1 00:00:27,000 --> 00:00:29,984 So, last time, we had driven damped 2 00:00:29,984 --> 00:00:33,143 oscillations, and we did steady-state 3 00:00:33,143 --> 00:00:36,654 solutions. And the steady-state solutions 4 00:00:36,654 --> 00:00:41,920 have no adjustable constants. That is strange because at time 5 00:00:41,920 --> 00:00:45,343 T equals zero, I can put the object in a 6 00:00:45,343 --> 00:00:48,853 certain position. I can give it a certain 7 00:00:48,853 --> 00:00:52,364 velocity, and so I have two free choices. 8 00:00:52,364 --> 00:00:57,805 And yet, those free choices did not show up in our steady-state 9 00:00:57,805 --> 00:01:01,422 solution. In other words, 10 00:01:01,422 --> 00:01:06,922 the system had lost its memory over what happened at time T 11 00:01:06,922 --> 00:01:09,387 equals zero. And therefore, 12 00:01:09,387 --> 00:01:14,318 I discussed with you that you have to wait to go into 13 00:01:14,318 --> 00:01:18,491 steady-state. And that is what I will address 14 00:01:18,491 --> 00:01:22,000 today. There must be something missing 15 00:01:22,000 --> 00:01:27,784 what we did so that the time T equals zero adjustables do show 16 00:01:27,784 --> 00:01:30,913 up. So, let's return to our spring 17 00:01:30,913 --> 00:01:35,509 system. So, we have here spring 18 00:01:35,509 --> 00:01:40,854 constant K mass M. And, this object can be driven. 19 00:01:40,854 --> 00:01:47,290 You could do that with a force directly on it here where you 20 00:01:47,290 --> 00:01:53,181 can shake on the left side, omega zero squared equals K 21 00:01:53,181 --> 00:01:56,563 over M. I will call it that way, 22 00:01:56,563 --> 00:02:03,000 and gamma equals B over M. So, that's the damping. 23 00:02:03,000 --> 00:02:09,043 Now, let us assume that I don't drive it, that I just give it a 24 00:02:09,043 --> 00:02:14,014 kick, time T equals zero. I let it do its own thing. 25 00:02:14,014 --> 00:02:19,570 We discussed that two lectures ago, so we get an un-driven 26 00:02:19,570 --> 00:02:23,956 situation, not driven. And then, the solution, 27 00:02:23,956 --> 00:02:28,732 you should remember, has this exponential decay in 28 00:02:28,732 --> 00:02:33,151 it. [Is it?] X equals some value X 29 00:02:33,151 --> 00:02:39,455 that follows from the initial conditions times E to the minus 30 00:02:39,455 --> 00:02:45,758 gamma over two times T times cosine omega prime T plus alpha? 31 00:02:45,758 --> 00:02:51,957 And these, the X and the alpha can be found if you know at T 32 00:02:51,957 --> 00:02:57,000 equals zero what X is, and what X dot is. 33 00:02:57,000 --> 00:03:02,425 Omega prime is a frequency which is the square root of 34 00:03:02,425 --> 00:03:07,338 omega zero squared minus gamma squared over four, 35 00:03:07,338 --> 00:03:12,047 just a hair under omega zero, but that depends, 36 00:03:12,047 --> 00:03:17,165 of course, on gamma. So, this is the situation when 37 00:03:17,165 --> 00:03:22,283 we are not driving. Let's now take a situation that 38 00:03:22,283 --> 00:03:24,842 we do drive. So, we drive, 39 00:03:24,842 --> 00:03:28,834 for instance, with a force that we apply 40 00:03:28,834 --> 00:03:36,000 directly to this object, which is one way of driving it. 41 00:03:36,000 --> 00:03:40,081 And, that force is F zero times cosine omega T. 42 00:03:40,081 --> 00:03:44,430 This omega is my omega. I force that omega on that 43 00:03:44,430 --> 00:03:47,713 system. I don't give a damn what omega 44 00:03:47,713 --> 00:03:51,262 prime is [or?] the term of what omega is. 45 00:03:51,262 --> 00:03:54,634 And, that's tough luck for this object. 46 00:03:54,634 --> 00:03:58,716 And, we know that in the steady-state solution, 47 00:03:58,716 --> 00:04:04,037 only this omega survives. That one will go. 48 00:04:04,037 --> 00:04:11,230 And, the solution that we have seen that we derived last time 49 00:04:11,230 --> 00:04:16,984 is some amplitude, A, times the cosine of omega T 50 00:04:16,984 --> 00:04:21,299 minus delta. That is my omega that is 51 00:04:21,299 --> 00:04:25,974 nonnegotiable. This is the steady-state. 52 00:04:25,974 --> 00:04:32,687 And, we derived that the tangent of delta was omega gamma 53 00:04:32,687 --> 00:04:40,000 divided by omega zero squared minus omega squared. 54 00:04:40,000 --> 00:04:45,923 And, we found for A this monstrous solution. 55 00:04:45,923 --> 00:04:52,122 Remember, upstairs we had F zero divided by M. 56 00:04:52,122 --> 00:04:59,974 And, downstairs we have the square root omega zero squared 57 00:04:59,974 --> 00:05:07,000 minus omega squared plus omega gamma squared. 58 00:05:07,000 --> 00:05:12,578 And we spent the entire lecture on dealing with this A. 59 00:05:12,578 --> 00:05:17,330 We changed omega, and we evaluated omega equals 60 00:05:17,330 --> 00:05:22,599 zero, omega at residents, omega at very high values. 61 00:05:22,599 --> 00:05:27,454 There is no adjustable constant in the solution. 62 00:05:27,454 --> 00:05:32,000 But, there was one in that solution. 63 00:05:32,000 --> 00:05:36,566 Now, let us look at the differential equations that we 64 00:05:36,566 --> 00:05:40,356 were solving. And so, first we go through the 65 00:05:40,356 --> 00:05:44,147 un-driven system. Un-driven means at T equals 66 00:05:44,147 --> 00:05:48,886 zero, you give it a kick, and you just let it do its own 67 00:05:48,886 --> 00:05:51,987 thing. The differential equation that 68 00:05:51,987 --> 00:05:57,156 we then had was X double lot plus gamma X dot plus omega zero 69 00:05:57,156 --> 00:06:01,977 squared X equal zero. That's the one we had. 70 00:06:01,977 --> 00:06:06,382 And we solved it and we found that solution there. 71 00:06:06,382 --> 00:06:10,426 Then, we were driving it. So, now we drive it. 72 00:06:10,426 --> 00:06:13,752 Now, what's the differential equation? 73 00:06:13,752 --> 00:06:18,786 Well, we had an X double dot, gamma X dot plus omega zero 74 00:06:18,786 --> 00:06:22,471 squared times X. And now, we had here this 75 00:06:22,471 --> 00:06:26,157 driving term. In the case that we directly 76 00:06:26,157 --> 00:06:31,460 put the force on the object, then we had here F zero divided 77 00:06:31,460 --> 00:06:36,404 by M times cosine omega T because the M comes in because 78 00:06:36,404 --> 00:06:40,000 you divide by M, right? 79 00:06:40,000 --> 00:06:42,681 You had Newton's Second Law as MA. 80 00:06:42,681 --> 00:06:47,558 But, in the case that you shake the left side with your hand, 81 00:06:47,558 --> 00:06:52,353 with this [eta?] zero cosine omega T, that is another way of 82 00:06:52,353 --> 00:06:56,416 effectively driving the object. We worked that out. 83 00:06:56,416 --> 00:07:00,968 Then we have here eta zero times omega zero squared times 84 00:07:00,968 --> 00:07:04,651 cosine omega T. But in any case, 85 00:07:04,651 --> 00:07:10,327 you see here a driving term, and we solve our equations. 86 00:07:10,327 --> 00:07:15,797 And, if you take this case, then this is the solution. 87 00:07:15,797 --> 00:07:20,957 If you take this case, then this takes the place of 88 00:07:20,957 --> 00:07:23,846 that. Suppose now I take this 89 00:07:23,846 --> 00:07:29,935 solution and a substitute that solution in this differential 90 00:07:29,935 --> 00:07:34,485 equation. Then I get the result is zero 91 00:07:34,485 --> 00:07:36,714 because, look, forget this. 92 00:07:36,714 --> 00:07:41,342 So, I put it in this part. I get a zero because it fits 93 00:07:41,342 --> 00:07:46,142 this differential equation. So, what is wrong with adding 94 00:07:46,142 --> 00:07:49,057 zero? Therefore, if I add these two 95 00:07:49,057 --> 00:07:54,114 solutions, it must be solution to this differential equation 96 00:07:54,114 --> 00:07:58,742 because you just add zero. And, if you take a 18.03 and 97 00:07:58,742 --> 00:08:03,139 they use very nice terms. They say, yes, 98 00:08:03,139 --> 00:08:07,699 of course, if you have the special solution which is this 99 00:08:07,699 --> 00:08:11,282 one, you have to add the homogenous solution. 100 00:08:11,282 --> 00:08:15,842 And this word homogeneous means that you put the zero in. 101 00:08:15,842 --> 00:08:20,483 And so, the general solution is really the sum of the two. 102 00:08:20,483 --> 00:08:24,066 By adding this one, you effectively add zero. 103 00:08:24,066 --> 00:08:27,648 So, you add nothing, but you get something in 104 00:08:27,648 --> 00:08:30,743 return. What you get in return are your 105 00:08:30,743 --> 00:08:36,952 two adjustable constants. So, now you can deal with the 106 00:08:36,952 --> 00:08:42,958 situation that at T equals zero, you know exactly where that 107 00:08:42,958 --> 00:08:46,419 object is and what its velocity is. 108 00:08:46,419 --> 00:08:51,305 And so, I'll write down out the general solution, 109 00:08:51,305 --> 00:08:57,209 which is the one that governs the major part of the lecture 110 00:08:57,209 --> 00:09:00,874 today. So, X now, a function of time, 111 00:09:00,874 --> 00:09:08,000 is the steady-state solution, A cosine omega T minus delta. 112 00:09:08,000 --> 00:09:11,855 This is my omega. This is my will. 113 00:09:11,855 --> 00:09:18,163 That's Walter Lewin's omega plus X times E to the minus 114 00:09:18,163 --> 00:09:25,172 gamma over two times T times cosine omega prime T plus alpha. 115 00:09:25,172 --> 00:09:29,728 And, this is the will of the oscillator. 116 00:09:29,728 --> 00:09:35,586 And this is my will. They are two different omegas. 117 00:09:35,586 --> 00:09:38,344 And now you can see what happens. 118 00:09:38,344 --> 00:09:41,879 You see that this term will never die out. 119 00:09:41,879 --> 00:09:45,327 It will last forever, and ever, and ever. 120 00:09:45,327 --> 00:09:50,500 But this one is going to die. It's a one over EDK time of two 121 00:09:50,500 --> 00:09:53,603 over gamma. And so, if two over gamma 122 00:09:53,603 --> 00:09:58,344 happens to be ten hours, then you have to wait ten hours 123 00:09:58,344 --> 00:10:03,000 for this one to be down by a factor of E. 124 00:10:03,000 --> 00:10:08,323 But, if T over gamma happens to be 1 ms, then all you have to do 125 00:10:08,323 --> 00:10:12,887 is rate 1 ms for that term to go down by a factor of E. 126 00:10:12,887 --> 00:10:16,014 So, this is the one that will die out. 127 00:10:16,014 --> 00:10:20,492 That's why we have to wait. And so, this is called the 128 00:10:20,492 --> 00:10:24,042 [trangent?]