WEBVTT 00:00:02.110 --> 00:00:04.480 The following content is provided under a Creative 00:00:04.480 --> 00:00:05.870 Commons license. 00:00:05.870 --> 00:00:08.080 Your support will help MIT OpenCourseWare 00:00:08.080 --> 00:00:12.170 continue to offer high quality educational resources for free. 00:00:12.170 --> 00:00:14.710 To make a donation or to view additional materials 00:00:14.710 --> 00:00:18.670 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:18.670 --> 00:00:19.870 at ocw.mit.edu. 00:00:23.000 --> 00:00:25.350 YEN-JIE LEE: OK, so welcome back everybody. 00:00:25.350 --> 00:00:27.850 Happy to see you again. 00:00:27.850 --> 00:00:33.210 So today, we are going to continue our exploration 00:00:33.210 --> 00:00:38.830 and understand single harmonic oscillator. 00:00:38.830 --> 00:00:42.690 And this is actually a list of our goals. 00:00:42.690 --> 00:00:45.420 And we would like to learn how to translate 00:00:45.420 --> 00:00:49.560 physical situations into mathematics so that we can 00:00:49.560 --> 00:00:51.670 actually solve the physical problem, 00:00:51.670 --> 00:00:54.510 so we actually have and predict what 00:00:54.510 --> 00:00:56.760 is going to happen afterward. 00:00:56.760 --> 00:01:03.210 And we also sort of started this course 00:01:03.210 --> 00:01:06.440 by solving really simple examples, 00:01:06.440 --> 00:01:09.000 single harmonic oscillators. 00:01:09.000 --> 00:01:11.400 And as a function of time, you will 00:01:11.400 --> 00:01:15.060 see that, for our next class, the next lecture, 00:01:15.060 --> 00:01:17.460 we are going to bring in more and more objects. 00:01:17.460 --> 00:01:21.150 And of course, more objects means more excitement also, 00:01:21.150 --> 00:01:24.060 in terms of phenomena, but also more complication 00:01:24.060 --> 00:01:26.610 on the mathematics. 00:01:26.610 --> 00:01:28.990 So we will see how things go. 00:01:28.990 --> 00:01:30.600 And then after that, we are going 00:01:30.600 --> 00:01:33.420 to go through infinite number of oscillators 00:01:33.420 --> 00:01:35.670 to see what will happen. 00:01:35.670 --> 00:01:38.340 Of course, we will produce waves. 00:01:38.340 --> 00:01:39.690 That's very exciting. 00:01:39.690 --> 00:01:42.390 Then we'll do all kinds of different tricks 00:01:42.390 --> 00:01:45.500 to do those waves. 00:01:45.500 --> 00:01:49.260 So what we have we learned last time? 00:01:49.260 --> 00:01:51.740 So last time we went over example, 00:01:51.740 --> 00:01:53.740 a simple harmonic oscillator. 00:01:53.740 --> 00:01:56.100 It says you have a rod fixed on the wall, 00:01:56.100 --> 00:01:58.750 and you can actually go back and forth. 00:01:58.750 --> 00:02:03.390 And we also introduced a model of the drag force, or drag 00:02:03.390 --> 00:02:07.200 torque, and that's actually proportionate to the velocity 00:02:07.200 --> 00:02:10.050 of the motion of that single particle. 00:02:10.050 --> 00:02:12.840 And the interesting thing we learned last time 00:02:12.840 --> 00:02:18.520 is that we have three completely different behaviors 00:02:18.520 --> 00:02:21.660 if we actually turn on the drag force. 00:02:21.660 --> 00:02:24.880 The first one is on that damped. 00:02:24.880 --> 00:02:26.790 Damping is actually very small. 00:02:26.790 --> 00:02:29.680 Then we have the solution in this form. 00:02:29.680 --> 00:02:31.580 It's oscillating. 00:02:31.580 --> 00:02:34.740 The amplitude is decaying exponentially. 00:02:38.790 --> 00:02:42.750 As we make the drag force larger and larger, 00:02:42.750 --> 00:02:47.850 you will pass a critical point, which actually give you 00:02:47.850 --> 00:02:51.290 a solution, which you don't have oscillation anymore. 00:02:51.290 --> 00:02:53.550 The cosine disappeared. 00:02:53.550 --> 00:02:59.760 Finally, if you actually put the whole system into water 00:02:59.760 --> 00:03:02.700 or introduce something really dramatic-- 00:03:02.700 --> 00:03:05.490 a very, very big drag force-- 00:03:05.490 --> 00:03:09.160 then you have overdamp situation. 00:03:09.160 --> 00:03:11.660 And there you see that the solution is actually 00:03:11.660 --> 00:03:15.490 a sum of two exponential functions. 00:03:15.490 --> 00:03:19.800 So this is actually the one equation 00:03:19.800 --> 00:03:23.850 which actually works for all the damped situation we discussed 00:03:23.850 --> 00:03:25.370 up to now. 00:03:25.370 --> 00:03:27.330 And this is actually the map. 00:03:27.330 --> 00:03:31.200 Basically, if gamma goes to zero-- 00:03:31.200 --> 00:03:34.100 gamma actually controls the size of the drag force. 00:03:34.100 --> 00:03:35.430 Then we got no damping. 00:03:35.430 --> 00:03:38.600 Then you have a pure, simple harmonic motion. 00:03:38.600 --> 00:03:40.950 And as we increase the gamma, then you 00:03:40.950 --> 00:03:44.780 get see that the behavior is changing 00:03:44.780 --> 00:03:48.690 as we increase the gamma. 00:03:48.690 --> 00:03:52.590 So you can see that we can use a quantity, which 00:03:52.590 --> 00:03:57.705 is called Q. Q is actually defined as a ratio of omega 00:03:57.705 --> 00:04:01.680 at zero, which is basically the natural angular 00:04:01.680 --> 00:04:04.500 frequency of the system. 00:04:04.500 --> 00:04:10.230 And gamma is a measure of how big the drag force is. 00:04:10.230 --> 00:04:13.110 If we make a ratio of this to quantity, 00:04:13.110 --> 00:04:17.959 then you'll see that, at Q equal to 0.5, 00:04:17.959 --> 00:04:22.125 it reaches a critical point, which actually the behavior 00:04:22.125 --> 00:04:24.570 of the whole system changed. 00:04:24.570 --> 00:04:27.150 And you can see that the oscillation completely 00:04:27.150 --> 00:04:29.400 disappeared. 00:04:29.400 --> 00:04:33.450 So that is actually what we have learned last time. 00:04:33.450 --> 00:04:37.280 So what are we going to do today? 00:04:37.280 --> 00:04:42.940 We have been really doing experiments really 00:04:42.940 --> 00:04:45.410 with our hands, hands-on, right? 00:04:45.410 --> 00:04:48.170 So basically we will prepare the system. 00:04:48.170 --> 00:04:49.850 Then we release it. 00:04:49.850 --> 00:04:53.330 Then we don't touch it again and see how this system actually 00:04:53.330 --> 00:04:55.380 evolves as a function of time. 00:04:55.380 --> 00:04:57.500 So that's what we have been doing. 00:04:57.500 --> 00:05:00.530 So today, what we are going to do 00:05:00.530 --> 00:05:04.100 is to start to drive this system. 00:05:04.100 --> 00:05:07.250 We can introduce some kind of driving force 00:05:07.250 --> 00:05:12.230 and see how the system will respond to this external force. 00:05:12.230 --> 00:05:14.320 So that is actually what we are going to do. 00:05:14.320 --> 00:05:17.030 And that will bring us to the situation 00:05:17.030 --> 00:05:21.380 of damped driven harmonic oscillator. 00:05:21.380 --> 00:05:23.220 So let's immediately get started. 00:05:27.810 --> 00:05:32.280 So we will use the example which we went through last time 00:05:32.280 --> 00:05:34.110 as a starting point. 00:05:34.110 --> 00:05:38.270 So set example from the last lecture 00:05:38.270 --> 00:05:42.730 is a rod, which is fixed on the wall. 00:05:42.730 --> 00:05:47.040 And the lens of this rod is over two. 00:05:47.040 --> 00:05:53.130 And I define a counter-clockwise direction to be positive. 00:05:53.130 --> 00:05:55.650 And I measure the position of the rod 00:05:55.650 --> 00:05:57.660 by this theta, which is the angle 00:05:57.660 --> 00:06:01.770 between the vertical direction and the pointing direction 00:06:01.770 --> 00:06:04.110 of the rod. 00:06:04.110 --> 00:06:06.890 And we have went through with the math, 00:06:06.890 --> 00:06:09.130 and we got the equation of motion 00:06:09.130 --> 00:06:12.660 without external force, which is already 00:06:12.660 --> 00:06:14.850 shown on the blackboard. 00:06:14.850 --> 00:06:17.560 So now, as I mentioned at the beginning, 00:06:17.560 --> 00:06:20.250 I would like to add a driving force, 00:06:20.250 --> 00:06:23.835 or driving torque, tau drive. 00:06:27.030 --> 00:06:36.490 This is equal to d0 cosine omega d t. 00:06:36.490 --> 00:06:43.494 So I am adding a driving torque. 00:06:43.494 --> 00:06:48.520 The amplitude of the torque is actually d0. 00:06:48.520 --> 00:06:52.530 And there's actually also a harmonic oscillating force, 00:06:52.530 --> 00:06:57.670 or torque, and the angle frequency of this torque 00:06:57.670 --> 00:06:58.270 is omega d. 00:07:01.570 --> 00:07:06.700 And that means our total torque, tau of t, 00:07:06.700 --> 00:07:10.120 will be equal to tau g t-- 00:07:10.120 --> 00:07:13.360 this is actually coming from the gravitational force-- 00:07:13.360 --> 00:07:19.495 plus tau drag, which is to account for the drag force. 00:07:22.810 --> 00:07:25.990 So this time we are adding in a tau drag. 00:07:30.440 --> 00:07:34.190 So I'm not going to go over all the calculations on how 00:07:34.190 --> 00:07:37.080 did the right from the beginning to the end. 00:07:37.080 --> 00:07:39.500 But I will just continue from what we actually 00:07:39.500 --> 00:07:40.940 started the last time. 00:07:40.940 --> 00:07:46.100 So if I have additional driving torque there, 00:07:46.100 --> 00:07:50.030 that means my equation of motion will be slightly modified. 00:07:50.030 --> 00:07:51.920 This time, my equation of motion will 00:07:51.920 --> 00:08:00.980 become theta double dot plus gamma theta dot plus omega 0 00:08:00.980 --> 00:08:06.330 squared theta, and that is equal to d0 divided 00:08:06.330 --> 00:08:11.540 by I. This is actually divided by I because, in order 00:08:11.540 --> 00:08:13.490 to get the acceleration, I'll need 00:08:13.490 --> 00:08:16.970 to divide my torque by a movement of inertia 00:08:16.970 --> 00:08:21.960 of this system and cosine omega d t. 00:08:21.960 --> 00:08:25.520 This is actually the oscillating frequency 00:08:25.520 --> 00:08:28.910 of the driving torque. 