1 00:00:24,000 --> 00:00:28,133 We have discussed free oscillations of harmonic 2 00:00:28,133 --> 00:00:32,986 oscillators without damping, and then we introduced the 3 00:00:32,986 --> 00:00:36,131 damping. But in each of those cases, 4 00:00:36,131 --> 00:00:40,983 we let the simple harmonic oscillator do its own thing. 5 00:00:40,983 --> 00:00:46,285 We did not interfere with it. Today, that's going to change. 6 00:00:46,285 --> 00:00:51,767 Today we're going to impose our will on to the simple harmonic 7 00:00:51,767 --> 00:00:55,451 oscillator. And, we can impose our will on 8 00:00:55,451 --> 00:01:00,663 to it by driving it with a force, and then see what the net 9 00:01:00,663 --> 00:01:05,493 result is. And let's start with a simple 10 00:01:05,493 --> 00:01:09,277 example that I have here. [NOISE OBSCURES] K, 11 00:01:09,277 --> 00:01:13,318 and an object of mass M. Let this be equilibrium 12 00:01:13,318 --> 00:01:17,101 position X equals zero. That will be damping. 13 00:01:17,101 --> 00:01:21,057 I will introduce, again, B over M equals gamma, 14 00:01:21,057 --> 00:01:24,324 and omega zero squared equals K over M. 15 00:01:24,324 --> 00:01:30,000 We've seen this before. There's the shorthand notation. 16 00:01:30,000 --> 00:01:34,265 So now, in addition to the fact that when the object is away 17 00:01:34,265 --> 00:01:37,445 from equilibrium, that there is here a spring 18 00:01:37,445 --> 00:01:41,277 force, I am now going to apply on that object a force. 19 00:01:41,277 --> 00:01:45,325 It may not be easy to do. We will get back to that how we 20 00:01:45,325 --> 00:01:48,144 do that. But I can apply a force on that 21 00:01:48,144 --> 00:01:52,481 maybe through magnetic fields, maybe through electric fields, 22 00:01:52,481 --> 00:01:55,951 and I'm going to, this force is not going to have 23 00:01:55,951 --> 00:02:00,000 the character F zero times cosine omega T. 24 00:02:00,000 --> 00:02:05,455 I impose on a system, now, that frequency omega. 25 00:02:05,455 --> 00:02:09,053 I can choose anything I want to. 26 00:02:09,053 --> 00:02:15,901 So now I can write down the differential equation of motion, 27 00:02:15,901 --> 00:02:21,705 Newton's second Law, MX double dot equals minus KX, 28 00:02:21,705 --> 00:02:26,464 nothing new. That's the spring force minus 29 00:02:26,464 --> 00:02:30,991 BX dot, nothing new. That's the damping. 30 00:02:30,991 --> 00:02:39,000 But now comes this external force at zero cosine omega T. 31 00:02:39,000 --> 00:02:42,577 What I'm going to do now: I'm going to move this to the 32 00:02:42,577 --> 00:02:44,962 complex plane, not that is absolutely 33 00:02:44,962 --> 00:02:47,214 necessary, but I'm so used to that. 34 00:02:47,214 --> 00:02:50,195 So, I'm going to write this now in terms of Z, 35 00:02:50,195 --> 00:02:53,110 and then we'll take the real part of Z later. 36 00:02:53,110 --> 00:02:56,488 That goes back to X. So, I'm going to write this now 37 00:02:56,488 --> 00:03:00,000 in terms of Z double dot. I divide M out. 38 00:03:00,000 --> 00:03:06,777 We get plus gamma times Z dot plus omega zero squared times Z. 39 00:03:06,777 --> 00:03:11,333 And, that now becomes F zero divided by M. 40 00:03:11,333 --> 00:03:17,666 Remember, I divided M out. And then we get cosine omega T, 41 00:03:17,666 --> 00:03:24,444 for which I will write E to the power J omega T because I work 42 00:03:24,444 --> 00:03:29,333 now in the complex plane. And, through Euler, 43 00:03:29,333 --> 00:03:35,000 I can always convert that back to cosine. 44 00:03:35,000 --> 00:03:41,666 My trial function for Z, which is a complex notation, 45 00:03:41,666 --> 00:03:49,358 is some amplitude times E to the power J omega T minus delta. 46 00:03:49,358 --> 00:03:56,153 Now, crucial is that you understand why this omega and 47 00:03:56,153 --> 00:04:02,084 this omega are the same. This is the omega of my driving 48 00:04:02,084 --> 00:04:05,714 decision that is why well that I impose on that system. 49 00:04:05,714 --> 00:04:08,605 Clearly, given enough time in the beginning, 50 00:04:08,605 --> 00:04:12,100 the system may be unhappy, and it may do all kinds of 51 00:04:12,100 --> 00:04:14,722 nasty things, which we will discuss next 52 00:04:14,722 --> 00:04:16,336 lecture. But, ultimately, 53 00:04:16,336 --> 00:04:19,831 I will come out to be the winner, and ultimately that 54 00:04:19,831 --> 00:04:23,462 system is bound to start oscillating with the frequency 55 00:04:23,462 --> 00:04:26,756 that I impose on it. If I start shaking you in the 56 00:04:26,756 --> 00:04:30,386 beginning, you may not like that, and you may oppose to 57 00:04:30,386 --> 00:04:33,902 that. But ultimately I will be the 58 00:04:33,902 --> 00:04:37,504 winner, and I will make you shake with that frequency, 59 00:04:37,504 --> 00:04:38,728 omega. So, clearly, 60 00:04:38,728 --> 00:04:42,805 the ultimate solution must have the same omega as the driver. 61 00:04:42,805 --> 00:04:45,048 What is the meaning of this delta? 62 00:04:45,048 --> 00:04:49,126 Well, it is not at all obvious that the object will have been 63 00:04:49,126 --> 00:04:53,203 in the same phase as the driver. It is possible that when the 64 00:04:53,203 --> 00:04:57,009 force is pointing in the direction that the object may be 65 00:04:57,009 --> 00:05:01,793 going in the other direction. And you will see that, 66 00:05:01,793 --> 00:05:05,708 indeed, can happen. And so, this delta is a phase 67 00:05:05,708 --> 00:05:10,682 angle which takes into account the possibility that the driver 68 00:05:10,682 --> 00:05:14,351 and object in motion are not exactly in phase. 69 00:05:14,351 --> 00:05:18,021 We call this solution a steady state solution, 70 00:05:18,021 --> 00:05:22,832 a steady state that you must wait long enough for the system 71 00:05:22,832 --> 00:05:27,399 not to fight you any longer. That will be part of my next 72 00:05:27,399 --> 00:05:32,128 lecture, the fighting issue. This is when I ultimately come 73 00:05:32,128 --> 00:05:38,000 out to be the winner and when the system follows my will. 74 00:05:38,000 --> 00:05:42,686 So now, I'm going to take the second derivative. 75 00:05:42,686 --> 00:05:47,971 So I get minus omega squared. J omega comes out twice. 76 00:05:47,971 --> 00:05:53,754 So I get minus omega squared. Then I get plus gamma times J 77 00:05:53,754 --> 00:05:56,945 omega. Then I get plus omega zero 78 00:05:56,945 --> 00:05:59,937 squared. And, that whole thing, 79 00:05:59,937 --> 00:06:05,122 multiplied by A to the power E J omega T minus delta, 80 00:06:05,122 --> 00:06:10,806 that now equals F zero divided by M times E to the power J 81 00:06:10,806 --> 00:06:15,251 omega T. The whole thing is now in the 82 00:06:15,251 --> 00:06:18,356 complex plane. And, you see that E to the 83 00:06:18,356 --> 00:06:21,228 power J omega T cancels on both sides. 84 00:06:21,228 --> 00:06:25,497 So, I lose my E to the power J omega T, and I'm going to 85 00:06:25,497 --> 00:06:30,000 multiply both sides by E to the power J plus delta. 86 00:06:30,000 --> 00:06:35,300 So, I lose my delta here, but it appears then here. 87 00:06:35,300 --> 00:06:41,024 And so, if I make that simple change, algebraic change, 88 00:06:41,024 --> 00:06:47,597 we are going to get minus omega squared plus gamma J omega plus 89 00:06:47,597 --> 00:06:52,685 omega zero squared. That multiplied by A must now 90 00:06:52,685 --> 00:07:00,000 be equal to F zero divided by M times E to the power J delta. 91 00:07:00,000 --> 00:07:04,578 Omega T is gone, and I have moved the delta to 92 00:07:04,578 --> 00:07:08,139 the right side. Can we live as that? 93 00:07:08,139 --> 00:07:14,447 And this can be written F zero divided by M times the cosine of 94 00:07:14,447 --> 00:07:18,110 delta plus J times the sine of delta. 95 00:07:18,110 --> 00:07:22,994 I'm still in the complex plane, but that's Euler. 96 00:07:22,994 --> 00:07:28,895 Now, let's compare the apples with apples, and oranges with 97 00:07:28,895 --> 00:07:31,438 oranges. This is an apple, 98 00:07:31,438 --> 00:07:36,996 and this is an apple. That means it's real. 99 00:07:36,996 --> 00:07:39,942 And that means this is an apple. 100 00:07:39,942 --> 00:07:44,885 But there are also oranges. This is an orange as a J, 101 00:07:44,885 --> 00:07:48,117 and this is an orange that has a J. 102 00:07:48,117 --> 00:07:53,346 And so, for this equation, to always hold at all moments 103 00:07:53,346 --> 00:07:58,479 in time, the apples must be equal to the apples on this 104 00:07:58,479 --> 00:08:02,814 side. And oranges on this side must 105 00:08:02,814 --> 00:08:06,308 be equal to the oranges on that side. 106 00:08:06,308 --> 00:08:12,326 So, it looks like one equation, but it really is two equations. 107 00:08:12,326 --> 00:08:18,344 So, we now get that minus omega squared plus omega zero squared 108 00:08:18,344 --> 00:08:24,458 times A must be equal to F zero divided by M times the cosine of 109 00:08:24,458 --> 00:08:29,797 delta apples on both sides are equal, and now we get the 110 00:08:29,797 --> 00:08:36,922 oranges on both sides. Gamma omega must be equal to 111 00:08:36,922 --> 00:08:41,545 times A. I still have this A here. 112 00:08:41,545 --> 00:08:49,811 So, gamma omega times A equals F zero divided by M times the 113 00:08:49,811 --> 00:08:54,855 sine of delta. Two equations with two 114 00:08:54,855 --> 00:09:02,000 unknowns: A is unknown and delta is unknown. 115 00:09:02,000 --> 00:09:07,117 And, they are easy to solve. If you square them, 116 00:09:07,117 --> 00:09:13,214 then you get sine squared delta and cosine squared delta. 