1 00:00:02,195 --> 00:00:04,520 The following content is provided under a Creative 2 00:00:04,520 --> 00:00:05,910 Commons license. 3 00:00:05,910 --> 00:00:08,119 Your support will help MIT OpenCourseWare 4 00:00:08,119 --> 00:00:12,210 continue to offer high-quality educational resources for free. 5 00:00:12,210 --> 00:00:14,750 To make a donation or to view additional materials 6 00:00:14,750 --> 00:00:18,710 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:18,710 --> 00:00:19,580 at ocw.mit.edu. 8 00:00:23,270 --> 00:00:27,110 YEN-JIE LEE: OK, so welcome back, everybody. 9 00:00:27,110 --> 00:00:29,980 Welcome back to 8.03. 10 00:00:29,980 --> 00:00:34,400 Today, we are going to continue the discussion 11 00:00:34,400 --> 00:00:36,600 of the harmonic oscillators. 12 00:00:36,600 --> 00:00:40,550 And also, we will add damping force into the game 13 00:00:40,550 --> 00:00:43,160 and see what will happen, OK? 14 00:00:43,160 --> 00:00:45,950 So this is actually what we have learned last time 15 00:00:45,950 --> 00:00:47,900 from this slide. 16 00:00:47,900 --> 00:00:52,700 We have analyzed the physics of a harmonic oscillator, which 17 00:00:52,700 --> 00:00:54,680 we actually demonstrated last time. 18 00:00:54,680 --> 00:00:57,590 And you can see the device still there. 19 00:00:57,590 --> 00:01:01,020 And Hooke's law, actually the Hooke's law 20 00:01:01,020 --> 00:01:05,180 is actually far more general than what we saw before. 21 00:01:05,180 --> 00:01:09,340 It works for all small oscillations 22 00:01:09,340 --> 00:01:16,170 around about a point of equilibrium position, OK? 23 00:01:16,170 --> 00:01:23,190 And that can be demonstrated by multiple different kinds 24 00:01:23,190 --> 00:01:24,960 of physical systems. 25 00:01:24,960 --> 00:01:30,630 For example here, I have a mass, which actually can only 26 00:01:30,630 --> 00:01:33,600 move along this track here. 27 00:01:33,600 --> 00:01:37,350 And if I put this mass set free, then this thing 28 00:01:37,350 --> 00:01:43,160 is actually exercising harmonic oscillation, OK? 29 00:01:43,160 --> 00:01:44,660 We can do this with large amplitude. 30 00:01:44,660 --> 00:01:48,540 We can also do it with small amplitude. 31 00:01:48,540 --> 00:01:52,680 And you see that, huh, really, it works. 32 00:01:52,680 --> 00:01:54,080 Hooke's law actually works. 33 00:01:54,080 --> 00:01:56,940 And it predicts exactly the same motion 34 00:01:56,940 --> 00:02:02,040 as to what you see on the slide, OK? 35 00:02:02,040 --> 00:02:06,690 And we also have a little bit more complicated system. 36 00:02:06,690 --> 00:02:11,009 For example, this is some kind of rod. 37 00:02:11,009 --> 00:02:14,980 And you can actually fix one point and make it oscillate. 38 00:02:14,980 --> 00:02:18,120 And you see that, huh, it also does 39 00:02:18,120 --> 00:02:21,430 some kind of harmonic oscillation. 40 00:02:21,430 --> 00:02:25,630 But now, what is actually oscillating is the amplitude. 41 00:02:25,630 --> 00:02:29,110 The amplitude is actually the angle with respect 42 00:02:29,110 --> 00:02:32,510 to the downward direction. 43 00:02:32,510 --> 00:02:36,340 And finally this is actually the vertical version 44 00:02:36,340 --> 00:02:40,420 of this spring mass system, which you will be 45 00:02:40,420 --> 00:02:43,080 analyzing that in your P-set. 46 00:02:43,080 --> 00:02:46,850 And you see that, huh, it actually oscillates up and down 47 00:02:46,850 --> 00:02:48,410 harmonically. 48 00:02:48,410 --> 00:02:51,400 So that's all very nice. 49 00:02:51,400 --> 00:02:54,280 And we also have learned one thing which 50 00:02:54,280 --> 00:02:55,930 is very, very interesting. 51 00:02:55,930 --> 00:03:02,020 It's that a complex exponential is actually a pretty beautiful 52 00:03:02,020 --> 00:03:04,390 way to present the solution. 53 00:03:04,390 --> 00:03:09,070 And you will see it works also when describing 54 00:03:09,070 --> 00:03:10,470 the damped oscillators. 55 00:03:10,470 --> 00:03:17,310 And we will see how it works in the lecture today. 56 00:03:17,310 --> 00:03:21,240 I received several questions during my office hour 57 00:03:21,240 --> 00:03:23,790 and through email or Piazza. 58 00:03:23,790 --> 00:03:29,640 There were some confusions about doing the Taylor expansion, OK? 59 00:03:29,640 --> 00:03:34,450 So in lecture last time, the equilibrium position 60 00:03:34,450 --> 00:03:36,630 is at x equal to 0. 61 00:03:36,630 --> 00:03:41,730 Therefore, I do Taylor expansion around 0, OK? 62 00:03:41,730 --> 00:03:45,210 But in this case, if the equilibrium position 63 00:03:45,210 --> 00:03:50,850 or the minima of the potential is at x equal to L, 64 00:03:50,850 --> 00:03:55,050 then what you need to do is to do a Taylor expansion 65 00:03:55,050 --> 00:03:58,700 around x equal to L, just to make that really, 66 00:03:58,700 --> 00:04:00,460 really clear, OK? 67 00:04:00,460 --> 00:04:04,950 OK, I hope that will help you with the P-set question. 68 00:04:04,950 --> 00:04:07,780 OK, so let's get started immediately. 69 00:04:07,780 --> 00:04:13,200 So let's continue the discussion of the equation of motion 70 00:04:13,200 --> 00:04:14,740 we arrived at last time. 71 00:04:14,740 --> 00:04:22,690 So we have M x double-dot and this is equal to minus kx, OK? 72 00:04:22,690 --> 00:04:25,120 That is actually the formula from last time. 73 00:04:25,120 --> 00:04:35,470 And we can actually calculate the kinetic energy 74 00:04:35,470 --> 00:04:38,230 of this spring and mass system. 75 00:04:38,230 --> 00:04:45,820 And basically, this is going to be equal to 1/2 M 76 00:04:45,820 --> 00:04:50,400 times x dot squared. 77 00:04:50,400 --> 00:04:53,740 OK, and we can also calculate the potential energy 78 00:04:53,740 --> 00:04:55,920 of the spring. 79 00:04:55,920 --> 00:05:10,760 Potential energy, and that is equal to 1/2 kx squared. 80 00:05:10,760 --> 00:05:13,940 We also know what would be the total energy. 81 00:05:13,940 --> 00:05:19,520 The total energy would be a sum of the kinetic energy 82 00:05:19,520 --> 00:05:21,360 and of the potential. 83 00:05:21,360 --> 00:05:25,280 Basically, you get this formula, 1/2 M x dot 84 00:05:25,280 --> 00:05:33,840 squared plus 1/2 kx squared. 