00:08:28.910 --> 00:08:37.100 And just a reminder, gamma is defined to be equal to 3b m l 00:08:37.100 --> 00:08:38.270 squared. 00:08:38.270 --> 00:08:41.270 And the omega 0 is actually defined 00:08:41.270 --> 00:08:48.350 to be square root of 3g over 2l. 00:08:48.350 --> 00:08:50.810 So as I mentioned in the beginning, 00:08:50.810 --> 00:08:52.490 this is actually giving you a sense 00:08:52.490 --> 00:08:55.850 of the size of the drag force. 00:08:55.850 --> 00:08:59.060 And the right hand side, the omega 0 00:08:59.060 --> 00:09:04.190 is actually the natural angular frequency. 00:09:04.190 --> 00:09:06.470 So, of course, we can actually simplify this 00:09:06.470 --> 00:09:12.170 by replacing this term, or this constant, by symbol. 00:09:12.170 --> 00:09:15.350 So the symbol I'm choosing is f0. 00:09:15.350 --> 00:09:23.390 And this is defined to be d0 divided by I. Therefore, 00:09:23.390 --> 00:09:26.930 I arrive at my final equation of motion-- 00:09:26.930 --> 00:09:35.390 theta double dot plus gamma theta dot plus omega 0 00:09:35.390 --> 00:09:44.294 squared theta, and that is equal to f0 cosine omega d t. 00:09:48.770 --> 00:09:51.810 So I hope this looks pretty straightforward to you. 00:09:55.330 --> 00:09:57.140 So this is our equation of motion 00:09:57.140 --> 00:09:59.650 you can see from this slide. 00:09:59.650 --> 00:10:02.890 So we have three terms in addition 00:10:02.890 --> 00:10:04.780 to the theta double dot. 00:10:04.780 --> 00:10:06.730 The first one is actually related 00:10:06.730 --> 00:10:09.700 drag force, or drag torque. 00:10:09.700 --> 00:10:12.820 The second one is actually related to so-called spring 00:10:12.820 --> 00:10:13.840 force. 00:10:13.840 --> 00:10:17.920 So that is actually be related to the spring constant 00:10:17.920 --> 00:10:21.790 or, because of the restoring force 00:10:21.790 --> 00:10:26.470 of the gravitational force. 00:10:26.470 --> 00:10:28.715 The third one is actually what we just add in 00:10:28.715 --> 00:10:31.540 as a driving force. 00:10:31.540 --> 00:10:37.600 So one question which I would like to ask you is-- 00:10:37.600 --> 00:10:42.440 so now I bring one more complication to this system. 00:10:42.440 --> 00:10:46.990 So now I am driving this system with a different frequency, 00:10:46.990 --> 00:10:49.232 which is omega d. 00:10:49.232 --> 00:10:52.920 The question is, what would be the resulting oscillation 00:10:52.920 --> 00:10:57.810 frequency of this driven harmonic oscillator? 00:10:57.810 --> 00:11:00.350 What is going to happen? 00:11:00.350 --> 00:11:03.950 Well, this system actually follows 00:11:03.950 --> 00:11:09.950 the original damped oscillator frequency, omega, which 00:11:09.950 --> 00:11:13.040 is actually close to omega 0. 00:11:13.040 --> 00:11:17.990 Or what this system actually follows the driving force 00:11:17.990 --> 00:11:19.890 frequency. 00:11:19.890 --> 00:11:23.270 Finally, maybe this system chooses 00:11:23.270 --> 00:11:25.520 to do something in between. 00:11:25.520 --> 00:11:30.760 We don't know what is going to happen. 00:11:30.760 --> 00:11:35.260 So our job today is to solve these equation of motion 00:11:35.260 --> 00:11:38.150 and to see what we can learn from the mathematics. 00:11:38.150 --> 00:11:43.660 Then we can actually check those results 00:11:43.660 --> 00:11:47.310 to see if that agrees with the experimental result, which 00:11:47.310 --> 00:11:50.580 is through those demos, OK? 00:11:50.580 --> 00:11:55.160 So as usual, I have this equation of motion here. 00:11:55.160 --> 00:11:58.420 So one trick, which I have been using, 00:11:58.420 --> 00:12:02.230 is to go to complex notation, right? 00:12:02.230 --> 00:12:08.440 Therefore, I can now re-write this thing to be Z double dot 00:12:08.440 --> 00:12:16.540 plus gamma Z dot plus omega 0 squared Z, 00:12:16.540 --> 00:12:25.750 and that is equal to f0 exponential i omega d t. 00:12:25.750 --> 00:12:30.300 So basically, I just go to the complex notation. 00:12:30.300 --> 00:12:33.765 And we would like to solve this equation. 00:12:36.430 --> 00:12:37.980 So in order to solve this equation, 00:12:37.980 --> 00:12:40.245 I make a guess, a test function. 00:12:44.850 --> 00:12:48.570 I guess Z of t has this functional form. 00:12:48.570 --> 00:12:51.156 This is equal to A-- 00:12:51.156 --> 00:12:53.940 some kind of amplitude-- 00:12:53.940 --> 00:13:01.600 exponential i omega d t minus delta. 00:13:01.600 --> 00:13:05.550 Delta is actually some kind of angle, 00:13:05.550 --> 00:13:11.070 which is actually to account for the possible delay 00:13:11.070 --> 00:13:12.620 of the system. 00:13:12.620 --> 00:13:14.370 So if I start to try-- 00:13:14.370 --> 00:13:17.730 for example, this is a system, which I am interested-- 00:13:17.730 --> 00:13:22.320 I start to drive this system, it may take some time 00:13:22.320 --> 00:13:26.520 for the system to react to your driving force. 00:13:26.520 --> 00:13:31.970 So that's actually accounted for by this delta constant. 00:13:31.970 --> 00:13:35.880 And the amplitude is actually what we were wondering, 00:13:35.880 --> 00:13:37.320 what would be the amplitude. 00:13:37.320 --> 00:13:41.420 Therefore, you have some kind of a constant in front 00:13:41.420 --> 00:13:43.720 of the exponential function. 00:13:43.720 --> 00:13:48.180 And you can see that this exponential is actually having 00:13:48.180 --> 00:13:50.550 angular frequency, omega d. 00:13:50.550 --> 00:13:55.410 And that is actually designed to cancel this exponential i omega 00:13:55.410 --> 00:13:58.950 d t here in the drag force. 00:13:58.950 --> 00:14:04.920 So now we can as you calculate what z dot of t 00:14:04.920 --> 00:14:12.600 would be equal to i omega d Z. Z double dot t will 00:14:12.600 --> 00:14:19.680 be equal to minus omega d squared Z. With those, 00:14:19.680 --> 00:14:24.000 we can now plug that back into this equation of motion 00:14:24.000 --> 00:14:27.360 and see what we can actually learn from there. 00:14:27.360 --> 00:14:29.340 So basically, what I'm going to do 00:14:29.340 --> 00:14:31.740 is to insert all those things back 00:14:31.740 --> 00:14:33.450 into the equation of motion. 00:14:33.450 --> 00:14:37.080 And that is actually going to be like this. 00:14:37.080 --> 00:14:39.690 Basically, the first term, the double dot, 00:14:39.690 --> 00:14:45.690 you get a minus omega d squared out of it. 00:14:45.690 --> 00:14:49.200 The second term, gamma Z dot-- 00:14:49.200 --> 00:14:51.060 I have Z dots here. 00:14:51.060 --> 00:14:58.485 Basically, I would get plus i omega d gamma out 00:14:58.485 --> 00:15:00.660 of the second term. 00:15:00.660 --> 00:15:06.470 That's third term, I get omega 0 squared out of it. 00:15:06.470 --> 00:15:09.660 And that is actually multiplied by Z. 00:15:09.660 --> 00:15:16.490 And this is equal to f0 exponential i omega d t. 00:15:23.070 --> 00:15:29.120 All right, and we also know from this expression 00:15:29.120 --> 00:15:33.500 Z is equal to A exponential i omega d t minus delta. 00:15:33.500 --> 00:15:35.020 That's the test function. 00:15:35.020 --> 00:15:42.030 So this is actually equal to A exponential i omega d 00:15:42.030 --> 00:15:45.730 t minus delta. 00:15:45.730 --> 00:15:49.150 So now what I can do is-- 00:15:49.150 --> 00:15:53.020 I have some constant in the front. 00:15:53.020 --> 00:15:56.290 Multiply it by exponential i omega d t. 00:15:56.290 --> 00:16:01.110 And now I can actually cancel this exponential i omega d t 00:16:01.110 --> 00:16:04.000 with the right hand side term. 00:16:04.000 --> 00:16:04.540 Very good. 00:16:07.710 --> 00:16:11.680 The whole equation is actually exponential free. 00:16:11.680 --> 00:16:14.560 Now I don't have any exponential function left. 00:16:14.560 --> 00:16:19.430 And exponential i delta is actually just a constant. 00:16:19.430 --> 00:16:23.764 So now this equation is actually independent of time. 00:16:26.290 --> 00:16:30.110 So what I getting is like this-- basically 00:16:30.110 --> 00:16:35.860 if I multiply the both sides by exponential i delta, 00:16:35.860 --> 00:16:45.610 then I get minus omega d squared plus i omega d t plus omega 0 00:16:45.610 --> 00:16:50.240 squared A. And this is going to be 00:16:50.240 --> 00:16:56.540 equal to f exponential i delta because I multiply both sides 00:16:56.540 --> 00:17:00.620 by exponential i delta. 00:17:00.620 --> 00:17:11.410 And this is equal to f cosine delta plus i f sine delta. 00:17:11.410 --> 00:17:14.115 Just your last equation. 00:17:14.115 --> 00:17:14.990 Any questions so far? 00:17:19.930 --> 00:17:21.410 So look at what I have been doing. 00:17:21.410 --> 00:17:25.485 So I have this equation of motion. 00:17:25.485 --> 00:17:30.040 As usual, I go to complex notation. 00:17:30.040 --> 00:17:35.620 Then I guess Z equal to A exponential i omega d 00:17:35.620 --> 00:17:39.910 t minus delta because my friends from the math department 00:17:39.910 --> 00:17:44.650 already solved this, and I'm just following it. 00:17:44.650 --> 00:17:49.660 Then I can calculate all those terms, plug in e, 00:17:49.660 --> 00:17:53.440 and basically, you will arrive at this equation. 00:17:53.440 --> 00:17:58.090 This equation is a complex equation. 00:17:58.090 --> 00:17:59.350 So what does that mean? 00:17:59.350 --> 00:18:03.250 That means one equation is equal to two equations 00:18:03.250 --> 00:18:09.360 because you have the real part, you have the imaginary part. 00:18:09.360 --> 00:18:13.980 Therefore, that's very nice because I have two unknowns. 00:18:13.980 --> 00:18:17.040 The first one is A, a constant. 00:18:17.040 --> 00:18:18.840 And the second one is delta. 00:18:18.840 --> 00:18:21.210 Now I have two equations I can solve 00:18:21.210 --> 00:18:24.660 what would be the functional form for A and the delta. 00:18:24.660 --> 00:18:30.910 And let me go immediately solve this equation. 