. And it will die out faster the 129 00:10:24,042 --> 00:10:27,000 higher gamma is. And this is called, 130 00:10:27,000 --> 00:10:31,056 then, the steady-state solution, which ultimately 131 00:10:31,056 --> 00:10:35,451 survives. Suppose I told you that at T 132 00:10:35,451 --> 00:10:40,180 equals zero, X equals zero, and X dot also equals zero. 133 00:10:40,180 --> 00:10:45,084 And, I was so nasty to say, oh, by the way when you solve 134 00:10:45,084 --> 00:10:49,288 for X and alpha would be a very nice thing to do. 135 00:10:49,288 --> 00:10:53,228 But, it would take you 15 minutes of grinding; 136 00:10:53,228 --> 00:10:57,694 not so fast because, remember, if you have an X dot, 137 00:10:57,694 --> 00:11:02,073 you have to take the time derivative of this entire 138 00:11:02,073 --> 00:11:05,987 function. You have a T here or you have a 139 00:11:05,987 --> 00:11:09,194 T there, and you have to substitute in there time T 140 00:11:09,194 --> 00:11:11,951 equals zero. And then, you have to make that 141 00:11:11,951 --> 00:11:14,517 equal zero. And, it takes you 15 minutes. 142 00:11:14,517 --> 00:11:16,569 And, out pops a [UNINTELLIGIBLE], 143 00:11:16,569 --> 00:11:18,877 a value for X, and a value for alpha. 144 00:11:18,877 --> 00:11:21,891 You haven't learned much physics when you do it, 145 00:11:21,891 --> 00:11:23,879 but you've learned some algebra. 146 00:11:23,879 --> 00:11:27,214 And so, I decided not to spend my time on doing that. 147 00:11:27,214 --> 00:11:30,356 But, in principle, you must agree with me now that 148 00:11:30,356 --> 00:11:33,691 if I specify the initial conditions, I don't have too 149 00:11:33,691 --> 00:11:37,948 call this zero. I could call this X zero. 150 00:11:37,948 --> 00:11:42,314 I can do anything I want to. I can give X dot any value I 151 00:11:42,314 --> 00:11:45,276 want to. And they get unique values for 152 00:11:45,276 --> 00:11:48,395 X and for alpha. And, that is ultimately, 153 00:11:48,395 --> 00:11:51,981 then, what the solution is. I must add the two. 154 00:11:51,981 --> 00:11:55,333 The bottom line, and that's really where the 155 00:11:55,333 --> 00:12:00,010 physics is, and that has to do with problem two-five that you 156 00:12:00,010 --> 00:12:05,000 have this week on your plate is then the following. 157 00:12:05,000 --> 00:12:09,102 If I make a plot of X as a function of time, 158 00:12:09,102 --> 00:12:13,681 then the solution is really the sum of these two. 159 00:12:13,681 --> 00:12:18,166 This is the steady-state solution which has been 160 00:12:18,166 --> 00:12:22,363 amplitude, capital A, which is nonnegotiable. 161 00:12:22,363 --> 00:12:26,943 It has nothing to do with the initial conditions. 162 00:12:26,943 --> 00:12:32,000 It has its own omega, Walter Lewin's omega. 163 00:12:32,000 --> 00:12:37,722 So, let's assume that this is the period, T. 164 00:12:37,722 --> 00:12:43,577 And so, out of that pops a nice cosinusoidal, 165 00:12:43,577 --> 00:12:48,768 sinusoidal curve. Let me put it in here. 166 00:12:48,768 --> 00:12:54,756 And, this never changes. This goes on forever, 167 00:12:54,756 --> 00:12:59,148 and ever, and ever. And this here, 168 00:12:59,148 --> 00:13:04,205 this time, T, is 2 pi divided by omega, 169 00:13:04,205 --> 00:13:10,528 my omega. However, there is also this 170 00:13:10,528 --> 00:13:17,977 one, but this one dies out. And so, I will now put in some 171 00:13:17,977 --> 00:13:24,380 kind of an exponential decay, something like this. 172 00:13:24,380 --> 00:13:30,000 It has its own frequency, omega prime. 173 00:13:30,000 --> 00:13:34,126 I can choose anything. I can make omega larger, 174 00:13:34,126 --> 00:13:39,059 omega prime larger than omega. I can make it smaller so, 175 00:13:39,059 --> 00:13:43,275 I just pick one, and let's suppose that the zero 176 00:13:43,275 --> 00:13:46,684 crossings are here, and here, and here, 177 00:13:46,684 --> 00:13:50,272 and here, and here. And so, for instance, 178 00:13:50,272 --> 00:13:55,475 the curve than that has to be added could be something like 179 00:13:55,475 --> 00:13:57,000 this. 180 00:13:57,000 --> 00:14:02,000 181 00:14:02,000 --> 00:14:07,463 And this time it's T prime is 2 pi divided by omega prime. 182 00:14:07,463 --> 00:14:13,501 And so, you see that the sum of the two, which I will not try to 183 00:14:13,501 --> 00:14:18,581 sketch, is the solution. But it is the pink one that's 184 00:14:18,581 --> 00:14:22,990 going to die out. That's why sometimes you have 185 00:14:22,990 --> 00:14:27,111 to be patient, except if it is 1 ms two over 186 00:14:27,111 --> 00:14:32,000 gamma, you don't have to be very patient. 187 00:14:32,000 --> 00:14:34,614 Now, let us look at the situation. 188 00:14:34,614 --> 00:14:38,814 And I can arrange that. And I'm going to arrange that, 189 00:14:38,814 --> 00:14:42,379 that omega and omega prime are close together. 190 00:14:42,379 --> 00:14:46,183 I can choose that. The system cannot choose omega 191 00:14:46,183 --> 00:14:48,243 prime. The system is stuck. 192 00:14:48,243 --> 00:14:51,808 This is omega prime. The system has no choice. 193 00:14:51,808 --> 00:14:55,691 But, I have a choice. I can make omega any value I 194 00:14:55,691 --> 00:14:58,781 want to. So, I can make it very close to 195 00:14:58,781 --> 00:15:03,325 omega prime. What do you think is going to 196 00:15:03,325 --> 00:15:06,372 happen now? When I turn this system on, 197 00:15:06,372 --> 00:15:10,943 all of a sudden the driver, and my own omega is very close 198 00:15:10,943 --> 00:15:14,712 to that omega prime. It's true that the trangent 199 00:15:14,712 --> 00:15:18,160 will die out, but let's say we take a system 200 00:15:18,160 --> 00:15:22,330 with a pretty high Q. So, it doesn't die out so fast. 201 00:15:22,330 --> 00:15:25,377 What do you expect you're going to see? 202 00:15:25,377 --> 00:15:28,183 Excellent. You're going to see beats 203 00:15:28,183 --> 00:15:32,674 because now you have two harmonic oscillations which have 204 00:15:32,674 --> 00:15:36,788 to be added. But, the frequencies are a 205 00:15:36,788 --> 00:15:40,022 little different. And, if this one survives long 206 00:15:40,022 --> 00:15:43,393 enough, there comes a time that they are in phase. 207 00:15:43,393 --> 00:15:46,144 There comes a time they are out of phase. 208 00:15:46,144 --> 00:15:50,066 You see very low amplitude. And so, you are going to see a 209 00:15:50,066 --> 00:15:53,093 bit phenomenon. And that is to do what I want 210 00:15:53,093 --> 00:15:55,638 you to see. For that, we need a system 211 00:15:55,638 --> 00:15:59,560 preferably with a high IQ. And then, the driving frequency 212 00:15:59,560 --> 00:16:03,000 has to be close to the omega prime. 213 00:16:03,000 --> 00:16:07,025 And for that, I have chosen this system here. 214 00:16:07,025 --> 00:16:11,507 This is an air track. And, we can make the damping 215 00:16:11,507 --> 00:16:16,813 very low, unpleasantly low, believe me, a very low value of 216 00:16:16,813 --> 00:16:22,028 gamma that when I turn on the airflow, so here is a spring 217 00:16:22,028 --> 00:16:25,687 constant, K. Here is one spring constant, 218 00:16:25,687 --> 00:16:30,078 K, and here is the mass. And, there's very little 219 00:16:30,078 --> 00:16:34,095 friction. And now I'm going to drive it 220 00:16:34,095 --> 00:16:37,997 just a little bit off resonance, a little bit below the 221 00:16:37,997 --> 00:16:41,610 resonance frequency. And, what you are going to see 222 00:16:41,610 --> 00:16:45,512 now is the sum of these two. But since gamma is so low, 223 00:16:45,512 --> 00:16:49,342 it will take a long time for this trangent to die out. 224 00:16:49,342 --> 00:16:52,377 And that is exactly what I want you to see. 225 00:16:52,377 --> 00:16:55,123 In addition, you're going to see beats. 226 00:16:55,123 --> 00:16:59,170 And as long as you see beating, you know that you haven't 227 00:16:59,170 --> 00:17:03,000 reached the steady-state solution yet. 228 00:17:03,000 --> 00:17:09,643 But, if you are patient and I'm patient, we probably will see it 229 00:17:09,643 --> 00:17:14,283 go into steady-state, but it may take several 230 00:17:14,283 --> 00:17:17,552 minutes. So, you ready for that? 231 00:17:17,552 --> 00:17:23,458 I'm going to drive it here. And, I start the driving out. 232 00:17:23,458 --> 00:17:28,730 So, relax and look at the amplitude of this object. 233 00:17:28,730 --> 00:17:34,392 And see what happens. Hey, the amplitude is going 234 00:17:34,392 --> 00:17:37,848 down, hey, hey, hey, I call that a beat. 235 00:17:37,848 --> 00:17:41,835 Did you see that? Did you see the amplitude go 236 00:17:41,835 --> 00:17:44,936 down? The two frequencies were being 237 00:17:44,936 --> 00:17:48,924 against each other. Now it's picking up again. 238 00:17:48,924 --> 00:17:53,177 It's nowhere near a steady state: very low gamma, 239 00:17:53,177 --> 00:17:55,835 very high Q. There we go again, 240 00:17:55,835 --> 00:18:00,000 amplitude way down picks up again. 241 00:18:00,000 --> 00:18:03,626 Just be patient. Let's be patient and see 242 00:18:03,626 --> 00:18:09,337 whether we have the privilege of seeing it go into steady-state. 243 00:18:09,337 --> 00:18:13,597 It's a very high Q system. Since I am just below 244 00:18:13,597 --> 00:18:19,399 resonance, the driver in the car will be in phase when I am below 245 00:18:19,399 --> 00:18:22,481 resonance in steady-state solution. 246 00:18:22,481 --> 00:18:26,198 So, this delta will be very close to zero, 247 00:18:26,198 --> 00:18:29,099 below resonance, above resonance, 248 00:18:29,099 --> 00:18:35,563 180° out of phase. But, I am just below resonance. 249 00:18:35,563 --> 00:18:39,241 So, when we go into steady-state, 250 00:18:39,241 --> 00:18:45,908 we also will see that we are very close to a delta of zero. 251 00:18:45,908 --> 00:18:50,505 Now, let's see what the amplitude is now, 252 00:18:50,505 --> 00:18:54,758 and whether the amplitude is changing. 