So this has an axis of symmetry.
00:31:51.680 --> 00:31:54.695
Everything is reflected
exactly just across the axes,
00:31:54.695 --> 00:31:58.090
you don't have to pick
any particular line.
00:31:58.090 --> 00:32:02.510
So what it's saying if
it's an axis of symmetry,
00:32:02.510 --> 00:32:05.630
then it is a principal axis,
and rotation about that
00:32:05.630 --> 00:32:08.440
axis-- the things will be
nice perfectly in balance.
00:32:08.440 --> 00:32:12.340
If you go through all the
hairy details of calculating
00:32:12.340 --> 00:32:16.490
the hard way, the Ixx,
Iyy, et cetera terms--
00:32:16.490 --> 00:32:18.600
all of the off diagonal
terms will come out zero.
00:32:18.600 --> 00:32:23.230
And the reason is that for
every mass particle over here,
00:32:23.230 --> 00:32:26.070
let's say there's a little
bit right here in the corner,
00:32:26.070 --> 00:32:30.130
there is one exactly like
it on the other side.
00:32:30.130 --> 00:32:33.450
So if this one we're trying
to bend this thing up as it
00:32:33.450 --> 00:32:37.080
spins around, this one over
here is telling it to come back.
00:32:37.080 --> 00:32:38.446
Yeah?
00:32:38.446 --> 00:32:42.293
AUDIENCE: How come this is
not an axis, if you hold it--
00:32:42.293 --> 00:32:42.793
yeah--
00:32:42.793 --> 00:32:43.759
[INTERPOSING VOICES]
00:32:43.759 --> 00:32:45.700
PROFESSOR: So if you do this.
00:32:45.700 --> 00:32:48.470
So I think it's kind of just
the definition we're getting to
00:32:48.470 --> 00:32:50.590
of an axis of symmetry.
00:32:50.590 --> 00:32:58.490
Because in simple terms,
what the object looks
00:32:58.490 --> 00:33:01.310
like this way is
it's a narrow object,
00:33:01.310 --> 00:33:04.170
it is in fact a mirror
image reflection.
00:33:04.170 --> 00:33:06.770
But it looks different
when you go to this angle.
00:33:06.770 --> 00:33:09.020
And it looks different
when you go to this angle
00:33:09.020 --> 00:33:13.370
so it's not an axis of symmetry,
that's a plane of symmetry.
00:33:13.370 --> 00:33:14.980
We're going talk
about that next.
00:33:14.980 --> 00:33:17.480
So axes of symmetry--
the thing essentially
00:33:17.480 --> 00:33:23.050
looks the same at all
orientations about that axis.
00:33:23.050 --> 00:33:27.130
But it is certainly
dynamically balanced.
00:33:27.130 --> 00:33:30.770
Every mass bit here balances one
over there, it doesn't wobble.
00:33:30.770 --> 00:33:31.640
All right.
00:33:31.640 --> 00:33:33.990
So that's for sure true.
00:33:33.990 --> 00:33:42.890
Now keep in mind that exists
a set of orthogonal axes.
00:33:42.890 --> 00:33:44.890
So that means once
you find one, you
00:33:44.890 --> 00:33:47.970
know that the other two
are perpendicular to it
00:33:47.970 --> 00:33:50.390
and perpendicular to each other.
00:33:50.390 --> 00:33:54.560
So for this system, when
I got the x and y here,
00:33:54.560 --> 00:33:56.810
there has to be an x
and y perpendicular
00:33:56.810 --> 00:34:00.690
that that are the other two.
00:34:00.690 --> 00:34:02.364
And because it's
actually symmetric,
00:34:02.364 --> 00:34:04.280
it doesn't matter where
you out them, you just
00:34:04.280 --> 00:34:07.800
could say it's these two,
these two, it doesn't matter.
00:34:07.800 --> 00:34:09.300
You just have to
decide where you're
00:34:09.300 --> 00:34:12.989
going to put them on the
object, they're all the same.
00:34:12.989 --> 00:34:17.900
So for this axis symmetry,
they're just two more.
00:34:17.900 --> 00:34:22.610
They're going to be in the body
perpendicular to the first.
00:34:22.610 --> 00:34:26.699
Now we're going to do
about g so the book uses g
00:34:26.699 --> 00:34:29.281
to talk about the center
of mass, so I'll use g.
00:34:29.281 --> 00:34:30.739
So the center of
mass of this thing
00:34:30.739 --> 00:34:35.469
is right in the middle of
this and in the middle here.
00:34:35.469 --> 00:34:38.010
So it's inside this body.
00:34:38.010 --> 00:34:41.370
Right in the dead center of it.
00:34:41.370 --> 00:34:44.585
So the three principal axes are
an orthogonal set, one of which
00:34:44.585 --> 00:34:45.960
goes through it
and the other two
00:34:45.960 --> 00:34:47.790
are embedded in it right angles.
00:34:47.790 --> 00:34:49.989
So this is the simplest rule.
00:34:49.989 --> 00:34:53.480
If you have an axis of symmetry,
you know right way just
00:34:53.480 --> 00:34:55.980
about everything you need to
know about the principal axes
00:34:55.980 --> 00:34:56.659
of the body.
00:35:09.060 --> 00:35:09.630
Shoot.
00:35:09.630 --> 00:35:12.782
AUDIENCE: It seems like
then there would only
00:35:12.782 --> 00:35:16.470
be axis of symmetry in
objects that are either round
00:35:16.470 --> 00:35:17.720
or spherical, is that correct?
00:35:17.720 --> 00:35:19.470
PROFESSOR: Pretty much.
00:35:19.470 --> 00:35:21.610
AUDIENCE: So it's a very
limited special case.
00:35:21.610 --> 00:35:25.870
PROFESSOR: Yeah but there sure
appear an awful lot in machines
00:35:25.870 --> 00:35:28.930
because they have this beautiful
property of just being balanced
00:35:28.930 --> 00:35:31.800
in all directions.
00:35:31.800 --> 00:35:34.180
So everything
rotating in the world
00:35:34.180 --> 00:35:38.595
tends to have an almost
perfect axial symmetry.