00:18:30.910 --> 00:18:37.110 So if I take the real part from this equation, 00:18:37.110 --> 00:18:41.970 basically what I'm going to get is omega 0 squared-- 00:18:41.970 --> 00:18:46.680 this is real-- minus omega d squared-- 00:18:46.680 --> 00:18:48.080 this is also real-- 00:18:48.080 --> 00:18:52.260 times A. A is actually some real number. 00:18:52.260 --> 00:18:56.621 This is actually equal to f cosine-- 00:18:56.621 --> 00:18:57.120 f0. 00:18:57.120 --> 00:19:00.690 Sorry, I missed a zero here. 00:19:00.690 --> 00:19:02.970 So that zero I missed. 00:19:02.970 --> 00:19:05.310 This should be f0. 00:19:05.310 --> 00:19:11.100 f0 cosine delta. 00:19:11.100 --> 00:19:14.580 And I can also collect all the terms, 00:19:14.580 --> 00:19:17.520 which is imaginary terms. 00:19:17.520 --> 00:19:22.170 Then I get only the second term from the left hand side 00:19:22.170 --> 00:19:24.900 is with i in front of it. 00:19:24.900 --> 00:19:33.000 Therefore, I get omega d gamma A from the left hand side. 00:19:33.000 --> 00:19:34.620 And from the right hand side, there's 00:19:34.620 --> 00:19:38.320 only one imaginary term. 00:19:38.320 --> 00:19:46.920 Therefore, I get-- this is equal to f0 sine delta. 00:19:46.920 --> 00:19:49.620 So now I have two equations. 00:19:49.620 --> 00:19:51.330 I have two unknowns. 00:19:51.330 --> 00:19:56.310 Therefore, I can easily solve A and delta. 00:19:56.310 --> 00:19:59.130 So I call this equation number one. 00:19:59.130 --> 00:20:02.310 I call this equation number two. 00:20:02.310 --> 00:20:08.760 So now I can-- sounding in quadrature the two equations-- 00:20:08.760 --> 00:20:10.010 in quadrature. 00:20:10.010 --> 00:20:14.700 And the left-hand side will give you 00:20:14.700 --> 00:20:21.200 A squared omega 0 squared minus omega 00:20:21.200 --> 00:20:29.250 d squared squared plus omega d squared gamma squared. 00:20:29.250 --> 00:20:31.940 That is actually coming from the second equation. 00:20:35.180 --> 00:20:38.030 That gives you the left hand side. 00:20:38.030 --> 00:20:45.140 It's a square of the sum the first and second equation. 00:20:45.140 --> 00:20:56.060 And the right hand side will become f0 square cosine delta 00:20:56.060 --> 00:20:59.963 cosine squared delta plus sine squared delta. 00:21:02.540 --> 00:21:05.752 And this is equal to 1. 00:21:05.752 --> 00:21:10.470 So that's actually the trick to get rid of delta. 00:21:10.470 --> 00:21:15.830 Then I can get what will be the resulting A. A is actually 00:21:15.830 --> 00:21:19.940 a function of omega d. 00:21:19.940 --> 00:21:22.350 Omega d is given to you. 00:21:22.350 --> 00:21:25.890 It's actually determined by you-- 00:21:25.890 --> 00:21:30.600 how fast do you want to oscillate this system. 00:21:30.600 --> 00:21:37.350 And this is equal to f0 divided by square root 00:21:37.350 --> 00:21:39.150 of this whole thing. 00:21:39.150 --> 00:21:42.510 So this will give you omega 0 squared 00:21:42.510 --> 00:21:48.920 minus omega d squared squared plus omega 00:21:48.920 --> 00:21:51.930 d squared gamma squared. 00:21:58.530 --> 00:22:02.910 Then we can also calculate what would be the delta. 00:22:02.910 --> 00:22:07.800 The trick is to take a ratio between equation number two 00:22:07.800 --> 00:22:10.080 and the equation number one-- 00:22:10.080 --> 00:22:11.790 2 divided by 1. 00:22:11.790 --> 00:22:16.820 Basically, you will get tangent delta. 00:22:16.820 --> 00:22:19.370 This is sine divided by cosine. 00:22:19.370 --> 00:22:22.260 f0 actually cancel. 00:22:22.260 --> 00:22:24.540 This is equal to what? 00:22:24.540 --> 00:22:28.030 Equal to the ratio of these two terms. 00:22:28.030 --> 00:22:31.480 After you take the ratio, A drops out. 00:22:31.480 --> 00:22:35.760 Basically, what you get is gamma omega 00:22:35.760 --> 00:22:43.790 d divided by omega 0 squared minus omega d squared. 00:22:47.150 --> 00:22:54.890 So we have solved A and the delta through this exercise. 00:22:54.890 --> 00:22:56.450 So what does that mean? 00:22:59.260 --> 00:23:06.280 Originally, I assume my solution to be A exponential i omega d 00:23:06.280 --> 00:23:09.500 t minus delta. 00:23:09.500 --> 00:23:17.100 Therefore, I would like to go back to the real world, which 00:23:17.100 --> 00:23:18.950 is actually theta. 00:23:18.950 --> 00:23:21.560 So basically, if I take the real part, 00:23:21.560 --> 00:23:29.150 I would get theta of t, which is actually the real part of Z. 00:23:29.150 --> 00:23:41.840 And that will give you A omega d cosine omega d t minus delta is 00:23:41.840 --> 00:23:43.749 also a function of omega d. 00:23:47.581 --> 00:23:50.490 So we have done this exercise. 00:23:50.490 --> 00:23:54.810 And you can see that the first thing which we see here 00:23:54.810 --> 00:24:01.220 is that there's no free parameter from this solution. 00:24:01.220 --> 00:24:05.310 A is decided by omega d. 00:24:05.310 --> 00:24:08.340 And delta is also decided by omega d. 00:24:08.340 --> 00:24:10.200 There's a lot of math, but actually we 00:24:10.200 --> 00:24:13.090 have overcome those and that we have a solution. 00:24:13.090 --> 00:24:18.310 But it is actually clear to you that this cannot be the full 00:24:18.310 --> 00:24:20.340 story. 00:24:20.340 --> 00:24:23.040 Because you have a second-order differential equation, 00:24:23.040 --> 00:24:27.940 you need to have two free parameters in the solution. 00:24:27.940 --> 00:24:29.220 What is actually missing? 00:24:29.220 --> 00:24:31.152 Anybody can tell me what is missing. 00:24:31.152 --> 00:24:32.610 AUDIENCE: The homogeneous solution. 00:24:32.610 --> 00:24:34.170 YEN-JIE LEE: Very good. 00:24:34.170 --> 00:24:36.810 The homogeneous solution is missing. 00:24:36.810 --> 00:24:39.960 So that's actually why we actually 00:24:39.960 --> 00:24:41.910 have no free parameter here. 00:24:41.910 --> 00:24:45.960 Once is the omega d is determined, 00:24:45.960 --> 00:24:51.120 once the f0 is given, then you have the functional form which 00:24:51.120 --> 00:24:54.930 decides what is actually theta. 00:24:54.930 --> 00:24:59.360 So what in actually the full solution? 00:24:59.360 --> 00:25:04.170 A full solution should be, as you said, 00:25:04.170 --> 00:25:07.290 a combination of homogeneous solution 00:25:07.290 --> 00:25:10.960 and the particular solution which we actually got here. 00:25:10.960 --> 00:25:17.990 So if I prepare the system to be in a situation of, for example, 00:25:17.990 --> 00:25:20.420 underdamped situation. 00:25:20.420 --> 00:25:23.780 Then what I'm going to do is actually pretty simple. 00:25:23.780 --> 00:25:30.350 What I am going to do is to just copy the underdamped solution 00:25:30.350 --> 00:25:33.300 from last lecture and combine that 00:25:33.300 --> 00:25:36.900 with my particular solution, which I obtained here. 00:25:36.900 --> 00:25:39.990 So that actually to see what actually 00:25:39.990 --> 00:25:41.870 the full solution looks like. 00:25:41.870 --> 00:25:49.545 I have A omega d cosine omega d t minus 00:25:49.545 --> 00:25:54.990 delta is a function of omega d. 00:25:54.990 --> 00:26:03.565 This is actually so-called steady-state solution. 00:26:07.380 --> 00:26:09.420 And, of course, as you mentioned, 00:26:09.420 --> 00:26:13.394 I need to also add the homogeneous solution 00:26:13.394 --> 00:26:15.060 and basically the no -- this actually -- 00:26:15.060 --> 00:26:17.610 according to what I wrote there, I 00:26:17.610 --> 00:26:21.150 have a functional form of exponential A exponential 00:26:21.150 --> 00:26:26.220 minus gamma over 2t equals sine omega t plus alpha. 00:26:26.220 --> 00:26:29.560 So I changed A to B because I already have the A there just 00:26:29.560 --> 00:26:31.490 to avoid confusion. 00:26:31.490 --> 00:26:36.125 Then basically, you get B exponential minus gamma 00:26:36.125 --> 00:26:45.540 over 2t cosine omega t plus alpha. 00:26:45.540 --> 00:26:52.120 Basically, they are two free parameters, B and alpha. 00:26:52.120 --> 00:26:56.070 Those two free parameters can be determined 00:26:56.070 --> 00:26:58.740 by initial conditions. 00:26:58.740 --> 00:27:05.290 So, for example, initially I actually release the rod 00:27:05.290 --> 00:27:10.050 at some fixed angle of theta initial. 00:27:10.050 --> 00:27:12.750 And also the initial velocity is 0. 00:27:12.750 --> 00:27:15.930 Then I can actually practice solution A 00:27:15.930 --> 00:27:22.200 using those initial conditions to solve B and alpha. 00:27:22.200 --> 00:27:24.431 Any questions so far? 00:27:24.431 --> 00:27:24.930 Yes. 00:27:24.930 --> 00:27:27.249 AUDIENCE: Are we assuming it's underdamped? 00:27:27.249 --> 00:27:29.290 YEN-JIE LEE: Yeah, I'm assuming it's underdamped, 00:27:29.290 --> 00:27:31.000 the situation. 00:27:31.000 --> 00:27:32.350 So that's the assumption. 00:27:32.350 --> 00:27:39.960 So it depends on the size of gamma and the omega 0. 00:27:39.960 --> 00:27:44.115 Then you have actually four different kinds of solution. 00:27:44.115 --> 00:27:48.460 If gamma is equal to 0, then what you are going to plug in 00:27:48.460 --> 00:27:55.890 is the solution from no damping as a your homogeneous solution. 00:27:55.890 --> 00:27:59.640 And if you prepare this system underwater, 00:27:59.640 --> 00:28:03.030 damping is colossal, it's huge, then 00:28:03.030 --> 00:28:08.940 you actually plug in the overdamped solution 00:28:08.940 --> 00:28:12.960 to be your homogeneous part of the solution. 00:28:12.960 --> 00:28:14.619 Any other questions? 00:28:14.619 --> 00:28:15.410 Very good question. 00:28:18.100 --> 00:28:24.020 So now maybe you got confused a bit. 00:28:24.020 --> 00:28:27.200 I have now omega d. 00:28:27.200 --> 00:28:30.050 I have also omega. 00:28:30.050 --> 00:28:33.830 And there's another one we just called omega 0. 00:28:33.830 --> 00:28:35.480 What are those? 00:28:35.480 --> 00:28:41.405 So omega 0 is the natural angular frequency 00:28:41.405 --> 00:28:44.260 without given the drag force. 