253 00:18:54,758 --> 00:19:02,000 Well, we are getting there. It pays off to be patient. 254 00:19:02,000 --> 00:19:05,727 Later today, I will do an experiment where 255 00:19:05,727 --> 00:19:08,818 two over gamma is two many seconds. 256 00:19:08,818 --> 00:19:12,000 So, all you have to do is wait 4 ms. 257 00:19:12,000 --> 00:19:17,090 So, I'll make up for the fact that now we have to wait to 258 00:19:17,090 --> 00:19:20,272 long. Let's take a look at this now. 259 00:19:20,272 --> 00:19:24,272 I think it looks terrific. It looks terrific. 260 00:19:24,272 --> 00:19:28,090 In phase, I don't see much beating anymore. 261 00:19:28,090 --> 00:19:33,000 It looks like the amplitude is constant. 262 00:19:33,000 --> 00:19:38,522 I think we've killed this one, and I think this one has 263 00:19:38,522 --> 00:19:42,306 survived. If you increase the damping, 264 00:19:42,306 --> 00:19:47,215 this would happen, of course, earlier that you go 265 00:19:47,215 --> 00:19:51,000 into the steady-state solution. 266 00:19:51,000 --> 00:20:00,000 267 00:20:00,000 --> 00:20:04,141 It looks great. I don't see any change anymore 268 00:20:04,141 --> 00:20:08,466 in the amplitude. So, that's the A that you have 269 00:20:08,466 --> 00:20:12,699 there, capital A, and they are nicely in phase. 270 00:20:12,699 --> 00:20:17,852 It's a very high Q system, so the change to go from delta 271 00:20:17,852 --> 00:20:23,098 zero to delta pi over two at resonance takes place over an 272 00:20:23,098 --> 00:20:26,319 extremely narrow range of frequency. 273 00:20:26,319 --> 00:20:32,867 So, they are still in phase. All right, if we are driving 274 00:20:32,867 --> 00:20:38,295 this system with a force, say, directly on the object, 275 00:20:38,295 --> 00:20:43,518 F zero cosine omega T, then in steady state there is 276 00:20:43,518 --> 00:20:48,024 energy dissipation because there is friction. 277 00:20:48,024 --> 00:20:52,427 And, where there is friction, there is heat. 278 00:20:52,427 --> 00:20:57,753 And that means energy. That means I had to do work to 279 00:20:57,753 --> 00:21:04,000 provide that energy to the steady-state situation. 280 00:21:04,000 --> 00:21:09,760 So, as the thing is never changing, it's A just going on 281 00:21:09,760 --> 00:21:13,530 forever and ever. While that happens, 282 00:21:13,530 --> 00:21:18,871 I must put in energy, which comes out in the form of 283 00:21:18,871 --> 00:21:22,432 heat. So, let us return to the good 284 00:21:22,432 --> 00:21:27,354 old days of 8.01. And, I want to remind you that 285 00:21:27,354 --> 00:21:33,429 work is the dot product between a force and a displacement, 286 00:21:33,429 --> 00:21:36,621 DX. It's a dot product. 287 00:21:36,621 --> 00:21:40,890 It's a scalar work. A little bit of work is done by 288 00:21:40,890 --> 00:21:44,475 this force if it moves over a distance, DX. 289 00:21:44,475 --> 00:21:49,426 If the two are perpendicular to each other, then no work is 290 00:21:49,426 --> 00:21:52,329 done. Satellite into circular orbit 291 00:21:52,329 --> 00:21:56,000 around the Earth: no work is done. 292 00:21:56,000 --> 00:22:00,698 And so, now I can calculate what the power is because the 293 00:22:00,698 --> 00:22:04,306 power is DW DT, how many joules per second I 294 00:22:04,306 --> 00:22:09,088 had to put into the system. And so, if I take a derivative 295 00:22:09,088 --> 00:22:13,031 times the derivative, then I get F dotted with V 296 00:22:13,031 --> 00:22:16,052 because DX DT is simply the velocity. 297 00:22:16,052 --> 00:22:19,324 Now, if I have a one-dimensional system, 298 00:22:19,324 --> 00:22:23,770 and what I mean by that is the force is either in this 299 00:22:23,770 --> 00:22:28,637 direction or in this direction in the velocity is either in 300 00:22:28,637 --> 00:22:33,000 this direction or in this direction. 301 00:22:33,000 --> 00:22:36,795 That's what I mean by a one-dimensional system. 302 00:22:36,795 --> 00:22:40,837 Then I can delete the dot, and then the signs will 303 00:22:40,837 --> 00:22:44,137 automatically take care of the direction. 304 00:22:44,137 --> 00:22:47,437 The minus V is then this. Plus V is that, 305 00:22:47,437 --> 00:22:50,984 and same for force. So, I can kill the dots. 306 00:22:50,984 --> 00:22:54,615 So, now, I have to know what the velocity is. 307 00:22:54,615 --> 00:22:58,492 In the steady-state solution. Well, that's easy, 308 00:22:58,492 --> 00:23:02,700 because I go to the steady-state solution here and I 309 00:23:02,700 --> 00:23:08,325 calculate what X dot is. So I'll put that here, 310 00:23:08,325 --> 00:23:13,189 X dot, which is V. So, the derivative of cosine 311 00:23:13,189 --> 00:23:17,101 omega T is minus omega times the sine. 312 00:23:17,101 --> 00:23:23,127 So, I get minus omega times A times the sine omega T minus 313 00:23:23,127 --> 00:23:25,876 delta. That's the velocity. 314 00:23:25,876 --> 00:23:33,162 But, I know what the force is. That is F zero cosine omega T. 315 00:23:33,162 --> 00:23:37,169 So, there we go, F zero cosine omega T, 316 00:23:37,169 --> 00:23:41,175 but I'm going to put these in also now. 317 00:23:41,175 --> 00:23:46,025 So, I get omega A, and then I put in the cosine 318 00:23:46,025 --> 00:23:51,929 omega T, and now I'm going to put in this one the sine of 319 00:23:51,929 --> 00:23:56,463 omega T minus delta. So, take a deep breath. 320 00:23:56,463 --> 00:24:00,047 F zero cosine omega T is the force. 321 00:24:00,047 --> 00:24:05,604 See that there? And the velocity has been 322 00:24:05,604 --> 00:24:11,020 derived from the first derivative of the steady-state 323 00:24:11,020 --> 00:24:17,270 solution, gives me a minus omega A, sine omega T minus delta. 324 00:24:17,270 --> 00:24:23,416 Now, this, what we have here between brackets can be written 325 00:24:23,416 --> 00:24:29,562 as the sine of omega T cosine delta minus the cosine omega T 326 00:24:29,562 --> 00:24:35,485 times the sine of delta. So, what you see here between 327 00:24:35,485 --> 00:24:39,085 brackets is the same as what you see there. 328 00:24:39,085 --> 00:24:43,714 And I really interested in knowing every moment in time 329 00:24:43,714 --> 00:24:46,885 what exactly the power is? Not really. 330 00:24:46,885 --> 00:24:51,771 Most of the time I'm really interested in knowing what the 331 00:24:51,771 --> 00:24:55,628 average power is, that is required to keep the 332 00:24:55,628 --> 00:25:00,771 thing going, average over one oscillation or over hundreds of 333 00:25:00,771 --> 00:25:05,188 oscillations. That's really what I'm 334 00:25:05,188 --> 00:25:08,471 interested in, not necessarily the 335 00:25:08,471 --> 00:25:11,952 instantaneous power. In other words, 336 00:25:11,952 --> 00:25:17,324 most of the time my interest is really in what this is, 337 00:25:17,324 --> 00:25:20,408 that is, the time average value. 338 00:25:20,408 --> 00:25:25,780 And, think of it as being one oscillation that is fine. 339 00:25:25,780 --> 00:25:29,361 But you can think of it also as days. 340 00:25:29,361 --> 00:25:33,041 Now, I need some experts and audience. 341 00:25:33,041 --> 00:25:39,109 I see here with cosine omega T, and I see here with sine omega 342 00:25:39,109 --> 00:25:44,113 T. What is the time average value 343 00:25:44,113 --> 00:25:48,990 of that product? The time average value [between 344 00:25:48,990 --> 00:25:53,245 the?] sine omega T and the cosine omega T, 345 00:25:53,245 --> 00:25:59,471 if I average it over one cycle or two cycles or three cycles. 346 00:25:59,471 --> 00:26:03,000 Come on, high school, yeah? 347 00:26:03,000 --> 00:26:08,003 What is the time average of sine omega T times cosine omega 348 00:26:08,003 --> 00:26:11,194 T, time average over one period? Zero. 349 00:26:11,194 --> 00:26:15,594 This one times this one times average gives me zero. 350 00:26:15,594 --> 00:26:19,389 You don't believe it, go back to high school. 351 00:26:19,389 --> 00:26:23,789 Ask your high school teacher. He will agree with me. 352 00:26:23,789 --> 00:26:28,447 Here I see a cosine omega T, and I see a cosine omega T 353 00:26:28,447 --> 00:26:32,314 there. What is the time average of 354 00:26:32,314 --> 00:26:36,599 cosine squared omega T? Ah, you guys are waking up. 355 00:26:36,599 --> 00:26:39,000 It's one half. So, therefore, 356 00:26:39,000 --> 00:26:42,514 for this product, sorry, for this product, 357 00:26:42,514 --> 00:26:45,857 I can write one half. And so now we get, 358 00:26:45,857 --> 00:26:51,000 notice there is a minus here. And so this minus picks up this 359 00:26:51,000 --> 00:26:53,400 minus. So, it becomes a plus. 360 00:26:53,400 --> 00:26:57,257 So, I get F zero. I get the half that is here. 361 00:26:57,257 --> 00:27:03,310 Then I get omega. I get my A, and then I get the 362 00:27:03,310 --> 00:27:07,393 sine of delta. Are we happy with that? 363 00:27:07,393 --> 00:27:11,696 So, you see, it collapses into something 364 00:27:11,696 --> 00:27:16,662 that is relatively simple. What is sine delta? 365 00:27:16,662 --> 00:27:20,855 Well, I remember what tangent delta is. 366 00:27:20,855 --> 00:27:26,262 If I know tangent delta, can I then calculate sine 367 00:27:26,262 --> 00:27:29,462 delta? And, the answer is yes, 368 00:27:29,462 --> 00:27:33,936 of course. If this is delta, 369 00:27:33,936 --> 00:27:40,430 and if this is omega gamma, and this is omega zero squared 370 00:27:40,430 --> 00:27:46,012 minus omega squared, then it only takes Pythagoras 371 00:27:46,012 --> 00:27:52,392 to calculate what this is. And so I know that the sine of 372 00:27:52,392 --> 00:27:58,544 delta must be omega gamma divided by the square root of 373 00:27:58,544 --> 00:28:03,898 omega zero squared minus omega squared, squared, 374 00:28:03,898 --> 00:28:11,167 plus omega gamma squared. So, yes, I do know what the 375 00:28:11,167 --> 00:28:16,681 sine of delta is. So, now, I can come to a close 376 00:28:16,681 --> 00:28:22,430 by substituting in here what the sine of delta is. 377 00:28:22,430 --> 00:28:27,005 And, I can substitute in here what A is. 378 00:28:27,005 --> 00:28:31,759 Here is A. You know what is nice? 379 00:28:31,759 --> 00:28:39,161 Look at the downstairs here. It's the same as the downstairs 380 00:28:39,161 --> 00:28:42,799 there. So, if I multiply them, 381 00:28:42,799 --> 00:28:49,448 the square root goes away. So, if I can write down now 382 00:28:49,448 --> 00:28:54,842 what P averages, go slowly, so we get F zero 383 00:28:54,842 --> 00:28:58,605 divided by two. I get an omega. 