1 00:00:02,490 --> 00:00:04,830 The following content is provided under a Creative 2 00:00:04,830 --> 00:00:06,250 Commons license. 3 00:00:06,250 --> 00:00:08,460 Your support will help MIT OpenCourseWare 4 00:00:08,460 --> 00:00:12,550 continue to offer high-quality educational resources for free. 5 00:00:12,550 --> 00:00:15,090 To make a donation or to view additional materials 6 00:00:15,090 --> 00:00:18,090 from hundreds of MIT courses, visit MIT 7 00:00:18,090 --> 00:00:23,566 OpenCourseWare at osw.mit.edu, 8 00:00:23,566 --> 00:00:26,370 YEN-JIE LEE: So welcome, everybody. 9 00:00:26,370 --> 00:00:27,740 My name is Yen-Jie Lee. 10 00:00:27,740 --> 00:00:32,210 I am a assistant professor of physics in the physics 11 00:00:32,210 --> 00:00:36,500 department, and I will be your instructor 12 00:00:36,500 --> 00:00:39,860 of this semester on 8.03. 13 00:00:39,860 --> 00:00:44,930 So of course, one first question you have is, 14 00:00:44,930 --> 00:00:50,240 why do we want to learn about vibrations and waves? 15 00:00:50,240 --> 00:00:52,460 Why do we learn about this? 16 00:00:52,460 --> 00:00:55,550 Why do we even care? 17 00:00:55,550 --> 00:00:58,350 The answer is really, simple. 18 00:00:58,350 --> 00:01:00,920 If you look at this slide, you can 19 00:01:00,920 --> 00:01:05,720 see that the reason you can follow this class 20 00:01:05,720 --> 00:01:11,270 is because I'm producing sound wave by oscillating the air, 21 00:01:11,270 --> 00:01:14,180 and you can receive those sound waves. 22 00:01:14,180 --> 00:01:17,120 And you can see me-- 23 00:01:17,120 --> 00:01:19,790 that's really pretty amazing by itself-- 24 00:01:19,790 --> 00:01:22,880 because there are a lot of photons 25 00:01:22,880 --> 00:01:25,580 or electromagnetic waves. 26 00:01:25,580 --> 00:01:27,700 They are bouncing around in this room, 27 00:01:27,700 --> 00:01:34,790 and your eye actually receive those electromagnetic waves. 28 00:01:34,790 --> 00:01:37,430 And that translates into your brain waves. 29 00:01:37,430 --> 00:01:41,075 You obviously, start to think about what this instructor is 30 00:01:41,075 --> 00:01:43,400 trying to tell you. 31 00:01:43,400 --> 00:01:47,030 And of course, all those things we learned from 8.03 32 00:01:47,030 --> 00:01:50,870 is closely connected to probability density 33 00:01:50,870 --> 00:01:57,500 waves, which you will learn from 8.04, quantum physics. 34 00:01:57,500 --> 00:01:59,660 And finally, it's also, of course, 35 00:01:59,660 --> 00:02:05,030 related to a recent discovery of the gravitational waves. 36 00:02:05,030 --> 00:02:08,050 When we are sitting here, maybe there 37 00:02:08,050 --> 00:02:12,800 are already some space-time distortion already passing 38 00:02:12,800 --> 00:02:16,370 through our body and you don't feel it. 39 00:02:16,370 --> 00:02:18,550 When I'm moving around like this, 40 00:02:18,550 --> 00:02:22,370 I am creating also the gravitational waves, 41 00:02:22,370 --> 00:02:25,430 but it's so small to be detected. 42 00:02:25,430 --> 00:02:26,960 So that's actually really cool. 43 00:02:26,960 --> 00:02:32,570 So the take-home message is that we cannot even recognize 44 00:02:32,570 --> 00:02:37,310 the universe without using waves and the vibrations. 45 00:02:37,310 --> 00:02:40,810 So that's actually why we care about this subject. 46 00:02:40,810 --> 00:02:42,575 And the last is actually why this subject 47 00:02:42,575 --> 00:02:49,170 is so cool even without quantum, without any fancy names. 48 00:02:49,170 --> 00:02:55,420 So what is actually the relation of 8.03 to other class 49 00:02:55,420 --> 00:02:57,280 or other field of studies? 50 00:02:57,280 --> 00:03:00,410 It's closely related to classical mechanics, which 51 00:03:00,410 --> 00:03:03,050 I will use it immediately, and I hope 52 00:03:03,050 --> 00:03:05,060 you will still remember what you have 53 00:03:05,060 --> 00:03:07,480 learned from 8.01 and 8.02. 54 00:03:07,480 --> 00:03:13,310 Electromagnetic force is actually closely related also, 55 00:03:13,310 --> 00:03:16,640 and we are going to use a technique we learned 56 00:03:16,640 --> 00:03:21,270 from this class to understand optics, quantum mechanics, 57 00:03:21,270 --> 00:03:24,800 and also there are many practical applications, which 58 00:03:24,800 --> 00:03:29,200 you will learn from this class. 59 00:03:29,200 --> 00:03:31,390 This is the concrete goal. 60 00:03:31,390 --> 00:03:36,100 We care about the future of our space time. 61 00:03:36,100 --> 00:03:40,450 We would like to predict what is going to happen 62 00:03:40,450 --> 00:03:42,790 when we set up an experiment. 63 00:03:42,790 --> 00:03:47,290 We would like to design experiments which can improve 64 00:03:47,290 --> 00:03:50,110 our understanding of nature. 65 00:03:50,110 --> 00:03:53,850 But without using the most powerful tool 66 00:03:53,850 --> 00:03:58,030 is very, very difficult to make progress. 67 00:03:58,030 --> 00:04:03,900 So the most powerful tool we have is mathematics. 68 00:04:03,900 --> 00:04:07,840 You will see that it really works in this class. 69 00:04:07,840 --> 00:04:09,610 But the first thing we have to learn 70 00:04:09,610 --> 00:04:15,610 is how to translate physical situations into mathematics 71 00:04:15,610 --> 00:04:19,959 so that we can actually include this really wonderful tool 72 00:04:19,959 --> 00:04:22,960 to help us to solve problems. 73 00:04:22,960 --> 00:04:25,180 Once we have done that, we will start 74 00:04:25,180 --> 00:04:29,180 to look at single harmonic oscillator, 75 00:04:29,180 --> 00:04:32,590 then we try to couple all those oscillators together 76 00:04:32,590 --> 00:04:36,590 to see how they interact with each other. 77 00:04:36,590 --> 00:04:41,860 Finally, we go to an infinite number of oscillators. 78 00:04:41,860 --> 00:04:46,080 Sounds scary, but it's actually not scary after all. 79 00:04:46,080 --> 00:04:49,810 And we will see waves because waves are actually 80 00:04:49,810 --> 00:04:52,990 coming from an infinite number of oscillating particles, 81 00:04:52,990 --> 00:04:55,360 if you think about it. 82 00:04:55,360 --> 00:04:57,880 Then we would do Fourier decomposition of waves 83 00:04:57,880 --> 00:04:59,890 to see what we can learn about it. 84 00:04:59,890 --> 00:05:04,390 We learn how to put together physical systems. 85 00:05:04,390 --> 00:05:08,800 That brings us to the issue of boundary conditions, 86 00:05:08,800 --> 00:05:12,520 and we will also enjoy what we have learned 87 00:05:12,520 --> 00:05:14,620 by looking at the phenomenon related 88 00:05:14,620 --> 00:05:18,610 to electromagnetic waves and practical application 89 00:05:18,610 --> 00:05:21,530 and optics. 90 00:05:21,530 --> 00:05:22,240 Any questions? 91 00:05:22,240 --> 00:05:26,080 If you have any questions, please stop me any time. 92 00:05:26,080 --> 00:05:29,040 So if you don't stop me, I'm going to continue talking. 93 00:05:31,600 --> 00:05:33,950 So that gets started. 94 00:05:33,950 --> 00:05:37,090 So the first example, the concrete example 95 00:05:37,090 --> 00:05:43,570 I'm going to talk about is a spring block, a massive block 96 00:05:43,570 --> 00:05:45,520 system. 97 00:05:45,520 --> 00:05:50,150 So this is actually what I have on that table. 