00:28:44.260 --> 00:28:47.465 If you remove everything just like 00:28:47.465 --> 00:28:51.230 without considering any drag force, et cetera, 00:28:51.230 --> 00:28:55.100 and that is actually the natural frequency of the system. 00:28:55.100 --> 00:28:57.470 And what is omega? 00:28:57.470 --> 00:29:00.050 Omega, according to the function, 00:29:00.050 --> 00:29:04.130 omega is defined to be omega 0 squared minus gamma squared 00:29:04.130 --> 00:29:06.740 over 4 square root of that. 00:29:06.740 --> 00:29:10.970 That is actually the oscillation frequency, which we actually 00:29:10.970 --> 00:29:14.480 discussed last lecture, after you 00:29:14.480 --> 00:29:18.680 add drag force into it again. 00:29:18.680 --> 00:29:25.560 Finally, omega d is how fast you actually drive this system. 00:29:25.560 --> 00:29:28.450 So that is actually the definition 00:29:28.450 --> 00:29:31.890 of these three omegas. 00:29:31.890 --> 00:29:38.070 So you can see that, if I prepare my solution 00:29:38.070 --> 00:29:43.055 to be underdamped situation, then basically you 00:29:43.055 --> 00:29:45.720 will see that this is actually so-called a steady-state 00:29:45.720 --> 00:29:50.730 solution because A omega d is a constant. 00:29:50.730 --> 00:29:52.165 So it's going to be there forever. 00:29:55.410 --> 00:30:00.210 And the second term is actually B exponential minus gamma 00:30:00.210 --> 00:30:02.146 over 2t. 00:30:02.146 --> 00:30:05.200 It's decaying as a function of time. 00:30:05.200 --> 00:30:09.690 So if you are patient enough, you wait, 00:30:09.690 --> 00:30:13.350 then this will be gone. 00:30:13.350 --> 00:30:16.260 So that is actually how we actually understand 00:30:16.260 --> 00:30:18.900 this mathematical result. And now, of course, you 00:30:18.900 --> 00:30:20.880 can actually take a look at this. 00:30:20.880 --> 00:30:25.770 This is actually just assuming some kind of initial condition 00:30:25.770 --> 00:30:30.870 and plug in the solution and plot it as a function of time. 00:30:30.870 --> 00:30:32.970 And you can see that this function looks really 00:30:32.970 --> 00:30:35.520 weird, looks a bit surprising. 00:30:35.520 --> 00:30:36.780 What does that mean? 00:30:36.780 --> 00:30:38.160 It looks really strange. 00:30:38.160 --> 00:30:40.890 But at some point, this superposition 00:30:40.890 --> 00:30:42.240 of these two functions-- 00:30:42.240 --> 00:30:46.050 because one of the functions actually dies out, disappears-- 00:30:46.050 --> 00:30:49.900 then you will see that, if you wait long enough, 00:30:49.900 --> 00:30:55.110 then you actually only see a very simple structure, 00:30:55.110 --> 00:30:58.950 which is oscillation frequency of omega d. 00:30:58.950 --> 00:31:02.520 And that means a large t. 00:31:02.520 --> 00:31:05.840 In the beginning, the system will not like it. 00:31:05.840 --> 00:31:08.110 You drive it, and the system don't like it. 00:31:08.110 --> 00:31:11.820 Like if I go and shake you, in the beginning, 00:31:11.820 --> 00:31:14.400 you would not like it-- 00:31:14.400 --> 00:31:14.970 maybe. 00:31:14.970 --> 00:31:18.950 And if I shake you long enough, and you say, come on, OK, fine. 00:31:18.950 --> 00:31:21.210 I accept that. 00:31:21.210 --> 00:31:25.350 So that is actually what is going to happen to the system. 00:31:25.350 --> 00:31:31.900 So now I would like to go through a short demonstration, 00:31:31.900 --> 00:31:35.480 which is actually the air cart, which you seen before. 00:31:35.480 --> 00:31:39.120 There's a mass and there are two springs in the front 00:31:39.120 --> 00:31:41.940 and the back of this cart. 00:31:41.940 --> 00:31:46.260 And, of course, as usual, I would turn on the air 00:31:46.260 --> 00:31:49.950 so that I make the friction smaller, 00:31:49.950 --> 00:31:54.240 but there's still some residual friction. 00:31:54.240 --> 00:31:57.630 And you will see that this mass is actually 00:31:57.630 --> 00:32:00.630 oscillating back and forth. 00:32:00.630 --> 00:32:04.290 And the amplitude can become smaller and smaller 00:32:04.290 --> 00:32:07.460 as a function of time. 00:32:07.460 --> 00:32:11.310 Now in the right hand side, I have a motor, 00:32:11.310 --> 00:32:14.220 which actually can drive this-- 00:32:14.220 --> 00:32:17.540 I can actually shorten or increase 00:32:17.540 --> 00:32:19.430 the length of the right hand side string. 00:32:19.430 --> 00:32:23.960 Then I actually introduce a driving force 00:32:23.960 --> 00:32:26.280 by the right hand side motor. 00:32:26.280 --> 00:32:32.670 If I turn it down, this is what is going to happen. 00:32:32.670 --> 00:32:38.850 So we can see now this motor is actually going back and forth. 00:32:38.850 --> 00:32:43.830 And it has a slightly higher frequency compared 00:32:43.830 --> 00:32:46.135 to the natural frequency. 00:32:46.135 --> 00:32:49.680 So the frequency of the motor is higher. 00:32:49.680 --> 00:32:55.320 And you can see that this cart is actually oscillating. 00:32:55.320 --> 00:33:00.404 But you can see that sometimes it pulls and sometimes it 00:33:00.404 --> 00:33:00.945 moves faster. 00:33:04.050 --> 00:33:05.840 So you can see that it's actually moving. 00:33:05.840 --> 00:33:09.410 And it stops a little bit because they 00:33:09.410 --> 00:33:16.290 are all superposition of two different kinds of oscillating 00:33:16.290 --> 00:33:17.930 functions come into play. 00:33:17.930 --> 00:33:19.550 You can see that now. 00:33:19.550 --> 00:33:21.600 It got slowed down, and it can become 00:33:21.600 --> 00:33:23.820 faster and slower and faster. 00:33:23.820 --> 00:33:30.100 But eventually, if you wait long enough, 00:33:30.100 --> 00:33:31.760 what is going to happen? 00:33:31.760 --> 00:33:33.960 What is going to happen? 00:33:33.960 --> 00:33:35.250 If we wait long enough-- 00:33:35.250 --> 00:33:36.648 AUDIENCE: [INAUDIBLE]. 00:33:39.450 --> 00:33:40.380 YEN-JIE LEE: Exactly. 00:33:40.380 --> 00:33:43.500 So basically, if you wait long enough, as you said, 00:33:43.500 --> 00:33:47.261 you will actually just oscillate at the frequency of the driving 00:33:47.261 --> 00:33:47.760 force. 00:33:47.760 --> 00:33:52.440 You can see that this motion looks really bizarre, right? 00:33:52.440 --> 00:33:54.120 Sometimes it stops, and sometimes 00:33:54.120 --> 00:33:57.600 it actually continues to move. 00:33:57.600 --> 00:34:00.217 And are you surprised? 00:34:04.690 --> 00:34:06.880 Probably you are not surprised anymore 00:34:06.880 --> 00:34:10.510 because we know math is the language to describe nature. 00:34:10.510 --> 00:34:14.260 And indeed it predicts this kind of behavior. 00:34:14.260 --> 00:34:16.120 That's really pretty cool. 00:34:19.120 --> 00:34:23.260 In order to help you to learn a bit how to actually translate 00:34:23.260 --> 00:34:26.679 a physical situation into mathematics, what I am going 00:34:26.679 --> 00:34:31.000 to do is to introduce you another example so 00:34:31.000 --> 00:34:35.420 that actually we can actually solve it together. 00:34:35.420 --> 00:34:38.830 So now I would like to drive a pendulum. 00:34:45.969 --> 00:34:50.395 So I prepare a pendulum at time equal to 0. 00:34:53.650 --> 00:35:01.210 This is a string attached to a ball with mass equal to m. 00:35:01.210 --> 00:35:04.870 And the length of the string is equal to l. 00:35:04.870 --> 00:35:08.410 And the angle between the vertical direction 00:35:08.410 --> 00:35:14.440 and the direction of the string is theta. 00:35:14.440 --> 00:35:18.000 And, of course, I can actually give you initial condition X 00:35:18.000 --> 00:35:21.010 initial, which is actually measured 00:35:21.010 --> 00:35:32.620 with respect to the vertical direction, and time equal to t. 00:35:32.620 --> 00:35:37.430 This is actually the original vertical direction, 00:35:37.430 --> 00:35:40.350 the same as this dashed line. 00:35:40.350 --> 00:35:48.800 And I can actually move the top of the string back and forth 00:35:48.800 --> 00:35:50.750 to some position. 00:35:50.750 --> 00:35:57.620 And, of course, this string is connected to the ball. 00:35:57.620 --> 00:36:02.960 And this system is actually driven from the top 00:36:02.960 --> 00:36:04.780 by the engine's hand, so the engine 00:36:04.780 --> 00:36:08.750 is actually shaking this system from the top. 00:36:08.750 --> 00:36:11.220 And I do it really nicely. 00:36:11.220 --> 00:36:16.440 So basically, I define that the displacement, d, 00:36:16.440 --> 00:36:22.650 as a function of time, is equal to delta sine omega d t. 00:36:25.820 --> 00:36:29.660 So that is actually what I'm going to do. 00:36:29.660 --> 00:36:31.740 OK, and I would like to see what is going 00:36:31.740 --> 00:36:34.910 to happen to this pendulum. 00:36:34.910 --> 00:36:40.200 So, as usual, the first step towards solving this problem 00:36:40.200 --> 00:36:44.702 is to define a coordinate system. 00:36:44.702 --> 00:36:46.410 So what is actually the coordinate system 00:36:46.410 --> 00:36:47.800 I'm going to use? 00:36:47.800 --> 00:36:53.760 So now I define pointing upward to be y. 00:36:53.760 --> 00:36:57.660 I define the horizontal direction 00:36:57.660 --> 00:37:02.220 pointing to the right hand side of the board to be x. 00:37:02.220 --> 00:37:03.810 So that's not good enough. 00:37:03.810 --> 00:37:06.480 I still need the origin, right? 00:37:06.480 --> 00:37:12.870 So now I also define my origin to be the original position 00:37:12.870 --> 00:37:21.270 of the ball which is actually completely addressed 00:37:21.270 --> 00:37:23.340 before I do the experiment. 00:37:23.340 --> 00:37:26.250 So that this is actually the equilibrium position 00:37:26.250 --> 00:37:28.150 of this system actually. 00:37:28.150 --> 00:37:33.130 Then I define here to be 0, 0. 