1 00:00:30,000 --> 00:00:33,966 I'm Walter Lewin. I will be your lecturer this 2 00:00:33,966 --> 00:00:36,964 term. Make sure you have a handout, 3 00:00:36,964 --> 00:00:41,636 and make sure you read it. It tells you everything you 4 00:00:41,636 --> 00:00:46,749 want to know about the course. The course is about waves and 5 00:00:46,749 --> 00:00:52,038 vibrations, about oscillations, periodic, and not so periodic 6 00:00:52,038 --> 00:00:55,123 events. And, when you look around in 7 00:00:55,123 --> 00:00:58,121 the world, you see them everywhere. 8 00:00:58,121 --> 00:01:02,000 For one thing: your heartbeat. 9 00:01:02,000 --> 00:01:07,272 That's a periodic oscillation, at least I hope that most of 10 00:01:07,272 --> 00:01:11,727 you it is periodic. Your breathing is some kind of 11 00:01:11,727 --> 00:01:15,727 a periodic motion. The blinking of your eyes, 12 00:01:15,727 --> 00:01:19,000 your daily routines, and your habits, 13 00:01:19,000 --> 00:01:21,363 your eating, your sleeping, 14 00:01:21,363 --> 00:01:24,000 taking a shower, your classes, 15 00:01:24,000 --> 00:01:29,000 and occasionally doing some work: all those are periodic 16 00:01:29,000 --> 00:01:32,377 actions. When you drink, 17 00:01:32,377 --> 00:01:37,428 I drink some orange juice. Notice as I tried to move the 18 00:01:37,428 --> 00:01:42,663 liquid down into my stomach that it's not a steady stream. 19 00:01:42,663 --> 00:01:46,795 But it's a periodic motion. Look at my throat. 20 00:01:46,795 --> 00:01:51,479 In fact, even if I don't want to swallow the liquid, 21 00:01:51,479 --> 00:01:56,346 but simply have a bottle of liquid and I turn it over, 22 00:01:56,346 --> 00:02:01,857 and we all know that the water doesn't come out like a steady 23 00:02:01,857 --> 00:02:07,000 stream, but it goes glop, glop, glop, glop. 24 00:02:07,000 --> 00:02:09,762 That's some kind of a periodic motion. 25 00:02:09,762 --> 00:02:14,168 I have here a toy which I use to entertain my dinner guests. 26 00:02:14,168 --> 00:02:17,080 Particularly, physicists is interesting. 27 00:02:17,080 --> 00:02:20,888 And, this liquid here, the idea is to get the liquid 28 00:02:20,888 --> 00:02:23,202 there. And then, the problem is, 29 00:02:23,202 --> 00:02:26,562 how can you do it in the fastest possible way? 30 00:02:26,562 --> 00:02:30,520 Well, if you turn it over, you'll see the phenomenon I 31 00:02:30,520 --> 00:02:32,984 just mentioned, which is the glop, 32 00:02:32,984 --> 00:02:37,083 glop, glop. It's not a steady stream. 33 00:02:37,083 --> 00:02:41,750 It's almost pathetic the way that it runs from one side to the other. 34 00:02:41,750 --> 00:02:44,916 It will take minutes before it's there. 35 00:02:44,916 --> 00:02:47,750 But, it can be done and 17 seconds. 36 00:02:47,750 --> 00:02:52,166 And, during the five minute intermission that we have, 37 00:02:52,166 --> 00:02:56,750 you may give that a try. And, I hope you won't break it, 38 00:02:56,750 --> 00:03:01,833 and see if any of you can think of a way that you can transfer 39 00:03:01,833 --> 00:03:07,637 the liquid in 17 seconds. You have breakfast in the 40 00:03:07,637 --> 00:03:13,544 morning, and you casually put your breakfast plate on the 41 00:03:13,544 --> 00:03:15,970 table. What do you hear? 42 00:03:15,970 --> 00:03:22,299 Some kind of a periodic motion. And, two things can happen to 43 00:03:22,299 --> 00:03:26,308 this plate. It can move as a musilly, I call this as a musilly 44 00:03:26,308 --> 00:03:32,478 because I'm an astronomer. But, it can also wobble. 45 00:03:32,478 --> 00:03:38,364 In fact, something can wobble without moving as a musilly, 46 00:03:38,364 --> 00:03:43,733 and something can move as a musilly without wobbling. 47 00:03:43,733 --> 00:03:46,418 In this case, it does both. 48 00:03:46,418 --> 00:03:52,510 And, a fabulous example of that is what's called the Euler's 49 00:03:52,510 --> 00:03:58,293 disk, which is a metal disc, you'll see it shortly there, 50 00:03:58,293 --> 00:04:04,076 and this metal disk we are going to wobble in the similar 51 00:04:04,076 --> 00:04:10,381 way that I wobble plates. And then, we will follow its 52 00:04:10,381 --> 00:04:14,378 motion as a musilly, and the wobbling frequency. 53 00:04:14,378 --> 00:04:17,865 And, what is interesting, as you will see, 54 00:04:17,865 --> 00:04:22,457 that as [a musile?] motion, which has a certain period, 55 00:04:22,457 --> 00:04:27,134 that period gets longer and time, but the wobble motion, 56 00:04:27,134 --> 00:04:31,642 the frequency goes up. So, I'll start it here and then 57 00:04:31,642 --> 00:04:37,000 I'll show it to you in a way that is more appealing. 58 00:04:37,000 --> 00:04:41,371 And you can follow that. It's an amazing toy to work out 59 00:04:41,371 --> 00:04:44,312 to physics. It's very, very difficult. 60 00:04:44,312 --> 00:04:48,684 I was told that Professor Wilczek at MIT once gave a 61 00:04:48,684 --> 00:04:53,612 one-hour lecture exclusively on the explanation of this Euler's 62 00:04:53,612 --> 00:04:56,474 disk. So, try to see this as a musile 63 00:04:56,474 --> 00:04:59,256 motion. It will become clearer as it 64 00:04:59,256 --> 00:05:02,912 slows down further. You may be able to hear the 65 00:05:02,912 --> 00:05:06,489 wobble motion. I will hold my microphone close 66 00:05:06,489 --> 00:05:10,322 up. Can you hear it? 67 00:05:10,322 --> 00:05:17,000 Very high frequency already. Did you hear it? 68 00:05:17,000 --> 00:05:24,000 69 00:05:24,000 --> 00:05:29,773 It's quite amazing, isn't it, when you look at 70 00:05:29,773 --> 00:05:33,237 this? The wobbling frequency 71 00:05:33,237 --> 00:05:40,294 increases quite rapidly. Look as the musile motion slows 72 00:05:40,294 --> 00:05:46,581 down, and how the frequency of the wobble goes up. 73 00:05:46,581 --> 00:05:53,766 And now it comes to a stop. That's a very difficult piece 74 00:05:53,766 --> 00:05:58,000 of physics right there. 75 00:05:58,000 --> 00:06:12,000 76 00:06:12,000 --> 00:06:16,542 If you take a tennis ball, this is a super ball, 77 00:06:16,542 --> 00:06:20,408 and you bounce it. This is called student 78 00:06:20,408 --> 00:06:25,531 involvement, thank you. Then you also get some kind of 79 00:06:25,531 --> 00:06:31,137 the periodic motion whereby, again, the frequency increases 80 00:06:31,137 --> 00:06:36,453 just like in the case of the Euler desk at the breakfast 81 00:06:36,453 --> 00:06:40,366 plate. Here's an object that is 82 00:06:40,366 --> 00:06:45,985 floating in a liquid in water. And if I push that a little 83 00:06:45,985 --> 00:06:49,732 farther in and I let it go, it wobbles. 84 00:06:49,732 --> 00:06:55,450 And, that is a very unique frequency that you'll be able to 85 00:06:55,450 --> 00:06:59,887 calculate in 8.03, a very unique period of one 86 00:06:59,887 --> 00:07:06,000 complete oscillation as this object goes up and down. 87 00:07:06,000 --> 00:07:10,447 Even winds, steady winds, can generate periodic or almost 88 00:07:10,447 --> 00:07:13,576 periodic motions, which all of you have 89 00:07:13,576 --> 00:07:15,964 experienced. You walk outside, 90 00:07:15,964 --> 00:07:19,258 it's windy, and your hair goes like this. 91 00:07:19,258 --> 00:07:22,141 Your hair doesn't go flat like this. 92 00:07:22,141 --> 00:07:26,917 It always has this tendency, just like a flag does the same 93 00:07:26,917 --> 00:07:29,388 thing. If I generate wind here, 94 00:07:29,388 --> 00:07:34,082 and I have here some aluminum, now, you will see that this 95 00:07:34,082 --> 00:07:38,282 wind doesn't make the aluminum just go straight out, 96 00:07:38,282 --> 00:07:46,860 but it wobbles. There is a certain period to 97 00:07:46,860 --> 00:07:54,395 that. After work, if you want to have 98 00:07:54,395 --> 00:08:08,000 some fun, what is more fun than riding your own rocking horse? 99 00:08:08,000 --> 00:08:13,482 That's a periodic motion. Falling in love can be a 100 00:08:13,482 --> 00:08:18,069 periodic event. Now, don't do it too often 101 00:08:18,069 --> 00:08:23,552 because as most of you know it's quite exhausting. 102 00:08:23,552 --> 00:08:28,139 The motion of electrons, atoms, molecules, 103 00:08:28,139 --> 00:08:33,286 periodic and oscillatory, the motion of the moon, 104 00:08:33,286 --> 00:08:36,195 the planets, and the stars: 105 00:08:36,195 --> 00:08:43,161 periodic, oscillatory. Sound is a beautiful example. 106 00:08:43,161 --> 00:08:48,175 I produce sound by oscillating my vocal chords. 107 00:08:48,175 --> 00:08:51,772 I produce there by pressure waves. 108 00:08:51,772 --> 00:08:57,004 My vocal chords push on the air, suck on the air, 109 00:08:57,004 --> 00:09:03,000 push on the air, which produces a pressure wave. 110 00:09:03,000 --> 00:09:08,298 And, the pressure wave propagates out in the lecture 111 00:09:08,298 --> 00:09:13,909 hall, reaches your eardrum. Your eardrum starts to move 112 00:09:13,909 --> 00:09:18,272 back and forth, and your brains tell you 113 00:09:18,272 --> 00:09:23,259 that you hear a sound. I have here a tuning fork, 114 00:09:23,259 --> 00:09:29,701 which is designed so that if I give it a hit at the prongs move 115 00:09:29,701 --> 00:09:35,000 256 times per second. Recall that 256 Hz. 116 00:09:35,000 --> 00:09:38,472 A hertz is one oscillation per second. 117 00:09:38,472 --> 00:09:42,415 [SOUND PLAYS] And all of you can hear that. 118 00:09:42,415 --> 00:09:48,046 Pressure waves are generated. We will deal with them in 8.03. 119 00:09:48,046 --> 00:09:52,458 They travel through the air, reach your eardrum, 120 00:09:52,458 --> 00:09:55,462 and your eardrum starts to shake. 121 00:09:55,462 --> 00:10:01,000 This is a higher frequency, 440 Hz [SOUND PLAYS]. 