98 00:05:50,150 --> 00:05:54,740 So basically, I have a highly-idealized spring. 99 00:05:54,740 --> 00:05:58,480 This is ideal spring with spring constant, k, 100 00:05:58,480 --> 00:06:01,480 and the natural length L0. 101 00:06:01,480 --> 00:06:04,150 So that is actually what I have. 102 00:06:04,150 --> 00:06:13,830 And at t equal to 0, what I am going to do is I am going to-- 103 00:06:13,830 --> 00:06:16,480 I should remove this mass a little bit, 104 00:06:16,480 --> 00:06:22,900 and I hold this mass still and release that really carefully. 105 00:06:22,900 --> 00:06:24,910 So that is actually the experiment, 106 00:06:24,910 --> 00:06:27,070 which I am going to do. 107 00:06:27,070 --> 00:06:33,390 And we were wondering what is going to happen afterward. 108 00:06:33,390 --> 00:06:36,940 Well, the mass as you move, will it stay there 109 00:06:36,940 --> 00:06:40,790 or it just disappear, I don't know 110 00:06:40,790 --> 00:06:42,220 before I solved this question. 111 00:06:44,890 --> 00:06:49,690 Now I have put together a concrete question to you, 112 00:06:49,690 --> 00:06:52,900 but I don't know how to proceed because you 113 00:06:52,900 --> 00:06:54,040 say everything works. 114 00:06:54,040 --> 00:06:55,190 What I am going to do? 115 00:06:55,190 --> 00:06:58,540 I mean, I don't know. 116 00:06:58,540 --> 00:07:03,160 So as I mentioned before, there is a pretty powerful tool, 117 00:07:03,160 --> 00:07:04,780 mathematics. 118 00:07:04,780 --> 00:07:08,140 So I'm going to use that, even though I don't 119 00:07:08,140 --> 00:07:11,587 know why mathematics can work. 120 00:07:11,587 --> 00:07:12,670 Have you thought about it? 121 00:07:15,340 --> 00:07:19,180 So let's try it and see how we can make progress. 122 00:07:19,180 --> 00:07:25,540 So the first thing which you can do in order to make progress 123 00:07:25,540 --> 00:07:29,170 is to define a coordinate system. 124 00:07:29,170 --> 00:07:32,950 So here I define a coordinate system, which 125 00:07:32,950 --> 00:07:34,630 is in the horizontal direction. 126 00:07:34,630 --> 00:07:36,580 It's the x direction. 127 00:07:36,580 --> 00:07:39,640 And the x equal to 0, the origin, 128 00:07:39,640 --> 00:07:44,800 is the place which the spring is not stressed, 129 00:07:44,800 --> 00:07:47,330 is at its natural length. 130 00:07:47,330 --> 00:07:51,490 That is actually what I define as x equal to 0. 131 00:07:51,490 --> 00:07:56,020 And once I define this, I can now 132 00:07:56,020 --> 00:08:01,720 express what is actually the initial position of the mass 133 00:08:01,720 --> 00:08:03,990 by these coordinates is x0. 134 00:08:03,990 --> 00:08:08,020 It can be expressed as x initial. 135 00:08:08,020 --> 00:08:11,830 And also, initially, I said that this mass is not moving. 136 00:08:11,830 --> 00:08:16,810 Therefore, the velocity at 0 is 0. 137 00:08:16,810 --> 00:08:20,000 So now I can also formulate my question really concretely 138 00:08:20,000 --> 00:08:21,460 with some mathematics. 139 00:08:21,460 --> 00:08:25,330 Basically, you can see that at time equal to t, 140 00:08:25,330 --> 00:08:28,690 I was wondering where is this mass. 141 00:08:28,690 --> 00:08:31,060 So actually, the question is, what is actually 142 00:08:31,060 --> 00:08:34,210 x as a function of t? 143 00:08:34,210 --> 00:08:38,150 So you can see that once I have the mathematics to help me, 144 00:08:38,150 --> 00:08:41,490 everything becomes pretty simple. 145 00:08:41,490 --> 00:08:44,770 So once I have those defined, I would 146 00:08:44,770 --> 00:08:48,460 like to predict what is going to happen at time equal to t. 147 00:08:48,460 --> 00:08:54,370 Therefore, I would like to make use physical laws to actually 148 00:08:54,370 --> 00:08:56,540 help me to solve this problem. 149 00:08:56,540 --> 00:09:01,200 So apparently what we are going to use is Newton's law. 150 00:09:01,200 --> 00:09:05,700 And I am going to go through this example 151 00:09:05,700 --> 00:09:10,100 really slowly so that everybody is on the same page. 152 00:09:10,100 --> 00:09:12,300 So the first thing which I usually do 153 00:09:12,300 --> 00:09:18,610 is now I would like to do a force diagram analysis. 154 00:09:18,610 --> 00:09:19,930 So I have this mass. 155 00:09:19,930 --> 00:09:25,080 This setup is on Earth, and the question is, 156 00:09:25,080 --> 00:09:28,240 how many forces are acting on this mass? 157 00:09:28,240 --> 00:09:32,185 Can anybody answer my question. 158 00:09:32,185 --> 00:09:33,151 AUDIENCE: Two. 159 00:09:33,151 --> 00:09:35,566 We got the-- 160 00:09:35,566 --> 00:09:38,685 So acceleration and the spring force. 161 00:09:38,685 --> 00:09:41,150 YEN-JIE LEE: OK, so your answer is two. 162 00:09:41,150 --> 00:09:43,210 Any different? 163 00:09:43,210 --> 00:09:44,850 Three. 164 00:09:44,850 --> 00:09:45,450 Very good. 165 00:09:45,450 --> 00:09:48,120 So we have two and three. 166 00:09:48,120 --> 00:09:52,750 And the answer actually is three. 167 00:09:52,750 --> 00:09:54,260 So look at this scene. 168 00:09:54,260 --> 00:09:57,120 I am drawing in and I have product here. 169 00:09:57,120 --> 00:10:00,105 So this is actually the most difficult part of the question, 170 00:10:00,105 --> 00:10:01,220 actually. 171 00:10:01,220 --> 00:10:05,140 So once you pass this step, everything is straightforward. 172 00:10:05,140 --> 00:10:06,660 It's just mathematics. 173 00:10:06,660 --> 00:10:09,150 It's not my problem any more, but the math department, 174 00:10:09,150 --> 00:10:12,300 they have problem, OK? 175 00:10:12,300 --> 00:10:13,320 All right. 176 00:10:13,320 --> 00:10:16,952 So now let's look at this mass. 177 00:10:16,952 --> 00:10:17,910 There are three forces. 178 00:10:17,910 --> 00:10:21,915 The first one as you mentioned correctly is F spring. 179 00:10:21,915 --> 00:10:26,190 It's pulling the mass. 180 00:10:26,190 --> 00:10:31,020 And since we are working on Earth, 181 00:10:31,020 --> 00:10:33,240 we have not yet moved the whole class 182 00:10:33,240 --> 00:10:36,810 to the moon or somewhere else, but there 183 00:10:36,810 --> 00:10:39,945 would be gravitational force pointing downward. 184 00:10:42,720 --> 00:10:49,440 But this whole setup is on a table of friction, this table. 185 00:10:49,440 --> 00:10:51,555 Therefore, there will be no more force. 186 00:10:54,708 --> 00:10:56,369 So don't forget this one. 187 00:10:56,369 --> 00:10:57,535 There will be no more force. 188 00:11:00,080 --> 00:11:02,995 So the answer is that we have three forces. 189 00:11:05,740 --> 00:11:08,265 The normal force is, actually, a complicated subject, 190 00:11:08,265 --> 00:11:09,640 which you will need to understand 191 00:11:09,640 --> 00:11:11,440 that will quantum physics. 192 00:11:15,160 --> 00:11:18,790 So now I have three force, and now I can actually 193 00:11:18,790 --> 00:11:24,670 calculate the total force, the total force, 194 00:11:24,670 --> 00:11:35,510 F. F is equal to Fs plus Fn plus Fg. 195 00:11:38,830 --> 00:11:43,000 So since we know that the mass is 196 00:11:43,000 --> 00:11:45,310 moving in the horizontal direction, 197 00:11:45,310 --> 00:11:49,900 the mass didn't suddenly jump and disappear. 198 00:11:49,900 --> 00:11:51,760 So it is there. 199 00:11:51,760 --> 00:11:59,020 Therefore, we know that the normal force is actually 200 00:11:59,020 --> 00:12:07,690 equal to minus Fg, which is actually Ng in the y direction. 