1 00:00:02,110 --> 00:00:04,480 The following content is provided under a Creative 2 00:00:04,480 --> 00:00:05,870 Commons license. 3 00:00:05,870 --> 00:00:08,080 Your support will help MIT OpenCourseWare 4 00:00:08,080 --> 00:00:12,170 continue to offer high quality educational resources for free. 5 00:00:12,170 --> 00:00:14,710 To make a donation or to view additional materials 6 00:00:14,710 --> 00:00:18,670 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:18,670 --> 00:00:19,870 at ocw.mit.edu. 8 00:00:23,000 --> 00:00:25,350 YEN-JIE LEE: OK, so welcome back everybody. 9 00:00:25,350 --> 00:00:27,850 Happy to see you again. 10 00:00:27,850 --> 00:00:33,210 So today, we are going to continue our exploration 11 00:00:33,210 --> 00:00:38,830 and understand single harmonic oscillator. 12 00:00:38,830 --> 00:00:42,690 And this is actually a list of our goals. 13 00:00:42,690 --> 00:00:45,420 And we would like to learn how to translate 14 00:00:45,420 --> 00:00:49,560 physical situations into mathematics so that we can 15 00:00:49,560 --> 00:00:51,670 actually solve the physical problem, 16 00:00:51,670 --> 00:00:54,510 so we actually have and predict what 17 00:00:54,510 --> 00:00:56,760 is going to happen afterward. 18 00:00:56,760 --> 00:01:03,210 And we also sort of started this course 19 00:01:03,210 --> 00:01:06,440 by solving really simple examples, 20 00:01:06,440 --> 00:01:09,000 single harmonic oscillators. 21 00:01:09,000 --> 00:01:11,400 And as a function of time, you will 22 00:01:11,400 --> 00:01:15,060 see that, for our next class, the next lecture, 23 00:01:15,060 --> 00:01:17,460 we are going to bring in more and more objects. 24 00:01:17,460 --> 00:01:21,150 And of course, more objects means more excitement also, 25 00:01:21,150 --> 00:01:24,060 in terms of phenomena, but also more complication 26 00:01:24,060 --> 00:01:26,610 on the mathematics. 27 00:01:26,610 --> 00:01:28,990 So we will see how things go. 28 00:01:28,990 --> 00:01:30,600 And then after that, we are going 29 00:01:30,600 --> 00:01:33,420 to go through infinite number of oscillators 30 00:01:33,420 --> 00:01:35,670 to see what will happen. 31 00:01:35,670 --> 00:01:38,340 Of course, we will produce waves. 32 00:01:38,340 --> 00:01:39,690 That's very exciting. 33 00:01:39,690 --> 00:01:42,390 Then we'll do all kinds of different tricks 34 00:01:42,390 --> 00:01:45,500 to do those waves. 35 00:01:45,500 --> 00:01:49,260 So what we have we learned last time? 36 00:01:49,260 --> 00:01:51,740 So last time we went over example, 37 00:01:51,740 --> 00:01:53,740 a simple harmonic oscillator. 38 00:01:53,740 --> 00:01:56,100 It says you have a rod fixed on the wall, 39 00:01:56,100 --> 00:01:58,750 and you can actually go back and forth. 40 00:01:58,750 --> 00:02:03,390 And we also introduced a model of the drag force, or drag 41 00:02:03,390 --> 00:02:07,200 torque, and that's actually proportionate to the velocity 42 00:02:07,200 --> 00:02:10,050 of the motion of that single particle. 43 00:02:10,050 --> 00:02:12,840 And the interesting thing we learned last time 44 00:02:12,840 --> 00:02:18,520 is that we have three completely different behaviors 45 00:02:18,520 --> 00:02:21,660 if we actually turn on the drag force. 46 00:02:21,660 --> 00:02:24,880 The first one is on that damped. 47 00:02:24,880 --> 00:02:26,790 Damping is actually very small. 48 00:02:26,790 --> 00:02:29,680 Then we have the solution in this form. 49 00:02:29,680 --> 00:02:31,580 It's oscillating. 50 00:02:31,580 --> 00:02:34,740 The amplitude is decaying exponentially. 51 00:02:38,790 --> 00:02:42,750 As we make the drag force larger and larger, 52 00:02:42,750 --> 00:02:47,850 you will pass a critical point, which actually give you 53 00:02:47,850 --> 00:02:51,290 a solution, which you don't have oscillation anymore. 54 00:02:51,290 --> 00:02:53,550 The cosine disappeared. 55 00:02:53,550 --> 00:02:59,760 Finally, if you actually put the whole system into water 56 00:02:59,760 --> 00:03:02,700 or introduce something really dramatic-- 57 00:03:02,700 --> 00:03:05,490 a very, very big drag force-- 58 00:03:05,490 --> 00:03:09,160 then you have overdamp situation. 59 00:03:09,160 --> 00:03:11,660 And there you see that the solution is actually 60 00:03:11,660 --> 00:03:15,490 a sum of two exponential functions. 61 00:03:15,490 --> 00:03:19,800 So this is actually the one equation 62 00:03:19,800 --> 00:03:23,850 which actually works for all the damped situation we discussed 63 00:03:23,850 --> 00:03:25,370 up to now. 64 00:03:25,370 --> 00:03:27,330 And this is actually the map. 65 00:03:27,330 --> 00:03:31,200 Basically, if gamma goes to zero-- 66 00:03:31,200 --> 00:03:34,100 gamma actually controls the size of the drag force. 67 00:03:34,100 --> 00:03:35,430 Then we got no damping. 68 00:03:35,430 --> 00:03:38,600 Then you have a pure, simple harmonic motion. 69 00:03:38,600 --> 00:03:40,950 And as we increase the gamma, then you 70 00:03:40,950 --> 00:03:44,780 get see that the behavior is changing 71 00:03:44,780 --> 00:03:48,690 as we increase the gamma. 72 00:03:48,690 --> 00:03:52,590 So you can see that we can use a quantity, which 73 00:03:52,590 --> 00:03:57,705 is called Q. Q is actually defined as a ratio of omega 74 00:03:57,705 --> 00:04:01,680 at zero, which is basically the natural angular 75 00:04:01,680 --> 00:04:04,500 frequency of the system. 76 00:04:04,500 --> 00:04:10,230 And gamma is a measure of how big the drag force is. 77 00:04:10,230 --> 00:04:13,110 If we make a ratio of this to quantity, 78 00:04:13,110 --> 00:04:17,959 then you'll see that, at Q equal to 0.5, 79 00:04:17,959 --> 00:04:22,125 it reaches a critical point, which actually the behavior 80 00:04:22,125 --> 00:04:24,570 of the whole system changed. 81 00:04:24,570 --> 00:04:27,150 And you can see that the oscillation completely 82 00:04:27,150 --> 00:04:29,400 disappeared. 83 00:04:29,400 --> 00:04:33,450 So that is actually what we have learned last time. 84 00:04:33,450 --> 00:04:37,280 So what are we going to do today? 85 00:04:37,280 --> 00:04:42,940 We have been really doing experiments really 86 00:04:42,940 --> 00:04:45,410 with our hands, hands-on, right? 87 00:04:45,410 --> 00:04:48,170 So basically we will prepare the system. 88 00:04:48,170 --> 00:04:49,850 Then we release it. 89 00:04:49,850 --> 00:04:53,330 Then we don't touch it again and see how this system actually 90 00:04:53,330 --> 00:04:55,380 evolves as a function of time. 91 00:04:55,380 --> 00:04:57,500 So that's what we have been doing. 92 00:04:57,500 --> 00:05:00,530 So today, what we are going to do 93 00:05:00,530 --> 00:05:04,100 is to start to drive this system. 94 00:05:04,100 --> 00:05:07,250 We can introduce some kind of driving force 95 00:05:07,250 --> 00:05:12,230 and see how the system will respond to this external force. 96 00:05:12,230 --> 00:05:14,320 So that is actually what we are going to do. 97 00:05:14,320 --> 00:05:17,030 And that will bring us to the situation 98 00:05:17,030 --> 00:05:21,380 of damped driven harmonic oscillator. 99 00:05:21,380 --> 00:05:23,220 So let's immediately get started. 100 00:05:27,810 --> 00:05:32,280 So we will use the example which we went through last time 101 00:05:32,280 --> 00:05:34,110 as a starting point. 102 00:05:34,110 --> 00:05:38,270 So set example from the last lecture 103 00:05:38,270 --> 00:05:42,730 is a rod, which is fixed on the wall. 104 00:05:42,730 --> 00:05:47,040 And the lens of this rod is over two. 105 00:05:47,040 --> 00:05:53,130 And I define a counter-clockwise direction to be positive. 106 00:05:53,130 --> 00:05:55,650 And I measure the position of the rod 107 00:05:55,650 --> 00:05:57,660 by this theta, which is the angle 108 00:05:57,660 --> 00:06:01,770 between the vertical direction and the pointing direction 109 00:06:01,770 --> 00:06:04,110 of the rod. 110 00:06:04,110 --> 00:06:06,890 And we have went through with the math, 111 00:06:06,890 --> 00:06:09,130 and we got the equation of motion 112 00:06:09,130 --> 00:06:12,660 without external force, which is already 113 00:06:12,660 --> 00:06:14,850 shown on the blackboard. 114 00:06:14,850 --> 00:06:17,560 So now, as I mentioned at the beginning, 115 00:06:17,560 --> 00:06:20,250 I would like to add a driving force, 116 00:06:20,250 --> 00:06:23,835 or driving torque, tau drive. 117 00:06:27,030 --> 00:06:36,490 This is equal to d0 cosine omega d t. 118 00:06:36,490 --> 00:06:43,494 So I am adding a driving torque. 119 00:06:43,494 --> 00:06:48,520 The amplitude of the torque is actually d0. 120 00:06:48,520 --> 00:06:52,530 And there's actually also a harmonic oscillating force, 121 00:06:52,530 --> 00:06:57,670 or torque, and the angle frequency of this torque 122 00:06:57,670 --> 00:06:58,270 is omega d. 123 00:07:01,570 --> 00:07:06,700 And that means our total torque, tau of t, 124 00:07:06,700 --> 00:07:10,120 will be equal to tau g t-- 125 00:07:10,120 --> 00:07:13,360 this is actually coming from the gravitational force-- 126 00:07:13,360 --> 00:07:19,495 plus tau drag, which is to account for the drag force. 127 00:07:22,810 --> 00:07:25,990 So this time we are adding in a tau drag. 128 00:07:30,440 --> 00:07:34,190 So I'm not going to go over all the calculations on how 129 00:07:34,190 --> 00:07:37,080 did the right from the beginning to the end. 130 00:07:37,080 --> 00:07:39,500 But I will just continue from what we actually 131 00:07:39,500 --> 00:07:40,940 started the last time. 132 00:07:40,940 --> 00:07:46,100 So if I have additional driving torque there, 133 00:07:46,100 --> 00:07:50,030 that means my equation of motion will be slightly modified. 134 00:07:50,030 --> 00:07:51,920 This time, my equation of motion will 135 00:07:51,920 --> 00:08:00,980 become theta double dot plus gamma theta dot plus omega 0 136 00:08:00,980 --> 00:08:06,330 squared theta, and that is equal to d0 divided 137 00:08:06,330 --> 00:08:11,540 by I. This is actually divided by I because, in order 138 00:08:11,540 --> 00:08:13,490 to get the acceleration, I'll need 139 00:08:13,490 --> 00:08:16,970 to divide my torque by a movement of inertia 140 00:08:16,970 --> 00:08:21,960 of this system and cosine omega d t. 141 00:08:21,960 --> 00:08:25,520 This is actually the oscillating frequency 142 00:08:25,520 --> 00:08:28,910 of the driving torque. 143 00:08:28,910 --> 00:08:37,100 And just a reminder, gamma is defined to be equal to 3b m l 144 00:08:37,100 --> 00:08:38,270 squared. 