85 00:05:33,840 --> 00:05:37,420 One last time, we have solved this equation of motion, right? 86 00:05:37,420 --> 00:05:48,890 So the solution we got is x equal to A cosine omega 0 t 87 00:05:48,890 --> 00:05:51,470 plus phi. 88 00:05:51,470 --> 00:05:59,160 Well, omega 0 is equal to a square root of k over M. 89 00:05:59,160 --> 00:06:01,460 Therefore, we can actually calculate 90 00:06:01,460 --> 00:06:06,790 what would be the total energy as a function of time, right? 91 00:06:06,790 --> 00:06:08,940 So if we calculate that, we'll get 92 00:06:08,940 --> 00:06:19,410 E will be equal to 1/2 M A squared omega 0 squared sine 93 00:06:19,410 --> 00:06:23,850 squared omega 0 t plus phi-- 94 00:06:23,850 --> 00:06:26,940 so this is actually the first term here-- 95 00:06:26,940 --> 00:06:39,210 plus 1/2 kA squared cosine squared omega 0 t plus phi, OK? 96 00:06:39,210 --> 00:06:44,570 Then, we also know that this coefficient here 97 00:06:44,570 --> 00:06:47,250 is just kA squared, right? 98 00:06:47,250 --> 00:06:50,970 Because omega 0 is actually equal to the square root of k 99 00:06:50,970 --> 00:06:58,470 over M. And if you replace this omega 0 squared by k over M, 100 00:06:58,470 --> 00:07:02,160 then you actually arrive at kA squared, OK? 101 00:07:02,160 --> 00:07:03,750 So that is actually very good. 102 00:07:03,750 --> 00:07:07,470 So that means I can simplify the total energy. 103 00:07:07,470 --> 00:07:12,690 And what we are going to get is 1/2 kA squared. 104 00:07:12,690 --> 00:07:15,180 I can take this factor out. 105 00:07:15,180 --> 00:07:19,680 And that will give me, inside these brackets, I will get sine 106 00:07:19,680 --> 00:07:27,910 squared omega 0 t plus phi plus cosine squared omega 107 00:07:27,910 --> 00:07:31,410 0 t plus phi. 108 00:07:31,410 --> 00:07:36,430 And this is actually equal to 1, right? 109 00:07:36,430 --> 00:07:39,990 Just a reminder, sine squared of theta plus cosine squared 110 00:07:39,990 --> 00:07:43,000 of theta is always equal to 1. 111 00:07:43,000 --> 00:07:45,360 So that gives me this result. This 112 00:07:45,360 --> 00:07:52,750 is actually 1/2 kA squared, OK? 113 00:07:52,750 --> 00:07:56,740 So that is actually the result. What does that mean? 114 00:07:56,740 --> 00:08:04,840 That means, if I actually pull this mass harder, 115 00:08:04,840 --> 00:08:09,600 so that initially it has larger amplitude, 116 00:08:09,600 --> 00:08:12,630 then the total energy is actually proportioned 117 00:08:12,630 --> 00:08:15,360 to amplitude squared, OK? 118 00:08:15,360 --> 00:08:18,630 So I am storing more and more energy. 119 00:08:18,630 --> 00:08:23,230 If I increase the amplitude even more, 120 00:08:23,230 --> 00:08:26,430 then I am storing the energy in this system. 121 00:08:26,430 --> 00:08:28,430 And it's proportional to A squared. 122 00:08:28,430 --> 00:08:32,200 And also, if the spring constant is larger, 123 00:08:32,200 --> 00:08:36,142 the same amplitude will give you more energy. 124 00:08:36,142 --> 00:08:38,475 So that means that you can store more energy if you have 125 00:08:38,475 --> 00:08:41,950 a larger string constant, OK? 126 00:08:41,950 --> 00:08:46,230 The most surprising thing is that actually this 127 00:08:46,230 --> 00:08:50,020 is actually a constant, right? 128 00:08:50,020 --> 00:08:51,610 What does that mean? 129 00:08:51,610 --> 00:08:55,450 The total energy is actually not variating 130 00:08:55,450 --> 00:08:56,873 as a function of time. 131 00:08:56,873 --> 00:08:57,740 You see? 132 00:08:57,740 --> 00:09:00,970 So total energy is constant, OK? 133 00:09:00,970 --> 00:09:06,970 So you can see from this slide the total energy is actually 134 00:09:06,970 --> 00:09:11,522 showing us the sum, which is the green curve. 135 00:09:11,522 --> 00:09:13,480 And the kinetic energy and the potential energy 136 00:09:13,480 --> 00:09:17,500 are shown as red and blue curves. 137 00:09:17,500 --> 00:09:23,050 You can see that the total energy is actually constant. 138 00:09:23,050 --> 00:09:25,900 But this system is very dynamical. 139 00:09:25,900 --> 00:09:26,410 You see? 140 00:09:26,410 --> 00:09:31,840 So that energy is actually going back and forth 141 00:09:31,840 --> 00:09:38,840 between the spring and the mass in the form of kinetic energy 142 00:09:38,840 --> 00:09:41,200 and in the form of potential energy. 143 00:09:41,200 --> 00:09:45,820 But they are doing it so well, such that the sum is actually 144 00:09:45,820 --> 00:09:47,125 a constant. 145 00:09:47,125 --> 00:09:50,200 So the energy is actually constant, OK? 146 00:09:50,200 --> 00:09:53,170 So that is actually pretty beautiful. 147 00:09:53,170 --> 00:09:59,520 And it can be described very well by these mathematics. 148 00:09:59,520 --> 00:10:00,550 Any questions from here? 149 00:10:04,380 --> 00:10:08,342 OK, so I would like to say simple harmonic motion, 150 00:10:08,342 --> 00:10:09,800 actually, what you are going to get 151 00:10:09,800 --> 00:10:12,780 is the energy is actually conserved and independent 152 00:10:12,780 --> 00:10:14,300 of the time. 153 00:10:14,300 --> 00:10:18,130 And later, you will see an example with damping. 154 00:10:18,130 --> 00:10:20,735 And you will see that energy conservation 155 00:10:20,735 --> 00:10:24,320 is now no longer the case, OK? 156 00:10:24,320 --> 00:10:29,040 So let's immediately jump to another example, 157 00:10:29,040 --> 00:10:33,270 which is actually involving simple harmonic motion. 158 00:10:33,270 --> 00:10:41,100 So let's take this rod and nail system as an example. 159 00:10:41,100 --> 00:10:44,660 If I actually slightly move this rod, 160 00:10:44,660 --> 00:10:49,490 and then I release that, then actually 161 00:10:49,490 --> 00:10:55,010 you will see simple harmonic motion, also for this system. 162 00:10:55,010 --> 00:11:00,620 So let's actually do the calculation as another example. 163 00:11:00,620 --> 00:11:02,720 So this is actually my system. 