122 00:10:01,000 --> 00:10:06,438 Most human beings can hear in the range from 20 Hz to 20 kHz. 123 00:10:06,438 --> 00:10:10,608 There are animals you can go way beyond 20 kHz. 124 00:10:10,608 --> 00:10:15,865 And, to be nice to you for the first time, this first lecture, 125 00:10:15,865 --> 00:10:18,856 I would like to test your hearing. 126 00:10:18,856 --> 00:10:21,666 And that will be free of charge. 127 00:10:21,666 --> 00:10:26,470 I'm not so much interested in knowing what your lowest 128 00:10:26,470 --> 00:10:30,459 frequency is, but what your highest frequency 129 00:10:30,459 --> 00:10:35,251 is. So, I'm going to generate here 130 00:10:35,251 --> 00:10:38,720 sounds. I will start with 100 Hz, 131 00:10:38,720 --> 00:10:45,548 and then we'll go up higher and higher, and we'll see where your 132 00:10:45,548 --> 00:10:50,101 hearing stops. So, let's start with 100 Hz. 133 00:10:50,101 --> 00:10:56,496 I'm not going to ask you who can hear it because clearly all 134 00:10:56,496 --> 00:11:00,506 of you can. Let's now go to kilohertz, 135 00:11:00,506 --> 00:11:05,779 1,000 hertz. Piece of cake, 136 00:11:05,779 --> 00:11:10,544 right? 2,000, no problem. 137 00:11:10,544 --> 00:11:16,698 I have to change, now, my scale. 138 00:11:16,698 --> 00:11:30,000 4,000, I didn't say that this was going to be a pleasant test. 139 00:11:30,000 --> 00:11:36,593 5,000, this is where the violins come in. 140 00:11:36,593 --> 00:11:41,703 6,000. Anyone in my audience who 141 00:11:41,703 --> 00:11:45,659 cannot hear 6,000? 7,000. 142 00:11:45,659 --> 00:11:52,747 Anyone in my audience who cannot hear 7,000? 143 00:11:52,747 --> 00:12:00,000 I cannot hear 7,000. I hear nothing. 144 00:12:00,000 --> 00:12:04,582 With age, you lose ability to hear higher frequencies. 145 00:12:04,582 --> 00:12:08,472 You all will experience that in your lifetime. 146 00:12:08,472 --> 00:12:12,968 You won't escape that. Now, for some people lose more 147 00:12:12,968 --> 00:12:16,340 than that. I cannot hear above 6,000 Hz. 148 00:12:16,340 --> 00:12:19,279 I hear nothing. OK, 10,000, 12,000, 149 00:12:19,279 --> 00:12:22,564 14,000. Now, I want to see hands if you 150 00:12:22,564 --> 00:12:26,801 cannot hear it any longer. Who cannot hear 14,000? 151 00:12:26,801 --> 00:12:31,886 Don't be ashamed of it. It's not your fault. 152 00:12:31,886 --> 00:12:35,283 14,000. All right, we're slowly going 153 00:12:35,283 --> 00:12:38,113 up. 15,000, who cannot hear it? 154 00:12:38,113 --> 00:12:42,641 Raise your hands. Ah, Professor [UNINTELLIGIBLE], 155 00:12:42,641 --> 00:12:48,018 you're also getting older. 16,000, who cannot hear 16,000? 156 00:12:48,018 --> 00:12:53,207 Of course, the ones who have already raised their hands, 157 00:12:53,207 --> 00:12:56,981 you don't have to raise your hands again. 158 00:12:56,981 --> 00:13:01,156 16,000, who cannot? 17,000? 159 00:13:01,156 --> 00:13:06,443 18,000. OK, so now we're going to 160 00:13:06,443 --> 00:13:12,721 change it. Now I want you to raise your 161 00:13:12,721 --> 00:13:21,643 hands if you can hear it. And so, I first go to 20,000, 162 00:13:21,643 --> 00:13:23,791 19,000. Sorry. 163 00:13:23,791 --> 00:13:29,078 I was only off by a factor of 10. 164 00:13:29,078 --> 00:13:34,695 19,000, who can hear it? Fantastic. 165 00:13:34,695 --> 00:13:39,923 20,000. 21,000. 166 00:13:39,923 --> 00:13:50,637 You see how it cut off? It's very sharp. 167 00:13:50,637 --> 00:13:55,307 22,000. Very good. 168 00:13:55,307 --> 00:13:59,153 23,000. 25,000. 169 00:13:59,153 --> 00:14:05,445 27,000. Some of you have amazing ears 170 00:14:05,445 --> 00:14:08,903 because I already turned it off at 21,000. 171 00:14:08,903 --> 00:14:10,000 [LAUGHTER] 172 00:14:10,000 --> 00:14:21,000 173 00:14:21,000 --> 00:14:27,000 All right, key absolutely key in this course will be simple 174 00:14:27,000 --> 00:14:33,333 harmonic oscillations because they are extremely common in 175 00:14:33,333 --> 00:14:37,444 nature. A simple harmonic oscillation, 176 00:14:37,444 --> 00:14:42,222 and you've seen this, of course, in 8.01, can be 177 00:14:42,222 --> 00:14:46,111 written as follows: x equals x zero, 178 00:14:46,111 --> 00:14:52,000 cosine (omega T plus phi). You can write a sign here if 179 00:14:52,000 --> 00:14:56,000 you want to. X zero is the amplitude. 180 00:14:56,000 --> 00:15:03,000 That's the largest displacement from equilibrium. 181 00:15:03,000 --> 00:15:09,198 Omega is the angular frequency. Omega, which we expressed in 182 00:15:09,198 --> 00:15:13,610 terms of radians per second, the period, T, 183 00:15:13,610 --> 00:15:19,599 2 pi divided by omega is an expressed in terms of seconds, 184 00:15:19,599 --> 00:15:24,642 and the frequency, F, which is one over T is what 185 00:15:24,642 --> 00:15:30,000 we call hertz: number of cycles per second. 186 00:15:30,000 --> 00:15:38,587 Do not confuse omega with F. There is a factor of 2 pi 187 00:15:38,587 --> 00:15:44,907 difference. If I have a uniform circular 188 00:15:44,907 --> 00:15:53,657 motion and I project that uniform circular motion on to 189 00:15:53,657 --> 00:16:02,569 any line on the blackboard, then I get a simple harmonic 190 00:16:02,569 --> 00:16:07,840 motion. So, I take for simplicity just 191 00:16:07,840 --> 00:16:12,291 this horizontal line. But I could take any other 192 00:16:12,291 --> 00:16:15,132 line. And let's call this the x 193 00:16:15,132 --> 00:16:18,825 direction. And let this point be X zero. 194 00:16:18,825 --> 00:16:23,181 And, I take an object which is rotating around. 195 00:16:23,181 --> 00:16:26,685 Here is the object. It's going around, 196 00:16:26,685 --> 00:16:33,275 uniform circular motion. If I project this onto the 197 00:16:33,275 --> 00:16:40,834 x-axis, and this angle is theta, then this position here is x 198 00:16:40,834 --> 00:16:47,007 zero cosine theta. And, if I make theta a function 199 00:16:47,007 --> 00:16:54,692 of prime, theta equals omega T. This omega is what we call not 200 00:16:54,692 --> 00:16:59,732 angular frequency, but we call it angular 201 00:16:59,732 --> 00:17:04,516 velocity. It's very awkward in physics 202 00:17:04,516 --> 00:17:10,269 that we have the same symbol for angular velocity and for angular 203 00:17:10,269 --> 00:17:12,337 frequency. In this case, 204 00:17:12,337 --> 00:17:16,921 it happens to be the same numerically because it's a 205 00:17:16,921 --> 00:17:20,606 uniform circular motion. That's excellent. 206 00:17:20,606 --> 00:17:25,460 So, now you see that x zero then becomes cosine omega T 207 00:17:25,460 --> 00:17:30,314 because the two are the same. I do not have to call the 208 00:17:30,314 --> 00:17:36,083 position T equals zero here. I can choose T equals zero 209 00:17:36,083 --> 00:17:38,647 anywhere along the circumference, 210 00:17:38,647 --> 00:17:42,012 and that introduces, then, phase angle phi. 211 00:17:42,012 --> 00:17:44,737 We call that the initial condition. 212 00:17:44,737 --> 00:17:49,464 So, x zero is the amplitude. Omega is the angular frequency, 213 00:17:49,464 --> 00:17:53,951 and phi has to be adjusted so that at time T equals zero, 214 00:17:53,951 --> 00:17:59,000 you get the right angle. You get the right position. 215 00:17:59,000 --> 00:18:04,000 216 00:18:04,000 --> 00:18:10,171 An easy example of a simple harmonic motion is a spring 217 00:18:10,171 --> 00:18:13,714 system. If I have here a spring, 218 00:18:13,714 --> 00:18:20,114 and this is in the relaxed position, a spring constant is 219 00:18:20,114 --> 00:18:24,685 K, the mass is M, and x equals zero here, 220 00:18:24,685 --> 00:18:30,628 and I bring it further out, I bring it to a position, 221 00:18:30,628 --> 00:18:34,645 X. Then there is a spring force 222 00:18:34,645 --> 00:18:38,612 that wants to drive it back to equilibrium. 223 00:18:38,612 --> 00:18:43,052 It's a restoring force. That's the spring force. 224 00:18:43,052 --> 00:18:47,398 That's arbitrarily called this direction force. 225 00:18:47,398 --> 00:18:52,594 The spring force we call minus KX, minus because if X is 226 00:18:52,594 --> 00:18:57,695 positive, then this force is in the opposite direction. 227 00:18:57,695 --> 00:19:03,080 If the mass of the spring can be ignored, if it is negligibly 228 00:19:03,080 --> 00:19:08,465 small compared to the mass of the object, I can write down 229 00:19:08,465 --> 00:19:13,000 Newton's Second Law: F equals MA. 230 00:19:13,000 --> 00:19:17,676 You may remember that from the good old days. 231 00:19:17,676 --> 00:19:22,671 And so, MA is MX double dot. It is now minus KX. 232 00:19:22,671 --> 00:19:27,560 It's really a vector notation, but since it's a 233 00:19:27,560 --> 00:19:32,980 one-dimensional problem, the minus takes care of the 234 00:19:32,980 --> 00:19:38,359 directions. And so, I can massage this a 235 00:19:38,359 --> 00:19:45,799 little further and I can write this as x double dot plus K over 236 00:19:45,799 --> 00:19:51,799 M times X equals zero. And, what is the solution to 237 00:19:51,799 --> 00:19:57,680 this differential equation? This is a differential 238 00:19:57,680 --> 00:20:04,195 equation: x double dot and x. This is the solution. 