117 00:09:13,214 --> 00:09:16,154 You add them up, that's one. 118 00:09:16,154 --> 00:09:21,707 And so, that immediately gives you, then, what A is. 119 00:09:21,707 --> 00:09:27,042 So, A is going to be F zero divided by M upstairs, 120 00:09:27,042 --> 00:09:33,358 and downstairs you're going to get omega zero squared minus 121 00:09:33,358 --> 00:09:37,604 omega squared, squared, plus omega gamma 122 00:09:37,604 --> 00:09:42,033 squared. So, this is the amplitude of 123 00:09:42,033 --> 00:09:44,211 the object. We'll massage that. 124 00:09:44,211 --> 00:09:48,204 We'll talk about this for at least the next ten minutes. 125 00:09:48,204 --> 00:09:50,528 It's a very complicated function. 126 00:09:50,528 --> 00:09:54,448 We want to see through that equation what that actually 127 00:09:54,448 --> 00:09:56,917 means. And, the tangent of delta is 128 00:09:56,917 --> 00:10:02,000 easy to find because you do find this equation by that one. 129 00:10:02,000 --> 00:10:05,727 You get immediately the tensions of delta. 130 00:10:05,727 --> 00:10:09,545 A's disappear, and F zero over M disappear, 131 00:10:09,545 --> 00:10:14,272 and so you will get that the tangent over that angle, 132 00:10:14,272 --> 00:10:19,363 delta, is omega gamma divided by omega zero squared minus 133 00:10:19,363 --> 00:10:23,272 omega squared. We can now return to the real 134 00:10:23,272 --> 00:10:26,363 world. And if we return to the real 135 00:10:26,363 --> 00:10:31,000 world, we have to change Z back into X. 136 00:10:31,000 --> 00:10:37,135 And so, our final solution, which I will put in color, 137 00:10:37,135 --> 00:10:43,617 would then be that X as a function of T is this amplitude 138 00:10:43,617 --> 00:10:48,363 A times the cosine of omega T minus delta. 139 00:10:48,363 --> 00:10:54,498 And, that is the omega. It's my will that I imposed on 140 00:10:54,498 --> 00:10:59,012 the system. Notice that there are no two 141 00:10:59,012 --> 00:11:05,032 adjustable constants, which we were so used to in the 142 00:11:05,032 --> 00:11:08,498 past. In the past we said, 143 00:11:08,498 --> 00:11:12,021 well, you can start the system at T equals zero. 144 00:11:12,021 --> 00:11:15,992 You can define the position. You can give it a certain 145 00:11:15,992 --> 00:11:18,840 velocity. So, you always expect that in 146 00:11:18,840 --> 00:11:21,988 your solution, there are two adjustables in 147 00:11:21,988 --> 00:11:24,686 order to meet the initial conditions. 148 00:11:24,686 --> 00:11:28,208 There are none here. And, the reason for that is 149 00:11:28,208 --> 00:11:32,629 that this is a steady state solution, which means the system 150 00:11:32,629 --> 00:11:36,826 doesn't even remember anymore what the situation was at T 151 00:11:36,826 --> 00:11:41,357 equals zero. It has lost all its memory. 152 00:11:41,357 --> 00:11:45,547 And so, A, which is the amplitude of that object, 153 00:11:45,547 --> 00:11:48,690 is not something that you may choose. 154 00:11:48,690 --> 00:11:53,841 A follows immediately from this equation, which is a complex 155 00:11:53,841 --> 00:11:57,246 function: omega zero, F zero, and so on. 156 00:11:57,246 --> 00:12:01,000 And delta is always nonnegotiable. 157 00:12:01,000 --> 00:12:04,685 Delta has nothing to do with your initial conditions. 158 00:12:04,685 --> 00:12:08,015 Delta follows from gamma, omega, and omega zero. 159 00:12:08,015 --> 00:12:12,055 So now we're going to look at A, and try to understand the 160 00:12:12,055 --> 00:12:15,102 complexity of that amplitude. For one thing, 161 00:12:15,102 --> 00:12:17,795 it is pleasing that F zero is upstairs. 162 00:12:17,795 --> 00:12:21,976 It is an intuitive pleasing that if the force that you apply 163 00:12:21,976 --> 00:12:25,165 becomes larger, then the amplitude will become 164 00:12:25,165 --> 00:12:28,000 larger. That's reasonable. 165 00:12:28,000 --> 00:12:33,279 It is also pleasing to see that there is a gamma here 166 00:12:33,279 --> 00:12:37,341 downstairs. That means if there is a huge 167 00:12:37,341 --> 00:12:42,113 amount of damping, you don't expect A to be very 168 00:12:42,113 --> 00:12:45,667 large. So, that's also pleasing that 169 00:12:45,667 --> 00:12:51,049 you see a gamma downstairs. Now, I want to evaluate in 170 00:12:51,049 --> 00:12:56,329 detail what's hidden in this very difficult equation. 171 00:12:56,329 --> 00:13:01,000 And let me try out your intuition. 172 00:13:01,000 --> 00:13:03,991 Common sense, without looking at my 173 00:13:03,991 --> 00:13:08,567 solutions, without looking at differential equations, 174 00:13:08,567 --> 00:13:13,144 without looking at the equation, A, just common sense 175 00:13:13,144 --> 00:13:17,895 now, suppose I apply a force here on the subject with a 176 00:13:17,895 --> 00:13:23,087 frequency which is near zero. So, it takes 100 million years 177 00:13:23,087 --> 00:13:27,663 for it to reach a maximum. And then, it takes another 178 00:13:27,663 --> 00:13:34,000 hundred million years for the force to go to zero and so on. 179 00:13:34,000 --> 00:13:39,236 When that force has a value, F zero, what do you think will 180 00:13:39,236 --> 00:13:44,381 be the position of that object? If you know that position, 181 00:13:44,381 --> 00:13:49,527 that may tell you what A is, what the amplitude is of that 182 00:13:49,527 --> 00:13:53,229 object without any differential equations. 183 00:13:53,229 --> 00:13:56,659 Any one of you able to immediately say, 184 00:13:56,659 --> 00:14:02,556 of course A has to be this? Maybe that is a little tougher. 185 00:14:02,556 --> 00:14:07,340 I see some hands there now. Or you're just doing your hair? 186 00:14:07,340 --> 00:14:08,000 Yeah? 187 00:14:08,000 --> 00:14:14,000 188 00:14:14,000 --> 00:14:17,225 Of course. If that force goes so slowly, 189 00:14:17,225 --> 00:14:21,526 then at all moments in time there must be equilibrium 190 00:14:21,526 --> 00:14:25,248 between the spring force, which is, of course, 191 00:14:25,248 --> 00:14:28,969 KX, and the force that you apply, which is UF. 192 00:14:28,969 --> 00:14:33,684 And so, if you do it extremely slowly, the two must always 193 00:14:33,684 --> 00:14:38,616 cancel each other. And so, I make the prediction 194 00:14:38,616 --> 00:14:43,578 now that when omega goes to zero, that A should become F 195 00:14:43,578 --> 00:14:46,556 zero divided by K. That is that X. 196 00:14:46,556 --> 00:14:49,443 Now, let's look at that equation. 197 00:14:49,443 --> 00:14:53,954 Let's see whether that is true. We make omega zero. 198 00:14:53,954 --> 00:15:00,000 So, this equation tells us that it is F zero divided by M. 199 00:15:00,000 --> 00:15:03,813 And then, downstairs we have omega zero squared. 200 00:15:03,813 --> 00:15:06,653 But, omega zero squared is K over M. 201 00:15:06,653 --> 00:15:09,899 And you see, indeed, that is exactly what 202 00:15:09,899 --> 00:15:12,414 you get. Now, without looking at 203 00:15:12,414 --> 00:15:17,282 equations, can you guess what the phase difference is between 204 00:15:17,282 --> 00:15:21,907 the driver and the follower? If it takes 100 million years 205 00:15:21,907 --> 00:15:26,938 for that force to slowly reach its maximum at 100 million years 206 00:15:26,938 --> 00:15:30,752 to go back again, what do you think will be your 207 00:15:30,752 --> 00:15:33,348 phase difference between the two? 208 00:15:33,348 --> 00:15:37,000 It will be zero, of course. 209 00:15:37,000 --> 00:15:41,181 There's plenty of time for that object to follow. 210 00:15:41,181 --> 00:15:44,492 So, you expect that delta becomes zero. 211 00:15:44,492 --> 00:15:49,284 Well, if omega becomes zero, this zero is omega squared. 212 00:15:49,284 --> 00:15:52,943 So, this goes away. He has a zero upstairs, 213 00:15:52,943 --> 00:15:58,431 so you see the tangent of delta is zero, and that indeed is what 214 00:15:58,431 --> 00:16:02,318 you see. So, the two follow each other. 215 00:16:02,318 --> 00:16:05,256 It's extremely boring, the whole thing, 216 00:16:05,256 --> 00:16:09,507 to watch, and the amplitude is exactly what you predict. 217 00:16:09,507 --> 00:16:14,067 Let's now do something more interesting, and let us drive it 218 00:16:14,067 --> 00:16:17,082 at what we call the resonance frequency. 219 00:16:17,082 --> 00:16:21,024 We give it that word. That is the frequency that the 220 00:16:21,024 --> 00:16:25,507 system really would love to oscillate in the absence of any 221 00:16:25,507 --> 00:16:29,990 damping, and in the absence of my doing this silly thing by 222 00:16:29,990 --> 00:16:34,915 driving it. So, now we are at what we call 223 00:16:34,915 --> 00:16:38,219 residents. So, this term goes away, 224 00:16:38,219 --> 00:16:42,301 and this term now becomes omega zero gamma. 225 00:16:42,301 --> 00:16:47,063 So you now get that A becomes F zero divided by M, 226 00:16:47,063 --> 00:16:51,631 and then downstairs you have F omega zero gamma. 227 00:16:51,631 --> 00:16:57,074 Well, if you remember that we introduced a quality factor 228 00:16:57,074 --> 00:17:02,127 omega zero divided by gamma, which is a dimensionless 229 00:17:02,127 --> 00:17:07,764 number, then you can also write this as F zero divided by K 230 00:17:07,764 --> 00:17:12,170 times Q. So, that's nice to remember 231 00:17:12,170 --> 00:17:15,305 that at resonance, if you define this as 232 00:17:15,305 --> 00:17:20,128 resonance, the amplitude of the object is Q times higher than 233 00:17:20,128 --> 00:17:23,745 what it would be at extremely low frequencies. 234 00:17:23,745 --> 00:17:28,086 Interesting to remember, so, this is the amplitude at a 235 00:17:28,086 --> 00:17:32,427 very low frequency that when you drive it at resonance, 236 00:17:32,427 --> 00:17:37,684 it is Q times higher. And then, I will put that here. 237 00:17:37,684 --> 00:17:42,533 When omega goes to infinity, everything goes so fast that 238 00:17:42,533 --> 00:17:46,256 the object has no time to follow the driver. 