164 00:11:02,720 --> 00:11:05,920 I have this rod, OK? 165 00:11:05,920 --> 00:11:09,680 Now, I am assume that the mass is actually uniformly 166 00:11:09,680 --> 00:11:15,370 distributed on this rod and is nailed on the wall, OK? 167 00:11:15,370 --> 00:11:20,150 And the length of this rod is actually l. 168 00:11:20,150 --> 00:11:25,430 So that means the center of mass is actually at l/2 169 00:11:25,430 --> 00:11:29,600 with respect to the nail, OK? 170 00:11:29,600 --> 00:11:36,270 And also, this whole system is set up on Earth, right? 171 00:11:36,270 --> 00:11:38,870 Therefore, there will be gravitational force 172 00:11:38,870 --> 00:11:41,570 pointing downward, OK? 173 00:11:41,570 --> 00:11:46,600 So that means you have gravitational force, Fg, 174 00:11:46,600 --> 00:11:49,670 pointing downward, OK? 175 00:11:49,670 --> 00:11:52,490 So this is actually the system, which 176 00:11:52,490 --> 00:11:54,290 I would like to understand. 177 00:11:54,290 --> 00:11:58,490 And just a reminder, what are we going to do afterwards in order 178 00:11:58,490 --> 00:12:02,420 to turn the whole system into a language we 179 00:12:02,420 --> 00:12:04,640 know describes the nature? 180 00:12:04,640 --> 00:12:08,330 What are we going to do? 181 00:12:08,330 --> 00:12:09,040 Anybody? 182 00:12:13,360 --> 00:12:18,010 We are going to define the coordinate system, 183 00:12:18,010 --> 00:12:20,980 so that I can translate everything 184 00:12:20,980 --> 00:12:22,600 into mathematics, right? 185 00:12:22,600 --> 00:12:25,450 So that's actually what we are always doing. 186 00:12:25,450 --> 00:12:26,950 And you will see that we are always 187 00:12:26,950 --> 00:12:32,290 doing this in this class, OK? 188 00:12:32,290 --> 00:12:35,060 So what is actually the coordinate system 189 00:12:35,060 --> 00:12:36,760 which I would like to use? 190 00:12:36,760 --> 00:12:42,100 Since this system is going to be rotating back and forth, 191 00:12:42,100 --> 00:12:47,660 therefore, I would like to define theta 192 00:12:47,660 --> 00:12:52,700 to be that angle with respect to the axis, which essentially 193 00:12:52,700 --> 00:12:55,190 pointing downward, OK? 194 00:12:55,190 --> 00:13:02,510 So the origin of this coordinate system uses theta equal to 0. 195 00:13:02,510 --> 00:13:14,100 This means that the rod is actually pointing downward, OK? 196 00:13:14,100 --> 00:13:21,750 And also, I need to define what is actually the positive value 197 00:13:21,750 --> 00:13:22,950 of the zeta, right? 198 00:13:22,950 --> 00:13:29,400 So I define anti-clockwise direction to be positive, OK? 199 00:13:29,400 --> 00:13:34,370 So it is actually important to actually first define that, 200 00:13:34,370 --> 00:13:38,170 then actually to translate everything into mathematics, 201 00:13:38,170 --> 00:13:40,530 OK? 202 00:13:40,530 --> 00:13:42,880 So the initial condition is the following. 203 00:13:42,880 --> 00:13:48,060 So I actually move this thing, rotate this thing slightly. 204 00:13:48,060 --> 00:13:50,880 Then, I actually release that really carefully 205 00:13:50,880 --> 00:13:55,140 without introducing any initial velocity, OK? 206 00:13:55,140 --> 00:13:58,380 Therefore, I have two initial conditions. 207 00:14:05,760 --> 00:14:11,750 OK, at t equal to 0, there are two initial conditions. 208 00:14:11,750 --> 00:14:16,780 The first one is theta 0 is equal to theta initial. 209 00:14:21,130 --> 00:14:23,290 The second condition is the same as what 210 00:14:23,290 --> 00:14:26,000 we have been doing last time. 211 00:14:26,000 --> 00:14:29,730 The initial velocity or angular velocity 212 00:14:29,730 --> 00:14:30,980 is actually equal to 0. 213 00:14:30,980 --> 00:14:36,610 So that gives you theta dot equal to 0, OK? 214 00:14:36,610 --> 00:14:43,030 Now, we have actually defined the coordinate system. 215 00:14:43,030 --> 00:14:49,660 Now, we can actually draw a force diagram, 216 00:14:49,660 --> 00:14:54,460 so that we can actually use our knowledge about the physics 217 00:14:54,460 --> 00:14:57,020 to obtain the equation of motion, right? 218 00:14:57,020 --> 00:14:59,680 So now, the force diagram looks like this. 219 00:15:05,570 --> 00:15:10,420 So this is actually the center of mass of this rod. 220 00:15:10,420 --> 00:15:14,300 And you have a force pointing downward, 221 00:15:14,300 --> 00:15:17,540 which is due to the gravitational force. 222 00:15:17,540 --> 00:15:21,660 Fg is equal to mg. 223 00:15:21,660 --> 00:15:23,620 It's pointing downward. 224 00:15:23,620 --> 00:15:28,960 The magnitude is actually equal to mg. 225 00:15:28,960 --> 00:15:36,160 And also, we know the R vector. 226 00:15:36,160 --> 00:15:40,880 This vector has a length, l/2. 227 00:15:40,880 --> 00:15:49,228 It's pointing from the center of mass of this rod to the nail, 228 00:15:49,228 --> 00:15:51,420 OK? 229 00:15:51,420 --> 00:15:57,720 And also, we know the angle between these vectors, 230 00:15:57,720 --> 00:16:00,960 pointing from the center of mass to the nail, 231 00:16:00,960 --> 00:16:04,920 and the vertical direction, which we have already defined, 232 00:16:04,920 --> 00:16:07,180 which is actually called theta. 233 00:16:07,180 --> 00:16:10,260 Therefore, now, we can actually calculate 234 00:16:10,260 --> 00:16:13,700 what would be the torque. 235 00:16:13,700 --> 00:16:19,970 Tau will be equal to this R vector 236 00:16:19,970 --> 00:16:27,300 cross the force, total force acting on the center mass. 237 00:16:27,300 --> 00:16:33,410 In this case, it's just Fg, OK? 238 00:16:33,410 --> 00:16:36,650 So now, we can actually write this down explicitly. 239 00:16:36,650 --> 00:16:39,050 Since the whole system is actually 240 00:16:39,050 --> 00:16:44,980 rotating on a single plane, so there's only one plane 241 00:16:44,980 --> 00:16:46,590 this is sitting on. 242 00:16:46,590 --> 00:16:51,990 And it's actually going back and forth only on this plane, OK? 243 00:16:51,990 --> 00:16:56,630 Therefore, actually, I can drop all the arrows 244 00:16:56,630 --> 00:17:01,550 and write down the magnitude of the tau directly. 245 00:17:01,550 --> 00:17:14,240 And this will be equal to minus mg l/2 sine theta t, OK? 246 00:17:14,240 --> 00:17:15,319 Any questions so far? 247 00:17:20,079 --> 00:17:22,720 OK, so now, we have the torque. 