00:37:33.130 --> 00:37:37.170 So once I have that defined, I can now 00:37:37.170 --> 00:37:42.840 express the position of this mass of this ball 00:37:42.840 --> 00:37:45.210 to be xt and yt. 00:37:48.150 --> 00:37:53.020 And see what we are going to get. 00:37:53.020 --> 00:37:55.780 Of course, as usual, we are going 00:37:55.780 --> 00:38:04.270 to analyze the force actually acting on this ball. 00:38:04.270 --> 00:38:08.455 So therefore, as usual, we will draw a force diagram. 00:38:13.540 --> 00:38:17.080 So basically, you have the little mass here, 00:38:17.080 --> 00:38:25.990 and you have actually two forces acting on this little mass, 00:38:25.990 --> 00:38:28.806 or little ball. 00:38:28.806 --> 00:38:31.980 This is Fg pointing downward. 00:38:31.980 --> 00:38:33.880 It's a gravitational force. 00:38:33.880 --> 00:38:38.290 And now this is actually equal to minus mg y. 00:38:41.950 --> 00:38:46.400 And there's also a string tension, 00:38:46.400 --> 00:38:54.790 T. Since we have this definition of theta here, basically 00:38:54.790 --> 00:38:57.640 I have a T which is actually pointing 00:38:57.640 --> 00:39:00.940 to the upper left direction of the board. 00:39:05.420 --> 00:39:07.890 Oh, don't forget-- actually there's 00:39:07.890 --> 00:39:10.110 a third force, which is actually the F drag. 00:39:13.860 --> 00:39:18.470 F drag is actually equal to minus bx 00:39:18.470 --> 00:39:21.294 dot in the x direction. 00:39:28.950 --> 00:39:33.450 Now I would like to write down the expression for also 00:39:33.450 --> 00:39:36.390 the string tension, T. The T is actually 00:39:36.390 --> 00:39:43.320 equal to minus T sine theta in the x direction 00:39:43.320 --> 00:39:47.430 because the T is pointing to upper left direction. 00:39:47.430 --> 00:39:50.390 So therefore, the position to x direction 00:39:50.390 --> 00:39:57.560 will be minus sine theta and plus T cosine 00:39:57.560 --> 00:40:01.260 theta in the y direction because the tension is actually 00:40:01.260 --> 00:40:02.725 pointing upper left. 00:40:06.210 --> 00:40:10.750 As usual, this is actually pretty complicated to solve. 00:40:10.750 --> 00:40:11.920 I have this cosine. 00:40:11.920 --> 00:40:13.540 I have this sine there, right? 00:40:13.540 --> 00:40:22.530 So what I'm going to do is to assume that this angle 00:40:22.530 --> 00:40:27.340 theta is very small, as usual. 00:40:27.340 --> 00:40:30.160 So I will take small angle approximation. 00:40:30.160 --> 00:40:32.940 Then basically, you have sine theta 00:40:32.940 --> 00:40:38.080 is roughly to be equal to theta, and that is actually 00:40:38.080 --> 00:40:40.060 equal to what? 00:40:40.060 --> 00:40:42.440 Equal to-- here. 00:40:42.440 --> 00:40:44.600 Basically, you can actually calculate 00:40:44.600 --> 00:40:45.820 what will be the theta. 00:40:49.360 --> 00:40:52.090 The sine theta, or theta, would be 00:40:52.090 --> 00:40:56.700 equal to x minus d divided by l. 00:41:00.300 --> 00:41:03.180 And of course, taking a small angle approximation 00:41:03.180 --> 00:41:09.940 will bring cosine theta to be 1. 00:41:09.940 --> 00:41:14.020 Then after this approximation, my T 00:41:14.020 --> 00:41:20.670 will become minus T x minus d divided 00:41:20.670 --> 00:41:30.280 by l in the x direction plus T in the y direction 00:41:30.280 --> 00:41:35.170 because sine theta is replaced by this approximated value 00:41:35.170 --> 00:41:39.440 because sine theta is actually replaced by 1. 00:41:39.440 --> 00:41:41.720 Any questions? 00:41:41.720 --> 00:41:42.601 Yes. 00:41:42.601 --> 00:41:54.702 AUDIENCE: Is that a constant or is that the change in sine? 00:41:54.702 --> 00:41:55.910 YEN-JIE LEE: It's a constant. 00:41:55.910 --> 00:41:57.180 Yeah, I was going too fast. 00:41:57.180 --> 00:42:02.149 So this is actually a constant of my amplitude. 00:42:02.149 --> 00:42:05.600 AUDIENCE: And the drag force is only in the x direction? 00:42:05.600 --> 00:42:08.700 YEN-JIE LEE: Yes, it's only in the x direction. 00:42:08.700 --> 00:42:13.170 So I'm only trying to actually move this point back and forth 00:42:13.170 --> 00:42:14.157 horizontally. 00:42:18.720 --> 00:42:26.010 So now I have all the components T, Fg, f drag. 00:42:26.010 --> 00:42:28.860 And, of course, you can see that I already ignored the drag 00:42:28.860 --> 00:42:32.460 force in the y direction from that formula 00:42:32.460 --> 00:42:36.090 because I am only considering the system to be 00:42:36.090 --> 00:42:38.940 moving in the x direction. 00:42:38.940 --> 00:42:44.790 Therefore, I can now collect all the terms in the x direction. 00:42:44.790 --> 00:42:49.060 Basically, you will have m x double dot. 00:42:49.060 --> 00:42:54.360 This is equal to minus b x dot, which is actually 00:42:54.360 --> 00:43:05.860 coming from the drag force, minus T x minus d divided by l. 00:43:05.860 --> 00:43:11.960 This is actually coming from this term in the x direction. 00:43:11.960 --> 00:43:15.140 Let's look at those in the y direction. 00:43:15.140 --> 00:43:20.320 And y double dot would be equal to minus mg 00:43:20.320 --> 00:43:26.830 plus T. The minus mg is from the gravitational force. 00:43:26.830 --> 00:43:30.940 And this T is coming from the y component of the string 00:43:30.940 --> 00:43:33.640 tension. 00:43:33.640 --> 00:43:36.910 And of course, since we are taking a very small angle 00:43:36.910 --> 00:43:42.401 approximation, there will be no vertical motion. 00:43:42.401 --> 00:43:42.900 Yes. 00:43:42.900 --> 00:43:46.316 AUDIENCE: Why did we use the small approximation 00:43:46.316 --> 00:43:49.244 for sine theta when we're going to use 00:43:49.244 --> 00:43:53.136 x minus d over l, which represents psi instead of just 00:43:53.136 --> 00:43:53.636 theta? 00:43:53.636 --> 00:43:55.990 YEN-JIE LEE: Yeah, so also in this case, 00:43:55.990 --> 00:43:59.230 they happen to be exactly the same. 00:43:59.230 --> 00:44:02.260 And why I care is actually the cosine theta. 00:44:02.260 --> 00:44:04.930 Otherwise, I would have to deal with cosine theta. 00:44:04.930 --> 00:44:07.090 And also, this y double dot would not 00:44:07.090 --> 00:44:12.340 be equal to 0, which is what I'm going to assume here. 00:44:12.340 --> 00:44:13.010 Good question. 00:44:13.010 --> 00:44:16.060 So the question was, why do I need to take an approximation? 00:44:16.060 --> 00:44:19.750 Because I want to get rid of cosine theta. 00:44:19.750 --> 00:44:22.800 So now from this y direction, I can 00:44:22.800 --> 00:44:29.050 solve T will be equal to mg because I assume that there's 00:44:29.050 --> 00:44:31.170 no y direction motion. 00:44:31.170 --> 00:44:33.780 And I can conclude that-- 00:44:33.780 --> 00:44:36.610 originally, I don't know what is actually the string tension. 00:44:36.610 --> 00:44:41.330 It's denoted by T. Now, from this second equation, 00:44:41.330 --> 00:44:45.100 I can conclude that T will be equal to mg, which 00:44:45.100 --> 00:44:46.260 is the gravitational force. 00:44:48.800 --> 00:44:54.670 Then, once I have that, I can go back to x direction. 00:44:54.670 --> 00:44:58.210 Basically, I get m x double dot. 00:44:58.210 --> 00:45:06.130 This is equal to minus b x dot minus mg over l x minus d. 00:45:10.430 --> 00:45:13.880 Everything is working very well. 00:45:13.880 --> 00:45:23.870 And I just have to really write down the d function explicitly. 00:45:23.870 --> 00:45:25.070 What is d? 00:45:25.070 --> 00:45:31.570 d is just a reminder, delta sine omega d t. 00:45:31.570 --> 00:45:33.980 So I will plug that into that equation. 00:45:33.980 --> 00:45:38.900 And also I will bring all the terms 00:45:38.900 --> 00:45:42.740 related to x to the left hand side 00:45:42.740 --> 00:45:49.060 just to match my convention. 00:45:49.060 --> 00:45:53.740 All right, so now I will be able to get the result, 00:45:53.740 --> 00:46:04.420 m x double dot plus b x dot plus mg over l x. 00:46:04.420 --> 00:46:11.290 And that is actually equal to mg over l d. 00:46:11.290 --> 00:46:20.740 And this is equal to mg over l delta sine omega dt. 00:46:20.740 --> 00:46:22.940 So basically, I collect all the terms, 00:46:22.940 --> 00:46:24.460 put it to the left hand side. 00:46:24.460 --> 00:46:29.980 And I write down T explicitly, which is this. 00:46:29.980 --> 00:46:32.770 Then I can divide everything by m. 00:46:32.770 --> 00:46:41.710 Then I get m x double dot plus b over m x dot plus g over l x. 00:46:41.710 --> 00:46:49.760 And that would be equal to g over l delta sine omega dt. 00:46:49.760 --> 00:46:54.960 Now, of course, as usual, I will define this to be gamma, 00:46:54.960 --> 00:46:58.170 define this to be omega 0 squared. 00:46:58.170 --> 00:47:03.840 And I would define this to be f0, which is 00:47:03.840 --> 00:47:06.630 equal to omega 0 squared delta. 00:47:06.630 --> 00:47:08.970 It happened to be equal like that. 00:47:08.970 --> 00:47:12.060 And then this actually becomes x double 00:47:12.060 --> 00:47:16.920 dot plus gamma x dot plus omega 0 00:47:16.920 --> 00:47:22.440 squared x equal to f0 sine omega dt. 00:47:25.266 --> 00:47:27.840 Am I going too fast? 00:47:27.840 --> 00:47:30.530 OK, everybody is following. 00:47:30.530 --> 00:47:32.910 So we see that ha! 00:47:32.910 --> 00:47:34.350 This equation-- I know that. 00:47:34.350 --> 00:47:36.510 I know this equation, right? 00:47:36.510 --> 00:47:42.210 Because we have just solved that a few minutes ago. 00:47:42.210 --> 00:47:47.020 Therefore, I know immediately what will be the solution. 00:47:47.020 --> 00:47:49.650 The solution is here already. 00:47:49.650 --> 00:47:55.070 I have A omega d and the tangent delta, the function of force 00:47:55.070 --> 00:47:56.550 there. 00:47:56.550 --> 00:48:03.010 Therefore, I can now write down what will be the A. 00:48:03.010 --> 00:48:08.800 So A is actually just equal to f0 divided 00:48:08.800 --> 00:48:12.790 by square root of omega 0 squared 00:48:12.790 --> 00:48:17.110 minus omega d squared squared plus omega 00:48:17.110 --> 00:48:19.300 d squared gamma squared. 