201 00:12:07,690 --> 00:12:13,480 And here I define y is actually pointing up, 202 00:12:13,480 --> 00:12:16,660 and the x is pointing to the right-hand side. 203 00:12:16,660 --> 00:12:18,760 Therefore, what is going to happen 204 00:12:18,760 --> 00:12:22,730 is that the total force is actually just Fs. 205 00:12:27,740 --> 00:12:31,660 And this is equal to minus k, which 206 00:12:31,660 --> 00:12:35,290 is the spring constant and x, which 207 00:12:35,290 --> 00:12:43,230 is the position of the little mass at time equal to t. 208 00:12:46,350 --> 00:12:55,670 So once we have those forces and the total force, 209 00:12:55,670 --> 00:12:57,770 actually, we can use Newton's law. 210 00:13:03,674 --> 00:13:09,240 So F is equal to m times a. 211 00:13:09,240 --> 00:13:16,760 And this is actually equal to m d squared xt dt squared 212 00:13:16,760 --> 00:13:20,060 in the x direction, and that is actually 213 00:13:20,060 --> 00:13:26,880 equal to mx double dot t x. 214 00:13:26,880 --> 00:13:28,190 So here is my notation. 215 00:13:28,190 --> 00:13:32,810 I'm going to use each of the dot is actually the differentiation 216 00:13:32,810 --> 00:13:34,630 with respect to t. 217 00:13:37,760 --> 00:13:45,600 So this is actually equal to minus kxt in the x direction. 218 00:13:45,600 --> 00:13:49,730 So you can see that here is actually what you already 219 00:13:49,730 --> 00:13:51,330 know about Newton's law. 220 00:13:51,330 --> 00:13:55,250 And that is actually coming from the force analysis. 221 00:13:55,250 --> 00:13:58,550 So in this example, it's simple enough such 222 00:13:58,550 --> 00:14:00,830 that you can write it down immediately, 223 00:14:00,830 --> 00:14:03,110 but in the later examples, things 224 00:14:03,110 --> 00:14:06,740 will become very complicated and things will be slightly more 225 00:14:06,740 --> 00:14:08,960 difficult. Therefore, you will really need help 226 00:14:08,960 --> 00:14:12,080 from the force diagram. 227 00:14:12,080 --> 00:14:15,320 So now we have everything in the x direction, 228 00:14:15,320 --> 00:14:18,140 therefore, I can drop the x hat. 229 00:14:18,140 --> 00:14:25,010 Therefore, finally, my equation of motion is x double dot t. 230 00:14:25,010 --> 00:14:32,720 And this is equal to minus k over n x of t. 231 00:14:32,720 --> 00:14:37,280 To make my life easier, I am going 232 00:14:37,280 --> 00:14:43,460 to define omega equal to square root of k over n. 233 00:14:43,460 --> 00:14:45,390 You will see why afterward. 234 00:14:45,390 --> 00:14:48,810 It looks really weird why professor Lee wants to do this, 235 00:14:48,810 --> 00:14:53,680 but afterward, you will see that omega really have a meaning, 236 00:14:53,680 --> 00:14:58,000 and that is equal to minus omega squared x. 237 00:15:00,710 --> 00:15:06,110 So we have solved this problem, actually, as a physicist. 238 00:15:06,110 --> 00:15:08,420 Now the problem is what is actually 239 00:15:08,420 --> 00:15:11,180 the solution to this differential 240 00:15:11,180 --> 00:15:13,440 second-order differential equation. 241 00:15:13,440 --> 00:15:15,570 And as I mentioned, this is actually 242 00:15:15,570 --> 00:15:17,530 not the content of 8.03, actually, 243 00:15:17,530 --> 00:15:20,930 it's a content of 18.03, maybe. 244 00:15:20,930 --> 00:15:24,650 How many of you actually have taken 18.03? 245 00:15:24,650 --> 00:15:27,380 Everybody knows the solution, so very good. 246 00:15:27,380 --> 00:15:30,475 I am safe. 247 00:15:30,475 --> 00:15:31,475 So what is the solution? 248 00:15:35,730 --> 00:15:46,940 The solution is x of t equal to a cosine of omega t plus b 249 00:15:46,940 --> 00:15:49,820 sine omega t. 250 00:15:49,820 --> 00:15:53,120 So my friends from the math department 251 00:15:53,120 --> 00:15:58,160 tell me secretly that this is actually the solution. 252 00:15:58,160 --> 00:16:00,470 And I trust him or her. 253 00:16:04,340 --> 00:16:07,130 So that's very nice. 254 00:16:07,130 --> 00:16:09,950 Now I have the solution, and how do I 255 00:16:09,950 --> 00:16:12,450 know this is the only solution? 256 00:16:12,450 --> 00:16:14,850 How do I know? 257 00:16:14,850 --> 00:16:17,330 Actually, there are two unknowns, 258 00:16:17,330 --> 00:16:20,964 just to remind you what you have learned. 259 00:16:20,964 --> 00:16:23,810 There are two unknowns. 260 00:16:23,810 --> 00:16:29,860 And if you plug this thing into this equation, 261 00:16:29,860 --> 00:16:32,000 you satisfy that equation. 262 00:16:32,000 --> 00:16:34,880 If you don't trust me, you can do it offline. 263 00:16:34,880 --> 00:16:37,610 It's always good to check to make 264 00:16:37,610 --> 00:16:39,290 sure I didn't make a mistake. 265 00:16:39,290 --> 00:16:40,980 But that's very good news. 266 00:16:40,980 --> 00:16:46,250 So that means we will have two unknowns, 267 00:16:46,250 --> 00:16:49,760 and those will satisfy the equation. 268 00:16:49,760 --> 00:16:54,900 So by uniqueness theorem, this is actually 269 00:16:54,900 --> 00:17:01,080 the one and the only one solution in my universe, 270 00:17:01,080 --> 00:17:07,800 also yours, which satisfy the equation because 271 00:17:07,800 --> 00:17:09,700 of the uniqueness theorem. 272 00:17:09,700 --> 00:17:12,690 So I hope I have convinced you that we 273 00:17:12,690 --> 00:17:16,810 have solved this equation. 274 00:17:16,810 --> 00:17:22,349 So now I take my physicist hat back and now 275 00:17:22,349 --> 00:17:24,400 it is actually my job again. 276 00:17:24,400 --> 00:17:27,030 So now we have the solution, and we 277 00:17:27,030 --> 00:17:29,290 need to determine what is actually 278 00:17:29,290 --> 00:17:32,440 these two unknown coefficients. 279 00:17:32,440 --> 00:17:37,770 So what I'm going to use is to use the two initial conditions. 280 00:17:44,840 --> 00:17:49,180 The first initial condition is x of 0 equal to x initial. 281 00:17:52,770 --> 00:17:57,540 The second one is that since I released this mass really 282 00:17:57,540 --> 00:18:03,180 carefully and the initial velocity is 0, therefore, 283 00:18:03,180 --> 00:18:09,160 I have x dot 0 equal to 0. 284 00:18:09,160 --> 00:18:11,610 From this, you can solve. 285 00:18:11,610 --> 00:18:16,000 Plug these two conditions into this equation. 286 00:18:16,000 --> 00:18:20,370 You can actually figure out that a is equal to x initial. 287 00:18:26,400 --> 00:18:29,690 And b is equal to 0. 288 00:18:33,000 --> 00:18:34,110 Any questions so far? 289 00:18:39,830 --> 00:18:41,460 Very good. 290 00:18:41,460 --> 00:18:43,530 So now we have the solution. 291 00:18:43,530 --> 00:18:46,640 So finally, what is actually the solution? 292 00:18:46,640 --> 00:19:03,660 The solution we get is x of t equal to x initial cosine omega 293 00:19:03,660 --> 00:19:04,160 t. 294 00:19:08,120 --> 00:19:12,620 So this is actually the amplitude of the oscillation, 295 00:19:12,620 --> 00:19:17,240 and this is actually the angular velocity. 296 00:19:17,240 --> 00:19:20,420 So you may be asking why angular? 297 00:19:20,420 --> 00:19:22,970 Where is the angular coming from? 298 00:19:22,970 --> 00:19:25,910 Because this is actually a one-dimensional motion. 