248 00:17:22,720 --> 00:17:26,660 And we can make use of the rotational version 249 00:17:26,660 --> 00:17:30,950 of Newton's Law to obtain the equation of motion, right? 250 00:17:30,950 --> 00:17:35,180 So that should be pretty straightforward. 251 00:17:35,180 --> 00:17:39,890 Tau will be equal to I, which is the moment of inertia 252 00:17:39,890 --> 00:17:48,320 of the system, times alpha t, OK? 253 00:17:48,320 --> 00:17:50,900 And just for your information, I already 254 00:17:50,900 --> 00:17:56,490 calculated the I for you. 255 00:17:56,490 --> 00:18:02,970 I is equal to 1/3 ml squared, OK? 256 00:18:02,970 --> 00:18:06,060 So you can actually go back home and actually do 257 00:18:06,060 --> 00:18:08,580 a check to see if I'm telling the truth. 258 00:18:08,580 --> 00:18:11,890 And if you trust me, then that's the answer, 259 00:18:11,890 --> 00:18:14,370 which is actually 1/3 ml squared, 260 00:18:14,370 --> 00:18:19,350 if the mass is actually uniformly distributed 261 00:18:19,350 --> 00:18:22,050 on this rod, OK? 262 00:18:22,050 --> 00:18:37,410 So that would give me minus mgl divided by 2 sine theta t, OK? 263 00:18:37,410 --> 00:18:43,140 So that is actually coming from this side, OK? 264 00:18:43,140 --> 00:18:46,740 So now, I can actually simplify this expression. 265 00:18:46,740 --> 00:18:50,550 I can now plug in the I value into this equation. 266 00:18:50,550 --> 00:19:01,390 And I will get 1/3 ml squared theta double-dot t, which 267 00:19:01,390 --> 00:19:02,880 is actually alpha, OK? 268 00:19:02,880 --> 00:19:05,250 Now, I write it as theta double-dot. 269 00:19:05,250 --> 00:19:14,670 And that will be equal to minus mgl over 2 sine theta, OK? 270 00:19:14,670 --> 00:19:17,470 I can move all the constants to the right-hand side. 271 00:19:17,470 --> 00:19:20,970 Therefore, I get theta double-dot t. 272 00:19:20,970 --> 00:19:23,610 This is equal to minus mgl. 273 00:19:27,380 --> 00:19:32,800 OK, actually, I can already simplify this, right? 274 00:19:32,800 --> 00:19:34,400 These actually cancel. 275 00:19:34,400 --> 00:19:36,920 And the 1/l actually cancels. 276 00:19:36,920 --> 00:19:44,150 So therefore, I get minus 3g over 2 sine theta t. 277 00:19:48,050 --> 00:19:53,110 OK, as you know, we actually defined omega 278 00:19:53,110 --> 00:19:57,380 to replace this constant to make our life easier. 279 00:19:57,380 --> 00:20:05,050 So I can now define omega 0 equal to square root of 3g 280 00:20:05,050 --> 00:20:09,760 over 2l, OK? 281 00:20:09,760 --> 00:20:12,390 And that will give you theta double-dot 282 00:20:12,390 --> 00:20:20,320 of t equal to minus omega 0 squared sin theta t. 283 00:20:24,610 --> 00:20:26,530 Any questions so far? 284 00:20:26,530 --> 00:20:28,820 A lot of calculations. 285 00:20:28,820 --> 00:20:31,480 But they should all be pretty straightforward. 286 00:20:31,480 --> 00:20:35,900 And actually, we are done now. 287 00:20:35,900 --> 00:20:36,740 We are done. 288 00:20:36,740 --> 00:20:39,090 Because we have the equation of motion. 289 00:20:39,090 --> 00:20:42,530 And the rest of the job is to solve just it. 290 00:20:42,530 --> 00:20:46,040 So it is actually now the problem of the math department. 291 00:20:46,040 --> 00:20:50,960 So can anybody actually tell me the solution of the theta of t? 292 00:20:50,960 --> 00:20:54,025 Anybody? 293 00:20:54,025 --> 00:20:56,947 AUDIENCE: Unfortunately, we'd have to approximate it. 294 00:20:56,947 --> 00:20:59,880 YEN-JIE LEE: That's very unfortunate. 295 00:20:59,880 --> 00:21:04,600 So now, we are facing a very difficult situation. 296 00:21:04,600 --> 00:21:08,970 We don't know how to solve this equation in front of you. 297 00:21:08,970 --> 00:21:13,080 I don't know, OK? 298 00:21:13,080 --> 00:21:18,870 Of course, you can actually solve it with a computer, 299 00:21:18,870 --> 00:21:23,020 or, if you want to go fancy, solve it with your cellphone, 300 00:21:23,020 --> 00:21:24,020 if it doesn't explode. 301 00:21:28,740 --> 00:21:34,710 But it's not really nice to do this in front of you. 302 00:21:34,710 --> 00:21:35,830 We don't learn too much. 303 00:21:35,830 --> 00:21:39,690 OK, so what are we going to do? 304 00:21:39,690 --> 00:21:45,720 So what we can do is actually to consider a special case. 305 00:21:45,720 --> 00:21:50,930 So we know that this equation of motion is exact, OK? 306 00:21:50,930 --> 00:21:53,780 So if you solve it, it would describe 307 00:21:53,780 --> 00:21:57,350 the motion of this rod. 308 00:22:00,440 --> 00:22:04,920 Even with a large angle, it works, OK? 309 00:22:04,920 --> 00:22:07,560 And now, in order to actually show 310 00:22:07,560 --> 00:22:11,100 you the math in the class, therefore 311 00:22:11,100 --> 00:22:14,090 actually I will do a small approximation. 312 00:22:14,090 --> 00:22:17,130 So actually, I would only work on the case 313 00:22:17,130 --> 00:22:21,390 that when the amplitude is very small 314 00:22:21,390 --> 00:22:23,710 and see what is going to happen. 315 00:22:23,710 --> 00:22:27,150 So now, I'm considering a special case. 316 00:22:27,150 --> 00:22:29,650 Up to now, everything is exact. 317 00:22:29,650 --> 00:22:34,340 And now, I am now going to a special case. 318 00:22:34,340 --> 00:22:39,390 Theta t goes to 0, OK? 319 00:22:39,390 --> 00:22:41,390 Then, we can actually get this. 320 00:22:41,390 --> 00:22:48,020 Sine theta t is roughly theta t, OK? 321 00:22:48,020 --> 00:22:50,000 Based on the Taylor expansion, you 322 00:22:50,000 --> 00:22:53,870 can actually verify this, OK? 323 00:22:53,870 --> 00:23:02,120 So in this case, if we take theta equal to 1 degree, 324 00:23:02,120 --> 00:23:06,626 then the ratio of the sine theta and the theta 325 00:23:06,626 --> 00:23:12,710 is actually equal to 99.99%, which is very good. 326 00:23:12,710 --> 00:23:23,930 If I take it as 5 degrees, then it's actually 99%. 327 00:23:23,930 --> 00:23:29,970 Even at 10 degrees, it's actually 99.5%. 328 00:23:29,970 --> 00:23:34,700 Now, that shows you that sine theta is so close to theta, OK? 329 00:23:34,700 --> 00:23:35,690 We are pretty safe. 330 00:23:35,690 --> 00:23:39,750 Because the difference is smaller than 1%. 