1 00:00:02,165 --> 00:00:04,460 The following content is provided under a Creative 2 00:00:04,460 --> 00:00:05,880 Commons license. 3 00:00:05,880 --> 00:00:08,090 Your support will help MIT OpenCourseWare 4 00:00:08,090 --> 00:00:12,180 continue to offer high-quality educational resources for free. 5 00:00:12,180 --> 00:00:14,720 To make a donation or to view additional materials 6 00:00:14,720 --> 00:00:18,680 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:18,680 --> 00:00:19,700 at ocw.mit.edu. 8 00:00:24,060 --> 00:00:25,980 BOLESLAW WYSLOUCH: Good morning, everybody. 9 00:00:25,980 --> 00:00:27,090 I'm Bolek Wyslouch. 10 00:00:27,090 --> 00:00:31,920 I'm a teacher substitute for Professor Lee, who is now 11 00:00:31,920 --> 00:00:34,770 at some conference in China, and he 12 00:00:34,770 --> 00:00:39,510 asked me to talk to you about coupled oscillators. 13 00:00:39,510 --> 00:00:43,290 I understand that he introduced the concept last time. 14 00:00:43,290 --> 00:00:45,430 You worked through some examples. 15 00:00:45,430 --> 00:00:47,775 So what we are going to do today is 16 00:00:47,775 --> 00:00:50,100 to basically go through one or two 17 00:00:50,100 --> 00:00:55,500 examples of very straightforward coupled oscillators, where 18 00:00:55,500 --> 00:00:59,790 I will introduce various kinds of systematic calculational 19 00:00:59,790 --> 00:01:01,380 techniques, how to set things up, 20 00:01:01,380 --> 00:01:05,129 how to prepare things for calculations. 21 00:01:05,129 --> 00:01:09,150 And also, we may, depending on how much time we have, 22 00:01:09,150 --> 00:01:14,560 start driving, have driven coupled oscillators. 23 00:01:14,560 --> 00:01:20,990 And we will work on two, again, simple physical systems, 24 00:01:20,990 --> 00:01:28,230 one that consists of two pendula driven by forces of gravity, 25 00:01:28,230 --> 00:01:29,067 each of them. 26 00:01:29,067 --> 00:01:30,900 And then they are connected with the spring. 27 00:01:30,900 --> 00:01:33,450 So each of those pendula, each of those masses, 28 00:01:33,450 --> 00:01:36,805 will feel the effects of gravity and effects of springs 29 00:01:36,805 --> 00:01:38,930 at the same time, and they will talk to each other. 30 00:01:38,930 --> 00:01:40,720 There will be coupling between them. 31 00:01:40,720 --> 00:01:44,520 So that's one physical example which we'll consider. 32 00:01:44,520 --> 00:01:47,220 The other physical example consists 33 00:01:47,220 --> 00:01:54,540 of two masses in the horizontal frictionless track connected 34 00:01:54,540 --> 00:01:55,590 by a set of springs. 35 00:01:55,590 --> 00:01:58,260 So they are driven by forces of spring. 36 00:01:58,260 --> 00:02:00,660 And those two systems are very similar to each other, 37 00:02:00,660 --> 00:02:03,420 almost identical in terms of calculations, 38 00:02:03,420 --> 00:02:05,610 and they exhibit the same phenomena, 39 00:02:05,610 --> 00:02:07,350 and I will be able to demonstrate 40 00:02:07,350 --> 00:02:09,340 several of the neat new things. 41 00:02:09,340 --> 00:02:11,580 And this particular system is set up 42 00:02:11,580 --> 00:02:14,640 to introduce external driving force, which will 43 00:02:14,640 --> 00:02:16,090 create a new set of phenomena. 44 00:02:16,090 --> 00:02:19,150 And we'll talk about it today. 45 00:02:19,150 --> 00:02:21,540 And what I would like to stress today 46 00:02:21,540 --> 00:02:24,240 when we go through all those calculations is, 47 00:02:24,240 --> 00:02:28,470 A, how do you convert a given physical system 48 00:02:28,470 --> 00:02:32,310 with all the forces, et cetera, into some sort of fixed form, 49 00:02:32,310 --> 00:02:36,690 fixed type of notation, with which you can treat all 50 00:02:36,690 --> 00:02:39,150 possible coupled oscillators? 51 00:02:39,150 --> 00:02:41,370 And also we will discuss various interesting-- 52 00:02:41,370 --> 00:02:43,650 even though the system is very simple, just 53 00:02:43,650 --> 00:02:47,980 two masses, a spring, a little bit of gravity on top of that, 54 00:02:47,980 --> 00:02:53,340 the way they behave could be extremely complex, 55 00:02:53,340 --> 00:02:55,500 but it can be understood in terms 56 00:02:55,500 --> 00:02:58,710 of very simple systematic way of looking things 57 00:02:58,710 --> 00:03:03,070 through normal modes and normal frequencies, 58 00:03:03,070 --> 00:03:05,650 so the characteristic frequencies of the system. 59 00:03:05,650 --> 00:03:08,070 So let's set things up. 60 00:03:08,070 --> 00:03:09,570 So we'll start. 61 00:03:09,570 --> 00:03:11,370 This will be our workhorse. 62 00:03:11,370 --> 00:03:15,300 And by the way, once we understand two, 63 00:03:15,300 --> 00:03:18,930 we will then generalize to infinite number of oscillators, 64 00:03:18,930 --> 00:03:22,770 which is actually-- so this model, which consists 65 00:03:22,770 --> 00:03:26,460 of weights hanging under the influence of gravity 66 00:03:26,460 --> 00:03:28,920 plus the springs will be then used 67 00:03:28,920 --> 00:03:33,160 for many applications of the concepts later in this course. 68 00:03:33,160 --> 00:03:39,430 So let's try to convert this physical system 69 00:03:39,430 --> 00:03:41,590 into a set of equations. 70 00:03:41,590 --> 00:03:47,920 So we have a mass, m, hanging from some sort of fixed 71 00:03:47,920 --> 00:03:52,360 support, another mass here, same mass for simplicity. 72 00:03:52,360 --> 00:03:55,150 We connect them with a string, and we know 73 00:03:55,150 --> 00:03:57,080 everything about this system. 74 00:03:57,080 --> 00:04:00,610 We know the length of each of those pendula, which 75 00:04:00,610 --> 00:04:01,840 is the same. 76 00:04:01,840 --> 00:04:03,280 We know masses. 77 00:04:03,280 --> 00:04:07,330 We know spring constant of a spring connecting those two 78 00:04:07,330 --> 00:04:08,860 things. 79 00:04:08,860 --> 00:04:11,695 The spring is initially at its rest position 80 00:04:11,695 --> 00:04:16,269 such that when the two pendula are hanging vertically, 81 00:04:16,269 --> 00:04:17,950 the spring is relaxed. 82 00:04:17,950 --> 00:04:20,209 But if you move it away from verticality, 83 00:04:20,209 --> 00:04:24,790 the spring either compresses or stretches. 84 00:04:24,790 --> 00:04:30,970 And everything is in Earth's gravitational field, g. 85 00:04:30,970 --> 00:04:35,460 We assume that this is an ideal system, highly idealized. 86 00:04:35,460 --> 00:04:40,180 We only consider motion with small angle approximation, only 87 00:04:40,180 --> 00:04:42,290 small displacement. 88 00:04:42,290 --> 00:04:44,680 There's no drag force assumed. 89 00:04:44,680 --> 00:04:47,560 The spring is ideal, et cetera, et cetera. 90 00:04:47,560 --> 00:04:51,670 Of course, this thing here is very far from being ideal, 91 00:04:51,670 --> 00:04:54,940 but hopefully basic behaviors are similar. 92 00:04:54,940 --> 00:04:57,890 It's approximately ideal. 93 00:04:57,890 --> 00:04:59,720 To study the motion of this thing, 94 00:04:59,720 --> 00:05:02,480 to understand how it works, let's try to-- 95 00:05:02,480 --> 00:05:06,770 let's try to parameterize it, and displace it 96 00:05:06,770 --> 00:05:08,780 from equilibrium, and look at the forces, 97 00:05:08,780 --> 00:05:11,550 and try to calculate equations of motion. 98 00:05:11,550 --> 00:05:13,520 So we will characterize this system 99 00:05:13,520 --> 00:05:16,760 by two position coordinates. 100 00:05:16,760 --> 00:05:18,080 We will have x. 101 00:05:18,080 --> 00:05:19,580 We'll give this one number one. 102 00:05:19,580 --> 00:05:21,500 This will be number two. 103 00:05:21,500 --> 00:05:26,060 And we will have x subscript 1, which in general would 104 00:05:26,060 --> 00:05:26,960 depend on the time. 105 00:05:26,960 --> 00:05:29,600 This is the position of this mass with respect 106 00:05:29,600 --> 00:05:31,320 to its equilibrium position. 107 00:05:31,320 --> 00:05:35,070 We will have x2 as a function of t. 108 00:05:35,070 --> 00:05:37,100 Again, this tells us everything. 109 00:05:37,100 --> 00:05:39,770 And the full description of the system 110 00:05:39,770 --> 00:05:42,920 is to know exactly what happens to x1 and x2 111 00:05:42,920 --> 00:05:46,910 for all possible times. 112 00:05:46,910 --> 00:05:49,910 And we will impose some initial conditions. 113 00:05:49,910 --> 00:05:52,650 We can come back to that later. 114 00:05:52,650 --> 00:05:55,550 So again, so the coordinate system is this. 115 00:05:55,550 --> 00:05:58,790 When we start talking about the system in principle 116 00:05:58,790 --> 00:06:01,910 in the case of somewhat larger angles, 117 00:06:01,910 --> 00:06:04,425 you have to worry about vertical positions as well. 118 00:06:04,425 --> 00:06:05,300 So we will introduce. 119 00:06:05,300 --> 00:06:09,500 So there is also a coordinate y, which we will need temporarily 120 00:06:09,500 --> 00:06:12,040 to set things up. 121 00:06:12,040 --> 00:06:15,690 So x is, as I say, x is measured from equilibrium. 122 00:06:15,690 --> 00:06:17,860 Y is positioned vertically. 123 00:06:17,860 --> 00:06:20,451 So to calculate the equations of motion, 124 00:06:20,451 --> 00:06:21,700 we have to look at the forces. 125 00:06:21,700 --> 00:06:24,010 So let's look at what are the forces acting, 126 00:06:24,010 --> 00:06:27,460 for example, on this mass, the mass, which 127 00:06:27,460 --> 00:06:31,450 is-- if it's displaced from a vertical position. 128 00:06:31,450 --> 00:06:37,420 Let's say this mass, mass 1, has moved by some distance 129 00:06:37,420 --> 00:06:41,500 away from thing Temporarily, let's introduce an angle here 130 00:06:41,500 --> 00:06:44,860 to characterize this displacement from vertical. 131 00:06:44,860 --> 00:06:47,590 And let's write down all the forces acting on this 132 00:06:47,590 --> 00:06:50,320 - force diagram acting on this mass. 133 00:06:50,320 --> 00:06:55,570 So there is a tension in the string or the rod. 134 00:06:55,570 --> 00:06:58,480 Let's call it T1. 135 00:06:58,480 --> 00:07:03,020 There is a force of spring acting 136 00:07:03,020 --> 00:07:04,680 in a horizontal direction. 137 00:07:04,680 --> 00:07:06,350 This is a vector. 138 00:07:06,350 --> 00:07:11,840 And there is a force of gravity acting on this 139 00:07:11,840 --> 00:07:14,130 in the vertical direction. 140 00:07:14,130 --> 00:07:16,610 We can write down those forces. 141 00:07:16,610 --> 00:07:18,050 We know a lot about them. 142 00:07:18,050 --> 00:07:23,810 This one is minus mg y-hat. 143 00:07:23,810 --> 00:07:33,660 This one is equal to k x2 minus x1 in the x-hat direction. 144 00:07:33,660 --> 00:07:37,520 So this is the force which, when the spring is displaced from 145 00:07:37,520 --> 00:07:40,730 equilibrium, there is a spring force, Hooke force, 146 00:07:40,730 --> 00:07:42,480 in the direction of -- 147 00:07:42,480 --> 00:07:43,940 in the usual direction. 148 00:07:43,940 --> 00:07:47,570 In this case, it's actually in the opposite direction. 149 00:07:47,570 --> 00:07:49,710 And then there is a tension the spring, 150 00:07:49,710 --> 00:07:53,370 which has to be calculated such that we understand 151 00:07:53,370 --> 00:07:56,340 the acceleration of this object. 152 00:07:56,340 --> 00:07:59,210 So let's write down the equations 153 00:07:59,210 --> 00:08:02,450 in the x-hat direction. 154 00:08:02,450 --> 00:08:07,990 This is m acceleration of object number 1 155 00:08:07,990 --> 00:08:16,560 in x direction is equal to minus T1 sine theta 1 plus k 156 00:08:16,560 --> 00:08:21,150 x2 minus x1. 157 00:08:21,150 --> 00:08:26,270 And in the y-hat direction, we have 158 00:08:26,270 --> 00:08:39,340 m y1 direction is equal to T cosine theta 1 minus mg. 159 00:08:39,340 --> 00:08:43,909 At the small angle for theta 1 much, much smaller than one, 160 00:08:43,909 --> 00:08:49,060 we can assume, that cos theta 1 is approximately equal to 1 161 00:08:49,060 --> 00:08:53,910 and sine theta 1 is equal to angle. 162 00:08:53,910 --> 00:08:55,710 We do the usual thing. 163 00:08:55,710 --> 00:08:59,250 So basically, in this approximation, 164 00:08:59,250 --> 00:09:01,200 and also by looking at the system, 165 00:09:01,200 --> 00:09:03,570 it's clear that the system does not move, 166 00:09:03,570 --> 00:09:06,270 and the vertical direction can be ignored. 167 00:09:06,270 --> 00:09:07,121 Yes? 168 00:09:07,121 --> 00:09:09,162 AUDIENCE: How do you know which way [INAUDIBLE]?? 