145 00:08:38,270 --> 00:08:41,270 And the omega 0 is actually defined 146 00:08:41,270 --> 00:08:48,350 to be square root of 3g over 2l. 147 00:08:48,350 --> 00:08:50,810 So as I mentioned in the beginning, 148 00:08:50,810 --> 00:08:52,490 this is actually giving you a sense 149 00:08:52,490 --> 00:08:55,850 of the size of the drag force. 150 00:08:55,850 --> 00:08:59,060 And the right hand side, the omega 0 151 00:08:59,060 --> 00:09:04,190 is actually the natural angular frequency. 152 00:09:04,190 --> 00:09:06,470 So, of course, we can actually simplify this 153 00:09:06,470 --> 00:09:12,170 by replacing this term, or this constant, by symbol. 154 00:09:12,170 --> 00:09:15,350 So the symbol I'm choosing is f0. 155 00:09:15,350 --> 00:09:23,390 And this is defined to be d0 divided by I. Therefore, 156 00:09:23,390 --> 00:09:26,930 I arrive at my final equation of motion-- 157 00:09:26,930 --> 00:09:35,390 theta double dot plus gamma theta dot plus omega 0 158 00:09:35,390 --> 00:09:44,294 squared theta, and that is equal to f0 cosine omega d t. 159 00:09:48,770 --> 00:09:51,810 So I hope this looks pretty straightforward to you. 160 00:09:55,330 --> 00:09:57,140 So this is our equation of motion 161 00:09:57,140 --> 00:09:59,650 you can see from this slide. 162 00:09:59,650 --> 00:10:02,890 So we have three terms in addition 163 00:10:02,890 --> 00:10:04,780 to the theta double dot. 164 00:10:04,780 --> 00:10:06,730 The first one is actually related 165 00:10:06,730 --> 00:10:09,700 drag force, or drag torque. 166 00:10:09,700 --> 00:10:12,820 The second one is actually related to so-called spring 167 00:10:12,820 --> 00:10:13,840 force. 168 00:10:13,840 --> 00:10:17,920 So that is actually be related to the spring constant 169 00:10:17,920 --> 00:10:21,790 or, because of the restoring force 170 00:10:21,790 --> 00:10:26,470 of the gravitational force. 171 00:10:26,470 --> 00:10:28,715 The third one is actually what we just add in 172 00:10:28,715 --> 00:10:31,540 as a driving force. 173 00:10:31,540 --> 00:10:37,600 So one question which I would like to ask you is-- 174 00:10:37,600 --> 00:10:42,440 so now I bring one more complication to this system. 175 00:10:42,440 --> 00:10:46,990 So now I am driving this system with a different frequency, 176 00:10:46,990 --> 00:10:49,232 which is omega d. 177 00:10:49,232 --> 00:10:52,920 The question is, what would be the resulting oscillation 178 00:10:52,920 --> 00:10:57,810 frequency of this driven harmonic oscillator? 179 00:10:57,810 --> 00:11:00,350 What is going to happen? 180 00:11:00,350 --> 00:11:03,950 Well, this system actually follows 181 00:11:03,950 --> 00:11:09,950 the original damped oscillator frequency, omega, which 182 00:11:09,950 --> 00:11:13,040 is actually close to omega 0. 183 00:11:13,040 --> 00:11:17,990 Or what this system actually follows the driving force 184 00:11:17,990 --> 00:11:19,890 frequency. 185 00:11:19,890 --> 00:11:23,270 Finally, maybe this system chooses 186 00:11:23,270 --> 00:11:25,520 to do something in between. 187 00:11:25,520 --> 00:11:30,760 We don't know what is going to happen. 188 00:11:30,760 --> 00:11:35,260 So our job today is to solve these equation of motion 189 00:11:35,260 --> 00:11:38,150 and to see what we can learn from the mathematics. 190 00:11:38,150 --> 00:11:43,660 Then we can actually check those results 191 00:11:43,660 --> 00:11:47,310 to see if that agrees with the experimental result, which 192 00:11:47,310 --> 00:11:50,580 is through those demos, OK? 193 00:11:50,580 --> 00:11:55,160 So as usual, I have this equation of motion here. 194 00:11:55,160 --> 00:11:58,420 So one trick, which I have been using, 195 00:11:58,420 --> 00:12:02,230 is to go to complex notation, right? 196 00:12:02,230 --> 00:12:08,440 Therefore, I can now re-write this thing to be Z double dot 197 00:12:08,440 --> 00:12:16,540 plus gamma Z dot plus omega 0 squared Z, 198 00:12:16,540 --> 00:12:25,750 and that is equal to f0 exponential i omega d t. 199 00:12:25,750 --> 00:12:30,300 So basically, I just go to the complex notation. 200 00:12:30,300 --> 00:12:33,765 And we would like to solve this equation. 201 00:12:36,430 --> 00:12:37,980 So in order to solve this equation, 202 00:12:37,980 --> 00:12:40,245 I make a guess, a test function. 203 00:12:44,850 --> 00:12:48,570 I guess Z of t has this functional form. 204 00:12:48,570 --> 00:12:51,156 This is equal to A-- 205 00:12:51,156 --> 00:12:53,940 some kind of amplitude-- 206 00:12:53,940 --> 00:13:01,600 exponential i omega d t minus delta. 207 00:13:01,600 --> 00:13:05,550 Delta is actually some kind of angle, 208 00:13:05,550 --> 00:13:11,070 which is actually to account for the possible delay 209 00:13:11,070 --> 00:13:12,620 of the system. 210 00:13:12,620 --> 00:13:14,370 So if I start to try-- 211 00:13:14,370 --> 00:13:17,730 for example, this is a system, which I am interested-- 212 00:13:17,730 --> 00:13:22,320 I start to drive this system, it may take some time 213 00:13:22,320 --> 00:13:26,520 for the system to react to your driving force. 214 00:13:26,520 --> 00:13:31,970 So that's actually accounted for by this delta constant. 215 00:13:31,970 --> 00:13:35,880 And the amplitude is actually what we were wondering, 216 00:13:35,880 --> 00:13:37,320 what would be the amplitude. 217 00:13:37,320 --> 00:13:41,420 Therefore, you have some kind of a constant in front 218 00:13:41,420 --> 00:13:43,720 of the exponential function. 219 00:13:43,720 --> 00:13:48,180 And you can see that this exponential is actually having 220 00:13:48,180 --> 00:13:50,550 angular frequency, omega d. 221 00:13:50,550 --> 00:13:55,410 And that is actually designed to cancel this exponential i omega 222 00:13:55,410 --> 00:13:58,950 d t here in the drag force. 223 00:13:58,950 --> 00:14:04,920 So now we can as you calculate what z dot of t 224 00:14:04,920 --> 00:14:12,600 would be equal to i omega d Z. Z double dot t will 225 00:14:12,600 --> 00:14:19,680 be equal to minus omega d squared Z. With those, 226 00:14:19,680 --> 00:14:24,000 we can now plug that back into this equation of motion 227 00:14:24,000 --> 00:14:27,360 and see what we can actually learn from there. 228 00:14:27,360 --> 00:14:29,340 So basically, what I'm going to do 229 00:14:29,340 --> 00:14:31,740 is to insert all those things back 230 00:14:31,740 --> 00:14:33,450 into the equation of motion. 231 00:14:33,450 --> 00:14:37,080 And that is actually going to be like this. 232 00:14:37,080 --> 00:14:39,690 Basically, the first term, the double dot, 233 00:14:39,690 --> 00:14:45,690 you get a minus omega d squared out of it. 234 00:14:45,690 --> 00:14:49,200 The second term, gamma Z dot-- 235 00:14:49,200 --> 00:14:51,060 I have Z dots here. 236 00:14:51,060 --> 00:14:58,485 Basically, I would get plus i omega d gamma out 237 00:14:58,485 --> 00:15:00,660 of the second term. 238 00:15:00,660 --> 00:15:06,470 That's third term, I get omega 0 squared out of it. 239 00:15:06,470 --> 00:15:09,660 And that is actually multiplied by Z. 240 00:15:09,660 --> 00:15:16,490 And this is equal to f0 exponential i omega d t. 241 00:15:23,070 --> 00:15:29,120 All right, and we also know from this expression 242 00:15:29,120 --> 00:15:33,500 Z is equal to A exponential i omega d t minus delta. 243 00:15:33,500 --> 00:15:35,020 That's the test function. 244 00:15:35,020 --> 00:15:42,030 So this is actually equal to A exponential i omega d 245 00:15:42,030 --> 00:15:45,730 t minus delta. 246 00:15:45,730 --> 00:15:49,150 So now what I can do is-- 247 00:15:49,150 --> 00:15:53,020 I have some constant in the front. 248 00:15:53,020 --> 00:15:56,290 Multiply it by exponential i omega d t. 249 00:15:56,290 --> 00:16:01,110 And now I can actually cancel this exponential i omega d t 250 00:16:01,110 --> 00:16:04,000 with the right hand side term. 251 00:16:04,000 --> 00:16:04,540 Very good. 252 00:16:07,710 --> 00:16:11,680 The whole equation is actually exponential free. 253 00:16:11,680 --> 00:16:14,560 Now I don't have any exponential function left. 254 00:16:14,560 --> 00:16:19,430 And exponential i delta is actually just a constant. 255 00:16:19,430 --> 00:16:23,764 So now this equation is actually independent of time. 256 00:16:26,290 --> 00:16:30,110 So what I getting is like this-- basically 257 00:16:30,110 --> 00:16:35,860 if I multiply the both sides by exponential i delta, 258 00:16:35,860 --> 00:16:45,610 then I get minus omega d squared plus i omega d t plus omega 0 259 00:16:45,610 --> 00:16:50,240 squared A. And this is going to be 260 00:16:50,240 --> 00:16:56,540 equal to f exponential i delta because I multiply both sides 261 00:16:56,540 --> 00:17:00,620 by exponential i delta. 262 00:17:00,620 --> 00:17:11,410 And this is equal to f cosine delta plus i f sine delta. 263 00:17:11,410 --> 00:17:14,115 Just your last equation. 264 00:17:14,115 --> 00:17:14,990 Any questions so far? 265 00:17:19,930 --> 00:17:21,410 So look at what I have been doing. 266 00:17:21,410 --> 00:17:25,485 So I have this equation of motion. 267 00:17:25,485 --> 00:17:30,040 As usual, I go to complex notation. 268 00:17:30,040 --> 00:17:35,620 Then I guess Z equal to A exponential i omega d 269 00:17:35,620 --> 00:17:39,910 t minus delta because my friends from the math department 270 00:17:39,910 --> 00:17:44,650 already solved this, and I'm just following it. 271 00:17:44,650 --> 00:17:49,660 Then I can calculate all those terms, plug in e, 272 00:17:49,660 --> 00:17:53,440 and basically, you will arrive at this equation. 273 00:17:53,440 --> 00:17:58,090 This equation is a complex equation. 274 00:17:58,090 --> 00:17:59,350 So what does that mean? 275 00:17:59,350 --> 00:18:03,250 That means one equation is equal to two equations 276 00:18:03,250 --> 00:18:09,360 because you have the real part, you have the imaginary part. 277 00:18:09,360 --> 00:18:13,980 Therefore, that's very nice because I have two unknowns. 278 00:18:13,980 --> 00:18:17,040 The first one is A, a constant. 279 00:18:17,040 --> 00:18:18,840 And the second one is delta. 280 00:18:18,840 --> 00:18:21,210 Now I have two equations I can solve 281 00:18:21,210 --> 00:18:24,660 what would be the functional form for A and the delta. 282 00:18:24,660 --> 00:18:30,910 And let me go immediately solve this equation. 