239 00:17:46,256 --> 00:17:50,673 The object goes nuts because of this high frequency. 240 00:17:50,673 --> 00:17:54,569 It can do anything. And so, A would then go to 241 00:17:54,569 --> 00:17:56,907 zero. And, let's check that. 242 00:17:56,907 --> 00:18:01,497 If omega goes to infinity, you see the downstairs here 243 00:18:01,497 --> 00:18:06,000 goes to infinity so, A goes in D2 zero. 244 00:18:06,000 --> 00:18:08,799 So, you have no amplitude at all. 245 00:18:08,799 --> 00:18:12,650 What is not so obvious that delta here is pi, 246 00:18:12,650 --> 00:18:17,900 and what is not so obvious here either, that delta goes to pi 247 00:18:17,900 --> 00:18:21,837 over two in case of resonance. In other words, 248 00:18:21,837 --> 00:18:25,687 at resonance, the driver and the follower are 249 00:18:25,687 --> 00:18:29,537 90° out of phase. The follower is 90° behind: 250 00:18:29,537 --> 00:18:34,000 very hard to imagine what that is like. 251 00:18:34,000 --> 00:18:37,560 But, I will demonstrate it. You will be able to see it. 252 00:18:37,560 --> 00:18:40,725 What it means here that at very high frequencies, 253 00:18:40,725 --> 00:18:43,362 the amplitude of the object goes to zero, 254 00:18:43,362 --> 00:18:47,450 but what I will be able to show you, that if the driver goes in 255 00:18:47,450 --> 00:18:50,285 this direction, that the object goes in this 256 00:18:50,285 --> 00:18:52,923 direction. So, they are 180° out of phase 257 00:18:52,923 --> 00:18:56,417 that I could show you. That's what it means when delta 258 00:18:56,417 --> 00:19:00,969 equals pi. So, now we can make a graph, 259 00:19:00,969 --> 00:19:05,000 a plot of A as a function of omega -- 260 00:19:05,000 --> 00:19:11,000 261 00:19:11,000 --> 00:19:14,466 So, here is omega, and here is A. 262 00:19:14,466 --> 00:19:19,991 And, let this be the resonance frequency omega zero. 263 00:19:19,991 --> 00:19:24,000 I'll make it a little straighter. 264 00:19:24,000 --> 00:19:30,000 265 00:19:30,000 --> 00:19:35,196 So, you start at very low frequencies if this is zero. 266 00:19:35,196 --> 00:19:39,019 You start here with F zero divided by K. 267 00:19:39,019 --> 00:19:42,352 We all agree that that was obvious. 268 00:19:42,352 --> 00:19:48,137 And then, the amplitude will build up, go through a maximum, 269 00:19:48,137 --> 00:19:51,862 goes down, and ultimately goes to zero. 270 00:19:51,862 --> 00:19:56,274 And at this value, omega zero, this value is Q 271 00:19:56,274 --> 00:20:02,947 times F zero divided by K. Now, for those of you who look 272 00:20:02,947 --> 00:20:07,269 very carefully, you may have noticed that the 273 00:20:07,269 --> 00:20:12,967 maximum here that I have drawn is not at omega equals omega 274 00:20:12,967 --> 00:20:16,896 zero, which may go against your instinct. 275 00:20:16,896 --> 00:20:22,103 This maximum occurs at a frequency, which we will call 276 00:20:22,103 --> 00:20:27,604 omega max, which is always a little bit below omega zero, 277 00:20:27,604 --> 00:20:32,516 but for high Q systems, as I will show you shortly, 278 00:20:32,516 --> 00:20:39,000 it is effectively the same. I will come back to this. 279 00:20:39,000 --> 00:20:45,450 The delta, the phase as a function of omega, 280 00:20:45,450 --> 00:20:54,150 if this is pi and this is pi over two, and if this is omega 281 00:20:54,150 --> 00:21:03,000 zero, then that delta will change in the following way. 282 00:21:03,000 --> 00:21:07,584 It is way harder to imagine than what A is doing you are in 283 00:21:07,584 --> 00:21:12,405 phase at very low frequencies, at resonance precisely at omega 284 00:21:12,405 --> 00:21:14,697 zero. You hit the pi over two, 285 00:21:14,697 --> 00:21:18,254 90° out of phase, and at very high frequencies 286 00:21:18,254 --> 00:21:21,573 you will see that the two are out of phase. 287 00:21:21,573 --> 00:21:26,000 And I will be able to demonstrate that to you. 288 00:21:26,000 --> 00:21:31,099 Coming back to this mysterious maximum, is also mysterious 289 00:21:31,099 --> 00:21:35,750 actually, where is this? At what frequency do we have 290 00:21:35,750 --> 00:21:40,492 really the maximum amplitude? Well, to calculate that, 291 00:21:40,492 --> 00:21:45,322 you would have to take the derivative of that monstrous 292 00:21:45,322 --> 00:21:48,364 equation. You will have to take DAD 293 00:21:48,364 --> 00:21:51,674 omega, and you [ask?] that to be zero. 294 00:21:51,674 --> 00:21:56,952 So, that's when the maximum occurs and I will leave you with 295 00:21:56,952 --> 00:22:00,798 that exercise. It may take you a few minutes 296 00:22:00,798 --> 00:22:05,361 to do that, and you will find, then, that omega max, 297 00:22:05,361 --> 00:22:10,817 so, where the real maximum is located, the maximum in terms of 298 00:22:10,817 --> 00:22:16,185 amplitude is omega zero squared minus gamma squared over two, 299 00:22:16,185 --> 00:22:22,000 not four, but two to the power of one half. 300 00:22:22,000 --> 00:22:25,202 Not so intuitive that it is there. 301 00:22:25,202 --> 00:22:29,666 And, if you'd like to write that in terms of Q, 302 00:22:29,666 --> 00:22:32,869 which is often done, then omega x, 303 00:22:32,869 --> 00:22:38,303 so that is the frequency at which the amplitude reaches a 304 00:22:38,303 --> 00:22:45,000 maximum is omega zero times one minus one over two Q squared. 305 00:22:45,000 --> 00:22:49,147 And then, you square root of that whole thing. 306 00:22:49,147 --> 00:22:53,940 And the reason why this is nice, you can immediately, 307 00:22:53,940 --> 00:22:57,811 if you know Q, you can immediately evaluate 308 00:22:57,811 --> 00:23:02,511 what the difference is percentage-wise between omega 309 00:23:02,511 --> 00:23:07,416 max and omega zero. If you want to know what the 310 00:23:07,416 --> 00:23:11,299 maximum amplitude itself is, so what A max is, 311 00:23:11,299 --> 00:23:15,959 so that's really this value. It must be very close to Q 312 00:23:15,959 --> 00:23:19,928 times F zero over Q. But, it's a little higher. 313 00:23:19,928 --> 00:23:23,898 Then, you can write that in the following form. 314 00:23:23,898 --> 00:23:28,213 And that's just a matter of algebraic manipulation. 315 00:23:28,213 --> 00:23:31,578 And you get a Q here, what you'd expect, 316 00:23:31,578 --> 00:23:35,634 and then downstairs you get something like this, 317 00:23:35,634 --> 00:23:41,782 one minus one over four. And then you get a Q squared 318 00:23:41,782 --> 00:23:47,645 through the power of one half. And so, now let's put in some 319 00:23:47,645 --> 00:23:53,509 numbers so that you get some feeling for the answers that we 320 00:23:53,509 --> 00:23:57,086 have. Suppose we have an example by Q 321 00:23:57,086 --> 00:24:00,863 equals five. It's a modest value for Q. 322 00:24:00,863 --> 00:24:06,627 Most pendulums that we have, the Q is way higher than five. 323 00:24:06,627 --> 00:24:11,000 So, I take a modest number for Q. 324 00:24:11,000 --> 00:24:16,243 If I go to this equation here, Q squared is 25. 325 00:24:16,243 --> 00:24:21,715 Two times 25 is 50. That's 2% that I have to take 326 00:24:21,715 --> 00:24:25,932 the square root. So, it's only 1% off. 327 00:24:25,932 --> 00:24:31,176 So, omega zero, omega max divided by omega zero 328 00:24:31,176 --> 00:24:34,790 is 0.99. It's only 1% lower. 329 00:24:34,790 --> 00:24:40,443 It's only 1% below omega zero. And then, if you want to know, 330 00:24:40,443 --> 00:24:44,966 now, what A max is, so you would think that A max 331 00:24:44,966 --> 00:24:49,960 is very close to Q times F zero over K, but it is not. 332 00:24:49,960 --> 00:24:55,425 Q times, it is a little larger. And so, if we defined A max 333 00:24:55,425 --> 00:24:59,854 divided by A zero, A zero now is meant to be the 334 00:24:59,854 --> 00:25:04,000 amplitude when omega equals zero. 335 00:25:04,000 --> 00:25:08,794 That is a shorthand notation. This number is not Q. 336 00:25:08,794 --> 00:25:12,150 It's a little higher. It's now 5.03. 337 00:25:12,150 --> 00:25:17,424 And, you can see it if Q is higher, then of course these 338 00:25:17,424 --> 00:25:23,465 numbers become even closer than omega max becomes even closer to 339 00:25:23,465 --> 00:25:28,931 omega zero, and then this maximum A becomes even closer to 340 00:25:28,931 --> 00:25:34,746 Q times F zero over K. Rarely ever will we be bothered 341 00:25:34,746 --> 00:25:39,265 too much with the fact that the resonance frequency, 342 00:25:39,265 --> 00:25:43,962 which we call omega zero, is not exactly the frequency 343 00:25:43,962 --> 00:25:48,126 whereby the response of the object is A maximum. 344 00:25:48,126 --> 00:25:51,848 Very rarely ever will that become an issue. 345 00:25:51,848 --> 00:25:56,278 I want to show you, now, the transparency from your 346 00:25:56,278 --> 00:26:00,000 own book. So, don't take notes. 