384 00:28:58,605 --> 00:29:04,000 I will go to the A very shortly. 385 00:29:04,000 --> 00:29:09,968 I will first pick omega gamma here, which is my sine delta. 386 00:29:09,968 --> 00:29:14,907 So, that makes this a square. And, I get a gamma. 387 00:29:14,907 --> 00:29:20,155 And, now I turn to the A, which is an F zero over M. 388 00:29:20,155 --> 00:29:24,786 So, I get a square here. And, I get an M here. 389 00:29:24,786 --> 00:29:30,137 And now, I get this downstairs times this downstairs. 390 00:29:30,137 --> 00:29:36,105 The square root disappears, and so I get omega zero squared 391 00:29:36,105 --> 00:29:40,736 minus omega squared, squared, plus omega gamma 392 00:29:40,736 --> 00:29:44,422 squared. Almost end of story. 393 00:29:44,422 --> 00:29:46,698 I'm going to rewrite it a little. 394 00:29:46,698 --> 00:29:48,690 And Tony French, in his book, 395 00:29:48,690 --> 00:29:51,393 rewrites it in, again, a different way. 396 00:29:51,393 --> 00:29:55,020 He loves to work with Q's. He puts the Q's in there. 397 00:29:55,020 --> 00:30:00,000 I'm going to divide upstairs and downstairs by omega squared. 398 00:30:00,000 --> 00:30:05,768 And, when I do that, I got F zero squared. 399 00:30:05,768 --> 00:30:11,256 I've got a gamma here. I get two M here. 400 00:30:11,256 --> 00:30:19,839 And then, I get here omega zero squared divided by omega minus 401 00:30:19,839 --> 00:30:24,341 omega squared plus gamma squared. 402 00:30:24,341 --> 00:30:30,367 That's what I get. That's not the only way you can 403 00:30:30,367 --> 00:30:33,804 write it, but that's one way you can write it. 404 00:30:33,804 --> 00:30:37,241 Let me check that. I have five zeroes squared. 405 00:30:37,241 --> 00:30:40,372 I have gamma, two M here, and we have this 406 00:30:40,372 --> 00:30:43,350 downstairs. So, the time has come now to 407 00:30:43,350 --> 00:30:45,794 try to see through this equation. 408 00:30:45,794 --> 00:30:50,071 You remember last lecture, we spent the whole lecture not 409 00:30:50,071 --> 00:30:54,042 to look at this dumb equation, but to see through it. 410 00:30:54,042 --> 00:31:00,000 And we were able to see through it, see all its idiosyncrasies. 411 00:31:00,000 --> 00:31:05,114 Let's look at the idiosyncrasies of this average 412 00:31:05,114 --> 00:31:10,229 power over one cycle, over a multiple of cycles. 413 00:31:10,229 --> 00:31:15,235 Well, let us first make gamma infinitely large. 414 00:31:15,235 --> 00:31:18,608 Don't look even at the equation. 415 00:31:18,608 --> 00:31:24,049 That means there is an infinite amount of friction. 416 00:31:24,049 --> 00:31:30,035 The system never gets going. There's no way it will ever 417 00:31:30,035 --> 00:31:33,735 move. So, clearly the average power 418 00:31:33,735 --> 00:31:37,985 must go to zero. And, indeed, 419 00:31:37,985 --> 00:31:41,022 you see that. The gamma is downstairs. 420 00:31:41,022 --> 00:31:44,634 If gamma goes to infinity, the power is zero. 421 00:31:44,634 --> 00:31:47,589 Let's make the mass infinitely large. 422 00:31:47,589 --> 00:31:52,268 If the mass in infinitely large, you have an infinity high 423 00:31:52,268 --> 00:31:55,223 inertia. Nothing will ever get going. 424 00:31:55,223 --> 00:31:58,261 No force will ever get the mass going. 425 00:31:58,261 --> 00:32:02,776 Well, that means you expect that the power goes to zero. 426 00:32:02,776 --> 00:32:04,828 And, indeed, you see here, 427 00:32:04,828 --> 00:32:10,000 if M goes to infinity, the power goes to zero. 428 00:32:10,000 --> 00:32:12,193 Let's say the force goes to zero. 429 00:32:12,193 --> 00:32:15,688 We are not even driving it. Well, if we are not even 430 00:32:15,688 --> 00:32:19,663 driving it, I hope you will agree with me you don't have to 431 00:32:19,663 --> 00:32:21,171 put in any work, right? 432 00:32:21,171 --> 00:32:24,049 Nothing gets going. So, clearly you expect, 433 00:32:24,049 --> 00:32:26,517 then, that the power will go to zero. 434 00:32:26,517 --> 00:32:28,915 And, indeed, if F zero goes to zero, 435 00:32:28,915 --> 00:32:34,390 you see that the power is zero. Suppose you make omega zero. 436 00:32:34,390 --> 00:32:38,317 So that means there will never be any velocity. 437 00:32:38,317 --> 00:32:43,097 You never pick up any velocity. It takes infinitely long. 438 00:32:43,097 --> 00:32:45,317 Omega is zero. So, clearly, 439 00:32:45,317 --> 00:32:49,329 if nothing ever moves the power will go to zero. 440 00:32:49,329 --> 00:32:51,890 Now, look. If omega makes zero, 441 00:32:51,890 --> 00:32:55,560 this is zero. But this one goes to infinity. 442 00:32:55,560 --> 00:33:00,000 And, therefore, the power goes to zero. 443 00:33:00,000 --> 00:33:05,076 So, you need no equations for that, just common sense to 444 00:33:05,076 --> 00:33:09,599 immediately conclude that this has to be the case. 445 00:33:09,599 --> 00:33:13,015 Suppose you go to infinity with omega. 446 00:33:13,015 --> 00:33:16,892 Very, very fast. Well, if you go infinitely 447 00:33:16,892 --> 00:33:22,615 fast, because of the inertia of the system, it can never react, 448 00:33:22,615 --> 00:33:26,861 can never get going. So I will predict that the 449 00:33:26,861 --> 00:33:32,064 power must go to zero. And, if you put omega infinity 450 00:33:32,064 --> 00:33:35,436 in here, this goes to zero. This goes to infinity, 451 00:33:35,436 --> 00:33:39,634 and so the power goes to zero. So, all this is complete common 452 00:33:39,634 --> 00:33:41,836 sense. All of this you could have 453 00:33:41,836 --> 00:33:45,896 predicted without that equation. But, isn't it nice that the 454 00:33:45,896 --> 00:33:49,888 equation supports my intuition? So, now comes the question, 455 00:33:49,888 --> 00:33:53,741 if omega goes to omega zero, and that is the reason why I 456 00:33:53,741 --> 00:33:57,320 wrote it in this way. Notice when omega goes to zero, 457 00:33:57,320 --> 00:34:02,000 when omega goes to omega zero, this one goes to zero. 458 00:34:02,000 --> 00:34:06,162 And therefore, that is the frequency at which 459 00:34:06,162 --> 00:34:12,216 the average power is the maximum [ever?], can never be any higher 460 00:34:12,216 --> 00:34:16,000 because it's independent of gamma, right, 461 00:34:16,000 --> 00:34:19,310 my omega? So, I have an omega equals 462 00:34:19,310 --> 00:34:23,189 omega zero. We reach the maximum value for 463 00:34:23,189 --> 00:34:26,310 power. We can never go any higher. 464 00:34:26,310 --> 00:34:31,920 It's exactly at omega zero. And, this is zero. 465 00:34:31,920 --> 00:34:36,613 You lose one gamma. And so you find that this 466 00:34:36,613 --> 00:34:42,480 value, then, becomes F zero squared divided by 2M gamma. 467 00:34:42,480 --> 00:34:47,813 And, you can write that, rewrite have a little bit. 468 00:34:47,813 --> 00:34:51,973 I do that because Tony French likes Q's. 469 00:34:51,973 --> 00:34:55,813 And so, I write it with a Q in there. 470 00:34:55,813 --> 00:35:02,000 Remember that Q is omega zero divided by gamma. 471 00:35:02,000 --> 00:35:09,514 So, I can rewrite this as Q times F zero squared upstairs. 472 00:35:09,514 --> 00:35:15,578 So, I get the Q in there, which is always nice, 473 00:35:15,578 --> 00:35:22,434 divided by 2M omega zero. So, this is the same thing. 474 00:35:22,434 --> 00:35:28,102 So, if I now plot, make a curve for you of P 475 00:35:28,102 --> 00:35:35,090 average, not P maximum, but P average as a function of 476 00:35:35,090 --> 00:35:43,000 frequency, so here is omega and here is P average. 477 00:35:43,000 --> 00:35:47,576 Here is omega zero. It goes through a maximum 478 00:35:47,576 --> 00:35:51,735 exactly at omega zero. It starts at zero, 479 00:35:51,735 --> 00:35:57,976 you see, and it ends at zero. And, it sweeps up to a maximum, 480 00:35:57,976 --> 00:36:03,903 and then it goes down again. And, for reasonable values of 481 00:36:03,903 --> 00:36:09,000 Q, these curves look extremely symmetric. 482 00:36:09,000 --> 00:36:17,460 And so, this value here is then the average max, 483 00:36:17,460 --> 00:36:23,039 which is that value, this value. 484 00:36:23,039 --> 00:36:33,840 If we look at the width of this curve, at half maximum of the 485 00:36:33,840 --> 00:36:38,542 power, so this is one half times P 486 00:36:38,542 --> 00:36:43,534 maximum, then you can show, and it is not so difficult 487 00:36:43,534 --> 00:36:47,301 algebraically, but I will not attempt it, 488 00:36:47,301 --> 00:36:52,858 you can show that the width at half maximum is very close to 489 00:36:52,858 --> 00:36:56,249 gamma. Remember, gamma and omega have 490 00:36:56,249 --> 00:37:00,110 the same unit. It's one divided by second. 491 00:37:00,110 --> 00:37:03,972 In other words, if you go to half maximum, 492 00:37:03,972 --> 00:37:10,000 this point here is omega zero minus gamma over two. 493 00:37:10,000 --> 00:37:15,447 And, this point here is omega zero plus gamma over two. 494 00:37:15,447 --> 00:37:21,096 And so, you see immediately, which of course makes sense, 495 00:37:21,096 --> 00:37:27,048 that if gamma is very small, that the peak gets very narrow. 496 00:37:27,048 --> 00:37:33,000 And, if gamma is very high, the peak gets very broad. 497 00:37:33,000 --> 00:37:36,300 That's intuitively quite pleasing. 498 00:37:36,300 --> 00:37:39,599 High Q systems, very narrow peaks. 