299 00:19:25,910 --> 00:19:28,490 Where is the angular velocity coming from? 300 00:19:28,490 --> 00:19:34,250 And I will explain that in the later lecture. 301 00:19:34,250 --> 00:19:38,220 And also this is actually a harmonic oscillation. 302 00:19:38,220 --> 00:19:40,700 So what we are actually predicting 303 00:19:40,700 --> 00:19:46,490 is that this mass is going to do this, have a fixed 304 00:19:46,490 --> 00:19:48,830 amplitude and it's actually going 305 00:19:48,830 --> 00:19:54,920 to go back and forth with the angular frequency of omega. 306 00:19:54,920 --> 00:20:00,230 So we can now do an experiment to verify if this is actually 307 00:20:00,230 --> 00:20:01,940 really the case. 308 00:20:01,940 --> 00:20:04,400 So there's a small difference. 309 00:20:04,400 --> 00:20:08,870 There's another spring here, but essentially, the solution 310 00:20:08,870 --> 00:20:10,290 will be very similar. 311 00:20:10,290 --> 00:20:15,860 You may get this in a p-set or exam. 312 00:20:15,860 --> 00:20:20,670 So now I can turn on the air so that I make this surface 313 00:20:20,670 --> 00:20:23,880 frictionless. 314 00:20:23,880 --> 00:20:29,120 And you can see that now I actually move this thing 315 00:20:29,120 --> 00:20:33,390 slightly away from the equilibrium position, 316 00:20:33,390 --> 00:20:36,410 and I release that carefully. 317 00:20:36,410 --> 00:20:42,030 So you can see that really it's actually going back and forth 318 00:20:42,030 --> 00:20:43,080 harmonically. 319 00:20:46,290 --> 00:20:50,880 I can change the amplitude and see what will happen. 320 00:20:50,880 --> 00:20:53,540 The amplitude is becoming bigger, 321 00:20:53,540 --> 00:20:59,120 and you can see that the oscillation amplitude really 322 00:20:59,120 --> 00:21:03,365 depends on where you put that initially with respect 323 00:21:03,365 --> 00:21:04,750 to the equilibrium position. 324 00:21:04,750 --> 00:21:08,330 I can actually make a small amplitude oscillation also. 325 00:21:08,330 --> 00:21:11,570 Now you can see that now the amplitude is small but still 326 00:21:11,570 --> 00:21:14,660 oscillating back and forth. 327 00:21:14,660 --> 00:21:16,401 So that's very encouraging. 328 00:21:20,730 --> 00:21:25,340 Let's take another example, which I actually rotate 329 00:21:25,340 --> 00:21:28,520 the whole thing by 90 degrees. 330 00:21:28,520 --> 00:21:32,150 You are going to get a question about this system 331 00:21:32,150 --> 00:21:33,510 in your p-set. 332 00:21:33,510 --> 00:21:38,500 The amazing thing is that the solution is the same. 333 00:21:38,500 --> 00:21:41,330 What is that? 334 00:21:41,330 --> 00:21:45,470 And you don't believe me, let me do the experiment. 335 00:21:45,470 --> 00:21:48,680 I actually shifted the position. 336 00:21:48,680 --> 00:21:53,220 I changed the position, and I release that really carefully. 337 00:21:53,220 --> 00:21:56,460 You see that this mass is oscillating up and down. 338 00:21:56,460 --> 00:22:00,410 The amplitude did not change. 339 00:22:00,410 --> 00:22:03,890 The frequency did not change as a function of time. 340 00:22:03,890 --> 00:22:08,960 It really matched with the solution we found here. 341 00:22:11,690 --> 00:22:14,060 It's truly amazing. 342 00:22:14,060 --> 00:22:16,040 No? 343 00:22:16,040 --> 00:22:19,940 The problem is that we are so used to this already. 344 00:22:19,940 --> 00:22:25,250 You have seen this maybe 100 times before my lecture, 345 00:22:25,250 --> 00:22:28,040 so therefore, you got so used to this. 346 00:22:28,040 --> 00:22:30,680 Therefore, when I say, OK, I make a prediction. 347 00:22:30,680 --> 00:22:34,120 This is what happened, you are just so used to this 348 00:22:34,120 --> 00:22:36,740 or you don't feel the excitement. 349 00:22:36,740 --> 00:22:41,510 But for me, after I teach this class so many times, 350 00:22:41,510 --> 00:22:46,030 I still find this thing really amazing. 351 00:22:46,030 --> 00:22:48,110 Why is that? 352 00:22:48,110 --> 00:22:53,120 This means that actually, mathematics 353 00:22:53,120 --> 00:22:56,520 really works, first of all. 354 00:22:56,520 --> 00:23:03,560 That means we can use the same tool for the understanding 355 00:23:03,560 --> 00:23:11,300 of gravitational waves, for the prediction of the Higgs boson, 356 00:23:11,300 --> 00:23:13,920 for the calculation of the property 357 00:23:13,920 --> 00:23:17,810 of the quark-gluon plasma in the early universe, 358 00:23:17,810 --> 00:23:22,280 and also at the same time the motion 359 00:23:22,280 --> 00:23:25,400 of this spring-mass system. 360 00:23:25,400 --> 00:23:30,350 We actually use always the same tool, the mathematics, 361 00:23:30,350 --> 00:23:32,840 to understand this system. 362 00:23:32,840 --> 00:23:35,360 And nobody will understands why. 363 00:23:35,360 --> 00:23:37,845 If you understand why, please tell me. 364 00:23:37,845 --> 00:23:38,720 I would like to know. 365 00:23:38,720 --> 00:23:39,886 I will be very proud of you. 366 00:23:43,780 --> 00:23:49,400 Rene Descartes said once, "But in my opinion, 367 00:23:49,400 --> 00:23:54,190 all things in nature occur mathematically." 368 00:23:54,190 --> 00:23:55,340 Apparently, he's right. 369 00:23:58,570 --> 00:24:03,670 Albert Einstein also once said, "The most incomprehensible 370 00:24:03,670 --> 00:24:08,810 thing about the universe is that it is comprehensible." 371 00:24:08,810 --> 00:24:13,870 So I would say this is really something 372 00:24:13,870 --> 00:24:17,470 we need to appreciate the need to think 373 00:24:17,470 --> 00:24:21,250 about why this is the case. 374 00:24:21,250 --> 00:24:22,325 Any questions? 375 00:24:26,790 --> 00:24:29,990 So you may say, oh, come on. 376 00:24:29,990 --> 00:24:33,990 We just solved the problem of an ideal spring. 377 00:24:33,990 --> 00:24:35,380 Who cares? 378 00:24:35,380 --> 00:24:39,120 It's so simple, so easy, and you are making really 379 00:24:39,120 --> 00:24:41,760 a big thing out of this. 380 00:24:41,760 --> 00:24:44,490 But actually, what we have been solving 381 00:24:44,490 --> 00:24:47,110 is really much more than that. 382 00:24:47,110 --> 00:24:53,850 This equation is much more than just a spring-mass system. 383 00:24:53,850 --> 00:24:58,800 Actually, if you think about this question carefully, 384 00:24:58,800 --> 00:25:02,130 there's really no Hooke's law forever. 385 00:25:02,130 --> 00:25:05,820 Hooke's law will give you a potential proportional 386 00:25:05,820 --> 00:25:07,800 to x squared. 387 00:25:07,800 --> 00:25:15,970 And if you are so far away, you pull the spring so really hard, 388 00:25:15,970 --> 00:25:19,080 you can store the energy of the whole universe. 389 00:25:19,080 --> 00:25:21,730 Does that make sense? 390 00:25:21,730 --> 00:25:22,980 No. 391 00:25:22,980 --> 00:25:25,620 At some point, it should break down. 392 00:25:25,620 --> 00:25:27,990 So there's really no Hook's law. 393 00:25:27,990 --> 00:25:31,860 But there's also Hook's law everywhere. 394 00:25:31,860 --> 00:25:37,230 If you look at this system, it follows the harmonic 395 00:25:37,230 --> 00:25:38,280 oscillation. 396 00:25:38,280 --> 00:25:40,950 If you look at this system I perturb this, 397 00:25:40,950 --> 00:25:42,150 it goes back and forth. 398 00:25:44,940 --> 00:25:46,560 It's almost like everywhere. 