169 00:09:09,162 --> 00:09:10,370 BOLESLAW WYSLOUCH: Excuse me? 170 00:09:10,370 --> 00:09:11,572 AUDIENCE: The [INAUDIBLE]. 171 00:09:11,572 --> 00:09:13,323 How do you know which way it [INAUDIBLE]?? 172 00:09:13,323 --> 00:09:14,697 BOLESLAW WYSLOUCH: How do I know? 173 00:09:14,697 --> 00:09:15,413 AUDIENCE: Yeah. 174 00:09:15,413 --> 00:09:16,395 [INAUDIBLE] 175 00:09:19,090 --> 00:09:21,920 BOLESLAW WYSLOUCH: The spring force is-- 176 00:09:21,920 --> 00:09:24,740 well, you have to look at the mass 1. 177 00:09:24,740 --> 00:09:26,930 You are just looking at mass 1. 178 00:09:26,930 --> 00:09:29,570 So the spring is connected to mass 1. 179 00:09:29,570 --> 00:09:32,930 And the force of the spring on mass 1 180 00:09:32,930 --> 00:09:39,390 is k times however the spring is squashed or stretched, 181 00:09:39,390 --> 00:09:41,210 all right? 182 00:09:41,210 --> 00:09:44,960 So it knows about the existence of mass 2, but only in a sense 183 00:09:44,960 --> 00:09:47,320 that you have to know the position of mass 2. 184 00:09:47,320 --> 00:09:50,930 So we just assume that x2 is something, 185 00:09:50,930 --> 00:09:53,820 and we just look where the spring is. 186 00:09:53,820 --> 00:09:55,980 So that's why-- the force of spring 187 00:09:55,980 --> 00:09:59,350 depends on the difference of position x1 minus x2. 188 00:10:02,560 --> 00:10:05,410 So this is written here. 189 00:10:05,410 --> 00:10:08,610 And in fact, interestingly, the position of the mass 1 190 00:10:08,610 --> 00:10:11,380 itself is a negative sign here. 191 00:10:11,380 --> 00:10:14,080 So if you move mass 1, the spring force 192 00:10:14,080 --> 00:10:17,926 is in the right direction, minus kx. 193 00:10:17,926 --> 00:10:19,360 All right? 194 00:10:19,360 --> 00:10:21,520 So there is no motion x1, so we can 195 00:10:21,520 --> 00:10:25,750 conclude from here the T cosine1 is approximately equal to 1. 196 00:10:25,750 --> 00:10:28,960 So T is simply equal to mg. 197 00:10:28,960 --> 00:10:33,250 So the tension in the spring can be assumed to be mg. 198 00:10:33,250 --> 00:10:35,000 We don't have to worry about it. 199 00:10:35,000 --> 00:10:37,390 And then we just plug in-- 200 00:10:37,390 --> 00:10:40,780 also the angle can be converted into position 201 00:10:40,780 --> 00:10:46,630 by realizing that the distance times the angle 202 00:10:46,630 --> 00:10:49,180 is equal to displacement, the usual geometry. 203 00:10:49,180 --> 00:10:52,120 The net result is that by simplifying things, 204 00:10:52,120 --> 00:10:58,990 I can write down equations for acceleration 205 00:10:58,990 --> 00:11:03,760 in the horizontal direction for mass 1 206 00:11:03,760 --> 00:11:18,920 is equal to minus mg x1 over l plus k x2 minus x1. 207 00:11:18,920 --> 00:11:19,900 OK? 208 00:11:19,900 --> 00:11:26,730 So this is an equation of motion for mass 1 209 00:11:26,730 --> 00:11:29,980 in our coupled system. 210 00:11:29,980 --> 00:11:33,750 And I could say most of the terms 211 00:11:33,750 --> 00:11:37,860 have to do with a motion of mass 1 itself. 212 00:11:37,860 --> 00:11:40,620 Mass 1 is its own pendulum. 213 00:11:40,620 --> 00:11:46,080 And mass 1 is feeling the effect of the spring force. 214 00:11:46,080 --> 00:11:48,420 But because the force of the spring 215 00:11:48,420 --> 00:11:50,670 depends on the difference between positions, 216 00:11:50,670 --> 00:11:53,010 there is this coupling-- 217 00:11:53,010 --> 00:11:58,200 so the motion of mass 1 knows of where mass 2 is. 218 00:11:58,200 --> 00:12:03,310 And motion of mass 2 influences the motion of mass 1. 219 00:12:03,310 --> 00:12:05,970 That's how the coupling shows up. 220 00:12:05,970 --> 00:12:08,760 So for most of those problems, what you do is 221 00:12:08,760 --> 00:12:13,220 you simply focus on the mass in question. 222 00:12:13,220 --> 00:12:15,690 You take all the forces, you calculate them, 223 00:12:15,690 --> 00:12:17,610 and then this coupling will somehow 224 00:12:17,610 --> 00:12:20,250 appear in the equations. 225 00:12:20,250 --> 00:12:26,080 So we can repeat exactly the same calculation 226 00:12:26,080 --> 00:12:29,020 focusing on mass 2. 227 00:12:29,020 --> 00:12:33,210 And then the equation which you will get will be very similar. 228 00:12:33,210 --> 00:12:36,820 Let me just slightly rewrite this equation here 229 00:12:36,820 --> 00:12:40,390 to kind of combine all the terms which 230 00:12:40,390 --> 00:12:42,420 depend on the position of mass 1 with terms 231 00:12:42,420 --> 00:12:44,172 that depend on mass 2. 232 00:12:44,172 --> 00:12:53,910 So where m x-acceleration is equal to minus k 233 00:12:53,910 --> 00:13:06,040 plus mg over l times x1 plus k times x2. 234 00:13:06,040 --> 00:13:07,740 So this is the coupling term. 235 00:13:13,680 --> 00:13:17,710 This is what makes those pendula coupled. 236 00:13:17,710 --> 00:13:18,370 All right? 237 00:13:18,370 --> 00:13:23,740 And then I can write almost exactly the same equation 238 00:13:23,740 --> 00:13:31,420 of mass 2 with the proper replacement of masses. 239 00:13:31,420 --> 00:13:37,080 So let me write this down in the following way-- kx1 240 00:13:37,080 --> 00:13:45,420 minus k plus mg over l times x2. 241 00:13:50,440 --> 00:13:54,650 So the motion of mass x1 depends on x1 242 00:13:54,650 --> 00:13:57,440 itself multiplied by something with a spring 243 00:13:57,440 --> 00:13:59,710 term and gravitational term and depends 244 00:13:59,710 --> 00:14:04,250 on the position of mass 2 only through the spring. 245 00:14:04,250 --> 00:14:10,970 Mass 2 also is mostly driven by its own gravitational force 246 00:14:10,970 --> 00:14:15,950 of itself plus the spring depends on the position of x2. 247 00:14:15,950 --> 00:14:18,410 But there is this coupling term that 248 00:14:18,410 --> 00:14:20,330 depends on position of mass 1. 249 00:14:20,330 --> 00:14:23,480 So both of them feel the neighbor on the other side, 250 00:14:23,480 --> 00:14:24,290 right? 251 00:14:24,290 --> 00:14:29,770 So if I keep this one steady of x2 equals 0, 252 00:14:29,770 --> 00:14:31,580 then basically the forces here is just 253 00:14:31,580 --> 00:14:34,310 the spring plus the gravity. 254 00:14:34,310 --> 00:14:37,910 If I move this one and keep this one at 0, the force on this 255 00:14:37,910 --> 00:14:40,000 spring spring and gravity. 256 00:14:40,000 --> 00:14:44,010 But if this one is displaced, and I move that guy, 257 00:14:44,010 --> 00:14:47,050 the forces on this one are affected by the fact 258 00:14:47,050 --> 00:14:49,500 that number 2 changed. 259 00:14:49,500 --> 00:14:50,000 OK? 260 00:14:50,000 --> 00:14:52,250 Again, I was able to determine those coupling 261 00:14:52,250 --> 00:14:56,842 terms by simply looking at mass 1 itself, mass 2 itself. 262 00:14:56,842 --> 00:15:00,350 All right, so this is the set of two coupled equations. 263 00:15:00,350 --> 00:15:03,440 I have accelerations here for x1, x2, 264 00:15:03,440 --> 00:15:05,240 and I have positions here. 265 00:15:05,240 --> 00:15:08,420 It's like an oscillator of position acceleration 266 00:15:08,420 --> 00:15:13,130 with a constant term except that things here are a little mixed. 267 00:15:13,130 --> 00:15:16,970 And the trick in this whole mathematics, 268 00:15:16,970 --> 00:15:19,670 and calculations, and the way we do 269 00:15:19,670 --> 00:15:25,820 things is how do you solve those coupled equations? 270 00:15:25,820 --> 00:15:26,860 OK? 271 00:15:26,860 --> 00:15:30,660 So what I would like to do is-- 272 00:15:30,660 --> 00:15:32,950 and there is multiple ways of doing that. 273 00:15:32,950 --> 00:15:34,840 So let me do everything. 274 00:15:34,840 --> 00:15:38,250 Let's write down everything in the matrix form, 275 00:15:38,250 --> 00:15:40,170 because it turns out that linear matrices are 276 00:15:40,170 --> 00:15:41,400 very useful for that. 277 00:15:41,400 --> 00:15:43,060 We will use them very, very-- 278 00:15:43,060 --> 00:15:44,410 in a very simple way. 279 00:15:44,410 --> 00:15:48,210 So let's introduce to them and show vector, 280 00:15:48,210 --> 00:15:53,790 which consists of x1 and x2. 281 00:15:53,790 --> 00:15:58,880 So basically, all the position x1 and x2 are here. 282 00:15:58,880 --> 00:16:02,900 So we will be monitoring the change of this x2 283 00:16:02,900 --> 00:16:04,460 as a function of time. 284 00:16:04,460 --> 00:16:09,900 We will introduce a force matrix k, 285 00:16:09,900 --> 00:16:25,440 which is equal to k plus mg over l minus k here, minus k here, 286 00:16:25,440 --> 00:16:29,600 k plus mg over l there. 287 00:16:29,600 --> 00:16:32,340 This is a two by two matrix. 288 00:16:32,340 --> 00:16:37,140 And then we need a third matrix, mass matrix, 289 00:16:37,140 --> 00:16:45,070 which simply says that masses are mass of first object is m 290 00:16:45,070 --> 00:16:46,610 and the other one is also m, right? 291 00:16:49,130 --> 00:16:52,520 So these are three matrices that basically 292 00:16:52,520 --> 00:16:56,640 contains exactly the same information as out there. 293 00:16:56,640 --> 00:16:58,110 I probably need another matrix. 294 00:16:58,110 --> 00:17:01,440 I need an inverse matrix for mass, which basically 295 00:17:01,440 --> 00:17:05,900 is 1 over m, 1 over m, 0 and 0. 296 00:17:05,900 --> 00:17:10,630 This is a inverted matrix. 297 00:17:10,630 --> 00:17:20,069 OK, and it turns out that after I introduced these matrices, 298 00:17:20,069 --> 00:17:23,810 this set of equations can be written simply 299 00:17:23,810 --> 00:17:30,700 as X, the second derivative of the vector capital X, 300 00:17:30,700 --> 00:17:36,890 is equal to minus m to the minus 1, 301 00:17:36,890 --> 00:17:43,480 this matrix, multiplying matrix k and then multiplying 302 00:17:43,480 --> 00:17:45,953 vectors x again. 303 00:17:49,801 --> 00:17:52,210 All right? 304 00:17:52,210 --> 00:17:58,880 So this is exactly the same as this, just written 305 00:17:58,880 --> 00:18:00,720 a different way. 306 00:18:00,720 --> 00:18:02,380 So it's only the question of notation. 307 00:18:02,380 --> 00:18:06,190 So it turns out it's very convenient 308 00:18:06,190 --> 00:18:11,240 to use matrix calculation to do things faster. 309 00:18:11,240 --> 00:18:14,510 So instead of repeating writing, all the x1s, x2, et cetera, 310 00:18:14,510 --> 00:18:19,480 instead I just stick them into one or two element objects. 311 00:18:19,480 --> 00:18:21,800 I use matrices to multiply things, 312 00:18:21,800 --> 00:18:23,540 and if I want to know x1 and x2, I 313 00:18:23,540 --> 00:18:26,540 can always go, OK, the top component 314 00:18:26,540 --> 00:18:30,450 of vector x, lower component of vector x gives me the solution. 315 00:18:30,450 --> 00:18:31,010 Simple. 316 00:18:31,010 --> 00:18:31,510 Right? 317 00:18:34,690 --> 00:18:42,210 So let's try to use this terminology to find solutions. 318 00:18:42,210 --> 00:18:46,550 So the question is how do we find solutions 319 00:18:46,550 --> 00:18:47,880 to coupled oscillations. 320 00:18:47,880 --> 00:18:53,150 What is the most efficient way of finding the most general 321 00:18:53,150 --> 00:18:56,060 motion of a coupled system? 322 00:18:56,060 --> 00:18:57,380 Anybody knows? 323 00:18:57,380 --> 00:18:59,690 What's the first thing? 324 00:18:59,690 --> 00:19:01,002 Yes? 325 00:19:01,002 --> 00:19:02,430 AUDIENCE: [INAUDIBLE]. 326 00:19:02,430 --> 00:19:03,940 BOLESLAW WYSLOUCH: Introduce what? 327 00:19:03,940 --> 00:19:06,031 AUDIENCE: [INAUDIBLE] using complex notation. 328 00:19:06,031 --> 00:19:07,156 BOLESLAW WYSLOUCH: Coupled? 329 00:19:07,156 --> 00:19:07,930 AUDIENCE: Complex. 330 00:19:07,930 --> 00:19:09,990 BOLESLAW WYSLOUCH: Complex oscillation. 331 00:19:09,990 --> 00:19:11,860 Yes, that's right. 332 00:19:11,860 --> 00:19:15,240 So all right, let's do it. 333 00:19:15,240 --> 00:19:17,310 But hold on. 334 00:19:17,310 --> 00:19:22,640 But what form of oscillation? 335 00:19:22,640 --> 00:19:28,100 OK, all kinds of complex numbers can write, but any particular-- 336 00:19:28,100 --> 00:19:28,975 AUDIENCE: [INAUDIBLE] 337 00:19:28,975 --> 00:19:30,475 BOLESLAW WYSLOUCH: That's something. 338 00:19:30,475 --> 00:19:32,450 That's the physics answer, all right? 339 00:19:32,450 --> 00:19:36,080 Complex notation is a mathematical answer, 340 00:19:36,080 --> 00:19:38,220 how to solve a mathematical equation. 