239 00:20:04,195 --> 00:20:10,048 It's a simple harmonic motion, provided that omega is the 240 00:20:10,048 --> 00:20:15,588 square root of K over M. So, I advise you to take this 241 00:20:15,588 --> 00:20:22,174 function, substitute it in here, and you will see that out pops, 242 00:20:22,174 --> 00:20:28,132 yes, you can satisfy this equation, provided that omega is 243 00:20:28,132 --> 00:20:33,916 the square root of K over M. Notice, which is not so 244 00:20:33,916 --> 00:20:37,750 intuitive, but this angular frequency of omega, 245 00:20:37,750 --> 00:20:41,416 and therefore also the period of oscillation, 246 00:20:41,416 --> 00:20:45,333 2 pi divided by omega, is independent of x zero. 247 00:20:45,333 --> 00:20:49,666 So, it's independent of how far and move it away from 248 00:20:49,666 --> 00:20:52,416 equilibrium. If I move it far out, 249 00:20:52,416 --> 00:20:57,666 it will take the same amount of time for one oscillation than if 250 00:20:57,666 --> 00:21:02,000 I move it out a teeny-weeny little bit. 251 00:21:02,000 --> 00:21:05,780 Not so intuitive. So, omega is independent of my 252 00:21:05,780 --> 00:21:09,801 initial conditions. It's independent on how I start 253 00:21:09,801 --> 00:21:13,260 the experiment off. It's independent of phi. 254 00:21:13,260 --> 00:21:16,879 It's independent of what I call T equals zero. 255 00:21:16,879 --> 00:21:21,061 Nature doesn't give a damn what I call T equals zero. 256 00:21:21,061 --> 00:21:24,198 Nature has one answer for the frequency. 257 00:21:24,198 --> 00:21:28,541 That's only determined by K and by M, not by my initial 258 00:21:28,541 --> 00:21:34,795 conditions, not so intuitive. If I take the same spring, 259 00:21:34,795 --> 00:21:40,817 and if I hang the spring vertically, there is the spring, 260 00:21:40,817 --> 00:21:47,053 due to gravity the object will come to a halt equilibrium a 261 00:21:47,053 --> 00:21:49,526 little lower, obviously. 262 00:21:49,526 --> 00:21:55,978 If now I displace it from this equilibrium position and I let 263 00:21:55,978 --> 00:21:59,741 it oscillate, I get exactly the same 264 00:21:59,741 --> 00:22:04,236 frequency. Maybe that's not so intuitive 265 00:22:04,236 --> 00:22:07,089 either. And, you can work that out for 266 00:22:07,089 --> 00:22:09,402 yourself. It's an 8.01 problem. 267 00:22:09,402 --> 00:22:13,874 What it means is that you can define this as X equals zero, 268 00:22:13,874 --> 00:22:18,038 ignore gravity completely, and set up your differential 269 00:22:18,038 --> 00:22:20,737 equation as if there was no gravity. 270 00:22:20,737 --> 00:22:24,361 And this x equals zero. So, you offset it over a 271 00:22:24,361 --> 00:22:27,677 distance, X, from that equilibrium position. 272 00:22:27,677 --> 00:22:31,069 You only allow for a spring force minus KX, 273 00:22:31,069 --> 00:22:35,515 and everything works. And, of course, 274 00:22:35,515 --> 00:22:40,363 you should be able to prove that that is correct. 275 00:22:40,363 --> 00:22:45,717 If a spring oscillates in the simple harmonic fashion, 276 00:22:45,717 --> 00:22:51,474 and we have such a spring here, Marcos, if you can do me a 277 00:22:51,474 --> 00:22:56,424 favor and get it up here, then I should be able to 278 00:22:56,424 --> 00:23:01,979 demonstrate that a uniform circular motion, projected on 279 00:23:01,979 --> 00:23:06,650 the wall, we call shadow projection, 280 00:23:06,650 --> 00:23:09,913 should be able, thank you Marcos, 281 00:23:09,913 --> 00:23:14,705 to be able to have the same motion as my spring, 282 00:23:14,705 --> 00:23:19,701 provided, of course, that we very carefully make a 283 00:23:19,701 --> 00:23:25,615 period of oscillation of the spring exactly the same as the 284 00:23:25,615 --> 00:23:30,000 time for this object to go around. 285 00:23:30,000 --> 00:23:34,229 We then shadow projected on there, and then I will even try 286 00:23:34,229 --> 00:23:37,364 to release this one, this is very difficult, 287 00:23:37,364 --> 00:23:40,135 at the same time that this one is here. 288 00:23:40,135 --> 00:23:43,781 And what you'll see, then, you will see the uniform 289 00:23:43,781 --> 00:23:47,645 circular motion projected, becomes the simple harmonic 290 00:23:47,645 --> 00:23:51,218 motion, and you'll see the spring simple harmonic. 291 00:23:51,218 --> 00:23:55,593 And so, we'll try to do that, and shadow projection will make 292 00:23:55,593 --> 00:23:58,000 it a little darker. 293 00:23:58,000 --> 00:24:03,000 294 00:24:03,000 --> 00:24:10,000 And for that, I need some light here. 295 00:24:10,000 --> 00:24:19,000 296 00:24:19,000 --> 00:24:21,872 OK, and somebody already turned it off. 297 00:24:21,872 --> 00:24:26,180 So here, you see the spring. In there, you see this object 298 00:24:26,180 --> 00:24:30,337 which is rotating in a circle. But you think it's simple 299 00:24:30,337 --> 00:24:34,808 harmonic motion. And that's, of course, 300 00:24:34,808 --> 00:24:40,297 my objective. And so, now that is difficult. 301 00:24:40,297 --> 00:24:45,659 I will have to block you for a few seconds. 302 00:24:45,659 --> 00:24:51,659 I will try now to release this at the same time, 303 00:24:51,659 --> 00:24:58,936 and also the same amplitude. Boy, that wasn't my best day, 304 00:24:58,936 --> 00:25:00,212 was it? No. 305 00:25:00,212 --> 00:25:02,000 No. 306 00:25:02,000 --> 00:25:07,000 307 00:25:07,000 --> 00:25:09,917 Oh, this is perhaps the best I can do today. 308 00:25:09,917 --> 00:25:12,903 So, they don't go exactly next to each other, 309 00:25:12,903 --> 00:25:17,110 but you see they have the same periods, and they both represent 310 00:25:17,110 --> 00:25:20,842 simple harmonic oscillation, the spring, because we just 311 00:25:20,842 --> 00:25:23,692 calculated that. And, the projection of the 312 00:25:23,692 --> 00:25:26,000 uniform circular motion. 313 00:25:26,000 --> 00:25:35,000 314 00:25:35,000 --> 00:25:40,742 So, if we return to the spring, and maybe we should remove 315 00:25:40,742 --> 00:25:46,182 this, if we return to the spring, then we have a period 316 00:25:46,182 --> 00:25:52,328 for the spring system which is 2 pi times the square root of M 317 00:25:52,328 --> 00:25:56,055 over K. And, I want to bring this to a 318 00:25:56,055 --> 00:26:02,000 test, to a quantitative test. How accurate is this? 319 00:26:02,000 --> 00:26:06,063 I'm going to double the mass that I'm going to hang on that 320 00:26:06,063 --> 00:26:08,375 spring. We're going to measure the 321 00:26:08,375 --> 00:26:10,757 periods. And then, I want the mass, 322 00:26:10,757 --> 00:26:14,331 which is twice as high. I want that period to be the 323 00:26:14,331 --> 00:26:18,184 square root of two times higher because that's what this 324 00:26:18,184 --> 00:26:21,547 equation predicts. Now, whenever you want to do a 325 00:26:21,547 --> 00:26:25,611 measurement in physics whereby you want to compare numbers, 326 00:26:25,611 --> 00:26:29,815 so you have a certain goal in mind, a measurement without the 327 00:26:29,815 --> 00:26:35,000 uncertainty in the measurements is completely meaningless. 328 00:26:35,000 --> 00:26:40,962 You must know the accuracy of your measurement. 329 00:26:40,962 --> 00:26:45,240 So, M1 is 500 plus or minus 0.2 g. 330 00:26:45,240 --> 00:26:50,425 And, M2 is thousands plus or minus 0.2 g. 331 00:26:50,425 --> 00:26:57,037 That's the best we can do. That's an extremely small 332 00:26:57,037 --> 00:27:03,000 error. This is an error of only 0.04%. 333 00:27:03,000 --> 00:27:07,597 And this is only half as large. So, now comes the question, 334 00:27:07,597 --> 00:27:11,798 if I measure the periods of oscillation with the 500 g 335 00:27:11,798 --> 00:27:15,841 hanging on the spring, how accurately can I do that? 336 00:27:15,841 --> 00:27:19,012 On a good day, I can do it to 0.1 seconds 337 00:27:19,012 --> 00:27:21,152 accuracy. I have to start it, 338 00:27:21,152 --> 00:27:24,957 and I had to stop it. And if I do that ten times, 339 00:27:24,957 --> 00:27:29,000 obviously you'll get different answers. 340 00:27:29,000 --> 00:27:34,358 And, they vary by about a tenth of a second on a good day. 341 00:27:34,358 --> 00:27:37,649 Now, on a bad day, 2/10 of a second. 342 00:27:37,649 --> 00:27:43,384 I don't know whether today is a good day or whether it's a bad 343 00:27:43,384 --> 00:27:49,025 day, but lets it's in-between. So, let's say I can do to 0.15 344 00:27:49,025 --> 00:27:52,504 seconds, which I cannot guarantee you, 345 00:27:52,504 --> 00:27:56,452 but I'll try. So, I can measure the periods 346 00:27:56,452 --> 00:28:02,000 to plus or minus 0.15 seconds. This is with M1. 347 00:28:02,000 --> 00:28:06,197 However, I can get a very accurate measurement for time, 348 00:28:06,197 --> 00:28:09,402 for the period, if I oscillate if I make 10 349 00:28:09,402 --> 00:28:12,913 oscillations because if I make 10 oscillations, 350 00:28:12,913 --> 00:28:17,415 the error in T goes down by a factor of ten because the 0.15 351 00:28:17,415 --> 00:28:20,086 is 0.15. That's not going to change. 352 00:28:20,086 --> 00:28:23,063 So, I'm going to oscillate it ten times. 353 00:28:23,063 --> 00:28:27,794 And then, we're going to make a prediction about what we should 354 00:28:27,794 --> 00:28:33,591 measure for the higher mass. So, let's first measure the 355 00:28:33,591 --> 00:28:39,253 period of, this is the spring, and here is the 500 g plus or 356 00:28:39,253 --> 00:28:42,708 minus 0.2. I'm going to oscillate it. 357 00:28:42,708 --> 00:28:47,507 We already know that it's independent of amplitude. 358 00:28:47,507 --> 00:28:51,922 And then, I'm going to start it when it's down. 359 00:28:51,922 --> 00:28:56,817 That's the easiest for me. And, I will count to ten. 360 00:28:56,817 --> 00:29:02,000 You will count to ten, and then we'll stop. 361 00:29:02,000 --> 00:29:09,750 So, let's give it an oscillation. 362 00:29:09,750 --> 00:29:16,289 Yeah, one, two, three, four, 363 00:29:16,289 --> 00:29:23,312 five, six, seven, eight, nine, 364 00:29:23,312 --> 00:29:25,734 ten. 14.96. 365 00:29:25,734 --> 00:29:34,555 Let's write this down. We have 14.96. 