1 00:00:00,060 --> 00:00:06,660 So far, we have described the motion of a point mass basically particle located at a singular point 2 00:00:06,660 --> 00:00:11,250 in space on a rotational or on a circular trajectory. 3 00:00:12,120 --> 00:00:17,220 And in the end, we have calculated the rotational energy, which is the same as the kinetic energy. 4 00:00:17,580 --> 00:00:21,990 And we have found out that its m half times omega squared times are square. 5 00:00:23,490 --> 00:00:27,420 So far, we have considered only a single point or a single mass. 6 00:00:27,780 --> 00:00:35,460 So now in the following, we will discuss the motion of real objects that have a spatial extent, but 7 00:00:35,460 --> 00:00:40,320 we could also see that this is the motion of multiple point masses. 8 00:00:41,370 --> 00:00:47,670 So the idea that we are using here is that we are combining several of these point masses to form a 9 00:00:47,670 --> 00:00:49,350 whole object. 10 00:00:50,250 --> 00:00:56,460 And if we assume that all of these point passes belong to the same object, then the only thing that 11 00:00:56,460 --> 00:01:03,540 we have to make sure is true is that all of these point masses will have the same angular velocity omega. 12 00:01:04,709 --> 00:01:08,280 And the radius, for example, can differ in principle. 13 00:01:08,280 --> 00:01:13,200 Also, the mass can differ if we have some density distribution in the sample. 14 00:01:13,500 --> 00:01:16,780 But here in this lecture, we are not going to consider this. 15 00:01:16,800 --> 00:01:22,320 So here also the mass will be uniform, and so only the radius will differ here. 16 00:01:23,250 --> 00:01:31,290 And so what we can do then is we can write down the rotational energy basically as being m half times 17 00:01:31,680 --> 00:01:35,710 square times the sum over all of the position. 18 00:01:35,940 --> 00:01:43,800 So all of the the the distances, the radii square and typically in physics, this is written down with 19 00:01:43,800 --> 00:01:52,380 this eye constant here, which is the moment of his or typo moment of inertia. 20 00:01:53,760 --> 00:02:01,590 So the moment of inertia, as you can see by these equations, will be m times the sum over all the 21 00:02:01,590 --> 00:02:02,850 radii squared. 22 00:02:04,170 --> 00:02:10,440 So just one second, and please really try to understand what the RBI means here. 23 00:02:10,800 --> 00:02:18,000 This is really the distance of the point in space that we are considering at the moment with respect 24 00:02:18,120 --> 00:02:20,370 to the rotational axis. 25 00:02:21,240 --> 00:02:26,520 So it doesn't necessarily have to be some type of radius, but you will see this in a second, I think. 26 00:02:27,360 --> 00:02:30,810 So our first example will be a rotating stick. 27 00:02:31,200 --> 00:02:38,550 So you basically have a stick and you rotated around one of its ends and we try to calculate the moment 28 00:02:38,550 --> 00:02:39,630 of inertia. 29 00:02:40,200 --> 00:02:46,440 And you see the analytical result that you can take from a textbook will be one third of the mass times 30 00:02:46,440 --> 00:02:47,670 the length squared. 31 00:02:48,390 --> 00:02:54,000 So first of all, we must define the length and for simplicity, I will define that it's equal to one. 32 00:02:55,170 --> 00:03:01,410 And here you see already the analytical result that I have derived, but we will come back to this in 33 00:03:01,410 --> 00:03:01,830 a second. 34 00:03:01,860 --> 00:03:10,350 First of all, I want to plot this motion so we can, for example, take this plot and copy it and then 35 00:03:10,350 --> 00:03:13,380 just slightly modify what's what's going on here. 36 00:03:14,370 --> 00:03:23,520 So what we're going to have here is the X coordinate, the Y coordinate, and then we must create some 37 00:03:23,520 --> 00:03:28,350 list of points that belong to the stick, basically. 38 00:03:29,130 --> 00:03:30,720 So I would call this our list. 39 00:03:31,530 --> 00:03:40,740 So the R list is a num pi in space starting from zero and going to S, which is the total length. 