341 00:19:38,220 --> 00:19:43,970 But the physics answer is to find fixed frequency modes us 342 00:19:43,970 --> 00:19:46,340 such that the system, the complete system, 343 00:19:46,340 --> 00:19:48,720 oscillates at one frequency. 344 00:19:48,720 --> 00:19:50,670 Everybody moves together. 345 00:19:50,670 --> 00:19:53,050 This is so-called normal mode. 346 00:19:53,050 --> 00:19:55,670 It turns out that every of the system, 347 00:19:55,670 --> 00:19:57,280 depending on number of dimensions, 348 00:19:57,280 --> 00:20:03,880 will have a certain number of frequencies, normal modes, that 349 00:20:03,880 --> 00:20:05,190 would-- 350 00:20:05,190 --> 00:20:08,050 the whole system oscillates at the same frequency, 351 00:20:08,050 --> 00:20:12,850 both x1 and x2, undergoing motion of the same frequency. 352 00:20:12,850 --> 00:20:14,670 We don't know what the frequency is. 353 00:20:14,670 --> 00:20:17,150 We don't know it's amplitude, et cetera. 354 00:20:17,150 --> 00:20:19,450 But it is the same. 355 00:20:19,450 --> 00:20:19,950 OK? 356 00:20:23,930 --> 00:20:28,850 So this means that I can write that the whole vector 357 00:20:28,850 --> 00:20:35,630 x, both x1 and x2, are undergoing the same oscillatory 358 00:20:35,630 --> 00:20:36,650 motion. 359 00:20:36,650 --> 00:20:40,151 So I propose that-- 360 00:20:40,151 --> 00:20:46,010 so of course, we use the usual trick that anytime 361 00:20:46,010 --> 00:20:50,960 we have a solution in complex variables, 362 00:20:50,960 --> 00:20:54,870 we can always get back to real things by taking a real part. 363 00:20:54,870 --> 00:20:58,430 So I understand you've done this before. 364 00:20:58,430 --> 00:21:02,660 So let's introduce variable z, just kind 365 00:21:02,660 --> 00:21:12,910 of a two-element vector, which has a complex term, a fixed 366 00:21:12,910 --> 00:21:16,690 frequency, plus a phase, a rhythm complex, 367 00:21:16,690 --> 00:21:24,700 multiplying vector A, a fixed vector A. OK? 368 00:21:24,700 --> 00:21:36,730 And vector A is simply has two components, A1, A2, or maybe 369 00:21:36,730 --> 00:21:40,290 I should write it differently. 370 00:21:40,290 --> 00:21:45,170 So vector A contains information about some sort 371 00:21:45,170 --> 00:21:51,910 of initial conditions for position x 1 2. 372 00:21:51,910 --> 00:21:55,040 Anyway, these are two constant numbers. 373 00:21:55,040 --> 00:21:59,690 And also, we will, because we have this phase here, 374 00:21:59,690 --> 00:22:02,480 because we keep phase in this expression, 375 00:22:02,480 --> 00:22:06,570 we can assume and require that is a real number. 376 00:22:06,570 --> 00:22:07,510 So A is real. 377 00:22:10,960 --> 00:22:13,980 It's a slightly different way of doing things, 378 00:22:13,980 --> 00:22:17,330 but we can assume this for now, right? 379 00:22:17,330 --> 00:22:20,830 So the solution which is written here-- 380 00:22:20,830 --> 00:22:25,710 it's some two numbers, oscillatory term, 381 00:22:25,710 --> 00:22:30,710 with both x1 and x2 oscillating with the same frequency, 382 00:22:30,710 --> 00:22:32,520 and this is our postulated solution. 383 00:22:32,520 --> 00:22:36,560 So we plug it into the equation, and we adjust things 384 00:22:36,560 --> 00:22:38,570 until it fits. 385 00:22:38,570 --> 00:22:44,060 So let's plug this into our matrix calculation. 386 00:22:44,060 --> 00:22:46,169 And what you see here is that-- 387 00:22:46,169 --> 00:22:46,960 so what do we have? 388 00:22:46,960 --> 00:22:50,870 So this is the term, which is second time 389 00:22:50,870 --> 00:22:54,340 the derivative vector X. And because vector-- 390 00:22:54,340 --> 00:22:55,960 or vector Z really. 391 00:22:55,960 --> 00:22:57,880 So I have to do-- 392 00:22:57,880 --> 00:22:58,770 so I plug this here. 393 00:22:58,770 --> 00:23:08,460 So Z double dot is simply equal minus omega squared times Z. 394 00:23:08,460 --> 00:23:09,498 Right? 395 00:23:09,498 --> 00:23:10,740 Like this. 396 00:23:10,740 --> 00:23:13,050 So this is a simple thing. 397 00:23:13,050 --> 00:23:16,080 When I plug this in here, my equation 398 00:23:16,080 --> 00:23:24,280 becomes an equation for A. So I have minus omega 399 00:23:24,280 --> 00:23:30,060 squared z-hat, which maybe I just write it immediately 400 00:23:30,060 --> 00:23:34,730 in terms of a complex term by times the vector A. 401 00:23:34,730 --> 00:23:45,570 So I have e to i omega t plus y times A is equal to minus M 402 00:23:45,570 --> 00:23:57,071 to minus 1 K times e to the i omega t plus phi times vector 403 00:23:57,071 --> 00:24:00,840 A. OK? 404 00:24:00,840 --> 00:24:04,680 And this term is a proportionality 405 00:24:04,680 --> 00:24:06,870 constant at any given moment of time. 406 00:24:06,870 --> 00:24:10,400 So it goes through the matrix multiplication. 407 00:24:10,400 --> 00:24:11,730 So you can just delete this. 408 00:24:11,730 --> 00:24:13,950 You can divide both sides. 409 00:24:13,950 --> 00:24:16,980 You have signs here. 410 00:24:16,980 --> 00:24:19,870 And then I have an equation which 411 00:24:19,870 --> 00:24:23,430 is a linear matrix equation, which 412 00:24:23,430 --> 00:24:30,570 is M minus 1 K times vector A. 413 00:24:30,570 --> 00:24:33,840 And I can rewrite it a little bit again. 414 00:24:33,840 --> 00:24:40,320 So I can rewrite in this minus 1 K minus omega squared 415 00:24:40,320 --> 00:24:47,860 times unity matrix times vector A is equal to 0. 416 00:24:47,860 --> 00:24:53,110 So this is the equation which we need 417 00:24:53,110 --> 00:25:01,879 to solve to obtain the solutions to at least one normal mode, 418 00:25:01,879 --> 00:25:04,420 and we expect that there will be two normal modes, because we 419 00:25:04,420 --> 00:25:05,086 have two masses. 420 00:25:09,100 --> 00:25:11,680 So now, this is-- 421 00:25:16,250 --> 00:25:19,390 so this is some matrix, two by two matrix, 422 00:25:19,390 --> 00:25:22,120 which we can know very easily how to write. 423 00:25:22,120 --> 00:25:26,070 Multiplying a vector gives you 0. 424 00:25:26,070 --> 00:25:32,790 It turns out that for this to work, there are two-- 425 00:25:32,790 --> 00:25:37,860 there is a criterion, which has to be satisfied, 426 00:25:37,860 --> 00:25:40,270 namely the determinant of the two by two matrix 427 00:25:40,270 --> 00:25:43,740 has to be equal to 0, because if you take 428 00:25:43,740 --> 00:25:47,550 the determinant on both sides, you have to have 0 on this side 429 00:25:47,550 --> 00:25:49,410 to be able to obtain 0 on the other side. 430 00:25:49,410 --> 00:25:53,250 So mathematically, the way to find out the oscillating 431 00:25:53,250 --> 00:25:58,730 frequency is you take a determinant of m minus 1 K 432 00:25:58,730 --> 00:26:04,590 minus i omega squared must be equal to 0. 433 00:26:04,590 --> 00:26:09,580 So let's try to see how to calculate things. 434 00:26:09,580 --> 00:26:19,970 So let's write down this matrix explicitly using this and that. 435 00:26:19,970 --> 00:26:23,840 So let's write this down. 436 00:26:23,840 --> 00:26:28,590 So I take a big object like this. 437 00:26:28,590 --> 00:26:34,490 And so in this element here, I have 438 00:26:34,490 --> 00:26:36,680 to multiply this matrix times that. 439 00:26:36,680 --> 00:26:40,640 If I multiply this matrix, I simply divide 440 00:26:40,640 --> 00:26:43,430 all those effectively multiplication of m 441 00:26:43,430 --> 00:26:47,500 minus 1 times this matrix divides all the elements here 442 00:26:47,500 --> 00:26:49,200 by m. 443 00:26:49,200 --> 00:26:50,290 That's all there is to it. 444 00:26:50,290 --> 00:26:52,210 I just divide everything by m. 445 00:26:54,800 --> 00:27:03,970 So the first M minus 1 K is k over m plus g over l. 446 00:27:03,970 --> 00:27:14,260 This is minus k over m minus k over m k over m plus g over l. 447 00:27:17,338 --> 00:27:18,790 So this is multiplication. 448 00:27:18,790 --> 00:27:21,500 This is this term here. 449 00:27:21,500 --> 00:27:24,450 And then I have to do minus unity matrix times omega 450 00:27:24,450 --> 00:27:25,360 squared. 451 00:27:25,360 --> 00:27:32,690 All this will do is it will subtract omega squared here. 452 00:27:32,690 --> 00:27:33,880 I should write this. 453 00:27:38,300 --> 00:27:39,480 OK? 454 00:27:39,480 --> 00:27:40,510 So this is in this one. 455 00:27:40,510 --> 00:27:46,610 Maybe it would be more clear if I move it over here. 456 00:27:46,610 --> 00:27:48,480 All right, so this is the matrix that 457 00:27:48,480 --> 00:27:51,760 contains all the information about our system, 458 00:27:51,760 --> 00:27:54,725 the mass, the gravitational acceleration, the length, 459 00:27:54,725 --> 00:27:57,090 the spring strength, et cetera. 460 00:27:57,090 --> 00:28:03,180 And we assumed they oscillate with a fixed frequency. 461 00:28:03,180 --> 00:28:09,540 So I have to find the determinant of this matrix 462 00:28:09,540 --> 00:28:11,790 equal to 0. 463 00:28:11,790 --> 00:28:13,330 So how do I get that? 464 00:28:13,330 --> 00:28:18,440 And by the way, you have a matrix, 465 00:28:18,440 --> 00:28:21,560 and you want to make sure that its determinant is 0. 466 00:28:21,560 --> 00:28:23,910 It turns out the only variable which 467 00:28:23,910 --> 00:28:27,870 we have to change parameters of this matrix-- 468 00:28:27,870 --> 00:28:30,440 you know, the spring constant and the mass this affects 469 00:28:30,440 --> 00:28:31,470 is given. 470 00:28:31,470 --> 00:28:34,440 The system has been built. It's hanging over there. 471 00:28:34,440 --> 00:28:36,400 I cannot change anything. 472 00:28:36,400 --> 00:28:40,590 So the only parameter here, which I can change, or adjust, 473 00:28:40,590 --> 00:28:43,560 or find is omega square. 474 00:28:43,560 --> 00:28:46,980 So I will try all possible matrices of this type 475 00:28:46,980 --> 00:28:52,740 until I find one or two that have a determinant equal to 0. 476 00:28:52,740 --> 00:28:55,430 But if I find them, this would correspond 477 00:28:55,430 --> 00:28:58,905 to the normal frequencies. 478 00:28:58,905 --> 00:29:01,230 OK? 479 00:29:01,230 --> 00:29:04,770 So how do I calculate the determinant of a two 480 00:29:04,770 --> 00:29:05,980 by two matrix? 481 00:29:05,980 --> 00:29:10,280 I do this by this minus this by that, right? 482 00:29:10,280 --> 00:29:15,590 So that of this matrix is equal to k over 483 00:29:15,590 --> 00:29:24,110 m minus g plus g over l minus omega squared. 484 00:29:24,110 --> 00:29:27,280 The two identical terms so I can put the square 485 00:29:27,280 --> 00:29:32,960 and then minus this minus k squared over m squared 486 00:29:32,960 --> 00:29:35,620 must be equal to 0. 487 00:29:35,620 --> 00:29:36,120 Right? 488 00:29:36,120 --> 00:29:40,710 So this is the equation which we need to solve. 489 00:29:40,710 --> 00:29:45,960 We need to find which parameter omega sets this to 0. 490 00:29:45,960 --> 00:29:53,440 And then this is a pretty straightforward calculation, 491 00:29:53,440 --> 00:29:57,610 except if I don't have-- 492 00:29:57,610 --> 00:29:59,220 I'll just use this one. 493 00:30:02,650 --> 00:30:05,080 OK, so let's rewrite this a little bit. 494 00:30:05,080 --> 00:30:08,680 So this is basically equivalent to the following equation 495 00:30:08,680 --> 00:30:16,120 g over l plus k over m minus omega squared 496 00:30:16,120 --> 00:30:21,080 must be equal either to plus or minus k over m. 497 00:30:21,080 --> 00:30:21,580 Right? 498 00:30:21,580 --> 00:30:23,960 I took a square root of both sides. 499 00:30:23,960 --> 00:30:25,690 If you take a square root, you have 500 00:30:25,690 --> 00:30:29,150 to worry about plus and minus signs, right? 501 00:30:29,150 --> 00:30:32,200 So there are two solutions which corresponds to plus here. 502 00:30:32,200 --> 00:30:34,660 The other one corresponds to minus here. 503 00:30:34,660 --> 00:30:37,840 So solution number 1, which corresponds 504 00:30:37,840 --> 00:30:42,760 to plus sign right here, it basically 505 00:30:42,760 --> 00:30:49,340 says that omega squared is equal to g over l. 506 00:30:49,340 --> 00:30:49,840 Right? 507 00:30:49,840 --> 00:30:53,620 So there is one solution, one oscillation, 508 00:30:53,620 --> 00:30:55,780 that does not depend on the spring constant, 509 00:30:55,780 --> 00:30:58,390 because the spring constant cancels. 