366 00:29:34,555 --> 00:29:41,555 If you want to see whether this is a good day or whether this is 367 00:29:41,555 --> 00:29:45,000 a bad day, measure it again. 368 00:29:45,000 --> 00:29:50,000 369 00:29:50,000 --> 00:29:55,090 One, two, three, four, five, six, 370 00:29:55,090 --> 00:29:58,750 seven, eight, nine, ten. 371 00:29:58,750 --> 00:30:03,840 14.98. So, this is not a bad day. 372 00:30:03,840 --> 00:30:11,954 But, it's luck that I comes out so close, of course. 373 00:30:11,954 --> 00:30:20,386 So, now, we can make a prediction that ten times TM2 374 00:30:20,386 --> 00:30:28,181 must be 1.414, which is the square root of two 375 00:30:28,181 --> 00:30:35,576 times 14.97. OK, so I take the 1.414. 376 00:30:35,576 --> 00:30:41,836 I multiply that by 14.97, and I find 21.17. 377 00:30:41,836 --> 00:30:48,245 And of course, process to be multiplied also 378 00:30:48,245 --> 00:30:55,548 by 1.4, so that becomes plus or minus 0.2 seconds. 379 00:30:55,548 --> 00:31:03,000 That is a prediction. This is predict. 380 00:31:03,000 --> 00:31:10,210 And now comes the observation. This is a thrilling moment for 381 00:31:10,210 --> 00:31:16,939 you because what is at stake is the integrity of physics. 382 00:31:16,939 --> 00:31:23,188 And, this is going to be measured plus or minus 0.15, 383 00:31:23,188 --> 00:31:26,193 right? Every time I make a 384 00:31:26,193 --> 00:31:32,000 measurement, plus or minus 0.15. I'm nervous. 385 00:31:32,000 --> 00:31:36,000 386 00:31:36,000 --> 00:31:44,000 So, I'm going to add 500 g. Period will indeed increase. 387 00:31:44,000 --> 00:31:50,000 388 00:31:50,000 --> 00:31:54,000 I'm going to oscillate it. 389 00:31:54,000 --> 00:32:02,000 390 00:32:02,000 --> 00:32:07,812 Yeah. One, two, three, 391 00:32:07,812 --> 00:32:15,839 four, five, six, seven, eight, 392 00:32:15,839 --> 00:32:21,098 nine, yeah. Oh, boy. 393 00:32:21,098 --> 00:32:33,000 Oh, boy, what have I done? [LAUGHTER] 394 00:32:33,000 --> 00:32:36,853 What have I done? What have I done? What is it? 395 00:32:36,853 --> 00:32:40,155 20.52. We have a problem. 396 00:32:40,155 --> 00:32:47,036 Physics is not working. Do any one of you have an 397 00:32:47,036 --> 00:32:54,467 idea whether there's something wrong with the equation, 398 00:32:54,467 --> 00:33:03,000 or whether there's something wrong with Walter Lewin? 399 00:33:03,000 --> 00:33:07,400 Any idea? Come on, give it a try, 400 00:33:07,400 --> 00:33:13,587 the worst case your suggestion is not correct, 401 00:33:13,587 --> 00:33:16,750 yeah? Ah, you accuse me, 402 00:33:16,750 --> 00:33:21,424 right? [LAUGHTER] What's your name? 403 00:33:21,424 --> 00:33:27,474 [LAUGHTER] Questioning my 0.15. You say, man, 404 00:33:27,474 --> 00:33:33,525 you couldn't even do better than 0.4 seconds, 405 00:33:33,525 --> 00:33:37,623 maybe. And then, of course, 406 00:33:37,623 --> 00:33:41,113 the two would be consistent with each other. 407 00:33:41,113 --> 00:33:43,792 Thank you, very nice of you. Yeah? 408 00:33:43,792 --> 00:33:46,876 OK, now, that's a very good suggestion. 409 00:33:46,876 --> 00:33:50,123 In other words, you begin to think like a 410 00:33:50,123 --> 00:33:53,207 physicist. Now, you also thought like a 411 00:33:53,207 --> 00:33:56,941 physicist because, indeed, if my uncertainty is 412 00:33:56,941 --> 00:34:01,000 higher than 0.15, you could be right. 413 00:34:01,000 --> 00:34:04,721 Friction in this case, we will deal with friction 414 00:34:04,721 --> 00:34:08,598 later in the course, has such a negligibly small 415 00:34:08,598 --> 00:34:13,018 effect that it couldn't be measured to either one of these 416 00:34:13,018 --> 00:34:14,259 two. In any case, 417 00:34:14,259 --> 00:34:18,446 it's almost the same for both because the shape has not 418 00:34:18,446 --> 00:34:21,625 changed much. It's a very good suggestion; 419 00:34:21,625 --> 00:34:25,425 friction doesn't come near the proper explanation, 420 00:34:25,425 --> 00:34:27,751 but you tried. And that's good. 421 00:34:27,751 --> 00:34:31,668 One more try. Mass of the spring: 422 00:34:31,668 --> 00:34:35,797 we have said earlier; you may not have heard it, 423 00:34:35,797 --> 00:34:39,310 but I did say it, we can replay the tape, 424 00:34:39,310 --> 00:34:43,439 I can prove it to you; I said if the mass of the 425 00:34:43,439 --> 00:34:47,567 spring is negligible, then that is the equation. 426 00:34:47,567 --> 00:34:51,959 Now, what do we do when the mass cannot be ignored? 427 00:34:51,959 --> 00:34:56,351 That's not so easy. But, I requested some mandatory 428 00:34:56,351 --> 00:35:01,445 reading, and I'm sure all of you have done that before this 429 00:35:01,445 --> 00:35:05,662 lecture, and the mandatory reading was French, 430 00:35:05,662 --> 00:35:11,540 p 60-61 among others. And, French says that the mass 431 00:35:11,540 --> 00:35:16,705 of the spring itself is capital M, and if capital M divided by 432 00:35:16,705 --> 00:35:21,616 3 is substantially less than the mass at the end of the 433 00:35:21,616 --> 00:35:25,935 spring, then a very, very good approximation is that 434 00:35:25,935 --> 00:35:30,000 the period of oscillation is, then, this. 435 00:35:30,000 --> 00:35:40,795 And, he actually derives it [SOUND OFF/THEN ON] period is 436 00:35:40,795 --> 00:35:47,542 higher. So, we can bring this to the 437 00:35:47,542 --> 00:35:52,168 test now. In other words, 438 00:35:52,168 --> 00:36:02,000 the mass of the spring, we have weighed that. 439 00:36:02,000 --> 00:36:09,708 In our case it's 175.6 plus or minus 0.2 g. 440 00:36:09,708 --> 00:36:19,803 And so, M divided by three is 58.5 plus or minus 0.07 g. 441 00:36:19,803 --> 00:36:26,594 That's a very small error, by the way. 442 00:36:26,594 --> 00:36:34,455 It's a 0.1% error. And so, we can now do the 443 00:36:34,455 --> 00:36:40,398 following test. We can now take the ratio of 444 00:36:40,398 --> 00:36:45,235 these two and eliminate, thereby, K. 445 00:36:45,235 --> 00:36:52,422 So, we can write ten T. M2 divided by ten T M1 is now 446 00:36:52,422 --> 00:37:00,439 the square root of M2 plus M divided by three divided by M1 447 00:37:00,439 --> 00:37:07,391 plus M divided by three. And, that number is easy to 448 00:37:07,391 --> 00:37:10,314 calculate because M2, you know M1, 449 00:37:10,314 --> 00:37:14,831 you know these numbers, and I have calculated it for 450 00:37:14,831 --> 00:37:16,691 you. And, it is 1.377. 451 00:37:16,691 --> 00:37:21,297 And, your uncertainty is so small compared to my tiny 452 00:37:21,297 --> 00:37:25,725 uncertainty that I don't even have to allow for any 453 00:37:25,725 --> 00:37:29,622 uncertainty in that number because, remember, 454 00:37:29,622 --> 00:37:36,000 the uncertainties in these masses was of the order of 0.1%. 455 00:37:36,000 --> 00:37:41,420 Compare that with the uncertainty in the observations 456 00:37:41,420 --> 00:37:45,069 of the time which were closer to 1%. 457 00:37:45,069 --> 00:37:48,718 So, we can bring this now to a test. 458 00:37:48,718 --> 00:37:54,347 And all I must do now is multiply if I want to find out 459 00:37:54,347 --> 00:37:57,891 ten, TM2. And, I take 1.377, 460 00:37:57,891 --> 00:38:03,000 and I multiply it by TM1, times ten TM1. 461 00:38:03,000 --> 00:38:09,986 So, I take this number. And now, I'm really getting 462 00:38:09,986 --> 00:38:16,414 nervous, not joking. 14.97 multiplied by 1.377, 463 00:38:16,414 --> 00:38:22,423 that is 20.61. And, the uncertainty would be 464 00:38:22,423 --> 00:38:30,528 the same uncertainty as in there which is a 1% uncertainty. 465 00:38:30,528 --> 00:38:37,480 So, that is 0.2 seconds. This number you can now compare 466 00:38:37,480 --> 00:38:41,480 with this number on the button. Within the error of 467 00:38:41,480 --> 00:38:43,719 measurements, they now agree. 468 00:38:43,719 --> 00:38:48,119 This is what we observed, and this is what we predict if 469 00:38:48,119 --> 00:38:53,239 we apply the proper relation and take the mass of the spring into 470 00:38:53,239 --> 00:38:56,360 account. So, you see that physics works, 471 00:38:56,360 --> 00:39:01,000 except that this equation was too simple to be used for our 472 00:39:01,000 --> 00:39:06,076 observation. Notice, by the way, 473 00:39:06,076 --> 00:39:10,609 that this 1.414, in our case, 474 00:39:10,609 --> 00:39:15,142 is lower. All right, in 8.03, 475 00:39:15,142 --> 00:39:20,323 we will often, though not always, 476 00:39:20,323 --> 00:39:28,580 use complex notation. And, the reason why we do that 477 00:39:28,580 --> 00:39:37,000 is that it can at times simplify your life. 478 00:39:37,000 --> 00:39:42,937 And you are completely free to choose when you want to use it 479 00:39:42,937 --> 00:39:46,203 and when you don't want to use it. 480 00:39:46,203 --> 00:39:50,953 You can be the judge. So, let's talk a little bit 481 00:39:50,953 --> 00:39:56,000 about complex numbers. I start with a circle. 482 00:39:56,000 --> 00:40:01,000 483 00:40:01,000 --> 00:40:06,785 And this is the complex plane. The blackboard is a complex 484 00:40:06,785 --> 00:40:10,235 plane. That's quite a promotion for 485 00:40:10,235 --> 00:40:14,802 the blackboard. And here, I call this axis the 486 00:40:14,802 --> 00:40:18,964 real axis. So, all the real numbers are on 487 00:40:18,964 --> 00:40:22,617 this axis. And, let this be plus one, 488 00:40:22,617 --> 00:40:27,286 let this be minus one, and I call this axis the 489 00:40:27,286 --> 00:40:31,143 imaginary axis. So, this one is plus J, 490 00:40:31,143 --> 00:40:38,398 and this one is minus J. And, J is the square root of 491 00:40:38,398 --> 00:40:43,131 minus one. We don't call it I in general 492 00:40:43,131 --> 00:40:48,349 because I is, stands for the current, so we pick J. 493 00:40:48,349 --> 00:40:55,266 I now pick a position here, which now represents a complex 494 00:40:55,266 --> 00:41:00,000 number called this angle, theta. 