1
00:00:00,300 --> 00:00:01,110
‫Welcome back.

2
00:00:01,590 --> 00:00:08,460
‫So I hope you tried this exercise yourself and let's see which tube will win the race.

3
00:00:08,970 --> 00:00:15,000
‫Well, both of them have the same diameter and both of them have the same mass.

4
00:00:15,390 --> 00:00:15,840
‫Right.

5
00:00:16,050 --> 00:00:18,240
‫The diameter is zero point three meters.

6
00:00:18,240 --> 00:00:21,210
‫In both cases, in the mass is five kilograms.

7
00:00:21,540 --> 00:00:27,160
‫And so that means that both of them experience a force of gravity down.

8
00:00:27,660 --> 00:00:36,870
‫And of course, you can also decompose that force in the direction normal to the slope and parallel

9
00:00:36,870 --> 00:00:39,790
‫to the slope in the same thing here.

10
00:00:40,290 --> 00:00:47,530
‫So both of them will have the same component from the force of gravity that is parallel to the slope.

11
00:00:48,000 --> 00:00:56,100
‫However, because of the fact that in this case, where you have the hollow tube, all this mass is

12
00:00:56,100 --> 00:01:01,080
‫concentrated further away from this central axis.

13
00:01:01,350 --> 00:01:05,760
‫Because of that, it has a higher mass moment of inertia.

14
00:01:06,330 --> 00:01:09,920
‫Then this lower tube that is completely filled.

15
00:01:10,410 --> 00:01:15,480
‫And remember, mass moment of inertia is resistance to rotation.

16
00:01:15,780 --> 00:01:21,400
‫So the higher the mass movement of inertia, the more difficult it is to rotate.

17
00:01:21,840 --> 00:01:30,510
‫Therefore, this tube that is empty in the middle, it will start rolling slower than this tube that

18
00:01:30,510 --> 00:01:37,980
‫is completely filled because this to here that is completely filled, has less mass movement of inertia.

19
00:01:38,310 --> 00:01:43,830
‫So it will roll down faster and it will reach the finish line faster.

20
00:01:44,340 --> 00:01:48,810
‫And you can also look at it from the equations point of view.

21
00:01:49,050 --> 00:01:55,950
‫So you have this force of gravity component parallel to the ramp and you have some kind of distance

22
00:01:55,950 --> 00:01:56,370
‫here.

23
00:01:56,670 --> 00:02:03,420
‫So therefore you will have some kind of moment here and you will have the same moment here as well.

24
00:02:03,990 --> 00:02:10,890
‫So the moments are equal, but the mass moment of inertia here is bigger than it is here.

25
00:02:11,400 --> 00:02:20,970
‫And since the moments are equal, then that means that the angular acceleration here needs to be smaller

26
00:02:21,300 --> 00:02:24,230
‫and here it will be bigger.

27
00:02:24,450 --> 00:02:29,730
‫So this guy here will angular really accelerate down faster.

28
00:02:30,480 --> 00:02:35,620
‫But let's go back to the original point that I want to make mass symmetry.

29
00:02:36,060 --> 00:02:43,860
‫Now, you know that the mass movement of inertia depends on the mass and it depends how far away that

30
00:02:43,860 --> 00:02:50,410
‫mass is from the axis, about which you're measuring your mass movement of inertia.

31
00:02:50,940 --> 00:02:58,680
‫So you need to take into account two things mass and the distance of mass from the axis.

32
00:02:59,190 --> 00:03:04,830
‫In this tube, we said that the mass is distributed equally.

33
00:03:05,340 --> 00:03:09,110
‫So the density here is the same everywhere.

34
00:03:09,600 --> 00:03:15,630
‫But if you look at it, then the radius is also the same everywhere.

35
00:03:15,990 --> 00:03:24,300
‫And we assume that the width of this tube, the width of this wall here is also the same.

36
00:03:24,840 --> 00:03:29,030
‫So let's just put it W here and W here.

37
00:03:29,280 --> 00:03:35,480
‫So even though it doesn't appear like that here in the drawing, actually the width is the same everywhere.

38
00:03:36,150 --> 00:03:42,160
‫That means that this tube has the same radius about the entire Z axis.

39
00:03:42,750 --> 00:03:52,530
‫So that means that if you look at this tube from the side like this and you have the Z axis here and

40
00:03:52,530 --> 00:04:02,460
‫you put two lines here like this, then this entire mass is distributed equally on all sides of these

41
00:04:02,460 --> 00:04:03,030
‫lines.