347 00:26:00,000 --> 00:26:04,016 This is from French. You see here that a function A, 348 00:26:04,016 --> 00:26:08,742 that it is divided by A zero, which is that F zero divided by 349 00:26:08,742 --> 00:26:11,183 K. So, that is the amplitude for 350 00:26:11,183 --> 00:26:14,491 zero frequency. And so, when you start off, 351 00:26:14,491 --> 00:26:17,484 that ratio is one by definition, right, 352 00:26:17,484 --> 00:26:20,477 because it's A omega divided by A zero. 353 00:26:20,477 --> 00:26:24,021 And, horizontally, you see of omega divided by 354 00:26:24,021 --> 00:26:26,305 omega zero. So, by definition, 355 00:26:26,305 --> 00:26:31,109 right here at the one sign is that point that I put omega zero 356 00:26:31,109 --> 00:26:35,655 there. And, you see here these various 357 00:26:35,655 --> 00:26:38,396 curves for different values of Q. 358 00:26:38,396 --> 00:26:42,336 And, the one that I made green is Q equals ten. 359 00:26:42,336 --> 00:26:47,219 And, no surprise that the height is at plus ten because we 360 00:26:47,219 --> 00:26:52,187 predicted that it is Q times higher than the amplitude when 361 00:26:52,187 --> 00:26:55,099 we have low frequency. And you see, 362 00:26:55,099 --> 00:26:58,954 indeed, that the red one is very close to ten. 363 00:26:58,954 --> 00:27:03,151 If you look at the one that a mark Q equals three, 364 00:27:03,151 --> 00:27:07,262 which is this one, if you look very carefully you 365 00:27:07,262 --> 00:27:12,231 may see that the maximum A shifted slightly below the value 366 00:27:12,231 --> 00:27:18,757 one, which is omega zero. But even for Q equals three, 367 00:27:18,757 --> 00:27:22,500 the differences in significantly small. 368 00:27:22,500 --> 00:27:26,931 And at the bottom, you see the delta function, 369 00:27:26,931 --> 00:27:31,917 the phase delay. The object follows the driver 370 00:27:31,917 --> 00:27:36,847 at very low frequencies. Precisely, delta is zero. 371 00:27:36,847 --> 00:27:40,972 At resonance, it is precisely pi over two, 372 00:27:40,972 --> 00:27:46,908 90°, very hard to imagine that I will try to show it to you. 373 00:27:46,908 --> 00:27:51,737 And, at very high frequencies, they go like this. 374 00:27:51,737 --> 00:27:57,170 They are 180° out of phase. So, now comes the question, 375 00:27:57,170 --> 00:28:02,000 how do you apply a force on a system? 376 00:28:02,000 --> 00:28:06,910 It's nice to say there's a force, but you have to think of 377 00:28:06,910 --> 00:28:09,926 a way that you can actually do that. 378 00:28:09,926 --> 00:28:13,544 And, I will just discuss one case with you, 379 00:28:13,544 --> 00:28:16,990 and then I'll try to demonstrate it also. 380 00:28:16,990 --> 00:28:21,643 If I have a pendulum and I want a force on this object, 381 00:28:21,643 --> 00:28:26,381 then I can do that as you will see in an indirect way by 382 00:28:26,381 --> 00:28:32,983 starting to move my hand here. You'll see how that translates 383 00:28:32,983 --> 00:28:37,458 into a force on this object by moving my hand. 384 00:28:37,458 --> 00:28:41,038 I am now the driver. My displacement, 385 00:28:41,038 --> 00:28:44,320 now, is in inches, not [a force?], 386 00:28:44,320 --> 00:28:48,099 but is in inches. Here is the pendulum, 387 00:28:48,099 --> 00:28:52,475 length L mass M. And, this is the equilibrium 388 00:28:52,475 --> 00:28:56,651 position of my hand. And here's the object. 389 00:28:56,651 --> 00:29:02,419 But, going to move my hand in a way, [eta?] equals eta zero 390 00:29:02,419 --> 00:29:08,444 times cosine omega T. That is the frequency that I 391 00:29:08,444 --> 00:29:11,674 decide. I impose that frequency on the 392 00:29:11,674 --> 00:29:16,039 top of the pendulum. And, the amplitude of my hand, 393 00:29:16,039 --> 00:29:21,015 in terms of inches or miles or light-years, that is linear 394 00:29:21,015 --> 00:29:23,896 scale, eta zero. It's not a force. 395 00:29:23,896 --> 00:29:28,000 It's not a force. It's a displacement. 396 00:29:28,000 --> 00:29:32,989 Well, I take a picture at one moment in time and what do I 397 00:29:32,989 --> 00:29:35,702 see? I see that this is what the 398 00:29:35,702 --> 00:29:39,204 pendulum looks like. This angle is theta. 399 00:29:39,204 --> 00:29:43,055 The pendulum is displaced over a distance, X, 400 00:29:43,055 --> 00:29:47,257 from equilibrium. And the top is displaced over a 401 00:29:47,257 --> 00:29:50,233 distance eta. This is Walter Lewin. 402 00:29:50,233 --> 00:29:53,734 I am doing that. I am there with my hand. 403 00:29:53,734 --> 00:29:56,798 I can't help it. This is where I am. 404 00:29:56,798 --> 00:30:01,000 And this is where the object is. 405 00:30:01,000 --> 00:30:06,487 So, now I want to put in all the forces that are at work. 406 00:30:06,487 --> 00:30:12,268 We'll put it up a little bit because I want to have a little 407 00:30:12,268 --> 00:30:17,069 bit of room for my forces. So, we'll make a little 408 00:30:17,069 --> 00:30:19,910 shorter. So here's the object, 409 00:30:19,910 --> 00:30:24,809 and here's the object. There are only two forces on 410 00:30:24,809 --> 00:30:29,219 this object, and that is gravity, which is MG, 411 00:30:29,219 --> 00:30:35,000 and that is the tangent. There is nothing else. 412 00:30:35,000 --> 00:30:40,209 I call this X equals zero. And I call this displacement X 413 00:30:40,209 --> 00:30:43,837 away from equilibrium. For small angles, 414 00:30:43,837 --> 00:30:47,744 I want to argue that T is very close to MG. 415 00:30:47,744 --> 00:30:51,930 For one thing, if you hold them vertically and 416 00:30:51,930 --> 00:30:57,511 he did nothing and there is no motion, it's obvious that T is 417 00:30:57,511 --> 00:31:00,488 MG. The two forces have to cancel 418 00:31:00,488 --> 00:31:04,413 each other out. That's clear. 419 00:31:04,413 --> 00:31:10,173 But I can show you that even if the angles are modest, 420 00:31:10,173 --> 00:31:13,760 that that should also be the case. 421 00:31:13,760 --> 00:31:20,173 Suppose I decompose T in two directions, vertical direction. 422 00:31:20,173 --> 00:31:24,413 So, this is T times the cosine of theta. 423 00:31:24,413 --> 00:31:29,847 And in the horizontal direction, so this is T times 424 00:31:29,847 --> 00:31:35,011 the sine of theta. If the angles are very small, 425 00:31:35,011 --> 00:31:38,619 the object is hardly moving at all in this direction. 426 00:31:38,619 --> 00:31:42,088 The motion is almost exclusively in this direction. 427 00:31:42,088 --> 00:31:45,349 So, there is no acceleration in the Y direction, 428 00:31:45,349 --> 00:31:48,818 or I should say the acceleration in the Y direction 429 00:31:48,818 --> 00:31:52,356 is negligibly small. So, that means to a high degree 430 00:31:52,356 --> 00:31:55,964 of accuracy, T cosine theta is always the same as MG: 431 00:31:55,964 --> 00:31:59,086 high degree of accuracy. But for small angles, 432 00:31:59,086 --> 00:32:04,299 cosine theta itself is one. Therefore, T equals MG. 433 00:32:04,299 --> 00:32:09,500 And so, the force that is driving this object back to 434 00:32:09,500 --> 00:32:15,200 equilibrium is T sine theta. And so, that force is MG sine 435 00:32:15,200 --> 00:32:18,599 theta to a high degree of accuracy. 436 00:32:18,599 --> 00:32:23,900 I'm going to introduce, again, that gamma is B over N. 437 00:32:23,900 --> 00:32:28,400 So, it is damping. And, I'm going to introduce 438 00:32:28,400 --> 00:32:33,000 that omega squared equals G over L. 439 00:32:33,000 --> 00:32:39,263 Omega zero squared is G over L, the square root of G over L 440 00:32:39,263 --> 00:32:46,067 being the resonance frequency of a pendulum length L independent 441 00:32:46,067 --> 00:32:51,036 of the mass of the object as we've seen before. 442 00:32:51,036 --> 00:32:56,544 So, now I'm going to write down Newton's Second Law. 443 00:32:56,544 --> 00:33:02,956 So, I get MX double dot. And then I get minus BX dot. 444 00:33:02,956 --> 00:33:06,335 That is the damping minus BX dot. 445 00:33:06,335 --> 00:33:12,565 And now comes this force which is the only one that wants to 446 00:33:12,565 --> 00:33:18,267 drive it back to equilibrium. It's the restoring force. 447 00:33:18,267 --> 00:33:23,124 And so, that's force, if you set my T being MG. 448 00:33:23,124 --> 00:33:26,714 That is MG times the sine of theta. 449 00:33:26,714 --> 00:33:31,993 That's the differential equation that I now have to 450 00:33:31,993 --> 00:33:36,531 solve. That is a driven system. 451 00:33:36,531 --> 00:33:42,544 Now, here in the driven system, and boy, I saw force here. 452 00:33:42,544 --> 00:33:46,341 I don't see anything like that there. 453 00:33:46,341 --> 00:33:52,248 Where on Earth does Walter Lewin come in to this picture? 454 00:33:52,248 --> 00:33:57,206 Who's doing something? Have I overlooked myself, 455 00:33:57,206 --> 00:33:59,000 perhaps? 456 00:33:59,000 --> 00:34:04,000 457 00:34:04,000 --> 00:34:07,091 Excuse me? I'd change nothing. 458 00:34:07,091 --> 00:34:12,529 Would I change anything? I've changed nothing that I 459 00:34:12,529 --> 00:34:17,007 don't see myself anymore. So, what's wrong? 460 00:34:17,007 --> 00:34:20,525 Is there anything wrong with this? 461 00:34:20,525 --> 00:34:24,257 Where do I show up in this equation? 462 00:34:24,257 --> 00:34:27,988 Yeah? So where in that equation do I 463 00:34:27,988 --> 00:34:33,000 show up? What is the sine of theta? 464 00:34:33,000 --> 00:34:40,599 What is the sine of theta? What is the sine of this angle? 465 00:34:40,599 --> 00:34:45,266 X minus [eta?]. That's Walter Lewin. 466 00:34:45,266 --> 00:34:50,066 X minus eta divided by L. There I am. 467 00:34:50,066 --> 00:34:55,933 And so, I'm going to substitute that in here, 468 00:34:55,933 --> 00:35:05,000 and I'm going to divide by M. No, let's not divide by M yet. 469 00:35:05,000 --> 00:35:09,857 Let's just say X double dot minus BX dot, 470 00:35:09,857 --> 00:35:14,471 and now we get minus MG times X over L. 471 00:35:14,471 --> 00:35:20,300 And, now we bring Walter Lewin to the other side. 472 00:35:20,300 --> 00:35:25,764 And so, we get plus MG times eta divided by L. 473 00:35:25,764 --> 00:35:32,199 And, eta is eta zero times cosine omega T because I am 474 00:35:32,199 --> 00:35:39,000 moving my hand that eta is a function of time. 475 00:35:39,000 --> 00:35:43,629 Let me write down MG in here, and then we'll check this. 476 00:35:43,629 --> 00:35:46,744 So, MG times sine theta has two terms. 