495 00:41:00,000 --> 00:41:03,326 And I protect this. This is position Z, 496 00:41:03,326 --> 00:41:07,391 complex number. This is the real part of that 497 00:41:07,391 --> 00:41:11,548 complex number, and this is the imaginary part 498 00:41:11,548 --> 00:41:15,336 of a complex number. So, you can see that, 499 00:41:15,336 --> 00:41:19,864 indeed, Z can be written since this length is one, 500 00:41:19,864 --> 00:41:24,945 is the cosine of theta equals J times the sine of theta. 501 00:41:24,945 --> 00:41:27,994 So, does this part, which is real. 502 00:41:27,994 --> 00:41:34,000 And, this is the sine of theta because this is one. 503 00:41:34,000 --> 00:41:39,918 I have to multiply that by J. And this now, 504 00:41:39,918 --> 00:41:46,259 according to Euler, great mathematician Euler, 505 00:41:46,259 --> 00:41:54,150 after whom this disk was also mentioned, already in 1748, 506 00:41:54,150 --> 00:42:02,040 he proved that this was the same as E to the power of Jphi, 507 00:42:02,040 --> 00:42:07,561 sorry, theta. This equality is mind-boggling. 508 00:42:07,561 --> 00:42:11,527 And, when I saw this equality for the first time, 509 00:42:11,527 --> 00:42:15,658 I didn't believe it, number one, and I could hardly 510 00:42:15,658 --> 00:42:19,128 sleep at night because I couldn't prove it. 511 00:42:19,128 --> 00:42:22,598 See, I hadn't had any Taylor expansion yet. 512 00:42:22,598 --> 00:42:26,729 So, I couldn't prove it. Because my teacher in high 513 00:42:26,729 --> 00:42:30,447 school said, this is the case and I said, why? 514 00:42:30,447 --> 00:42:36,058 He said this is the way it is. But we now can prove this. 515 00:42:36,058 --> 00:42:40,097 You can do the Taylor expansion of the cosine theta, 516 00:42:40,097 --> 00:42:44,769 Taylor expansion of the sine theta, and the Taylor expansion 517 00:42:44,769 --> 00:42:48,807 of E to the power J theta, and it's exactly correct, 518 00:42:48,807 --> 00:42:52,766 not an approximation. So, why would we ever want to 519 00:42:52,766 --> 00:42:55,934 use this? Well, if you make this thing go 520 00:42:55,934 --> 00:43:00,131 around, going back to my uniform circular motion here, 521 00:43:00,131 --> 00:43:04,407 and if I make that point go around, and I only look at the 522 00:43:04,407 --> 00:43:09,000 real part, I have a simple harmonic motion. 523 00:43:09,000 --> 00:43:14,699 And so, if I change theta into omega T, then I get that Z 524 00:43:14,699 --> 00:43:20,601 equals the cosine of omega T plus J times the sine of omega 525 00:43:20,601 --> 00:43:25,995 T, the real part of which is a simple harmonic motion. 526 00:43:25,995 --> 00:43:30,473 And, of course, I'm not stuck to an amplitude 527 00:43:30,473 --> 00:43:35,858 of one. I can easily make the amplitude 528 00:43:35,858 --> 00:43:40,091 A times larger. And, of course, 529 00:43:40,091 --> 00:43:47,560 there's nothing wrong depending upon my initial conditions to 530 00:43:47,560 --> 00:43:51,045 have here a phase angle, phi. 531 00:43:51,045 --> 00:43:56,398 And, this, then, is A times E to the power J 532 00:43:56,398 --> 00:44:02,000 omega T plus phi according to Euler. 533 00:44:02,000 --> 00:44:06,500 So, what that means is that if you use this as your trial 534 00:44:06,500 --> 00:44:11,642 function to solve a differential equation, and you can manipulate 535 00:44:11,642 --> 00:44:14,616 this very easily, you could take first 536 00:44:14,616 --> 00:44:19,437 derivative, second derivative of exponentials extremely easy, 537 00:44:19,437 --> 00:44:23,776 and then when you're done, you take the real part of Z, 538 00:44:23,776 --> 00:44:26,669 and out pops X as a function of time. 539 00:44:26,669 --> 00:44:31,665 And, you're done. As I said, it's up to you when 540 00:44:31,665 --> 00:44:37,261 you want to use it next lecture. I will give you an example that 541 00:44:37,261 --> 00:44:42,325 it's clearly the way to go. I wouldn't even know how to do 542 00:44:42,325 --> 00:44:45,968 it any other way. But, often you do have a 543 00:44:45,968 --> 00:44:48,544 choice. So, we are interested, 544 00:44:48,544 --> 00:44:53,785 then, in the real part of that, which is then our acceptable 545 00:44:53,785 --> 00:44:56,717 solution. So, if we have a complex 546 00:44:56,717 --> 00:45:01,870 number, Z equals A plus JB, but we should always be able to 547 00:45:01,870 --> 00:45:08,000 write it as an the torqueamplitude times E to the power J theta. 548 00:45:08,000 --> 00:45:13,620 And then, the amplitude, A, is the square root of A 549 00:45:13,620 --> 00:45:19,802 squared plus B squared and tangent of theta is B over A. 550 00:45:19,802 --> 00:45:24,410 That follows immediately from that figure. 551 00:45:24,410 --> 00:45:30,368 And so, in problem set one you will get some chance to 552 00:45:30,368 --> 00:45:33,852 practice. It will give you a few 553 00:45:33,852 --> 00:45:39,664 interesting cases. And a classic case that all of 554 00:45:39,664 --> 00:45:44,814 you in your lifetime have to be able to do once is the very 555 00:45:44,814 --> 00:45:48,189 non-intuitive problem J to the power J. 556 00:45:48,189 --> 00:45:52,274 When I saw for the first time J to the power J, 557 00:45:52,274 --> 00:45:56,536 I said to myself, well, what on Earth can be more 558 00:45:56,536 --> 00:46:01,397 complex than J to the power J? But, it's real. 559 00:46:01,397 --> 00:46:05,496 It is not complex. And, you will wrestle with 560 00:46:05,496 --> 00:46:08,757 this. There is an infinite number of 561 00:46:08,757 --> 00:46:11,925 solutions. Not one, all of them are 562 00:46:11,925 --> 00:46:15,186 correct. And, I will help you little 563 00:46:15,186 --> 00:46:19,565 because the first time I want to be nice to you. 564 00:46:19,565 --> 00:46:25,062 But it's only the first time. I can also write J as E to the 565 00:46:25,062 --> 00:46:30,000 power J times pi over two; do you agree? 566 00:46:30,000 --> 00:46:35,954 Because it simply means that the angle theta is pi over two. 567 00:46:35,954 --> 00:46:40,091 It's here, so I end up here. But that's J. 568 00:46:40,091 --> 00:46:45,339 I'm not saying it is a very nice way of expressing J, 569 00:46:45,339 --> 00:46:48,871 but it is J. But not only is this J, 570 00:46:48,871 --> 00:46:53,412 I can also rotate an integer number times 360° 571 00:46:53,412 --> 00:46:58,761 whereby N zero, one, two, three rotate either 572 00:46:58,761 --> 00:47:06,285 clockwise or counterclockwise. And, it's again J because if I 573 00:47:06,285 --> 00:47:11,652 rotate 90°, it's J. But, if I rotate another 360°, 574 00:47:11,652 --> 00:47:15,814 it's again J, or if I rotate back 360°. 575 00:47:15,814 --> 00:47:21,290 And so, you see that this is also a way to write J. 576 00:47:21,290 --> 00:47:26,000 And, that will help you, believe me. 577 00:47:26,000 --> 00:47:32,000 578 00:47:32,000 --> 00:47:36,235 I will always have a five-minute break during this 85 579 00:47:36,235 --> 00:47:40,146 minute lecture so that you can stretch your legs. 580 00:47:40,146 --> 00:47:44,952 If you can manage to make it back and forth to the bathroom, 581 00:47:44,952 --> 00:47:47,803 that's fine but that's your problem. 582 00:47:47,803 --> 00:47:50,980 I will start exactly after five minutes. 583 00:47:50,980 --> 00:47:54,890 However, every Tuesday, during part of these five 584 00:47:54,890 --> 00:47:57,497 minutes we will have a mini quiz. 585 00:47:57,497 --> 00:48:01,000 It's really mini, this small. 586 00:48:01,000 --> 00:48:06,000 And we will collect it after the lecture, and you'll even get 587 00:48:06,000 --> 00:48:09,583 some credit for that. Before, that's only on 588 00:48:09,583 --> 00:48:13,416 Tuesdays, but not today. Before we go into this 589 00:48:13,416 --> 00:48:17,916 five-minute break today, I want you to see something so 590 00:48:17,916 --> 00:48:21,083 that you have something to think about. 591 00:48:21,083 --> 00:48:25,416 Believe me, it's healthy for an MIT student to sleep, 592 00:48:25,416 --> 00:48:30,000 but is also healthy sometimes to not sleep. 593 00:48:30,000 --> 00:48:33,898 Sleepless nights and worry, just the way that I had 594 00:48:33,898 --> 00:48:38,109 sleepless nights in high school about Euler's equation: 595 00:48:38,109 --> 00:48:41,384 it's healthy. The reason why that's healthy 596 00:48:41,384 --> 00:48:44,892 is because once you see the solution, you say, 597 00:48:44,892 --> 00:48:47,855 ah, of course, and you never forget it, 598 00:48:47,855 --> 00:48:51,832 whereas if someone tells you from the start you say, 599 00:48:51,832 --> 00:48:54,561 yeah, of course. And, you forget it. 600 00:48:54,561 --> 00:48:57,290 And the next day you don't remember. 601 00:48:57,290 --> 00:49:01,578 So, what I want you to see is a remarkable example of an 602 00:49:01,578 --> 00:49:06,335 oscillation that can be produced not by wind, as we have seen, 603 00:49:06,335 --> 00:49:12,297 but by heat and by cooling. I have here a nice pipe, 604 00:49:12,297 --> 00:49:16,127 and there is a grid here. I can touch it. 605 00:49:16,127 --> 00:49:19,957 I'm touching it now. That's all there is. 606 00:49:19,957 --> 00:49:23,978 It's an open pipe, and there's a grid here. 607 00:49:23,978 --> 00:49:29,627 And, when I heat that grid and cool it, somehow it generates 608 00:49:29,627 --> 00:49:32,978 110 Hz oscillation, a pressure wave, 609 00:49:32,978 --> 00:49:39,477 which you will be able to hear. And, I'll give you until the 610 00:49:39,477 --> 00:49:42,575 end of December, maybe mid-December, 611 00:49:42,575 --> 00:49:47,000 to come up with a solution why it's doing that. 612 00:49:47,000 --> 00:49:57,000 613 00:49:57,000 --> 00:49:58,000 I'm heating the grid now. 614 00:49:58,000 --> 00:50:17,000 615 00:50:17,000 --> 00:50:20,153 110 Hz, roughly. If you want to play with this, 616 00:50:20,153 --> 00:50:23,538 don't break it. Try to transfer the liquid in 617 00:50:23,538 --> 00:50:25,384 17 seconds. I will start. 