40 00:03:41,280 --> 00:03:46,170 And now we must decide a numbers of points that we want to use. 41 00:03:47,490 --> 00:03:49,530 So I don't know. 42 00:03:49,560 --> 00:03:53,040 I would say, let's use 100 for the beginning. 43 00:03:53,250 --> 00:03:55,410 We can increase the number later on. 44 00:03:56,790 --> 00:03:57,990 So like this 45 00:04:00,810 --> 00:04:02,940 now I have to change this one here. 46 00:04:03,330 --> 00:04:12,300 So I write our list and then for the Y coordinate for the beginning, I just want to plot here two starting 47 00:04:12,300 --> 00:04:12,690 points. 48 00:04:12,730 --> 00:04:18,810 I want to say, let's assume that our stick is pointing along to along the X direction, so the Y coordinate 49 00:04:18,810 --> 00:04:19,620 will be zero. 50 00:04:20,070 --> 00:04:23,520 So we just have to generate here a list of zeros. 51 00:04:23,680 --> 00:04:25,170 And this you can do pretty easily. 52 00:04:25,290 --> 00:04:32,100 You could also do it by alien space, but it's even easier to just write NP zeros and then you just 53 00:04:32,100 --> 00:04:34,650 have to provide the number of points that you want to have. 54 00:04:34,950 --> 00:04:38,970 And of course, this must be the same number of points that you have in our list. 55 00:04:40,290 --> 00:04:46,680 So I think now I can run this, so we have the stick and I just want to change here. 56 00:04:46,680 --> 00:04:56,100 The limits of the plots x limb from minus one to one and the same goes for Y limits. 57 00:04:57,840 --> 00:04:59,250 So it will be like this. 58 00:05:00,350 --> 00:05:02,300 And here I did a mistake. 59 00:05:02,670 --> 00:05:06,050 Yeah, this is, of course, a thought, not an underscore. 60 00:05:06,620 --> 00:05:09,410 And you see, maybe we make this a bit larger. 61 00:05:14,130 --> 00:05:14,520 Yes. 62 00:05:15,600 --> 00:05:20,460 So you have to make it really precise, we should probably right, 1.2 times has. 63 00:05:21,240 --> 00:05:25,260 And then here as well, but in our case, it wouldn't matter because as this one. 64 00:05:26,010 --> 00:05:33,210 So you see, this is the stick, and over time, the stick will be rotated around the center coordinate. 65 00:05:33,780 --> 00:05:36,570 And now we have a way to really visualize this. 66 00:05:37,430 --> 00:05:41,940 So I want to make this new sound just so that it looks nicer. 67 00:05:42,450 --> 00:05:46,530 So, yeah, this will be the points and this will be the actual plot. 68 00:05:48,000 --> 00:05:48,480 All right. 69 00:05:49,110 --> 00:05:57,690 So before we use our numerical methods to calculate the moment of inertia, I I want to show you how 70 00:05:57,690 --> 00:06:00,630 you how you do it, when you want to do it analytically. 71 00:06:01,560 --> 00:06:07,770 So of course, a physicist would say doing it, doing it analytically is always the best thing because 72 00:06:07,770 --> 00:06:11,700 then you really understand what's going on and you get really the precise results. 73 00:06:12,300 --> 00:06:15,780 But you can see already here that it can be a bit difficult. 74 00:06:16,530 --> 00:06:23,190 So I will just briefly go over this because this is not a course about theoretical or mathematical physics, 75 00:06:23,190 --> 00:06:26,610 it's about Python and using Python for physics. 76 00:06:26,880 --> 00:06:31,260 But still, I think here it's quite nice if you understand what is happening. 77 00:06:31,710 --> 00:06:39,630 So you see the rotational energy we have to find before this was i of a two times I'll make a square. 78 00:06:39,930 --> 00:06:42,960 Let's go up here just to make sure here it is. 79 00:06:44,070 --> 00:06:51,000 But then we have also seen that the kinetic energy or the rotational energy for a single point mass 80 00:06:51,000 --> 00:06:55,680 was Imhoff times omega squared times our square. 81 00:06:56,730 --> 00:07:04,290 So what's happening here is we have to now integrate or add up all the points to describe the whole 82 00:07:04,290 --> 00:07:06,150 object that is rotating here. 83 00:07:06,690 --> 00:07:11,310 So it's basically one-half omega squared times to sum over all these points. 84 00:07:11,610 --> 00:07:13,650 And then, of course, over the radius square. 85 00:07:14,400 --> 00:07:16,130 So why do we want to write this as an integral? 