399 00:25:46,560 --> 00:25:48,540 Why is this the case? 400 00:25:48,540 --> 00:25:51,580 I'm going to answer this question immediately. 401 00:25:54,280 --> 00:25:58,860 So let's take a look at an example. 402 00:25:58,860 --> 00:26:01,060 So if I consider a potential, this 403 00:26:01,060 --> 00:26:03,870 is an artificial potential, which 404 00:26:03,870 --> 00:26:07,110 you can find in Georgi's book, so v 405 00:26:07,110 --> 00:26:11,780 is equal to E times L over x plus x over L. 406 00:26:11,780 --> 00:26:14,640 And if you practice as a function of x, then 407 00:26:14,640 --> 00:26:16,800 basically you get this funny shape. 408 00:26:16,800 --> 00:26:20,300 It's not proportional to x squared. 409 00:26:20,300 --> 00:26:23,130 Therefore, you will see that, OK, 410 00:26:23,130 --> 00:26:28,710 the resulting motion for the particle in this potential, 411 00:26:28,710 --> 00:26:31,950 it's not going to be harmonic motion. 412 00:26:31,950 --> 00:26:35,370 But if I zoom in, zoom in, and zoom in 413 00:26:35,370 --> 00:26:38,550 and basically, you will see that if I am patient enough, 414 00:26:38,550 --> 00:26:44,440 I zoom in enough, you'll see that this is a parabola. 415 00:26:44,440 --> 00:26:50,100 Again, you follow Hooke's law. 416 00:26:50,100 --> 00:26:52,590 So that is actually really cool. 417 00:26:52,590 --> 00:26:59,100 So if I consider an arbitrary v of x, 418 00:26:59,100 --> 00:27:05,740 we can do a Taylor expansion to this potential. 419 00:27:05,740 --> 00:27:10,920 So basically v of x will be equal to v of 0 420 00:27:10,920 --> 00:27:16,890 plus v prime 0 divided by 1 factorial times 421 00:27:16,890 --> 00:27:24,900 x plus v double prime 0 over 2 factorial 422 00:27:24,900 --> 00:27:30,670 x squared plus v triple prime 0 divided 423 00:27:30,670 --> 00:27:37,110 by 3 factorial x to the third plus infinite number of terms. 424 00:27:37,110 --> 00:27:47,010 v 0 is the position of where you have minimum potential. 425 00:27:47,010 --> 00:27:50,550 So that's actually where the equilibrium position 426 00:27:50,550 --> 00:27:54,020 is in my coordinate system. 427 00:27:54,020 --> 00:27:55,770 It's the standard, the coordinate system 428 00:27:55,770 --> 00:28:02,070 I used for the solving the spring-mass question. 429 00:28:02,070 --> 00:28:06,900 So if I calculate the force, the force, f of x, 430 00:28:06,900 --> 00:28:12,690 will be equal to minus d dx v of x. 431 00:28:12,690 --> 00:28:15,150 And that will be equal to minus v 432 00:28:15,150 --> 00:28:24,000 prime 0 minus v double prime 0 x minus 1 433 00:28:24,000 --> 00:28:34,320 over 2 v triple prime 0 x squared plus many other terms. 434 00:28:34,320 --> 00:28:39,980 Since I have mentioned that v of 0-- 435 00:28:46,108 --> 00:28:50,160 this will be x. 436 00:28:50,160 --> 00:28:55,260 v of 0 is actually the position of the minima. 437 00:28:55,260 --> 00:29:01,002 Therefore, v prime of 0 will be equal to 0. 438 00:29:03,540 --> 00:29:05,100 Therefore. 439 00:29:05,100 --> 00:29:07,610 This term is gone. 440 00:29:11,660 --> 00:29:16,490 So what essentially is left over is the remaining terms here. 441 00:29:19,910 --> 00:29:26,450 Now, if I assume that x is very small, what is going to happen? 442 00:29:26,450 --> 00:29:31,090 Anybody know when x is very small, what is going to happen? 443 00:29:31,090 --> 00:29:32,140 Anybody have the answer? 444 00:29:32,140 --> 00:29:34,720 AUDIENCE: [INAUDIBLE]. 445 00:29:34,720 --> 00:29:36,110 YEN-JIE LEE: Exactly. 446 00:29:36,110 --> 00:29:41,630 So when x is very small, he said that the higher order terms all 447 00:29:41,630 --> 00:29:43,095 become negligible. 448 00:29:43,095 --> 00:29:43,910 OK? 449 00:29:43,910 --> 00:29:45,290 So that is essentially correct. 450 00:29:45,290 --> 00:29:50,060 So when x is very small, then I only 451 00:29:50,060 --> 00:29:55,130 need to consider the leading order term. 452 00:29:55,130 --> 00:29:59,030 But how small is the question. 453 00:29:59,030 --> 00:30:00,900 How small is small? 454 00:30:03,760 --> 00:30:07,630 Actually, what you can do is to take the ratio 455 00:30:07,630 --> 00:30:12,517 between these two terms. 456 00:30:12,517 --> 00:30:14,350 So if you take the ratio, then basically you 457 00:30:14,350 --> 00:30:19,300 would get a condition xv triple dot 458 00:30:19,300 --> 00:30:29,940 0, which will be much smaller than v double prime 0. 459 00:30:29,940 --> 00:30:32,050 So that is essentially the condition 460 00:30:32,050 --> 00:30:37,630 which is required to satisfy it so that we actually can ignore 461 00:30:37,630 --> 00:30:39,340 all the higher-order terms. 462 00:30:39,340 --> 00:30:42,160 Then the whole question becomes f 463 00:30:42,160 --> 00:30:48,910 of x equal to minus v double prime 0 x. 464 00:30:48,910 --> 00:30:52,100 And that essentially, Hooke's law. 465 00:30:52,100 --> 00:30:54,760 So you can see that first of all, there's 466 00:30:54,760 --> 00:30:57,190 no Hooke's law in general. 467 00:30:57,190 --> 00:31:01,240 Secondly, Hook's law essentially applicable 468 00:31:01,240 --> 00:31:06,490 almost everywhere when you have a well-behaved potential 469 00:31:06,490 --> 00:31:10,860 and if you only perturb the system really slightly 470 00:31:10,860 --> 00:31:14,960 with very small amplitude, then it always works. 471 00:31:14,960 --> 00:31:19,120 So what I would like to say is that after we 472 00:31:19,120 --> 00:31:23,620 have done this exercise, you will see that, actually, we 473 00:31:23,620 --> 00:31:28,960 have solved all the possible systems, which 474 00:31:28,960 --> 00:31:33,630 have a well-behaved potential. 475 00:31:33,630 --> 00:31:39,830 It has a minima, and if I have the amplitude small enough, 476 00:31:39,830 --> 00:31:46,400 then the system is going to do simple harmonic oscillation. 477 00:31:46,400 --> 00:31:47,860 Any questions? 478 00:31:54,540 --> 00:31:56,065 No question, then we'll continue. 479 00:32:02,060 --> 00:32:07,040 So let's come back to this equation of motion. 480 00:32:07,040 --> 00:32:13,360 x double dot plus omega squared x, this is equal to 0. 481 00:32:16,380 --> 00:32:19,600 There are two important properties 482 00:32:19,600 --> 00:32:22,930 of this linear equation of motion. 483 00:32:26,950 --> 00:32:48,130 The first one is that if x1 of t and x2 of t are both solutions, 484 00:32:48,130 --> 00:32:54,280 then x12, which is the superposition 485 00:32:54,280 --> 00:33:01,340 of the first and second solution, is also a solution. 486 00:33:13,930 --> 00:33:17,690 The second thing, which is very interesting about this equation 487 00:33:17,690 --> 00:33:27,235 of motion, is that there's a time translation invariance. 488 00:33:31,990 --> 00:33:38,830 So this means that if x of t is a solution, 489 00:33:38,830 --> 00:33:49,285 then xt prime equal to xt plus a is also a solution. 490 00:33:52,190 --> 00:33:55,520 So that is really cool, because that 491 00:33:55,520 --> 00:34:04,370 means if I change t equal to 0, so I shift the 0-th time, 492 00:34:04,370 --> 00:34:08,090 the whole physics did not change. 