477 00:35:46,744 --> 00:35:51,710 It has an MG times X over L, but it has also an MG times eta 478 00:35:51,710 --> 00:35:54,909 over L. And, I bring that eta L over on 479 00:35:54,909 --> 00:35:59,117 this side, but I know that eta is changing in time. 480 00:35:59,117 --> 00:36:04,000 And so, you see now Walter Lewin is right there. 481 00:36:04,000 --> 00:36:07,660 And now divide by M, and I substitute omega zero 482 00:36:07,660 --> 00:36:10,697 squared over here. Oh, I have an M here, 483 00:36:10,697 --> 00:36:13,345 too. You should have screamed there 484 00:36:13,345 --> 00:36:16,772 was an M there. I decided not to divide by M, 485 00:36:16,772 --> 00:36:19,732 remember? Now I'm going to divide by M, 486 00:36:19,732 --> 00:36:23,626 so I get X double dot. There is an equal sign here. 487 00:36:23,626 --> 00:36:29,000 You should not be sleeping. You're not supposed to sleep. 488 00:36:29,000 --> 00:36:34,312 This is an equal sign minus BX dot, right, minus B, 489 00:36:34,312 --> 00:36:37,181 yeah? We're in business now. 490 00:36:37,181 --> 00:36:42,812 And so, you are all sleeping. All of you are sleeping, 491 00:36:42,812 --> 00:36:46,849 my goodness. MX double dot minus BX dot 492 00:36:46,849 --> 00:36:51,631 minus MG KX over L, and then the minus and the 493 00:36:51,631 --> 00:36:56,199 minus becomes plus. Right, try not to sleep. 494 00:36:56,199 --> 00:37:01,087 So, X double not, now we get plus gamma times X 495 00:37:01,087 --> 00:37:05,969 dot. And now we get plus omega zero 496 00:37:05,969 --> 00:37:10,324 squared times, because G over L is omega zero 497 00:37:10,324 --> 00:37:14,184 squared times X. So, I divide the M out, 498 00:37:14,184 --> 00:37:20,023 and now I get equals omega zero squared times eta zero times 499 00:37:20,023 --> 00:37:24,378 cosine omega T. I will move this L up a teeny 500 00:37:24,378 --> 00:37:30,218 little bit, and on going to look now at that equation at the 501 00:37:30,218 --> 00:37:34,666 bottom. And I am now overjoyed, 502 00:37:34,666 --> 00:37:41,391 happiness, because this one looks almost like a carbon copy 503 00:37:41,391 --> 00:37:47,768 of the one that I had here was an F zero cosine omega T. 504 00:37:47,768 --> 00:37:54,376 And now, instead of an F zero divided by M cosine omega T, 505 00:37:54,376 --> 00:37:59,710 I now have this. So, this takes the place of my 506 00:37:59,710 --> 00:38:05,714 earlier F zero divided by M. F zero divided by M is an 507 00:38:05,714 --> 00:38:07,360 acceleration, by the way. 508 00:38:07,360 --> 00:38:09,622 Now, it better be an acceleration, 509 00:38:09,622 --> 00:38:11,748 because this is an acceleration. 510 00:38:11,748 --> 00:38:15,657 And apples have to be apples. So, this is an acceleration. 511 00:38:15,657 --> 00:38:18,605 This is an exhilaration, and this is also an 512 00:38:18,605 --> 00:38:21,211 acceleration. Multiply omega squared by 513 00:38:21,211 --> 00:38:23,817 distance. Then you get distance divided 514 00:38:23,817 --> 00:38:26,491 by time squared. So, you see now how the 515 00:38:26,491 --> 00:38:30,400 connection between the two go, where originally I got an F 516 00:38:30,400 --> 00:38:34,651 zero over M times cosine omega T, now because of Walter Lewin's 517 00:38:34,651 --> 00:38:40,000 motion, I'm going to get an omega squared times eta zero. 518 00:38:40,000 --> 00:38:44,482 And so, you see now how this motion of my hand indeed 519 00:38:44,482 --> 00:38:47,672 translates into a force on the object. 520 00:38:47,672 --> 00:38:52,327 Well, I have the solution. I don't have to do anything. 521 00:38:52,327 --> 00:38:57,672 All I have to do is change this by omega zero squared times eta 522 00:38:57,672 --> 00:39:02,041 zero, and I'm done. Differential equations are 523 00:39:02,041 --> 00:39:05,260 identical. I don't even have to change the 524 00:39:05,260 --> 00:39:07,851 tension of Delta. Nothing changes. 525 00:39:07,851 --> 00:39:10,599 This is the only thing that changes. 526 00:39:10,599 --> 00:39:13,425 So, we are done. We can now make some 527 00:39:13,425 --> 00:39:16,644 predictions. The prediction is that if I'm 528 00:39:16,644 --> 00:39:21,198 going to shake this pendulum, and are going to do that very 529 00:39:21,198 --> 00:39:26,066 slowly, taking one hour to go to the left and taking one are to 530 00:39:26,066 --> 00:39:28,971 the right. And, if my amplitude is eta 531 00:39:28,971 --> 00:39:32,504 zero, what do you think that the amplitude, A, 532 00:39:32,504 --> 00:39:35,723 the solution of the differential equation, 533 00:39:35,723 --> 00:39:39,042 will be? In other words, 534 00:39:39,042 --> 00:39:43,071 I'm going to shake very slowly. What do you think A will be 535 00:39:43,071 --> 00:39:46,198 without looking at the differential equations? 536 00:39:46,198 --> 00:39:50,089 So, I just go with my hand like this: amplitude eta zero, 537 00:39:50,089 --> 00:39:53,076 amplitude eta zero, and I do it very slowly. 538 00:39:53,076 --> 00:39:55,300 What would be the amplitude of A? 539 00:39:55,300 --> 00:39:57,940 Eta zero. So, you expect that this goes 540 00:39:57,940 --> 00:40:00,858 to eta zero. Well, if you don't believe it, 541 00:40:00,858 --> 00:40:03,915 go to this equation, substitute in here zero, 542 00:40:03,915 --> 00:40:07,634 in here zero. You get omega zero squared. 543 00:40:07,634 --> 00:40:10,782 Each of us: omega zero squared, and you see eta zero. 544 00:40:10,782 --> 00:40:13,204 It's exactly what that equation predicts. 545 00:40:13,204 --> 00:40:15,747 But, your common sense says the same thing. 546 00:40:15,747 --> 00:40:19,137 Now, what do you think delta is if I'm going to move this 547 00:40:19,137 --> 00:40:22,104 pendulum very slowly to the left and to the right? 548 00:40:22,104 --> 00:40:24,768 Of course the object [UNINTELLIGIBLE] will be 549 00:40:24,768 --> 00:40:27,977 ridiculous if I take one week to go from here to here. 550 00:40:27,977 --> 00:40:29,914 The object would be there, right? 551 00:40:29,914 --> 00:40:33,365 The object is always here. So, you also predict that delta 552 00:40:33,365 --> 00:40:36,982 is zero. And so, now we can make 553 00:40:36,982 --> 00:40:40,862 [UNINTELLIGIBLE] prediction that at resonance, 554 00:40:40,862 --> 00:40:45,517 you probably get Q times eta zero, and then delta would 555 00:40:45,517 --> 00:40:49,655 become pi over two. And, when you go to very high 556 00:40:49,655 --> 00:40:52,586 frequency, then A would go to zero. 557 00:40:52,586 --> 00:40:57,413 And then delta would go to pi. And this is what I want to 558 00:40:57,413 --> 00:41:02,524 demonstrate to you. So, the final solution of this 559 00:41:02,524 --> 00:41:07,487 pendulum, which I will write down in red is going to be that 560 00:41:07,487 --> 00:41:12,536 X equals A times the cosine of omega T minus delta just as we 561 00:41:12,536 --> 00:41:15,060 had before. A is nonnegotiable. 562 00:41:15,060 --> 00:41:19,098 It has nothing to do with the initial conditions. 563 00:41:19,098 --> 00:41:23,473 Delta is nonnegotiable. It has nothing to do with the 564 00:41:23,473 --> 00:41:27,091 initial conditions. This is the steady state 565 00:41:27,091 --> 00:41:30,456 solution. All right, let me take my shoes 566 00:41:30,456 --> 00:41:35,000 off because then you can see it better. 567 00:41:35,000 --> 00:41:40,593 All right, here's the pendulum. Here's a pendulum. 568 00:41:40,593 --> 00:41:46,643 It's going to be very exciting. I'm going to tell you. 569 00:41:46,643 --> 00:41:52,579 I'm going to move this with omega very close to zero. 570 00:41:52,579 --> 00:41:56,803 Very exciting. I'm doing it right now. 571 00:41:56,803 --> 00:42:01,944 Aren't you thrilled? No, you're not thrilled? 572 00:42:01,944 --> 00:42:05,994 [You're not?] moving. And now I'm going to go back. 573 00:42:05,994 --> 00:42:09,802 Do we agree that A, the amplitude of that object 574 00:42:09,802 --> 00:42:13,772 it's exactly the same as the amplitude of my hand? 575 00:42:13,772 --> 00:42:16,040 Do we agree? Do you see that? 576 00:42:16,040 --> 00:42:20,658 That is why that A is eta zero, and that follows from that 577 00:42:20,658 --> 00:42:25,032 rather complicated equation. Did you see that delta was 578 00:42:25,032 --> 00:42:27,787 zero? Did you see that we went hand 579 00:42:27,787 --> 00:42:32,000 in hand, so to speak, no pun implied? 580 00:42:32,000 --> 00:42:35,466 We're going hand in hand. That one follows exactly my 581 00:42:35,466 --> 00:42:37,000 hand. So, delta is zero. 582 00:42:37,000 --> 00:42:40,400 Let's now go to high frequency, very high frequency, 583 00:42:40,400 --> 00:42:43,666 way above resonance. And, what you see now is that 584 00:42:43,666 --> 00:42:45,933 the object is not moving very much. 585 00:42:45,933 --> 00:42:49,266 But, if you look very carefully, you'll see when my 586 00:42:49,266 --> 00:42:52,066 hand as here, the object tends to go there. 587 00:42:52,066 --> 00:42:55,400 When my hand is here, the object tends to go there. 588 00:42:55,400 --> 00:42:58,000 That is that pi. Ready? 589 00:42:58,000 --> 00:43:02,000 590 00:43:02,000 --> 00:43:04,583 You see that there's almost no motion? 591 00:43:04,583 --> 00:43:07,795 A is near zero, but can you really see that the 592 00:43:07,795 --> 00:43:10,867 phase difference is pi? Can you see the 180°? 593 00:43:10,867 --> 00:43:14,778 You see, if A is exactly zero, of course you cannot tell. 594 00:43:14,778 --> 00:43:17,221 So, I try not to go infinitely fast. 595 00:43:17,221 --> 00:43:20,433 I go a little slower than [OVERLAPPING VOICES]. 596 00:43:20,433 --> 00:43:23,157 Can you see it? OK, now comes resonance. 597 00:43:23,157 --> 00:43:26,788 And now it will be very difficult to see this pi over 598 00:43:26,788 --> 00:43:30,000 two. That's almost impossible. 