86 00:07:16,140 --> 00:07:22,920 We have to write integral over the M, which is the integration of all infinitesimal elements of the 87 00:07:22,920 --> 00:07:27,300 mass and then the corresponding radius square of this element. 88 00:07:28,320 --> 00:07:34,470 So the item at the moment of inertia is then just the integral D.M. our square. 89 00:07:35,610 --> 00:07:42,600 So now you have to think of it, and this is maybe always a bit difficult because, yeah, you have 90 00:07:42,600 --> 00:07:48,750 to just think what is happening and how can you transform your variables to be able to solve this whole 91 00:07:48,750 --> 00:07:49,260 problem? 92 00:07:49,770 --> 00:07:56,610 And of course, the mass element will be equal to some kind of length density. 93 00:07:57,210 --> 00:07:59,430 So the sigma here is called a length. 94 00:07:59,430 --> 00:08:03,930 Density is just like the real density, which is mass divided by volume. 95 00:08:04,260 --> 00:08:09,840 But here, since we basically have a one dimensional problem and not three dimensional, it is mass 96 00:08:09,840 --> 00:08:11,940 divided by the length. 97 00:08:12,210 --> 00:08:17,910 So therefore, we call it only a length density, but it's a similar property to an actual density. 98 00:08:18,840 --> 00:08:25,920 So the dam will be just signaler times R and the Zygmunt itself, as I said, will be the mass, the 99 00:08:25,920 --> 00:08:29,010 total mass divided by the total length. 100 00:08:30,000 --> 00:08:40,440 So now we can just substitute this one in here, so we get integral Sigma Deora, our square and sigma 101 00:08:40,440 --> 00:08:47,250 is m over s and since M and S to not depend on our, we can just pull it out of the integral. 102 00:08:47,250 --> 00:08:53,160 So we have amassed times this integral D, our times are square. 103 00:08:54,330 --> 00:08:57,160 So this is a pretty easy integral to solve. 104 00:08:57,180 --> 00:09:01,980 You just have to think, OK, which function do I have to drive to end up with our square? 105 00:09:02,190 --> 00:09:04,130 And this is, of course, one third. 106 00:09:04,240 --> 00:09:06,150 Times are two to three. 107 00:09:06,810 --> 00:09:13,320 Then use set out and you plug in the limits so you go from R2 zero to our equal to s. 108 00:09:13,860 --> 00:09:21,450 So this term will be zero and you just substitute R with S and you get finally the result m of S times 109 00:09:21,450 --> 00:09:23,970 one third times as to the power of three. 110 00:09:24,840 --> 00:09:32,100 And then this s and this three cancels, or we just end up with one third and times as square. 111 00:09:33,480 --> 00:09:40,530 So this is one of the easiest analytical derivations of the moments of inertia for almost all other 112 00:09:40,530 --> 00:09:40,950 problems. 113 00:09:40,950 --> 00:09:42,300 It will be way more difficult. 114 00:09:42,330 --> 00:09:48,000 And I have to say, even here, it can be a bit difficult if you're doing this for the first time. 115 00:09:48,000 --> 00:09:53,220 And maybe if you're not so good with solving integrals and doing these transformations here, it can 116 00:09:53,220 --> 00:09:56,430 be quite tricky to yeah, to get the result. 117 00:09:57,270 --> 00:10:01,290 So what we are going to do here is we are just solving this numerically. 118 00:10:02,100 --> 00:10:06,030 And first of all, so that we can check if we did it correctly. 119 00:10:06,030 --> 00:10:11,460 Let me add here a cell and let me just calculate the value from the analytical results. 120 00:10:11,770 --> 00:10:21,310 Parametres, so this will basically just be one third times. 121 00:10:24,560 --> 00:10:28,900 Times, mass, Times Square. 122 00:10:28,990 --> 00:10:30,580 This will just be the value for AI. 123 00:10:31,000 --> 00:10:35,680 And then we could also calculate the value for the rotational energy, which is maybe better to compare 124 00:10:35,680 --> 00:10:35,800 it. 125 00:10:36,220 --> 00:10:45,180 So basically, we have to then divide by two and have to multiply by omega squared. 126 00:10:46,300 --> 00:10:47,800 OK, so this will be the value. 