283 00:18:30,910 --> 00:18:37,110 So if I take the real part from this equation, 284 00:18:37,110 --> 00:18:41,970 basically what I'm going to get is omega 0 squared-- 285 00:18:41,970 --> 00:18:46,680 this is real-- minus omega d squared-- 286 00:18:46,680 --> 00:18:48,080 this is also real-- 287 00:18:48,080 --> 00:18:52,260 times A. A is actually some real number. 288 00:18:52,260 --> 00:18:56,621 This is actually equal to f cosine-- 289 00:18:56,621 --> 00:18:57,120 f0. 290 00:18:57,120 --> 00:19:00,690 Sorry, I missed a zero here. 291 00:19:00,690 --> 00:19:02,970 So that zero I missed. 292 00:19:02,970 --> 00:19:05,310 This should be f0. 293 00:19:05,310 --> 00:19:11,100 f0 cosine delta. 294 00:19:11,100 --> 00:19:14,580 And I can also collect all the terms, 295 00:19:14,580 --> 00:19:17,520 which is imaginary terms. 296 00:19:17,520 --> 00:19:22,170 Then I get only the second term from the left hand side 297 00:19:22,170 --> 00:19:24,900 is with i in front of it. 298 00:19:24,900 --> 00:19:33,000 Therefore, I get omega d gamma A from the left hand side. 299 00:19:33,000 --> 00:19:34,620 And from the right hand side, there's 300 00:19:34,620 --> 00:19:38,320 only one imaginary term. 301 00:19:38,320 --> 00:19:46,920 Therefore, I get-- this is equal to f0 sine delta. 302 00:19:46,920 --> 00:19:49,620 So now I have two equations. 303 00:19:49,620 --> 00:19:51,330 I have two unknowns. 304 00:19:51,330 --> 00:19:56,310 Therefore, I can easily solve A and delta. 305 00:19:56,310 --> 00:19:59,130 So I call this equation number one. 306 00:19:59,130 --> 00:20:02,310 I call this equation number two. 307 00:20:02,310 --> 00:20:08,760 So now I can-- sounding in quadrature the two equations-- 308 00:20:08,760 --> 00:20:10,010 in quadrature. 309 00:20:10,010 --> 00:20:14,700 And the left-hand side will give you 310 00:20:14,700 --> 00:20:21,200 A squared omega 0 squared minus omega 311 00:20:21,200 --> 00:20:29,250 d squared squared plus omega d squared gamma squared. 312 00:20:29,250 --> 00:20:31,940 That is actually coming from the second equation. 313 00:20:35,180 --> 00:20:38,030 That gives you the left hand side. 314 00:20:38,030 --> 00:20:45,140 It's a square of the sum the first and second equation. 315 00:20:45,140 --> 00:20:56,060 And the right hand side will become f0 square cosine delta 316 00:20:56,060 --> 00:20:59,963 cosine squared delta plus sine squared delta. 317 00:21:02,540 --> 00:21:05,752 And this is equal to 1. 318 00:21:05,752 --> 00:21:10,470 So that's actually the trick to get rid of delta. 319 00:21:10,470 --> 00:21:15,830 Then I can get what will be the resulting A. A is actually 320 00:21:15,830 --> 00:21:19,940 a function of omega d. 321 00:21:19,940 --> 00:21:22,350 Omega d is given to you. 322 00:21:22,350 --> 00:21:25,890 It's actually determined by you-- 323 00:21:25,890 --> 00:21:30,600 how fast do you want to oscillate this system. 324 00:21:30,600 --> 00:21:37,350 And this is equal to f0 divided by square root 325 00:21:37,350 --> 00:21:39,150 of this whole thing. 326 00:21:39,150 --> 00:21:42,510 So this will give you omega 0 squared 327 00:21:42,510 --> 00:21:48,920 minus omega d squared squared plus omega 328 00:21:48,920 --> 00:21:51,930 d squared gamma squared. 329 00:21:58,530 --> 00:22:02,910 Then we can also calculate what would be the delta. 330 00:22:02,910 --> 00:22:07,800 The trick is to take a ratio between equation number two 331 00:22:07,800 --> 00:22:10,080 and the equation number one-- 332 00:22:10,080 --> 00:22:11,790 2 divided by 1. 333 00:22:11,790 --> 00:22:16,820 Basically, you will get tangent delta. 334 00:22:16,820 --> 00:22:19,370 This is sine divided by cosine. 335 00:22:19,370 --> 00:22:22,260 f0 actually cancel. 336 00:22:22,260 --> 00:22:24,540 This is equal to what? 337 00:22:24,540 --> 00:22:28,030 Equal to the ratio of these two terms. 338 00:22:28,030 --> 00:22:31,480 After you take the ratio, A drops out. 339 00:22:31,480 --> 00:22:35,760 Basically, what you get is gamma omega 340 00:22:35,760 --> 00:22:43,790 d divided by omega 0 squared minus omega d squared. 341 00:22:47,150 --> 00:22:54,890 So we have solved A and the delta through this exercise. 342 00:22:54,890 --> 00:22:56,450 So what does that mean? 343 00:22:59,260 --> 00:23:06,280 Originally, I assume my solution to be A exponential i omega d 344 00:23:06,280 --> 00:23:09,500 t minus delta. 345 00:23:09,500 --> 00:23:17,100 Therefore, I would like to go back to the real world, which 346 00:23:17,100 --> 00:23:18,950 is actually theta. 347 00:23:18,950 --> 00:23:21,560 So basically, if I take the real part, 348 00:23:21,560 --> 00:23:29,150 I would get theta of t, which is actually the real part of Z. 349 00:23:29,150 --> 00:23:41,840 And that will give you A omega d cosine omega d t minus delta is 350 00:23:41,840 --> 00:23:43,749 also a function of omega d. 351 00:23:47,581 --> 00:23:50,490 So we have done this exercise. 352 00:23:50,490 --> 00:23:54,810 And you can see that the first thing which we see here 353 00:23:54,810 --> 00:24:01,220 is that there's no free parameter from this solution. 354 00:24:01,220 --> 00:24:05,310 A is decided by omega d. 355 00:24:05,310 --> 00:24:08,340 And delta is also decided by omega d. 356 00:24:08,340 --> 00:24:10,200 There's a lot of math, but actually we 357 00:24:10,200 --> 00:24:13,090 have overcome those and that we have a solution. 358 00:24:13,090 --> 00:24:18,310 But it is actually clear to you that this cannot be the full 359 00:24:18,310 --> 00:24:20,340 story. 360 00:24:20,340 --> 00:24:23,040 Because you have a second-order differential equation, 361 00:24:23,040 --> 00:24:27,940 you need to have two free parameters in the solution. 362 00:24:27,940 --> 00:24:29,220 What is actually missing? 363 00:24:29,220 --> 00:24:31,152 Anybody can tell me what is missing. 364 00:24:31,152 --> 00:24:32,610 AUDIENCE: The homogeneous solution. 365 00:24:32,610 --> 00:24:34,170 YEN-JIE LEE: Very good. 366 00:24:34,170 --> 00:24:36,810 The homogeneous solution is missing. 367 00:24:36,810 --> 00:24:39,960 So that's actually why we actually 368 00:24:39,960 --> 00:24:41,910 have no free parameter here. 369 00:24:41,910 --> 00:24:45,960 Once is the omega d is determined, 370 00:24:45,960 --> 00:24:51,120 once the f0 is given, then you have the functional form which 371 00:24:51,120 --> 00:24:54,930 decides what is actually theta. 372 00:24:54,930 --> 00:24:59,360 So what in actually the full solution? 373 00:24:59,360 --> 00:25:04,170 A full solution should be, as you said, 374 00:25:04,170 --> 00:25:07,290 a combination of homogeneous solution 375 00:25:07,290 --> 00:25:10,960 and the particular solution which we actually got here. 376 00:25:10,960 --> 00:25:17,990 So if I prepare the system to be in a situation of, for example, 377 00:25:17,990 --> 00:25:20,420 underdamped situation. 378 00:25:20,420 --> 00:25:23,780 Then what I'm going to do is actually pretty simple. 379 00:25:23,780 --> 00:25:30,350 What I am going to do is to just copy the underdamped solution 380 00:25:30,350 --> 00:25:33,300 from last lecture and combine that 381 00:25:33,300 --> 00:25:36,900 with my particular solution, which I obtained here. 382 00:25:36,900 --> 00:25:39,990 So that actually to see what actually 383 00:25:39,990 --> 00:25:41,870 the full solution looks like. 384 00:25:41,870 --> 00:25:49,545 I have A omega d cosine omega d t minus 385 00:25:49,545 --> 00:25:54,990 delta is a function of omega d. 386 00:25:54,990 --> 00:26:03,565 This is actually so-called steady-state solution. 387 00:26:07,380 --> 00:26:09,420 And, of course, as you mentioned, 388 00:26:09,420 --> 00:26:13,394 I need to also add the homogeneous solution 389 00:26:13,394 --> 00:26:15,060 and basically the no -- this actually -- 390 00:26:15,060 --> 00:26:17,610 according to what I wrote there, I 391 00:26:17,610 --> 00:26:21,150 have a functional form of exponential A exponential 392 00:26:21,150 --> 00:26:26,220 minus gamma over 2t equals sine omega t plus alpha. 393 00:26:26,220 --> 00:26:29,560 So I changed A to B because I already have the A there just 394 00:26:29,560 --> 00:26:31,490 to avoid confusion. 395 00:26:31,490 --> 00:26:36,125 Then basically, you get B exponential minus gamma 396 00:26:36,125 --> 00:26:45,540 over 2t cosine omega t plus alpha. 397 00:26:45,540 --> 00:26:52,120 Basically, they are two free parameters, B and alpha. 398 00:26:52,120 --> 00:26:56,070 Those two free parameters can be determined 399 00:26:56,070 --> 00:26:58,740 by initial conditions. 400 00:26:58,740 --> 00:27:05,290 So, for example, initially I actually release the rod 401 00:27:05,290 --> 00:27:10,050 at some fixed angle of theta initial. 402 00:27:10,050 --> 00:27:12,750 And also the initial velocity is 0. 403 00:27:12,750 --> 00:27:15,930 Then I can actually practice solution A 404 00:27:15,930 --> 00:27:22,200 using those initial conditions to solve B and alpha. 405 00:27:22,200 --> 00:27:24,431 Any questions so far? 406 00:27:24,431 --> 00:27:24,930 Yes. 407 00:27:24,930 --> 00:27:27,249 AUDIENCE: Are we assuming it's underdamped? 408 00:27:27,249 --> 00:27:29,290 YEN-JIE LEE: Yeah, I'm assuming it's underdamped, 409 00:27:29,290 --> 00:27:31,000 the situation. 410 00:27:31,000 --> 00:27:32,350 So that's the assumption. 411 00:27:32,350 --> 00:27:39,960 So it depends on the size of gamma and the omega 0. 412 00:27:39,960 --> 00:27:44,115 Then you have actually four different kinds of solution. 413 00:27:44,115 --> 00:27:48,460 If gamma is equal to 0, then what you are going to plug in 414 00:27:48,460 --> 00:27:55,890 is the solution from no damping as a your homogeneous solution. 415 00:27:55,890 --> 00:27:59,640 And if you prepare this system underwater, 416 00:27:59,640 --> 00:28:03,030 damping is colossal, it's huge, then 417 00:28:03,030 --> 00:28:08,940 you actually plug in the overdamped solution 418 00:28:08,940 --> 00:28:12,960 to be your homogeneous part of the solution. 419 00:28:12,960 --> 00:28:14,619 Any other questions? 420 00:28:14,619 --> 00:28:15,410 Very good question. 421 00:28:18,100 --> 00:28:24,020 So now maybe you got confused a bit. 422 00:28:24,020 --> 00:28:27,200 I have now omega d. 423 00:28:27,200 --> 00:28:30,050 I have also omega. 424 00:28:30,050 --> 00:28:33,830 And there's another one we just called omega 0. 425 00:28:33,830 --> 00:28:35,480 What are those? 426 00:28:35,480 --> 00:28:41,405 So omega 0 is the natural angular frequency 427 00:28:41,405 --> 00:28:44,260 without given the drag force. 428 00:28:44,260 --> 00:28:47,465 If you remove everything just like 429 00:28:47,465 --> 00:28:51,230 without considering any drag force, et cetera, 430 00:28:51,230 --> 00:28:55,100 and that is actually the natural frequency of the system. 