510 00:30:58,390 --> 00:31:01,960 And there's a second solution which 511 00:31:01,960 --> 00:31:09,340 corresponds to minus, where omega squared is equal to g 512 00:31:09,340 --> 00:31:14,640 over l plus 2k over m. 513 00:31:17,630 --> 00:31:18,440 Right? 514 00:31:18,440 --> 00:31:21,740 Because there are two possible solutions. 515 00:31:21,740 --> 00:31:23,340 And this is what we have. 516 00:31:23,340 --> 00:31:26,970 So we have a-- 517 00:31:26,970 --> 00:31:33,420 so what this says is that if I set my frequency to g over l, 518 00:31:33,420 --> 00:31:36,450 if I set the system to oscillate to this frequency, 519 00:31:36,450 --> 00:31:40,260 then it will be-- 520 00:31:40,260 --> 00:31:45,030 I will be able to set things up such that it oscillates forever 521 00:31:45,030 --> 00:31:48,730 at this frequency, one fixed frequency forever. 522 00:31:48,730 --> 00:31:49,730 And this is interesting. 523 00:31:49,730 --> 00:31:50,420 This is a frequency. 524 00:31:50,420 --> 00:31:52,530 It does not depend on the strength of the spring. 525 00:31:52,530 --> 00:31:54,500 How is it possible? 526 00:31:54,500 --> 00:31:57,380 Somehow spring is irrelevant for this motion. 527 00:31:57,380 --> 00:32:00,060 And it turns out that there is a very simple oscillation, 528 00:32:00,060 --> 00:32:02,570 easy to see, if basically that this 529 00:32:02,570 --> 00:32:05,160 is a frequency of a single pendulum. 530 00:32:05,160 --> 00:32:06,680 So basically, you got both pendula 531 00:32:06,680 --> 00:32:10,520 going together, each of them happily 532 00:32:10,520 --> 00:32:13,160 oscillating by themselves. 533 00:32:13,160 --> 00:32:16,020 And the spring is completely irrelevant for this motion. 534 00:32:16,020 --> 00:32:18,700 If I cut it off, the motion will not change. 535 00:32:18,700 --> 00:32:22,030 It just happens that two identical pendula are going 536 00:32:22,030 --> 00:32:24,570 at their own natural frequency. 537 00:32:24,570 --> 00:32:26,600 So the force of spring is irrelevant. 538 00:32:26,600 --> 00:32:27,360 Nothing happens. 539 00:32:27,360 --> 00:32:28,730 This is a normal mode. 540 00:32:28,730 --> 00:32:32,770 And it can go forever at this particular frequency. 541 00:32:32,770 --> 00:32:33,690 OK? 542 00:32:33,690 --> 00:32:38,120 The other option is usually symmetrically. 543 00:32:38,120 --> 00:32:41,280 I move them away from each other. 544 00:32:41,280 --> 00:32:45,200 And this is the motion where, again, it's not exactly 545 00:32:45,200 --> 00:32:50,710 ideal small angle oscillation, but let me try again, 546 00:32:50,710 --> 00:32:52,730 I guess with less. 547 00:32:52,730 --> 00:32:54,830 So this is the situation where the spring really 548 00:32:54,830 --> 00:32:56,920 comes in at full force. 549 00:32:56,920 --> 00:33:00,144 It's being stretched maximally, because they go away 550 00:33:00,144 --> 00:33:00,810 from each other. 551 00:33:00,810 --> 00:33:03,360 So very quickly, the spring is stretched. 552 00:33:03,360 --> 00:33:07,060 And they go together so it's stretch from both sides. 553 00:33:07,060 --> 00:33:10,940 And the whole system oscillates at the same frequency, 554 00:33:10,940 --> 00:33:15,760 and because of this additional force of spring, 555 00:33:15,760 --> 00:33:19,670 the frequency is actually higher, it's larger. 556 00:33:19,670 --> 00:33:23,260 It oscillates faster. 557 00:33:23,260 --> 00:33:25,210 All right, so that's the first step 558 00:33:25,210 --> 00:33:26,590 in understanding the system. 559 00:33:26,590 --> 00:33:30,370 We now know that there are two oscillations and two 560 00:33:30,370 --> 00:33:31,880 normal frequencies. 561 00:33:31,880 --> 00:33:34,600 And the next step to finish our understanding 562 00:33:34,600 --> 00:33:37,560 of the system in a mathematical way, to describe it fully, 563 00:33:37,560 --> 00:33:40,810 I have to know what is the shape of oscillations. 564 00:33:40,810 --> 00:33:44,140 I simply showed you here so you know what to expect. 565 00:33:44,140 --> 00:33:48,910 But I have to be able to dig it out from the equations. 566 00:33:48,910 --> 00:33:53,800 And the way to dig it out is to find vector A. See, 567 00:33:53,800 --> 00:34:01,360 our real equation of motion is up here. 568 00:34:01,360 --> 00:34:02,950 This is an equation of motion. 569 00:34:02,950 --> 00:34:08,710 This is, I have to now find the vector A, which 570 00:34:08,710 --> 00:34:11,620 when you plug it in, it works-- 571 00:34:11,620 --> 00:34:13,810 it satisfies this equation. 572 00:34:13,810 --> 00:34:16,929 So I already know what are the two possible omegas-- 573 00:34:16,929 --> 00:34:21,219 they can do it, but still I have to find vector A. 574 00:34:21,219 --> 00:34:23,650 So I have to solve two separate independent problems. 575 00:34:23,650 --> 00:34:26,600 One is finding vector A for this situation 576 00:34:26,600 --> 00:34:28,900 and then find the vector A for that situation 577 00:34:28,900 --> 00:34:30,070 and see if it works. 578 00:34:30,070 --> 00:34:31,969 So I had to plug in the whole. 579 00:34:31,969 --> 00:34:34,980 I had to plug it into the whole equation. 580 00:34:34,980 --> 00:34:38,440 And you can show that if you set-- 581 00:34:38,440 --> 00:34:45,730 if you set omega squared to g over l, and you plug it into-- 582 00:34:45,730 --> 00:34:47,860 if you plug it into this equation, what 583 00:34:47,860 --> 00:34:50,230 you get is a matrix equation which 584 00:34:50,230 --> 00:34:56,600 looks like this-- k over m minus k over m 585 00:34:56,600 --> 00:35:01,050 minus k over m k over m. 586 00:35:01,050 --> 00:35:03,320 And this is because-- 587 00:35:03,320 --> 00:35:08,760 [I try to-- so if you plug omega squared here equal to g over 588 00:35:08,760 --> 00:35:14,120 l, then this cancels out, and this cancels out. 589 00:35:14,120 --> 00:35:16,420 So you plug it in here, and you get 590 00:35:16,420 --> 00:35:22,660 this very simple, very simple matrix that has k over m terms. 591 00:35:22,660 --> 00:35:25,210 So the question is what sort of thing 592 00:35:25,210 --> 00:35:29,660 can you put here to get 0. 593 00:35:29,660 --> 00:35:34,750 What kind of vector you can plug into those two places 594 00:35:34,750 --> 00:35:40,450 such that the matrix times vector will end up with 0? 595 00:35:40,450 --> 00:35:45,250 One example is that basically amplitude is the same. 596 00:35:45,250 --> 00:35:47,070 Both of them move together. 597 00:35:47,070 --> 00:35:51,131 So you plug 1 here and 1 here. 598 00:35:51,131 --> 00:35:51,630 Right? 599 00:35:51,630 --> 00:35:54,890 So this is a good solution. 600 00:35:54,890 --> 00:35:58,880 And every other solution is a linear multiplication 601 00:35:58,880 --> 00:36:01,220 of this one for this frequency, right? 602 00:36:01,220 --> 00:36:04,790 There is k over m times 1 minus k over m gives you 0. 603 00:36:04,790 --> 00:36:07,280 So this is a good solution for-- 604 00:36:07,280 --> 00:36:10,250 so this is solution number 1. 605 00:36:10,250 --> 00:36:12,350 What about this thing here? 606 00:36:12,350 --> 00:36:23,650 If I plug this omega squared into this matrix, 607 00:36:23,650 --> 00:36:26,300 it's g over l plus 2k over m. 608 00:36:26,300 --> 00:36:32,330 If I plug it in here, then this matrix is way more complicated. 609 00:36:32,330 --> 00:36:34,470 It will actually look very similar, 610 00:36:34,470 --> 00:36:37,780 but with important differences. 611 00:36:37,780 --> 00:36:42,030 So this one will look minus k over m 612 00:36:42,030 --> 00:36:49,346 minus k over m minus k over m minus k over m. 613 00:36:49,346 --> 00:36:52,240 OK? 614 00:36:52,240 --> 00:36:58,430 And then again, for this second possible normal frequency, 615 00:36:58,430 --> 00:37:01,310 I have to find the vector A, which corresponds 616 00:37:01,310 --> 00:37:02,510 to that frequency motion. 617 00:37:02,510 --> 00:37:07,260 And it turns out that they are the same, but the sign changes. 618 00:37:07,260 --> 00:37:13,620 So one possible solution is 1 and minus 1. 619 00:37:13,620 --> 00:37:18,450 If I plug in 1 minus 1, then this matrix times the vector 620 00:37:18,450 --> 00:37:20,620 gives you automatically 0. 621 00:37:20,620 --> 00:37:23,680 So this is the second possible normal mode. 622 00:37:23,680 --> 00:37:24,640 All right? 623 00:37:24,640 --> 00:37:28,390 So this is a systematic way to solve equations. 624 00:37:28,390 --> 00:37:31,090 You plug in all the information you 625 00:37:31,090 --> 00:37:34,240 know about the system into a two by two matrix. 626 00:37:34,240 --> 00:37:38,110 And then you calculate the normal mode. 627 00:37:38,110 --> 00:37:41,284 And then you calculate a shape of a normal mode. 628 00:37:43,950 --> 00:37:47,770 Is that clear? 629 00:37:47,770 --> 00:37:51,570 Any questions at this time? 630 00:37:51,570 --> 00:37:52,710 Right? 631 00:37:52,710 --> 00:37:55,240 So in principle, we know, now, at the end of the day, 632 00:37:55,240 --> 00:37:58,824 I still want to know how much 1 moves, how much 2 moves. 633 00:37:58,824 --> 00:38:00,240 So we have to put it all together. 634 00:38:00,240 --> 00:38:04,560 We have identified the frequency and the kind of, in the matrix 635 00:38:04,560 --> 00:38:06,420 notation, shape of the node. 636 00:38:06,420 --> 00:38:08,130 But of course, the final solution 637 00:38:08,130 --> 00:38:11,790 is a linear superposition of all possible normal modes 638 00:38:11,790 --> 00:38:14,560 with described position of mass 1, 639 00:38:14,560 --> 00:38:15,810 position of mass 2, et cetera. 640 00:38:15,810 --> 00:38:19,590 So let's do a little bit of-- 641 00:38:19,590 --> 00:38:23,700 so maybe graphically I can write down that this is the-- 642 00:38:23,700 --> 00:38:27,300 this is the oscillation that corresponds 643 00:38:27,300 --> 00:38:32,130 to this type of mode, to those two masses move together. 644 00:38:32,130 --> 00:38:34,920 And this is oscillation that corresponds to the mode 645 00:38:34,920 --> 00:38:40,240 where masses move in opposite directions. 646 00:38:40,240 --> 00:38:45,120 At any moment of time, in this normal mode, 647 00:38:45,120 --> 00:38:49,360 at any moment of time, wherever mass 1 is, 648 00:38:49,360 --> 00:38:53,830 mass 2 is minus the distance away from its own equilibrium. 649 00:38:53,830 --> 00:38:56,080 So if this one is plus 1 centimeter here, 650 00:38:56,080 --> 00:38:57,580 the other one is minus 1 centimeter. 651 00:38:57,580 --> 00:38:58,770 This one is minus 5. 652 00:38:58,770 --> 00:39:01,250 This one is plus 5 and so on. 653 00:39:01,250 --> 00:39:05,396 Whereas in this mode, both of them move together. 654 00:39:05,396 --> 00:39:06,140 All right? 655 00:39:06,140 --> 00:39:11,730 So let's try to go back to the-- 656 00:39:11,730 --> 00:39:13,510 you can get rid of this one. 657 00:39:13,510 --> 00:39:16,740 Let's try to go back, and now with this knowledge, 658 00:39:16,740 --> 00:39:24,154 let's write down x1 and x2 for positions of the two masses. 659 00:39:33,270 --> 00:39:45,306 So x1-- so basically, the x will have to be-- 660 00:39:45,306 --> 00:39:46,630 I used z there. 661 00:39:46,630 --> 00:39:52,510 So x will be real of vector z. 662 00:39:52,510 --> 00:39:55,850 So I take my complex numbers and take a real of them. 663 00:39:55,850 --> 00:39:59,470 So from an exponent, I will end up with a cosine 664 00:39:59,470 --> 00:40:00,640 appropriately and so on. 665 00:40:00,640 --> 00:40:03,980 And then I will use the [INAUDIBLE].. 666 00:40:03,980 --> 00:40:10,030 So this is real part of e to the i omega 667 00:40:10,030 --> 00:40:14,530 plus phi where omega is one of the two possibilities. 668 00:40:14,530 --> 00:40:21,980 Omega t plus phi times vector A, which we've identified here, 669 00:40:21,980 --> 00:40:26,200 and times some additional-- 670 00:40:26,200 --> 00:40:30,700 these are those vectors A this is one possible amplitude 671 00:40:30,700 --> 00:40:31,830 of notation. 672 00:40:31,830 --> 00:40:34,090 But in general, it can be anything. 673 00:40:34,090 --> 00:40:34,810 You can multiply. 674 00:40:34,810 --> 00:40:38,210 You can have small oscillations, large oscillations. 675 00:40:38,210 --> 00:40:40,000 So there is some overall amplitude. 676 00:40:40,000 --> 00:40:41,660 But the shape always has to be simple. 677 00:40:41,660 --> 00:40:45,430 They either go together, or they go opposite. 678 00:40:45,430 --> 00:40:46,880 So to make it more general, I have 679 00:40:46,880 --> 00:40:48,680 to give some multiplicative factor there. 