499 00:37:39,599 --> 00:37:45,500 And, that's the way that Tony French likes to plot his data. 500 00:37:45,500 --> 00:37:49,599 I will show you that on the overhead here, 501 00:37:49,599 --> 00:37:53,300 this is just a picture from your book. 502 00:37:53,300 --> 00:37:57,699 What Tony does here, he plots not omega here, 503 00:37:57,699 --> 00:38:01,699 but he plots omega divided by omega zero. 504 00:38:01,699 --> 00:38:07,000 So, that means the resonance is at one. 505 00:38:07,000 --> 00:38:10,837 And, he doesn't plot the average value for P here, 506 00:38:10,837 --> 00:38:15,066 but he plots it into strange units into the unit F zero 507 00:38:15,066 --> 00:38:17,572 squared divided by 2M omega zero. 508 00:38:17,572 --> 00:38:22,036 So, now, he effectively can compare the vertical axis with 509 00:38:22,036 --> 00:38:26,891 the Q value because he likes the fact that it is Q times higher 510 00:38:26,891 --> 00:38:30,180 than something. And, he has plotted this in 511 00:38:30,180 --> 00:38:35,410 terms of that something. And so, if you take a curve for 512 00:38:35,410 --> 00:38:38,442 Q equals ten, which has the peak here in 513 00:38:38,442 --> 00:38:42,018 power, you see, indeed, that he finds that very 514 00:38:42,018 --> 00:38:45,050 close on his scale to ten. Notice, also, 515 00:38:45,050 --> 00:38:48,705 the nice symmetry. And, you see for lower values 516 00:38:48,705 --> 00:38:51,815 of Q, which are curves here that, indeed, 517 00:38:51,815 --> 00:38:55,780 the peak gets broader. The width of this peak is one 518 00:38:55,780 --> 00:38:59,123 over Q because, remember, this axis is omega 519 00:38:59,123 --> 00:39:03,981 divided by omega zero. So, if the width is gamma on 520 00:39:03,981 --> 00:39:07,872 that peak, it is now gamma divided by omega zero in this 521 00:39:07,872 --> 00:39:10,136 plot. And, gamma divided by omega 522 00:39:10,136 --> 00:39:13,391 zero is one over Q. So, here, the width in this 523 00:39:13,391 --> 00:39:17,070 presentation is directly inversely proportional to Q. 524 00:39:17,070 --> 00:39:20,113 So, if Q is ten, then the width there is one 525 00:39:20,113 --> 00:39:22,660 tenth. He then shows you another plot 526 00:39:22,660 --> 00:39:24,924 whereby he does what I did there. 527 00:39:24,924 --> 00:39:28,816 He plots it as a function of omega, not as a function of 528 00:39:28,816 --> 00:39:34,739 omega divided by omega zero. And then, he emphasizes the 529 00:39:34,739 --> 00:39:41,028 fact that the width here is that gamma that I mentioned at half 530 00:39:41,028 --> 00:39:46,811 the maximum power you get here [the width/with?] of gamma. 531 00:39:46,811 --> 00:39:52,797 And, this is a picture that I chose verbatim from your book. 532 00:39:52,797 --> 00:39:56,550 This is the best moment for the break. 533 00:39:56,550 --> 00:40:02,148 That means the mini quiz. I realize it's a bit early; 534 00:40:02,148 --> 00:40:05,251 we are only 40 minutes into the lecture. 535 00:40:05,251 --> 00:40:09,071 But it's a natural point. You will see what comes 536 00:40:09,071 --> 00:40:12,175 afterwards. It's better that we make the 537 00:40:12,175 --> 00:40:14,084 break now. So, therefore, 538 00:40:14,084 --> 00:40:18,779 I need some help from people who are willing to hand out the 539 00:40:18,779 --> 00:40:21,962 mini quiz. It would be nice if I can find 540 00:40:21,962 --> 00:40:24,668 the mini quizzes. I have them here, 541 00:40:24,668 --> 00:40:30,000 but someone took them. Oh, no, they are still there. 542 00:40:30,000 --> 00:40:35,578 I have a nice conspiracy. Afterwards, after the break, 543 00:40:35,578 --> 00:40:41,684 we will collect them this time in some boxes so that it's a 544 00:40:41,684 --> 00:40:47,684 little bit more organized. And so, I'm returning to an RLC 545 00:40:47,684 --> 00:40:53,578 circuit, which we discussed earlier, the good old days of 546 00:40:53,578 --> 00:40:56,842 8.02. I'm going to drive it now, 547 00:40:56,842 --> 00:41:04,000 not with the battery but with an alternating power supply. 548 00:41:04,000 --> 00:41:11,000 549 00:41:11,000 --> 00:41:14,000 V zero cosine omega T. 550 00:41:14,000 --> 00:41:18,000 551 00:41:18,000 --> 00:41:23,579 Yeah, put it in here, thank you. 552 00:41:23,579 --> 00:41:33,119 So, here's the circuit, resistor R, self inductance L, 553 00:41:33,119 --> 00:41:39,157 capacitor C. And, I have to write down, 554 00:41:39,157 --> 00:41:46,688 now, the differential equation. I will adopt a positive current 555 00:41:46,688 --> 00:41:51,789 in this direction. That will be my positive 556 00:41:51,789 --> 00:41:56,283 current. The charge here on this right 557 00:41:56,283 --> 00:42:00,534 plate I will call Q. And, therefore, 558 00:42:00,534 --> 00:42:06,000 by that definition, I is then DQ DT. 559 00:42:06,000 --> 00:42:10,680 Sign sensitive. I call the potential difference 560 00:42:10,680 --> 00:42:15,870 over this capacitor, in going from the right side to 561 00:42:15,870 --> 00:42:19,228 the left side, I call that V of C. 562 00:42:19,228 --> 00:42:25,333 That, then, is Q divided by C. All of that is sign sensitive. 563 00:42:25,333 --> 00:42:31,336 I go a round this circuit and I want to calculate the closed 564 00:42:31,336 --> 00:42:38,000 loop integral of E dot DL. And that closed loop integral 565 00:42:38,000 --> 00:42:43,172 of E dot DL is not zero, which many books tell you, 566 00:42:43,172 --> 00:42:47,827 even many professors tell you. It is not zero, 567 00:42:47,827 --> 00:42:53,413 but it is minus D phi DT. This is [UNINTELLIGIBLE] Law, 568 00:42:53,413 --> 00:42:59,000 and this runs our economy. Because of the magnetic flux 569 00:42:59,000 --> 00:43:04,586 change in closed loops, we can generate induced [EMS?], 570 00:43:04,586 --> 00:43:09,980 which run our economy. Look at the lights. 571 00:43:09,980 --> 00:43:15,818 Luckily, this is not zero. This [fires?] in magnetic flux 572 00:43:15,818 --> 00:43:21,969 that goes through a surface, any surface that you can attach 573 00:43:21,969 --> 00:43:26,764 to this closed loop. So, I've done this before. 574 00:43:26,764 --> 00:43:29,996 So, I can do it a little faster. 575 00:43:29,996 --> 00:43:35,000 I go from here to here. So, that is IR. 576 00:43:35,000 --> 00:43:39,653 There is no electric field inside this ideal self inductor 577 00:43:39,653 --> 00:43:44,061 because a superconducting wire cannot be an [E?] field. 578 00:43:44,061 --> 00:43:48,877 So, that is zero going from here to there when I go over the 579 00:43:48,877 --> 00:43:50,673 capacitor. I get my VC, 580 00:43:50,673 --> 00:43:55,571 and here, depending upon the phase, if I assume this plus and 581 00:43:55,571 --> 00:43:58,510 this minus, but you can reverse that. 582 00:43:58,510 --> 00:44:01,693 Then, I would get, when I walk into this 583 00:44:01,693 --> 00:44:07,000 direction, I would get minus V zero cosine omega T. 584 00:44:07,000 --> 00:44:11,284 But, if you feel like reversing it, I have no problem with that. 585 00:44:11,284 --> 00:44:13,596 That's just a matter of 180° phase. 586 00:44:13,596 --> 00:44:15,705 [It's known here from?] physics. 587 00:44:15,705 --> 00:44:18,085 And, this now equals minus D phi DT. 588 00:44:18,085 --> 00:44:21,962 The only thing where you apply [UNINTELLIGIBLE] law is you 589 00:44:21,962 --> 00:44:25,906 should always integrate in the direction that you have your 590 00:44:25,906 --> 00:44:28,695 current assumed, that it is minus L DI DT. 591 00:44:28,695 --> 00:44:32,707 If you do it in the opposite direction, then it is plus L DI 592 00:44:32,707 --> 00:44:36,047 DT. I have learned a certain 593 00:44:36,047 --> 00:44:39,802 discipline in my life. It took me many years. 594 00:44:39,802 --> 00:44:44,239 So, you have a long way to go. And I always go in the 595 00:44:44,239 --> 00:44:47,994 direction of I, so I never have to think that 596 00:44:47,994 --> 00:44:52,517 this is minus L DI DT. This now covers the minus D phi 597 00:44:52,517 --> 00:44:54,650 DT. So, now, what do I do? 598 00:44:54,650 --> 00:44:58,576 I bring the L in. And, I take one more time the 599 00:44:58,576 --> 00:45:02,074 derivative. And so, I get L times I double 600 00:45:02,074 --> 00:45:08,786 dot plus R times I dot plus VC. But I take the time derivative. 601 00:45:08,786 --> 00:45:13,191 So, the Q dot becomes I. So, I get I divided by C. 602 00:45:13,191 --> 00:45:18,404 And that now becomes the timed derivative of this function. 603 00:45:18,404 --> 00:45:23,707 But, it goes to the right side, which makes the minus sign a 604 00:45:23,707 --> 00:45:26,943 plus. But, when I take the derivative 605 00:45:26,943 --> 00:45:31,078 of the cosine omega T, I get a minus omega out. 606 00:45:31,078 --> 00:45:36,112 So, I get here minus P zero times omega times the sign of 607 00:45:36,112 --> 00:45:41,000 omega T. This is the differential 608 00:45:41,000 --> 00:45:48,625 equation that has to be solved. And, I will divide this out by 609 00:45:48,625 --> 00:45:52,625 L. I will divide everything by L. 610 00:45:52,625 --> 00:45:59,250 I will put the C a little higher, and so with R over L 611 00:45:59,250 --> 00:46:05,875 gamma, and with omega zero squared equals one over LC, 612 00:46:05,875 --> 00:46:11,125 this becomes, then, I double dot plus gamma 613 00:46:11,125 --> 00:46:18,000 times I dot plus omega zero squared times I. 614 00:46:18,000 --> 00:46:25,483 That now equals minus V zero divided by L times omega times 615 00:46:25,483 --> 00:46:30,645 the sign of omega T. But, here, you see a 616 00:46:30,645 --> 00:46:36,571 differential equation. And that differential equation 617 00:46:36,571 --> 00:46:39,571 looks amazingly similar to this one. 618 00:46:39,571 --> 00:46:43,000 And so, you should be able to solve that. 619 00:46:43,000 --> 00:46:46,685 In fact, you wouldn't even want to solve it. 620 00:46:46,685 --> 00:46:50,199 You can write down immediately the answer. 621 00:46:50,199 --> 00:46:53,971 You're going to get an I, which is an I zero, 622 00:46:53,971 --> 00:46:57,142 which takes the place of that A there. 623 00:46:57,142 --> 00:47:01,000 This is a steady state solution. 624 00:47:01,000 --> 00:47:06,767 I only go for steady state solution times the sign of omega 625 00:47:06,767 --> 00:47:10,546 T minus delta: no adjustable constants. 626 00:47:10,546 --> 00:47:15,917 It's a steady-state solution that I have, steady state. 