127 00:10:47,800 --> 00:10:52,360 And in case you don't know, this is equal to one over six. 128 00:10:54,440 --> 00:11:02,360 OK, so now let's finally come to the numerical results, so we have already generated a list of points 129 00:11:02,360 --> 00:11:04,550 that represent our stake. 130 00:11:04,940 --> 00:11:10,940 So let's work with this list and maybe just to make it look better. 131 00:11:10,940 --> 00:11:15,500 I just right here our list so that we see what's going on. 132 00:11:16,010 --> 00:11:22,850 So we have these 100 points describing our stick and an equidistant manner from zero where it starts 133 00:11:23,210 --> 00:11:24,920 to one word ends. 134 00:11:25,800 --> 00:11:33,500 And now what we have to do is we have to basically calculate the expression for the kinetic energy. 135 00:11:34,280 --> 00:11:38,110 So we have to calculate em so. 136 00:11:39,040 --> 00:11:48,170 So, so basically, we have to say em half times omega squared times. 137 00:11:50,120 --> 00:11:53,840 Our square, this was our equation from the beginning, let me scroll up. 138 00:11:54,260 --> 00:11:56,750 This was this one here that we have derived. 139 00:11:57,530 --> 00:12:04,280 But now, of course, every point in this list has a different radius, so we have to add up all the 140 00:12:04,280 --> 00:12:07,580 points, all the contribution from these points. 141 00:12:08,330 --> 00:12:12,170 And furthermore, since we are adding up, we have to. 142 00:12:12,680 --> 00:12:17,880 He have to consider here not the total mass, but the mass of every of these elements. 143 00:12:17,880 --> 00:12:22,850 So this will be the mass divided by the number of points. 144 00:12:23,450 --> 00:12:28,020 So there's all the and divided by no points. 145 00:12:28,040 --> 00:12:28,400 Yes. 146 00:12:31,940 --> 00:12:38,720 And here we have to say this will be the Sun and the Comanche will be and taught some. 147 00:12:39,680 --> 00:12:46,310 And now we have to take every element after this, our list and square it. 148 00:12:47,300 --> 00:12:52,340 And since this one here is not actually a list, as I just said, but it's an array we have seen in 149 00:12:52,340 --> 00:12:58,370 our crash course that when we apply some mathematical operation like squaring it, this will be applied 150 00:12:58,370 --> 00:13:00,050 to every individual element. 151 00:13:00,350 --> 00:13:01,940 And this is exactly what we want. 152 00:13:02,510 --> 00:13:05,620 So this year, our list square will be the list. 153 00:13:05,630 --> 00:13:08,000 Ah, but every element is squared. 154 00:13:08,580 --> 00:13:12,110 And now we just have to calculate the sum of all of these elements. 155 00:13:12,740 --> 00:13:16,250 So I think this expression here is already correct. 156 00:13:17,240 --> 00:13:23,120 And you see, the result is quite close to the numerical sorry to the analytical result. 157 00:13:23,990 --> 00:13:27,160 And this really didn't take much effort. 158 00:13:27,170 --> 00:13:29,000 It was just a single line of code. 159 00:13:30,230 --> 00:13:38,440 So now we can check what happens if we increase the number of points, so we could say 1000 that we 160 00:13:38,450 --> 00:13:39,830 rerun all the cells. 161 00:13:40,250 --> 00:13:45,050 And finally, this one and you see it gets even closer. 162 00:13:45,800 --> 00:13:51,590 So now there's only a very small difference, and you can further improve the results by increasing 163 00:13:51,590 --> 00:13:52,430 the number here. 164 00:13:52,880 --> 00:13:57,890 Or we could also be a bit more clever about the choice here because it's not so clever to start from 165 00:13:57,890 --> 00:13:59,330 zero to go to S. 166 00:13:59,810 --> 00:14:03,290 It would be better to shift all the points by one half. 167 00:14:03,940 --> 00:14:09,200 So, yeah, but I think it's not really necessary here because our result is already pretty good. 168 00:14:10,840 --> 00:14:19,180 So you see at this example that it can be very helpful to use NUMÉRIQUES to calculate such things. 169 00:14:19,840 --> 00:14:27,820 So of course, you could just look up this result in a textbook or you could derive it yourself analytically, 170 00:14:27,820 --> 00:14:28,660 as we have done. 171 00:14:29,020 --> 00:14:35,410 But you could also just tried a single line of code and then it calculates you the moment of inertia.