680 00:40:51,670 --> 00:40:56,560 So if I do everything, I end up with x, the mode 1 681 00:40:56,560 --> 00:41:00,610 will in general have some sort of overall constant C, 682 00:41:00,610 --> 00:41:08,900 cosine omega 1 t plus phi 1 times the vector 1, 1. 683 00:41:08,900 --> 00:41:10,220 This will be for x1. 684 00:41:10,220 --> 00:41:12,020 This will be for x2. 685 00:41:12,020 --> 00:41:21,660 And the mode number 2 will be C2 cosine omega 2 times 686 00:41:21,660 --> 00:41:27,170 t plus phi 2 times 1 minus 1. 687 00:41:27,170 --> 00:41:28,970 All right? 688 00:41:28,970 --> 00:41:32,150 So let's see what things are adjustable 689 00:41:32,150 --> 00:41:34,110 and what things are fixed. 690 00:41:34,110 --> 00:41:37,370 So the omega 1 and omega 2 are fixed 691 00:41:37,370 --> 00:41:41,900 given by the construction of the two coupled oscillators. 692 00:41:41,900 --> 00:41:44,060 This shape, 1 and 1, and 1 minus 1 693 00:41:44,060 --> 00:41:47,060 is fixed, because these are the shape of normal modes, which 694 00:41:47,060 --> 00:41:48,930 corresponds to those frequencies. 695 00:41:48,930 --> 00:41:52,670 So we have only four constants-- overall amplitude c1 696 00:41:52,670 --> 00:41:54,040 for normal mode 1. 697 00:41:54,040 --> 00:41:56,930 Overall amplitude c2 for normal mode 2 and then 698 00:41:56,930 --> 00:42:01,010 the relative phase of those two normal modes. 699 00:42:01,010 --> 00:42:03,260 And the superposition of x1 plus x2 700 00:42:03,260 --> 00:42:08,790 gives you the most general combination of possible motion. 701 00:42:08,790 --> 00:42:11,090 So if I write this down now in terms 702 00:42:11,090 --> 00:42:14,540 of position of number 1 and number 2, 703 00:42:14,540 --> 00:42:18,332 so I have a position of x1 as a function of time. 704 00:42:18,332 --> 00:42:19,790 In general, it will look like this. 705 00:42:19,790 --> 00:42:24,690 It will be some sort of constant alpha, cosine omega 1 t 706 00:42:24,690 --> 00:42:32,720 plus phi plus constant beta cosine omega 2 times 707 00:42:32,720 --> 00:42:36,610 t plus phi 2 plus phi 1. 708 00:42:36,610 --> 00:42:40,670 So mass number 1, this is position of mass 1, 709 00:42:40,670 --> 00:42:45,400 will in general be a superposition of the two 710 00:42:45,400 --> 00:42:47,740 possible oscillations. 711 00:42:47,740 --> 00:42:54,000 The position of mass 2 will be very similar, 712 00:42:54,000 --> 00:42:55,970 but there will be a very important difference 713 00:42:55,970 --> 00:43:00,820 between the alpha cosine omega 1 t 714 00:43:00,820 --> 00:43:08,340 plus phi 1 minus beta cosine omega 2t plus phi 2. 715 00:43:10,870 --> 00:43:13,900 This is very important to understand exactly how 716 00:43:13,900 --> 00:43:16,350 this equation came about. 717 00:43:16,350 --> 00:43:19,470 You see, this is the influence of the symmetric mode, 718 00:43:19,470 --> 00:43:21,550 where the two things are together. 719 00:43:21,550 --> 00:43:23,880 So they are multiplied by alpha, some sort 720 00:43:23,880 --> 00:43:30,550 of arbitrary constant, but with exactly the same sign. 721 00:43:30,550 --> 00:43:32,290 And this is the part which corresponds 722 00:43:32,290 --> 00:43:37,040 to a second mode, which is with different frequencies. 723 00:43:37,040 --> 00:43:39,600 And there is an opposite sign between this amplitude 724 00:43:39,600 --> 00:43:41,250 and that amplitude. 725 00:43:41,250 --> 00:43:47,100 So you have only four coefficients-- 726 00:43:47,100 --> 00:43:53,970 alpha, beta, phi 1, and phi 2, which are determined, 727 00:43:53,970 --> 00:43:56,028 which need initial conditions. 728 00:44:03,210 --> 00:44:06,770 So any arbitrary mode-- this is the most general motion 729 00:44:06,770 --> 00:44:10,550 of the two coupled oscillator systems. 730 00:44:10,550 --> 00:44:14,100 And to describe it in specifically-- defined 731 00:44:14,100 --> 00:44:16,250 for a specific configuration, you 732 00:44:16,250 --> 00:44:20,540 will have to determine the values of alphas and phis. 733 00:44:20,540 --> 00:44:22,560 OK? 734 00:44:22,560 --> 00:44:28,940 So what I want to do is I want to write down a specific motion 735 00:44:28,940 --> 00:44:31,790 for the following situation. 736 00:44:31,790 --> 00:44:37,040 So I keep position of x1 at 0. 737 00:44:37,040 --> 00:44:41,410 It's not moving, so the velocity is 0. 738 00:44:41,410 --> 00:44:45,760 I displaced this one by a small positive amount. 739 00:44:45,760 --> 00:44:51,270 So the position of number 2 at t equals 0 is different than 0-- 740 00:44:51,270 --> 00:44:53,970 some displacement x0 or something. 741 00:44:53,970 --> 00:44:55,645 And its velocity is 0. 742 00:44:55,645 --> 00:44:56,520 And then I let it go. 743 00:44:59,230 --> 00:45:03,690 Again, this is not the ideal decoupled oscillator, right? 744 00:45:03,690 --> 00:45:08,220 OK, and then you see the things start moving. 745 00:45:08,220 --> 00:45:11,980 Let me try to show it again, because it's not exactly here, 746 00:45:11,980 --> 00:45:13,520 so this one will be going on. 747 00:45:13,520 --> 00:45:15,580 So let's say this one is running, 748 00:45:15,580 --> 00:45:18,550 and then I let this one go. 749 00:45:18,550 --> 00:45:23,390 And what you see here is that this one is moving, 750 00:45:23,390 --> 00:45:24,740 and then that starts to move. 751 00:45:24,740 --> 00:45:26,170 This one stops. 752 00:45:26,170 --> 00:45:27,630 That starts moving. 753 00:45:27,630 --> 00:45:29,460 It starts being complicated, right? 754 00:45:29,460 --> 00:45:31,860 It's kind of complicated motion. 755 00:45:31,860 --> 00:45:35,070 But whatever this motion is, we know 756 00:45:35,070 --> 00:45:38,880 that it's simply those cosines which are kind of adding up 757 00:45:38,880 --> 00:45:42,330 to give you this impression of rather a complicated motion, 758 00:45:42,330 --> 00:45:43,360 right? 759 00:45:43,360 --> 00:45:46,515 So again, I let this one out. 760 00:45:46,515 --> 00:45:47,710 I let it go. 761 00:45:47,710 --> 00:45:50,870 This might be 0. 762 00:45:50,870 --> 00:45:52,160 So this one slows down. 763 00:45:52,160 --> 00:45:55,630 This starts going. 764 00:45:55,630 --> 00:45:57,280 And this one then slows down. 765 00:45:57,280 --> 00:45:59,290 The other one starts going. 766 00:45:59,290 --> 00:46:02,600 They kind of talk to each other. 767 00:46:02,600 --> 00:46:05,670 And it's this combination of cosines. 768 00:46:05,670 --> 00:46:08,120 All right, so let's try to write to simplify 769 00:46:08,120 --> 00:46:11,530 this for a specific case of specific initial conditions. 770 00:46:26,910 --> 00:46:34,000 So I said x1 equals 0, to equal 0 x1 velocity at 0 771 00:46:34,000 --> 00:46:35,480 is equal to 0. 772 00:46:35,480 --> 00:46:37,790 So those ones are not moving. 773 00:46:37,790 --> 00:46:46,470 X2 at 0 is equal to some sort of x0 and x2 velocity at 0 774 00:46:46,470 --> 00:46:47,780 is equal to 0. 775 00:46:47,780 --> 00:46:49,110 So this one is displaced. 776 00:46:49,110 --> 00:46:50,510 They are all stationary. 777 00:46:50,510 --> 00:46:51,670 This one is at position 0. 778 00:46:51,670 --> 00:46:56,540 If I plug this in, it turns out without lots of details 779 00:46:56,540 --> 00:47:02,060 that what you will get to is that alpha will 780 00:47:02,060 --> 00:47:06,690 be equal to x0 divided by 2. 781 00:47:06,690 --> 00:47:10,850 Beta will be equal to minus x0 divided by 2. 782 00:47:10,850 --> 00:47:14,840 And phi 1 will be equal to phi 2 equal to 0. 783 00:47:14,840 --> 00:47:16,100 You can check. 784 00:47:16,100 --> 00:47:18,380 If you plug it into those equations, 785 00:47:18,380 --> 00:47:23,120 if you plug t equals 0, phi is equal to 0, et cetera, 786 00:47:23,120 --> 00:47:24,650 you will see that it works. 787 00:47:24,650 --> 00:47:28,850 So you can write down the specific case of x1 of t 788 00:47:28,850 --> 00:47:41,726 to be x0 over 2 cosine omega1 t minus cosine omega2 t. 789 00:47:41,726 --> 00:47:44,640 It's because beta has a negative sign. 790 00:47:44,640 --> 00:47:57,770 And x2 of t will be equal to x0 over 2 cosine omega1 t 791 00:47:57,770 --> 00:48:01,860 plus cosine omega2 t. 792 00:48:01,860 --> 00:48:03,640 OK? 793 00:48:03,640 --> 00:48:06,580 So each of those objects effectively feels 794 00:48:06,580 --> 00:48:09,460 the effects of omega 1 and omega 2, 795 00:48:09,460 --> 00:48:11,860 but in a slightly different way. 796 00:48:11,860 --> 00:48:14,470 That's why their relative motions are different. 797 00:48:14,470 --> 00:48:19,600 So what I will do now is I will show you an animation. 798 00:48:19,600 --> 00:48:22,180 Hopefully, it works. 799 00:48:22,180 --> 00:48:24,970 And we will have time-- 800 00:48:24,970 --> 00:48:28,850 since on the computer, you can make things perfect. 801 00:48:28,850 --> 00:48:29,380 Let's do it. 802 00:48:29,380 --> 00:48:33,640 So I'll have-- running a MatLab simulation. 803 00:48:33,640 --> 00:48:36,210 Let's see how it goes. 804 00:48:36,210 --> 00:48:38,560 Large. 805 00:48:38,560 --> 00:48:40,680 So what is going on here is the following. 806 00:48:40,680 --> 00:48:42,050 I took some initial conditions. 807 00:48:42,050 --> 00:48:43,850 I'm not sure if it's exactly the same. 808 00:48:43,850 --> 00:48:47,050 This was for the course that I taught some time ago. 809 00:48:47,050 --> 00:48:49,840 What you see here is the following-- 810 00:48:49,840 --> 00:48:56,660 you have the green is the normal mode, number 1. 811 00:48:59,570 --> 00:49:03,810 The magenta is normal mode number 2. 812 00:49:03,810 --> 00:49:07,080 And blue and the red are the actual pendula. 813 00:49:09,900 --> 00:49:10,740 All right? 814 00:49:10,740 --> 00:49:13,970 And the motion of blue and red is simply 815 00:49:13,970 --> 00:49:16,920 a linear sum of the two. 816 00:49:16,920 --> 00:49:18,510 And what you see here is-- 817 00:49:18,510 --> 00:49:21,590 and then I plot the position of the blue and red 818 00:49:21,590 --> 00:49:24,390 in color, the function of time. 819 00:49:24,390 --> 00:49:25,190 So you see this-- 820 00:49:25,190 --> 00:49:28,750 the fact that let's say red is now stopped, 821 00:49:28,750 --> 00:49:30,640 and the blue is at maximum. 822 00:49:30,640 --> 00:49:34,890 And now, the red is picking up. 823 00:49:34,890 --> 00:49:38,430 And now the blue stopped, and the red 824 00:49:38,430 --> 00:49:40,890 is going full swing, et cetera. 825 00:49:40,890 --> 00:49:43,070 And this is exactly what-- 826 00:49:43,070 --> 00:49:46,269 this is the computer simulation that shows you that one of them 827 00:49:46,269 --> 00:49:48,060 is going up, the other one down, et cetera. 828 00:49:48,060 --> 00:49:49,874 And this is for the certain combination 829 00:49:49,874 --> 00:49:50,790 of initial conditions. 830 00:49:50,790 --> 00:49:54,210 I could go change initial conditions in my program 831 00:49:54,210 --> 00:49:55,830 and have a different behavior. 832 00:49:55,830 --> 00:49:59,850 But whatever happens, I would be able to-- 833 00:49:59,850 --> 00:50:05,430 it will always be a combination of the two motions. 834 00:50:05,430 --> 00:50:14,150 Now, is there a way to disable one of the normal modes? 835 00:50:14,150 --> 00:50:16,204 How would you disable one of the normal modes? 836 00:50:19,452 --> 00:50:21,400 Is there a quick way to set things 837 00:50:21,400 --> 00:50:26,700 up such that the second normal mode, whichever you choose, 838 00:50:26,700 --> 00:50:28,450 doesn't show up in their equations at all? 839 00:50:34,294 --> 00:50:36,729 AUDIENCE: You said [INAUDIBLE]. 840 00:50:36,729 --> 00:50:37,703 BOLESLAW WYSLOUCH: Hmm? 841 00:50:37,703 --> 00:50:40,138 AUDIENCE: [INAUDIBLE] 842 00:50:44,934 --> 00:50:46,600 BOLESLAW WYSLOUCH: Yeah, so what you do, 843 00:50:46,600 --> 00:50:48,500 is you just change the initial conditions. 844 00:50:48,500 --> 00:50:50,170 So you set it up at T equal to 0. 845 00:50:50,170 --> 00:50:55,690 I have initial conditions that basically favor or demand 846 00:50:55,690 --> 00:50:59,320 that only in this general equation 847 00:50:59,320 --> 00:51:02,080 either alpha or beta is equal to 0. 848 00:51:02,080 --> 00:51:08,740 So for example, one possibility is I move both of them 849 00:51:08,740 --> 00:51:11,680 at the same distance, and I just let them go like this such 850 00:51:11,680 --> 00:51:15,010 that the spring is irrelevant, right? 