1 00:00:00,270 --> 00:00:06,280 ‫And so I also want to give you general formulas for mass moment of inertia. 2 00:00:06,690 --> 00:00:09,450 ‫I don't want to stop here for a long time. 3 00:00:09,900 --> 00:00:15,420 ‫However, you can read something out of these formulas. 4 00:00:15,690 --> 00:00:23,760 ‫In fact, if you think about it, it's a pretty good art and it's pretty good skill to know how to translate 5 00:00:23,760 --> 00:00:27,250 ‫mathematical formulas into a human language. 6 00:00:27,690 --> 00:00:31,110 ‫So if you look at these formulas, what do you see? 7 00:00:31,530 --> 00:00:33,030 ‫Will you see an integral? 8 00:00:33,510 --> 00:00:36,140 ‫And that means that you have continuous summation. 9 00:00:36,600 --> 00:00:43,890 ‫So in our case, let's focus on this guy here, because this is what we are computing, where computing 10 00:00:43,890 --> 00:00:47,100 ‫mass moment of inertia about the Z axis. 11 00:00:47,590 --> 00:00:57,600 ‫And so you have this D.M., which is this infinitesimal mass here, and then you have this are Z squared. 12 00:00:57,990 --> 00:01:04,630 ‫And that's the radius of that particular D.M. from the Axis. 13 00:01:05,280 --> 00:01:14,940 ‫And so essentially what is happening here is that this integral goes through this entire shape, takes 14 00:01:15,240 --> 00:01:23,690 ‫each individual D.M. let's say D.M. one here and then later maybe it's D.M. two here. 15 00:01:24,270 --> 00:01:30,510 ‫And for each the M, it measures the distance from the Z axis. 16 00:01:31,050 --> 00:01:40,710 ‫Now notice that you have squared here, which means that the effect of increasing the distance of that 17 00:01:40,710 --> 00:01:45,960 ‫small mass element from the axis is not linear, it's squared. 18 00:01:46,440 --> 00:01:56,400 ‫So if you increase the distances of all the DM's from their axis by two times, then the mass movement 19 00:01:56,400 --> 00:02:01,650 ‫of inertia about this axis will not increase by two times. 20 00:02:02,070 --> 00:02:04,860 ‫It will increase more by four times. 21 00:02:05,340 --> 00:02:14,550 ‫And so you go through the entire shape and you sum up each product of an infinitesimal mass element 22 00:02:15,000 --> 00:02:18,910 ‫and its distance squared from the axis. 23 00:02:19,440 --> 00:02:26,610 ‫All these products you sum up through continuous summation, which is the integral, and that's how 24 00:02:26,610 --> 00:02:28,870 ‫you will get your mass moment of inertia. 25 00:02:29,520 --> 00:02:38,880 ‫And if this one here was the Z axis like you see here, well, then here you will have the X axis and 26 00:02:38,880 --> 00:02:40,850 ‫here you will have the Y axis. 27 00:02:41,250 --> 00:02:54,120 ‫And if that's the case, then you can rewrite this R Z squared like this because R Z equals square root 28 00:02:54,120 --> 00:02:59,400 ‫of X squared plus Y squared is the Pythagorean theorem. 29 00:02:59,400 --> 00:02:59,730 ‫Right. 30 00:03:00,150 --> 00:03:08,970 ‫So this distance here can be computed by taking this side here and this side here. 31 00:03:09,510 --> 00:03:18,090 ‫So this would be X and this would be Y, and then you compute this length here using the Pythagorean 32 00:03:18,090 --> 00:03:19,700 ‫theorem like this. 33 00:03:20,070 --> 00:03:28,290 ‫And since you have your R Z squared here, then instead of this, you can write like this and that will 34 00:03:28,290 --> 00:03:31,890 ‫give you X squared plus Y squared. 35 00:03:32,460 --> 00:03:35,310 ‫And then D.M. can be a constant number. 36 00:03:35,700 --> 00:03:45,630 ‫Or if you don't have a density that is constant, then you can rewrite D.M. Like this, the M equals 37 00:03:46,170 --> 00:03:55,410 ‫the density times, the V where D.V. is a differential volume, a differential volume essentially is 38 00:03:55,770 --> 00:03:59,520 ‫the X Times DUI times. 