42
00:04:03,660 --> 00:04:07,790
‫So in this case, this tube is mass symmetric.

43
00:04:08,340 --> 00:04:10,190
‫You can think of it like this.

44
00:04:10,440 --> 00:04:13,550
‫It has equal mass on all sides.

45
00:04:14,610 --> 00:04:26,670
‫And if that's the case about all axis, about the Z axis, x axis and Y axis, then all product of inertia

46
00:04:26,940 --> 00:04:32,360
‫will become zeros and you will have this matrix here.

47
00:04:32,940 --> 00:04:37,170
‫So you will only have these three elements here diagonally.

48
00:04:37,830 --> 00:04:48,030
‫However, if I start changing the densities in some parts of the tube or you have a completely different

49
00:04:48,030 --> 00:04:55,860
‫shape, something like this, and then let's say that this is your axis here and the mass is distributed

50
00:04:56,220 --> 00:04:58,110
‫equally in the walls.

51
00:04:58,800 --> 00:04:59,940
‫However, that this.

52
00:05:00,020 --> 00:05:08,250
‫This is from the Axis are different like this, and if you look at this object from the side and you

53
00:05:08,250 --> 00:05:16,260
‫have again these lines here, then you will see that on this side you will have less mass than, for

54
00:05:16,260 --> 00:05:18,430
‫example, on this side.

55
00:05:19,020 --> 00:05:24,180
‫So this object about this axis is not mass symmetric.

56
00:05:24,990 --> 00:05:31,830
‫So if you want to make this object mass symmetric, then you need to consider another axis somewhere.

57
00:05:32,340 --> 00:05:33,330
‫Maybe here.

58
00:05:33,900 --> 00:05:34,520
‫Who knows?

59
00:05:35,070 --> 00:05:36,600
‫It's a complicated shape.

60
00:05:37,140 --> 00:05:44,190
‫But in our case with this axis, this object is not mass symmetric about this axis.

61
00:05:44,730 --> 00:05:53,490
‫And when that's the case, then you have this matrix here with these products of inertia.

62
00:05:54,060 --> 00:06:01,770
‫That is, of course, if there is no mass symmetry about all the X, Y and Z axis, if that's the case,

63
00:06:02,190 --> 00:06:05,870
‫then there are no elements in this matrix that are zero.

64
00:06:06,420 --> 00:06:15,630
‫And by the way, from mathematics, when you compute products of inertia, you'll find that I x y always

65
00:06:15,630 --> 00:06:28,650
‫equals I y x, then I always equals two, i, x, z and I y, z always equals izi y.

66
00:06:29,190 --> 00:06:36,450
‫There are formulas to compute these products of inertia and when you follow those formulas, then that's

67
00:06:36,450 --> 00:06:39,170
‫the fact that you will find.

68
00:06:39,900 --> 00:06:47,550
‫And by the way, when you compute the mass movement of inertia in the product of inertia, then of course

69
00:06:47,550 --> 00:06:53,550
‫when you computed about the Axis Z, then you don't only look at the H here.

70
00:06:53,970 --> 00:06:56,670
‫You have to go inside the object.

71
00:06:56,850 --> 00:07:06,990
‫And then you also need to measure mass here and here and here and of course, its distance to this Z

72
00:07:07,020 --> 00:07:07,750
‫axis.

73
00:07:08,280 --> 00:07:16,080
‫So let's draw a Z axis here that goes through the tube or through this weird looking object.

74
00:07:16,510 --> 00:07:23,880
‫And so you need to take into account the distance of, for example, this mass to this axis here and

75
00:07:23,880 --> 00:07:25,620
‫here and here.

76
00:07:25,980 --> 00:07:35,100
‫And in fact, what it really comes down to is that you will have infinitesimally small mass elements

77
00:07:35,100 --> 00:07:40,860
‫called D.M. that have their own distance from the axis.

78
00:07:41,370 --> 00:07:47,240
‫And you have to sum up all the DBMS and the distances from the axis.

79
00:07:47,700 --> 00:07:55,950
‫So you can already see that when you calculate mass moments of inertia, products of inertia, then

80
00:07:55,950 --> 00:08:01,680
‫you're going to have to use integrals because integrals are continuous summations.

81
00:08:02,160 --> 00:08:09,420
‫You would be summing up those DBMS combined with the distance to the axis.