00:48:27.980 --> 00:48:31.610 So now the question is-- 00:48:31.610 --> 00:48:36.650 what does the result actually mean. 00:48:36.650 --> 00:48:38.930 I have this function. 00:48:38.930 --> 00:48:42.070 I have that function, tangent delta. 00:48:42.070 --> 00:48:43.210 It's solved. 00:48:43.210 --> 00:48:48.860 It's actually the amplitude of the steady-state solution 00:48:48.860 --> 00:48:53.180 and also the phase difference between the drag force 00:48:53.180 --> 00:48:59.570 phase and the steady-state oscillation phase. 00:48:59.570 --> 00:49:02.690 So that's actually the amount of lag 00:49:02.690 --> 00:49:05.900 and the size of the amplitude. 00:49:05.900 --> 00:49:10.820 But through this equation it is very difficult to understand. 00:49:10.820 --> 00:49:14.105 So what I'm going to do is to take some limit 00:49:14.105 --> 00:49:18.080 so that actually we can help you to understand what is going on. 00:49:18.080 --> 00:49:25.550 So suppose I assume that omega d goes to 0. 00:49:25.550 --> 00:49:27.260 So what does that mean? 00:49:27.260 --> 00:49:32.060 This is the engine's hand and is moving really slowly 00:49:32.060 --> 00:49:35.420 and see what is going to happen. 00:49:35.420 --> 00:49:42.090 If I do this, then you will find that A omega d-- 00:49:42.090 --> 00:49:47.720 since omega goes to 0, this is gone, this is gone. 00:49:47.720 --> 00:49:51.680 Therefore, you will see that omega A will be 00:49:51.680 --> 00:49:53.930 equal to omega 0 divided by-- 00:49:53.930 --> 00:49:58.860 I'm sorry-- of f0 divided by omega 0 squared. 00:49:58.860 --> 00:50:02.420 And that is actually equal to g delta 00:50:02.420 --> 00:50:05.720 over l divded by g over l. 00:50:05.720 --> 00:50:09.320 And that will give you delta. 00:50:09.320 --> 00:50:10.860 So what does that mean? 00:50:10.860 --> 00:50:16.430 This means that, if I drive this thing really slowly, 00:50:16.430 --> 00:50:21.530 then the amplitude of the mass will 00:50:21.530 --> 00:50:27.722 be equal to how much I actually move, which is delta. 00:50:27.722 --> 00:50:28.222 OK. 00:50:28.222 --> 00:50:30.820 Do you get it? 00:50:30.820 --> 00:50:37.360 In addition to that, tangent delta-- 00:50:37.360 --> 00:50:42.980 since I am taking the limit omega d goes to 0. 00:50:42.980 --> 00:50:46.460 Therefore, tangent delta will be equal to 0, 00:50:46.460 --> 00:50:52.020 and that means delta will be equal 0. 00:50:52.020 --> 00:50:55.080 Any questions? 00:50:55.080 --> 00:50:58.540 So that means there will be no phase difference. 00:50:58.540 --> 00:51:02.220 The system has enough time to keep up with my speed. 00:51:05.160 --> 00:51:11.410 The second limit, obviously, omega d goes to infinity. 00:51:11.410 --> 00:51:12.670 What does that mean? 00:51:12.670 --> 00:51:16.730 That means I'm going to hold this as a string 00:51:16.730 --> 00:51:21.220 and shake it like crazy really fast 00:51:21.220 --> 00:51:23.920 and see what will happen, OK? 00:51:23.920 --> 00:51:27.490 So in that case, you will get A omega 00:51:27.490 --> 00:51:31.950 d, and that one goes to 0, because omega d goes 00:51:31.950 --> 00:51:33.260 to infinity. 00:51:33.260 --> 00:51:36.090 This one goes to 0. 00:51:36.090 --> 00:51:41.930 And also, tangent delta will go to infinity. 00:51:41.930 --> 00:51:45.430 Therefore, delta will go to pi. 00:51:45.430 --> 00:51:48.100 So that means they will be out of phase. 00:51:50.682 --> 00:51:52.390 Any questions so far in these two limits? 00:51:55.530 --> 00:51:58.760 OK, so what I'm going to do now is 00:51:58.760 --> 00:52:03.580 to take a small toy, which I made for my son, who 00:52:03.580 --> 00:52:06.170 is one-year-old now. 00:52:06.170 --> 00:52:09.200 Because I would like him to learn wavelength vibration 00:52:09.200 --> 00:52:11.200 before he goes to quantum, right? 00:52:11.200 --> 00:52:12.050 Hey? 00:52:12.050 --> 00:52:13.700 So I made this toy. 00:52:13.700 --> 00:52:17.420 And he looked at it. 00:52:17.420 --> 00:52:20.240 So you can see now, I can demonstrate 00:52:20.240 --> 00:52:26.872 what is going to happen when omega is approaching to 0. 00:52:26.872 --> 00:52:27.372 OK? 00:52:30.844 --> 00:52:33.460 I am already doing it. 00:52:33.460 --> 00:52:35.640 Can you see it? 00:52:35.640 --> 00:52:37.290 No? 00:52:37.290 --> 00:52:40.686 It's a very exciting experiment. 00:52:40.686 --> 00:52:42.550 Can you see that? 00:52:42.550 --> 00:52:48.420 You see that this is the origin vertical direction. 00:52:48.420 --> 00:52:53.700 If I do it really, really, really slowly, 00:52:53.700 --> 00:52:59.140 you can see that the amplitude of the ball 00:52:59.140 --> 00:53:04.200 is actually exactly the same as the displacement I introduced. 00:53:04.200 --> 00:53:08.160 So that's kind of obvious. 00:53:08.160 --> 00:53:13.170 So now, let's see what is going to happen if I 00:53:13.170 --> 00:53:15.950 drive this system like crazy. 00:53:15.950 --> 00:53:18.350 OK, not going up and down. 00:53:18.350 --> 00:53:21.180 Eeeee-- that's the maximum speed I can do. 00:53:21.180 --> 00:53:23.290 Maybe you can do it faster. 00:53:23.290 --> 00:53:27.900 But you can see that nothing happened. 00:53:27.900 --> 00:53:31.660 So amplitude is close to 0, because what 00:53:31.660 --> 00:53:34.690 you have been doing is-- 00:53:34.690 --> 00:53:40.050 disappear, you sort cancelling each other. 00:53:40.050 --> 00:53:43.590 And it's actually not going to contribute 00:53:43.590 --> 00:53:47.040 to the motion of this ball. 00:53:47.040 --> 00:53:50.960 So now, you can see that I can also test, what is actually 00:53:50.960 --> 00:53:52.580 the natural frequency? 00:53:52.580 --> 00:53:55.890 And what I am going to do is to oscillate 00:53:55.890 --> 00:54:01.900 at around the natural frequency to see what is going to happen. 00:54:01.900 --> 00:54:04.990 Let's see what is going to happen. 00:54:04.990 --> 00:54:08.130 You can see that the delta is really small, right? 00:54:08.130 --> 00:54:09.535 Can you see the delta. 00:54:09.535 --> 00:54:12.910 It's really small-- very small-- 00:54:12.910 --> 00:54:16.480 very small. 00:54:16.480 --> 00:54:21.145 But you can see that the amplitude, the A, is huge. 00:54:23.760 --> 00:54:26.370 What does that tell us? 00:54:26.370 --> 00:54:29.841 What does that tell us? 00:54:29.841 --> 00:54:30.340 Yeah? 00:54:30.340 --> 00:54:32.348 AUDIENCE: Well, we're experiencing resonance. 00:54:32.348 --> 00:54:35.810 YEN-JIE LEE: Yes, we are experiencing resonance. 00:54:35.810 --> 00:54:40.230 And also, that also tells you that the system 00:54:40.230 --> 00:54:43.440 is under-damped very much. 00:54:43.440 --> 00:54:47.940 The Q value is very big. 00:54:47.940 --> 00:54:53.470 So if I calculate the amplitude, A-- 00:54:53.470 --> 00:54:59.144 now, I can calculate the amplitude, A, 00:54:59.144 --> 00:55:02.760 at natural frequency. 00:55:02.760 --> 00:55:04.170 What I'm going to get is-- 00:55:07.230 --> 00:55:13.440 now, I can actually plug in omega d equal to omega 0. 00:55:13.440 --> 00:55:15.200 So if I plug in omega d-- 00:55:20.000 --> 00:55:23.460 omega d equal to omega 0-- 00:55:23.460 --> 00:55:25.290 then what is going to happen? 00:55:25.290 --> 00:55:27.500 So this term is working. 00:55:27.500 --> 00:55:40.880 So you have A is equal to f-0 divided by omega 0 gamma. 00:55:40.880 --> 00:55:44.630 Omega-d is now equal to omega 0. 00:55:44.630 --> 00:55:48.770 And that is going to give you-- 00:55:48.770 --> 00:55:55.890 so f-0 is actually omega 0 square delta 00:55:55.890 --> 00:56:00.190 divided by omega 0 gamma. 00:56:00.190 --> 00:56:02.760 And the one omega 0 actually cancels. 00:56:02.760 --> 00:56:10.830 Then, basically, you will get Q times delta. 00:56:10.830 --> 00:56:12.060 What is Q? 00:56:12.060 --> 00:56:15.640 Just a reminder, it's actually the ratio 00:56:15.640 --> 00:56:18.330 of omega 0 and the gamma. 00:56:18.330 --> 00:56:21.850 When the Q is very large, what does that mean? 00:56:21.850 --> 00:56:25.800 That means it's so close to an idealized situation 00:56:25.800 --> 00:56:27.780 that direct force is very small. 00:56:27.780 --> 00:56:32.390 You can see that in the example which I have been doing. 00:56:32.390 --> 00:56:37.050 So you can see that, ah, it is really the case. 00:56:37.050 --> 00:56:42.570 So you can see that if my delta is something like 1 centimeter, 00:56:42.570 --> 00:56:48.767 but the amplitude is actually at the order of 1 meter, maybe. 00:56:48.767 --> 00:56:49.600 What does that mean? 00:56:49.600 --> 00:56:55.200 That means that Q is actually, roughly, 100. 00:56:55.200 --> 00:57:00.410 So you can actually even get a Q out of this experiment. 00:57:00.410 --> 00:57:01.451 Any questions so far? 00:57:04.890 --> 00:57:07.350 OK, that's very good. 00:57:07.350 --> 00:57:09.990 So now, we can go ahead and take a look 00:57:09.990 --> 00:57:15.720 at the structure of the A and the delta. 00:57:15.720 --> 00:57:18.030 As we demonstrated before, we make 00:57:18.030 --> 00:57:23.140 sense of those three different kinds of situations-- 00:57:23.140 --> 00:57:27.670 omega d goes to 0, omega d goes to infinity. 00:57:27.670 --> 00:57:30.060 And of course, I would like to know the force 00:57:30.060 --> 00:57:33.480 structure of A and delta. 00:57:33.480 --> 00:57:36.685 Therefore, what I'm going to do is to plug A omega 00:57:36.685 --> 00:57:41.850 d as a function of omega d. 00:57:44.670 --> 00:57:50.010 So what I'm going to get is this will be equal to delta 00:57:50.010 --> 00:57:52.810 when omega d goes to 0. 00:57:52.810 --> 00:57:54.990 We just demonstrated that. 00:57:54.990 --> 00:57:58.230 And this will increase to a large value 00:57:58.230 --> 00:58:06.060 and drop down to 0, when omega d goes to infinity. 00:58:06.060 --> 00:58:10.410 And you can see that this is around omega 0. 00:58:10.410 --> 00:58:16.440 And you are going to get a huge amplitude at around omega 0. 00:58:16.440 --> 00:58:17.220 But not quite. 00:58:17.220 --> 00:58:22.020 The maxima is actually slightly smaller than omega 0. 