431 00:28:55,100 --> 00:28:57,470 And what is omega? 432 00:28:57,470 --> 00:29:00,050 Omega, according to the function, 433 00:29:00,050 --> 00:29:04,130 omega is defined to be omega 0 squared minus gamma squared 434 00:29:04,130 --> 00:29:06,740 over 4 square root of that. 435 00:29:06,740 --> 00:29:10,970 That is actually the oscillation frequency, which we actually 436 00:29:10,970 --> 00:29:14,480 discussed last lecture, after you 437 00:29:14,480 --> 00:29:18,680 add drag force into it again. 438 00:29:18,680 --> 00:29:25,560 Finally, omega d is how fast you actually drive this system. 439 00:29:25,560 --> 00:29:28,450 So that is actually the definition 440 00:29:28,450 --> 00:29:31,890 of these three omegas. 441 00:29:31,890 --> 00:29:38,070 So you can see that, if I prepare my solution 442 00:29:38,070 --> 00:29:43,055 to be underdamped situation, then basically you 443 00:29:43,055 --> 00:29:45,720 will see that this is actually so-called a steady-state 444 00:29:45,720 --> 00:29:50,730 solution because A omega d is a constant. 445 00:29:50,730 --> 00:29:52,165 So it's going to be there forever. 446 00:29:55,410 --> 00:30:00,210 And the second term is actually B exponential minus gamma 447 00:30:00,210 --> 00:30:02,146 over 2t. 448 00:30:02,146 --> 00:30:05,200 It's decaying as a function of time. 449 00:30:05,200 --> 00:30:09,690 So if you are patient enough, you wait, 450 00:30:09,690 --> 00:30:13,350 then this will be gone. 451 00:30:13,350 --> 00:30:16,260 So that is actually how we actually understand 452 00:30:16,260 --> 00:30:18,900 this mathematical result. And now, of course, you 453 00:30:18,900 --> 00:30:20,880 can actually take a look at this. 454 00:30:20,880 --> 00:30:25,770 This is actually just assuming some kind of initial condition 455 00:30:25,770 --> 00:30:30,870 and plug in the solution and plot it as a function of time. 456 00:30:30,870 --> 00:30:32,970 And you can see that this function looks really 457 00:30:32,970 --> 00:30:35,520 weird, looks a bit surprising. 458 00:30:35,520 --> 00:30:36,780 What does that mean? 459 00:30:36,780 --> 00:30:38,160 It looks really strange. 460 00:30:38,160 --> 00:30:40,890 But at some point, this superposition 461 00:30:40,890 --> 00:30:42,240 of these two functions-- 462 00:30:42,240 --> 00:30:46,050 because one of the functions actually dies out, disappears-- 463 00:30:46,050 --> 00:30:49,900 then you will see that, if you wait long enough, 464 00:30:49,900 --> 00:30:55,110 then you actually only see a very simple structure, 465 00:30:55,110 --> 00:30:58,950 which is oscillation frequency of omega d. 466 00:30:58,950 --> 00:31:02,520 And that means a large t. 467 00:31:02,520 --> 00:31:05,840 In the beginning, the system will not like it. 468 00:31:05,840 --> 00:31:08,110 You drive it, and the system don't like it. 469 00:31:08,110 --> 00:31:11,820 Like if I go and shake you, in the beginning, 470 00:31:11,820 --> 00:31:14,400 you would not like it-- 471 00:31:14,400 --> 00:31:14,970 maybe. 472 00:31:14,970 --> 00:31:18,950 And if I shake you long enough, and you say, come on, OK, fine. 473 00:31:18,950 --> 00:31:21,210 I accept that. 474 00:31:21,210 --> 00:31:25,350 So that is actually what is going to happen to the system. 475 00:31:25,350 --> 00:31:31,900 So now I would like to go through a short demonstration, 476 00:31:31,900 --> 00:31:35,480 which is actually the air cart, which you seen before. 477 00:31:35,480 --> 00:31:39,120 There's a mass and there are two springs in the front 478 00:31:39,120 --> 00:31:41,940 and the back of this cart. 479 00:31:41,940 --> 00:31:46,260 And, of course, as usual, I would turn on the air 480 00:31:46,260 --> 00:31:49,950 so that I make the friction smaller, 481 00:31:49,950 --> 00:31:54,240 but there's still some residual friction. 482 00:31:54,240 --> 00:31:57,630 And you will see that this mass is actually 483 00:31:57,630 --> 00:32:00,630 oscillating back and forth. 484 00:32:00,630 --> 00:32:04,290 And the amplitude can become smaller and smaller 485 00:32:04,290 --> 00:32:07,460 as a function of time. 486 00:32:07,460 --> 00:32:11,310 Now in the right hand side, I have a motor, 487 00:32:11,310 --> 00:32:14,220 which actually can drive this-- 488 00:32:14,220 --> 00:32:17,540 I can actually shorten or increase 489 00:32:17,540 --> 00:32:19,430 the length of the right hand side string. 490 00:32:19,430 --> 00:32:23,960 Then I actually introduce a driving force 491 00:32:23,960 --> 00:32:26,280 by the right hand side motor. 492 00:32:26,280 --> 00:32:32,670 If I turn it down, this is what is going to happen. 493 00:32:32,670 --> 00:32:38,850 So we can see now this motor is actually going back and forth. 494 00:32:38,850 --> 00:32:43,830 And it has a slightly higher frequency compared 495 00:32:43,830 --> 00:32:46,135 to the natural frequency. 496 00:32:46,135 --> 00:32:49,680 So the frequency of the motor is higher. 497 00:32:49,680 --> 00:32:55,320 And you can see that this cart is actually oscillating. 498 00:32:55,320 --> 00:33:00,404 But you can see that sometimes it pulls and sometimes it 499 00:33:00,404 --> 00:33:00,945 moves faster. 500 00:33:04,050 --> 00:33:05,840 So you can see that it's actually moving. 501 00:33:05,840 --> 00:33:09,410 And it stops a little bit because they 502 00:33:09,410 --> 00:33:16,290 are all superposition of two different kinds of oscillating 503 00:33:16,290 --> 00:33:17,930 functions come into play. 504 00:33:17,930 --> 00:33:19,550 You can see that now. 505 00:33:19,550 --> 00:33:21,600 It got slowed down, and it can become 506 00:33:21,600 --> 00:33:23,820 faster and slower and faster. 507 00:33:23,820 --> 00:33:30,100 But eventually, if you wait long enough, 508 00:33:30,100 --> 00:33:31,760 what is going to happen? 509 00:33:31,760 --> 00:33:33,960 What is going to happen? 510 00:33:33,960 --> 00:33:35,250 If we wait long enough-- 511 00:33:35,250 --> 00:33:36,648 AUDIENCE: [INAUDIBLE]. 512 00:33:39,450 --> 00:33:40,380 YEN-JIE LEE: Exactly. 513 00:33:40,380 --> 00:33:43,500 So basically, if you wait long enough, as you said, 514 00:33:43,500 --> 00:33:47,261 you will actually just oscillate at the frequency of the driving 515 00:33:47,261 --> 00:33:47,760 force. 516 00:33:47,760 --> 00:33:52,440 You can see that this motion looks really bizarre, right? 517 00:33:52,440 --> 00:33:54,120 Sometimes it stops, and sometimes 518 00:33:54,120 --> 00:33:57,600 it actually continues to move. 519 00:33:57,600 --> 00:34:00,217 And are you surprised? 520 00:34:04,690 --> 00:34:06,880 Probably you are not surprised anymore 521 00:34:06,880 --> 00:34:10,510 because we know math is the language to describe nature. 522 00:34:10,510 --> 00:34:14,260 And indeed it predicts this kind of behavior. 523 00:34:14,260 --> 00:34:16,120 That's really pretty cool. 524 00:34:19,120 --> 00:34:23,260 In order to help you to learn a bit how to actually translate 525 00:34:23,260 --> 00:34:26,679 a physical situation into mathematics, what I am going 526 00:34:26,679 --> 00:34:31,000 to do is to introduce you another example so 527 00:34:31,000 --> 00:34:35,420 that actually we can actually solve it together. 528 00:34:35,420 --> 00:34:38,830 So now I would like to drive a pendulum. 529 00:34:45,969 --> 00:34:50,395 So I prepare a pendulum at time equal to 0. 530 00:34:53,650 --> 00:35:01,210 This is a string attached to a ball with mass equal to m. 531 00:35:01,210 --> 00:35:04,870 And the length of the string is equal to l. 532 00:35:04,870 --> 00:35:08,410 And the angle between the vertical direction 533 00:35:08,410 --> 00:35:14,440 and the direction of the string is theta. 534 00:35:14,440 --> 00:35:18,000 And, of course, I can actually give you initial condition X 535 00:35:18,000 --> 00:35:21,010 initial, which is actually measured 536 00:35:21,010 --> 00:35:32,620 with respect to the vertical direction, and time equal to t. 537 00:35:32,620 --> 00:35:37,430 This is actually the original vertical direction, 538 00:35:37,430 --> 00:35:40,350 the same as this dashed line. 539 00:35:40,350 --> 00:35:48,800 And I can actually move the top of the string back and forth 540 00:35:48,800 --> 00:35:50,750 to some position. 541 00:35:50,750 --> 00:35:57,620 And, of course, this string is connected to the ball. 542 00:35:57,620 --> 00:36:02,960 And this system is actually driven from the top 543 00:36:02,960 --> 00:36:04,780 by the engine's hand, so the engine 544 00:36:04,780 --> 00:36:08,750 is actually shaking this system from the top. 545 00:36:08,750 --> 00:36:11,220 And I do it really nicely. 546 00:36:11,220 --> 00:36:16,440 So basically, I define that the displacement, d, 547 00:36:16,440 --> 00:36:22,650 as a function of time, is equal to delta sine omega d t. 548 00:36:25,820 --> 00:36:29,660 So that is actually what I'm going to do. 549 00:36:29,660 --> 00:36:31,740 OK, and I would like to see what is going 550 00:36:31,740 --> 00:36:34,910 to happen to this pendulum. 551 00:36:34,910 --> 00:36:40,200 So, as usual, the first step towards solving this problem 552 00:36:40,200 --> 00:36:44,702 is to define a coordinate system. 553 00:36:44,702 --> 00:36:46,410 So what is actually the coordinate system 554 00:36:46,410 --> 00:36:47,800 I'm going to use? 555 00:36:47,800 --> 00:36:53,760 So now I define pointing upward to be y. 556 00:36:53,760 --> 00:36:57,660 I define the horizontal direction 557 00:36:57,660 --> 00:37:02,220 pointing to the right hand side of the board to be x. 558 00:37:02,220 --> 00:37:03,810 So that's not good enough. 559 00:37:03,810 --> 00:37:06,480 I still need the origin, right? 560 00:37:06,480 --> 00:37:12,870 So now I also define my origin to be the original position 561 00:37:12,870 --> 00:37:21,270 of the ball which is actually completely addressed 562 00:37:21,270 --> 00:37:23,340 before I do the experiment. 563 00:37:23,340 --> 00:37:26,250 So that this is actually the equilibrium position 564 00:37:26,250 --> 00:37:28,150 of this system actually. 565 00:37:28,150 --> 00:37:33,130 Then I define here to be 0, 0. 566 00:37:33,130 --> 00:37:37,170 So once I have that defined, I can now 567 00:37:37,170 --> 00:37:42,840 express the position of this mass of this ball 568 00:37:42,840 --> 00:37:45,210 to be xt and yt. 569 00:37:48,150 --> 00:37:53,020 And see what we are going to get. 570 00:37:53,020 --> 00:37:55,780 Of course, as usual, we are going 571 00:37:55,780 --> 00:38:04,270 to analyze the force actually acting on this ball. 572 00:38:04,270 --> 00:38:08,455 So therefore, as usual, we will draw a force diagram. 