599 00:43:30,000 --> 00:43:34,005 That's not my objective. My objective is to show you 600 00:43:34,005 --> 00:43:38,010 that enormously small, very small eta zero here will 601 00:43:38,010 --> 00:43:42,015 give an amplitude there are which is Q times higher. 602 00:43:42,015 --> 00:43:46,806 So, you get a huge swing when my hand is hardly moving at all. 603 00:43:46,806 --> 00:43:49,790 That's the power of Q. So, there we go. 604 00:43:49,790 --> 00:43:52,696 First [carried?] into it; there it is. 605 00:43:52,696 --> 00:43:55,759 Now, this is resonance. Would you agree? 606 00:43:55,759 --> 00:44:00,000 This is resonance. Now, look at my hand. 607 00:44:00,000 --> 00:44:03,876 My hand is moving, probably no more there is an 608 00:44:03,876 --> 00:44:06,067 amplitude of 3 mm, no more. 609 00:44:06,067 --> 00:44:09,606 And yet, I see an amplitude there of 60 cm. 610 00:44:09,606 --> 00:44:14,241 That would mean that very roughly, this pendulum has a Q 611 00:44:14,241 --> 00:44:17,359 of 200, namely, 60 cm divided by 3 mm. 612 00:44:17,359 --> 00:44:21,994 So, this is even a way to make an extremely rough guess, 613 00:44:21,994 --> 00:44:25,112 admittedly very rough, of the Q value. 614 00:44:25,112 --> 00:44:27,808 You cannot even see my hand move. 615 00:44:27,808 --> 00:44:32,536 Be honest. You can't even see my hand 616 00:44:32,536 --> 00:44:36,439 move, but I know I am moving it a little. 617 00:44:36,439 --> 00:44:40,341 You're lying. [LAUGHTER] No you were not. 618 00:44:40,341 --> 00:44:44,829 No you were not. OK, so you see all the goodies 619 00:44:44,829 --> 00:44:50,975 that we have calculated actually can be demonstrated and show up 620 00:44:50,975 --> 00:44:55,658 quite dramatically. Suppose I had a spring system 621 00:44:55,658 --> 00:45:02,000 like this, and I wanted the force on that object here. 622 00:45:02,000 --> 00:45:06,711 Well, what I can do is just shake it here [NOISE OBSCURES]. 623 00:45:06,711 --> 00:45:10,448 And, when I shake it there, we can make certain 624 00:45:10,448 --> 00:45:13,616 predictions. We can make predictions now 625 00:45:13,616 --> 00:45:16,459 based on the knowledge that we have. 626 00:45:16,459 --> 00:45:20,114 Suppose I shake it with an amplitude eta zero. 627 00:45:20,114 --> 00:45:23,770 No differential equations, no nothing for now. 628 00:45:23,770 --> 00:45:28,725 But I know that somehow it will come out in terms of the force 629 00:45:28,725 --> 00:45:33,127 at the object. So, I know that when I write 630 00:45:33,127 --> 00:45:37,686 down the differential equation, of course it shows up exactly 631 00:45:37,686 --> 00:45:42,017 this way again in omega zero squared times eta zero except 632 00:45:42,017 --> 00:45:44,904 the omega zero squared is now K over M. 633 00:45:44,904 --> 00:45:49,159 So, what do you think if I shake it at omega equals zero? 634 00:45:49,159 --> 00:45:52,502 What is, then, the amplitude that this object 635 00:45:52,502 --> 00:45:55,541 will have relative to my motion eta zero? 636 00:45:55,541 --> 00:45:58,960 If I move my hands, eta zero infinitely won't, 637 00:45:58,960 --> 00:46:03,685 what will this object do? It will just follow it. 638 00:46:03,685 --> 00:46:07,129 So, you get this answer. What will be the delta? 639 00:46:07,129 --> 00:46:09,840 It will be zero. When I get resonance, 640 00:46:09,840 --> 00:46:14,090 what will be the amplitude of that object hanging from this 641 00:46:14,090 --> 00:46:16,801 spring? It will be Q times higher than 642 00:46:16,801 --> 00:46:19,293 my eta zero. What will be the phase 643 00:46:19,293 --> 00:46:20,392 difference? 90°. 644 00:46:20,392 --> 00:46:23,469 When I shake like crazy, A will go to zero. 645 00:46:23,469 --> 00:46:26,987 So, there is a spring. If you shake it like this, 646 00:46:26,987 --> 00:46:31,237 which is part of your problem set, you will see exactly the 647 00:46:31,237 --> 00:46:36,000 same results that we have done for a pendulum. 648 00:46:36,000 --> 00:46:40,140 This, now, I want to independently demonstrate to 649 00:46:40,140 --> 00:46:42,642 you. I have here in air track. 650 00:46:42,642 --> 00:46:47,645 I can blow out air so that the object here starts floating. 651 00:46:47,645 --> 00:46:52,476 So, we can make the damping very slow by making it float. 652 00:46:52,476 --> 00:46:57,392 But, if we lower the airflow, the damping becomes a little 653 00:46:57,392 --> 00:47:00,153 higher. I have a spring here with 654 00:47:00,153 --> 00:47:06,322 [spring?] constant K. And I have another spring here 655 00:47:06,322 --> 00:47:08,869 with spring constant, K. 656 00:47:08,869 --> 00:47:12,523 They both have spring constant, K. 657 00:47:12,523 --> 00:47:18,281 And now I'm going to drive this here at extremely low 658 00:47:18,281 --> 00:47:23,486 frequencies over a distance eta zero at maximum. 659 00:47:23,486 --> 00:47:29,687 Eta zero, cosine omega T. What do you think the amplitude 660 00:47:29,687 --> 00:47:36,000 of this object will be at that very low, yeah? 661 00:47:36,000 --> 00:47:38,662 Very, very good. Not eta zero, 662 00:47:38,662 --> 00:47:42,977 but why is it half? Because we have two springs. 663 00:47:42,977 --> 00:47:47,016 So, effectively, the spring constant is twice 664 00:47:47,016 --> 00:47:50,504 that, exactly. So, if I go very slowly, 665 00:47:50,504 --> 00:47:55,186 you'll see that this displacement here will be twice 666 00:47:55,186 --> 00:48:00,603 as high as this displacement. But what I really want to show 667 00:48:00,603 --> 00:48:06,171 you is they are in phase. This one will go to the right 668 00:48:06,171 --> 00:48:09,990 and this one goes to the right. Now comes the catch. 669 00:48:09,990 --> 00:48:13,958 I showed you earlier there is a steady state solution. 670 00:48:13,958 --> 00:48:17,253 In the beginning, the system doesn't like me. 671 00:48:17,253 --> 00:48:19,125 It hates me. It fights me. 672 00:48:19,125 --> 00:48:22,944 It doesn't like that omega. It wants to do something 673 00:48:22,944 --> 00:48:26,463 different, which is part of next week's lecture. 674 00:48:26,463 --> 00:48:29,758 And, you'll see that in the beginning and so, 675 00:48:29,758 --> 00:48:35,000 we have to be a little patient before my will survives. 676 00:48:35,000 --> 00:48:38,264 Ready for that? So, I'm going to start now to 677 00:48:38,264 --> 00:48:42,418 drive the system at a frequency which is below resonance. 678 00:48:42,418 --> 00:48:46,498 I want you to see two things, that they go hand-in-hand. 679 00:48:46,498 --> 00:48:50,875 And, I want you to see that. See, you're going to a very low 680 00:48:50,875 --> 00:48:53,323 frequency. Here, this is twice the 681 00:48:53,323 --> 00:48:56,216 amplitude of the driver. Now it is here, 682 00:48:56,216 --> 00:49:00,000 the spring, and now the spring is here. 683 00:49:00,000 --> 00:49:03,469 So, only this much is two eta zero. 684 00:49:03,469 --> 00:49:08,979 So, eta zero is no more than three quarters of an inch. 685 00:49:08,979 --> 00:49:14,795 And now we're going to let that object the exposed to this 686 00:49:14,795 --> 00:49:18,673 driver. And, we'll give it a little bit 687 00:49:18,673 --> 00:49:24,081 of time to recognize me. It takes a little bit of time 688 00:49:24,081 --> 00:49:27,551 to reach the steady state solution. 689 00:49:27,551 --> 00:49:35,000 And, next time we will learn how much time it actually takes. 690 00:49:35,000 --> 00:49:40,000 So, if you want to be a little bit patient, then you'll see -- 691 00:49:40,000 --> 00:49:46,000 692 00:49:46,000 --> 00:49:49,853 -- if we give it too much damping, too little air, 693 00:49:49,853 --> 00:49:53,000 then of course it starts to get stuck. 694 00:49:53,000 --> 00:49:58,000 695 00:49:58,000 --> 00:50:00,286 Yeah, we're close. We are close, 696 00:50:00,286 --> 00:50:02,942 for me close enough. Now, look at it. 697 00:50:02,942 --> 00:50:07,295 They're going both to the left for me, and both to the right 698 00:50:07,295 --> 00:50:10,098 for me. For you, they're going now both 699 00:50:10,098 --> 00:50:13,196 to the right, and they're going both to the 700 00:50:13,196 --> 00:50:15,852 left. They're going both to the right 701 00:50:15,852 --> 00:50:19,467 and both to the left. Now, this was the amplitude, 702 00:50:19,467 --> 00:50:21,901 twice the amplitude of the driver. 703 00:50:21,901 --> 00:50:24,704 And then, you look very carefully here, 704 00:50:24,704 --> 00:50:27,508 it's less. It is that eta zero over two 705 00:50:27,508 --> 00:50:31,786 that this gentleman immediately noticed because we have two 706 00:50:31,786 --> 00:50:36,583 springs. So, you see here apart from the 707 00:50:36,583 --> 00:50:39,666 factor of two, you see the delta zero, 708 00:50:39,666 --> 00:50:44,750 and you see that the amplitude indeed is half of the amplitude 709 00:50:44,750 --> 00:50:48,083 of the driver because of the two springs. 710 00:50:48,083 --> 00:50:51,500 Now, we're going to resonance, omega zero. 711 00:50:51,500 --> 00:50:55,000 And now, nasty things may happen. 712 00:50:55,000 --> 00:51:01,000 713 00:51:01,000 --> 00:51:05,182 It may break. We have to give it time. 714 00:51:05,182 --> 00:51:08,800 See what funny things it's doing? 715 00:51:08,800 --> 00:51:14,000 Not in steady-state yet. You have to wait. 716 00:51:14,000 --> 00:51:22,000 717 00:51:22,000 --> 00:51:23,000 Just a little patient. 718 00:51:23,000 --> 00:51:40,000 719 00:51:40,000 --> 00:51:41,000 Give it more time. 720 00:51:41,000 --> 00:51:53,000 721 00:51:53,000 --> 00:51:56,104 Notice, also, that remember this is only 722 00:51:56,104 --> 00:51:59,766 moving this much. Look how much this is moving. 