1
00:00:03,760 --> 00:00:06,220
I would now like to calculate
the moment of inertia

2
00:00:06,220 --> 00:00:08,245
of a uniform sphere.

3
00:00:12,440 --> 00:00:15,390
And it has a mass
m and radius r.

4
00:00:15,390 --> 00:00:18,810
I'm going to look at three axes.

5
00:00:18,810 --> 00:00:24,640
So I'll call this the x-axis,
the y-axis, and the z-axis.

6
00:00:24,640 --> 00:00:29,130
And first, let's calculate
the moment about the z-axis.

7
00:00:29,130 --> 00:00:31,770
So if I write down
our definition,

8
00:00:31,770 --> 00:00:34,470
and I'm going to calculate
it about the center of mass,

9
00:00:34,470 --> 00:00:38,110
so the moment about the z-axis.

10
00:00:38,110 --> 00:00:39,040
How do we do that?

11
00:00:39,040 --> 00:00:42,270
Well, we take a mass element,
and have to be a little bit

12
00:00:42,270 --> 00:00:43,920
careful here.

13
00:00:43,920 --> 00:00:46,680
Because if you
think about what is

14
00:00:46,680 --> 00:00:49,890
our [? perp ?] for
this mass element,

15
00:00:49,890 --> 00:00:54,870
it's actually x squared
plus y squared, squared.

16
00:00:54,870 --> 00:00:58,230
So the distance here, because
it's going in a circle,

17
00:00:58,230 --> 00:01:02,200
the radius of that circle
is x squared plus y squared.

18
00:01:02,200 --> 00:01:05,160
So we have x squared
plus y squared,

19
00:01:05,160 --> 00:01:07,620
and we're integrating
over the sphere.

20
00:01:07,620 --> 00:01:10,050
Now, this looks like
a tough integral.

21
00:01:10,050 --> 00:01:13,210
But let's now look at what
would the moment of inertia

22
00:01:13,210 --> 00:01:15,820
be about the x-axis?

23
00:01:15,820 --> 00:01:19,320
Well, the only difference here
is I'm integrating now, first,

24
00:01:19,320 --> 00:01:21,780
if it's rotating
about this axis,

25
00:01:21,780 --> 00:01:24,390
and I had my mass
element, instead

26
00:01:24,390 --> 00:01:27,330
of x squared plus y
squared about the z-axis,

27
00:01:27,330 --> 00:01:31,890
it's y squared plus z
squared about the x-axis.

28
00:01:31,890 --> 00:01:34,570
That's the
perpendicular distance.

29
00:01:34,570 --> 00:01:38,820
And if I calculate the moment
of inertia about the y-axis,

30
00:01:38,820 --> 00:01:44,970
then same argument-- I have
z squared plus x squared.

31
00:01:44,970 --> 00:01:47,931
Now, the beauty
of this problem--

32
00:01:47,931 --> 00:01:49,680
and in physics, when
we talk about beauty,

33
00:01:49,680 --> 00:01:51,630
we often talk
about symmetry-- is

34
00:01:51,630 --> 00:01:55,259
that by the symmetry of the
sphere, all of these moments

35
00:01:55,259 --> 00:01:55,800
are equal.

36
00:02:00,460 --> 00:02:05,320
And let's call that I cm.

37
00:02:05,320 --> 00:02:11,350
Now, if I add these three
pieces together, what do I get?

38
00:02:11,350 --> 00:02:19,900
So if I add I cm z plus
I cm x plus I cm y,

39
00:02:19,900 --> 00:02:24,620
I get 3 times the moment
about the center of the axes.

40
00:02:24,620 --> 00:02:28,000
So what happens when I
add these three integrals?

41
00:02:28,000 --> 00:02:29,170
I get dm.

42
00:02:29,170 --> 00:02:31,300
If you'll notice, x
squared appears twice,

43
00:02:31,300 --> 00:02:35,150
y squared appears twice,
and z squared appears twice.

44
00:02:35,150 --> 00:02:41,740
So we get 2 times x squared
plus y squared plus z squared.

45
00:02:41,740 --> 00:02:45,430
But x squared plus y
squared plus z squared

46
00:02:45,430 --> 00:02:55,900
is the radius of a small
sphere of thickness dr.

47
00:02:55,900 --> 00:03:00,790
And my mass element-- now I have
to integrate over the sphere.

48
00:03:00,790 --> 00:03:07,750
And so now our mass element,
dm, is the volume density

49
00:03:07,750 --> 00:03:14,370
of this times the volume.

50
00:03:14,370 --> 00:03:16,260
Now what is the volume density?

51
00:03:16,260 --> 00:03:21,079
Well, that's the total
mass over 4/3 pi r cubed.

52
00:03:21,079 --> 00:03:23,150
And what is the
volume of a sphere

53
00:03:23,150 --> 00:03:26,150
of radius r and thickness dr?

54
00:03:26,150 --> 00:03:32,120
That's 4 pi r squared dr.

55
00:03:32,120 --> 00:03:37,220
So if I put that into
my expression, what

56
00:03:37,220 --> 00:03:39,380
I get-- let's just
get rid of the 4

57
00:03:39,380 --> 00:03:49,550
pi's, and I get 3m over r
cubed times r squared dr.

58
00:03:49,550 --> 00:04:01,580
And so our 3 I cm, I
have a factor of 2,

59
00:04:01,580 --> 00:04:10,550
this integral becomes 2
times dm times r squared.

60
00:04:10,550 --> 00:04:18,200
And the dm is 3m r
cubed r squares dr times

61
00:04:18,200 --> 00:04:20,190
another r squared.

62
00:04:20,190 --> 00:04:24,220
And what are we integrating
over are r variable from?

63
00:04:24,220 --> 00:04:28,120
These spherical shells that
we're integrating outward

64
00:04:28,120 --> 00:04:32,450
go from r equal 0 to
r equal capital R.

65
00:04:32,450 --> 00:04:36,800
Notice that our 3 m's cancel.

66
00:04:36,800 --> 00:04:40,490
And we'll just
write this is I cm

67
00:04:40,490 --> 00:04:49,340
equals factor 2 times m over r
cubed times the integral of r

68
00:04:49,340 --> 00:04:53,930
to the fourth dr from zero to
R. And that's a simple integral

69
00:04:53,930 --> 00:04:54,430
to do.

70
00:04:54,430 --> 00:05:00,840
r to the fourth is r to
the fifth divided by 5.

71
00:05:00,840 --> 00:05:08,270
And so we get 2 over 5 m R
to the fifth over R cubed.

72
00:05:08,270 --> 00:05:10,760
And we conclude that
the moment of inertia

73
00:05:10,760 --> 00:05:18,470
about any of the axes of the
sphere is 2/5 m R squared.

74
00:05:18,470 --> 00:05:21,740
And in this calculation,
it's a beautiful example

75
00:05:21,740 --> 00:05:25,130
of how we use the symmetry
of the sphere to simplify

76
00:05:25,130 --> 00:05:28,030
very complicated integrals.