627 00:47:15,917 --> 00:47:21,883 And, I will leave you to find me I zero, and you can work out 628 00:47:21,883 --> 00:47:26,060 what delta is. That is part of your problem 629 00:47:26,060 --> 00:47:29,939 set anyhow. But, with the knowledge that 630 00:47:29,939 --> 00:47:34,116 you have here, you could write it down in a 631 00:47:34,116 --> 00:47:39,480 manner seconds. So, without my telling you what 632 00:47:39,480 --> 00:47:43,320 I zero is, at least working it out algebraically, 633 00:47:43,320 --> 00:47:47,159 we can talk 8.02. And then, we can make all kinds 634 00:47:47,159 --> 00:47:51,239 of predictions without even looking at the equation. 635 00:47:51,239 --> 00:47:55,400 So, that's interesting. So, everything that I'm going 636 00:47:55,400 --> 00:47:59,480 to tell you now I do without knowing what I zero is. 637 00:47:59,480 --> 00:48:03,000 And, it better work out that way. 638 00:48:03,000 --> 00:48:06,753 Suppose I make omega go to zero. 639 00:48:06,753 --> 00:48:11,717 Remember 8.02? Remember the word reactants 640 00:48:11,717 --> 00:48:18,618 that a capacitor has a certain reactant, which is one over 641 00:48:18,618 --> 00:48:22,493 omega C, which has units of ohms? 642 00:48:22,493 --> 00:48:30,000 If omega goes to zero, these reactants go to infinity. 643 00:48:30,000 --> 00:48:36,547 No current can ever flow. So, I predict that I zero will 644 00:48:36,547 --> 00:48:41,428 go to zero. Suppose my omega goes to omega 645 00:48:41,428 --> 00:48:45,476 zero. Now, your memory may fail you 646 00:48:45,476 --> 00:48:49,880 here on 8.02. But, we have a wonderful 647 00:48:49,880 --> 00:48:56,666 demonstration in 8.02 that at resonance, one over omega C, 648 00:48:56,666 --> 00:49:02,857 which is the reactant of the capacitor minus omega L, 649 00:49:02,857 --> 00:49:10,000 which is the reactance of the inductor, is zero. 650 00:49:10,000 --> 00:49:13,450 That determines, actually, the resonance. 651 00:49:13,450 --> 00:49:18,282 And, when this is the case, perhaps you remember that the 652 00:49:18,282 --> 00:49:22,164 system doesn't even know there is a capacitor, 653 00:49:22,164 --> 00:49:26,392 and it doesn't even know there is a self inductor. 654 00:49:26,392 --> 00:49:32,000 The two at all moments in time exactly cancel each other. 655 00:49:32,000 --> 00:49:35,346 And therefore, Ohm's law holds. 656 00:49:35,346 --> 00:49:38,469 There is no L. There is no C. 657 00:49:38,469 --> 00:49:43,711 There is only the power supply and the resistor. 658 00:49:43,711 --> 00:49:50,403 And so, since Ohm's law says V equals IR, you must get I zero 659 00:49:50,403 --> 00:49:56,538 equals V zero divided by R. That is what you must get at 660 00:49:56,538 --> 00:50:01,000 resonance. It's nonnegotiable. 661 00:50:01,000 --> 00:50:05,253 Now, if omega goes to infinity, omega L says, 662 00:50:05,253 --> 00:50:08,250 ha, ha, yeah, over my dead body. 663 00:50:08,250 --> 00:50:12,601 No current ever, imagine, a very fast changing 664 00:50:12,601 --> 00:50:15,791 signal. That's what the whole self 665 00:50:15,791 --> 00:50:20,432 inductance is about. It doesn't want any changes. 666 00:50:20,432 --> 00:50:23,622 It's conservative like you and me. 667 00:50:23,622 --> 00:50:28,939 And so, the self inductance says, sorry, the reactant is 668 00:50:28,939 --> 00:50:33,000 infinitely high, no current. 669 00:50:33,000 --> 00:50:37,214 And so, I zero goes to zero. And so, I make these 670 00:50:37,214 --> 00:50:40,990 predictions. And that's always nice that you 671 00:50:40,990 --> 00:50:46,082 can use the knowledge to make predictions without even ever 672 00:50:46,082 --> 00:50:49,858 having looked at this differential equation. 673 00:50:49,858 --> 00:50:55,302 Any of these predictions I made to not come out of my knowledge 674 00:50:55,302 --> 00:51:00,395 of that differential equation. So, if we make now a plot of 675 00:51:00,395 --> 00:51:04,170 the current I zero, that is not the current, 676 00:51:04,170 --> 00:51:09,000 but that is the maximum possible current. 677 00:51:09,000 --> 00:51:15,327 It is that I zero that you see here without having solved it. 678 00:51:15,327 --> 00:51:21,654 I can look now what it's going to do exactly at I omega zero. 679 00:51:21,654 --> 00:51:26,505 It will go to a maximum. It will start at zero. 680 00:51:26,505 --> 00:51:31,883 I will go to a maximum, and then it will fall off at 681 00:51:31,883 --> 00:51:35,574 zero. And, this value here is V zero 682 00:51:35,574 --> 00:51:39,931 divided by R. This is not power. 683 00:51:39,931 --> 00:51:45,080 I have now plotted current. The power, of course, 684 00:51:45,080 --> 00:51:50,337 would go as I squared R. That is the heat that you 685 00:51:50,337 --> 00:51:56,344 dissipate in the resistor. So, that will go as I squared. 686 00:51:56,344 --> 00:52:01,065 I have plotted here I as a function of not T, 687 00:52:01,065 --> 00:52:06,000 but this is I as a function of omega. 688 00:52:06,000 --> 00:52:10,503 You know when you catch an error that I make, 689 00:52:10,503 --> 00:52:14,393 you get partial credit for this course. 690 00:52:14,393 --> 00:52:19,409 So please, when you see me make a mistake, scream. 691 00:52:19,409 --> 00:52:22,582 So, this is omega. Here is zero. 692 00:52:22,582 --> 00:52:27,086 And, this is what I want to demonstrate, now. 693 00:52:27,086 --> 00:52:32,000 I'm not going to show you I zero only. 694 00:52:32,000 --> 00:52:36,325 But, what I'm going to do is the following. 695 00:52:36,325 --> 00:52:41,887 I'm going to show you what I is as a function of omega, 696 00:52:41,887 --> 00:52:47,139 here being the resonance. Let us suppose I pick this 697 00:52:47,139 --> 00:52:50,023 omega. Well, then, this is my 698 00:52:50,023 --> 00:52:53,730 solution. So, yes, the amplitude is I 699 00:52:53,730 --> 00:52:57,335 zero. Here is the sine omega T minus 700 00:52:57,335 --> 00:53:00,116 delta. So, all you will see, 701 00:53:00,116 --> 00:53:03,000 then, is this. 702 00:53:03,000 --> 00:53:07,000 703 00:53:07,000 --> 00:53:10,507 Plus zero, minus plus zero, if I do it here, 704 00:53:10,507 --> 00:53:15,156 then it goes [MAKES NOISE]. And, the demonstration that we 705 00:53:15,156 --> 00:53:20,049 have prepared for you is one whereby we will sweep omega from 706 00:53:20,049 --> 00:53:25,269 zero to a value which I remember in terms of Hertz is about 2,000 707 00:53:25,269 --> 00:53:30,000 hertz, and we'll do that in one sixth of a second. 708 00:53:30,000 --> 00:53:35,940 And so, what you see is you see this as an envelope, 709 00:53:35,940 --> 00:53:41,997 which is the I zero envelope. But, you will see this. 710 00:53:41,997 --> 00:53:48,403 And then, [it speeds?] back. And so, you see two things. 711 00:53:48,403 --> 00:53:52,130 You'll see the sine omega T term. 712 00:53:52,130 --> 00:53:57,721 But, as omega changes, you will see it go through 713 00:53:57,721 --> 00:54:02,380 resonance. And then you'll see it go over 714 00:54:02,380 --> 00:54:07,039 resonance. So, I will give you the values 715 00:54:07,039 --> 00:54:13,950 that we have chosen. R equals 50 ohms. 716 00:54:13,950 --> 00:54:22,107 L equals 50 millihenry. And, we choose C 0.5 µF, 717 00:54:22,107 --> 00:54:32,000 which is substantially higher than what we did before. 718 00:54:32,000 --> 00:54:38,043 And, we choose C so high because we want a low Q system. 719 00:54:38,043 --> 00:54:42,879 Omega zero is now 6.3 times ten to the third. 720 00:54:42,879 --> 00:54:49,142 It is in radians per second. So, F zero is about 1,000 Hz. 721 00:54:49,142 --> 00:54:54,527 And, we're going to sweep it from zero over 1,000, 722 00:54:54,527 --> 00:54:58,043 which is resonant to about 2,000. 723 00:54:58,043 --> 00:55:04,382 And then we sweep it back. And so, the Q of this system, 724 00:55:04,382 --> 00:55:08,490 which is omega zero divided by gamma, is about 6.3. 725 00:55:08,490 --> 00:55:13,337 And, what the V zero is of that circuit is not so important. 726 00:55:13,337 --> 00:55:17,526 But it is [four fold?], but that's not so important. 727 00:55:17,526 --> 00:55:21,716 And so, what we're going to show you, we measure the 728 00:55:21,716 --> 00:55:25,577 potential difference over a very small resistor, 729 00:55:25,577 --> 00:55:31,000 which is somewhere in that circuit, 1.7 ohms I believe. 730 00:55:31,000 --> 00:55:36,219 And so, that potential difference over that resistor is 731 00:55:36,219 --> 00:55:38,539 IR. And, R is a constant, 732 00:55:38,539 --> 00:55:42,309 and that's what we're going to show you. 733 00:55:42,309 --> 00:55:47,433 So, we're going to show you something that is directly 734 00:55:47,433 --> 00:55:51,493 linear proportional with I. It's not power. 735 00:55:51,493 --> 00:55:54,296 It's I. We want to know power. 736 00:55:54,296 --> 00:56:00,000 You have to square it. I square R is the power. 737 00:56:00,000 --> 00:56:05,851 And, we're going to sweep it 1/6 of a second this way, 738 00:56:05,851 --> 00:56:11,261 and 1/6 of a second back. Why did I only take into 739 00:56:11,261 --> 00:56:14,905 account the steady state solution? 740 00:56:14,905 --> 00:56:20,867 Why don't I have to also include the trangent solution, 741 00:56:20,867 --> 00:56:27,160 which with this experiment took us five minutes to finally 742 00:56:27,160 --> 00:56:31,025 arrive at the steady state solution? 743 00:56:31,025 --> 00:56:36,647 Why am I leaving it out? I can't hear you. 744 00:56:36,647 --> 00:56:40,038 Where is the sound coming from? Yes. 745 00:56:40,038 --> 00:56:44,593 What is two over gamma, which is the decay time? 746 00:56:44,593 --> 00:56:49,244 One over EDK time? Well, there is two over gamma. 747 00:56:49,244 --> 00:56:52,441 I didn't write down what gamma is. 748 00:56:52,441 --> 00:56:55,639 Gamma is 1,000. Gamma is R over L, 749 00:56:55,639 --> 00:57:00,000 right? Gamma is R over L is 1,000. 750 00:57:00,000 --> 00:57:05,863 So, two over gamma, which is the one over EDK time, 751 00:57:05,863 --> 00:57:10,201 is 2 ms. Or, another way of putting it 752 00:57:10,201 --> 00:57:14,071 is that in about two oscillations, 753 00:57:14,071 --> 00:57:16,768 I'm down by a factor, E. 754 00:57:16,768 --> 00:57:22,280 Remember, Q over pi oscillations will reduce the 755 00:57:22,280 --> 00:57:28,260 amplitude by a factor of E. So, in two oscillations, 756 00:57:28,260 --> 00:57:36,000 already the trangent phenomenon is effectively killed. 