1 00:00:00,120 --> 00:00:00,610 All right. 2 00:00:00,630 --> 00:00:02,730 So let's get started with our next lecture. 3 00:00:03,180 --> 00:00:09,570 So here we will discuss some rotational motion, and we will begin really easy by considering just a 4 00:00:09,570 --> 00:00:10,440 point mass. 5 00:00:10,860 --> 00:00:15,720 And it's basically circling or orbiting around a specific point in space. 6 00:00:16,379 --> 00:00:24,720 And we will make it most simple for us by choosing the zero point of the coordinate system as our point 7 00:00:24,720 --> 00:00:27,210 around which to point mass will be circulating. 8 00:00:28,170 --> 00:00:35,100 So the first thing, as always, that we will do is we will import Nampai because we will use Nampai 9 00:00:35,100 --> 00:00:37,320 for many mathematical commands. 10 00:00:37,830 --> 00:00:44,730 And also, we will import much plot lib and especially the pie plot module, because we will also create 11 00:00:44,730 --> 00:00:45,840 a few plots. 12 00:00:46,740 --> 00:00:56,340 So to begin our discussion, we have to think a bit about the units and the characterizing quantities 13 00:00:56,460 --> 00:00:58,140 that describe our motion. 14 00:00:59,040 --> 00:01:02,640 So our point mass will not really have a volume. 15 00:01:02,640 --> 00:01:07,860 It's just a singular point in space, which makes the integration and everything much easier. 16 00:01:08,280 --> 00:01:11,910 And still, it will have a well-defined mass. 17 00:01:12,600 --> 00:01:20,580 And since this is here in numerical course and Python is working with NUMÉRIQUES, it's a bit difficult 18 00:01:20,580 --> 00:01:21,780 to deal with units. 19 00:01:21,780 --> 00:01:25,350 So we have to think in advance what we are going to do here. 20 00:01:25,950 --> 00:01:33,780 And as often done in theoretical physics or in numerical physics, we will just define the mass as being 21 00:01:33,780 --> 00:01:34,650 equal to one. 22 00:01:35,580 --> 00:01:40,440 So we have here a particle with a mass of one kilogram, we could say. 23 00:01:41,730 --> 00:01:46,290 Then we also have to think about the radius of our orbit, of our circular motion. 24 00:01:46,680 --> 00:01:50,220 And likewise, I will just say the radius is equal to one. 25 00:01:51,030 --> 00:01:54,960 Of course, we can later on change these values and see how the results change. 26 00:01:56,460 --> 00:01:59,950 And also, we have to define the velocity. 27 00:01:59,970 --> 00:02:09,419 So basically, the time it takes for the particle to really do one whole orbit, one complete circular 28 00:02:09,419 --> 00:02:17,700 motion and instead of defining here at a time, I define the angular velocity and the angular velocity 29 00:02:17,700 --> 00:02:18,660 I call omega. 30 00:02:19,080 --> 00:02:22,170 And I also define it as being equal to one. 31 00:02:22,620 --> 00:02:29,220 So before we continue to calculate the rotational or kinetic energy, I want to visualize the motion 32 00:02:29,220 --> 00:02:30,450 that we are describing here. 33 00:02:30,990 --> 00:02:38,100 And this you can see as an exercise in creating arrays or lists and also an exercise in plotting. 34 00:02:39,210 --> 00:02:45,770 So what we're going to do first is we are going to create an array, a list of points of time. 35 00:02:45,780 --> 00:02:53,250 So I call this tiara and we will then calculate the coordinates for these points of time and plot these. 36 00:02:54,300 --> 00:03:02,610 So the tiara will be a continuous list of numbers, and we can create this with the non-power module 37 00:03:02,610 --> 00:03:05,550 in space or the command line space. 38 00:03:06,090 --> 00:03:15,000 And since the angle of velocity omega is equal to one, I simulate this from zero to two times PI because 39 00:03:15,000 --> 00:03:19,960 this will be the time it takes for one whole hour circular orbit. 