618 00:50:25,384 --> 00:50:30,000 I will resume this lecture exactly 5 minutes from now. 619 00:50:30,000 --> 00:50:38,622 [SOUND OFF/THEN ON] If you turn this in a tornado, 620 00:50:38,622 --> 00:50:42,293 you rotate it; then you open up a funnel of 621 00:50:42,293 --> 00:50:45,264 air. And so it is never the problem 622 00:50:45,264 --> 00:50:48,149 that the liquid cannot go through. 623 00:50:48,149 --> 00:50:51,907 As always, pressure equilibrium, and I don't 624 00:50:51,907 --> 00:50:56,277 remember how long it takes, but I thought it was 17 625 00:50:56,277 --> 00:50:58,724 seconds. But, if you want to, 626 00:50:58,724 --> 00:51:01,958 we can time that. It may even be less. 627 00:51:01,958 --> 00:51:07,290 I now want to address the issue of simple harmonic oscillation 628 00:51:07,290 --> 00:51:13,099 of a pendulum. As you will remember from 8.01, 629 00:51:13,099 --> 00:51:18,900 if you have a pendulum length L, mass M, and if the mass of 630 00:51:18,900 --> 00:51:24,699 the string is negligibly small compared to the mass that is 631 00:51:24,699 --> 00:51:28,900 hanging here, then a period of oscillations 632 00:51:28,900 --> 00:51:34,000 is two pi times the square root of L over G. 633 00:51:34,000 --> 00:51:40,082 G in the Boston area being to a high degree of accuracy 9.80 m 634 00:51:40,082 --> 00:51:43,970 per second squared. If you simply take L 635 00:51:43,970 --> 00:51:48,856 approximately 1 m, then you can see that you get a 636 00:51:48,856 --> 00:51:54,340 period of about two seconds. And, if you make the length 637 00:51:54,340 --> 00:52:00,322 about 25 cm, that is four times shorter than you would expect 638 00:52:00,322 --> 00:52:06,105 this period, which is two times shorter, which is about one 639 00:52:06,105 --> 00:52:11,013 second. And, without any pretense of 640 00:52:11,013 --> 00:52:16,825 accuracy, just eyeballing, not really testing if I just 641 00:52:16,825 --> 00:52:20,269 eyeball this to be about a meter. 642 00:52:20,269 --> 00:52:26,080 And, if I oscillate this back and forth, it's about two 643 00:52:26,080 --> 00:52:30,600 seconds for one oscillation: one, two, one, 644 00:52:30,600 --> 00:52:33,471 two. If I, however, 645 00:52:33,471 --> 00:52:39,043 make a 25 cm four times shorter, then it is very close 646 00:52:39,043 --> 00:52:42,723 to one second. No interference here: 647 00:52:42,723 --> 00:52:45,666 one, one, one, one, one, one. 648 00:52:45,666 --> 00:52:51,764 Remarkable when you look at this equation is that just like 649 00:52:51,764 --> 00:52:57,020 in the case of the spring, it is independent of the 650 00:52:57,020 --> 00:52:59,648 amplitude. In other words, 651 00:52:59,648 --> 00:53:05,220 whether I have a large amplitude or a small amplitude, 652 00:53:05,220 --> 00:53:13,000 it would take the same amount of time to go back and forth. 653 00:53:13,000 --> 00:53:18,420 Well, not quite for a pendulum. When we derive this period, 654 00:53:18,420 --> 00:53:24,028 you remember that you have to assume what we call small angle 655 00:53:24,028 --> 00:53:28,327 approximations. You'll see that again and again 656 00:53:28,327 --> 00:53:33,000 with 8.03 called small angle approximations. 657 00:53:33,000 --> 00:53:37,309 With small angle approximations, 658 00:53:37,309 --> 00:53:43,704 the final theta is always the sames as theta in radians. 659 00:53:43,704 --> 00:53:52,461 Now, if you ask me how small is small, that's a matter of taste. 660 00:53:52,461 --> 00:53:58,717 In 26-100, we have the mother of all pendulums. 661 00:53:58,717 --> 00:54:04,000 5.18 m long, quite impressive. 662 00:54:04,000 --> 00:54:08,600 So, we have a pendulum whose L is 5.1 plus or minus 0.05 m. 663 00:54:08,600 --> 00:54:13,439 We cannot measure it any better than 5 cm because it has to be 664 00:54:13,439 --> 00:54:15,977 under stretch when we measure it. 665 00:54:15,977 --> 00:54:20,022 And then, you have to go all the way to the ceiling, 666 00:54:20,022 --> 00:54:23,036 and all the way down. Marcos does that, 667 00:54:23,036 --> 00:54:26,764 risking his life, and he claims that the best he 668 00:54:26,764 --> 00:54:30,413 can do is 5 cm. We have 31 pounds hanging under 669 00:54:30,413 --> 00:54:34,114 there. We tried during the summer, 670 00:54:34,114 --> 00:54:37,167 believe me. We tried with technicians of 671 00:54:37,167 --> 00:54:41,787 MIT to have that pendulum here. And, one day it looked good. 672 00:54:41,787 --> 00:54:44,919 But finally they said, no we can't do it. 673 00:54:44,919 --> 00:54:48,208 We can't install it here as a safety issue. 674 00:54:48,208 --> 00:54:52,749 So, unfortunately we don't have the mother of all pendulums 675 00:54:52,749 --> 00:54:55,176 here. In 26-100, when I lectured 676 00:54:55,176 --> 00:54:58,308 Newtonian mechanics, I demonstrated that. 677 00:54:58,308 --> 00:55:03,163 That this pendulum produces is extremely close within the error 678 00:55:03,163 --> 00:55:08,465 of measurement what you predict. In other words, 679 00:55:08,465 --> 00:55:14,131 the mass of the string is indeed negligibly small compared 680 00:55:14,131 --> 00:55:19,015 to the mass of the object. We read the string ones. 681 00:55:19,015 --> 00:55:24,876 I don't remember what it was, it was such a small fraction of 682 00:55:24,876 --> 00:55:30,151 M that indeed could be ignored. And so, the prediction, 683 00:55:30,151 --> 00:55:35,817 then, is if you simply put this L in there, Tpredicted 684 00:55:35,817 --> 00:55:41,580 purely on the basis of that simply equation equals 4.57 plus 685 00:55:41,580 --> 00:55:47,758 or minus 0.02 seconds. And, this 0.02 is the result of 686 00:55:47,758 --> 00:55:51,050 the 0.05. There is a 1% error in here, 687 00:55:51,050 --> 00:55:53,542 right? Five out of 518 is 1%. 688 00:55:53,542 --> 00:55:58,525 And so, the error in T is half a percent because it's the 689 00:55:58,525 --> 00:56:01,728 square root. And so, you get a half a 690 00:56:01,728 --> 00:56:06,000 percent error. And, I rounded that off. 691 00:56:06,000 --> 00:56:10,681 So, that is the prediction. And then, I made two 692 00:56:10,681 --> 00:56:13,768 measurements: one at a 5° angle, 693 00:56:13,768 --> 00:56:18,549 and one at a 10° angle. And I did that ten times. 694 00:56:18,549 --> 00:56:22,434 So, 10T at five degrees, and 10T at 10°. 695 00:56:22,434 --> 00:56:26,916 Now, this was in 1999. Those were my good days. 696 00:56:26,916 --> 00:56:31,000 They were my good times, right? 697 00:56:31,000 --> 00:56:35,173 Past is always the good. And so, I then claimed that I 698 00:56:35,173 --> 00:56:38,559 could do this to an accuracy of 0.1 seconds. 699 00:56:38,559 --> 00:56:41,393 I had a lot of courage in those days. 700 00:56:41,393 --> 00:56:44,779 And, I measured the 5°, and what did I find? 701 00:56:44,779 --> 00:56:47,299 Unbelievable: truly unbelievable, 702 00:56:47,299 --> 00:56:50,527 purely lucky. I found exactly that number, 703 00:56:50,527 --> 00:56:54,070 which is, of course, is an accident because my 704 00:56:54,070 --> 00:56:58,952 accuracy was no better than 0.1 seconds, and then I did it at a 705 00:56:58,952 --> 00:57:03,114 10° angle. And then I found this. 706 00:57:03,114 --> 00:57:08,495 And so, I demonstrated that, indeed, five and 10° I still 707 00:57:08,495 --> 00:57:12,915 considered small angles for that approximation. 708 00:57:12,915 --> 00:57:17,720 And, it is within the uncertainty of my measurement 709 00:57:17,720 --> 00:57:22,141 what you expect. Then, I wanted to demonstrate, 710 00:57:22,141 --> 00:57:27,522 which is not so intuitive, that the period is independent 711 00:57:27,522 --> 00:57:33,000 of mass, which is not the case for the spring. 712 00:57:33,000 --> 00:57:38,631 So now, if you change the mass, and you don't change L, 713 00:57:38,631 --> 00:57:45,096 you expect no change in period. And that's what I really wanted 714 00:57:45,096 --> 00:57:48,120 to show you here. But I can't. 715 00:57:48,120 --> 00:57:52,812 And therefore, I have decided to show you what 716 00:57:52,812 --> 00:57:57,505 I did in 1999. If you can show that two minute 717 00:57:57,505 --> 00:58:03,762 version of my video lectures, then you can judge for yourself 718 00:58:03,762 --> 00:58:10,122 to what extent the mass does not influence the [SOUND OFF/THEN 719 00:58:10,122 --> 00:58:13,991 ON]. One of the most remarkable 720 00:58:13,991 --> 00:58:18,205 things I just mentioned to you is that the period of the 721 00:58:18,205 --> 00:58:22,266 oscillations is independent of the mass of the object. 722 00:58:22,266 --> 00:58:25,713 That would mean, if I joined the ball and I 723 00:58:25,713 --> 00:58:31,000 swing down with the ball that you should get that same period. 724 00:58:31,000 --> 00:58:34,720 Or, should you not? I'm asking you a question 725 00:58:34,720 --> 00:58:37,594 before we do this awful experiment. 