331 00:23:39,750 --> 00:23:42,890 OK, so that's very nice. 332 00:23:42,890 --> 00:23:50,800 After this approximation, I get my final equation of motion. 333 00:23:50,800 --> 00:24:00,250 Theta double-dot t equal to minus omega 0 squared theta t. 334 00:24:00,250 --> 00:24:04,375 Just a reminder, omega 0 is equal to square root 335 00:24:04,375 --> 00:24:11,400 of 3g over 2l, OK? 336 00:24:11,400 --> 00:24:17,870 We have solved this equation last time, last lecture, right? 337 00:24:17,870 --> 00:24:19,200 It's exactly the same. 338 00:24:19,200 --> 00:24:21,790 OK, it happened to be exactly the same. 339 00:24:21,790 --> 00:24:23,550 Therefore, I know the solution will 340 00:24:23,550 --> 00:24:30,660 be theta of t equal to A cosine omega 0 t plus phi. 341 00:24:34,000 --> 00:24:39,750 From the initial conditions, which I have one and two, 342 00:24:39,750 --> 00:24:43,650 I am not going to go over these calculation again. 343 00:24:43,650 --> 00:24:46,560 But again, we can actually plug in 1 and 2 344 00:24:46,560 --> 00:24:50,850 to solve the unknown A and the phi. 345 00:24:50,850 --> 00:24:53,400 If you do this exercise, you will 346 00:24:53,400 --> 00:24:58,620 conclude that A is equal to theta initial. 347 00:25:01,740 --> 00:25:08,900 And phi is equal to 0, OK? 348 00:25:08,900 --> 00:25:18,340 So the solution would be theta of t equal to theta 349 00:25:18,340 --> 00:25:24,280 initial cosine omega 0 t. 350 00:25:27,400 --> 00:25:33,560 You can see that this actually works for this system. 351 00:25:33,560 --> 00:25:35,940 Simple harmonic oscillation actually 352 00:25:35,940 --> 00:25:41,540 described the motion of this system as a function of time. 353 00:25:41,540 --> 00:25:45,470 You can also see a few more examples shown here. 354 00:25:45,470 --> 00:25:49,270 Two of them you are going to really work on in your P-set 355 00:25:49,270 --> 00:25:53,330 and also another one involving circuits. 356 00:25:53,330 --> 00:25:57,290 If you have a capacitor and you have an inductor, 357 00:25:57,290 --> 00:26:03,290 actually the size of the current is also 358 00:26:03,290 --> 00:26:06,740 doing a simple harmonic motion, OK? 359 00:26:06,740 --> 00:26:09,560 And as we actually discussed before, 360 00:26:09,560 --> 00:26:12,570 the energy is always conserved. 361 00:26:12,570 --> 00:26:15,410 And that is actually stored in different components 362 00:26:15,410 --> 00:26:16,890 of the system, OK? 363 00:26:20,520 --> 00:26:22,510 So we have done this. 364 00:26:22,510 --> 00:26:26,550 What is actually new today? 365 00:26:26,550 --> 00:26:30,100 What we are going to do today is let's actually 366 00:26:30,100 --> 00:26:34,140 observe this phenomenon here. 367 00:26:34,140 --> 00:26:38,010 So this thing is actually going to go back and forth. 368 00:26:38,010 --> 00:26:41,240 But it's actually not going to do that forever, right? 369 00:26:41,240 --> 00:26:48,340 Something is happening, which actually slows the motion down. 370 00:26:48,340 --> 00:26:52,060 I can also make use of this system, OK? 371 00:26:52,060 --> 00:26:54,180 I start from here. 372 00:26:54,180 --> 00:26:56,320 And I'm not worried that this actually 373 00:26:56,320 --> 00:26:58,330 goes out of this track. 374 00:26:58,330 --> 00:27:03,160 Because I know for sure it will stop there. 375 00:27:03,160 --> 00:27:04,240 Why? 376 00:27:04,240 --> 00:27:09,780 Because the initial amplitude is not going to-- 377 00:27:09,780 --> 00:27:11,434 the amplitude is not going to be larger 378 00:27:11,434 --> 00:27:12,850 than the initial amplitude, right? 379 00:27:12,850 --> 00:27:15,480 So I'm not worried at all, OK? 380 00:27:15,480 --> 00:27:17,710 But you can see that the amplitude is 381 00:27:17,710 --> 00:27:20,680 changing as a function of time. 382 00:27:20,680 --> 00:27:23,590 Apparently, something is missing. 383 00:27:23,590 --> 00:27:28,050 And that is actually a direct force, or friction, 384 00:27:28,050 --> 00:27:32,720 which is actually not included in our calculation. 385 00:27:32,720 --> 00:27:40,860 So let's actually try to make the calculation more realistic 386 00:27:40,860 --> 00:27:43,900 and see what is going to happen. 387 00:27:43,900 --> 00:27:51,500 So now, I will introduce a drag force, 388 00:27:51,500 --> 00:27:58,300 which actually introduces a torque tau drag, t, 389 00:27:58,300 --> 00:28:01,360 which is equal to minus b-- 390 00:28:01,360 --> 00:28:03,670 b is actually some kind of constant, 391 00:28:03,670 --> 00:28:06,970 which is given to you-- 392 00:28:06,970 --> 00:28:14,680 theta dot t, which is actually proportional to angular 393 00:28:14,680 --> 00:28:19,490 velocity of that rod, OK? 394 00:28:19,490 --> 00:28:24,350 And also of course, I keep the original approximation. 395 00:28:24,350 --> 00:28:27,590 The theta is very small, such that I 396 00:28:27,590 --> 00:28:30,810 don't have to deal with the integration of sine theta, OK? 397 00:28:30,810 --> 00:28:33,880 So solving this, theta double-dot equal to minus 398 00:28:33,880 --> 00:28:37,520 omega 0 squared sine theta is a complicated function. 399 00:28:40,360 --> 00:28:46,030 You may ask, why do I actually introduce a drag force 400 00:28:46,030 --> 00:28:49,770 proportional to the velocity? 401 00:28:49,770 --> 00:28:53,460 And why do I put a minus sign there? 402 00:28:53,460 --> 00:28:58,380 That is actually because, if you have a minus sign, that means, 403 00:28:58,380 --> 00:29:03,870 when this mass or that rod is actually going downward, 404 00:29:03,870 --> 00:29:06,030 then the drag force is really dragging it. 405 00:29:06,030 --> 00:29:09,750 Because it's actually in the opposite direction 406 00:29:09,750 --> 00:29:14,460 of the velocity of the mass or the angular 407 00:29:14,460 --> 00:29:16,470 velocity of the rod, OK? 408 00:29:16,470 --> 00:29:18,750 So I need a minus sign there, OK? 409 00:29:18,750 --> 00:29:21,550 Otherwise, it's not a drag force anymore. 410 00:29:21,550 --> 00:29:25,580 It's actually accelerating the whole thing. 411 00:29:25,580 --> 00:29:33,878 Secondly, why do I choose that to be proportional to theta dot 412 00:29:33,878 --> 00:29:37,750 or velocity? 413 00:29:37,750 --> 00:29:42,150 There's really no much deeper reason. 