40 00:03:21,150 --> 00:03:27,650 And then we have to define how many points do we have in this in this array between zero and two PI 41 00:03:27,660 --> 00:03:29,880 and let's try 100 here. 42 00:03:30,660 --> 00:03:33,950 And so when I run this, then we have to find the theory. 43 00:03:34,410 --> 00:03:39,420 And now we're going ahead and we are calculate getting an X-ray and a white array. 44 00:03:40,380 --> 00:03:44,910 And we are not really solving here the equation of motion numerically. 45 00:03:45,150 --> 00:03:47,850 We already know that we want to describe a circular motion. 46 00:03:48,180 --> 00:03:51,210 So I just go ahead and write down what is the X coordinate? 47 00:03:52,170 --> 00:03:56,970 If you have some experience with polar coordinates, then you know that we can describe this really 48 00:03:56,970 --> 00:03:59,070 easily with these polar coordinates. 49 00:03:59,400 --> 00:04:04,320 So it's just the radius, which is, in our case, one times the cosine of the angle. 50 00:04:04,890 --> 00:04:10,800 And this is also why we have to find the angular velocity, because now we can just write Omega Times 51 00:04:10,800 --> 00:04:11,190 team. 52 00:04:12,060 --> 00:04:17,589 And of course, instead of T, we have to write T array here so that this one will also be in the right. 53 00:04:19,589 --> 00:04:26,550 So this will be basically the projection onto the x axis and the sine instead is the projected on the 54 00:04:26,550 --> 00:04:27,270 y axis. 55 00:04:27,840 --> 00:04:36,270 And I think we are good to go now so we can now go ahead and as well as incorrect, wanted here and 56 00:04:36,270 --> 00:04:41,970 we can now go ahead and plot these two arrays in a scatterplot, for example, or in the regular plot. 57 00:04:42,990 --> 00:04:48,600 So what I'm going to do first is I'm just going to plot this. 58 00:04:48,600 --> 00:04:53,220 I'm going to write the polka dot plot and I'll write X three. 59 00:04:53,760 --> 00:04:55,530 Come on, y. 60 00:04:55,980 --> 00:04:56,400 All right. 61 00:04:57,300 --> 00:04:58,050 And this works. 62 00:04:58,770 --> 00:05:01,110 So you see, we have such an oval shape here. 63 00:05:01,380 --> 00:05:03,260 We could also do scatterplot. 64 00:05:04,410 --> 00:05:08,550 Then you would see individual points that we have created here, 100 of them. 65 00:05:09,420 --> 00:05:12,110 But I think here I will use plots. 66 00:05:13,200 --> 00:05:17,580 So of course, we could leave it as it is, but I want to make it just a bit nicer. 67 00:05:17,580 --> 00:05:22,800 So this is also a practice for you to learn a bit more about plotting. 68 00:05:24,090 --> 00:05:30,510 We can now right here, for example, for the ex-Labour or X coordinate and for the Y label, we can 69 00:05:30,510 --> 00:05:32,460 write y coordinate. 70 00:05:34,230 --> 00:05:41,070 And another thing that's a bit ugly here is that we have this typical aspect ratio of a plot, which 71 00:05:41,070 --> 00:05:42,750 is typically not a square plot. 72 00:05:42,750 --> 00:05:50,400 But since we want to plot here a circle, it would make much more sense to have a aspect ratio of one 73 00:05:50,400 --> 00:05:53,640 where this length and this length would be the same. 74 00:05:54,510 --> 00:06:01,590 And I found this to be a bit tricky in Python or in my plot loop because there is not a straightforward 75 00:06:01,590 --> 00:06:03,510 command for setting the aspect ratio. 76 00:06:04,320 --> 00:06:07,740 So what we have to do first, we have to create a real figure. 77 00:06:08,640 --> 00:06:11,680 So we have to first define some additional stuff. 78 00:06:11,700 --> 00:06:22,320 So we have to create a figure first by the command's plot figure and then I give it the name just fig. 79 00:06:22,500 --> 00:06:23,760 That is how you typically do it. 80 00:06:23,760 --> 00:06:27,060 You find many online examples that use exactly the same commands. 81 00:06:27,690 --> 00:06:32,640 And then we define another thing, which is a subplot. 