726 00:58:37,594 --> 00:58:41,568 Would the period come out to be the same or not? 727 00:58:41,568 --> 00:58:46,641 Some of you think is the same. Have you thought about it that 728 00:58:46,641 --> 00:58:52,052 I'm a little bit taller than the object and that therefore maybe 729 00:58:52,052 --> 00:58:57,125 effectively the length of the string has become a little less 730 00:58:57,125 --> 00:59:02,468 if I sit up like this? And, if the lengths of the 731 00:59:02,468 --> 00:59:07,406 string is a little less, the period would be a little 732 00:59:07,406 --> 00:59:10,729 shorter, yeah? Be prepared for that. 733 00:59:10,729 --> 00:59:15,097 On the other hand, well, I'm not quite prepared 734 00:59:15,097 --> 00:59:18,516 for it. I will try to hold my body as 735 00:59:18,516 --> 00:59:24,498 horizontal as I possibly can in order to be at the same level as 736 00:59:24,498 --> 00:59:28,106 the bulb. I will start when I come to a 737 00:59:28,106 --> 00:59:30,195 halt here. There we go. 738 00:59:30,195 --> 00:59:34,105 Now. You count. 739 00:59:34,105 --> 00:59:41,894 (CROWD: One.) This hurts. (CROWD: Two. 740 00:59:41,894 --> 00:59:48,421 Three.) I want to hear you loud. 741 00:59:48,421 --> 00:59:56,000 (CROWD: Four. Five.) Thank you. 742 00:59:56,000 --> 1:00:03,159 (CROWD: Six. Seven. 743 1:00:03,159 --> 1:00:11,909 Eight. [LAUGHTER] Nine. 744 1:00:11,909 --> 1:00:31,000 Ten.) [APPLAUSE] Ten T with Walter Lewin. 745 1:00:31,000 --> 1:00:34,808 45.6 plus or minus 0.1 seconds. Physics works, 746 1:00:34,808 --> 1:00:38,701 I'm telling you. [SOUND OFF/THEN ON] All right, 747 1:00:38,701 --> 1:00:43,440 so I think it was convincing at least for the freshman at 748 1:00:43,440 --> 1:00:48,095 indeed, the period of the pendulum is independent of the 749 1:00:48,095 --> 1:00:52,750 mass provided that you can ignore the mass of the string 750 1:00:52,750 --> 1:00:56,389 itself, which is the case for that pendulum. 751 1:00:56,389 --> 1:01:00,282 Many pendulums, as some we will see in 8.03 are 752 1:01:00,282 --> 1:01:02,821 more complex, more complicated, 753 1:01:02,821 --> 1:01:09,000 then simply a mass-less string with an object at the end. 754 1:01:09,000 --> 1:01:12,101 And, those pendulums we call a physical pendulum. 755 1:01:12,101 --> 1:01:14,556 For instance, I could have this pair of 756 1:01:14,556 --> 1:01:17,529 compasses, and just let it oscillate like this. 757 1:01:17,529 --> 1:01:19,726 That is not just a simple pendulum. 758 1:01:19,726 --> 1:01:23,538 Or, I could have a ruler like this with a pole through here, 759 1:01:23,538 --> 1:01:26,316 and a have a pin, and have it oscillate like 760 1:01:26,316 --> 1:01:28,126 this. But, I can also have it 761 1:01:28,126 --> 1:01:31,884 oscillate here. It's a different period. 762 1:01:31,884 --> 1:01:36,047 If I oscillate it right in the middle, then it doesn't 763 1:01:36,047 --> 1:01:39,502 oscillate at all. So, now comes the question: 764 1:01:39,502 --> 1:01:43,821 how do we deal with that? And most of you must have seen 765 1:01:43,821 --> 1:01:47,120 that in 8.01. But I do want to address that 766 1:01:47,120 --> 1:01:50,654 in quite some detail. So, a physical pendulum, 767 1:01:50,654 --> 1:01:53,795 then, looks like this. This is an object, 768 1:01:53,795 --> 1:01:56,544 and I drill a hole in here, point P. 769 1:01:56,544 --> 1:02:02,312 And, I put a pin in the wall. And it can, without friction it can 770 1:02:02,312 --> 1:02:07,022 oscillate back and forth. And, let the center of mass be 771 1:02:07,022 --> 1:02:10,876 here, position O, and the separation between P 772 1:02:10,876 --> 1:02:14,302 and O is B. And, O is the center of mass. 773 1:02:14,302 --> 1:02:17,471 You can choose P anywhere you want to. 774 1:02:17,471 --> 1:02:22,010 There's no restriction on P. so, you can see that this 775 1:02:22,010 --> 1:02:25,350 pendulum is offset over an angle, theta, 776 1:02:25,350 --> 1:02:29,204 and it will start to oscillate back and forth. 777 1:02:29,204 --> 1:02:34,000 And, the question is, what is the period? 778 1:02:34,000 --> 1:02:36,545 So, clearly, we may put the entire 779 1:02:36,545 --> 1:02:40,556 gravitational force at point O in the center of mass. 780 1:02:40,556 --> 1:02:43,796 So, this is the force acting at that point. 781 1:02:43,796 --> 1:02:47,884 And now comes the question: are there any other forces 782 1:02:47,884 --> 1:02:51,432 acting on this object, or is this the only one? 783 1:02:51,432 --> 1:02:55,674 Because when you study it, you've got to take all forces 784 1:02:55,674 --> 1:02:59,068 into account. Who is happy that we have taken 785 1:02:59,068 --> 1:03:03,140 all forces into account? Raise your hand. 786 1:03:03,140 --> 1:03:05,776 Most of you are getting scared, right? 787 1:03:05,776 --> 1:03:08,698 Who says, no, it has to be least one other 788 1:03:08,698 --> 1:03:10,836 force? And which force is that? 789 1:03:10,836 --> 1:03:13,401 Yeah, where does it act? 790 1:03:13,401 --> 1:03:16,536 What location? Yeah, so there must be somehow 791 1:03:16,536 --> 1:03:20,598 a force at P to hold it up. Otherwise, it would just start 792 1:03:20,598 --> 1:03:24,090 to accelerate down. Now, I'm not even sure that it 793 1:03:24,090 --> 1:03:26,085 is straight up. I doubt that. 794 1:03:26,085 --> 1:03:28,294 It may simply be at a direction. 795 1:03:28,294 --> 1:03:32,000 I don't want to think about that. 796 1:03:32,000 --> 1:03:36,554 But surely, there has to be a force up. 797 1:03:36,554 --> 1:03:39,670 Now, remember, F equals MA. 798 1:03:39,670 --> 1:03:47,101 When you deal with rotation of objects, and this is going to be 799 1:03:47,101 --> 1:03:52,973 rotational, then this equation changes into Tau, 800 1:03:52,973 --> 1:03:59,685 the torque is moment of inertia times alpha whereby 801 1:03:59,685 --> 1:04:07,387 alpha is theta double dot. It's the angular acceleration. 802 1:04:07,387 --> 1:04:13,935 And so, if I pick P as my point of origin, then the torque, 803 1:04:13,935 --> 1:04:18,790 due to this force, does not contribute to my 804 1:04:18,790 --> 1:04:24,096 torque equation because the torque is R cross F. 805 1:04:24,096 --> 1:04:30,306 It's a cross product between the position vector and the 806 1:04:30,306 --> 1:04:34,478 force. And, this is the position 807 1:04:34,478 --> 1:04:40,128 vector to a center of mass, and the position vector from P 808 1:04:40,128 --> 1:04:44,390 to P is zero. So, if we deal with the torque 809 1:04:44,390 --> 1:04:48,256 relative to point P, that force is of no 810 1:04:48,256 --> 1:04:52,320 consequence. So, I'm going to take P as my 811 1:04:52,320 --> 1:04:55,790 origin. And so, now is the question, 812 1:04:55,790 --> 1:04:59,556 what is the torque relative to point P? 813 1:04:59,556 --> 1:05:06,000 Well, it's R cross F. R is this distance which is B. 814 1:05:06,000 --> 1:05:09,668 F is MG. But, I have a cross product, 815 1:05:09,668 --> 1:05:15,070 so I have to take the sign of this angle into account. 816 1:05:15,070 --> 1:05:19,044 So, that is the magnitude of the torque. 817 1:05:19,044 --> 1:05:23,019 And, the magnitude of that torque, then, 818 1:05:23,019 --> 1:05:28,929 according to my rotational equivalent of F equals MA equals 819 1:05:28,929 --> 1:05:34,127 the moment of inertia for rotation about that point, 820 1:05:34,127 --> 1:05:40,193 P, times theta double dot. However, it is a restoring 821 1:05:40,193 --> 1:05:43,062 torque. The torque, you can do that 822 1:05:43,062 --> 1:05:47,449 with your right hand, whatever way you've learned how 823 1:05:47,449 --> 1:05:50,993 to do that. The torque is in the blackboard 824 1:05:50,993 --> 1:05:55,128 perpendicular to the blackboard in the blackboard. 825 1:05:55,128 --> 1:05:59,178 R cross F is in the blackboard. I have rotated it 826 1:05:59,178 --> 1:06:02,974 counterclockwise, which is a vector out of the 827 1:06:02,974 --> 1:06:07,491 blackboard. So, one is like this and the 828 1:06:07,491 --> 1:06:11,674 others like this. That is like saying the torque 829 1:06:11,674 --> 1:06:15,590 is restoring. Same reason why we wrote down F 830 1:06:15,590 --> 1:06:21,107 equals minus KX with the spring is why we now write this equals 831 1:06:21,107 --> 1:06:24,845 minus this. Take into account the direction 832 1:06:24,845 --> 1:06:27,871 of the vectors. And so, this is the 833 1:06:27,871 --> 1:06:32,676 differential equation that you would have to solve and, 834 1:06:32,676 --> 1:06:36,325 if now we go to small angle approximation, 835 1:06:36,325 --> 1:06:43,000 then the sine of theta goes to theta if theta is in radians. 836 1:06:43,000 --> 1:06:47,980 And so, I can rewrite this theta double dot plus BMG 837 1:06:47,980 --> 1:06:54,230 divided by the moment of inertia about point P times theta equals 838 1:06:54,230 --> 1:06:56,769 zero. And, now, small angle 839 1:06:56,769 --> 1:07:01,164 approximation, we have a differential equation 840 1:07:01,164 --> 1:07:04,289 which is, again, a piece of cake, 841 1:07:04,289 --> 1:07:10,599 simple harmonic oscillation. And so, the simple harmonic 842 1:07:10,599 --> 1:07:16,281 oscillation, the solution must be that theta is some maximum 843 1:07:16,281 --> 1:07:21,385 angle theta zero times the cosine of omega T plus phi. 844 1:07:21,385 --> 1:07:27,162 But this omega is the square root of this number just like we 845 1:07:27,162 --> 1:07:34,000 earlier had the square root of K over M, omega now must be this. 846 1:07:34,000 --> 1:07:39,616 Omega is the square root of BMG divided by I of P. 847 1:07:39,616 --> 1:07:44,316 That means T. The period of oscillation is 848 1:07:44,316 --> 1:07:50,047 the moment of inertia about point P divided by BMG. 849 1:07:50,047 --> 1:07:54,173 I want to repeat what I said earlier. 850 1:07:54,173 --> 1:07:58,529 This omega is called angular frequency. 851 1:07:58,529 --> 1:08:05,668 Angular frequency is a given. That's the angular frequency. 852 1:08:05,668 --> 1:08:11,190 Do not confuse that with theta dot, which we also call omega, 853 1:08:11,190 --> 1:08:14,134 which is called angular velocity. 