723 00:51:59,766 --> 00:52:02,791 I may not even be exactly at resonance. 724 00:52:02,791 --> 00:52:05,577 We can only do the best we can here. 725 00:52:05,577 --> 00:52:09,000 It may not be exactly at resonance. 726 00:52:09,000 --> 00:52:20,000 727 00:52:20,000 --> 00:52:23,596 Oh boy, I'm close to resonance now. 728 00:52:23,596 --> 00:52:26,769 Oh yeah. Oh, man, look at that. 729 00:52:26,769 --> 00:52:31,000 Am I at resonance! I think I got it. 730 00:52:31,000 --> 00:52:36,325 You see, they are neither in phase nor out of phase. 731 00:52:36,325 --> 00:52:41,443 Now you see the 90°. And, look at this teeny-weeny 732 00:52:41,443 --> 00:52:46,768 little displacement here. And, look what this man is 733 00:52:46,768 --> 00:52:49,275 doing. That is resonance. 734 00:52:49,275 --> 00:52:51,990 Remarkable. Where's Marcos? 735 00:52:51,990 --> 00:52:53,974 We hit it. Right on! 736 00:52:53,974 --> 00:52:58,465 [LAUGHTER] Now, I will oscillate it way over 737 00:52:58,465 --> 00:53:04,000 resonance, not way, but over resonance. 738 00:53:04,000 --> 00:53:06,767 The system must first calm down. 739 00:53:06,767 --> 00:53:11,857 And now I will change the frequency of a resonance so that 740 00:53:11,857 --> 00:53:17,214 now you will see the phenomenon that I discussed earlier that 741 00:53:17,214 --> 00:53:21,857 the amplitude is very small. Again, we have to wait a 742 00:53:21,857 --> 00:53:24,714 little. Look how fast it's going. 743 00:53:24,714 --> 00:53:29,000 And, then I will go 180° out of phase. 744 00:53:29,000 --> 00:53:35,000 745 00:53:35,000 --> 00:53:39,882 Now, look, this is going this much back and forth. 746 00:53:39,882 --> 00:53:43,370 And this one is not doing very much. 747 00:53:43,370 --> 00:53:46,160 Now, you can't see it. I can. 748 00:53:46,160 --> 00:53:49,548 It looks like this. Can you see it? 749 00:53:49,548 --> 00:53:53,932 That's what 180° out of phase is, very clear. 750 00:53:53,932 --> 00:53:58,516 Five minute break, see you back here in exactly 751 00:53:58,516 --> 00:54:03,000 five minutes. [SOUND OFF/THEN ON] 752 00:54:03,000 --> 00:54:06,755 So, we have discussed today some simple systems: 753 00:54:06,755 --> 00:54:10,031 pendulum, one object, springs, one object, 754 00:54:10,031 --> 00:54:14,266 one resonance frequency, but soon in 8.03 will discuss 755 00:54:14,266 --> 00:54:16,903 systems with more than one object. 756 00:54:16,903 --> 00:54:20,099 For instance, if I put three cars on here 757 00:54:20,099 --> 00:54:23,774 with four springs, three resonance frequencies, 758 00:54:23,774 --> 00:54:28,409 if I have a trouble pendulum, which I will demonstrate next 759 00:54:28,409 --> 00:54:32,484 week, when pendulum below the other below the other, 760 00:54:32,484 --> 00:54:37,826 three resonance frequencies. Five cars on there, 761 00:54:37,826 --> 00:54:42,826 five resonance frequencies. So, simple objects like a 762 00:54:42,826 --> 00:54:48,788 dinner plate or just a regular glass have an enormous number of 763 00:54:48,788 --> 00:54:53,307 resonance frequencies. It can oscillate in many, 764 00:54:53,307 --> 00:54:57,346 many different ways. If you drive your car, 765 00:54:57,346 --> 00:55:02,250 your wheels turn around, that's certain oscillation, 766 00:55:02,250 --> 00:55:08,231 a certain period underlying. And, you may notice at certain 767 00:55:08,231 --> 00:55:11,950 speeds that something in your car begins to rattle: 768 00:55:11,950 --> 00:55:14,776 very annoying. All you have to do is go 769 00:55:14,776 --> 00:55:17,752 slower, go a little faster, and it stops. 770 00:55:17,752 --> 00:55:20,429 You go off resonance for that object. 771 00:55:20,429 --> 00:55:23,925 Now, you may go on resonance for another object, 772 00:55:23,925 --> 00:55:27,198 of course, and some cars rattle at any speed. 773 00:55:27,198 --> 00:55:31,512 You have a radiator in your room which rotates that is also 774 00:55:31,512 --> 00:55:34,190 an underlying oscillation and period. 775 00:55:34,190 --> 00:55:37,611 That may start to cause residents in the frame. 776 00:55:37,611 --> 00:55:42,000 You may hear some awful noise sometimes. 777 00:55:42,000 --> 00:55:44,418 Unfortunately, these fans, you cannot change 778 00:55:44,418 --> 00:55:46,948 the speed so easily. But you can go from state 779 00:55:46,948 --> 00:55:49,422 three, to two, to one, and then this terrible 780 00:55:49,422 --> 00:55:52,122 noise will go away. You take your washing machine 781 00:55:52,122 --> 00:55:54,315 or dryer. They used to remember a friend 782 00:55:54,315 --> 00:55:56,452 of mine in the Netherlands had a dryer. 783 00:55:56,452 --> 00:55:59,376 And, when he started the dryer at a very early phase, 784 00:55:59,376 --> 00:56:02,356 when it was a certain frequency, the whole dryer would 785 00:56:02,356 --> 00:56:06,358 start to walk through the room. [LAUGHTER] It would. 786 00:56:06,358 --> 00:56:09,755 And then, at higher frequencies, it of course would 787 00:56:09,755 --> 00:56:11,861 stop. Resonances are everywhere, 788 00:56:11,861 --> 00:56:15,258 and they often occur when you don't expect them to. 789 00:56:15,258 --> 00:56:18,519 You open a faucet; you think it's a steady stream 790 00:56:18,519 --> 00:56:22,051 of water which I'm sure it is. But sometimes you hear 791 00:56:22,051 --> 00:56:26,059 [SCREAMS] an unbelievable sound that drives you almost nuts. 792 00:56:26,059 --> 00:56:30,000 I'm sure all of you have heard that sometime. 793 00:56:30,000 --> 00:56:33,946 If it isn't in your dormitory, maybe at hotels or at home. 794 00:56:33,946 --> 00:56:38,238 All you have to do is open the faucet a little more or a little 795 00:56:38,238 --> 00:56:41,769 less and it goes away. And, it's really an extremely 796 00:56:41,769 --> 00:56:45,853 loud and annoying resonance. If you take something as simple 797 00:56:45,853 --> 00:56:48,969 as a wine glass, which has a tremendous number 798 00:56:48,969 --> 00:56:52,084 of resonances, then I can make you listen to a 799 00:56:52,084 --> 00:56:55,615 well-known resonance, which is by rubbing the rim of 800 00:56:55,615 --> 00:56:58,038 the glass. When I rub the rim of the 801 00:56:58,038 --> 00:57:01,776 glass, I'm not exciting it at one particular frequency, 802 00:57:01,776 --> 00:57:06,000 and certainly not at a resonance frequency. 803 00:57:06,000 --> 00:57:09,229 I am exciting it at lots and lots of frequencies. 804 00:57:09,229 --> 00:57:12,189 I dump on it a whole spectrum of frequencies. 805 00:57:12,189 --> 00:57:15,755 But the glass is mean. It just picks out the one which 806 00:57:15,755 --> 00:57:18,715 is its resonance. That's where it builds up a 807 00:57:18,715 --> 00:57:21,877 large value for [A?]. It ignores all the others. 808 00:57:21,877 --> 00:57:25,577 And that's why I can make it resonate at that particular 809 00:57:25,577 --> 00:57:28,000 frequency. Listen to it. 810 00:57:28,000 --> 00:57:34,000 811 00:57:34,000 --> 00:57:36,876 This is not one frequency what I'm doing. 812 00:57:36,876 --> 00:57:40,759 It has nothing to do with the time for me to go around. 813 00:57:40,759 --> 00:57:43,707 It's a very high pitch. It's about 420 Hz. 814 00:57:43,707 --> 00:57:48,238 So, the rubbing is like dumping a spectrum of frequencies on it. 815 00:57:48,238 --> 00:57:50,898 And it selects what it likes the most. 816 00:57:50,898 --> 00:57:55,141 When I was a student I remember we often had an after dinner 817 00:57:55,141 --> 00:57:57,226 speaker. We had dinners at the 818 00:57:57,226 --> 00:58:00,534 fraternity, and we had an after dinner speaker. 819 00:58:00,534 --> 00:58:04,130 And more often than not we didn't like after dinner 820 00:58:04,130 --> 00:58:08,121 speaker. We didn't like the speech. 821 00:58:08,121 --> 00:58:10,650 And so, we made that very clear. 822 00:58:10,650 --> 00:58:14,402 And the way we do that is all our wine glasses, 823 00:58:14,402 --> 00:58:17,421 an enormous sound in that dining hall. 824 00:58:17,421 --> 00:58:21,010 And the speaker very quickly got the message, 825 00:58:21,010 --> 00:58:24,273 of course. That is an enormous sound that 826 00:58:24,273 --> 00:58:28,026 you can generate. And most of these wineglasses 827 00:58:28,026 --> 00:58:32,186 were roughly the same. So, it was always a tone that 828 00:58:32,186 --> 00:58:37,000 was loud and clear and almost one frequency. 829 00:58:37,000 --> 00:58:41,388 You've seen footage lately of the storms, three storms in a 830 00:58:41,388 --> 00:58:43,960 row. And you must remember sometime 831 00:58:43,960 --> 00:58:48,121 that you saw a traffic sign. You see a pole and then the 832 00:58:48,121 --> 00:58:51,375 traffic sign. And then even though there was 833 00:58:51,375 --> 00:58:55,763 some kind of a crazy when going, the traffic sign goes like 834 00:58:55,763 --> 00:59:00,000 this: all resonance frequencies. It can break. 835 00:59:00,000 --> 00:59:03,611 Even though the wind appears to be relatively steady, 836 00:59:03,611 --> 00:59:07,291 the wind then generates in a way the whole spectrum of 837 00:59:07,291 --> 00:59:11,111 frequencies and this traffic sign picks out the one that 838 00:59:11,111 --> 00:59:13,958 likes the most. And then it goes nuts at a 839 00:59:13,958 --> 00:59:16,666 frequency that is a resonance frequency. 840 00:59:16,666 --> 00:59:20,069 And, resonances can become destructive, of course, 841 00:59:20,069 --> 00:59:22,291 if these amplitudes are too high. 842 00:59:22,291 --> 00:59:25,972 Then things can break down. You may have noticed I was 843 00:59:25,972 --> 00:59:30,000 worried here when amplitude became so large that the spring 844 00:59:30,000 --> 00:59:35,000 might even break or the car might jump off that track. 