00:35:44.230 --> 00:35:45.650
Just thinking about order here.
00:35:45.650 --> 00:35:48.850
Let's do this and then I'm going
to go do a couple of examples.
00:35:48.850 --> 00:36:09.170
The second rule-- if you
have one plane of symmetry,
00:36:09.170 --> 00:36:11.070
a plane of symmetry.
00:36:11.070 --> 00:36:13.420
So this is kind of the
opposite direction,
00:36:13.420 --> 00:36:15.670
in this you have the
least information.
00:36:15.670 --> 00:36:18.867
Here's an object, does it
have a plane of symmetry?
00:36:18.867 --> 00:36:19.450
AUDIENCE: Yes.
00:36:19.450 --> 00:36:20.158
PROFESSOR: Where?
00:36:20.158 --> 00:36:21.990
AUDIENCE: Straight
through [INAUDIBLE].
00:36:21.990 --> 00:36:23.082
PROFESSOR: Show me.
00:36:23.082 --> 00:36:24.151
That cut.
00:36:24.151 --> 00:36:24.650
All right.
00:36:24.650 --> 00:36:27.620
So I've got these little
dotted marks on this thing.
00:36:27.620 --> 00:36:32.320
So if I slice through
that, I create
00:36:32.320 --> 00:36:33.920
two pieces that are identical.
00:36:33.920 --> 00:36:35.320
So that's a plane of symmetry.
00:36:35.320 --> 00:36:41.710
There is image match across
that plane at every point.
00:36:41.710 --> 00:36:45.350
So if I have a
plane of symmetry,
00:36:45.350 --> 00:36:48.430
what do you think you can
say about a principal axis?
00:36:48.430 --> 00:36:50.085
One of the principal axis?
00:36:50.085 --> 00:36:51.540
AUDIENCE: It'll
be on that plane.
00:36:51.540 --> 00:36:52.980
PROFESSOR: It'll
be on that plane.
00:36:52.980 --> 00:36:55.229
That turns out to be true,
but that's the second point
00:36:55.229 --> 00:36:56.268
I want to make.
00:36:56.268 --> 00:36:58.042
AUDIENCE: It might
be perpendicular.
00:36:58.042 --> 00:37:00.500
PROFESSOR: She says there might
be one perpendicular to it.
00:37:00.500 --> 00:37:02.125
And that's the one
I was searching for,
00:37:02.125 --> 00:37:03.680
you're right too.
00:37:03.680 --> 00:37:07.000
There's going to
be a principal axis
00:37:07.000 --> 00:37:09.380
that's perpendicular to
that plane of symmetry,
00:37:09.380 --> 00:37:12.470
and since we want to define
our moments of inertia
00:37:12.470 --> 00:37:14.230
for this to get started
with with respect
00:37:14.230 --> 00:37:15.817
to g, where will
it pass through?
00:37:15.817 --> 00:37:16.650
AUDIENCE: Through g.
00:37:16.650 --> 00:37:18.910
PROFESSOR: G, kind
of by definition.
00:37:18.910 --> 00:37:20.800
So we're going to have
it pass through g.
00:37:20.800 --> 00:37:33.530
If there's a plane of
symmetry, then there
00:37:33.530 --> 00:37:47.820
is a principal axis
perpendicular to it.
00:37:53.550 --> 00:37:58.030
And we'll0 define it, for the
purposes of our discussion,
00:37:58.030 --> 00:38:00.080
we'll just let it
pass through g.
00:38:03.540 --> 00:38:06.190
It doesn't have to,
but that's how we're
00:38:06.190 --> 00:38:07.820
going about this discussion.
00:38:07.820 --> 00:38:14.720
Let it pass through the center
of mass, this point we call g.
00:38:14.720 --> 00:38:18.440
All right, so that
means that there's
00:38:18.440 --> 00:38:21.690
a center of mass in this thing.
00:38:21.690 --> 00:38:24.990
And I'm just guessing
roughly where it is.
00:38:24.990 --> 00:38:28.040
But in fact if I hung this
thing up here like this,
00:38:28.040 --> 00:38:34.140
and let gravity find it's
natural hanging angle,
00:38:34.140 --> 00:38:36.100
and I drew a plumb
line down here,
00:38:36.100 --> 00:38:40.320
just drew a line that the string
would take with a plumb bob.
00:38:40.320 --> 00:38:43.310
Then I went to some other
point and did it again
00:38:43.310 --> 00:38:46.140
and hung a plumb bob on it drew
the line-- where they intersect
00:38:46.140 --> 00:38:48.487
is the center of mass.
00:38:48.487 --> 00:38:50.570
And then you know since
it's got symmetry this way
00:38:50.570 --> 00:38:51.990
that's in the middle.
00:38:51.990 --> 00:38:54.980
So I guesses that that's
about where it is.
00:38:54.980 --> 00:38:58.750
And there then is
a principal axis
00:38:58.750 --> 00:39:02.960
of this object that's
perpendicular to it,
00:39:02.960 --> 00:39:06.060
passing through the plane--
perpendicular to the plane.
00:39:06.060 --> 00:39:08.180
And that means now
there's two more.
00:39:08.180 --> 00:39:12.540
But this gets a little more
difficult, and I have no clue.
00:39:12.540 --> 00:39:15.390
There are two more, because
the principal axes always
00:39:15.390 --> 00:39:16.840
come in an orthogonal set.
00:39:16.840 --> 00:39:18.830
So now that I know one,
I know that there's
00:39:18.830 --> 00:39:24.330
two more somewhere
around oriented
00:39:24.330 --> 00:39:25.880
this way, someplace
in this plane.
00:39:25.880 --> 00:39:28.210
Maybe one like that,
maybe one like this,
00:39:28.210 --> 00:39:31.670
such that if you spun it
about one of those axes,
00:39:31.670 --> 00:39:33.809
it'd be in balance.
00:39:33.809 --> 00:39:35.350
You can see that
gets a little messy,
00:39:35.350 --> 00:39:37.535
I can't guess where it is.