00:58:22.020 --> 00:58:26.670 You can actually calculate that as part of the homework. 00:58:26.670 --> 00:58:28.320 So that makes sense. 00:58:30.870 --> 00:58:36.540 Now, I can also plug the delta, which is the phase difference-- 00:58:36.540 --> 00:58:39.270 and you can see that this phase difference 00:58:39.270 --> 00:58:44.560 will be, originally, 0, when the omega d is very small. 00:58:44.560 --> 00:58:47.530 And this is actually the omega 0. 00:58:47.530 --> 00:58:49.920 I hope you can see it. 00:58:49.920 --> 00:58:53.820 And this will be increasing rapidly here 00:58:53.820 --> 00:58:57.990 and approaching to pi. 00:58:57.990 --> 00:59:01.320 So that means, when you are shaking this system 00:59:01.320 --> 00:59:03.040 like crazy-- 00:59:03.040 --> 00:59:06.880 very high frequency-- then the system cannot keep up with 00:59:06.880 --> 00:59:08.680 the speed. 00:59:08.680 --> 00:59:11.500 The amplitude will be very small. 00:59:11.500 --> 00:59:16.530 And also, the amplitude will be out of phase completely. 00:59:16.530 --> 00:59:20.920 So let's actually do us another demonstration, 00:59:20.920 --> 00:59:26.260 using this little device here. 00:59:26.260 --> 00:59:28.640 This is actually what you see before, 00:59:28.640 --> 00:59:31.800 the ball with a Mexican hat. 00:59:31.800 --> 00:59:36.260 And you can see that there is a spring attached to this system. 00:59:36.260 --> 00:59:39.360 And on the top, what I am going to do 00:59:39.360 --> 00:59:46.430 is to use this motor to drive this system up and down, 00:59:46.430 --> 00:59:49.140 as a direct force. 00:59:49.140 --> 00:59:55.410 So now, what am I going to do is to come from a very 00:59:55.410 --> 00:59:58.720 low-frequency oscillation. 00:59:58.720 --> 01:00:03.770 So you can see that the natural frequency is sort of like this. 01:00:03.770 --> 01:00:07.880 And you can see that, now, I am driving this system really 01:00:07.880 --> 01:00:10.306 slowly. 01:00:10.306 --> 01:00:13.720 You can see, this is actually going up and down 01:00:13.720 --> 01:00:15.180 really slowly. 01:00:18.110 --> 01:00:19.630 And you see that-- huh-- 01:00:19.630 --> 01:00:24.330 the amplitude is actually pretty small. 01:00:26.840 --> 01:00:28.480 There's no excitement for the moment. 01:00:33.450 --> 01:00:35.570 All right, so what I'm going to do is, 01:00:35.570 --> 01:00:42.090 now, I increase the speed of the motor and see what will happen. 01:00:42.090 --> 01:00:45.620 So you can see, now, it's actually 01:00:45.620 --> 01:00:48.470 driving it with higher and higher frequency. 01:00:51.320 --> 01:00:53.830 You see that-- huh-- something is happening. 01:00:53.830 --> 01:00:58.280 You can see the amplitude is getting larger and larger. 01:00:58.280 --> 01:01:02.010 I'm still increasing the frequency-- 01:01:02.010 --> 01:01:07.230 increasing, increasing-- until something-- something happened! 01:01:07.230 --> 01:01:08.210 Right? 01:01:08.210 --> 01:01:09.780 Did you see that? 01:01:09.780 --> 01:01:13.950 It starts to oscillate up and down. 01:01:13.950 --> 01:01:16.170 Because right now, you can see that-- 01:01:16.170 --> 01:01:18.540 look at the top. 01:01:18.540 --> 01:01:21.120 The frequency of the motor is now 01:01:21.120 --> 01:01:28.980 really close to the natural frequency of this system. 01:01:28.980 --> 01:01:32.550 So a resonance behavior will happen. 01:01:32.550 --> 01:01:36.890 And what you are going to get is that, OK, 01:01:36.890 --> 01:01:38.640 omega d around omega 0. 01:01:38.640 --> 01:01:42.680 Then you are going to get large time amplitude. 01:01:42.680 --> 01:01:47.360 So now, what I am going to do is to continue 01:01:47.360 --> 01:01:52.950 to increase the driving frequency 01:01:52.950 --> 01:01:55.110 to a very large value. 01:01:55.110 --> 01:01:57.850 OK, now it's actually doing the "mmmmm"-- 01:01:57.850 --> 01:01:59.290 doing it really fast. 01:01:59.290 --> 01:02:00.750 You can see on the top-- 01:02:00.750 --> 01:02:01.440 very fast. 01:02:01.440 --> 01:02:03.910 OK, I even get it even faster. 01:02:06.545 --> 01:02:10.940 You see that-- huh-- indeed, this system 01:02:10.940 --> 01:02:15.440 is now oscillating at a larger frequency. 01:02:15.440 --> 01:02:17.800 It's trying to keep up with the driving force. 01:02:17.800 --> 01:02:21.510 But you can see that the amplitude is actually 01:02:21.510 --> 01:02:25.650 much smaller than what had happened before. 01:02:25.650 --> 01:02:29.610 So before the class, you may actually think that, 01:02:29.610 --> 01:02:31.500 OK, drive it really fast. 01:02:31.500 --> 01:02:33.870 Maybe we'll increase the amplitude. 01:02:33.870 --> 01:02:36.720 But in reality, actually, it will give you 01:02:36.720 --> 01:02:38.877 a very small amplitude. 01:02:38.877 --> 01:02:40.710 Another thing, which is interesting to know, 01:02:40.710 --> 01:02:46.300 is that you can see that, when the driving force is actually 01:02:46.300 --> 01:02:49.350 at the maximum. 01:02:49.350 --> 01:02:52.835 And actually the position of this mass 01:02:52.835 --> 01:02:54.660 is actually at the minimum. 01:02:54.660 --> 01:02:57.720 So they are actually out of phase. 01:02:57.720 --> 01:02:59.340 I hope you can see it. 01:02:59.340 --> 01:03:00.230 It's like this. 01:03:09.350 --> 01:03:17.480 OK, so what you can see is that, when I understand the system 01:03:17.480 --> 01:03:22.280 and I try to drive it with the natural frequency, what 01:03:22.280 --> 01:03:25.210 is going to happen is that I'm exciting 01:03:25.210 --> 01:03:28.880 this system to a state of resonance. 01:03:28.880 --> 01:03:32.420 So basically, you'll get some resonance behavior. 01:03:32.420 --> 01:03:35.510 So I have shown you that this works 01:03:35.510 --> 01:03:40.220 for driven mechanical oscillator. 01:03:40.220 --> 01:03:44.240 It also works for the spring-mass system. 01:03:44.240 --> 01:03:48.470 And there are many other things which also work, 01:03:48.470 --> 01:03:50.930 which is around you. 01:03:50.930 --> 01:03:55.550 For example, if you happen to be my office hour, 01:03:55.550 --> 01:04:00.880 you would notice that the air-condition in my office 01:04:00.880 --> 01:04:03.590 is actually creating a resonance behavior. 01:04:03.590 --> 01:04:05.853 You'll see low frequency sound-- 01:04:05.853 --> 01:04:10.150 "mm mm mm"-- something like that. 01:04:10.150 --> 01:04:13.590 And that is because the pipe actually 01:04:13.590 --> 01:04:17.061 happens to have the frequency match with the resonance 01:04:17.061 --> 01:04:17.560 frequen-- 01:04:17.560 --> 01:04:23.230 OK, the airflow actually happened to excite the pipe, 01:04:23.230 --> 01:04:27.930 so that it's actually oscillating up and down 01:04:27.930 --> 01:04:28.680 at that frequency. 01:04:28.680 --> 01:04:32.950 So what I did was I tried to turn it down to low 01:04:32.950 --> 01:04:34.300 and see what happened. 01:04:34.300 --> 01:04:39.760 But unfortunately, it actually excited another resonance. 01:04:39.760 --> 01:04:42.640 I see, now, not a low-frequency sound, but a very 01:04:42.640 --> 01:04:43.890 high-frequency sound. 01:04:43.890 --> 01:04:46.480 I will post a video, actually online. 01:04:46.480 --> 01:04:49.660 So my life is hard, right? 01:04:49.660 --> 01:04:51.670 But I'm a physicist. 01:04:51.670 --> 01:04:55.110 Is So I choose to use the median. 01:04:55.110 --> 01:04:58.590 Then I actually stay between the two resonances. 01:04:58.590 --> 01:05:04.900 Then I don't hear the additional sound, which bothers me. 01:05:04.900 --> 01:05:11.050 Another example is that, when I was in Taiwan as a undergrad, 01:05:11.050 --> 01:05:13.700 I was living outside in a apartment. 01:05:13.700 --> 01:05:20.650 And with my flat-mate, we owned a very old washing machine. 01:05:20.650 --> 01:05:23.920 So in the middle of the night, the washing machine 01:05:23.920 --> 01:05:29.680 would started to walk around, like my flat-mate. 01:05:29.680 --> 01:05:31.845 And we are not scared. 01:05:31.845 --> 01:05:35.740 That is because the oscillation frequency-- actually, 01:05:35.740 --> 01:05:36.840 the rotation-- 01:05:36.840 --> 01:05:39.490 happened to match with the frequency 01:05:39.490 --> 01:05:41.050 of the washing machine. 01:05:41.050 --> 01:05:43.690 Therefore, when we started to wash our clothes, 01:05:43.690 --> 01:05:47.830 it start to walk around in the room. 01:05:47.830 --> 01:05:51.400 So as a physicist, what we have decided 01:05:51.400 --> 01:05:57.940 is to make the spin slightly slower, or even faster. 01:05:57.940 --> 01:05:59.710 Then, actually, you can see that, 01:05:59.710 --> 01:06:03.370 when you do that, then you get rid of the resonance behavior. 01:06:03.370 --> 01:06:05.590 So it's not walking around any more. 01:06:05.590 --> 01:06:06.580 We can control it. 01:06:09.100 --> 01:06:10.630 Another thing which is interesting 01:06:10.630 --> 01:06:13.930 is that the resonance behavior is not only 01:06:13.930 --> 01:06:16.600 in the physical objects, which we actually 01:06:16.600 --> 01:06:18.530 deal with these days also. 01:06:18.530 --> 01:06:21.340 But either you learn quantum mechanics 01:06:21.340 --> 01:06:23.440 and upon the field theory, you will 01:06:23.440 --> 01:06:30.160 find that there are resonance also in a mass wave function. 01:06:30.160 --> 01:06:34.360 So basically, you can see that these are examples 01:06:34.360 --> 01:06:38.080 of the Z boson resonance peak. 01:06:38.080 --> 01:06:44.800 So if you scatter a electron and positron then, 01:06:44.800 --> 01:06:49.840 basically, you'll see that the cross-section have a resonance 01:06:49.840 --> 01:06:53.980 peak at around 90 GeV. 01:06:53.980 --> 01:06:58.570 And that is actually another very interesting example 01:06:58.570 --> 01:07:01.475 of a resonance in particle physics. 01:07:04.370 --> 01:07:08.780 Finally, the last example, which I am going to go through 01:07:08.780 --> 01:07:14.330 is an example involving a glass. 