493 00:34:10,989 --> 00:34:13,650 So this is actually because of the chain law. 494 00:34:13,650 --> 00:34:19,929 So if you have chain law dx t plus a dt, that 495 00:34:19,929 --> 00:34:35,690 is equal to d t plus a dt, dx t prime dt prime 496 00:34:35,690 --> 00:34:39,949 evaluated at t prime equal to t plus a. 497 00:34:39,949 --> 00:34:52,969 And that is equal to dx t prime dt, t prime equal to t plus a. 498 00:34:52,969 --> 00:35:00,020 So that means if I have changed the t equal to 0 499 00:35:00,020 --> 00:35:04,670 to other place, the whole equation of motion 500 00:35:04,670 --> 00:35:06,860 is still the same. 501 00:35:06,860 --> 00:35:10,880 On the other hand, if the k, or say the potential, 502 00:35:10,880 --> 00:35:14,640 is time dependent, then that may break this symmetry. 503 00:35:17,970 --> 00:35:18,932 Any questions? 504 00:35:22,150 --> 00:35:27,670 So before we take a five minute break, 505 00:35:27,670 --> 00:35:33,640 I would like to discuss further about this point, this linear 506 00:35:33,640 --> 00:35:34,990 and nonlinear event. 507 00:35:34,990 --> 00:35:38,230 So you can see that the force is actually 508 00:35:38,230 --> 00:35:42,760 linearly dependent on x. 509 00:35:42,760 --> 00:35:49,420 But what will happen if I increase x more? 510 00:35:49,420 --> 00:35:50,350 Something will happen. 511 00:35:50,350 --> 00:35:54,850 That means the higher-ordered term should also be 512 00:35:54,850 --> 00:35:57,370 taken into account carefully. 513 00:35:57,370 --> 00:36:01,210 So that means the solution of this kind, x 514 00:36:01,210 --> 00:36:07,070 initial cosine omega t, will not work perfectly. 515 00:36:07,070 --> 00:36:12,690 In 8.03, we only consider the linear term most of the time. 516 00:36:12,690 --> 00:36:15,700 But actually, I would like to make sure 517 00:36:15,700 --> 00:36:17,450 that everybody can at this point, 518 00:36:17,450 --> 00:36:21,220 the higher-order contribution is actually 519 00:36:21,220 --> 00:36:23,290 visible in our daily life. 520 00:36:23,290 --> 00:36:28,710 So let me actually give you a concrete example. 521 00:36:28,710 --> 00:36:33,520 So here I have two pendulums. 522 00:36:33,520 --> 00:36:36,550 So I can now perturb this pendulum slightly. 523 00:36:36,550 --> 00:36:42,970 And you you'll see that it goes back and forth and following 524 00:36:42,970 --> 00:36:45,610 simple harmonic emotion. 525 00:36:45,610 --> 00:36:48,670 So if I have both things slightly 526 00:36:48,670 --> 00:36:52,810 oscillating with small amplitude, what 527 00:36:52,810 --> 00:36:58,140 is going to happen is that both pendulums reach maxima 528 00:36:58,140 --> 00:37:00,800 amplitude at the same time. 529 00:37:00,800 --> 00:37:05,360 You can see that very clearly. 530 00:37:05,360 --> 00:37:08,916 I don't need to do this carefully. 531 00:37:08,916 --> 00:37:14,150 You see that they always reach maxima at the same time 532 00:37:14,150 --> 00:37:16,040 when the amplitude is small. 533 00:37:16,040 --> 00:37:17,250 Why? 534 00:37:17,250 --> 00:37:24,000 That is because the higher-order terms are not important. 535 00:37:24,000 --> 00:37:26,350 So now let's do a experiment. 536 00:37:26,350 --> 00:37:29,010 And now I go crazy. 537 00:37:29,010 --> 00:37:33,840 I make the amplitude very large so that I 538 00:37:33,840 --> 00:37:37,800 break that approximation. 539 00:37:37,800 --> 00:37:39,300 So let's see what will happen. 540 00:37:39,300 --> 00:37:41,805 So now I do this then. 541 00:37:41,805 --> 00:37:47,170 I release at the same time and see what will happen. 542 00:37:47,170 --> 00:37:49,410 You see that originally they are in phase. 543 00:37:49,410 --> 00:37:53,830 They are reaching maxima at the same time. 544 00:37:53,830 --> 00:37:57,110 But if we are patient enough, you see that now? 545 00:37:57,110 --> 00:38:01,020 They are is oscillating, actually, 546 00:38:01,020 --> 00:38:02,920 at different frequencies. 547 00:38:02,920 --> 00:38:06,150 Originally, the solution, the omega, 548 00:38:06,150 --> 00:38:09,810 is really independent of the amplitude. 549 00:38:09,810 --> 00:38:11,970 So they should, actually, be isolating 550 00:38:11,970 --> 00:38:13,810 at the same frequency. 551 00:38:13,810 --> 00:38:16,260 But clearly you can see here, when 552 00:38:16,260 --> 00:38:19,380 you increase the amplitude, then you 553 00:38:19,380 --> 00:38:24,600 need to consider also the nonlinear effects. 554 00:38:24,600 --> 00:38:30,067 So any questions before we take a five-minute break. 555 00:38:30,067 --> 00:38:32,150 So if not, then we would take a five-minute break, 556 00:38:32,150 --> 00:38:34,850 and we come back at 25. 557 00:38:40,590 --> 00:38:43,290 So welcome back, everybody. 558 00:38:43,290 --> 00:38:45,630 So we will continue the discussion 559 00:38:45,630 --> 00:38:51,170 of this equation of motion, x double dot plus omega square x 560 00:38:51,170 --> 00:38:54,120 equal to 0. 561 00:38:54,120 --> 00:38:58,750 So there are three possible way to like 562 00:38:58,750 --> 00:39:01,935 the solution to this equation. 563 00:39:01,935 --> 00:39:04,860 So the first one as I mentioned before, 564 00:39:04,860 --> 00:39:15,960 x of t equal to a cosine omega t plus b sine omega t. 565 00:39:15,960 --> 00:39:17,800 So this is actually the functional form 566 00:39:17,800 --> 00:39:21,010 we have been using before. 567 00:39:21,010 --> 00:39:26,520 And we can actually also rewrite it in a different way. 568 00:39:26,520 --> 00:39:39,600 So x or t equal to capital A cosine omega t plus phi. 569 00:39:39,600 --> 00:39:41,640 You may say, wait a second. 570 00:39:41,640 --> 00:39:44,190 You just promised me that this is the first one, 571 00:39:44,190 --> 00:39:47,190 the one is the one and only one solution 572 00:39:47,190 --> 00:39:51,750 in the universe, which actually satisfy the equation of motion. 573 00:39:51,750 --> 00:39:53,960 Now you write another one. 574 00:39:53,960 --> 00:39:56,210 What is going on? 575 00:39:56,210 --> 00:39:57,680 Why? 576 00:39:57,680 --> 00:39:59,060 But actually, they are the same. 577 00:40:01,770 --> 00:40:09,740 This is actually A cosine phi cosine omega t 578 00:40:09,740 --> 00:40:16,250 minus A sine phi sine omega t. 579 00:40:19,250 --> 00:40:23,390 So the good thing is that A and phi 580 00:40:23,390 --> 00:40:27,770 are arbitrary constant so that it should be you 581 00:40:27,770 --> 00:40:31,040 can use two initial conditions to determine 582 00:40:31,040 --> 00:40:32,570 the arbitrary constant. 583 00:40:32,570 --> 00:40:39,050 So you can see that one and two are completely equivalent. 584 00:40:39,050 --> 00:40:44,390 So I hope that solves some of the questions 585 00:40:44,390 --> 00:40:48,320 because you really find it confusing 586 00:40:48,320 --> 00:40:53,960 why we have different presentations of the solution. 587 00:40:53,960 --> 00:40:59,600 So there's a third one, which is actually much more fancier. 588 00:40:59,600 --> 00:41:04,270 The third one is that I have x of t. 589 00:41:04,270 --> 00:41:10,575 This is actually a real part of A-- 590 00:41:10,575 --> 00:41:16,650 again, the amplitude-- exponential i omega t 591 00:41:16,650 --> 00:41:24,030 plus phi, where i is equal to the square root of minus 1. 592 00:41:27,800 --> 00:41:28,460 Wait a second. 593 00:41:28,460 --> 00:41:32,530 We will say, well, professor, why are you writing 594 00:41:32,530 --> 00:41:34,975 such a horrible solution? 595 00:41:38,820 --> 00:41:40,150 Right? 596 00:41:40,150 --> 00:41:41,150 Really strange. 