00:39:37.535 --> 00:39:39.410
And so there are ways
to find it, one of them
00:39:39.410 --> 00:39:41.570
would be doing an
experiment, seeing which
00:39:41.570 --> 00:39:43.570
axis it spins nicely around.
00:39:43.570 --> 00:39:45.154
So that's the second rule.
00:39:47.820 --> 00:39:50.290
But even just with that
one plane of symmetry,
00:39:50.290 --> 00:39:53.600
you get some pretty
good insight.
00:39:53.600 --> 00:39:58.744
Now what if there are
two planes of symmetry?
00:39:58.744 --> 00:39:59.505
Yeah?
00:39:59.505 --> 00:40:01.630
AUDIENCE: You said that
the principal axis does not
00:40:01.630 --> 00:40:05.290
have to pass through
the center of mass?
00:40:05.290 --> 00:40:09.606
PROFESSOR: No, I'm saying
it doesn't have to pass
00:40:09.606 --> 00:40:10.730
through the center of mass.
00:40:10.730 --> 00:40:12.590
The question is?
00:40:12.590 --> 00:40:15.285
AUDIENCE: How could
it not because it
00:40:15.285 --> 00:40:17.990
becomes stable [INAUDIBLE].
00:40:17.990 --> 00:40:20.634
PROFESSOR: So she's saying
it'd be unstable if it's not
00:40:20.634 --> 00:40:21.550
passing through there.
00:40:21.550 --> 00:40:23.965
Like if I put an axle
through this wheel,
00:40:23.965 --> 00:40:27.799
and I spun it about
some point out here,
00:40:27.799 --> 00:40:29.840
you don't know because I
had that shaker in here,
00:40:29.840 --> 00:40:32.520
that thing is going
to shake like crazy.
00:40:32.520 --> 00:40:36.540
But does it produce
unbalanced torques
00:40:36.540 --> 00:40:39.070
about my rotation point?
00:40:41.850 --> 00:40:44.770
It produces centrifugal force
that you'll feel like crazy,
00:40:44.770 --> 00:40:48.030
but does it produce
a torque that you'd
00:40:48.030 --> 00:40:51.380
have to resist with
some static torque?
00:40:51.380 --> 00:40:52.130
What do you think?
00:40:55.030 --> 00:40:56.270
It won't.
00:40:56.270 --> 00:40:58.870
So there's a nuance to
this unbalanced thing,
00:40:58.870 --> 00:41:02.060
and I was going to get to
it, probably next lecture,
00:41:02.060 --> 00:41:07.140
but that is the when we say an
object is dynamically balanced,
00:41:07.140 --> 00:41:12.170
we mean that it doesn't
have any unbalanced torques.
00:41:12.170 --> 00:41:18.900
If it is statically
balanced, it's
00:41:18.900 --> 00:41:22.250
rotating about its
center of mass.
00:41:22.250 --> 00:41:25.600
But you can be
statically unbalanced
00:41:25.600 --> 00:41:29.580
and let it go around this axis,
but it'll produce no torques.
00:41:29.580 --> 00:41:31.210
It's still dynamically balanced.
00:41:31.210 --> 00:41:42.430
And the angular momentum of
an object which is rotating--
00:41:42.430 --> 00:41:44.490
And this we know has
a principal axis here,
00:41:44.490 --> 00:41:46.550
I just moved it off to the side.
00:41:46.550 --> 00:41:50.290
It's rotating about this.
00:41:50.290 --> 00:41:55.440
It is dynamically balanced,
and if you computed
00:41:55.440 --> 00:42:01.440
about this point now the
Ixy, Ixz off-diagonal entries
00:42:01.440 --> 00:42:05.360
in that moment of inertia
matrix, they're all 0.
00:42:05.360 --> 00:42:08.640
You'd get no unbalanced
torques, but you
00:42:08.640 --> 00:42:10.860
do have an unbalanced
centrifugal force
00:42:10.860 --> 00:42:12.090
as this thing goes around.
00:42:12.090 --> 00:42:13.631
And we'll talk a
bit more about that.
00:42:13.631 --> 00:42:15.595
AUDIENCE: Is that
sort of analogous
00:42:15.595 --> 00:42:19.032
to the homework problem
we had a few weeks back
00:42:19.032 --> 00:42:22.085
with the motorcycle wheel and
you just had mass on one side
00:42:22.085 --> 00:42:22.960
and not on the other?
00:42:22.960 --> 00:42:23.720
PROFESSOR: Right.
00:42:23.720 --> 00:42:25.510
So she's asking about the
motorcycle wheel problem
00:42:25.510 --> 00:42:27.500
where we had that little
mass that got there.
00:42:27.500 --> 00:42:30.510
I think in the next lecture
I'm going to come back
00:42:30.510 --> 00:42:32.790
to that problem
just so we could tie
00:42:32.790 --> 00:42:37.500
a bow around this whole
thing and understand why it's
00:42:37.500 --> 00:42:39.030
unbalanced, how
you can balance it,
00:42:39.030 --> 00:42:41.170
and the difference
between static unbalance
00:42:41.170 --> 00:42:42.880
and dynamic unbalance.
00:42:42.880 --> 00:42:45.510
But today, we're talking
about symmetry rules.
00:42:48.124 --> 00:42:50.290
Finally, so I was saying,
let's talk about something
00:42:50.290 --> 00:42:51.890
that has two planes of symmetry.
00:42:51.890 --> 00:42:53.810
This actually has three
planes of symmetry,
00:42:53.810 --> 00:42:55.060
but we'll settle for two.
00:42:59.820 --> 00:43:02.700
Pick a plane of symmetry
for this object.
00:43:02.700 --> 00:43:07.444
If I pick-- OK, so she picked
the one-- slice it this way.
00:43:07.444 --> 00:43:08.360
What about the second?
00:43:11.135 --> 00:43:13.057
Like that.
00:43:13.057 --> 00:43:13.890
Then a third, right?
00:43:13.890 --> 00:43:15.014
There's even one like that.
00:43:15.014 --> 00:43:16.980
So this has three
planes of symmetry.