414 00:29:42,150 --> 00:29:46,140 I choose this form because I can actually 415 00:29:46,140 --> 00:29:48,460 solve it in front of you, OK? 416 00:29:48,460 --> 00:29:52,560 The reality is actually between proportional 417 00:29:52,560 --> 00:29:57,210 to theta dot and theta dot squared, for example, OK? 418 00:29:57,210 --> 00:29:59,940 This is actually a model which I introduced here, 419 00:29:59,940 --> 00:30:04,440 which I can actually solve it in front of you. 420 00:30:04,440 --> 00:30:07,110 On the other hand, you'll see that it's actually not 421 00:30:07,110 --> 00:30:09,690 bad at all. 422 00:30:09,690 --> 00:30:12,900 It actually works and describes the system, 423 00:30:12,900 --> 00:30:17,910 which will actually work to perform the demo here, OK? 424 00:30:17,910 --> 00:30:24,390 And once we have introduced this, the equation of motion 425 00:30:24,390 --> 00:30:26,810 will be modified. 426 00:30:26,810 --> 00:30:29,070 So let's come back to the equation of motion. 427 00:30:29,070 --> 00:30:32,890 So you have to theta double-dot t 428 00:30:32,890 --> 00:30:42,490 originally would be equal to tau total t divided by I, OK? 429 00:30:42,490 --> 00:30:57,350 And now, this will become tau t plus tau drag t divided by I. 430 00:30:57,350 --> 00:30:59,540 So there's an additional time here. 431 00:30:59,540 --> 00:31:03,010 OK, if I simplify this whole equation, 432 00:31:03,010 --> 00:31:12,320 then I get minus mgl over 2 sine theta. 433 00:31:12,320 --> 00:31:16,960 And this is actually roughly theta minus 434 00:31:16,960 --> 00:31:27,140 b theta dot divided by 1/3 of ml squared, OK? 435 00:31:27,140 --> 00:31:30,520 So you can see that I still make this approximation sine 436 00:31:30,520 --> 00:31:34,940 theta roughly equal to theta. 437 00:31:34,940 --> 00:31:40,170 Then, I can actually write this equation in the small angle 438 00:31:40,170 --> 00:31:40,670 case. 439 00:31:44,740 --> 00:31:53,000 OK, I get minus 3g over 2l theta t 440 00:31:53,000 --> 00:31:58,740 minus 3b over ml squared theta dot t. 441 00:32:01,660 --> 00:32:10,490 OK, and now, as usual, I define omega 0 squared equal to 3g 442 00:32:10,490 --> 00:32:11,960 over 2l. 443 00:32:11,960 --> 00:32:18,410 And I can also define gamma is equal to 3b over 444 00:32:18,410 --> 00:32:25,050 ml squared, just to make my life easier, right? 445 00:32:25,050 --> 00:32:32,740 Finally, we will arrive at this expression, theta double-dot 446 00:32:32,740 --> 00:32:42,260 plus gamma theta dot plus omega 0 squared theta. 447 00:32:42,260 --> 00:32:44,217 And that is equal to 0. 448 00:32:48,120 --> 00:32:56,340 So what you can see from here is that we have actually derived 449 00:32:56,340 --> 00:32:58,950 the equation of motion, OK? 450 00:32:58,950 --> 00:33:01,140 We have derived the equation of motion. 451 00:33:01,140 --> 00:33:05,660 And actually, part of the work is actually really 452 00:33:05,660 --> 00:33:08,250 just solving this equation of motion. 453 00:33:08,250 --> 00:33:09,840 And you don't really have to solve it. 454 00:33:09,840 --> 00:33:13,650 Because you already get the result from 18.03 455 00:33:13,650 --> 00:33:16,130 actually, if you remember. 456 00:33:16,130 --> 00:33:18,610 And we are going to discuss the result. 457 00:33:18,610 --> 00:33:22,830 But before that, before I really try to solve this equation, 458 00:33:22,830 --> 00:33:28,460 I would like to take a vote, OK? 459 00:33:28,460 --> 00:33:34,540 So here, I have two different systems. 460 00:33:34,540 --> 00:33:37,980 They have equal amounts of mass. 461 00:33:37,980 --> 00:33:40,450 They are attached to a spring. 462 00:33:40,450 --> 00:33:44,440 If you do the same equation of motion derivation, 463 00:33:44,440 --> 00:33:51,850 you will actually get exactly the same equation of motion 464 00:33:51,850 --> 00:33:54,730 in that format, OK? 465 00:33:54,730 --> 00:33:56,380 So the form of the equation of motion 466 00:33:56,380 --> 00:34:01,850 will be the same between this system and that system, OK? 467 00:34:01,850 --> 00:34:06,380 I would like to ask you a question about the oscillation 468 00:34:06,380 --> 00:34:07,950 frequency. 469 00:34:07,950 --> 00:34:12,230 So you can see that one of them is actually a better mass. 470 00:34:12,230 --> 00:34:14,270 It's like a point-like particle. 471 00:34:14,270 --> 00:34:17,730 And the other one is wearing a hat, OK? 472 00:34:17,730 --> 00:34:21,469 What is going to happen is that this Mexican hat is 473 00:34:21,469 --> 00:34:26,270 going to be trying to push the air away, right? 474 00:34:26,270 --> 00:34:30,710 Then, you may think, OK, this Mexican thing 475 00:34:30,710 --> 00:34:33,290 is not really very important. 476 00:34:33,290 --> 00:34:35,150 Therefore, the oscillation frequency 477 00:34:35,150 --> 00:34:37,120 may be the same, right? 478 00:34:37,120 --> 00:34:40,040 How many of you think the oscillation frequency, 479 00:34:40,040 --> 00:34:42,889 if I actually tried to perturb these two systems, 480 00:34:42,889 --> 00:34:44,740 would be the same? 481 00:34:44,740 --> 00:34:45,870 Raise your hands. 482 00:34:45,870 --> 00:34:49,730 1, 2, 3, 4, 5, 6, 7, 8-- 483 00:34:49,730 --> 00:34:51,040 OK, we have 11. 484 00:34:53,639 --> 00:34:57,630 So the omega, the predicted omega, 485 00:34:57,630 --> 00:35:00,903 will be equal to omega 0-- 486 00:35:00,903 --> 00:35:02,670 11 of you. 487 00:35:02,670 --> 00:35:10,120 How many of you will think that, because of this hat, 488 00:35:10,120 --> 00:35:16,050 this pushing this air away, it's a lot of work to be done. 489 00:35:16,050 --> 00:35:21,810 Therefore, this is going to slow down the oscillation. 490 00:35:21,810 --> 00:35:24,930 How many of you think that is going to happen? 491 00:35:24,930 --> 00:35:26,040 1, 2, 3-- 492 00:35:31,540 --> 00:35:32,490 OK, 17. 493 00:35:45,790 --> 00:35:48,190 It may happen to you that you think 494 00:35:48,190 --> 00:35:54,100 this idea of wearing a hat is really fashionable. 495 00:35:54,100 --> 00:35:55,950 Therefore, it got really exciting 496 00:35:55,950 --> 00:35:58,950 and it oscillates faster. 497 00:35:58,950 --> 00:36:00,780 Can that happen? 