573 00:38:13,540 --> 00:38:17,080 So basically, you have the little mass here, 574 00:38:17,080 --> 00:38:25,990 and you have actually two forces acting on this little mass, 575 00:38:25,990 --> 00:38:28,806 or little ball. 576 00:38:28,806 --> 00:38:31,980 This is Fg pointing downward. 577 00:38:31,980 --> 00:38:33,880 It's a gravitational force. 578 00:38:33,880 --> 00:38:38,290 And now this is actually equal to minus mg y. 579 00:38:41,950 --> 00:38:46,400 And there's also a string tension, 580 00:38:46,400 --> 00:38:54,790 T. Since we have this definition of theta here, basically 581 00:38:54,790 --> 00:38:57,640 I have a T which is actually pointing 582 00:38:57,640 --> 00:39:00,940 to the upper left direction of the board. 583 00:39:05,420 --> 00:39:07,890 Oh, don't forget-- actually there's 584 00:39:07,890 --> 00:39:10,110 a third force, which is actually the F drag. 585 00:39:13,860 --> 00:39:18,470 F drag is actually equal to minus bx 586 00:39:18,470 --> 00:39:21,294 dot in the x direction. 587 00:39:28,950 --> 00:39:33,450 Now I would like to write down the expression for also 588 00:39:33,450 --> 00:39:36,390 the string tension, T. The T is actually 589 00:39:36,390 --> 00:39:43,320 equal to minus T sine theta in the x direction 590 00:39:43,320 --> 00:39:47,430 because the T is pointing to upper left direction. 591 00:39:47,430 --> 00:39:50,390 So therefore, the position to x direction 592 00:39:50,390 --> 00:39:57,560 will be minus sine theta and plus T cosine 593 00:39:57,560 --> 00:40:01,260 theta in the y direction because the tension is actually 594 00:40:01,260 --> 00:40:02,725 pointing upper left. 595 00:40:06,210 --> 00:40:10,750 As usual, this is actually pretty complicated to solve. 596 00:40:10,750 --> 00:40:11,920 I have this cosine. 597 00:40:11,920 --> 00:40:13,540 I have this sine there, right? 598 00:40:13,540 --> 00:40:22,530 So what I'm going to do is to assume that this angle 599 00:40:22,530 --> 00:40:27,340 theta is very small, as usual. 600 00:40:27,340 --> 00:40:30,160 So I will take small angle approximation. 601 00:40:30,160 --> 00:40:32,940 Then basically, you have sine theta 602 00:40:32,940 --> 00:40:38,080 is roughly to be equal to theta, and that is actually 603 00:40:38,080 --> 00:40:40,060 equal to what? 604 00:40:40,060 --> 00:40:42,440 Equal to-- here. 605 00:40:42,440 --> 00:40:44,600 Basically, you can actually calculate 606 00:40:44,600 --> 00:40:45,820 what will be the theta. 607 00:40:49,360 --> 00:40:52,090 The sine theta, or theta, would be 608 00:40:52,090 --> 00:40:56,700 equal to x minus d divided by l. 609 00:41:00,300 --> 00:41:03,180 And of course, taking a small angle approximation 610 00:41:03,180 --> 00:41:09,940 will bring cosine theta to be 1. 611 00:41:09,940 --> 00:41:14,020 Then after this approximation, my T 612 00:41:14,020 --> 00:41:20,670 will become minus T x minus d divided 613 00:41:20,670 --> 00:41:30,280 by l in the x direction plus T in the y direction 614 00:41:30,280 --> 00:41:35,170 because sine theta is replaced by this approximated value 615 00:41:35,170 --> 00:41:39,440 because sine theta is actually replaced by 1. 616 00:41:39,440 --> 00:41:41,720 Any questions? 617 00:41:41,720 --> 00:41:42,601 Yes. 618 00:41:42,601 --> 00:41:54,702 AUDIENCE: Is that a constant or is that the change in sine? 619 00:41:54,702 --> 00:41:55,910 YEN-JIE LEE: It's a constant. 620 00:41:55,910 --> 00:41:57,180 Yeah, I was going too fast. 621 00:41:57,180 --> 00:42:02,149 So this is actually a constant of my amplitude. 622 00:42:02,149 --> 00:42:05,600 AUDIENCE: And the drag force is only in the x direction? 623 00:42:05,600 --> 00:42:08,700 YEN-JIE LEE: Yes, it's only in the x direction. 624 00:42:08,700 --> 00:42:13,170 So I'm only trying to actually move this point back and forth 625 00:42:13,170 --> 00:42:14,157 horizontally. 626 00:42:18,720 --> 00:42:26,010 So now I have all the components T, Fg, f drag. 627 00:42:26,010 --> 00:42:28,860 And, of course, you can see that I already ignored the drag 628 00:42:28,860 --> 00:42:32,460 force in the y direction from that formula 629 00:42:32,460 --> 00:42:36,090 because I am only considering the system to be 630 00:42:36,090 --> 00:42:38,940 moving in the x direction. 631 00:42:38,940 --> 00:42:44,790 Therefore, I can now collect all the terms in the x direction. 632 00:42:44,790 --> 00:42:49,060 Basically, you will have m x double dot. 633 00:42:49,060 --> 00:42:54,360 This is equal to minus b x dot, which is actually 634 00:42:54,360 --> 00:43:05,860 coming from the drag force, minus T x minus d divided by l. 635 00:43:05,860 --> 00:43:11,960 This is actually coming from this term in the x direction. 636 00:43:11,960 --> 00:43:15,140 Let's look at those in the y direction. 637 00:43:15,140 --> 00:43:20,320 And y double dot would be equal to minus mg 638 00:43:20,320 --> 00:43:26,830 plus T. The minus mg is from the gravitational force. 639 00:43:26,830 --> 00:43:30,940 And this T is coming from the y component of the string 640 00:43:30,940 --> 00:43:33,640 tension. 641 00:43:33,640 --> 00:43:36,910 And of course, since we are taking a very small angle 642 00:43:36,910 --> 00:43:42,401 approximation, there will be no vertical motion. 643 00:43:42,401 --> 00:43:42,900 Yes. 644 00:43:42,900 --> 00:43:46,316 AUDIENCE: Why did we use the small approximation 645 00:43:46,316 --> 00:43:49,244 for sine theta when we're going to use 646 00:43:49,244 --> 00:43:53,136 x minus d over l, which represents psi instead of just 647 00:43:53,136 --> 00:43:53,636 theta? 648 00:43:53,636 --> 00:43:55,990 YEN-JIE LEE: Yeah, so also in this case, 649 00:43:55,990 --> 00:43:59,230 they happen to be exactly the same. 650 00:43:59,230 --> 00:44:02,260 And why I care is actually the cosine theta. 651 00:44:02,260 --> 00:44:04,930 Otherwise, I would have to deal with cosine theta. 652 00:44:04,930 --> 00:44:07,090 And also, this y double dot would not 653 00:44:07,090 --> 00:44:12,340 be equal to 0, which is what I'm going to assume here. 654 00:44:12,340 --> 00:44:13,010 Good question. 655 00:44:13,010 --> 00:44:16,060 So the question was, why do I need to take an approximation? 656 00:44:16,060 --> 00:44:19,750 Because I want to get rid of cosine theta. 657 00:44:19,750 --> 00:44:22,800 So now from this y direction, I can 658 00:44:22,800 --> 00:44:29,050 solve T will be equal to mg because I assume that there's 659 00:44:29,050 --> 00:44:31,170 no y direction motion. 660 00:44:31,170 --> 00:44:33,780 And I can conclude that-- 661 00:44:33,780 --> 00:44:36,610 originally, I don't know what is actually the string tension. 662 00:44:36,610 --> 00:44:41,330 It's denoted by T. Now, from this second equation, 663 00:44:41,330 --> 00:44:45,100 I can conclude that T will be equal to mg, which 664 00:44:45,100 --> 00:44:46,260 is the gravitational force. 665 00:44:48,800 --> 00:44:54,670 Then, once I have that, I can go back to x direction. 666 00:44:54,670 --> 00:44:58,210 Basically, I get m x double dot. 667 00:44:58,210 --> 00:45:06,130 This is equal to minus b x dot minus mg over l x minus d. 668 00:45:10,430 --> 00:45:13,880 Everything is working very well. 669 00:45:13,880 --> 00:45:23,870 And I just have to really write down the d function explicitly. 670 00:45:23,870 --> 00:45:25,070 What is d? 671 00:45:25,070 --> 00:45:31,570 d is just a reminder, delta sine omega d t. 672 00:45:31,570 --> 00:45:33,980 So I will plug that into that equation. 673 00:45:33,980 --> 00:45:38,900 And also I will bring all the terms 674 00:45:38,900 --> 00:45:42,740 related to x to the left hand side 675 00:45:42,740 --> 00:45:49,060 just to match my convention. 676 00:45:49,060 --> 00:45:53,740 All right, so now I will be able to get the result, 677 00:45:53,740 --> 00:46:04,420 m x double dot plus b x dot plus mg over l x. 678 00:46:04,420 --> 00:46:11,290 And that is actually equal to mg over l d. 679 00:46:11,290 --> 00:46:20,740 And this is equal to mg over l delta sine omega dt. 680 00:46:20,740 --> 00:46:22,940 So basically, I collect all the terms, 681 00:46:22,940 --> 00:46:24,460 put it to the left hand side. 682 00:46:24,460 --> 00:46:29,980 And I write down T explicitly, which is this. 683 00:46:29,980 --> 00:46:32,770 Then I can divide everything by m. 684 00:46:32,770 --> 00:46:41,710 Then I get m x double dot plus b over m x dot plus g over l x. 685 00:46:41,710 --> 00:46:49,760 And that would be equal to g over l delta sine omega dt. 686 00:46:49,760 --> 00:46:54,960 Now, of course, as usual, I will define this to be gamma, 687 00:46:54,960 --> 00:46:58,170 define this to be omega 0 squared. 688 00:46:58,170 --> 00:47:03,840 And I would define this to be f0, which is 689 00:47:03,840 --> 00:47:06,630 equal to omega 0 squared delta. 690 00:47:06,630 --> 00:47:08,970 It happened to be equal like that. 691 00:47:08,970 --> 00:47:12,060 And then this actually becomes x double 692 00:47:12,060 --> 00:47:16,920 dot plus gamma x dot plus omega 0 693 00:47:16,920 --> 00:47:22,440 squared x equal to f0 sine omega dt. 694 00:47:25,266 --> 00:47:27,840 Am I going too fast? 695 00:47:27,840 --> 00:47:30,530 OK, everybody is following. 696 00:47:30,530 --> 00:47:32,910 So we see that ha! 697 00:47:32,910 --> 00:47:34,350 This equation-- I know that. 698 00:47:34,350 --> 00:47:36,510 I know this equation, right? 699 00:47:36,510 --> 00:47:42,210 Because we have just solved that a few minutes ago. 700 00:47:42,210 --> 00:47:47,020 Therefore, I know immediately what will be the solution. 701 00:47:47,020 --> 00:47:49,650 The solution is here already. 702 00:47:49,650 --> 00:47:55,070 I have A omega d and the tangent delta, the function of force 703 00:47:55,070 --> 00:47:56,550 there. 704 00:47:56,550 --> 00:48:03,010 Therefore, I can now write down what will be the A. 705 00:48:03,010 --> 00:48:08,800 So A is actually just equal to f0 divided 706 00:48:08,800 --> 00:48:12,790 by square root of omega 0 squared 707 00:48:12,790 --> 00:48:17,110 minus omega d squared squared plus omega 708 00:48:17,110 --> 00:48:19,300 d squared gamma squared. 709 00:48:27,980 --> 00:48:31,610 So now the question is-- 710 00:48:31,610 --> 00:48:36,650 what does the result actually mean. 711 00:48:36,650 --> 00:48:38,930 I have this function. 712 00:48:38,930 --> 00:48:42,070 I have that function, tangent delta. 