00:43:16.980 --> 00:43:25.510
But if you have two planes
of symmetry that intersect,
00:43:25.510 --> 00:43:29.820
that are orthogonal
to one another,
00:43:29.820 --> 00:43:33.000
what do you think you
can say about that line
00:43:33.000 --> 00:43:34.894
of intersection?
00:43:34.894 --> 00:43:36.310
AUDIENCE: It's the
principal axis.
00:43:36.310 --> 00:43:39.100
PROFESSOR: It sure is.
00:43:39.100 --> 00:43:41.670
And it's probably
right through g.
00:43:41.670 --> 00:43:57.490
So if you have two
planes of symmetry--
00:43:57.490 --> 00:43:59.352
Now make them orthogonal.
00:43:59.352 --> 00:44:01.060
You can make all sorts
of symmetry rules,
00:44:01.060 --> 00:44:04.230
and I'm just picking these
three to help you out.
00:44:04.230 --> 00:44:07.340
This just to help you
see principal axes.
00:44:07.340 --> 00:44:12.990
If you have two
orthogonal planes
00:44:12.990 --> 00:44:46.050
of symmetry their intersection--
and once you know that,
00:44:46.050 --> 00:44:48.110
then you go back to rule two.
00:44:48.110 --> 00:44:50.236
And it tells you everything
else you need to know.
00:44:50.236 --> 00:44:51.860
Because you have one
plane of symmetry,
00:44:51.860 --> 00:44:57.080
you know there is a principal
axis perpendicular to it.
00:44:57.080 --> 00:44:59.720
Well if you have two planes of
symmetry, the rule still holds.
00:44:59.720 --> 00:45:01.710
There's one perpendicular
to each one.
00:45:01.710 --> 00:45:05.010
The intersection, let's say,
of this plane of symmetry
00:45:05.010 --> 00:45:07.910
and this plane of
symmetry is a line
00:45:07.910 --> 00:45:11.200
which goes right through the
center of this thing that way.
00:45:11.200 --> 00:45:13.710
So there's a principal
axis this way.
00:45:13.710 --> 00:45:16.200
But since there is a
plane of symmetry here,
00:45:16.200 --> 00:45:19.350
there must also be a principal
axis perpendicular to it.
00:45:19.350 --> 00:45:23.730
So sure enough, three
principal axes for this thing
00:45:23.730 --> 00:45:28.420
are through the center,
perpendicular this way,
00:45:28.420 --> 00:45:29.550
perpendicular that way.
00:45:29.550 --> 00:45:30.450
You instantly know.
00:45:30.450 --> 00:45:32.210
Two planes of
symmetry-- you instantly
00:45:32.210 --> 00:45:37.400
know where the three orthogonal
principal axes are that past
00:45:37.400 --> 00:45:38.783
through the center of mass.
00:45:38.783 --> 00:45:39.769
Yeah?
00:45:39.769 --> 00:45:40.269
[?
00:45:40.269 --> 00:45:42.094
AUDIENCE: Does this all apply
?] just like a constant mass
00:45:42.094 --> 00:45:43.040
throughout?
00:45:43.040 --> 00:45:48.150
PROFESSOR: Not
constant, symmetrically
00:45:48.150 --> 00:45:51.150
distributed density.
00:45:51.150 --> 00:45:55.280
Right so I'm choosing
my words carefully so
00:45:55.280 --> 00:45:57.970
that I succeed in
the following--
00:45:57.970 --> 00:46:01.130
that the planes
defining mass symmetry
00:46:01.130 --> 00:46:05.479
will be the same as the planes
defining geometric similarity.
00:46:05.479 --> 00:46:07.770
But you actually don't have
to have a constant density,
00:46:07.770 --> 00:46:11.420
it just has to be distributed so
that what I just said is true.
00:46:11.420 --> 00:46:13.290
So that the geometric
symmetries are
00:46:13.290 --> 00:46:17.527
the same as the mass
distribution symmetries.
00:46:17.527 --> 00:46:19.610
All right so those are my
three rules of symmetry.
00:46:19.610 --> 00:46:20.651
You could make up others.
00:46:20.651 --> 00:46:22.630
Those are the three
that I've made up
00:46:22.630 --> 00:46:24.315
to help you see objects.
00:46:31.230 --> 00:46:35.110
That object, it's
a circular disk
00:46:35.110 --> 00:46:42.430
put on top of another object
such that their centers
00:46:42.430 --> 00:46:45.290
of mass line up.
00:46:45.290 --> 00:46:50.151
Where are the principal axes of
this object using those rules?
00:46:55.930 --> 00:46:58.180
If you think you
know one, tell me.
00:47:05.124 --> 00:47:06.674
AUDIENCE: Through the middle.
00:47:06.674 --> 00:47:08.590
PROFESSOR: Through the
middle of both of them.
00:47:08.590 --> 00:47:11.498
Probably, good guess.
00:47:11.498 --> 00:47:12.426
How about another one?
00:47:16.610 --> 00:47:20.280
Where does this thing
have planes of symmetry?
00:47:20.280 --> 00:47:23.640
AUDIENCE: So there's a plane of
symmetry if you cut it in half.
00:47:23.640 --> 00:47:26.520
[INAUDIBLE] cut it
in half, [INAUDIBLE].
00:47:26.520 --> 00:47:27.960
PROFESSOR: OK, and?
00:47:27.960 --> 00:47:30.360
AUDIENCE: The other way.
00:47:30.360 --> 00:47:32.965
PROFESSOR: One like that,
we've got all three.
00:47:32.965 --> 00:47:35.590
And if we're going to want it to
go through the center of mass,
00:47:35.590 --> 00:47:37.964
then we're going to have to
find where the center of mass
00:47:37.964 --> 00:47:40.470
is this way, but
it's about there.
00:47:40.470 --> 00:47:42.500
So just using the
symmetry ideas,
00:47:42.500 --> 00:47:45.800
you can right away figure out
where these principal axes be.
