1 00:00:00,420 --> 00:00:02,760 ‫However, that is not all. 2 00:00:03,210 --> 00:00:08,470 ‫Did you notice that there were a lot of zeros in the inertia matrix? 3 00:00:08,970 --> 00:00:13,770 ‫Well, in fact, the most general case is this. 4 00:00:14,700 --> 00:00:19,710 ‫This inertia matrix is actually called the inertia tensor. 5 00:00:20,250 --> 00:00:22,830 ‫All right, the inertia tensor. 6 00:00:23,250 --> 00:00:26,580 ‫And so what are these red terms then? 7 00:00:26,970 --> 00:00:31,470 ‫They're called products of inertia. 8 00:00:31,920 --> 00:00:40,860 ‫The product of inertia is a measure that captures and objects mass symmetry about a certain axis. 9 00:01:00,940 --> 00:01:02,710 ‫So what is mass symmetry? 10 00:01:03,040 --> 00:01:11,410 ‫Let's take a simple example, let's take a hollow tube here and let's call this axis as the axis. 11 00:01:11,860 --> 00:01:20,800 ‫And so this axis goes through the tube and so the tube is empty inside and all the mass is concentrated 12 00:01:21,010 --> 00:01:23,920 ‫in the edges here and this tube. 13 00:01:24,190 --> 00:01:26,760 ‫Let's assume that it's homogeneous. 14 00:01:27,280 --> 00:01:36,780 ‫It means that its density is constant and density was kg per meter cute. 15 00:01:37,510 --> 00:01:44,250 ‫It means then mass is evenly distributed along the entire tube. 16 00:01:44,590 --> 00:01:50,250 ‫So wherever you have matter, the mass there is evenly distributed. 17 00:01:50,950 --> 00:01:56,340 ‫So the density is the same here and here and here everywhere. 18 00:01:56,920 --> 00:02:02,280 ‫And so the mass moment of inertia depends on two things. 19 00:02:02,650 --> 00:02:07,300 ‫It first of all, depends on how much mass you have in our body. 20 00:02:07,900 --> 00:02:13,810 ‫But it also depends how far that mass is from the object. 21 00:02:14,530 --> 00:02:20,440 ‫The more mass there is in the object, the more kilograms you have. 22 00:02:20,740 --> 00:02:27,970 ‫Then obviously that contributes to more mass movement of inertia about this Z axis. 23 00:02:28,480 --> 00:02:34,210 ‫And the less mass you have, in other words, the less kilograms you have, the less mass moment of 24 00:02:34,210 --> 00:02:35,190 ‫inertia you have. 25 00:02:35,770 --> 00:02:45,340 ‫But then also the further away from the axis this mass is, the higher your mass moment of inertia. 26 00:02:45,880 --> 00:02:52,730 ‫And the closer the mass is concentrated to the axis, the less mass movement of inertia you have. 27 00:02:53,230 --> 00:03:02,890 ‫So just as a quick side note, if I have another tube here and it's not hello, it's completely filled 28 00:03:03,010 --> 00:03:06,070 ‫inside and there masses are equal. 29 00:03:06,070 --> 00:03:14,380 ‫Let's say this one is five kilograms and also this one is five kilograms. 30 00:03:15,040 --> 00:03:17,660 ‫And both of them have the same diameter. 31 00:03:17,950 --> 00:03:24,820 ‫For example, this one has the diameter of zero point three meters. 32 00:03:25,180 --> 00:03:31,260 ‫And also this one has the diameter of zero point three meters. 33 00:03:31,720 --> 00:03:35,110 ‫And so now I have a small exercise for you. 34 00:03:35,590 --> 00:03:43,620 ‫You have two ramps and in the first ramp you have this hollow tube that has five kilograms of mass. 35 00:03:44,110 --> 00:03:53,440 ‫And in the second ramp you have this completely filled tube and both of them have the same angle and 36 00:03:53,440 --> 00:03:56,950 ‫the same distance from this point to this point. 37 00:03:57,400 --> 00:03:59,950 ‫And now your exercise is the following. 38 00:04:00,340 --> 00:04:04,060 ‫Which tube will reach the ground faster? 39 00:04:04,750 --> 00:04:07,990 ‫Both of them will start rolling at the same time. 40 00:04:08,500 --> 00:04:11,800 ‫Time equals, let's say, zero seconds and it doesn't matter. 41 00:04:12,110 --> 00:04:13,450 ‫Don't think about time. 42 00:04:13,870 --> 00:04:16,270 ‫Just think in terms of concepts. 43 00:04:16,840 --> 00:04:19,540 ‫Both of them start rolling at the same time. 44 00:04:19,810 --> 00:04:27,910 ‫And which tube will reach the ground sooner, considering that you have the same distance to roll and 45 00:04:27,910 --> 00:04:31,990 ‫the angles are the same, which tube will win the race? 46 00:04:32,410 --> 00:04:36,070 ‫So try it yourself and let's talk about it in the next video.