757 00:57:36,000 --> 00:57:39,373 And so, I don't have to take it into account. 758 00:57:39,373 --> 00:57:43,054 You won't even notice it. So, I told you earlier, 759 00:57:43,054 --> 00:57:45,968 when the decay time here was very long. 760 00:57:45,968 --> 00:57:49,955 I'll show you another experiment where the decay time 761 00:57:49,955 --> 00:57:52,638 is extremely short. While I'm at it, 762 00:57:52,638 --> 00:57:56,626 the experiment is set up there. You will see it here. 763 00:57:56,626 --> 00:58:00,000 I'm going to make this 100 ohms. 764 00:58:00,000 --> 00:58:05,740 I'm going to double it to show you that this point will exactly 765 00:58:05,740 --> 00:58:09,814 go down by a factor of two because, remember, 766 00:58:09,814 --> 00:58:12,685 the peak is V zero divided by R. 767 00:58:12,685 --> 00:58:17,592 And, I'm not changing P zero. And so, when I double R, 768 00:58:17,592 --> 00:58:22,870 you will see it come down to here, and you will see it get 769 00:58:22,870 --> 00:58:26,018 broader. And then, I will go to 150 770 00:58:26,018 --> 00:58:30,000 ohms. That's easy to do for us. 771 00:58:30,000 --> 00:58:34,324 So, we go to 150 ohms. So, at 100 ohms, 772 00:58:34,324 --> 00:58:37,851 the Q is 3.15. And, at 150 ohms, 773 00:58:37,851 --> 00:58:42,858 the Q is three times smaller. It's about 2.1. 774 00:58:42,858 --> 00:58:47,979 So, at 150 ohms, it would be one third of this 775 00:58:47,979 --> 00:58:53,896 height, somewhere here. And the peak will be broader. 776 00:58:53,896 --> 00:59:00,382 And, all of that you get today very easily by changing the 777 00:59:00,382 --> 00:59:06,229 resistance. Again, this is not power. 778 00:59:06,229 --> 00:59:13,205 Power is I squared times R. This is simply the current. 779 00:59:13,205 --> 00:59:21,215 All right, I will give you the correct light setting because we 780 00:59:21,215 --> 00:59:27,028 have to make it a little darker. And, then we, 781 00:59:27,028 --> 00:59:32,181 and there it is. So, noticed the vertical 782 00:59:32,181 --> 00:59:37,272 oscillations go so fast that your eyes cannot even follow 783 00:59:37,272 --> 00:59:40,363 them. But, that's the sine omega T. 784 00:59:40,363 --> 00:59:45,090 And then, when it sweeps over resonance, you see very 785 00:59:45,090 --> 00:59:49,818 dramatically this value which is V zero divided by R. 786 00:59:49,818 --> 00:59:55,181 Now, notice on the scale here that Marcos has set the I zero 787 00:59:55,181 --> 00:59:58,181 at resonance at three scale units. 788 00:59:58,181 --> 1:00:03,454 So, it's nice when we go from our 50 ohms to 100 ohms to go 789 1:00:03,454 --> 1:00:09,013 down by a factor of two. And, you can check that. 790 1:00:09,013 --> 1:00:13,627 And when we go to 150 ohms, it should go down from three 791 1:00:13,627 --> 1:00:17,402 units to one unit. So, it could quantitatively 792 1:00:17,402 --> 1:00:20,842 check this. And, you'll see that the curve 793 1:00:20,842 --> 1:00:23,778 gets broader because it has a low Q. 794 1:00:23,778 --> 1:00:26,379 So, now I will make it 100 ohms. 795 1:00:26,379 --> 1:00:32,000 You see, it's down by a factor of two from here to here. 796 1:00:32,000 --> 1:00:36,446 And, you may have noticed that it also gets broader. 797 1:00:36,446 --> 1:00:40,196 And now, I will go to 150 ohms. And you see, 798 1:00:40,196 --> 1:00:45,079 it's down by a factor of three. And again, it is broader. 799 1:00:45,079 --> 1:00:50,136 So, this is an amazing way how with RLC circuits you can do 800 1:00:50,136 --> 1:00:55,193 wonderful things because you can manipulate omega zero very 801 1:00:55,193 --> 1:00:59,901 easily by changing L and C. And, you can manipulate the 802 1:00:59,901 --> 1:01:04,000 driving frequency also very easily. 803 1:01:04,000 --> 1:01:09,682 So, it is clear that systems respond strongly when they're 804 1:01:09,682 --> 1:01:13,271 exposed to their resonance frequency. 805 1:01:13,271 --> 1:01:19,052 We've seen that for pendulums. We've seen that for springs. 806 1:01:19,052 --> 1:01:24,436 We've seen at 401 glass. And we've seen this now for an 807 1:01:24,436 --> 1:01:28,623 RLC circuit. So, these systems at resonance 808 1:01:28,623 --> 1:01:36,000 absorb a large amount of energy per unit time out of the driver. 809 1:01:36,000 --> 1:01:43,533 I have here two tuning forks, which have an extremely high Q. 810 1:01:43,533 --> 1:01:50,690 My attempts to measure it, I conclude it's way larger than 811 1:01:50,690 --> 1:01:55,838 even 1,000. And they have exactly the same 812 1:01:55,838 --> 1:02:01,488 frequency: both 256 Hz, this one and this one, 813 1:02:01,488 --> 1:02:06,374 both 256 Hz. That's' the way they are 814 1:02:06,374 --> 1:02:11,717 designed to a high degree of accuracy, to better than a 815 1:02:11,717 --> 1:02:16,268 fraction of 1 Hz. But, the Q's are so high that 816 1:02:16,268 --> 1:02:21,314 if you were to plot, if you drive these tuning forks 817 1:02:21,314 --> 1:02:27,151 and you are to plot here this average power as a function of 818 1:02:27,151 --> 1:02:32,000 omega, then you get something like this. 819 1:02:32,000 --> 1:02:36,468 It means you have to drive it exactly at the right frequency. 820 1:02:36,468 --> 1:02:39,446 Otherwise, it will not go into resonance. 821 1:02:39,446 --> 1:02:42,053 Well, we know how to get this going. 822 1:02:42,053 --> 1:02:45,478 You just bang it. That means you dump the whole 823 1:02:45,478 --> 1:02:48,904 spectrum on it. It picks out the frequency that 824 1:02:48,904 --> 1:02:53,148 likes, and now I'm going to show you something remarkable. 825 1:02:53,148 --> 1:02:56,276 When this one generates 256 pressure waves, 826 1:02:56,276 --> 1:03:00,000 this one feels those pressure waves. 827 1:03:00,000 --> 1:03:05,914 And it loves it because it's just at the right frequency. 828 1:03:05,914 --> 1:03:11,933 And so, it starts oscillate. And so, when I stop this one, 829 1:03:11,933 --> 1:03:17,003 you'll hear this one. And that's called resonance 830 1:03:17,003 --> 1:03:20,277 absorption. Let's do that first. 831 1:03:20,277 --> 1:03:26,613 Now, you must understand that the sound waves go from here to 832 1:03:26,613 --> 1:03:30,099 there. Not much power reaches that 833 1:03:30,099 --> 1:03:33,785 point. So, when I stop this one, 834 1:03:33,785 --> 1:03:36,642 you hear sound but it's not overwhelming. 835 1:03:36,642 --> 1:03:39,000 So, you have to be very quiet. 836 1:03:39,000 --> 1:03:48,000 837 1:03:48,000 --> 1:03:49,000 Hear it? 838 1:03:49,000 --> 1:03:53,000 839 1:03:53,000 --> 1:03:55,928 And I can do the same by hitting this one, 840 1:03:55,928 --> 1:03:59,000 and then this one will start to resonate. 841 1:03:59,000 --> 1:04:06,000 842 1:04:06,000 --> 1:04:12,059 Now, if the driving frequency is off by a fraction of the 843 1:04:12,059 --> 1:04:17,578 hertz, 1 Hz is enough, 1 Hz difference because the Q 844 1:04:17,578 --> 1:04:23,962 is so high that this system will not be able to get this one 845 1:04:23,962 --> 1:04:27,858 going. And, I can make this frequency 846 1:04:27,858 --> 1:04:35,000 a little lower than 256 Hz by putting this weight on here. 847 1:04:35,000 --> 1:04:39,607 And, we have measured the frequency at this loaded way. 848 1:04:39,607 --> 1:04:43,617 It's roughly 255 Hz. And, you are somewhere here 849 1:04:43,617 --> 1:04:48,054 because it's so narrow. The resonance absorption peak 850 1:04:48,054 --> 1:04:51,382 is very sharp. By the way, this is power 851 1:04:51,382 --> 1:04:55,307 because what reaches here is joules per second. 852 1:04:55,307 --> 1:05:00,000 That's what gets it going, energy per second. 853 1:05:00,000 --> 1:05:03,467 So it's really a power transfer. 854 1:05:03,467 --> 1:05:08,723 So now, I change the frequency, and there we go. 855 1:05:08,723 --> 1:05:12,750 Nothing. But just change this by 1 Hz 856 1:05:12,750 --> 1:05:15,657 and you hear nothing. Dead. 857 1:05:15,657 --> 1:05:20,690 So, now you see, you get some respect for high 858 1:05:20,690 --> 1:05:24,493 Q's. If you want to get a resonance 859 1:05:24,493 --> 1:05:31,315 absorption in the high Q system. You've got to be dead on that 860 1:05:31,315 --> 1:05:36,536 frequency. So, if you, for instance, 861 1:05:36,536 --> 1:05:42,014 banged all the keys on the piano, and this one would be 862 1:05:42,014 --> 1:05:48,304 nearby, it would only start to resonate if one of those strings 863 1:05:48,304 --> 1:05:51,550 would produce exactly the 256 Hz. 864 1:05:51,550 --> 1:05:56,217 Otherwise, it would not. It ignores everything. 865 1:05:56,217 --> 1:06:02,000 It's only sensitive to that resonance frequency. 866 1:06:02,000 --> 1:06:07,289 You probably in high school have learned a little bit about 867 1:06:07,289 --> 1:06:11,120 atomic physics. And, you probably know that 868 1:06:11,120 --> 1:06:16,592 electrons have discrete energy levels and discrete orbits and 869 1:06:16,592 --> 1:06:19,785 atoms. And, you can excite the atom. 870 1:06:19,785 --> 1:06:23,706 You can bring an electron in a higher orbit, 871 1:06:23,706 --> 1:06:30,000 discrete orbits which cost you a discrete amount of energy. 872 1:06:30,000 --> 1:06:34,308 And, when the atom recombines, when the electron falls back, 873 1:06:34,308 --> 1:06:38,178 you get that energy back, exactly the same amount that 874 1:06:38,178 --> 1:06:41,610 you had to put in. And, that energy that you get 875 1:06:41,610 --> 1:06:45,845 back comes out most of the time in the form of what we call 876 1:06:45,845 --> 1:06:49,643 electromagnetic radiation. I know that in 8.03 we are 877 1:06:49,643 --> 1:06:53,294 going to deal with electromagnetic radiation in the 878 1:06:53,294 --> 1:06:55,922 future. But, it's enough for now that 879 1:06:55,922 --> 1:06:58,332 you know that light, infrared, UV, 880 1:06:58,332 --> 1:07:01,691 gamma rays, x-rays, all of that radio emission, 881 1:07:01,691 --> 1:07:06,000 all of that is electromagnetic radiation. 