00:47:45.800 --> 00:47:48.680
And that means from a
dynamic point of view,
00:47:48.680 --> 00:47:50.960
if you spin it about
one of those axes,
00:47:50.960 --> 00:47:52.530
it's nice and
dynamically balanced.
00:47:52.530 --> 00:47:56.000
If you spin it off in
some other weird direction
00:47:56.000 --> 00:47:59.830
is it necessarily
dynamically balanced
00:47:59.830 --> 00:48:01.750
about that axis of spin?
00:48:08.730 --> 00:48:10.330
So let me restate that question.
00:48:16.620 --> 00:48:21.120
We know that this thing has
an axis of symmetry principal
00:48:21.120 --> 00:48:25.690
axis through the center,
and another one this way,
00:48:25.690 --> 00:48:26.890
and another one this way.
00:48:26.890 --> 00:48:28.650
And if I spin it about
any one of those,
00:48:28.650 --> 00:48:30.240
it's dynamically balanced.
00:48:30.240 --> 00:48:35.940
But if I pick some other
strange direction for the spin,
00:48:35.940 --> 00:48:40.180
and I spin it about that axis,
will I feel unbalanced torques
00:48:40.180 --> 00:48:42.040
on this axle, on
the bearings having
00:48:42.040 --> 00:48:44.080
to hold this thing in place?
00:48:44.080 --> 00:48:46.900
Yeah, you better believe it,
this thing wobbles like crazy.
00:48:46.900 --> 00:48:56.030
So the principal axes are
a property of the object,
00:48:56.030 --> 00:48:59.430
they're not a property
of the angular momentum.
00:48:59.430 --> 00:49:04.567
The angular momentum comes
then from multiplying
00:49:04.567 --> 00:49:06.150
the mass moment of
inertia that you've
00:49:06.150 --> 00:49:11.930
determined times the
actual rotation vector.
00:49:11.930 --> 00:49:15.590
And you'll find out then you get
angular components of angular
00:49:15.590 --> 00:49:18.220
momentum that are not in
the direction of spin,
00:49:18.220 --> 00:49:21.160
and as soon as that happens,
you have unbalanced terms.
00:49:25.600 --> 00:49:28.235
I've got to get on to something
else to help you do homework.
00:50:09.730 --> 00:50:18.960
So for my disk, with z coming
out of the board, the Izz--
00:50:18.960 --> 00:50:24.680
so let's say here's x, y, z
coming out of the board-- Izz,
00:50:24.680 --> 00:50:28.530
the mass moment of
inertia about this z axis
00:50:28.530 --> 00:50:35.070
is, from the basic definition,
the summation of the mis,
00:50:35.070 --> 00:50:41.510
xi squared plus yi squared.
00:50:41.510 --> 00:50:43.900
It's just that for every
little mass particle
00:50:43.900 --> 00:50:48.790
it's the radius squared away
from the center of rotation.
00:50:48.790 --> 00:50:51.320
That's what the x squared
plus y squared is.
00:50:51.320 --> 00:50:54.370
And we can turn this
into an integral.
00:50:54.370 --> 00:50:57.890
It's the integral of r
squared, that distance,
00:50:57.890 --> 00:51:01.570
times the little mass
bit that's there.
00:51:01.570 --> 00:51:04.660
And that's the same.
00:51:04.660 --> 00:51:09.740
If you wanted to do the integral
as x squared plus y squared dm.
00:51:12.880 --> 00:51:19.360
But to do this integral for a
nice circular, symmetric disk,
00:51:19.360 --> 00:51:30.506
you can pick a little mass bit
that has thickness dr and width
00:51:30.506 --> 00:51:32.510
rd theta.
00:51:32.510 --> 00:51:35.380
And this angle here is d theta.
00:51:38.390 --> 00:51:40.360
And that's a little bit of area.
00:51:40.360 --> 00:51:44.973
That's a little dA which
has area r dr d theta.
00:51:48.400 --> 00:51:50.520
It's just length times width.
00:51:50.520 --> 00:51:53.280
When it's small enough,
it's a little rectangle,
00:51:53.280 --> 00:51:56.280
and it has that area.
00:51:56.280 --> 00:52:02.500
And it has a volume, dV--
the volume of that thing
00:52:02.500 --> 00:52:05.960
is just the area times
the thickness of it.
00:52:05.960 --> 00:52:10.500
So here's our disk here,
but it has some thickness,
00:52:10.500 --> 00:52:12.840
and I'll call that h.
00:52:12.840 --> 00:52:20.500
So the volume is
just h r dr d theta.
00:52:20.500 --> 00:52:25.355
And the mass, dm, is
a density times dV.
00:52:31.330 --> 00:52:34.260
So I want to
integrate this, all I
00:52:34.260 --> 00:52:37.450
have to integrate the
integral then of r
00:52:37.450 --> 00:52:50.700
squared dm is the integral from
0 to 2pi, 0 to r of rho dV.
00:52:50.700 --> 00:52:57.890
Rho h-- oh, and I need
an r squared-- r squared
00:52:57.890 --> 00:53:04.830
dV is rho h r dr d theta.
00:53:04.830 --> 00:53:08.170
So this is 1802
integrals, right?
00:53:08.170 --> 00:53:12.790
So is any of this a
function of theta?
00:53:12.790 --> 00:53:14.334
No, so it's a
trivial integral, you
00:53:14.334 --> 00:53:16.250
integrate that over
theta, you just get theta,
00:53:16.250 --> 00:53:18.140
evaluate it 0 to 2pi.
00:53:18.140 --> 00:53:25.980
So this is 2pi rho h, can all
come to the outside, integral 0
00:53:25.980 --> 00:53:33.550
to R of r cubed dr.
And that ends up--
00:53:33.550 --> 00:53:37.090
the r cubed goes to
r to the 4th over 4.
00:53:37.090 --> 00:53:48.490
And the final result of this
one is 2pi r to the 4th over 4
00:53:48.490 --> 00:53:54.300
rho h, and when new account
for h times pi r squared
00:53:54.300 --> 00:53:57.030
is the volume times
rho is the mass.
00:53:57.030 --> 00:54:03.260
This all works out to
be m r squared over 2.