713 00:48:42,070 --> 00:48:43,210 It's solved. 714 00:48:43,210 --> 00:48:48,860 It's actually the amplitude of the steady-state solution 715 00:48:48,860 --> 00:48:53,180 and also the phase difference between the drag force 716 00:48:53,180 --> 00:48:59,570 phase and the steady-state oscillation phase. 717 00:48:59,570 --> 00:49:02,690 So that's actually the amount of lag 718 00:49:02,690 --> 00:49:05,900 and the size of the amplitude. 719 00:49:05,900 --> 00:49:10,820 But through this equation it is very difficult to understand. 720 00:49:10,820 --> 00:49:14,105 So what I'm going to do is to take some limit 721 00:49:14,105 --> 00:49:18,080 so that actually we can help you to understand what is going on. 722 00:49:18,080 --> 00:49:25,550 So suppose I assume that omega d goes to 0. 723 00:49:25,550 --> 00:49:27,260 So what does that mean? 724 00:49:27,260 --> 00:49:32,060 This is the engine's hand and is moving really slowly 725 00:49:32,060 --> 00:49:35,420 and see what is going to happen. 726 00:49:35,420 --> 00:49:42,090 If I do this, then you will find that A omega d-- 727 00:49:42,090 --> 00:49:47,720 since omega goes to 0, this is gone, this is gone. 728 00:49:47,720 --> 00:49:51,680 Therefore, you will see that omega A will be 729 00:49:51,680 --> 00:49:53,930 equal to omega 0 divided by-- 730 00:49:53,930 --> 00:49:58,860 I'm sorry-- of f0 divided by omega 0 squared. 731 00:49:58,860 --> 00:50:02,420 And that is actually equal to g delta 732 00:50:02,420 --> 00:50:05,720 over l divded by g over l. 733 00:50:05,720 --> 00:50:09,320 And that will give you delta. 734 00:50:09,320 --> 00:50:10,860 So what does that mean? 735 00:50:10,860 --> 00:50:16,430 This means that, if I drive this thing really slowly, 736 00:50:16,430 --> 00:50:21,530 then the amplitude of the mass will 737 00:50:21,530 --> 00:50:27,722 be equal to how much I actually move, which is delta. 738 00:50:27,722 --> 00:50:28,222 OK. 739 00:50:28,222 --> 00:50:30,820 Do you get it? 740 00:50:30,820 --> 00:50:37,360 In addition to that, tangent delta-- 741 00:50:37,360 --> 00:50:42,980 since I am taking the limit omega d goes to 0. 742 00:50:42,980 --> 00:50:46,460 Therefore, tangent delta will be equal to 0, 743 00:50:46,460 --> 00:50:52,020 and that means delta will be equal 0. 744 00:50:52,020 --> 00:50:55,080 Any questions? 745 00:50:55,080 --> 00:50:58,540 So that means there will be no phase difference. 746 00:50:58,540 --> 00:51:02,220 The system has enough time to keep up with my speed. 747 00:51:05,160 --> 00:51:11,410 The second limit, obviously, omega d goes to infinity. 748 00:51:11,410 --> 00:51:12,670 What does that mean? 749 00:51:12,670 --> 00:51:16,730 That means I'm going to hold this as a string 750 00:51:16,730 --> 00:51:21,220 and shake it like crazy really fast 751 00:51:21,220 --> 00:51:23,920 and see what will happen, OK? 752 00:51:23,920 --> 00:51:27,490 So in that case, you will get A omega 753 00:51:27,490 --> 00:51:31,950 d, and that one goes to 0, because omega d goes 754 00:51:31,950 --> 00:51:33,260 to infinity. 755 00:51:33,260 --> 00:51:36,090 This one goes to 0. 756 00:51:36,090 --> 00:51:41,930 And also, tangent delta will go to infinity. 757 00:51:41,930 --> 00:51:45,430 Therefore, delta will go to pi. 758 00:51:45,430 --> 00:51:48,100 So that means they will be out of phase. 759 00:51:50,682 --> 00:51:52,390 Any questions so far in these two limits? 760 00:51:55,530 --> 00:51:58,760 OK, so what I'm going to do now is 761 00:51:58,760 --> 00:52:03,580 to take a small toy, which I made for my son, who 762 00:52:03,580 --> 00:52:06,170 is one-year-old now. 763 00:52:06,170 --> 00:52:09,200 Because I would like him to learn wavelength vibration 764 00:52:09,200 --> 00:52:11,200 before he goes to quantum, right? 765 00:52:11,200 --> 00:52:12,050 Hey? 766 00:52:12,050 --> 00:52:13,700 So I made this toy. 767 00:52:13,700 --> 00:52:17,420 And he looked at it. 768 00:52:17,420 --> 00:52:20,240 So you can see now, I can demonstrate 769 00:52:20,240 --> 00:52:26,872 what is going to happen when omega is approaching to 0. 770 00:52:26,872 --> 00:52:27,372 OK? 771 00:52:30,844 --> 00:52:33,460 I am already doing it. 772 00:52:33,460 --> 00:52:35,640 Can you see it? 773 00:52:35,640 --> 00:52:37,290 No? 774 00:52:37,290 --> 00:52:40,686 It's a very exciting experiment. 775 00:52:40,686 --> 00:52:42,550 Can you see that? 776 00:52:42,550 --> 00:52:48,420 You see that this is the origin vertical direction. 777 00:52:48,420 --> 00:52:53,700 If I do it really, really, really slowly, 778 00:52:53,700 --> 00:52:59,140 you can see that the amplitude of the ball 779 00:52:59,140 --> 00:53:04,200 is actually exactly the same as the displacement I introduced. 780 00:53:04,200 --> 00:53:08,160 So that's kind of obvious. 781 00:53:08,160 --> 00:53:13,170 So now, let's see what is going to happen if I 782 00:53:13,170 --> 00:53:15,950 drive this system like crazy. 783 00:53:15,950 --> 00:53:18,350 OK, not going up and down. 784 00:53:18,350 --> 00:53:21,180 Eeeee-- that's the maximum speed I can do. 785 00:53:21,180 --> 00:53:23,290 Maybe you can do it faster. 786 00:53:23,290 --> 00:53:27,900 But you can see that nothing happened. 787 00:53:27,900 --> 00:53:31,660 So amplitude is close to 0, because what 788 00:53:31,660 --> 00:53:34,690 you have been doing is-- 789 00:53:34,690 --> 00:53:40,050 disappear, you sort cancelling each other. 790 00:53:40,050 --> 00:53:43,590 And it's actually not going to contribute 791 00:53:43,590 --> 00:53:47,040 to the motion of this ball. 792 00:53:47,040 --> 00:53:50,960 So now, you can see that I can also test, what is actually 793 00:53:50,960 --> 00:53:52,580 the natural frequency? 794 00:53:52,580 --> 00:53:55,890 And what I am going to do is to oscillate 795 00:53:55,890 --> 00:54:01,900 at around the natural frequency to see what is going to happen. 796 00:54:01,900 --> 00:54:04,990 Let's see what is going to happen. 797 00:54:04,990 --> 00:54:08,130 You can see that the delta is really small, right? 798 00:54:08,130 --> 00:54:09,535 Can you see the delta. 799 00:54:09,535 --> 00:54:12,910 It's really small-- very small-- 800 00:54:12,910 --> 00:54:16,480 very small. 801 00:54:16,480 --> 00:54:21,145 But you can see that the amplitude, the A, is huge. 802 00:54:23,760 --> 00:54:26,370 What does that tell us? 803 00:54:26,370 --> 00:54:29,841 What does that tell us? 804 00:54:29,841 --> 00:54:30,340 Yeah? 805 00:54:30,340 --> 00:54:32,348 AUDIENCE: Well, we're experiencing resonance. 806 00:54:32,348 --> 00:54:35,810 YEN-JIE LEE: Yes, we are experiencing resonance. 807 00:54:35,810 --> 00:54:40,230 And also, that also tells you that the system 808 00:54:40,230 --> 00:54:43,440 is under-damped very much. 809 00:54:43,440 --> 00:54:47,940 The Q value is very big. 810 00:54:47,940 --> 00:54:53,470 So if I calculate the amplitude, A-- 811 00:54:53,470 --> 00:54:59,144 now, I can calculate the amplitude, A, 812 00:54:59,144 --> 00:55:02,760 at natural frequency. 813 00:55:02,760 --> 00:55:04,170 What I'm going to get is-- 814 00:55:07,230 --> 00:55:13,440 now, I can actually plug in omega d equal to omega 0. 815 00:55:13,440 --> 00:55:15,200 So if I plug in omega d-- 816 00:55:20,000 --> 00:55:23,460 omega d equal to omega 0-- 817 00:55:23,460 --> 00:55:25,290 then what is going to happen? 818 00:55:25,290 --> 00:55:27,500 So this term is working. 819 00:55:27,500 --> 00:55:40,880 So you have A is equal to f-0 divided by omega 0 gamma. 820 00:55:40,880 --> 00:55:44,630 Omega-d is now equal to omega 0. 821 00:55:44,630 --> 00:55:48,770 And that is going to give you-- 822 00:55:48,770 --> 00:55:55,890 so f-0 is actually omega 0 square delta 823 00:55:55,890 --> 00:56:00,190 divided by omega 0 gamma. 824 00:56:00,190 --> 00:56:02,760 And the one omega 0 actually cancels. 825 00:56:02,760 --> 00:56:10,830 Then, basically, you will get Q times delta. 826 00:56:10,830 --> 00:56:12,060 What is Q? 827 00:56:12,060 --> 00:56:15,640 Just a reminder, it's actually the ratio 828 00:56:15,640 --> 00:56:18,330 of omega 0 and the gamma. 829 00:56:18,330 --> 00:56:21,850 When the Q is very large, what does that mean? 830 00:56:21,850 --> 00:56:25,800 That means it's so close to an idealized situation 831 00:56:25,800 --> 00:56:27,780 that direct force is very small. 832 00:56:27,780 --> 00:56:32,390 You can see that in the example which I have been doing. 833 00:56:32,390 --> 00:56:37,050 So you can see that, ah, it is really the case. 834 00:56:37,050 --> 00:56:42,570 So you can see that if my delta is something like 1 centimeter, 835 00:56:42,570 --> 00:56:48,767 but the amplitude is actually at the order of 1 meter, maybe. 836 00:56:48,767 --> 00:56:49,600 What does that mean? 837 00:56:49,600 --> 00:56:55,200 That means that Q is actually, roughly, 100. 838 00:56:55,200 --> 00:57:00,410 So you can actually even get a Q out of this experiment. 839 00:57:00,410 --> 00:57:01,451 Any questions so far? 840 00:57:04,890 --> 00:57:07,350 OK, that's very good. 841 00:57:07,350 --> 00:57:09,990 So now, we can go ahead and take a look 842 00:57:09,990 --> 00:57:15,720 at the structure of the A and the delta. 843 00:57:15,720 --> 00:57:18,030 As we demonstrated before, we make 844 00:57:18,030 --> 00:57:23,140 sense of those three different kinds of situations-- 845 00:57:23,140 --> 00:57:27,670 omega d goes to 0, omega d goes to infinity. 846 00:57:27,670 --> 00:57:30,060 And of course, I would like to know the force 847 00:57:30,060 --> 00:57:33,480 structure of A and delta. 848 00:57:33,480 --> 00:57:36,685 Therefore, what I'm going to do is to plug A omega 849 00:57:36,685 --> 00:57:41,850 d as a function of omega d. 850 00:57:44,670 --> 00:57:50,010 So what I'm going to get is this will be equal to delta 851 00:57:50,010 --> 00:57:52,810 when omega d goes to 0. 852 00:57:52,810 --> 00:57:54,990 We just demonstrated that. 853 00:57:54,990 --> 00:57:58,230 And this will increase to a large value 854 00:57:58,230 --> 00:58:06,060 and drop down to 0, when omega d goes to infinity. 855 00:58:06,060 --> 00:58:10,410 And you can see that this is around omega 0. 856 00:58:10,410 --> 00:58:16,440 And you are going to get a huge amplitude at around omega 0. 857 00:58:16,440 --> 00:58:17,220 But not quite. 858 00:58:17,220 --> 00:58:22,020 The maxima is actually slightly smaller than omega 0. 859 00:58:22,020 --> 00:58:26,670 You can actually calculate that as part of the homework. 860 00:58:26,670 --> 00:58:28,320 So that makes sense. 