851 00:51:15,010 --> 00:51:17,679 How would I do it in my program? 852 00:51:17,679 --> 00:51:18,220 I don't know. 853 00:51:18,220 --> 00:51:21,440 I can, for example-- 854 00:51:21,440 --> 00:51:28,510 I can, for example, set one of the initial conditions to-- 855 00:51:31,730 --> 00:51:32,670 this is still running. 856 00:51:32,670 --> 00:51:35,230 The old one is still running. 857 00:51:35,230 --> 00:51:40,151 So this is the moment. 858 00:51:40,151 --> 00:51:42,400 So what I did is I just changed the initial condition. 859 00:51:44,960 --> 00:51:49,370 And you see, this is the type of motion where one of the modes 860 00:51:49,370 --> 00:51:52,415 has stopped, just you switched it off, 861 00:51:52,415 --> 00:51:54,540 and the other one is going on, and then, of course, 862 00:51:54,540 --> 00:51:57,030 the total motion is equal to that. 863 00:51:57,030 --> 00:51:59,720 And both of them happily go with a constant amplitude. 864 00:51:59,720 --> 00:52:05,340 There is no shifting of energy from one to another. 865 00:52:05,340 --> 00:52:10,130 So you can have all kinds of motions by simply adjusting 866 00:52:10,130 --> 00:52:11,060 initial conditions. 867 00:52:11,060 --> 00:52:16,640 And those motions can be done a very different way. 868 00:52:16,640 --> 00:52:19,810 So do you know-- 869 00:52:19,810 --> 00:52:23,780 so this is how we can have different shape of motion, 870 00:52:23,780 --> 00:52:26,300 depending on the initial condition. 871 00:52:26,300 --> 00:52:30,830 Is there another way for me to change 872 00:52:30,830 --> 00:52:34,551 the way this system behaves? 873 00:52:34,551 --> 00:52:35,300 Let's say I take-- 874 00:52:35,300 --> 00:52:37,740 I have exactly this system, and I 875 00:52:37,740 --> 00:52:41,220 want to change, for example, the frequency of oscillations. 876 00:52:41,220 --> 00:52:44,200 How will I do it? 877 00:52:44,200 --> 00:52:46,838 It could be a very expensive proposition, yes? 878 00:52:46,838 --> 00:52:47,786 AUDIENCE: Drive it? 879 00:52:47,786 --> 00:52:51,290 BOLESLAW WYSLOUCH: Yes, but I don't want to drive it yet. 880 00:52:51,290 --> 00:52:54,100 I just want to have it free oscillation. 881 00:52:54,100 --> 00:52:55,014 Yes? 882 00:52:55,014 --> 00:52:57,309 AUDIENCE: [INAUDIBLE] 883 00:52:57,309 --> 00:52:58,850 BOLESLAW WYSLOUCH: Yeah, I could come 884 00:52:58,850 --> 00:53:00,930 and scratch it away a little bit. 885 00:53:00,930 --> 00:53:03,580 And yes, the equations depends on the mass. 886 00:53:03,580 --> 00:53:04,720 But I don't want to touch. 887 00:53:04,720 --> 00:53:06,494 I want to just have this thing. 888 00:53:06,494 --> 00:53:08,410 I don't want to make any physical modification 889 00:53:08,410 --> 00:53:09,610 to the system. 890 00:53:09,610 --> 00:53:12,302 However, I can move it into different places, 891 00:53:12,302 --> 00:53:14,260 any place you can think of where I could really 892 00:53:14,260 --> 00:53:16,440 modify the solution. 893 00:53:16,440 --> 00:53:17,083 Yeah? 894 00:53:17,083 --> 00:53:18,030 AUDIENCE: To the moon. 895 00:53:18,030 --> 00:53:19,696 BOLESLAW WYSLOUCH: To the moon, exactly. 896 00:53:19,696 --> 00:53:22,250 I could put it with me some spaceship, and go to a place 897 00:53:22,250 --> 00:53:25,160 where the gravity is different, right? 898 00:53:25,160 --> 00:53:26,690 Why not? 899 00:53:26,690 --> 00:53:28,480 So what would happen? 900 00:53:28,480 --> 00:53:35,440 So if gravity changes, then basically what will happen 901 00:53:35,440 --> 00:53:38,044 is both this term and that term will change. 902 00:53:38,044 --> 00:53:39,460 The spring will remained the same. 903 00:53:39,460 --> 00:53:40,960 The mass will remain the same. 904 00:53:40,960 --> 00:53:44,560 So the relative magnitude of omega 1 and omega 2 905 00:53:44,560 --> 00:53:46,900 will change. 906 00:53:46,900 --> 00:53:48,130 OK? 907 00:53:48,130 --> 00:53:51,910 So let's say, in fact, do I have it in this one here? 908 00:53:51,910 --> 00:53:52,430 Yes. 909 00:53:52,430 --> 00:53:56,060 So let's say I do again. 910 00:53:56,060 --> 00:53:59,840 So this is what I had before, right? 911 00:53:59,840 --> 00:54:03,770 So this is the one here which is operating here on earth, 912 00:54:03,770 --> 00:54:05,690 and I let it go. 913 00:54:05,690 --> 00:54:07,610 I displaced it by a certain distance. 914 00:54:07,610 --> 00:54:10,310 Let's say 1 millimeter, and that's how it's gone. 915 00:54:10,310 --> 00:54:12,985 So now, let's take it to, for example, Jupiter. 916 00:54:15,560 --> 00:54:17,860 So what do you think will happen when we go to Jupiter. 917 00:54:20,510 --> 00:54:24,710 Jupiter, g, is much larger. 918 00:54:24,710 --> 00:54:26,560 OK? 919 00:54:26,560 --> 00:54:27,810 So what would happen to those? 920 00:54:32,350 --> 00:54:35,320 So the frequency would be larger. 921 00:54:35,320 --> 00:54:37,860 Things will be faster, right? 922 00:54:37,860 --> 00:54:39,760 That's the higher frequency. 923 00:54:39,760 --> 00:54:43,474 But also the difference between two frequencies 924 00:54:43,474 --> 00:54:44,140 will be smaller. 925 00:54:46,419 --> 00:54:48,460 And what happens when the difference in frequency 926 00:54:48,460 --> 00:54:50,459 is smaller? 927 00:54:50,459 --> 00:54:52,500 You saw that there's the fact that the energy was 928 00:54:52,500 --> 00:54:55,140 moving from one to the other. 929 00:54:55,140 --> 00:54:56,940 The thing would take-- 930 00:54:56,940 --> 00:54:59,760 so one of them was oscillating, the other one is stationary, 931 00:54:59,760 --> 00:55:02,251 then the other one would pick up, et cetera. 932 00:55:02,251 --> 00:55:03,750 Do you think this transfer of energy 933 00:55:03,750 --> 00:55:04,800 will be faster or slower? 934 00:55:08,650 --> 00:55:13,030 Two omegas closer to each other. 935 00:55:13,030 --> 00:55:15,540 Any guesses? 936 00:55:15,540 --> 00:55:16,290 AUDIENCE: Smaller. 937 00:55:16,290 --> 00:55:17,915 BOLESLAW WYSLOUCH: Take kind of longer. 938 00:55:17,915 --> 00:55:19,560 Let's see what happens, right? 939 00:55:19,560 --> 00:55:24,489 So we go on the rocket, and nowadays, you 940 00:55:24,489 --> 00:55:25,780 don't have to go to the rocket. 941 00:55:25,780 --> 00:55:26,779 Just remove one comment. 942 00:55:29,400 --> 00:55:32,220 And I went from about 10 meters per square second 943 00:55:32,220 --> 00:55:37,539 to 25 meters per square second, and this is what is happening. 944 00:55:37,539 --> 00:55:38,080 Look at this. 945 00:55:38,080 --> 00:55:41,992 So first of all, this identical system-- everything 946 00:55:41,992 --> 00:55:42,700 at the same time. 947 00:55:42,700 --> 00:55:43,283 It's the same. 948 00:55:43,283 --> 00:55:47,420 And so you see that oscillations are much faster. 949 00:55:47,420 --> 00:55:51,940 So a number of amplitude changes per second is larger. 950 00:55:51,940 --> 00:55:55,860 But it takes much longer for the energy. 951 00:55:55,860 --> 00:55:57,340 So the red one is now stopping. 952 00:55:57,340 --> 00:56:01,480 It's now slowly coming up. 953 00:56:01,480 --> 00:56:04,470 So because the two frequencies are closer to each other, 954 00:56:04,470 --> 00:56:07,560 they stay-- 955 00:56:07,560 --> 00:56:12,890 it takes longer for them to shift from one to the other. 956 00:56:12,890 --> 00:56:14,030 OK? 957 00:56:14,030 --> 00:56:16,670 So we are done at Jupiter. 958 00:56:16,670 --> 00:56:21,350 Let's now go to the Moon, which has much lower 959 00:56:21,350 --> 00:56:23,772 gravitational acceleration. 960 00:56:23,772 --> 00:56:24,730 Let's see what happens. 961 00:56:27,990 --> 00:56:33,530 Again by logical argument-- if something-- 962 00:56:33,530 --> 00:56:36,110 so the smaller gravitation accelerations 963 00:56:36,110 --> 00:56:39,830 means that the frequency is now lower. 964 00:56:39,830 --> 00:56:43,370 So the pendula will move slower. 965 00:56:43,370 --> 00:56:45,779 However, the difference between frequency 966 00:56:45,779 --> 00:56:47,570 will be larger, because the spring is still 967 00:56:47,570 --> 00:56:48,981 the same strength. 968 00:56:48,981 --> 00:56:51,230 So it turns out that even though everything is slower, 969 00:56:51,230 --> 00:56:54,840 but the energy transfer will actually be faster. 970 00:56:54,840 --> 00:56:59,130 So let's try to see what happens on the Moon. 971 00:56:59,130 --> 00:57:00,425 It's OK. 972 00:57:03,964 --> 00:57:09,040 It's a little bit not completely clear what's going on, 973 00:57:09,040 --> 00:57:14,067 but you see, actually the motion is kind of a little strange. 974 00:57:14,067 --> 00:57:14,900 Look at the red one. 975 00:57:14,900 --> 00:57:16,170 The red one is stopping. 976 00:57:16,170 --> 00:57:19,885 Then it's going halfway out. 977 00:57:19,885 --> 00:57:22,450 It looks kind of messy, doesn't it? 978 00:57:22,450 --> 00:57:24,610 And so it doesn't show up here very well, 979 00:57:24,610 --> 00:57:28,530 because the parameters have changed so much that I have-- 980 00:57:28,530 --> 00:57:39,990 I have those fixed pictures which are-- 981 00:57:39,990 --> 00:57:40,830 just a second. 982 00:57:40,830 --> 00:57:41,530 I'll show you. 983 00:57:45,790 --> 00:57:47,540 So this is the picture on the-- 984 00:57:47,540 --> 00:57:51,530 some sort of stationary picture on the Earth. 985 00:57:51,530 --> 00:57:53,730 I saw one of them up, the other one-- 986 00:57:53,730 --> 00:57:56,489 you see them shift from one to the other. 987 00:57:56,489 --> 00:57:58,780 And you can see kind of the frequency of how the energy 988 00:57:58,780 --> 00:58:00,400 shifts from one to the other. 989 00:58:00,400 --> 00:58:03,790 And also you can see the frequency going up and down 990 00:58:03,790 --> 00:58:06,430 for the same exact conditions. 991 00:58:06,430 --> 00:58:10,060 This is now, just a moment, this is a Jupiter. 992 00:58:10,060 --> 00:58:12,580 So Jupiter, you see that the frequency itself it's 993 00:58:12,580 --> 00:58:13,970 much higher. 994 00:58:13,970 --> 00:58:20,190 And the energy transfer between the two things takes longer. 995 00:58:20,190 --> 00:58:23,520 And on the Moon however, the oscillations 996 00:58:23,520 --> 00:58:25,170 actually look really weird. 997 00:58:25,170 --> 00:58:27,180 This is an example of one of them. 998 00:58:27,180 --> 00:58:32,370 It's kind of, you know, the two frequencies are so far away, 999 00:58:32,370 --> 00:58:36,960 and it's really not even a nice oscillatory motion. 1000 00:58:36,960 --> 00:58:39,471 It's some sort of-- 1001 00:58:39,471 --> 00:58:43,530 it's much less obvious that this is 1002 00:58:43,530 --> 00:58:47,250 a superposition of two cosines, because they kind of are 1003 00:58:47,250 --> 00:58:48,920 exactly out of phase. 1004 00:58:48,920 --> 00:58:51,270 So the motion is kind of complete. 1005 00:58:51,270 --> 00:58:52,620 Anyway, so this is-- 1006 00:58:52,620 --> 00:58:56,190 actually, so the lesson is that the exact shape, 1007 00:58:56,190 --> 00:58:59,810 the exact motion, we know that can always be 1008 00:58:59,810 --> 00:59:01,950 decomposed into simple motions. 1009 00:59:01,950 --> 00:59:03,520 If you put them together, things may 1010 00:59:03,520 --> 00:59:05,340 get really interesting and complicated, 1011 00:59:05,340 --> 00:59:08,970 depending on what sort of frequencies we are running 1012 00:59:08,970 --> 00:59:10,620 and what sort of-- 1013 00:59:10,620 --> 00:59:14,670 what sort of initial conditions we have. 1014 00:59:14,670 --> 00:59:15,340 All right? 1015 00:59:15,340 --> 00:59:15,970 Yes? 1016 00:59:15,970 --> 00:59:18,490 Any questions? 1017 00:59:18,490 --> 00:59:19,940 Yes? 1018 00:59:19,940 --> 00:59:23,093 AUDIENCE: It's talking about the center mass of the system 1019 00:59:23,093 --> 00:59:24,920 or just one of the two --? 1020 00:59:24,920 --> 00:59:26,461 BOLESLAW WYSLOUCH: This one, I think, 1021 00:59:26,461 --> 00:59:28,650 this one is just one of them. 1022 00:59:28,650 --> 00:59:30,990 Actually, the one-- on the difference-- it 1023 00:59:30,990 --> 00:59:32,890 normally doesn't matter. 1024 00:59:32,890 --> 00:59:35,790 What matters this is the frequency 1025 00:59:35,790 --> 00:59:39,840 and how these move to the other. 