00:54:03.260 --> 00:54:08.800
So Izz-- so I needed to
do this once for you.
00:54:08.800 --> 00:54:13.230
For simple things
integrate, Izz in this case,
00:54:13.230 --> 00:54:15.650
you just integrate
it out, account
00:54:15.650 --> 00:54:18.650
for all little mass bits, that
is the mass moment of inertia
00:54:18.650 --> 00:54:25.390
with respect to the axis passing
through the center like this.
00:54:25.390 --> 00:54:26.662
Pardon?
00:54:26.662 --> 00:54:28.150
AUDIENCE: [INAUDIBLE]
00:54:28.150 --> 00:54:30.358
PROFESSOR: It's this, this
is what I'm talking about.
00:54:42.610 --> 00:54:44.220
Moving on to the last bit.
00:54:53.010 --> 00:54:57.720
So we need to know how
to be able rotate things
00:54:57.720 --> 00:55:01.160
about places other than
their centers of mass.
00:55:01.160 --> 00:55:05.210
So this is a stick, I can
rotate about the center of mass,
00:55:05.210 --> 00:55:06.890
but it's more interesting
if I rotate it
00:55:06.890 --> 00:55:08.520
about some other point.
00:55:08.520 --> 00:55:10.926
It makes it a pendulum
when I do it around here.
00:55:10.926 --> 00:55:12.300
So I need to be
able to calculate
00:55:12.300 --> 00:55:14.710
mass moments of inertia
about a point that's
00:55:14.710 --> 00:55:16.730
not through the center of mass.
00:55:16.730 --> 00:55:21.300
I know you've seen this
before in [? 8.01, ?]
00:55:21.300 --> 00:55:23.435
so this is going to
be a quick reminder.
00:55:26.500 --> 00:55:29.090
But I'll show you
where it comes from.
00:55:29.090 --> 00:55:30.070
So here's my stick.
00:55:39.980 --> 00:55:48.190
And it has a center of mass, and
that's where G is located here.
00:55:48.190 --> 00:55:50.155
It has a total length l.
00:55:53.260 --> 00:56:04.220
I'm going to give it a
thickness b, a width a.
00:56:04.220 --> 00:56:07.780
So it's a stick.
00:56:07.780 --> 00:56:12.120
a wide, b thick, l long.
00:56:12.120 --> 00:56:15.720
Uniform has a center
gravity right in the middle.
00:56:15.720 --> 00:56:23.130
And I'm going to
attach to this stick--
00:56:23.130 --> 00:56:25.080
and this point is
kind of hard to draw.
00:56:25.080 --> 00:56:28.140
This point is at the
center of the stick, OK?
00:56:28.140 --> 00:56:30.750
I'm going to put my
coordinate system attached
00:56:30.750 --> 00:56:33.730
at the center of
gravity, center of mass,
00:56:33.730 --> 00:56:40.060
and I'm going to make it
the-- that's x prime downward,
00:56:40.060 --> 00:56:45.680
z prime, and y prime is
then going off that way.
00:56:45.680 --> 00:56:47.960
So this is a body
set of coordinates
00:56:47.960 --> 00:56:49.790
at the center of mass.
00:56:49.790 --> 00:56:51.700
x prime, y prime, z prime.
00:56:51.700 --> 00:56:53.890
And x happens to be down.
00:56:53.890 --> 00:56:58.270
And I want to calculate
my mass moment of inertia
00:56:58.270 --> 00:57:06.050
with respect to a point up
here that is d, this distance.
00:57:06.050 --> 00:57:10.750
I've moved up the
x-axis an amount d.
00:57:10.750 --> 00:57:14.120
I'm going to set a new
coordinate system up here.
00:57:14.120 --> 00:57:18.010
So if this was z prime,
my new z is here.
00:57:18.010 --> 00:57:19.370
It's getting a little messy.
00:57:19.370 --> 00:57:24.220
Maybe I'll do just a face view.
00:57:27.660 --> 00:57:33.400
If my previously y
prime and x prime
00:57:33.400 --> 00:57:36.120
were like that, z
coming out of the board,
00:57:36.120 --> 00:57:39.420
now I have a new
system that is y
00:57:39.420 --> 00:57:42.960
and x like this, z still
coming out of the board.
00:57:47.350 --> 00:57:48.856
Now the coordinate.
00:57:51.500 --> 00:57:56.500
So how do I calculate
mass moment of inertia?
00:57:56.500 --> 00:58:02.670
Well I want Izz.
00:58:06.400 --> 00:58:08.770
I probably know Iz'z'.
00:58:08.770 --> 00:58:13.480
Iz'z' is the mass moment of
inertia about this point.
00:58:13.480 --> 00:58:15.790
I know it's a principal
axis from all the things we
00:58:15.790 --> 00:58:18.290
just-- that square block
is the same as this.
00:58:18.290 --> 00:58:20.590
That's a principal axis
in the z prime direction.
00:58:20.590 --> 00:58:24.715
I know the Izz'
with respect to G,
00:58:24.715 --> 00:58:27.960
I want to know what with
respect to this point.
00:58:27.960 --> 00:58:33.510
So well Izz, which
is my new location
00:58:33.510 --> 00:58:40.700
up here, and we'll call it A.
So Izz here with respect to A
00:58:40.700 --> 00:58:44.609
is the integral of r squared dm.
00:58:44.609 --> 00:58:46.192
We've got to do the
same integral now.
00:58:48.980 --> 00:58:57.515
But that's the integral of
x squared plus y squared dm.
00:59:01.300 --> 00:59:04.420
Now I can look at this and
I can say oh, well, these
00:59:04.420 --> 00:59:07.440
are d-- this is separated by d.
00:59:07.440 --> 00:59:08.740
I only moved it in the x.
00:59:08.740 --> 00:59:11.810
The ys haven't moved and
the zs didn't change.
00:59:11.810 --> 00:59:15.360
I just moved my point
only in the x direction.