39 00:04:00,030 --> 00:04:09,420 ‫DEASEY So if you imagine a small, infinitesimal cube and then you have your DKs here, then you will 40 00:04:09,420 --> 00:04:18,120 ‫have the Y here and then these here, then this entire thing will be your differential volume. 41 00:04:18,690 --> 00:04:27,750 ‫And so you might have a case where your density is not constant, but rather it depends, for example, 42 00:04:28,260 --> 00:04:30,150 ‫on X and Y. 43 00:04:30,720 --> 00:04:35,520 ‫And then instead of D.M., you will just write this one here. 44 00:04:36,060 --> 00:04:42,630 ‫And the same concept would apply to this shape as well, even though here it would be more complicated 45 00:04:42,630 --> 00:04:46,230 ‫because your answer would change a lot. 46 00:04:46,710 --> 00:04:54,510 ‫And the same exact concept would apply to mass movement of inertia about the Y axis and the x axis. 47 00:04:55,320 --> 00:04:57,480 ‫Now, one more thing that I want to say. 48 00:04:57,930 --> 00:04:59,400 ‫If you take this D.M.. 49 00:04:59,970 --> 00:05:09,750 ‫And you write it down like this, the density that self depends on X and Y and perhaps even Z and 10 50 00:05:09,750 --> 00:05:17,370 ‫times the differential volume where the differential volume is nothing but the product of the DUI and 51 00:05:17,370 --> 00:05:18,180 ‫the Z. 52 00:05:18,810 --> 00:05:27,930 ‫And you know that this mass moment of inertia is computed by summing up all the products of this differential 53 00:05:27,930 --> 00:05:32,690 ‫element and its distance squared from the axis. 54 00:05:33,150 --> 00:05:44,580 ‫And now knowing this and this, you can rewrite this equation like this is easy, equals X squared plus 55 00:05:44,580 --> 00:05:45,750 ‫Y squared. 56 00:05:46,170 --> 00:05:56,580 ‫And now instead of D.M., I'm going to write this Times Road that depends on X, Y and Z, the most 57 00:05:56,580 --> 00:05:57,600 ‫general case. 58 00:05:57,960 --> 00:06:05,700 ‫And instead of Dve, I'm going to write the X Times DUI Times Dizzie. 59 00:06:06,180 --> 00:06:16,440 ‫And now you have three differential elements here, which means that this integral converts into a triple 60 00:06:16,440 --> 00:06:17,100 ‫integral. 61 00:06:17,760 --> 00:06:26,430 ‫So if you choose this approach, then it's like going through the entire object and taking an infinitesimally 62 00:06:26,430 --> 00:06:35,800 ‫small volume that you then multiply by the density at this specific location. 63 00:06:36,180 --> 00:06:43,260 ‫And remember now, since the density is not constant, then in every single location, the density might 64 00:06:43,260 --> 00:06:44,070 ‫be different. 65 00:06:44,610 --> 00:06:54,480 ‫So you take an infinitesimally small volume divi one and then you multiplied by a raw one, which is 66 00:06:54,480 --> 00:06:58,190 ‫the density valid only here. 67 00:06:58,770 --> 00:07:02,130 ‫And then you multiply it by the squared distance. 68 00:07:02,550 --> 00:07:07,730 ‫That is the distance from the axis to this particular point. 69 00:07:08,400 --> 00:07:14,850 ‫And you do it for all infinitesimally small volumes in this object. 70 00:07:15,490 --> 00:07:25,530 ‫But since you could rewrite DV as DKS Times divides, then you could express this entire equation like 71 00:07:25,530 --> 00:07:25,980 ‫this. 72 00:07:26,460 --> 00:07:32,840 ‫And from calculus, you know, or you can learn how to compute triple integrals. 73 00:07:33,330 --> 00:07:38,130 ‫And so that's how mathematically you can compute your mass moment of inertia. 74 00:07:38,820 --> 00:07:44,100 ‫But that's not all, because you also need to know how to compute the product of inertia.