597 00:41:41,150 --> 00:41:42,400 But that will explain you why. 598 00:41:45,030 --> 00:41:48,120 So three is actually a mathematical trick. 599 00:41:48,120 --> 00:41:53,220 I'm not going to prove anything here because I'm a physicist, 600 00:41:53,220 --> 00:41:58,600 but I would like to share with you what I think is going on. 601 00:41:58,600 --> 00:42:01,410 I think three is really just a mathematical trick 602 00:42:01,410 --> 00:42:05,310 from the math department. 603 00:42:05,310 --> 00:42:10,780 In principle, I can drive it an even more horrible way. 604 00:42:10,780 --> 00:42:24,486 x of t equal to a real part of A cosine omega t plus phi plus i 605 00:42:24,486 --> 00:42:29,501 f of t. 606 00:42:29,501 --> 00:42:36,910 And f of t is a real function. 607 00:42:36,910 --> 00:42:38,730 In principle, I can do that. 608 00:42:38,730 --> 00:42:41,870 It's even more horrible. 609 00:42:41,870 --> 00:42:43,500 Why is that? 610 00:42:43,500 --> 00:42:46,470 Because I now have this function. 611 00:42:46,470 --> 00:42:48,870 I take the real part, and I actually 612 00:42:48,870 --> 00:42:56,200 take the two out of this operation. 613 00:42:56,200 --> 00:42:59,900 So f of t is actually the real function. 614 00:42:59,900 --> 00:43:02,170 It can be something arbitrary. 615 00:43:02,170 --> 00:43:11,690 And i can now plot the locus of this function, the solution 616 00:43:11,690 --> 00:43:15,280 on the complex print. 617 00:43:15,280 --> 00:43:19,800 Now I'm plotting this solution on this complex print. 618 00:43:19,800 --> 00:43:23,780 What is going to happen is that you're going to have-- 619 00:43:31,250 --> 00:43:33,310 That is what you are going to get. 620 00:43:33,310 --> 00:43:36,520 If I am lucky, if this f of t is confined 621 00:43:36,520 --> 00:43:39,250 in some specific region, if I not lucky, 622 00:43:39,250 --> 00:43:41,160 then it goes out of the print there. 623 00:43:41,160 --> 00:43:42,490 I couldn't see it. 624 00:43:42,490 --> 00:43:44,490 Maybe it go to the moon or something. 625 00:43:47,150 --> 00:43:50,090 But if you are smart enough, and I'm 626 00:43:50,090 --> 00:44:00,080 sure you are, if I choose f of t equal to A sine omega 627 00:44:00,080 --> 00:44:06,410 t plus phi, can anybody tell me what is going to happen? 628 00:44:09,314 --> 00:44:10,766 AUDIENCE: [INAUDIBLE]. 629 00:44:15,122 --> 00:44:16,950 YEN-JIE LEE: Would you count a circle? 630 00:44:19,920 --> 00:44:22,770 Very good. 631 00:44:22,770 --> 00:44:27,700 If I plot the locus again of this function, 632 00:44:27,700 --> 00:44:33,970 the real axis, imaginary axis, then you should get a circle. 633 00:44:33,970 --> 00:44:37,030 Some miracle happened. 634 00:44:37,030 --> 00:44:41,530 If you choose the f of t correctly, 635 00:44:41,530 --> 00:44:50,400 wisely, then you can actually turn all this mess into order. 636 00:44:50,400 --> 00:44:51,649 Any questions? 637 00:44:56,440 --> 00:44:59,590 So I can now follow up about this. 638 00:45:09,380 --> 00:45:20,110 So now I have x of t is equal to the real part of A cosine omega 639 00:45:20,110 --> 00:45:28,260 t plus phi plus iA sine omega t plus phi. 640 00:45:31,390 --> 00:45:36,120 And just a reminder, exponential i theta 641 00:45:36,120 --> 00:45:43,830 is equal to cosine theta plus i sine theta. 642 00:45:43,830 --> 00:45:45,920 Therefore, I arrive this. 643 00:45:45,920 --> 00:45:54,020 This is a real part of A exponential i omega t plus phi. 644 00:45:58,840 --> 00:46:01,090 So if I do this really carefully, 645 00:46:01,090 --> 00:46:08,680 I look at this the position of the point at a specific time. 646 00:46:08,680 --> 00:46:13,500 So now time is equal to t. 647 00:46:13,500 --> 00:46:16,915 And this is the real axis, and this is the imaginary axis. 648 00:46:16,915 --> 00:46:21,100 So I have this circle here. 649 00:46:21,100 --> 00:46:24,520 So at time equal to t, what you are getting 650 00:46:24,520 --> 00:46:27,400 is that x is actually-- 651 00:46:27,400 --> 00:46:31,430 before taking the real part, A, exponential i omega t plus phi, 652 00:46:31,430 --> 00:46:33,730 it's actually here. 653 00:46:33,730 --> 00:46:39,380 And this vector actually shows the amplitude. 654 00:46:39,380 --> 00:46:46,540 Amplitude is A. And the angle between this vector pointing 655 00:46:46,540 --> 00:46:54,460 to the position of this function is omega t plus phi. 656 00:46:54,460 --> 00:47:00,100 So this is actually the angle between this vector 657 00:47:00,100 --> 00:47:03,500 and the real axis. 658 00:47:03,500 --> 00:47:06,060 So that's pretty cool. 659 00:47:06,060 --> 00:47:06,770 Why? 660 00:47:06,770 --> 00:47:12,410 Because now I understand why I call this omega angular 661 00:47:12,410 --> 00:47:15,830 velocity or angular frequency. 662 00:47:15,830 --> 00:47:22,140 Because the solution to the equation of motion, 663 00:47:22,140 --> 00:47:26,325 which we have actually derived before, 664 00:47:26,325 --> 00:47:35,610 is actually the real part of rotation in a complex print. 665 00:47:35,610 --> 00:47:39,540 If you think about it, that means now 666 00:47:39,540 --> 00:47:45,910 I see this particle going up and down. 667 00:47:45,910 --> 00:47:49,160 I see this particle going up and down. 668 00:47:49,160 --> 00:47:51,950 You can think about that, this is Earth. 669 00:47:51,950 --> 00:47:55,370 If there is an extra dimension mention, 670 00:47:55,370 --> 00:47:58,620 which you couldn't see. 671 00:47:58,620 --> 00:48:02,620 Actually, this particle in the dimension 672 00:48:02,620 --> 00:48:06,200 where we can see into the extra dimension, which is hidden 673 00:48:06,200 --> 00:48:09,180 is actually rotating. 674 00:48:09,180 --> 00:48:12,470 And while we see that reality, it's 675 00:48:12,470 --> 00:48:16,040 a projection to the real axis. 676 00:48:16,040 --> 00:48:16,650 You see? 677 00:48:16,650 --> 00:48:25,210 So in reality, this particle is actually rotating, 678 00:48:25,210 --> 00:48:29,450 if you add the image and the extra dimension. 679 00:48:29,450 --> 00:48:33,620 So that is actually pretty cool, but the purity artificial. 680 00:48:33,620 --> 00:48:37,580 So you can see that I can choose f of t 681 00:48:37,580 --> 00:48:42,530 to be a different function, and then this whole picture 682 00:48:42,530 --> 00:48:43,970 is different. 683 00:48:43,970 --> 00:48:46,490 But I also would create a lot of trouble 684 00:48:46,490 --> 00:48:49,250 because then the mathematics become complicated. 685 00:48:49,250 --> 00:48:50,880 I didn't gain anything. 686 00:48:50,880 --> 00:48:58,240 But by choosing this functional form, 687 00:48:58,240 --> 00:49:02,280 you actually write a very beautiful picture. 688 00:49:02,280 --> 00:49:05,920 Another thing, which is very cool about this 689 00:49:05,920 --> 00:49:12,480 is that if I write this thing in the exponential functional 690 00:49:12,480 --> 00:49:18,340 form, since we are dealing with differential equations, 691 00:49:18,340 --> 00:49:22,170 there is a very good property about exponential function. 692 00:49:22,170 --> 00:49:26,856 That is it is essentially a phoenix function. 693 00:49:26,856 --> 00:49:30,480 Do you know what is a phoenix? 694 00:49:30,480 --> 00:49:36,060 Phoenix is actually some kind of animal, a long-beaked bird, 695 00:49:36,060 --> 00:49:41,100 which is cyclically called the regenerated or reborn. 