82 00:06:33,060 --> 00:06:41,010 So basically, we take this figure, which is empty now and we are adding a subplot and. 83 00:06:41,250 --> 00:06:47,530 The subplot is a really useful command, especially if you want to create a figure with several subplots. 84 00:06:47,550 --> 00:06:51,540 So several individual figures are combined to a full figure. 85 00:06:52,440 --> 00:06:54,300 So that is not what we are doing here. 86 00:06:54,300 --> 00:06:56,310 We are just adding a single subplot. 87 00:06:56,640 --> 00:06:57,900 But we could add more. 88 00:06:58,740 --> 00:07:06,900 And now here the subplot has basically the syntax down to you to find how many plots are you going to 89 00:07:06,900 --> 00:07:08,460 have in your total plots? 90 00:07:08,970 --> 00:07:13,260 And we are going to have a grid here of plots that is just one times one. 91 00:07:13,350 --> 00:07:14,850 We just want to have a single plot. 92 00:07:15,540 --> 00:07:20,310 And now we have to index our plots, so it will be the first and only one. 93 00:07:20,310 --> 00:07:24,690 So yeah, here is a bit useless because we just are going to have one plot. 94 00:07:24,690 --> 00:07:26,070 So it will be one one one. 95 00:07:27,270 --> 00:07:30,210 And then I give this subplot a name called X. 96 00:07:30,750 --> 00:07:39,780 And now finally, I tend to find the aspect ratio of I see X set past back and then can give it here 97 00:07:39,780 --> 00:07:40,320 value. 98 00:07:40,340 --> 00:07:42,540 For example, I could write to that. 99 00:07:42,590 --> 00:07:46,500 I would get something like this could also make it even more extreme. 100 00:07:46,980 --> 00:07:48,120 But would we want to have? 101 00:07:48,120 --> 00:07:50,520 Here is, of course, an aspect ratio of one. 102 00:07:50,970 --> 00:07:54,930 So we have really the circle here now. 103 00:07:55,230 --> 00:08:02,790 Another exercise that I want to show you is how can you really see what time does for this motion? 104 00:08:02,790 --> 00:08:08,790 Because here basically we just have the trajectory of one full oscillation, so it's just a circle. 105 00:08:09,390 --> 00:08:15,170 And of course, we could say, I only want to plot this to 1.5 Pi. 106 00:08:15,660 --> 00:08:19,080 So then we would have this as our trajectory. 107 00:08:19,770 --> 00:08:23,700 But still, it's a bit hard to see how this changes precisely under time. 108 00:08:24,690 --> 00:08:29,340 So what we are going to do next is we are going to make a three dimensional plot. 109 00:08:30,240 --> 00:08:34,710 And for this, we have practiced is already in the crash course. 110 00:08:35,850 --> 00:08:38,760 First of all, we must enable D projection 3D object. 111 00:08:39,390 --> 00:08:41,400 So our rights plotted on axes. 112 00:08:42,630 --> 00:08:49,980 Projection is equal to 3D and then we can also give the plot a name. 113 00:08:50,280 --> 00:08:59,780 So multi 3D will be the same as we used in the crash course, and now we can all at the data set that 114 00:08:59,790 --> 00:09:00,870 we are going to plot here. 115 00:09:01,680 --> 00:09:07,790 And for this, I will use a scatterplot plus 3D dot scatter 3D. 116 00:09:10,170 --> 00:09:11,040 Yes, that's correct. 117 00:09:11,060 --> 00:09:16,560 And then Ray Y array. 118 00:09:17,100 --> 00:09:20,070 And for the z axis, I'm going to plot the time. 119 00:09:21,090 --> 00:09:21,690 So T. 120 00:09:23,610 --> 00:09:25,320 So we will discuss this in a moment. 121 00:09:25,320 --> 00:09:36,510 But let me quickly add a few axes to this plot, so I will add an x axis label p o t three d dot set 122 00:09:36,750 --> 00:09:44,430 underscore x label and then i write x coordinates that. 123 00:09:44,430 --> 00:09:52,860 Of course, we do the same thing for Y and Z, so y, z, y and time. 124 00:09:54,510 --> 00:09:55,800 So now you see the plots. 125 00:09:56,580 --> 00:10:00,120 So the z axis is now visualizing the time. 126 00:10:00,450 --> 00:10:02,340 In our case, this will be in seconds. 