861 00:58:30,870 --> 00:58:36,540 Now, I can also plug the delta, which is the phase difference-- 862 00:58:36,540 --> 00:58:39,270 and you can see that this phase difference 863 00:58:39,270 --> 00:58:44,560 will be, originally, 0, when the omega d is very small. 864 00:58:44,560 --> 00:58:47,530 And this is actually the omega 0. 865 00:58:47,530 --> 00:58:49,920 I hope you can see it. 866 00:58:49,920 --> 00:58:53,820 And this will be increasing rapidly here 867 00:58:53,820 --> 00:58:57,990 and approaching to pi. 868 00:58:57,990 --> 00:59:01,320 So that means, when you are shaking this system 869 00:59:01,320 --> 00:59:03,040 like crazy-- 870 00:59:03,040 --> 00:59:06,880 very high frequency-- then the system cannot keep up with 871 00:59:06,880 --> 00:59:08,680 the speed. 872 00:59:08,680 --> 00:59:11,500 The amplitude will be very small. 873 00:59:11,500 --> 00:59:16,530 And also, the amplitude will be out of phase completely. 874 00:59:16,530 --> 00:59:20,920 So let's actually do us another demonstration, 875 00:59:20,920 --> 00:59:26,260 using this little device here. 876 00:59:26,260 --> 00:59:28,640 This is actually what you see before, 877 00:59:28,640 --> 00:59:31,800 the ball with a Mexican hat. 878 00:59:31,800 --> 00:59:36,260 And you can see that there is a spring attached to this system. 879 00:59:36,260 --> 00:59:39,360 And on the top, what I am going to do 880 00:59:39,360 --> 00:59:46,430 is to use this motor to drive this system up and down, 881 00:59:46,430 --> 00:59:49,140 as a direct force. 882 00:59:49,140 --> 00:59:55,410 So now, what am I going to do is to come from a very 883 00:59:55,410 --> 00:59:58,720 low-frequency oscillation. 884 00:59:58,720 --> 01:00:03,770 So you can see that the natural frequency is sort of like this. 885 01:00:03,770 --> 01:00:07,880 And you can see that, now, I am driving this system really 886 01:00:07,880 --> 01:00:10,306 slowly. 887 01:00:10,306 --> 01:00:13,720 You can see, this is actually going up and down 888 01:00:13,720 --> 01:00:15,180 really slowly. 889 01:00:18,110 --> 01:00:19,630 And you see that-- huh-- 890 01:00:19,630 --> 01:00:24,330 the amplitude is actually pretty small. 891 01:00:26,840 --> 01:00:28,480 There's no excitement for the moment. 892 01:00:33,450 --> 01:00:35,570 All right, so what I'm going to do is, 893 01:00:35,570 --> 01:00:42,090 now, I increase the speed of the motor and see what will happen. 894 01:00:42,090 --> 01:00:45,620 So you can see, now, it's actually 895 01:00:45,620 --> 01:00:48,470 driving it with higher and higher frequency. 896 01:00:51,320 --> 01:00:53,830 You see that-- huh-- something is happening. 897 01:00:53,830 --> 01:00:58,280 You can see the amplitude is getting larger and larger. 898 01:00:58,280 --> 01:01:02,010 I'm still increasing the frequency-- 899 01:01:02,010 --> 01:01:07,230 increasing, increasing-- until something-- something happened! 900 01:01:07,230 --> 01:01:08,210 Right? 901 01:01:08,210 --> 01:01:09,780 Did you see that? 902 01:01:09,780 --> 01:01:13,950 It starts to oscillate up and down. 903 01:01:13,950 --> 01:01:16,170 Because right now, you can see that-- 904 01:01:16,170 --> 01:01:18,540 look at the top. 905 01:01:18,540 --> 01:01:21,120 The frequency of the motor is now 906 01:01:21,120 --> 01:01:28,980 really close to the natural frequency of this system. 907 01:01:28,980 --> 01:01:32,550 So a resonance behavior will happen. 908 01:01:32,550 --> 01:01:36,890 And what you are going to get is that, OK, 909 01:01:36,890 --> 01:01:38,640 omega d around omega 0. 910 01:01:38,640 --> 01:01:42,680 Then you are going to get large time amplitude. 911 01:01:42,680 --> 01:01:47,360 So now, what I am going to do is to continue 912 01:01:47,360 --> 01:01:52,950 to increase the driving frequency 913 01:01:52,950 --> 01:01:55,110 to a very large value. 914 01:01:55,110 --> 01:01:57,850 OK, now it's actually doing the "mmmmm"-- 915 01:01:57,850 --> 01:01:59,290 doing it really fast. 916 01:01:59,290 --> 01:02:00,750 You can see on the top-- 917 01:02:00,750 --> 01:02:01,440 very fast. 918 01:02:01,440 --> 01:02:03,910 OK, I even get it even faster. 919 01:02:06,545 --> 01:02:10,940 You see that-- huh-- indeed, this system 920 01:02:10,940 --> 01:02:15,440 is now oscillating at a larger frequency. 921 01:02:15,440 --> 01:02:17,800 It's trying to keep up with the driving force. 922 01:02:17,800 --> 01:02:21,510 But you can see that the amplitude is actually 923 01:02:21,510 --> 01:02:25,650 much smaller than what had happened before. 924 01:02:25,650 --> 01:02:29,610 So before the class, you may actually think that, 925 01:02:29,610 --> 01:02:31,500 OK, drive it really fast. 926 01:02:31,500 --> 01:02:33,870 Maybe we'll increase the amplitude. 927 01:02:33,870 --> 01:02:36,720 But in reality, actually, it will give you 928 01:02:36,720 --> 01:02:38,877 a very small amplitude. 929 01:02:38,877 --> 01:02:40,710 Another thing, which is interesting to know, 930 01:02:40,710 --> 01:02:46,300 is that you can see that, when the driving force is actually 931 01:02:46,300 --> 01:02:49,350 at the maximum. 932 01:02:49,350 --> 01:02:52,835 And actually the position of this mass 933 01:02:52,835 --> 01:02:54,660 is actually at the minimum. 934 01:02:54,660 --> 01:02:57,720 So they are actually out of phase. 935 01:02:57,720 --> 01:02:59,340 I hope you can see it. 936 01:02:59,340 --> 01:03:00,230 It's like this. 937 01:03:09,350 --> 01:03:17,480 OK, so what you can see is that, when I understand the system 938 01:03:17,480 --> 01:03:22,280 and I try to drive it with the natural frequency, what 939 01:03:22,280 --> 01:03:25,210 is going to happen is that I'm exciting 940 01:03:25,210 --> 01:03:28,880 this system to a state of resonance. 941 01:03:28,880 --> 01:03:32,420 So basically, you'll get some resonance behavior. 942 01:03:32,420 --> 01:03:35,510 So I have shown you that this works 943 01:03:35,510 --> 01:03:40,220 for driven mechanical oscillator. 944 01:03:40,220 --> 01:03:44,240 It also works for the spring-mass system. 945 01:03:44,240 --> 01:03:48,470 And there are many other things which also work, 946 01:03:48,470 --> 01:03:50,930 which is around you. 947 01:03:50,930 --> 01:03:55,550 For example, if you happen to be my office hour, 948 01:03:55,550 --> 01:04:00,880 you would notice that the air-condition in my office 949 01:04:00,880 --> 01:04:03,590 is actually creating a resonance behavior. 950 01:04:03,590 --> 01:04:05,853 You'll see low frequency sound-- 951 01:04:05,853 --> 01:04:10,150 "mm mm mm"-- something like that. 952 01:04:10,150 --> 01:04:13,590 And that is because the pipe actually 953 01:04:13,590 --> 01:04:17,061 happens to have the frequency match with the resonance 954 01:04:17,061 --> 01:04:17,560 frequen-- 955 01:04:17,560 --> 01:04:23,230 OK, the airflow actually happened to excite the pipe, 956 01:04:23,230 --> 01:04:27,930 so that it's actually oscillating up and down 957 01:04:27,930 --> 01:04:28,680 at that frequency. 958 01:04:28,680 --> 01:04:32,950 So what I did was I tried to turn it down to low 959 01:04:32,950 --> 01:04:34,300 and see what happened. 960 01:04:34,300 --> 01:04:39,760 But unfortunately, it actually excited another resonance. 961 01:04:39,760 --> 01:04:42,640 I see, now, not a low-frequency sound, but a very 962 01:04:42,640 --> 01:04:43,890 high-frequency sound. 963 01:04:43,890 --> 01:04:46,480 I will post a video, actually online. 964 01:04:46,480 --> 01:04:49,660 So my life is hard, right? 965 01:04:49,660 --> 01:04:51,670 But I'm a physicist. 966 01:04:51,670 --> 01:04:55,110 Is So I choose to use the median. 967 01:04:55,110 --> 01:04:58,590 Then I actually stay between the two resonances. 968 01:04:58,590 --> 01:05:04,900 Then I don't hear the additional sound, which bothers me. 969 01:05:04,900 --> 01:05:11,050 Another example is that, when I was in Taiwan as a undergrad, 970 01:05:11,050 --> 01:05:13,700 I was living outside in a apartment. 971 01:05:13,700 --> 01:05:20,650 And with my flat-mate, we owned a very old washing machine. 972 01:05:20,650 --> 01:05:23,920 So in the middle of the night, the washing machine 973 01:05:23,920 --> 01:05:29,680 would started to walk around, like my flat-mate. 974 01:05:29,680 --> 01:05:31,845 And we are not scared. 975 01:05:31,845 --> 01:05:35,740 That is because the oscillation frequency-- actually, 976 01:05:35,740 --> 01:05:36,840 the rotation-- 977 01:05:36,840 --> 01:05:39,490 happened to match with the frequency 978 01:05:39,490 --> 01:05:41,050 of the washing machine. 979 01:05:41,050 --> 01:05:43,690 Therefore, when we started to wash our clothes, 980 01:05:43,690 --> 01:05:47,830 it start to walk around in the room. 981 01:05:47,830 --> 01:05:51,400 So as a physicist, what we have decided 982 01:05:51,400 --> 01:05:57,940 is to make the spin slightly slower, or even faster. 983 01:05:57,940 --> 01:05:59,710 Then, actually, you can see that, 984 01:05:59,710 --> 01:06:03,370 when you do that, then you get rid of the resonance behavior. 985 01:06:03,370 --> 01:06:05,590 So it's not walking around any more. 986 01:06:05,590 --> 01:06:06,580 We can control it. 987 01:06:09,100 --> 01:06:10,630 Another thing which is interesting 988 01:06:10,630 --> 01:06:13,930 is that the resonance behavior is not only 989 01:06:13,930 --> 01:06:16,600 in the physical objects, which we actually 990 01:06:16,600 --> 01:06:18,530 deal with these days also. 991 01:06:18,530 --> 01:06:21,340 But either you learn quantum mechanics 992 01:06:21,340 --> 01:06:23,440 and upon the field theory, you will 993 01:06:23,440 --> 01:06:30,160 find that there are resonance also in a mass wave function. 994 01:06:30,160 --> 01:06:34,360 So basically, you can see that these are examples 995 01:06:34,360 --> 01:06:38,080 of the Z boson resonance peak. 996 01:06:38,080 --> 01:06:44,800 So if you scatter a electron and positron then, 997 01:06:44,800 --> 01:06:49,840 basically, you'll see that the cross-section have a resonance 998 01:06:49,840 --> 01:06:53,980 peak at around 90 GeV. 999 01:06:53,980 --> 01:06:58,570 And that is actually another very interesting example 1000 01:06:58,570 --> 01:07:01,475 of a resonance in particle physics. 1001 01:07:04,370 --> 01:07:08,780 Finally, the last example, which I am going to go through 1002 01:07:08,780 --> 01:07:14,330 is an example involving a glass. 1003 01:07:14,330 --> 01:07:18,290 We have prepared a very high-quality glass here. 1004 01:07:21,910 --> 01:07:26,430 Maybe you have seen this glass before. 