882 1:07:06,000 --> 1:07:12,139 And so, here I have the energy level, energy increasing, 883 1:07:12,139 --> 1:07:18,390 here with an electron in orbit. That's the energy of that 884 1:07:18,390 --> 1:07:22,855 electron. A higher energy state is when I 885 1:07:22,855 --> 1:07:28,213 bring this electron here. I cannot do anything in 886 1:07:28,213 --> 1:07:34,078 between. Quantum mechanics says it's one 887 1:07:34,078 --> 1:07:39,210 or the other. And, if this difference is 888 1:07:39,210 --> 1:07:45,394 delta E in energy, this E stands now for energy. 889 1:07:45,394 --> 1:07:52,236 Then, when the electron falls back from here to here, 890 1:07:52,236 --> 1:08:00,000 it emits electromagnetic radiation with this energy. 891 1:08:00,000 --> 1:08:06,781 But, if I radiate onto this atom, electromagnetic radiation 892 1:08:06,781 --> 1:08:13,212 with exactly that energy, then this electron can go from 893 1:08:13,212 --> 1:08:18,123 here to there. And that is called resonance 894 1:08:18,123 --> 1:08:21,513 absorption. Now, let us stick, 895 1:08:21,513 --> 1:08:28,178 for now, to visible light. The higher the energy the bluer 896 1:08:28,178 --> 1:08:33,089 the light is, or as modern physicists would 897 1:08:33,089 --> 1:08:40,777 say, the higher the frequency. And the lower the energy, 898 1:08:40,777 --> 1:08:45,777 the redder the light, the lower the frequency. 899 1:08:45,777 --> 1:08:52,333 So, our visible light that we can see with our eyes goes all 900 1:08:52,333 --> 1:08:57,555 the way from the red, low energy, to the violet, 901 1:08:57,555 --> 1:09:02,000 high-energy. Let's go to the sun. 902 1:09:02,000 --> 1:09:07,561 The sun radiates in the visible spectrum all the way from the 903 1:09:07,561 --> 1:09:11,918 red to the violet. But, in the solar atmosphere, 904 1:09:11,918 --> 1:09:15,533 R elements. And, when these elements see 905 1:09:15,533 --> 1:09:20,817 just the right energy from that spectrum to which they are 906 1:09:20,817 --> 1:09:26,471 exposed, they love to take out of that spectrum just the right 907 1:09:26,471 --> 1:09:30,457 energy that gets them into an excited state, 908 1:09:30,457 --> 1:09:35,000 which is called resonance absorption. 909 1:09:35,000 --> 1:09:39,651 And so, that energy is removed from the spectrum. 910 1:09:39,651 --> 1:09:45,658 So, when you look at the solar spectrum, there are bands in the 911 1:09:45,658 --> 1:09:49,147 spectrum, but the colors are missing. 912 1:09:49,147 --> 1:09:54,379 Absorption in the spectrum, dark bends in the spectrum. 913 1:09:54,379 --> 1:10:00,000 They were discovered in 1802 by William Wollaston. 914 1:10:00,000 --> 1:10:04,300 And in 1814, Fraunhofer had cataloged 475 of 915 1:10:04,300 --> 1:10:08,399 these lines. And, they are now referred to 916 1:10:08,399 --> 1:10:13,899 as Fraunhofer absorption lines. Even though they did not 917 1:10:13,899 --> 1:10:18,899 understand the physics, this is a quantum mechanics 918 1:10:18,899 --> 1:10:23,600 picture that came from Niels Bohr, 20th Century, 919 1:10:23,600 --> 1:10:28,899 even though they did not understand what happened they 920 1:10:28,899 --> 1:10:34,399 had noticed that these black lines in the solar spectrum 921 1:10:34,399 --> 1:10:40,199 coincided with emission lines in the spectrum that they can 922 1:10:40,199 --> 1:10:48,000 generate in the laboratory by heating up the various elements. 923 1:10:48,000 --> 1:10:52,765 And so, without understanding why, they were able to say, 924 1:10:52,765 --> 1:10:57,617 ah, I see magnesium in the sun. I see aluminum in the sun, 925 1:10:57,617 --> 1:11:02,991 in the solar atmosphere. And so, that opened a whole new 926 1:11:02,991 --> 1:11:08,975 industry of spectroscopy which allowed astronomers to determine 927 1:11:08,975 --> 1:11:13,994 the chemical composition of the atmospheres of stars. 928 1:11:13,994 --> 1:11:19,978 And, it was in 1868 that Joseph Lockyer found at least one dark 929 1:11:19,978 --> 1:11:25,383 line which did not coincide with any emission line in the 930 1:11:25,383 --> 1:11:29,147 laboratory. There is no element on Earth 931 1:11:29,147 --> 1:11:33,780 that he could say, that must be the cause of that 932 1:11:33,780 --> 1:11:38,710 dark line. And so, he calls it helium 933 1:11:38,710 --> 1:11:43,296 because the Greek word for the sun is helios. 934 1:11:43,296 --> 1:11:49,446 So, it helium is an element that was first discovered on the 935 1:11:49,446 --> 1:11:53,407 sun before it was later found on Earth. 936 1:11:53,407 --> 1:11:59,348 I want to show you resonance absorption on the scale of an 937 1:11:59,348 --> 1:12:02,788 atom. And, the way I'm going to do 938 1:12:02,788 --> 1:12:08,000 that, the setup is here, is as follows. 939 1:12:08,000 --> 1:12:13,230 We have a carbon [arc?]. Think of that as being the sun, 940 1:12:13,230 --> 1:12:17,794 which produces a spectrum, a beautiful continuous 941 1:12:17,794 --> 1:12:21,407 spectrum. I will show you that spectrum 942 1:12:21,407 --> 1:12:25,021 all the way from the red to the violet. 943 1:12:25,021 --> 1:12:30,631 And then we have here a burner. And we're going to put table 944 1:12:30,631 --> 1:12:35,291 salt here on the grid, which dissociates the table 945 1:12:35,291 --> 1:12:41,943 salts, gives me sodium gas. That's what I want because 946 1:12:41,943 --> 1:12:48,283 sodium, when you heat it, can produce an emission line in 947 1:12:48,283 --> 1:12:52,132 yellow. But if you can produce that 948 1:12:52,132 --> 1:12:58,358 emission line when the electron goes from here to there, 949 1:12:58,358 --> 1:13:03,000 it's the 11th electron by the way. 950 1:13:03,000 --> 1:13:07,135 It's the most outer electron of sodium, 11 protons, 951 1:13:07,135 --> 1:13:10,195 11 electrons. So, if it can produce an 952 1:13:10,195 --> 1:13:13,834 emission line when it goes from here to here, 953 1:13:13,834 --> 1:13:16,563 it can also, resonance absorption, 954 1:13:16,563 --> 1:13:20,864 namely when it sees that yellow line, the energy that 955 1:13:20,864 --> 1:13:25,330 corresponds with the yellow line, it sucks it up and it 956 1:13:25,330 --> 1:13:28,804 produces, then, a dark line because when it 957 1:13:28,804 --> 1:13:31,864 absorbs out of here, this yellow line, 958 1:13:31,864 --> 1:13:36,000 it re-emits it almost immediately. 959 1:13:36,000 --> 1:13:39,333 But the re-emission will be in all directions. 960 1:13:39,333 --> 1:13:43,703 And so, what's left over here is very little of that yellow. 961 1:13:43,703 --> 1:13:46,888 And so, a dark line appears in the spectrum. 962 1:13:46,888 --> 1:13:49,259 That then is the absorption line. 963 1:13:49,259 --> 1:13:53,037 Its power, what I'm going to show you, because light 964 1:13:53,037 --> 1:13:57,777 intensity, which I will show you there on that screen is how many 965 1:13:57,777 --> 1:14:02,988 joules per second? So, it is resonance absorption 966 1:14:02,988 --> 1:14:06,077 of power. Now, there is a catch. 967 1:14:06,077 --> 1:14:12,254 And the catch is that probably only you up here will be able to 968 1:14:12,254 --> 1:14:15,741 see it. And, others could come down. 969 1:14:15,741 --> 1:14:21,121 You're going to see the spectrum here first of the sun, 970 1:14:21,121 --> 1:14:27,000 which is my carbon arc. Then I will put in sodium. 971 1:14:27,000 --> 1:14:30,594 And you will see an unbelievable, 972 1:14:30,594 --> 1:14:35,311 unimaginable, beautiful, sharp light like a 973 1:14:35,311 --> 1:14:40,590 razor in the yellow. But you've got to be close. 974 1:14:40,590 --> 1:14:45,644 So, let's first, Marcos, if you manage to open 975 1:14:45,644 --> 1:14:51,148 the gas, he knows exactly where that gas valve is, 976 1:14:51,148 --> 1:14:58,000 then I will ignite it. We won't put it yet in beam. 977 1:14:58,000 --> 1:15:04,000 978 1:15:04,000 --> 1:15:08,684 OK, so we're going to make it completely dark. 979 1:15:08,684 --> 1:15:12,118 So, here are the [sold?] crystals. 980 1:15:12,118 --> 1:15:17,011 And, we're going to show you the spectrum there. 981 1:15:17,011 --> 1:15:21,070 We try to actually make you see it here. 982 1:15:21,070 --> 1:15:25,442 [UNINTELLIGIBLE] that didn't work out well. 983 1:15:25,442 --> 1:15:29,085 OK, so I'm going to turn on the sun. 984 1:15:29,085 --> 1:15:34,199 There's the sun. OK, now we make it completely 985 1:15:34,199 --> 1:15:38,448 dark, and I will give you a minute or so for your eyes to 986 1:15:38,448 --> 1:15:41,103 adjust. So, you see a spectrum here; 987 1:15:41,103 --> 1:15:45,275 you see a spectrum there. How we do that is our problem. 988 1:15:45,275 --> 1:15:49,675 And, you will know how we do that in a month or so when you 989 1:15:49,675 --> 1:15:53,393 will learn about gradings. You will get a grading, 990 1:15:53,393 --> 1:15:56,579 actually, from us. There is a grading here, 991 1:15:56,579 --> 1:16:00,979 a wonderful piece of physics, which decomposes the light in 992 1:16:00,979 --> 1:16:05,000 colors, works way better than a prism. 993 1:16:05,000 --> 1:16:08,585 And, you get one on the right side. 994 1:16:08,585 --> 1:16:13,225 And, you get a mirror image on the left side. 995 1:16:13,225 --> 1:16:16,810 Look here and let your eyes adjust. 996 1:16:16,810 --> 1:16:20,396 And then comes the moment of truth. 997 1:16:20,396 --> 1:16:24,509 I'm going to put in here now the sodium. 998 1:16:24,509 --> 1:16:27,778 Unbelievable. I see a line here, 999 1:16:27,778 --> 1:16:31,785 sharp as a razor blade. Can you see it, 1000 1:16:31,785 --> 1:16:36,000 Nicole? Isn't it incredible? 1001 1:16:36,000 --> 1:16:39,600 Come closer. All of you, come closer; 1002 1:16:39,600 --> 1:16:43,899 look at that line. Come on; come out of your 1003 1:16:43,899 --> 1:16:46,300 seats. Look at that line, 1004 1:16:46,300 --> 1:16:52,000 and I will move the sodium out. Now I'm at the sodium out. 1005 1:16:52,000 --> 1:16:56,300 And now I move it in again. And there it is. 1006 1:16:56,300 --> 1:17:00,000 There it is. You see that? 1007 1:17:00,000 --> 1:17:03,843 Isn't that amazing? And now I move it out. 1008 1:17:03,843 --> 1:17:06,000 And now I move it in. 1009 1:17:06,000 --> 1:17:12,000 1010 1:17:12,000 --> 1:17:47,550 Isn't that a fantastic line? Look at that line. 1011 1:17:47,550 --> 1:18:21,555 Look at that line. Resonance absorption of an 1012 1:18:21,555 --> 1:18:54,015 extremely high Q system on an atomic scale. 1013 1:18:54,015 --> 1:19:16,427 I hope you can sleep tonight. [LAUGHTER] See you Thursday.