1026 00:59:39,840 --> 00:59:42,780 OK? 1027 00:59:42,780 --> 00:59:45,891 Let's just forget about it. 1028 00:59:45,891 --> 00:59:48,500 Just keep it. 1029 00:59:48,500 --> 00:59:53,120 So let me now talk about this thing, which 1030 00:59:53,120 --> 00:59:57,650 is called beat phenomenon, because when you look 1031 00:59:57,650 --> 01:00:01,660 at the motion of one of those objects, 1032 01:00:01,660 --> 01:00:04,430 or the difference between them or whatever, there's 1033 01:00:04,430 --> 01:00:08,907 something kind of interesting which can be extracted 1034 01:00:08,907 --> 01:00:09,740 for those equations. 1035 01:00:09,740 --> 01:00:11,400 Let's look at these equations here. 1036 01:00:11,400 --> 01:00:12,380 Let's look at mass 1. 1037 01:00:16,960 --> 01:00:19,610 This is mass 1 and mass 2. 1038 01:00:19,610 --> 01:00:23,143 So I can rewrite those solutions a little bit different. 1039 01:00:30,390 --> 01:00:34,030 And so what I want to do is I want to-- 1040 01:00:34,030 --> 01:00:38,660 you see, this is a difference of two cosines. 1041 01:00:38,660 --> 01:00:40,700 This is a sum of two cosines. 1042 01:00:40,700 --> 01:00:43,602 There are lots of neat trigonometrical identities 1043 01:00:43,602 --> 01:00:44,310 which we can use. 1044 01:00:44,310 --> 01:00:46,520 So we just-- we do zero physics here. 1045 01:00:46,520 --> 01:00:50,790 We just rewrite the trigonometrical formulas. 1046 01:00:50,790 --> 01:00:54,540 So I do exactly this, but I rewrite it. 1047 01:00:54,540 --> 01:00:59,650 I use, for example, some of-- you have cosine alpha plus 1048 01:00:59,650 --> 01:01:04,005 cosine beta is equal to-- 1049 01:01:04,005 --> 01:01:13,330 two cosine-- is equal to two cosine alpha plus beta divided 1050 01:01:13,330 --> 01:01:20,000 by 2 multiplied by cosine alpha minus beta divided by 2. 1051 01:01:20,000 --> 01:01:20,500 Right? 1052 01:01:20,500 --> 01:01:23,950 That's the trigonometric identity. 1053 01:01:23,950 --> 01:01:24,940 Right? 1054 01:01:24,940 --> 01:01:27,520 So let's just use this to write this down and what you get 1055 01:01:27,520 --> 01:01:29,060 is x1-- 1056 01:01:29,060 --> 01:01:40,560 x1 of t is equal to minus x0 sine of omega 1 1057 01:01:40,560 --> 01:01:50,430 plus omega 2 divided by 2 times sine omega 1 minus omega 2 1058 01:01:50,430 --> 01:01:54,240 divided by 2 times t. 1059 01:01:54,240 --> 01:02:03,970 And x2 t is equal to x0, some amplitude cosine omega 1 1060 01:02:03,970 --> 01:02:11,410 plus omega 2 divided by 2 cosine omega 1 minus omega 2 1061 01:02:11,410 --> 01:02:13,660 divided by t. 1062 01:02:13,660 --> 01:02:17,230 So again, we did zero physics here. 1063 01:02:17,230 --> 01:02:20,252 We just rewrote the simple trigonometric equations. 1064 01:02:20,252 --> 01:02:22,210 But what you see is something interesting here. 1065 01:02:22,210 --> 01:02:27,580 So there is-- we have those two frequencies which 1066 01:02:27,580 --> 01:02:29,170 are playing a role. 1067 01:02:29,170 --> 01:02:31,847 And for example, at Jupiter, those two frequencies 1068 01:02:31,847 --> 01:02:33,430 are actually very close to each other, 1069 01:02:33,430 --> 01:02:35,780 because everything is dominated by the gravity, 1070 01:02:35,780 --> 01:02:37,960 and we have a very weak spring. 1071 01:02:37,960 --> 01:02:39,850 So the omega 1 and omega 2 actually 1072 01:02:39,850 --> 01:02:42,800 are very close to each other. 1073 01:02:42,800 --> 01:02:45,650 So this thing, this term here, kind of 1074 01:02:45,650 --> 01:02:47,840 goes omega 1 plus omega 2 divided by 2 1075 01:02:47,840 --> 01:02:50,860 is like omega, right? 1076 01:02:50,860 --> 01:02:55,520 100 plus 105 divided by 2 is about 100. 1077 01:02:55,520 --> 01:02:57,400 Whereas this one here carries information 1078 01:02:57,400 --> 01:03:00,420 about the difference of frequencies-- 1079 01:03:00,420 --> 01:03:04,260 100, 102, the difference is 2, which is very small. 1080 01:03:04,260 --> 01:03:07,260 So how would this look like? 1081 01:03:07,260 --> 01:03:10,910 So if you make a plot under some conditions, 1082 01:03:10,910 --> 01:03:17,750 you can, let's say, so the two frequencies 1083 01:03:17,750 --> 01:03:19,250 are close to each other. 1084 01:03:24,810 --> 01:03:29,900 So if omega 1 is close to omega 2-- 1085 01:03:29,900 --> 01:03:37,350 for example, omega 1 is 0.9 times omega 2, right? 1086 01:03:37,350 --> 01:03:41,420 This is roughly what we have on Earth in case of our system 1087 01:03:41,420 --> 01:03:43,630 here. 1088 01:03:43,630 --> 01:03:47,770 Then omega 1 plus omega 2 divided 2 1089 01:03:47,770 --> 01:03:55,400 would be about 0.95 omega 1, omega 2, I think, 1090 01:03:55,400 --> 01:03:57,660 which is approximately equal to omega 2 1091 01:03:57,660 --> 01:04:02,880 or omega 1 and omega 1 minus omega 2 divided by 2 1092 01:04:02,880 --> 01:04:07,340 will be about minus 0.05 times omega 2-- 1093 01:04:07,340 --> 01:04:12,100 much, much smaller than that. 1094 01:04:12,100 --> 01:04:16,420 So we have-- so this term here-- 1095 01:04:16,420 --> 01:04:21,790 it basically oscillates at the frequency of omega, 1096 01:04:21,790 --> 01:04:24,900 of the frequency of the individual pendulum. 1097 01:04:24,900 --> 01:04:28,060 And the other term is much, much smaller. 1098 01:04:28,060 --> 01:04:29,010 How does this look? 1099 01:04:29,010 --> 01:04:35,300 Well, it turns out that if you make a sketch of this, 1100 01:04:35,300 --> 01:04:38,972 if you do signs, for example, it looks like this. 1101 01:04:44,268 --> 01:04:44,768 OK? 1102 01:04:47,666 --> 01:04:50,622 So there are in fact two-- 1103 01:04:50,622 --> 01:04:53,080 when you look at this picture, you can see two frequencies. 1104 01:04:53,080 --> 01:04:56,470 One which is clear the oscillation of the-- 1105 01:04:56,470 --> 01:05:00,700 high-frequency oscillation of things moving up and down. 1106 01:05:00,700 --> 01:05:05,830 But there's also this kind of overarching frequency 1107 01:05:05,830 --> 01:05:08,800 of much smaller frequency, and this 1108 01:05:08,800 --> 01:05:12,290 is what corresponds to a difference of two things. 1109 01:05:12,290 --> 01:05:16,990 So in a sense, if you look at this formula here, 1110 01:05:16,990 --> 01:05:18,970 you have oscillation, which is happening 1111 01:05:18,970 --> 01:05:22,930 very quickly with a typical oscillation of the system. 1112 01:05:22,930 --> 01:05:26,780 But this is like a modulation of the amplitude. 1113 01:05:26,780 --> 01:05:29,440 So the amplitude of the signal is changing. 1114 01:05:29,440 --> 01:05:31,240 And this is what you see here. 1115 01:05:31,240 --> 01:05:34,410 This is exactly the picture out there. 1116 01:05:34,410 --> 01:05:35,810 So the system oscillates. 1117 01:05:35,810 --> 01:05:41,980 So one of those pendula, either of them, is moving fast. 1118 01:05:41,980 --> 01:05:43,240 But it's going faster. 1119 01:05:43,240 --> 01:05:46,030 It's amplitude is larger, and after some time, 1120 01:05:46,030 --> 01:05:47,230 it slows down to 0. 1121 01:05:47,230 --> 01:05:50,590 It goes higher and slows down to 0. 1122 01:05:50,590 --> 01:05:51,550 And you've seen this. 1123 01:05:51,550 --> 01:05:57,700 We can do it again here that both of them 1124 01:05:57,700 --> 01:05:59,950 oscillate at roughly the same frequency, 1125 01:05:59,950 --> 01:06:03,430 but their individual amplitudes are changing. 1126 01:06:03,430 --> 01:06:06,020 And this transmission of-- 1127 01:06:06,020 --> 01:06:09,415 you know, one of them moving full blast, the other one 1128 01:06:09,415 --> 01:06:10,660 moving full blast. 1129 01:06:10,660 --> 01:06:14,500 There's this kind of frequency of energy 1130 01:06:14,500 --> 01:06:19,720 moving from one to the other, which is something called beat. 1131 01:06:19,720 --> 01:06:23,470 This a beat system, beat phenomenon somehow 1132 01:06:23,470 --> 01:06:27,850 that energy is moving from one place to another one. 1133 01:06:27,850 --> 01:06:32,230 And we can have some demonstration 1134 01:06:32,230 --> 01:06:33,680 of how this happens. 1135 01:06:33,680 --> 01:06:34,960 So we see this here. 1136 01:06:34,960 --> 01:06:37,660 We see it on the pendula. 1137 01:06:37,660 --> 01:06:41,864 We saw it on the computer simulation. 1138 01:06:41,864 --> 01:06:43,280 But now what we are going to do is 1139 01:06:43,280 --> 01:06:45,950 we're going to try to hear it, right? 1140 01:06:45,950 --> 01:06:50,030 So this is a demonstration which maybe it works, maybe not. 1141 01:06:50,030 --> 01:06:52,820 So let me-- it will work, OK? 1142 01:06:52,820 --> 01:06:54,770 So let me explain what we have. 1143 01:06:54,770 --> 01:06:58,860 So we have two speakers. 1144 01:06:58,860 --> 01:07:06,675 And they basically go on very, very similar frequencies, 1145 01:07:06,675 --> 01:07:07,470 all right? 1146 01:07:07,470 --> 01:07:11,340 So they both work at similar frequencies. 1147 01:07:14,436 --> 01:07:17,130 And so when I switched on, you should hear-- 1148 01:07:17,130 --> 01:07:18,628 hear the sound. 1149 01:07:18,628 --> 01:07:19,562 [HUM SOUND] 1150 01:07:20,062 --> 01:07:20,969 OK? 1151 01:07:20,969 --> 01:07:22,010 So this is the frequency. 1152 01:07:22,010 --> 01:07:26,170 I believe it's just one of them is working, and you know, 1153 01:07:26,170 --> 01:07:28,680 this is just one pendulum that is going 1154 01:07:28,680 --> 01:07:31,350 on that given frequency, right? 1155 01:07:31,350 --> 01:07:33,420 Then I will switch a second loudspeaker. 1156 01:07:36,390 --> 01:07:37,875 [HUM SOUND] 1157 01:08:05,595 --> 01:08:08,070 Can you hear this kind of-- 1158 01:08:08,070 --> 01:08:09,060 wiggle? 1159 01:08:09,060 --> 01:08:11,535 We'll change the frequency a little. 1160 01:08:15,495 --> 01:08:18,960 This is another frequency of the original sound. 1161 01:08:18,960 --> 01:08:22,490 And it's kind of the loudness of the sound overall is changing. 1162 01:08:26,482 --> 01:08:29,476 All right? 1163 01:08:29,476 --> 01:08:31,971 This is faster. 1164 01:08:31,971 --> 01:08:35,960 This is kind of extra, extra sound which 1165 01:08:35,960 --> 01:08:40,137 you hear is the difference of mainly the frequency is not 1166 01:08:40,137 --> 01:08:42,089 stable here, so I'll change it. 1167 01:08:46,490 --> 01:08:48,319 Right? 1168 01:08:48,319 --> 01:08:52,180 So this is, again, this is a single one, perfectly constant 1169 01:08:52,180 --> 01:08:54,939 frequency, no change in amplitude, 1170 01:08:54,939 --> 01:08:55,870 no change in loudness. 1171 01:08:58,560 --> 01:09:01,892 Put them together, right? 1172 01:09:01,892 --> 01:09:02,850 That's what they do. 1173 01:09:02,850 --> 01:09:06,140 So if you have two, and I can adjust the frequency, 1174 01:09:06,140 --> 01:09:10,670 and the frequency is close, then this frequency of changing 1175 01:09:10,670 --> 01:09:12,000 is very slow. 1176 01:09:12,000 --> 01:09:14,590 So you can actually hear it. 1177 01:09:14,590 --> 01:09:16,819 Let me switch it off. 1178 01:09:16,819 --> 01:09:18,500 So this is the effect of beats. 1179 01:09:18,500 --> 01:09:26,359 I can maybe show you another simulation of this works. 1180 01:09:26,359 --> 01:09:29,200 Let's See. 1181 01:09:29,200 --> 01:09:36,120 This one is oops, just a second. 1182 01:09:36,120 --> 01:09:37,779 Let's see what it is. 1183 01:09:42,170 --> 01:09:47,649 OK, so this is just a single frequency. 1184 01:09:47,649 --> 01:09:50,220 OK, again, I plot some pendulum. 1185 01:09:50,220 --> 01:09:53,069 Then I can plot-- 1186 01:09:53,069 --> 01:09:56,618 sorry, no this one is this. 1187 01:09:56,618 --> 01:10:03,923 I can-- this one. 1188 01:10:03,923 --> 01:10:05,675 OK, we'll just plot it here. 1189 01:10:09,680 --> 01:10:10,770 Maybe we can see. 1190 01:10:10,770 --> 01:10:13,820 So there's a red one, and there's a blue one. 1191 01:10:13,820 --> 01:10:15,860 And I plot two plots independently 1192 01:10:15,860 --> 01:10:16,740 on top of each other. 1193 01:10:16,740 --> 01:10:18,480 So they have an amplitude of 1. 1194 01:10:18,480 --> 01:10:21,360 And clearly, you see that they have a different frequency. 1195 01:10:21,360 --> 01:10:23,810 So the red one is going with some frequency. 1196 01:10:23,810 --> 01:10:25,880 The blue one is going with some other frequency. 1197 01:10:25,880 --> 01:10:27,980 Sometimes they agree. 1198 01:10:27,980 --> 01:10:30,170 Sometimes they do not agree, right? 1199 01:10:30,170 --> 01:10:33,250 And the places where they meet-- 1200 01:10:33,250 --> 01:10:34,680 they are on top of each other. 1201 01:10:34,680 --> 01:10:38,400 This is where when you add them up together, 1202 01:10:38,400 --> 01:10:40,430 this is where they will be large. 