127 00:10:02,730 --> 00:10:08,700 So maybe let me read this here, time from seconds and coordinate in meter. 128 00:10:13,200 --> 00:10:17,550 So you see at zero seconds time to the corresponding coordinates. 129 00:10:17,790 --> 00:10:23,670 This one here and then wants to time increases, it goes along to circle circular trajectory. 130 00:10:24,690 --> 00:10:28,680 So you really have to know how to read this charge chart. 131 00:10:29,100 --> 00:10:32,490 So it's not moving on a spiral. 132 00:10:32,610 --> 00:10:33,780 It's moving on a circle. 133 00:10:34,050 --> 00:10:37,650 The spiral just comes because we are plotting here to time on the z axis. 134 00:10:38,580 --> 00:10:44,460 So with these two graphs here, I think it's pretty clear what's happening, and I hope that you like 135 00:10:44,460 --> 00:10:48,360 this exercise and learning a bit more on how to plot such a motion. 136 00:10:49,620 --> 00:10:56,370 So the next thing that we are going to do is we are going to calculate the energy, and this is pretty 137 00:10:56,370 --> 00:11:00,780 easy because we are just using the equation that everyone knows. 138 00:11:01,170 --> 00:11:08,340 So we are going to start with, I would say, the finding the period it's a period of one motion is 139 00:11:08,700 --> 00:11:09,480 two PI. 140 00:11:12,360 --> 00:11:13,230 Over Omega. 141 00:11:13,980 --> 00:11:19,590 So this is how the this is actually how the Omega, how the angular velocity is defined. 142 00:11:20,430 --> 00:11:25,530 Omega is actually two pi over the periods, and so the period is two pi over omega. 143 00:11:26,610 --> 00:11:36,390 And then the path that you know, the path that that the particle has taken is basically the circumference 144 00:11:36,390 --> 00:11:37,140 of the circle. 145 00:11:38,100 --> 00:11:46,680 So of course, this is two pi times the radius, so two times PI, times the radius. 146 00:11:48,180 --> 00:11:51,660 And now we can calculate the velocity. 147 00:11:53,570 --> 00:12:00,380 So the velocity is, of course, you know, this path divided by the period as an F3 uniform motion. 148 00:12:00,710 --> 00:12:04,790 You take the total path of total time and this is the velocity. 149 00:12:04,790 --> 00:12:08,030 And in our case, this is the path divided by the period. 150 00:12:08,600 --> 00:12:11,090 And you see, in both cases, we have the two PI. 151 00:12:11,510 --> 00:12:15,650 So we have here two PI are divided by two pi, divided by omega. 152 00:12:16,100 --> 00:12:22,910 So it is equal to velocity is equal to our times omega. 153 00:12:25,940 --> 00:12:30,950 So, of course, we could run this now, but I just leave it here as a comment. 154 00:12:31,250 --> 00:12:32,780 But we can also run. 155 00:12:32,820 --> 00:12:33,680 It doesn't really matter. 156 00:12:35,840 --> 00:12:41,960 And then the total energy the kinetic energy will be, yeah, will be the rotational energy. 157 00:12:41,960 --> 00:12:42,980 It's exactly the same. 158 00:12:43,400 --> 00:12:50,270 So the kinetic energy is, of course, the mass divided by two times the velocity squared. 159 00:12:50,720 --> 00:12:52,880 So velocity is squared. 160 00:12:54,890 --> 00:12:58,340 But in our case, I just want to take this and plug it in already. 161 00:12:58,940 --> 00:13:00,800 So this will be just a commentaire. 162 00:13:01,190 --> 00:13:08,390 And the actual solution that we are going to use in the following will be omega squared times radius 163 00:13:08,510 --> 00:13:09,110 squared. 164 00:13:10,100 --> 00:13:16,310 And you see, in our case, if omega is one and if the radius is run one and the mass is one, then 165 00:13:16,310 --> 00:13:22,230 our rotational energy, which is the same as the kinetic energy, will be zero point five. 166 00:13:22,730 --> 00:13:28,610 And since we have used as our unit so far, we know that this will be zero point five Joule.