498 00:36:00,780 --> 00:36:04,700 How many of you actually think that is going to happen? 499 00:36:04,700 --> 00:36:09,150 OK, one-- you think so? 500 00:36:09,150 --> 00:36:10,990 Two. 501 00:36:10,990 --> 00:36:13,330 Very good, we have 2. 502 00:36:13,330 --> 00:36:14,080 What do you think? 503 00:36:14,080 --> 00:36:15,830 Where are the rest? 504 00:36:15,830 --> 00:36:21,700 Only 30 of you actually think that is going to happen. 505 00:36:24,320 --> 00:36:27,600 OK, all the rest think of the class 506 00:36:27,600 --> 00:36:29,960 think that this one is going to-- pew! 507 00:36:29,960 --> 00:36:32,600 Disappear to the moon, OK? 508 00:36:32,600 --> 00:36:34,330 So that is actually the opinion. 509 00:36:34,330 --> 00:36:36,280 And we have completed the poll. 510 00:36:36,280 --> 00:36:40,130 And what we are going to do is that we 511 00:36:40,130 --> 00:36:42,050 are going to solve this system and see 512 00:36:42,050 --> 00:36:44,160 what is going to happen. 513 00:36:44,160 --> 00:36:48,410 And we will do that experiment in front of you, OK? 514 00:36:48,410 --> 00:36:51,000 All right, so that's very nice. 515 00:36:51,000 --> 00:36:53,520 So now, we have this question of motion. 516 00:36:53,520 --> 00:36:57,040 And now, I will pretend that I'm from the math 517 00:36:57,040 --> 00:37:02,600 department for a bit and help guide you through the solution. 518 00:37:02,600 --> 00:37:06,470 So now, I can use this trick. 519 00:37:06,470 --> 00:37:08,570 I can actually say theta is actually 520 00:37:08,570 --> 00:37:14,730 the real part of the z, which is a complex function. 521 00:37:14,730 --> 00:37:18,620 And as we learned before, z of t, 522 00:37:18,620 --> 00:37:23,630 and I assume that to be exponential I alpha t. 523 00:37:23,630 --> 00:37:26,130 So alpha is actually some kind of constant, 524 00:37:26,130 --> 00:37:29,810 which I don't really know what is the constant yet. 525 00:37:29,810 --> 00:37:36,590 OK, I can now actually write the equation of motion 526 00:37:36,590 --> 00:37:38,390 in the form of z. 527 00:37:38,390 --> 00:37:47,127 Then basically, what I get is z double-dot t plus gamma z dot 528 00:37:47,127 --> 00:37:54,290 t plus omega 0 squared z of t. 529 00:37:54,290 --> 00:37:58,250 And this is equal to 0, OK? 530 00:37:58,250 --> 00:38:02,060 So remember, exponential function cannot be killed 531 00:38:02,060 --> 00:38:04,220 by differentiation, right? 532 00:38:04,220 --> 00:38:06,180 Therefore, it's really convenient. 533 00:38:06,180 --> 00:38:07,410 You can see from here. 534 00:38:07,410 --> 00:38:10,640 Now, I can plug in this expression-- 535 00:38:10,640 --> 00:38:16,440 which I did this and guessed to this equation of motion. 536 00:38:16,440 --> 00:38:20,910 Then what I am going to get is minus alpha squared. 537 00:38:20,910 --> 00:38:24,130 Because you take I alpha I alpha out 538 00:38:24,130 --> 00:38:28,640 of this exponential function, right? 539 00:38:28,640 --> 00:38:32,630 Because you do double differentiation. 540 00:38:32,630 --> 00:38:40,160 So you get minus alpha squared plus i gamma alpha-- 541 00:38:40,160 --> 00:38:43,680 because this is only differentiated one time-- 542 00:38:43,680 --> 00:38:47,120 plus omega 0 squared. 543 00:38:47,120 --> 00:38:51,570 And all those things are actually 544 00:38:51,570 --> 00:38:56,480 multiplying this exponential function, exponential i alpha t 545 00:38:56,480 --> 00:38:59,540 equal to 0, OK? 546 00:38:59,540 --> 00:39:01,455 So we will write this expression. 547 00:39:01,455 --> 00:39:03,350 That is very nice. 548 00:39:03,350 --> 00:39:07,370 And we also know that, this expression is 549 00:39:07,370 --> 00:39:09,680 going to be valid all the time. 550 00:39:09,680 --> 00:39:14,270 No matter what t you put in, it should be valid, right? 551 00:39:14,270 --> 00:39:16,130 Because this is the equation of motion. 552 00:39:16,130 --> 00:39:19,790 And we hope that this solution will survive this test. 553 00:39:22,380 --> 00:39:28,110 So I can easily conclude that this one is actually not 554 00:39:28,110 --> 00:39:29,960 equal to 0. 555 00:39:29,960 --> 00:39:32,640 It can be some value, not 0. 556 00:39:32,640 --> 00:39:35,400 So what is actually equal to 0? 557 00:39:35,400 --> 00:39:41,790 This first term is actually equal to 0, OK? 558 00:39:41,790 --> 00:39:46,560 Therefore, I can now solve this equation; 559 00:39:46,560 --> 00:39:50,370 minus alpha squared plus i gamma alpha plus omega 0 560 00:39:50,370 --> 00:39:52,530 squared equal to 0. 561 00:39:52,530 --> 00:39:54,780 I can solve it, OK? 562 00:39:54,780 --> 00:39:59,650 If I do that, then I would get alpha 563 00:39:59,650 --> 00:40:08,310 is equal to i gamma plus/minus square root of 4 omega 0 564 00:40:08,310 --> 00:40:13,840 squared minus gamma squared divided by 2. 565 00:40:13,840 --> 00:40:17,940 This is actually the second order polynomial. 566 00:40:17,940 --> 00:40:21,110 And that is actually equal to 0. 567 00:40:21,110 --> 00:40:23,800 Therefore, you can actually solve it easily. 568 00:40:23,800 --> 00:40:26,490 And this is actually the solution. 569 00:40:26,490 --> 00:40:30,810 And I can write it down in a slightly different form. 570 00:40:30,810 --> 00:40:36,780 i gamma over 2 plus/minus square root of omega 0 571 00:40:36,780 --> 00:40:43,180 squared minus gamma squared over 4, OK? 572 00:40:43,180 --> 00:40:44,160 Any questions so far? 573 00:40:44,160 --> 00:40:45,060 Am I going too fast? 574 00:40:48,450 --> 00:40:49,290 Everything's OK? 575 00:40:53,484 --> 00:41:00,005 OK, So you can see that alpha is equal to this expression. 576 00:41:00,005 --> 00:41:03,880 And I would like to consider a situation 577 00:41:03,880 --> 00:41:13,020 where omega 0 is much, much larger than gamma, OK? 578 00:41:13,020 --> 00:41:15,310 Just a reminder of what is gamma, OK? 579 00:41:15,310 --> 00:41:18,010 Maybe you've got already a bit confused. 580 00:41:18,010 --> 00:41:19,480 What is gamma? 581 00:41:19,480 --> 00:41:24,630 Gamma is related to the strength of the direct force, right? 582 00:41:24,630 --> 00:41:27,760 It is actually 3b over ml squared, OK? 583 00:41:27,760 --> 00:41:37,090 b is actually determining the size of the direct force, OK? 584 00:41:37,090 --> 00:41:40,816 So I would like to consider a situation. 