01:07:14.330 --> 01:07:18.290 We have prepared a very high-quality glass here. 01:07:21.910 --> 01:07:26.430 Maybe you have seen this glass before. 01:07:26.430 --> 01:07:28.250 They are pretty nice. 01:07:28.250 --> 01:07:38.590 And I usually use it to enjoy my red wines, which you cannot, 01:07:38.590 --> 01:07:39.780 enjoy, probably now. 01:07:42.530 --> 01:07:45.980 So you can see that this is the glass. 01:07:45.980 --> 01:07:49.300 And if I put a little bit of water on my hand and I rub it-- 01:07:53.584 --> 01:07:54.536 [VIBRATING TONE] 01:07:54.536 --> 01:07:59.860 --carefully, I can actually excite one 01:07:59.860 --> 01:08:03.250 of the resonance frequencies. 01:08:03.250 --> 01:08:06.040 So you can see that we have all everything working 01:08:06.040 --> 01:08:08.890 on a single particle. 01:08:08.890 --> 01:08:13.330 And that will give you one resonance frequency. 01:08:13.330 --> 01:08:15.550 If I work on two particles, which 01:08:15.550 --> 01:08:18.100 you will see that in the next lecture, 01:08:18.100 --> 01:08:20.710 I would get two resonance frequencies. 01:08:20.710 --> 01:08:27.220 And this glass is made of infinite number of particles. 01:08:27.220 --> 01:08:30.490 Therefore, I will have infinite number 01:08:30.490 --> 01:08:33.250 of resonance frequencies. 01:08:33.250 --> 01:08:36.160 When I'm rubbing it, I'm actually 01:08:36.160 --> 01:08:42.220 giving input of all kinds of different frequencies. 01:08:42.220 --> 01:08:46.330 But the glass will be smart enough so 01:08:46.330 --> 01:08:50.260 that it will pick up the one it likes the most, which 01:08:50.260 --> 01:08:53.420 is the resonance frequency. 01:08:53.420 --> 01:08:58.741 So you can see that the sound is actually, roughly, 683 Hertz. 01:09:01.630 --> 01:09:06.279 And you can actually measure it with your phone. 01:09:06.279 --> 01:09:10.810 So on the TV commercial, you may have seen 01:09:10.810 --> 01:09:13.760 that there's a lady singing. 01:09:13.760 --> 01:09:17.710 And she's singing so loudly such that the glass-- 01:09:17.710 --> 01:09:20.689 "bragh!"-- breaks. 01:09:20.689 --> 01:09:24.590 Can we get a volunteer today to sing in front of us? 01:09:24.590 --> 01:09:28.210 Oh-- singing. 01:09:28.210 --> 01:09:30.240 Can you sing it-- 01:09:30.240 --> 01:09:31.479 high frequencies? 01:09:31.479 --> 01:09:33.834 "Ahhhh." 01:09:33.834 --> 01:09:37.140 [LAUGHTER & APPLAUSE] 01:09:37.140 --> 01:09:38.069 Very good try. 01:09:38.069 --> 01:09:41.540 But it didn't work. 01:09:41.540 --> 01:09:45.350 OK, I guess it's really difficult 01:09:45.350 --> 01:09:49.819 to perform that in front of so many people unprepared. 01:09:49.819 --> 01:09:52.430 But fortunately, we are MIT. 01:09:52.430 --> 01:09:56.270 So we have designed a device, which actually 01:09:56.270 --> 01:09:59.990 can help us to achieve this. 01:09:59.990 --> 01:10:06.140 So this device actually contain a amplifier here. 01:10:06.140 --> 01:10:12.770 And I can now control the frequency of the sound 01:10:12.770 --> 01:10:14.330 through this scope. 01:10:14.330 --> 01:10:18.410 And this amplifier will actually amplify the signal 01:10:18.410 --> 01:10:26.750 and produce a sound wave and try to actually isolate this glass. 01:10:26.750 --> 01:10:28.820 So we are going to do this experiment. 01:10:28.820 --> 01:10:32.780 So we will need to change the loud setting a bit. 01:10:32.780 --> 01:10:36.710 Because the sound is going to be, probably, too loud. 01:10:36.710 --> 01:10:42.527 Just for safety-- some of you may not survive. 01:10:42.527 --> 01:10:43.840 [LAUGHTER] 01:10:43.840 --> 01:10:45.830 So I'm handing out these. 01:10:52.410 --> 01:10:55.110 OK, who is closer? 01:10:55.110 --> 01:10:56.768 OK, maybe you. 01:10:56.768 --> 01:10:58.672 AUDIENCE: [INAUDIBLE] 01:10:58.672 --> 01:10:59.616 YEN-JIE LEE: Oh. 01:10:59.616 --> 01:11:00.977 Oh, sorry. 01:11:00.977 --> 01:11:02.260 [LAUGHTER] 01:11:02.260 --> 01:11:03.960 I'm so sorry. 01:11:03.960 --> 01:11:05.170 What? 01:11:05.170 --> 01:11:07.080 I don't need that. 01:11:07.080 --> 01:11:07.580 OK. 01:11:10.760 --> 01:11:15.530 So just for safety, I will put this on. 01:11:15.530 --> 01:11:20.540 And what I am going to do is also put these glass on. 01:11:20.540 --> 01:11:22.290 OK, maybe I'll do this first. 01:11:30.586 --> 01:11:39.890 AUDIENCE: [INAUDIBLE] 01:11:39.890 --> 01:11:41.960 YEN-JIE LEE: OK, so what I am going to do now 01:11:41.960 --> 01:11:45.512 is to start producing sound wave. 01:11:45.512 --> 01:11:47.356 [LOUD TONE] 01:11:48.278 --> 01:11:50.550 So through the camera, you should 01:11:50.550 --> 01:11:55.916 be able to see what is actually shown on the screen. 01:11:55.916 --> 01:12:04.715 So you can see that, if this glass is actually moving, 01:12:04.715 --> 01:12:07.770 the wood inside would also vibrate. 01:12:07.770 --> 01:12:11.730 So you can see that, clearly, we don't have resonance yet. 01:12:11.730 --> 01:12:17.440 So what I am going to do is to increase the frequency 01:12:17.440 --> 01:12:19.590 and see what happens. 01:12:19.590 --> 01:12:24.240 So now, it's actually 643. 01:12:24.240 --> 01:12:28.585 It's actually still below the resonance frequency. 01:12:28.585 --> 01:12:31.650 Now, I have measured the frequency. 01:12:31.650 --> 01:12:34.860 It should be 684. 01:12:34.860 --> 01:12:36.296 So now, it's actually 653-- 01:12:39.595 --> 01:12:45.840 663 Hertz-- 673 Hertz. 01:12:45.840 --> 01:12:46.900 Can you see the movement? 01:12:46.900 --> 01:12:51.200 You cannot see the movement yet. 01:12:51.200 --> 01:12:54.410 683-- you see? 01:12:54.410 --> 01:12:55.350 You see, now-- 01:12:55.350 --> 01:12:55.960 AUDIENCE: Yes. 01:12:55.960 --> 01:12:58.512 YEN-JIE LEE: --the frequency of the sound 01:12:58.512 --> 01:13:03.200 is actually matching with one of the natural frequencies 01:13:03.200 --> 01:13:05.020 of the glass. 01:13:05.020 --> 01:13:07.340 Apparently, the glass likes it. 01:13:10.070 --> 01:13:14.990 And now, you can see that it is still vibrating. 01:13:14.990 --> 01:13:17.990 And the next step, which we are going to do, 01:13:17.990 --> 01:13:21.920 is to try to increase the amplitude, 01:13:21.920 --> 01:13:25.250 increase the volume of the sound, and see what happens. 01:13:25.250 --> 01:13:29.041 Maybe you want to cover your ears, just for safety. 01:13:29.041 --> 01:13:30.845 [LOUD TONE] 01:13:30.845 --> 01:13:37.852 OK, then the glass may break, if we are lucky. 01:13:37.852 --> 01:13:41.788 Let us see what is going to happen. 01:13:41.788 --> 01:13:44.248 [INCREASINGLY LOUDER TONE] 01:13:53.104 --> 01:13:55.564 Oh! 01:13:55.564 --> 01:13:57.532 [APPLAUSE] 01:13:59.034 --> 01:13:59.992 TECH SUPPORT: Good job. 01:13:59.992 --> 01:14:00.730 YEN-JIE LEE: Very good. 01:14:00.730 --> 01:14:01.046 TECH SUPPORT: Perfect. 01:14:01.046 --> 01:14:02.462 That's the quickest one we've had. 01:14:02.462 --> 01:14:04.260 YEN-JIE LEE: Yeah, thank you very much. 01:14:04.260 --> 01:14:05.830 Thank you, glass. 01:14:05.830 --> 01:14:07.120 [LAUGHTER] 01:14:07.980 --> 01:14:11.440 So you can see the power of resonance. 01:14:11.440 --> 01:14:19.110 So if I tune down the frequency slightly more, 01:14:19.110 --> 01:14:22.800 then you will be where? 01:14:22.800 --> 01:14:24.570 You'll be here. 01:14:24.570 --> 01:14:31.050 Then you will not have enough amplitude to break the glass. 01:14:31.050 --> 01:14:35.100 And also, as we discussed before, 01:14:35.100 --> 01:14:39.350 the quality of the glass should be really, really high, such 01:14:39.350 --> 01:14:42.880 that the resulting amplitude will be very large. 01:14:42.880 --> 01:14:45.940 Then you can actually break it with a external sound wave. 01:14:45.940 --> 01:14:49.580 And if we go above the resonance frequency, 01:14:49.580 --> 01:14:52.020 then you would not also move a bit. 01:14:52.020 --> 01:14:56.940 Because if you go to very large omega d, 01:14:56.940 --> 01:14:59.690 then amplitude will be pretty small. 01:15:02.660 --> 01:15:09.116 OK, let me try to switch back to my presentation. 01:15:09.116 --> 01:15:10.062 I think we did. 01:15:10.062 --> 01:15:10.562 Sure. 01:15:17.250 --> 01:15:20.200 So this is actually what we have learned today. 01:15:20.200 --> 01:15:27.540 We have learned the behavior of a damped driven oscillator. 01:15:27.540 --> 01:15:30.600 We have learned the transient behavior. 01:15:30.600 --> 01:15:33.420 So what is actually transient behavior 01:15:33.420 --> 01:15:42.420 is a mixture of steady state solution, which 01:15:42.420 --> 01:15:44.850 was coming from the driving force, 01:15:44.850 --> 01:15:48.900 and the homogeneous solution. 01:15:48.900 --> 01:15:53.530 If you wait long enough, this will decay and disappear. 01:15:53.530 --> 01:15:57.770 Is 01:15:57.770 --> 01:15:59.960 And we have learned resonance. 01:15:59.960 --> 01:16:04.040 So an IOC circuit, which you actually solved that 01:16:04.040 --> 01:16:08.870 in your P-set, in pendulum, which I just show you, 01:16:08.870 --> 01:16:13.370 which helped my son to learn wavelength vibrations-- 01:16:13.370 --> 01:16:20.030 and air condition, washing machine, glass-- 01:16:20.030 --> 01:16:21.140 particle physics. 01:16:21.140 --> 01:16:24.830 We can see damped os-- 01:16:24.830 --> 01:16:29.300 driven oscillator or resonance almost everywhere. 01:16:29.300 --> 01:16:32.650 So I hope that you enjoyed the lecture today. 01:16:32.650 --> 01:16:35.690 And what we are going to do next time 01:16:35.690 --> 01:16:39.710 is to put multiple objects together so that you 01:16:39.710 --> 01:16:44.320 see the interaction between one particle to the other particle 01:16:44.320 --> 01:16:48.020 and see how we can actually make sense of this kind of system. 01:16:48.020 --> 01:16:49.250 Thank you very much. 01:16:49.250 --> 01:16:52.640 And I will be here if you have additional questions related 01:16:52.640 --> 01:16:54.350 to the lecture.