00:59:15.360 --> 00:59:20.820
So I can now say that in
terms of my new coordinate,
00:59:20.820 --> 00:59:26.670
it's the same as x prime plus
d, the distance from here
00:59:26.670 --> 00:59:32.750
to a point down her,
some arbitrary mass point
00:59:32.750 --> 00:59:38.500
xi is going to be xi' plus d.
00:59:38.500 --> 00:59:43.390
So to do this integral
in the new coordinates,
00:59:43.390 --> 00:59:51.070
this is going to be the integral
of x prime plus d squared
00:59:51.070 --> 00:59:53.560
plus y.
00:59:53.560 --> 01:00:01.714
Now y prime equals y
and z prime equals z.
01:00:01.714 --> 01:00:02.630
Those haven't changed.
01:00:02.630 --> 01:00:05.720
I didn't move my new
coordinate system
01:00:05.720 --> 01:00:07.940
in the y direction
or the z direction.
01:00:07.940 --> 01:00:11.200
So the coordinate in
the new system in y
01:00:11.200 --> 01:00:12.390
is the same as before.
01:00:12.390 --> 01:00:15.030
So this is just y prime squared.
01:00:15.030 --> 01:00:18.910
And this whole thing
times d, integrated times
01:00:18.910 --> 01:00:21.835
every little mass bit.
01:00:21.835 --> 01:00:26.690
If I square this, I get x
prime squared, 2x'd, d squared,
01:00:26.690 --> 01:00:27.960
plus y squared.
01:00:27.960 --> 01:00:31.870
So this integral,
Izz with respect
01:00:31.870 --> 01:00:35.500
to A, when you
rearrange it, looks
01:00:35.500 --> 01:00:43.780
like x prime squared
plus y prime squared dm
01:00:43.780 --> 01:00:47.690
and the integral of a sum
is the sum of the integrals.
01:00:47.690 --> 01:00:49.730
So I break it into
bits here, there's
01:00:49.730 --> 01:00:53.340
a d squared, which
is a constant, dm.
01:00:53.340 --> 01:01:02.252
And then the last term is
plus 2d and it's x prime dm.
01:01:05.410 --> 01:01:08.550
Just multiply this out,
rewrite it, break it apart.
01:01:11.932 --> 01:01:12.890
Well let's do this one.
01:01:12.890 --> 01:01:16.360
Integral of x prime squared
plus y prime squared dm.
01:01:16.360 --> 01:01:20.620
That's something that we
already have a name for.
01:01:20.620 --> 01:01:22.175
This is IGzz.
01:01:27.640 --> 01:01:30.450
It's the original
mass moment of inertia
01:01:30.450 --> 01:01:34.980
with respect to the original
coordinate system at G
01:01:34.980 --> 01:01:35.950
in the z direction.
01:01:35.950 --> 01:01:39.810
So it's [? IzzG, ?]
we already know that.
01:01:39.810 --> 01:01:41.840
That's given for the object.
01:01:41.840 --> 01:01:45.180
Plus, this is the integral
of dm over the whole extent
01:01:45.180 --> 01:01:47.654
of the object?
01:01:47.654 --> 01:01:48.820
Just the mass of the object.
01:01:53.320 --> 01:01:58.330
This integral, this
is the integral
01:01:58.330 --> 01:02:00.990
in terms of the x-coordinate.
01:02:00.990 --> 01:02:07.210
And every mass bit from here,
if I go out here and find one,
01:02:07.210 --> 01:02:09.470
there's an equal and
opposite one up here.
01:02:09.470 --> 01:02:11.840
This is the definition
of the center of mass.
01:02:11.840 --> 01:02:14.540
This integral, if I'm
at the center of mass
01:02:14.540 --> 01:02:18.890
integrating out from it, this is
zero because of the definition
01:02:18.890 --> 01:02:21.220
the center of mass.
01:02:21.220 --> 01:02:26.540
And I've just proven the
parallel axis theorem.
01:02:26.540 --> 01:02:32.740
Izz about this new point is
I about G plus Md Squared,
01:02:32.740 --> 01:02:35.710
where d squared is
the distance I've
01:02:35.710 --> 01:02:39.160
moved this z-axis to a
new place parallel to it.
01:03:41.040 --> 01:03:48.710
So Izz with respect to G,
the original mass moment
01:03:48.710 --> 01:04:01.630
of inertia for Izz is m L
squared plus a squared over 12.
01:04:04.450 --> 01:04:20.570
And Ixx m L squared
plus b squared over 12.
01:04:20.570 --> 01:04:29.420
And Izz-- oh, we
already know that one.
01:04:32.980 --> 01:04:34.460
Wait a minute.
01:04:34.460 --> 01:04:36.440
I haven't told you what that is.
01:04:36.440 --> 01:04:38.510
That's Izz, Ixx, Iyy.
01:04:38.510 --> 01:04:42.180
Just a little messy here.
01:04:42.180 --> 01:04:56.990
Iyy for this problem is--
I have made a mistake.
01:04:56.990 --> 01:04:59.750
Ixx is a squared plus b squared.
01:04:59.750 --> 01:05:08.660
Iyy is m L squared
plus b squared.
01:05:08.660 --> 01:05:10.940
So those are the
three-- all with respect
01:05:10.940 --> 01:05:13.880
to G-- for this stick stick.
01:05:13.880 --> 01:05:14.785
And I'm going to--
01:05:17.550 --> 01:05:19.014
AUDIENCE: [INAUDIBLE]?
01:05:19.014 --> 01:05:20.788
Is that also divided by 12?
01:05:20.788 --> 01:05:21.454
PROFESSOR: Yeah.
01:05:28.800 --> 01:05:33.190
That kind of sets us up where
I can pick up next time.
01:05:33.190 --> 01:05:42.800
So let's finish by asking
ourselves the question, what
01:05:42.800 --> 01:05:45.770
do we think about-- if
I've moved to this new put
01:05:45.770 --> 01:05:52.320
new position, and I'm not
rotating about the center,
01:05:52.320 --> 01:05:58.000
is this new axis-- this one
around the center before,
01:05:58.000 --> 01:06:01.840
that one we know is
a principal axis.