696 00:49:41,100 --> 00:49:44,100 So basically, when this phoenix die, 697 00:49:44,100 --> 00:49:48,820 you will lay the eggs in the fire and you were reborn. 698 00:49:48,820 --> 00:49:50,670 This is actually the same as this function. 699 00:49:54,240 --> 00:49:59,650 I can do differentiation, still an exponential function, 700 00:49:59,650 --> 00:50:02,460 and differentiate, differentiate, differentiate. 701 00:50:02,460 --> 00:50:05,460 Still exponential function. 702 00:50:05,460 --> 00:50:07,620 So that is very nice because when 703 00:50:07,620 --> 00:50:13,350 we deal with differential equation, 704 00:50:13,350 --> 00:50:16,590 then you can actually remove all those dots 705 00:50:16,590 --> 00:50:20,940 and make them become just exponential function. 706 00:50:20,940 --> 00:50:25,770 So essentially, a very nice property. 707 00:50:25,770 --> 00:50:32,910 So the first property, which is very nice is that it cannot be 708 00:50:32,910 --> 00:50:39,480 killed by differentiation. 709 00:50:39,480 --> 00:50:46,980 You will see how useful this is in the following lectures. 710 00:50:49,620 --> 00:50:52,230 The second thing, which is really nice 711 00:50:52,230 --> 00:50:55,170 is that it has a very nice property. 712 00:50:55,170 --> 00:51:02,090 So basically the exponential i theta 1 times exponential i 713 00:51:02,090 --> 00:51:07,410 theta 2, and that will give you exponential i theta 714 00:51:07,410 --> 00:51:08,980 1 plus theta 2. 715 00:51:11,610 --> 00:51:13,420 So what does that mean? 716 00:51:13,420 --> 00:51:22,110 That means if I have a solution in this form, A exponential i 717 00:51:22,110 --> 00:51:25,400 omega t plus phi. 718 00:51:29,040 --> 00:51:40,200 And I do a times translation, t become t plus A. Then 719 00:51:40,200 --> 00:51:47,650 this become A exponential i omega t plus A plus phi. 720 00:51:51,350 --> 00:51:59,430 So this means that times translation in this rotation 721 00:51:59,430 --> 00:52:04,880 is just a rotation in complex print. 722 00:52:04,880 --> 00:52:05,730 You see? 723 00:52:05,730 --> 00:52:08,900 So now t becomes t plus A. Then you are actually 724 00:52:08,900 --> 00:52:16,190 just changing the angle between this vector and the x-axis. 725 00:52:16,190 --> 00:52:18,140 So as time goes on, what is going 726 00:52:18,140 --> 00:52:21,300 to happen is that this thing will go around and around 727 00:52:21,300 --> 00:52:28,100 and around and the physics is always the set, no matter when 728 00:52:28,100 --> 00:52:32,340 you start counting, and the translation is just 729 00:52:32,340 --> 00:52:35,790 the rotation in this print. 730 00:52:35,790 --> 00:52:37,165 Any questions? 731 00:52:44,940 --> 00:52:50,160 So I think this is actually a basic slide just to remind you 732 00:52:50,160 --> 00:52:52,710 about Euler's formula. 733 00:52:52,710 --> 00:52:55,460 So basically, the explanation i phi 734 00:52:55,460 --> 00:52:58,920 is equal to cosine phi plus i sine phi. 735 00:52:58,920 --> 00:53:03,300 And I think it will be useful if you are not familiar with this. 736 00:53:03,300 --> 00:53:06,180 It is useful to actually review a little bit 737 00:53:06,180 --> 00:53:10,230 about exponential function, which will 738 00:53:10,230 --> 00:53:12,050 be very useful for this class. 739 00:53:17,350 --> 00:53:20,260 So I'm running a bit faster today. 740 00:53:20,260 --> 00:53:25,930 So let's take a look at what we have learned today. 741 00:53:25,930 --> 00:53:31,060 We have analyzed the physics of a harmonic oscillator. 742 00:53:31,060 --> 00:53:37,830 So basically, we start by asking really just a verbal question, 743 00:53:37,830 --> 00:53:40,650 what is going to happen to this mass 744 00:53:40,650 --> 00:53:44,310 on the table attached to a spring. 745 00:53:44,310 --> 00:53:49,620 And what we have learned is that we actually use mathematics. 746 00:53:49,620 --> 00:53:57,750 Basically, we translate all what we have learned about this mass 747 00:53:57,750 --> 00:54:03,480 into mathematics by first define a coordinate system. 748 00:54:03,480 --> 00:54:09,130 Then I'd write everything using that coordinate system. 749 00:54:09,130 --> 00:54:13,125 Then I use Newton's law to help us to solve this question. 750 00:54:15,870 --> 00:54:19,710 And we have analyzed the physics of this harmonic oscillator. 751 00:54:19,710 --> 00:54:23,700 And Hooke's law, we found that he actually, not only 752 00:54:23,700 --> 00:54:30,380 works for this spring-mass system, 753 00:54:30,380 --> 00:54:37,260 it also works for all kinds of different small oscillations 754 00:54:37,260 --> 00:54:40,530 about a point of equilibrium. 755 00:54:40,530 --> 00:54:44,700 So basically, it's actually a universal solution 756 00:54:44,700 --> 00:54:47,070 what we have been doing. 757 00:54:47,070 --> 00:54:51,990 And we have found out a complex exponential function 758 00:54:51,990 --> 00:54:55,450 is actually a beautiful way to present 759 00:54:55,450 --> 00:54:59,910 the solution to the equation of motion we have been studying. 760 00:54:59,910 --> 00:55:02,240 So everything is nice and good. 761 00:55:02,240 --> 00:55:06,960 However, life is hard because there 762 00:55:06,960 --> 00:55:13,010 are many things which actually, we ignored in this example. 763 00:55:13,010 --> 00:55:16,620 One apparent thing, which we actually ignore, 764 00:55:16,620 --> 00:55:18,840 is the direct force. 765 00:55:18,840 --> 00:55:24,420 So you can see that before I was actually making this pendulum 766 00:55:24,420 --> 00:55:27,480 oscillate back and forth. 767 00:55:27,480 --> 00:55:29,430 What is happening now? 768 00:55:29,430 --> 00:55:31,930 There are not oscillating anymore. 769 00:55:31,930 --> 00:55:33,660 Why? 770 00:55:33,660 --> 00:55:37,020 Well, they stopped being. 771 00:55:37,020 --> 00:55:40,440 Apparently, something is missing. 772 00:55:40,440 --> 00:55:46,930 When I actually moved this system, 773 00:55:46,930 --> 00:55:51,490 if I turn off the air so that there's friction, 774 00:55:51,490 --> 00:55:53,590 then it doesn't really move. 775 00:55:53,590 --> 00:55:58,820 If I increase a bit, the air so that the slide have 776 00:55:58,820 --> 00:56:02,330 some slight freedom, then actually, you 777 00:56:02,330 --> 00:56:05,990 can see that you move a bit then you stop. 778 00:56:05,990 --> 00:56:12,625 If I increase this some more, you 779 00:56:12,625 --> 00:56:18,660 can see that the amplitude becomes smaller and smaller. 780 00:56:18,660 --> 00:56:22,400 So in the following lecture, what we are going to do 781 00:56:22,400 --> 00:56:27,110 is to study how to actually include a direct force into it 782 00:56:27,110 --> 00:56:31,210 again and of course, using the same machinery which we have 783 00:56:31,210 --> 00:56:34,160 learned from here and see if we can actually 784 00:56:34,160 --> 00:56:35,890 solve this problem. 785 00:56:35,890 --> 00:56:36,950 Thank you very much. 786 00:56:36,950 --> 00:56:39,360 We actually end up earlier today. 787 00:56:39,360 --> 00:56:40,520 Sorry for that. 788 00:56:40,520 --> 00:56:43,280 And maybe I will make the lecture longer next time. 789 00:56:45,890 --> 00:56:48,340 And if you have any questions about what 790 00:56:48,340 --> 00:56:55,480 we have covered today, I'm here available to help you.