1005 01:07:26,430 --> 01:07:28,250 They are pretty nice. 1006 01:07:28,250 --> 01:07:38,590 And I usually use it to enjoy my red wines, which you cannot, 1007 01:07:38,590 --> 01:07:39,780 enjoy, probably now. 1008 01:07:42,530 --> 01:07:45,980 So you can see that this is the glass. 1009 01:07:45,980 --> 01:07:49,300 And if I put a little bit of water on my hand and I rub it-- 1010 01:07:53,584 --> 01:07:54,536 [VIBRATING TONE] 1011 01:07:54,536 --> 01:07:59,860 --carefully, I can actually excite one 1012 01:07:59,860 --> 01:08:03,250 of the resonance frequencies. 1013 01:08:03,250 --> 01:08:06,040 So you can see that we have all everything working 1014 01:08:06,040 --> 01:08:08,890 on a single particle. 1015 01:08:08,890 --> 01:08:13,330 And that will give you one resonance frequency. 1016 01:08:13,330 --> 01:08:15,550 If I work on two particles, which 1017 01:08:15,550 --> 01:08:18,100 you will see that in the next lecture, 1018 01:08:18,100 --> 01:08:20,710 I would get two resonance frequencies. 1019 01:08:20,710 --> 01:08:27,220 And this glass is made of infinite number of particles. 1020 01:08:27,220 --> 01:08:30,490 Therefore, I will have infinite number 1021 01:08:30,490 --> 01:08:33,250 of resonance frequencies. 1022 01:08:33,250 --> 01:08:36,160 When I'm rubbing it, I'm actually 1023 01:08:36,160 --> 01:08:42,220 giving input of all kinds of different frequencies. 1024 01:08:42,220 --> 01:08:46,330 But the glass will be smart enough so 1025 01:08:46,330 --> 01:08:50,260 that it will pick up the one it likes the most, which 1026 01:08:50,260 --> 01:08:53,420 is the resonance frequency. 1027 01:08:53,420 --> 01:08:58,741 So you can see that the sound is actually, roughly, 683 Hertz. 1028 01:09:01,630 --> 01:09:06,279 And you can actually measure it with your phone. 1029 01:09:06,279 --> 01:09:10,810 So on the TV commercial, you may have seen 1030 01:09:10,810 --> 01:09:13,760 that there's a lady singing. 1031 01:09:13,760 --> 01:09:17,710 And she's singing so loudly such that the glass-- 1032 01:09:17,710 --> 01:09:20,689 "bragh!"-- breaks. 1033 01:09:20,689 --> 01:09:24,590 Can we get a volunteer today to sing in front of us? 1034 01:09:24,590 --> 01:09:28,210 Oh-- singing. 1035 01:09:28,210 --> 01:09:30,240 Can you sing it-- 1036 01:09:30,240 --> 01:09:31,479 high frequencies? 1037 01:09:31,479 --> 01:09:33,834 "Ahhhh." 1038 01:09:33,834 --> 01:09:37,140 [LAUGHTER & APPLAUSE] 1039 01:09:37,140 --> 01:09:38,069 Very good try. 1040 01:09:38,069 --> 01:09:41,540 But it didn't work. 1041 01:09:41,540 --> 01:09:45,350 OK, I guess it's really difficult 1042 01:09:45,350 --> 01:09:49,819 to perform that in front of so many people unprepared. 1043 01:09:49,819 --> 01:09:52,430 But fortunately, we are MIT. 1044 01:09:52,430 --> 01:09:56,270 So we have designed a device, which actually 1045 01:09:56,270 --> 01:09:59,990 can help us to achieve this. 1046 01:09:59,990 --> 01:10:06,140 So this device actually contain a amplifier here. 1047 01:10:06,140 --> 01:10:12,770 And I can now control the frequency of the sound 1048 01:10:12,770 --> 01:10:14,330 through this scope. 1049 01:10:14,330 --> 01:10:18,410 And this amplifier will actually amplify the signal 1050 01:10:18,410 --> 01:10:26,750 and produce a sound wave and try to actually isolate this glass. 1051 01:10:26,750 --> 01:10:28,820 So we are going to do this experiment. 1052 01:10:28,820 --> 01:10:32,780 So we will need to change the loud setting a bit. 1053 01:10:32,780 --> 01:10:36,710 Because the sound is going to be, probably, too loud. 1054 01:10:36,710 --> 01:10:42,527 Just for safety-- some of you may not survive. 1055 01:10:42,527 --> 01:10:43,840 [LAUGHTER] 1056 01:10:43,840 --> 01:10:45,830 So I'm handing out these. 1057 01:10:52,410 --> 01:10:55,110 OK, who is closer? 1058 01:10:55,110 --> 01:10:56,768 OK, maybe you. 1059 01:10:56,768 --> 01:10:58,672 AUDIENCE: [INAUDIBLE] 1060 01:10:58,672 --> 01:10:59,616 YEN-JIE LEE: Oh. 1061 01:10:59,616 --> 01:11:00,977 Oh, sorry. 1062 01:11:00,977 --> 01:11:02,260 [LAUGHTER] 1063 01:11:02,260 --> 01:11:03,960 I'm so sorry. 1064 01:11:03,960 --> 01:11:05,170 What? 1065 01:11:05,170 --> 01:11:07,080 I don't need that. 1066 01:11:07,080 --> 01:11:07,580 OK. 1067 01:11:10,760 --> 01:11:15,530 So just for safety, I will put this on. 1068 01:11:15,530 --> 01:11:20,540 And what I am going to do is also put these glass on. 1069 01:11:20,540 --> 01:11:22,290 OK, maybe I'll do this first. 1070 01:11:30,586 --> 01:11:39,890 AUDIENCE: [INAUDIBLE] 1071 01:11:39,890 --> 01:11:41,960 YEN-JIE LEE: OK, so what I am going to do now 1072 01:11:41,960 --> 01:11:45,512 is to start producing sound wave. 1073 01:11:45,512 --> 01:11:47,356 [LOUD TONE] 1074 01:11:48,278 --> 01:11:50,550 So through the camera, you should 1075 01:11:50,550 --> 01:11:55,916 be able to see what is actually shown on the screen. 1076 01:11:55,916 --> 01:12:04,715 So you can see that, if this glass is actually moving, 1077 01:12:04,715 --> 01:12:07,770 the wood inside would also vibrate. 1078 01:12:07,770 --> 01:12:11,730 So you can see that, clearly, we don't have resonance yet. 1079 01:12:11,730 --> 01:12:17,440 So what I am going to do is to increase the frequency 1080 01:12:17,440 --> 01:12:19,590 and see what happens. 1081 01:12:19,590 --> 01:12:24,240 So now, it's actually 643. 1082 01:12:24,240 --> 01:12:28,585 It's actually still below the resonance frequency. 1083 01:12:28,585 --> 01:12:31,650 Now, I have measured the frequency. 1084 01:12:31,650 --> 01:12:34,860 It should be 684. 1085 01:12:34,860 --> 01:12:36,296 So now, it's actually 653-- 1086 01:12:39,595 --> 01:12:45,840 663 Hertz-- 673 Hertz. 1087 01:12:45,840 --> 01:12:46,900 Can you see the movement? 1088 01:12:46,900 --> 01:12:51,200 You cannot see the movement yet. 1089 01:12:51,200 --> 01:12:54,410 683-- you see? 1090 01:12:54,410 --> 01:12:55,350 You see, now-- 1091 01:12:55,350 --> 01:12:55,960 AUDIENCE: Yes. 1092 01:12:55,960 --> 01:12:58,512 YEN-JIE LEE: --the frequency of the sound 1093 01:12:58,512 --> 01:13:03,200 is actually matching with one of the natural frequencies 1094 01:13:03,200 --> 01:13:05,020 of the glass. 1095 01:13:05,020 --> 01:13:07,340 Apparently, the glass likes it. 1096 01:13:10,070 --> 01:13:14,990 And now, you can see that it is still vibrating. 1097 01:13:14,990 --> 01:13:17,990 And the next step, which we are going to do, 1098 01:13:17,990 --> 01:13:21,920 is to try to increase the amplitude, 1099 01:13:21,920 --> 01:13:25,250 increase the volume of the sound, and see what happens. 1100 01:13:25,250 --> 01:13:29,041 Maybe you want to cover your ears, just for safety. 1101 01:13:29,041 --> 01:13:30,845 [LOUD TONE] 1102 01:13:30,845 --> 01:13:37,852 OK, then the glass may break, if we are lucky. 1103 01:13:37,852 --> 01:13:41,788 Let us see what is going to happen. 1104 01:13:41,788 --> 01:13:44,248 [INCREASINGLY LOUDER TONE] 1105 01:13:53,104 --> 01:13:55,564 Oh! 1106 01:13:55,564 --> 01:13:57,532 [APPLAUSE] 1107 01:13:59,034 --> 01:13:59,992 TECH SUPPORT: Good job. 1108 01:13:59,992 --> 01:14:00,730 YEN-JIE LEE: Very good. 1109 01:14:00,730 --> 01:14:01,046 TECH SUPPORT: Perfect. 1110 01:14:01,046 --> 01:14:02,462 That's the quickest one we've had. 1111 01:14:02,462 --> 01:14:04,260 YEN-JIE LEE: Yeah, thank you very much. 1112 01:14:04,260 --> 01:14:05,830 Thank you, glass. 1113 01:14:05,830 --> 01:14:07,120 [LAUGHTER] 1114 01:14:07,980 --> 01:14:11,440 So you can see the power of resonance. 1115 01:14:11,440 --> 01:14:19,110 So if I tune down the frequency slightly more, 1116 01:14:19,110 --> 01:14:22,800 then you will be where? 1117 01:14:22,800 --> 01:14:24,570 You'll be here. 1118 01:14:24,570 --> 01:14:31,050 Then you will not have enough amplitude to break the glass. 1119 01:14:31,050 --> 01:14:35,100 And also, as we discussed before, 1120 01:14:35,100 --> 01:14:39,350 the quality of the glass should be really, really high, such 1121 01:14:39,350 --> 01:14:42,880 that the resulting amplitude will be very large. 1122 01:14:42,880 --> 01:14:45,940 Then you can actually break it with a external sound wave. 1123 01:14:45,940 --> 01:14:49,580 And if we go above the resonance frequency, 1124 01:14:49,580 --> 01:14:52,020 then you would not also move a bit. 1125 01:14:52,020 --> 01:14:56,940 Because if you go to very large omega d, 1126 01:14:56,940 --> 01:14:59,690 then amplitude will be pretty small. 1127 01:15:02,660 --> 01:15:09,116 OK, let me try to switch back to my presentation. 1128 01:15:09,116 --> 01:15:10,062 I think we did. 1129 01:15:10,062 --> 01:15:10,562 Sure. 1130 01:15:17,250 --> 01:15:20,200 So this is actually what we have learned today. 1131 01:15:20,200 --> 01:15:27,540 We have learned the behavior of a damped driven oscillator. 1132 01:15:27,540 --> 01:15:30,600 We have learned the transient behavior. 1133 01:15:30,600 --> 01:15:33,420 So what is actually transient behavior 1134 01:15:33,420 --> 01:15:42,420 is a mixture of steady state solution, which 1135 01:15:42,420 --> 01:15:44,850 was coming from the driving force, 1136 01:15:44,850 --> 01:15:48,900 and the homogeneous solution. 1137 01:15:48,900 --> 01:15:53,530 If you wait long enough, this will decay and disappear. 1138 01:15:53,530 --> 01:15:57,770 Is 1139 01:15:57,770 --> 01:15:59,960 And we have learned resonance. 1140 01:15:59,960 --> 01:16:04,040 So an IOC circuit, which you actually solved that 1141 01:16:04,040 --> 01:16:08,870 in your P-set, in pendulum, which I just show you, 1142 01:16:08,870 --> 01:16:13,370 which helped my son to learn wavelength vibrations-- 1143 01:16:13,370 --> 01:16:20,030 and air condition, washing machine, glass-- 1144 01:16:20,030 --> 01:16:21,140 particle physics. 1145 01:16:21,140 --> 01:16:24,830 We can see damped os-- 1146 01:16:24,830 --> 01:16:29,300 driven oscillator or resonance almost everywhere. 1147 01:16:29,300 --> 01:16:32,650 So I hope that you enjoyed the lecture today. 1148 01:16:32,650 --> 01:16:35,690 And what we are going to do next time 1149 01:16:35,690 --> 01:16:39,710 is to put multiple objects together so that you 1150 01:16:39,710 --> 01:16:44,320 see the interaction between one particle to the other particle 1151 01:16:44,320 --> 01:16:48,020 and see how we can actually make sense of this kind of system. 1152 01:16:48,020 --> 01:16:49,250 Thank you very much. 1153 01:16:49,250 --> 01:16:52,640 And I will be here if you have additional questions related 1154 01:16:52,640 --> 01:16:54,350 to the lecture.