WEBVTT

00:00:03.777 --> 00:00:05.860
We would now like to compare
the moment of inertia

00:00:05.860 --> 00:00:06.820
for a rigid body.

00:00:06.820 --> 00:00:11.950
Let's take an arbitrary rigid
body about the center of mass.

00:00:11.950 --> 00:00:16.770
So let's say the rigid body
is rotating about this axis.

00:00:16.770 --> 00:00:18.940
And what we'd like
to compare that is

00:00:18.940 --> 00:00:22.990
to the moment of inertia, say,
about a parallel axis that's

00:00:22.990 --> 00:00:25.270
also going through
the rigid body.

00:00:25.270 --> 00:00:28.720
Now, let's recall how we
define moment of inertia.

00:00:28.720 --> 00:00:31.750
We first choose a
mass element dm.

00:00:31.750 --> 00:00:33.670
What I'd like to show
is if this object

00:00:33.670 --> 00:00:37.990
is rotating about this axis,
then what is that object doing.

00:00:37.990 --> 00:00:40.150
Well, that object is
undergoing a little bit

00:00:40.150 --> 00:00:41.650
of circular motion.

00:00:41.650 --> 00:00:45.940
And this distance here is
what we call the perpendicular

00:00:45.940 --> 00:00:49.250
distance about that axis.

00:00:49.250 --> 00:00:53.590
And let's indicate this
is for our element dm.

00:00:53.590 --> 00:00:56.260
So this perpendicular
distance is

00:00:56.260 --> 00:01:00.580
what shows up in our definition
for the center of mass moment

00:01:00.580 --> 00:01:05.980
of inertia about that axis--
it's the interval of dm r cm

00:01:05.980 --> 00:01:07.030
perp.

00:01:07.030 --> 00:01:09.594
Now again, quantity squared.

00:01:09.594 --> 00:01:10.510
What is this distance?

00:01:10.510 --> 00:01:13.570
This is the perpendicular
distance from our dm

00:01:13.570 --> 00:01:15.850
and to the axis of rotation.

00:01:15.850 --> 00:01:18.160
Imagine it's doing
a circle and that's

00:01:18.160 --> 00:01:20.480
the radius of that circle.

00:01:20.480 --> 00:01:23.230
So if we were to calculate
the moment of inertia

00:01:23.230 --> 00:01:26.860
about another axis,
then about this

00:01:26.860 --> 00:01:31.430
axis the perpendicular
distance here

00:01:31.430 --> 00:01:35.332
that I'll write as rs perp.

00:01:35.332 --> 00:01:37.539
And you can see these
perpendicular distances are not

00:01:37.539 --> 00:01:38.660
the same.

00:01:38.660 --> 00:01:40.390
And the moment of
inertia about that

00:01:40.390 --> 00:01:48.190
other axis is equal to the
integral of dm rs perp quantity

00:01:48.190 --> 00:01:49.700
squared.

00:01:49.700 --> 00:01:54.759
Now, how do we relate those
perpendicular distances?

00:01:54.759 --> 00:01:56.830
Well, there's a couple
of ways to do it.

00:01:56.830 --> 00:02:01.420
And notice that the
distance between these axes

00:02:01.420 --> 00:02:03.460
is given by d.

00:02:03.460 --> 00:02:08.560
And I'm going to call
this the distance r cm.

00:02:08.560 --> 00:02:15.780
Now, let's just call this-- the
x direction-- I'll call that x.

00:02:15.780 --> 00:02:19.980
So how do I relate
these distances?

00:02:19.980 --> 00:02:25.960
Well, d is a fixed distance.

00:02:25.960 --> 00:02:29.990
And you can see from
my diagram that rs perp

00:02:29.990 --> 00:02:35.280
is equal to d plus r cm x.

00:02:35.280 --> 00:02:44.310
And if I square this, I
get d-squared plus 2d rcm x

00:02:44.310 --> 00:02:48.390
plus rcm x-squared.

00:02:48.390 --> 00:02:51.990
And that r cm x-squared
is precisely what we're

00:02:51.990 --> 00:02:56.800
calling perpendicular distance.

00:02:56.800 --> 00:03:01.290
So when I put those into
my moment of inertia Is,

00:03:01.290 --> 00:03:16.079
I get dm times d-squared plus 2d
times rcm x plus, parentheses,

00:03:16.079 --> 00:03:20.340
rcm perp squared.

00:03:20.340 --> 00:03:23.170
Now, I'll separate
this into three terms.

00:03:23.170 --> 00:03:27.120
The first term is
dm times d-squared.

00:03:27.120 --> 00:03:29.950
This is an integral
over the body.

00:03:29.950 --> 00:03:34.260
The second term is
2d-- and I'm going

00:03:34.260 --> 00:03:36.240
to hold off on the
interval, because the 2d is

00:03:36.240 --> 00:03:41.490
the same for every
piece-- dm rcm x.

00:03:41.490 --> 00:03:49.876
And the third piece is
integral over the body of dm

00:03:49.876 --> 00:03:54.880
r-- since rcm x is
the r perp, I'll

00:03:54.880 --> 00:03:57.600
write it as r perp squared.

00:03:57.600 --> 00:04:00.690
And you can see that
this term is precisely

00:04:00.690 --> 00:04:03.930
the moment of inertia
about the center of mass.

00:04:03.930 --> 00:04:06.000
Now, what I'd like
to focus on is

00:04:06.000 --> 00:04:11.280
this terribly, in
particular, dm rcm

00:04:11.280 --> 00:04:14.280
x that appears in our
integral expression.

00:04:14.280 --> 00:04:16.740
Recall, that we
define center of mass.

00:04:16.740 --> 00:04:24.330
We had the condition that
the sum of mj rcmj was 0.

00:04:24.330 --> 00:04:27.120
Now for an integral
relationship,

00:04:27.120 --> 00:04:34.170
this is dm rcmj cm equal to 0.

00:04:34.170 --> 00:04:37.409
So when you sum up the position
of every object with respect

00:04:37.409 --> 00:04:41.590
to the vector from the center
of mass to your dm element, 0.

00:04:41.590 --> 00:04:44.220
What does this say in
terms of components?

00:04:44.220 --> 00:04:47.040
In terms of components,
each component

00:04:47.040 --> 00:04:51.780
separately vanishes so we have
the condition that cm x is 0.

00:04:51.780 --> 00:04:57.530
So that term is 0, which is
precisely this term-- that's 0.

00:04:57.530 --> 00:05:01.710
And so we can
conclude that Is-- now

00:05:01.710 --> 00:05:06.460
in this term, where d is the
same piece for every object--

00:05:06.460 --> 00:05:09.130
so we're just pulling
out the total mass.

00:05:09.130 --> 00:05:12.250
So it's m total d-squared.

00:05:12.250 --> 00:05:14.880
And let's remind
ourselves that d

00:05:14.880 --> 00:05:17.760
is the distance between
the two parallel axes

00:05:17.760 --> 00:05:22.710
plus the moment of inertia
about the center of mass.

00:05:22.710 --> 00:05:29.660
And this result is called
the parallel axis theorem.