854 1:08:14,134 --> 1:08:19,288 The angular velocity in this case is a strong function of 855 1:08:19,288 --> 1:08:22,141 time. When the object comes to a 856 1:08:22,141 --> 1:08:27,662 halt, the angular velocity is zero because theta dot is zero. 857 1:08:27,662 --> 1:08:33,000 It is unfortunate that we give them the same symbol. 858 1:08:33,000 --> 1:08:35,804 So, this is independent of time. 859 1:08:35,804 --> 1:08:38,879 But, theta dot does depend on time. 860 1:08:38,879 --> 1:08:42,316 And, theta dot is the angular velocity. 861 1:08:42,316 --> 1:08:46,567 And, in the case of the uniform circular motion, 862 1:08:46,567 --> 1:08:51,180 the two omegas are the same. So now, we have all the 863 1:08:51,180 --> 1:08:56,698 ingredients in hand to calculate for absurd looking objects of 864 1:08:56,698 --> 1:09:02,306 what the period of oscillation is, provided that we are able to 865 1:09:02,306 --> 1:09:09,000 calculate the moment of inertia about the point of rotation. 866 1:09:09,000 --> 1:09:13,449 And of course, we have to know B, 867 1:09:13,449 --> 1:09:21,513 and the mass of the object. You have a wonderful example in 868 1:09:21,513 --> 1:09:27,631 your problem set. I will solve that equation, 869 1:09:27,631 --> 1:09:31,524 calculate this T, for a hoop. 870 1:09:31,524 --> 1:09:36,988 This is the hoop. All the mass is that the 871 1:09:36,988 --> 1:09:40,468 circumference, so it should be very easy to 872 1:09:40,468 --> 1:09:45,439 calculate the moment of inertia. And, we have a hole in here. 873 1:09:45,439 --> 1:09:49,831 And so, we are going to oscillate it right at the rim. 874 1:09:49,831 --> 1:09:54,802 And so, our geometry is easy. But, we should be able to bring 875 1:09:54,802 --> 1:09:59,442 this equation to a rigid test, provided that we take into 876 1:09:59,442 --> 1:10:04,000 account the uncertainty of our measurements. 877 1:10:04,000 --> 1:10:10,071 And so, let me put here this circle, this hoop. 878 1:10:10,071 --> 1:10:16,934 So, all the mass to very good approximation is at the 879 1:10:16,934 --> 1:10:22,741 circumference. And, the oscillation is about 880 1:10:22,741 --> 1:10:30,000 an axis perpendicular to the blackboard, point P. 881 1:10:30,000 --> 1:10:36,162 This is the center of mass O, and I'm going to offset this 882 1:10:36,162 --> 1:10:39,405 hoop. So, this is when it is in 883 1:10:39,405 --> 1:10:43,837 equilibrium. This is offset over an angle, 884 1:10:43,837 --> 1:10:47,081 theta. So, point O is now here. 885 1:10:47,081 --> 1:10:52,810 We'll call it O prime. And so, in analogy with what 886 1:10:52,810 --> 1:10:56,702 we did there, we have here the force, 887 1:10:56,702 --> 1:10:59,405 MG. And, the derivation is 888 1:10:59,405 --> 1:11:04,879 identical. We don't have to go over that 889 1:11:04,879 --> 1:11:07,460 again. And the radius is R. 890 1:11:07,460 --> 1:11:11,531 M is given if you need it, and R is given. 891 1:11:11,531 --> 1:11:15,702 I'll show you what these numbers are later. 892 1:11:15,702 --> 1:11:21,163 So, all I have to do now is go through this equation and 893 1:11:21,163 --> 1:11:27,021 calculate the moment of inertia for rotation of an axis like 894 1:11:27,021 --> 1:11:30,000 this to a point, P. 895 1:11:30,000 --> 1:11:32,833 Who remembers how to do that? 8.01. 896 1:11:32,833 --> 1:11:35,916 Come on, in the worst case it's wrong. 897 1:11:35,916 --> 1:11:38,833 I see one hand there. Who remembers? 898 1:11:38,833 --> 1:11:42,583 Let me ask you this. Suppose you were rotating 899 1:11:42,583 --> 1:11:46,583 through an axis right through the center of mass. 900 1:11:46,583 --> 1:11:51,000 That's difficult because there's nothing to hold onto. 901 1:11:51,000 --> 1:11:55,083 Would you know then what the moment of inertia is? 902 1:11:55,083 --> 1:11:56,833 What is it then? Yeah? 903 1:11:56,833 --> 1:12:01,000 You say yes, but now you're quiet. 904 1:12:01,000 --> 1:12:04,268 OK, moment of inertia is never MR. 905 1:12:04,268 --> 1:12:09,616 It's dimensionally wrong, but you tried which is better 906 1:12:09,616 --> 1:12:13,974 than not trying. Yeah, MR squared is what the 907 1:12:13,974 --> 1:12:19,817 moment of inertia would be if the axis were straight through 908 1:12:19,817 --> 1:12:22,987 O. I'm slowly working you up now. 909 1:12:22,987 --> 1:12:26,156 Now we move the axis from O to P. 910 1:12:26,156 --> 1:12:32,000 What happens now? What do we call that theorem? 911 1:12:32,000 --> 1:12:36,874 Parallel axis theorem. Now, we have to add the mass 912 1:12:36,874 --> 1:12:42,626 times the distance between the center of mass and that point 913 1:12:42,626 --> 1:12:45,746 squared. That's the parallel axis 914 1:12:45,746 --> 1:12:49,353 theorem. And so, the moment of inertia 915 1:12:49,353 --> 1:12:54,910 about point P is M R squared for rotation about this point. 916 1:12:54,910 --> 1:12:58,615 We take the same axis. We move it to P, 917 1:12:58,615 --> 1:13:02,222 and we have to add M distance squared. 918 1:13:02,222 --> 1:13:07,000 So, we have to add plus MR squared. 919 1:13:07,000 --> 1:13:12,753 So, we get 2MR squared. And then we have B. 920 1:13:12,753 --> 1:13:18,232 What is B? What is the distance from P to 921 1:13:18,232 --> 1:13:22,068 the center of mass? That's R. 922 1:13:22,068 --> 1:13:29,602 So, we come now with the prediction that P is 2 pi times 923 1:13:29,602 --> 1:13:37,000 the square root of 2MR squared divided by RMG. 924 1:13:37,000 --> 1:13:40,659 M goes. M always goes with pendulums. 925 1:13:40,659 --> 1:13:45,740 You never have to worry about M if you do it right. 926 1:13:45,740 --> 1:13:48,891 M always goes, not with springs, 927 1:13:48,891 --> 1:13:53,973 but with pendulums. One R goes, and so you get 2 pi 928 1:13:53,973 --> 1:13:57,429 times the square root of 2R over G. 929 1:13:57,429 --> 1:14:02,816 Before we bring these to a test, there is a remarkable 930 1:14:02,816 --> 1:14:08,000 answer. What does it make you think of? 931 1:14:08,000 --> 1:14:11,796 Excuse me? It makes you think of a single 932 1:14:11,796 --> 1:14:16,922 pendulum whereby the length is 2R, which is by no means 933 1:14:16,922 --> 1:14:19,769 obvious, is it? In other words, 934 1:14:19,769 --> 1:14:24,230 if I had pendulum here, and I would hang here in 935 1:14:24,230 --> 1:14:29,735 object, M, that would have the same period because it has a 936 1:14:29,735 --> 1:14:33,627 length, 2R. So, T is 2 pi times the length 937 1:14:33,627 --> 1:14:37,731 divided by G. By no means obvious, 938 1:14:37,731 --> 1:14:40,501 absolutely not clear why that is. 939 1:14:40,501 --> 1:14:45,349 But that's the way it is. So, now comes the acid test. 940 1:14:45,349 --> 1:14:48,898 And so, we don't have to measure the mass. 941 1:14:48,898 --> 1:14:53,486 But we did measure as accurately as we can the radius. 942 1:14:53,486 --> 1:14:56,170 That's really all we have to do. 943 1:14:56,170 --> 1:15:00,931 And, the measurement of the radius is a little uncertain 944 1:15:00,931 --> 1:15:05,000 because it's not a perfect circle. 945 1:15:05,000 --> 1:15:14,384 So, we measured it at various places and we find that R equals 946 1:15:14,384 --> 1:15:22,846 40.0 plus or minus 0.5 cm. But, that's a 1% uncertainty. 947 1:15:22,846 --> 1:15:29,000 And so, we make a prediction now, T. 948 1:15:29,000 --> 1:15:34,578 We get the square root of R, so the 1% uncertainty becomes a 949 1:15:34,578 --> 1:15:38,455 half a percent because of the square root. 950 1:15:38,455 --> 1:15:41,386 You take 2R. You divide it by G, 951 1:15:41,386 --> 1:15:45,830 and you will find that the prediction, this is a 952 1:15:45,830 --> 1:15:49,991 prediction, is that T is 1.795 plus or minus, 953 1:15:49,991 --> 1:15:53,962 that is, your half a percent, 0.01 seconds. 954 1:15:53,962 --> 1:15:57,083 That's because of the square root. 955 1:15:57,083 --> 1:16:02,000 So, this becomes half a percent error. 956 1:16:02,000 --> 1:16:06,961 And now, we do the observation, and you guessed it, 957 1:16:06,961 --> 1:16:10,137 of course. We're going to do 10T. 958 1:16:10,137 --> 1:16:13,213 And, if this is a good day, 0.1. 959 1:16:13,213 --> 1:16:18,572 But, we'll give myself a little bit extra leeway today, 960 1:16:18,572 --> 1:16:21,748 0.15. I'm fairly sure I should be 961 1:16:21,748 --> 1:16:26,312 able to do that. And so, we bring this now to a 962 1:16:26,312 --> 1:16:30,130 test. If you're ready for this, 963 1:16:30,130 --> 1:16:33,879 oh, it's still on. I always like to start the 964 1:16:33,879 --> 1:16:37,032 timer when the object comes to a halt. 965 1:16:37,032 --> 1:16:41,293 That is a better criteria than when it goes through 966 1:16:41,293 --> 1:16:44,616 equilibrium. And I will not look at the, 967 1:16:44,616 --> 1:16:49,814 even if I did look at it as no way I can stop that when I want 968 1:16:49,814 --> 1:16:52,456 to. So, we'll give it an offset. 969 1:16:52,456 --> 1:16:57,057 I first wanted to swing in a way that it's not wobbling 970 1:16:57,057 --> 1:17:02,000 because I've make a very strong prediction. 971 1:17:02,000 --> 1:17:06,017 I want to get that number: 17.95. 972 1:17:06,017 --> 1:17:12,547 So, I better make sure that it's oscillating happily. 973 1:17:12,547 --> 1:17:19,076 No, this is not happy. I don't want any wobbling like 974 1:17:19,076 --> 1:17:22,215 this. Maybe a little more. 975 1:17:22,215 --> 1:17:27,363 OK, I think this looks good. You're ready; 976 1:17:27,363 --> 1:17:33,750 I am ready. One, two, three, 977 1:17:33,750 --> 1:17:40,546 four, five, six, seven, eight, 978 1:17:40,546 --> 1:17:46,640 nine: 17.80. Ah, man, 0.15. 979 1:17:46,640 --> 1:17:56,953 [LAUGHTER] So, it wasn't such a bad day after all 980 1:17:56,953 --> 1:18:04,669 for me OK, see you Tuesday, 981 1:18:04,669 and work on your problem sets.