1203 01:10:40,430 --> 01:10:42,710 In the places where they're out of phase, 1204 01:10:42,710 --> 01:10:44,560 they will cancel each other. 1205 01:10:44,560 --> 01:10:47,210 So if you take two of those together, 1206 01:10:47,210 --> 01:10:50,680 same amplitude, just slightly different frequency, 1207 01:10:50,680 --> 01:10:52,056 and you simply make a linear-- 1208 01:10:57,020 --> 01:11:01,780 superposition of the two, you will get exactly the beating 1209 01:11:01,780 --> 01:11:02,280 effect. 1210 01:11:02,280 --> 01:11:04,400 So I just took two of those pictures 1211 01:11:04,400 --> 01:11:07,040 before I added them together and got exactly that. 1212 01:11:07,040 --> 01:11:09,080 You have a maximum, minima, et cetera. 1213 01:11:09,080 --> 01:11:13,760 And you see this overall beat frequency, and the carrier, 1214 01:11:13,760 --> 01:11:16,100 it's called carrier frequency. 1215 01:11:16,100 --> 01:11:20,390 And this is something that, again, happens very often. 1216 01:11:20,390 --> 01:11:23,700 There's another demonstration here. 1217 01:11:23,700 --> 01:11:27,470 I have two tuning forks, and they 1218 01:11:27,470 --> 01:11:29,390 are very similar frequency. 1219 01:11:29,390 --> 01:11:33,900 So first, I will show you that they are coupled. 1220 01:11:33,900 --> 01:11:38,420 They are coupled because I gave this guy 1221 01:11:38,420 --> 01:11:40,810 some initial condition. 1222 01:11:40,810 --> 01:11:41,640 It's going. 1223 01:11:41,640 --> 01:11:43,440 Then I stop it. 1224 01:11:43,440 --> 01:11:46,140 But there's still sound, because the second one picked up 1225 01:11:46,140 --> 01:11:48,350 some energy, and it took off. 1226 01:11:48,350 --> 01:11:50,100 Of course, you don't see them. 1227 01:11:50,100 --> 01:11:53,640 So basically, what I'm saying is that I [TONE] give this energy. 1228 01:11:53,640 --> 01:11:55,610 This one is completely stationary. 1229 01:11:55,610 --> 01:11:58,926 Now energy is slowly moving to the other one. 1230 01:11:58,926 --> 01:12:00,800 I stop this guy, and this guy is still going. 1231 01:12:03,160 --> 01:12:05,320 So the energy is being transferred 1232 01:12:05,320 --> 01:12:07,720 by this air oscillating here. 1233 01:12:07,720 --> 01:12:12,070 The coupling goes through the air to the sound here, right? 1234 01:12:12,070 --> 01:12:13,960 And they have very similar frequency. 1235 01:12:13,960 --> 01:12:17,260 So they are nicely coupled. 1236 01:12:17,260 --> 01:12:19,999 But what we can also do-- 1237 01:12:19,999 --> 01:12:21,478 we can [TONE]. 1238 01:12:25,261 --> 01:12:25,760 Right? 1239 01:12:25,760 --> 01:12:27,620 So they're both going. 1240 01:12:27,620 --> 01:12:28,702 Do you hear the beats? 1241 01:12:31,594 --> 01:12:32,558 [TONE] 1242 01:12:34,968 --> 01:12:36,900 Not really. 1243 01:12:36,900 --> 01:12:41,040 In fact, if they would have exactly identical frequency, 1244 01:12:41,040 --> 01:12:41,540 right? 1245 01:12:41,540 --> 01:12:43,790 If they will be perfectly the same, 1246 01:12:43,790 --> 01:12:46,290 then the difference would be 0, and there 1247 01:12:46,290 --> 01:12:47,400 will be no beats at all. 1248 01:12:47,400 --> 01:12:49,290 The period of beats will be infinitely long, 1249 01:12:49,290 --> 01:12:52,186 so it will take forever for us to hear anything. 1250 01:12:52,186 --> 01:12:54,060 So what we can do-- we can break one of them. 1251 01:12:54,060 --> 01:12:58,750 We can add some sort of weight. 1252 01:12:58,750 --> 01:12:59,800 Some are here. 1253 01:12:59,800 --> 01:13:02,450 There's some magic place where it works best. 1254 01:13:02,450 --> 01:13:04,370 So what I would do is I will break this one. 1255 01:13:04,370 --> 01:13:06,590 I will modify its frequency. 1256 01:13:06,590 --> 01:13:07,910 That's another way to modify. 1257 01:13:07,910 --> 01:13:09,701 I don't have to go to Jupiter to modify it, 1258 01:13:09,701 --> 01:13:13,980 because this one is just a little mass here, right? 1259 01:13:13,980 --> 01:13:14,900 [TONE] 1260 01:13:18,770 --> 01:13:19,440 Ah, cool. 1261 01:13:19,440 --> 01:13:23,160 AUDIENCE: Is that [INAUDIBLE]? 1262 01:13:23,160 --> 01:13:27,278 BOLESLAW WYSLOUCH: Really, this is actually a huge effect. 1263 01:13:27,278 --> 01:13:28,262 [TONE] 1264 01:13:30,230 --> 01:13:32,930 You can clearly see that they are going up and down, up 1265 01:13:32,930 --> 01:13:36,560 and down, because the frequency is slightly different. 1266 01:13:36,560 --> 01:13:39,260 So now, this thing is probably-- 1267 01:13:39,260 --> 01:13:42,070 I know it's a period, a fraction of a second, right? 1268 01:13:42,070 --> 01:13:42,752 Yes? 1269 01:13:42,752 --> 01:13:45,112 AUDIENCE: Should both of those sine and cosines 1270 01:13:45,112 --> 01:13:47,460 have Ts in their arguments? 1271 01:13:47,460 --> 01:13:51,005 BOLESLAW WYSLOUCH: Of course always. 1272 01:13:51,005 --> 01:13:54,280 They are both time dependent, yeah. 1273 01:13:54,280 --> 01:13:57,080 This is the fast thing, and this is 1274 01:13:57,080 --> 01:14:02,100 this time-dependent modulation, yeah. 1275 01:14:02,100 --> 01:14:07,287 All right, so where are my notes? 1276 01:14:07,287 --> 01:14:11,110 So this is the-- 1277 01:14:11,110 --> 01:14:13,790 this is how the-- 1278 01:14:13,790 --> 01:14:15,910 so we were able to set up the system, 1279 01:14:15,910 --> 01:14:17,770 put in some of the matrix equation, 1280 01:14:17,770 --> 01:14:20,310 kind of solved it, found two frequencies, et cetera. 1281 01:14:20,310 --> 01:14:21,450 There is one more-- 1282 01:14:21,450 --> 01:14:23,500 one additional trick, which you can 1283 01:14:23,500 --> 01:14:29,140 do to describe the motion of a coupled pendula. 1284 01:14:29,140 --> 01:14:34,940 And that is, in a sense, force mathematically, 1285 01:14:34,940 --> 01:14:39,340 force the normal modes from sort of early on, to 1286 01:14:39,340 --> 01:14:42,720 instead of, so far, when we talked about pendula, 1287 01:14:42,720 --> 01:14:48,300 we describe their motion in terms of motion of number 1288 01:14:48,300 --> 01:14:50,503 1, motion of number 2. 1289 01:14:50,503 --> 01:14:52,970 It turns out we can rewrite the equation 1290 01:14:52,970 --> 01:14:56,370 into some sort of new variables, where, 1291 01:14:56,370 --> 01:15:00,980 so-called normal coordinates, where you'll simultaneously 1292 01:15:00,980 --> 01:15:05,120 describe both of them and then kind of mix 1293 01:15:05,120 --> 01:15:08,470 them together to have a new formula, 1294 01:15:08,470 --> 01:15:11,090 just rewrite the equation in terms of new variables. 1295 01:15:11,090 --> 01:15:13,565 So you do change of variables. 1296 01:15:13,565 --> 01:15:19,520 So instead of keeping track of x1 and x2 independently, 1297 01:15:19,520 --> 01:15:22,550 you define something which I called 1298 01:15:22,550 --> 01:15:30,300 u1, which is simply x1 plus x2, and I define 1299 01:15:30,300 --> 01:15:36,020 u2, which is x1 minus x2. 1300 01:15:36,020 --> 01:15:38,670 So instead of talking about x1 and x2 independently, 1301 01:15:38,670 --> 01:15:41,340 I have a sum of them and difference. 1302 01:15:41,340 --> 01:15:42,190 Why not? 1303 01:15:42,190 --> 01:15:42,800 Right? 1304 01:15:42,800 --> 01:15:43,960 Two variables. 1305 01:15:43,960 --> 01:15:47,360 I can always go back and get x1 and x2 if I want to. 1306 01:15:47,360 --> 01:15:51,620 So if one tells me that u1 is 1 centimeter and u2 2 1307 01:15:51,620 --> 01:15:54,230 centimeters, I can always go and get x1 and x2 1308 01:15:54,230 --> 01:15:55,700 if I want to, right? 1309 01:15:55,700 --> 01:15:57,060 So I can do it. 1310 01:15:57,060 --> 01:16:01,865 And it turns out that if I plot those variables 1311 01:16:01,865 --> 01:16:05,150 in, in other words, I take the original equations, which 1312 01:16:05,150 --> 01:16:09,660 I conveniently erased and make a sum or difference, 1313 01:16:09,660 --> 01:16:12,470 it turns out that this coupling kind of separates. 1314 01:16:12,470 --> 01:16:17,450 So I will end up having two separate equations 1315 01:16:17,450 --> 01:16:18,290 for this one. 1316 01:16:18,290 --> 01:16:20,990 So in general, the equation of motion 1317 01:16:20,990 --> 01:16:23,870 would be-- would look like, so let's say 1318 01:16:23,870 --> 01:16:33,620 I can write down m x1 plus x2 is equal to minus m g 1319 01:16:33,620 --> 01:16:44,110 over l times x1 plus x2. 1320 01:16:44,110 --> 01:16:47,780 OK, this is when I add two equations. 1321 01:16:47,780 --> 01:16:52,370 And the other equation when I subtract them-- 1322 01:16:52,370 --> 01:17:06,676 minus x2 is equal to minus mg over l plus 2k x1 minus x2. 1323 01:17:06,676 --> 01:17:09,120 I think that's what is coming out. 1324 01:17:09,120 --> 01:17:15,520 So if I add and subtract the two original equations of motion, 1325 01:17:15,520 --> 01:17:17,530 which I don't know if I have them somewhere, 1326 01:17:17,530 --> 01:17:19,680 and you can look back, then you end up 1327 01:17:19,680 --> 01:17:25,170 having those crossed terms drop out. 1328 01:17:25,170 --> 01:17:28,110 And you have one, which has only this coefficient, the other one 1329 01:17:28,110 --> 01:17:30,930 which has that coefficient. 1330 01:17:30,930 --> 01:17:33,480 And this immediately-- and it looks-- 1331 01:17:33,480 --> 01:17:37,050 if I now write it in terms of normal coordinates, 1332 01:17:37,050 --> 01:17:44,190 then I have that m u1 double dot is equal to simply minus mg 1333 01:17:44,190 --> 01:17:53,910 over l, u1, and m u2 double dot is equal to minus mg over 1334 01:17:53,910 --> 01:17:58,610 l plus 2k times u2. 1335 01:17:58,610 --> 01:18:03,050 And if you look at those two equations, 1336 01:18:03,050 --> 01:18:04,990 it turns out that they are not coupled. 1337 01:18:04,990 --> 01:18:13,380 Each of them is a question of a one-dimensional harmonic 1338 01:18:13,380 --> 01:18:15,150 oscillator. 1339 01:18:15,150 --> 01:18:17,120 The first part one only depends on u1. 1340 01:18:17,120 --> 01:18:20,940 The second one only depends on u2. 1341 01:18:20,940 --> 01:18:26,230 And you can see the oscillating frequency with your own eyes. 1342 01:18:26,230 --> 01:18:29,860 So no, the determinants needed no matrices, no nothing. 1343 01:18:29,860 --> 01:18:32,710 We just added and subtracted the two equations, 1344 01:18:32,710 --> 01:18:35,790 and things magically separated. 1345 01:18:35,790 --> 01:18:37,060 All right? 1346 01:18:37,060 --> 01:18:41,350 So sometimes, especially in case of very simple and symmetric 1347 01:18:41,350 --> 01:18:43,420 systems, if you introduce new variables, 1348 01:18:43,420 --> 01:18:45,200 you can simplify your life tremendously, 1349 01:18:45,200 --> 01:18:51,418 and these are called normal variables, normal coordinates. 1350 01:18:58,390 --> 01:19:01,380 And it turns out that you can always do that. 1351 01:19:01,380 --> 01:19:03,450 So you can always have a linear combination 1352 01:19:03,450 --> 01:19:07,470 of parameters for arbitrary size coupled oscillators system 1353 01:19:07,470 --> 01:19:11,510 where you combine different coordinates, 1354 01:19:11,510 --> 01:19:15,600 and you basically force the system to behave 1355 01:19:15,600 --> 01:19:21,630 in a way in which it induces the single oscillation, single 1356 01:19:21,630 --> 01:19:22,560 frequency. 1357 01:19:22,560 --> 01:19:25,800 So this is, again, a very powerful trick, 1358 01:19:25,800 --> 01:19:28,470 but usually for most cases, you can do that 1359 01:19:28,470 --> 01:19:31,620 only after you have solved it, after you've found out 1360 01:19:31,620 --> 01:19:32,760 normal modes, et cetera. 1361 01:19:32,760 --> 01:19:35,410 So after you know your normal mode, then you can say, ha, ha, 1362 01:19:35,410 --> 01:19:38,100 I can I can introduce normal variables 1363 01:19:38,100 --> 01:19:39,480 and make things simpler. 1364 01:19:39,480 --> 01:19:42,660 But at the end of the day for complicated systems 1365 01:19:42,660 --> 01:19:44,520 that work is the same. 1366 01:19:44,520 --> 01:19:47,910 But for simple systems like this one where there is 1367 01:19:47,910 --> 01:19:51,040 a good symmetry, you can do it. 1368 01:19:51,040 --> 01:19:54,870 Anyway, so I think we are done for today. 1369 01:19:54,870 --> 01:19:59,370 And on Tuesday, we'll continue with forced oscillators. 1370 01:19:59,370 --> 01:19:59,870 All right? 1371 01:19:59,870 --> 01:20:01,720 Thank you.