01:06:01.840 --> 01:06:03.990
If I rotate about
this new place which
01:06:03.990 --> 01:06:08.650
I've defined the mass moment
of inertia about that place?
01:06:08.650 --> 01:06:10.454
Is this a principal axis?
01:06:10.454 --> 01:06:11.342
AUDIENCE: Yes.
01:06:11.342 --> 01:06:12.230
AUDIENCE: Yes.
01:06:12.230 --> 01:06:15.320
PROFESSOR: How many think yes?
01:06:15.320 --> 01:06:18.330
How many think no?
01:06:18.330 --> 01:06:22.410
OK, so lots of people not sure.
01:06:22.410 --> 01:06:27.930
So my dynamic definition
of principal axis
01:06:27.930 --> 01:06:31.580
is if you can rotate the
object about that axis
01:06:31.580 --> 01:06:36.280
and produce no unbalanced
torques, it's a principal axis.
01:06:36.280 --> 01:06:39.180
And I can do that and this thing
will just spin all day long.
01:06:39.180 --> 01:06:43.390
Now there is a force, you could
think of a fictitious force.
01:06:43.390 --> 01:06:45.840
There's a center
of mass out here.
01:06:45.840 --> 01:06:49.610
As it spins around, there's
a centripetal acceleration
01:06:49.610 --> 01:06:53.270
making it go in the circle, that
means that fictitious force is
01:06:53.270 --> 01:06:56.790
like there is a centrifugal
force pulling out on it.
01:06:56.790 --> 01:06:57.960
Do I feel that?
01:06:57.960 --> 01:07:00.176
Do I have to this
resist that force
01:07:00.176 --> 01:07:01.300
as it goes round and round?
01:07:01.300 --> 01:07:02.530
Yes.
01:07:02.530 --> 01:07:05.960
So that is an
unbalance of a kind
01:07:05.960 --> 01:07:09.600
we know as a static imbalance,
but it doesn't produce torques
01:07:09.600 --> 01:07:17.118
about my axis right
lined up on the center.
01:07:17.118 --> 01:07:19.986
AUDIENCE: Doesn't gravity
pull on the center of mass
01:07:19.986 --> 01:07:21.541
[INAUDIBLE].
01:07:21.541 --> 01:07:23.415
PROFESSOR: Sure, gravity
does, but that's now
01:07:23.415 --> 01:07:24.248
a different problem.
01:07:24.248 --> 01:07:28.240
That's what makes this thing
act like an oscillator.
01:07:28.240 --> 01:07:30.040
The torques of the
kind I'm talking about
01:07:30.040 --> 01:07:34.390
is if I compute the angular
momentum of this thing
01:07:34.390 --> 01:07:38.590
and compute dh/dt-- the time
rate of change of the angular
01:07:38.590 --> 01:07:41.430
momentum is a torque
on the system, right?
01:07:41.430 --> 01:07:44.680
I will get the term that
makes it spin faster,
01:07:44.680 --> 01:07:46.640
and I will get, if
they exist, terms
01:07:46.640 --> 01:07:51.640
that make it want to bend
this way or bend back.
01:07:51.640 --> 01:07:58.650
It only happens if-- if I hold
this thing over here, and spin
01:07:58.650 --> 01:08:02.790
it, get it spinning, and I
compute the angle momentum
01:08:02.790 --> 01:08:06.030
with respect to this
point, will I get torques?
01:08:06.030 --> 01:08:12.360
Yeah, but that's not how I--
that's a different problem.
01:08:12.360 --> 01:08:18.569
The IG is as if I were
computing the angular momentum.
01:08:18.569 --> 01:08:22.660
Remember I started defining
mass moment of inertia matrix
01:08:22.660 --> 01:08:26.520
based on an angular
momentum computation at G.
01:08:26.520 --> 01:08:29.029
So it's right there in
the center of this object,
01:08:29.029 --> 01:08:31.560
there's no moment
arm that is causing
01:08:31.560 --> 01:08:35.830
torques that's trying to twist
this thing about that point.
01:08:35.830 --> 01:08:39.952
So the answer to the question
is this is a principal axis.
01:08:39.952 --> 01:08:40.859
Yeah?
01:08:40.859 --> 01:08:44.446
AUDIENCE: So if you take the
derivative of the angular
01:08:44.446 --> 01:08:49.370
momentum would you get torques
that are not in that direction?
01:08:49.370 --> 01:08:52.450
PROFESSOR: If you get torques
that aren't in that direction,
01:08:52.450 --> 01:08:54.979
either you made a
mistake doing the math,
01:08:54.979 --> 01:08:59.140
or you were in
error in identifying
01:08:59.140 --> 01:09:02.055
the mass moment of inertia
matrix to begin with.
01:09:02.055 --> 01:09:03.430
Because if you
get torques, there
01:09:03.430 --> 01:09:07.270
must be non-zero off-diagonal
terms and in mass moment
01:09:07.270 --> 01:09:08.109
of inertia matrix.
01:09:08.109 --> 01:09:12.140
They are what account
for the torques.
01:09:12.140 --> 01:09:15.420
So by this parallel axis
theorem, any other axis
01:09:15.420 --> 01:09:18.569
you go to-- if you started
at a principal axis,
01:09:18.569 --> 01:09:23.149
any other axis you create
is also a principal axis.
01:09:23.149 --> 01:09:26.870
That's the movement
of just one axis.
01:09:26.870 --> 01:09:30.490
If you do two, if you move
this way and this way,
01:09:30.490 --> 01:09:31.220
all bets are off.
01:09:31.220 --> 01:09:32.386
You get a difference answer.
01:09:32.386 --> 01:09:34.870
And if you're interested in
that more complicated problem,
01:09:34.870 --> 01:09:37.250
read that Williams
thing because he
01:09:37.250 --> 01:09:40.359
does the complete parallel
axis, parallel planes
01:09:40.359 --> 01:09:43.410
and comes up with a
super compact little way
01:09:43.410 --> 01:09:45.740
of calculating them.
01:09:45.740 --> 01:09:47.690
See you on Thursday.