845 00:59:35,000 --> 00:59:37,788 And, of course, a classic example of this 846 00:59:37,788 --> 00:59:41,832 district of resonance is the destruction of the very famous 847 00:59:41,832 --> 00:59:46,084 bridge in this country in 1940. The [common arrow is bridge?], 848 00:59:46,084 --> 00:59:48,803 Washington State, was destroyed by wind. 849 00:59:48,803 --> 00:59:52,638 And this bridge had many different resonant frequencies: 850 00:59:52,638 --> 00:59:55,008 won't like this and some like this. 851 00:59:55,008 --> 00:59:59,052 And depending upon the wind strengths, different resonances 852 00:59:59,052 --> 1:00:01,910 were excited at different moments in time, 853 1:00:01,910 --> 1:00:07,000 but that ultimately led to the destruction of the bridge. 854 1:00:07,000 --> 1:00:10,990 Now, most of you have seen this movie, but just for the few that 855 1:00:10,990 --> 1:00:13,714 haven't I really want you to see this movie. 856 1:00:13,714 --> 1:00:17,515 You cannot go to MIT and not have seen the destruction of the 857 1:00:17,515 --> 1:00:21,000 [Coma Bridge?] movie. [SOUND OFF/THEN ON] 858 1:00:21,000 --> 1:00:29,000 859 1:00:29,000 --> 1:00:32,829 There are countries where soldiers are not allowed to 860 1:00:32,829 --> 1:00:36,069 cross a bridge when they're marching in step. 861 1:00:36,069 --> 1:00:39,825 That's the case in the Netherlands, my home country. 862 1:00:39,825 --> 1:00:43,139 It's also the case in many European countries. 863 1:00:43,139 --> 1:00:46,895 The story has it that in England I think more than a 864 1:00:46,895 --> 1:00:51,313 hundred years ago when soldiers went over the bridge and step 865 1:00:51,313 --> 1:00:55,437 that the bridge collapsed. Whether that was the result of 866 1:00:55,437 --> 1:00:57,794 the soldiers, we will never know. 867 1:00:57,794 --> 1:01:01,182 But in any case, from those days can order that 868 1:01:01,182 --> 1:01:07,000 soldiers have to go out of step before they cross the bridge. 869 1:01:07,000 --> 1:01:12,544 There is a rumor that most of you have heard that there are 870 1:01:12,544 --> 1:01:18,566 women who are capable of singing with such a loud voice they can 871 1:01:18,566 --> 1:01:23,058 break a wine glass. They have to tune exactly at 872 1:01:23,058 --> 1:01:28,889 the resonance frequency of the wineglass, and they come out at 873 1:01:28,889 --> 1:01:33,000 a huge volume and the glass breaks. 874 1:01:33,000 --> 1:01:36,732 I don't believe it, but it's a rumor. 875 1:01:36,732 --> 1:01:42,849 There was a commercial many years ago, some of you may never 876 1:01:42,849 --> 1:01:45,441 have seen it, for Memorex. 877 1:01:45,441 --> 1:01:51,869 Memorex was a special tape for audio tape recorders this guy is 878 1:01:51,869 --> 1:01:57,053 going to a concert. And there is his woman singing, 879 1:01:57,053 --> 1:02:00,993 loud voice, beautiful frequency: bingo. 880 1:02:00,993 --> 1:02:06,484 The glass breaks. He comes home and he tells his 881 1:02:06,484 --> 1:02:09,714 wife the story. Well, his wife was smart 882 1:02:09,714 --> 1:02:12,696 enough, of course, not to believe it. 883 1:02:12,696 --> 1:02:16,257 And he says, well, it just so happens that I 884 1:02:16,257 --> 1:02:21,309 recorded it on my Memorex tape. So, he plays the tape at home, 885 1:02:21,309 --> 1:02:25,533 and at the moment that the woman's voice goes crack, 886 1:02:25,533 --> 1:02:28,929 what happens? The glasses break at home in 887 1:02:28,929 --> 1:02:32,748 his cabinet. So much for the physics of 888 1:02:32,748 --> 1:02:36,984 Memorex, because you can imagine that the resonance frequency of 889 1:02:36,984 --> 1:02:40,885 the glasses at home were very different than the glasses in 890 1:02:40,885 --> 1:02:43,373 the orchestra. So, it's all a swindle, 891 1:02:43,373 --> 1:02:46,937 but that's, of course, what commercials are all about. 892 1:02:46,937 --> 1:02:49,492 [LAUGHTER] What is now the bottom line? 893 1:02:49,492 --> 1:02:53,258 The bottom line was that if you buy Memorex, these tapes, 894 1:02:53,258 --> 1:02:57,091 then the reproduction is so perfect that you can even take 895 1:02:57,091 --> 1:02:59,579 it home and you see the glasses break. 896 1:02:59,579 --> 1:03:02,000 [SOUND OFF/THEN ON] 897 1:03:02,000 --> 1:03:07,000 898 1:03:07,000 --> 1:03:10,884 This brings up, now, the $64 million question, 899 1:03:10,884 --> 1:03:14,597 and that is, can it be done or can it not be 900 1:03:14,597 --> 1:03:17,446 done? So, any women in my audience 901 1:03:17,446 --> 1:03:21,762 you want to give it a try? That would be wonderful. 902 1:03:21,762 --> 1:03:25,733 [LAUGHTER] We've asked ourselves that question. 903 1:03:25,733 --> 1:03:31,000 Can this be done or can this not be done by person? 904 1:03:31,000 --> 1:03:35,231 And I think we came to the conclusion that a person alone 905 1:03:35,231 --> 1:03:39,688 without all kinds of electronic equipment would not do that. 906 1:03:39,688 --> 1:03:44,071 And I still believe that today. But, Professor Michael Felt 907 1:03:44,071 --> 1:03:48,679 [SP?] here at MIT with one of his graduate students many years 908 1:03:48,679 --> 1:03:52,835 ago developed some powerful equipment which you see here 909 1:03:52,835 --> 1:03:57,217 which was designed to make an attempt to break a wineglass. 910 1:03:57,217 --> 1:04:00,391 It doesn't always work, but it works often, 911 1:04:00,391 --> 1:04:05,000 the idea being then that here is the wineglass. 912 1:04:05,000 --> 1:04:09,763 It's almost a carbon copy of this one when they finally made 913 1:04:09,763 --> 1:04:14,366 this to work at near 440 Hz. He went to Crate & Barrel and 914 1:04:14,366 --> 1:04:18,242 asked, how many of these one glasses do you have? 915 1:04:18,242 --> 1:04:22,198 They said we have 5,000. And they bought all 5,000 916 1:04:22,198 --> 1:04:26,881 [LAUGHTER] because it's not obvious if you have to [check?] 917 1:04:26,881 --> 1:04:31,000 the glasses that it will ever work again. 918 1:04:31,000 --> 1:04:34,116 And so, here's one of those glasses. 919 1:04:34,116 --> 1:04:38,835 Here is a loudspeaker. The sound comes from this side, 920 1:04:38,835 --> 1:04:43,821 and what is nice about this arrangement: that we can make 921 1:04:43,821 --> 1:04:48,986 you see the distortion of the glass in this resonance mode. 922 1:04:48,986 --> 1:04:53,082 The glass oscillates like this, goes from oval, 923 1:04:53,082 --> 1:04:56,020 to circular, to oval, to circular. 924 1:04:56,020 --> 1:05:01,184 And, the way we can make you see that it is by strobing the 925 1:05:01,184 --> 1:05:06,527 glass at a frequency which is a little bit different from the 926 1:05:06,527 --> 1:05:11,125 frequency of the sound. Think about it: 927 1:05:11,125 --> 1:05:15,474 two frequencies almost the same give you a beat phenomenon. 928 1:05:15,474 --> 1:05:20,050 So, what that comes down to is you see the motion of the glass 929 1:05:20,050 --> 1:05:22,750 very slowly. And then, if we had that 930 1:05:22,750 --> 1:05:26,349 resonance frequency, we will increase the volume. 931 1:05:26,349 --> 1:05:30,324 And then maybe it will break. Now, I have to warn you, 932 1:05:30,324 --> 1:05:34,525 the sound will be unbearably high, and so for some of you 933 1:05:34,525 --> 1:05:38,500 here in the front row, you may even want to cover your 934 1:05:38,500 --> 1:05:42,290 ears. Or, you may want to move back. 935 1:05:42,290 --> 1:05:45,648 But, be careful, this is really an enormously 936 1:05:45,648 --> 1:05:48,625 strong signal that you're going to hear. 937 1:05:48,625 --> 1:05:52,366 So, I'll be careful. You can move back if you want 938 1:05:52,366 --> 1:05:54,122 to. I have hearing aids, 939 1:05:54,122 --> 1:05:57,786 as you have noticed, and I have the option that I 940 1:05:57,786 --> 1:06:02,307 can turn them off. But in spite of that, 941 1:06:02,307 --> 1:06:08,131 I'm not deaf without them. I will still cover my ears. 942 1:06:08,131 --> 1:06:14,285 So, let me first give you the light setting that we want, 943 1:06:14,285 --> 1:06:20,439 so we make it a little dark. And then I will show you the 944 1:06:20,439 --> 1:06:24,175 glass. I have to change the setting 945 1:06:24,175 --> 1:06:26,813 here. There is the glass. 946 1:06:26,813 --> 1:06:32,143 There's no sounds. It's not oscillating. 947 1:06:32,143 --> 1:06:38,167 And now I turn on the speaker. So, I am now going to turn my 948 1:06:38,167 --> 1:06:41,433 hearing aids off and put this on. 949 1:06:41,433 --> 1:06:45,109 And I'm going to increase the volume. 950 1:06:45,109 --> 1:06:50,723 And if it doesn't want to break, I will change the sound 951 1:06:50,723 --> 1:06:56,542 frequency little bit to sweep over that resonance curve so 952 1:06:56,542 --> 1:07:01,851 that I get onto the maximum because of it's only off, 953 1:07:01,851 --> 1:07:07,568 the cue is so high as the system that if I'm a little bit 954 1:07:07,568 --> 1:07:14,000 off in frequency, then the amplitude will below. 955 1:07:14,000 --> 1:07:21,034 So, let's first see what happens when I increase the 956 1:07:21,034 --> 1:07:25,448 volume. Did I do something wrong? 957 1:07:25,448 --> 1:07:30,000 Yes, I changed the frequency. 958 1:07:30,000 --> 1:08:37,000 959 1:08:37,000 --> 1:08:39,000 Getting very close. 960 1:08:39,000 --> 1:08:56,000 961 1:08:56,000 --> 1:10:41,581 [APPLAUSE] I think I've convinced you that a woman 962 1:10:41,581 --> 1:12:07,770 cannot do this. I have some last words of 963 1:12:07,770 --> 1:13:44,732 wisdom for you, and that is falling in love is 964 1:13:44,732 --> 1:15:43,242 also a form of resonance. And the two can be destructive 965 1:15:43,242 --> 1:17:52,525 because it can break your heart. So try to remember that next 966 1:17:52,525 --> 1:18:03,298 time. Have a good weekend.