585 00:41:40,816 --> 00:41:48,790 The first situation is if omega 0 squared is larger 586 00:41:48,790 --> 00:41:53,320 than gamma squared over 4. 587 00:41:53,320 --> 00:42:02,540 So in that case, the drag force is small. 588 00:42:02,540 --> 00:42:04,150 It is not huge. 589 00:42:04,150 --> 00:42:06,540 It's small, OK? 590 00:42:06,540 --> 00:42:12,140 If that is the case, this is actually real, right? 591 00:42:12,140 --> 00:42:14,420 Because omega 0 squared is larger 592 00:42:14,420 --> 00:42:16,370 than gamma squared over 4. 593 00:42:16,370 --> 00:42:18,950 Therefore, this is real, OK? 594 00:42:18,950 --> 00:42:23,870 So now, I can actually define omega squared, 595 00:42:23,870 --> 00:42:31,210 define that as omega 0 squared minus gamma squared over 4, OK? 596 00:42:33,820 --> 00:42:49,090 And this will become i gamma over 2 plus/minus omega, OK? 597 00:42:49,090 --> 00:42:53,110 So that means I would have two solutions coming 598 00:42:53,110 --> 00:42:57,040 from this exercise. 599 00:42:57,040 --> 00:43:04,345 Z plus of t is equal to exponential minus gamma over 2 600 00:43:04,345 --> 00:43:11,010 t exponential i omega t, OK? 601 00:43:11,010 --> 00:43:19,740 And the second solution, if I take one of the plus sign 602 00:43:19,740 --> 00:43:22,080 and one of the minus sign solutions, 603 00:43:22,080 --> 00:43:24,570 then the second solution would be exponential 604 00:43:24,570 --> 00:43:34,150 minus i gamma over 2 t exponential minus i omega t, 605 00:43:34,150 --> 00:43:35,240 OK? 606 00:43:35,240 --> 00:43:36,124 Any questions so far? 607 00:43:40,000 --> 00:43:46,160 OK, so we would like to go back to theta, right? 608 00:43:46,160 --> 00:43:47,670 So what would be the theta? 609 00:43:50,210 --> 00:43:55,040 So that means I would have a theta 1 of t, which is actually 610 00:43:55,040 --> 00:43:56,630 taking the real part. 611 00:43:56,630 --> 00:44:02,630 So it's theta plus maybe, taking the real part of z plus. 612 00:44:02,630 --> 00:44:06,890 And that will give you exponential minus gamma 613 00:44:06,890 --> 00:44:16,610 over 2 t cosine omega t, OK? 614 00:44:16,610 --> 00:44:20,640 I'm just plugging in the solution to this equation, OK? 615 00:44:25,240 --> 00:44:29,790 Theta minus t would be equal to exponential, 616 00:44:29,790 --> 00:44:38,050 and this gamma over 2 t sine omega t, OK? 617 00:44:38,050 --> 00:44:43,480 Finally, the full solution of theta of t 618 00:44:43,480 --> 00:44:49,200 would be a linear combination of these two solution, right? 619 00:44:49,200 --> 00:44:54,180 Therefore, you will get theta of t equal to exponential 620 00:44:54,180 --> 00:45:01,080 minus gamma over 2 t a (is some kind of constant) 621 00:45:01,080 --> 00:45:08,280 times cosine omega t plus b sine omega t. 622 00:45:12,401 --> 00:45:16,450 And of course, from the last time, as you will know, 623 00:45:16,450 --> 00:45:28,030 this can also be written as A cosine omega t plus phi, OK? 624 00:45:28,030 --> 00:45:32,440 Any questions so far? 625 00:45:32,440 --> 00:45:36,410 OK, very good. 626 00:45:36,410 --> 00:45:40,500 So we have actually already solved this equation. 627 00:45:40,500 --> 00:45:44,400 And of course, we can actually plug this back 628 00:45:44,400 --> 00:45:49,710 into this equation of motion. 629 00:45:49,710 --> 00:45:51,716 And you will see that it really works. 630 00:45:51,716 --> 00:45:53,090 And I'm not going to do that now. 631 00:45:53,090 --> 00:45:55,610 But you can actually go back home and check. 632 00:45:55,610 --> 00:45:59,250 And if you believe me, it works. 633 00:45:59,250 --> 00:46:04,650 And also at the same time, it got two undetermined constants, 634 00:46:04,650 --> 00:46:08,880 since this is a second order differential equation. 635 00:46:08,880 --> 00:46:11,835 Therefore, huh, this thing actually works. 636 00:46:11,835 --> 00:46:14,220 It has two arbitrary constants. 637 00:46:14,220 --> 00:46:16,530 Therefore, that is actually the one and only 638 00:46:16,530 --> 00:46:19,680 one solution in the universe which 639 00:46:19,680 --> 00:46:25,220 satisfies the equation of motion or satisfies that differential 640 00:46:25,220 --> 00:46:27,020 question, OK? 641 00:46:27,020 --> 00:46:33,090 So this thing actually has dramatic consequences. 642 00:46:33,090 --> 00:46:34,500 The first thing which we learn is 643 00:46:34,500 --> 00:46:39,300 that, as a function of time, what is going to happen? 644 00:46:39,300 --> 00:46:46,390 The amplitude is now becoming exponential minus gamma 645 00:46:46,390 --> 00:46:52,140 over 2 t times A. This is actually the amplitude. 646 00:46:52,140 --> 00:46:57,840 The amplitude is decreasing exponentially. 647 00:46:57,840 --> 00:47:00,570 So that is actually the first prediction 648 00:47:00,570 --> 00:47:03,150 coming from this exercise, OK? 649 00:47:03,150 --> 00:47:07,480 The second prediction is that this thing is still 650 00:47:07,480 --> 00:47:08,260 oscillating. 651 00:47:08,260 --> 00:47:12,400 Because you've got the cosine omega t plus phi there, 652 00:47:12,400 --> 00:47:13,760 you see? 653 00:47:13,760 --> 00:47:16,720 So the damping motion is going to be 654 00:47:16,720 --> 00:47:21,550 like going up and down, up and down, and get tired. 655 00:47:21,550 --> 00:47:25,030 Therefore, the amplitude becomes smaller, and smaller, 656 00:47:25,030 --> 00:47:26,320 and smaller. 657 00:47:26,320 --> 00:47:29,960 But it's never 0, right? 658 00:47:29,960 --> 00:47:31,190 It's never 0, OK? 659 00:47:31,190 --> 00:47:34,160 It's actually going to be oscillating down, down, down, 660 00:47:34,160 --> 00:47:35,550 so small I couldn't see it. 661 00:47:35,550 --> 00:47:39,590 But it's still oscillating, OK? 662 00:47:39,590 --> 00:47:44,610 Finally, we actually have also the answer 663 00:47:44,610 --> 00:47:49,050 to the original question we posed, OK? 664 00:47:49,050 --> 00:47:54,200 So now, you can see that the oscillation frequency is omega, 665 00:47:54,200 --> 00:47:55,080 OK? 666 00:47:55,080 --> 00:48:01,060 Originally, before we introduced the drag force, 667 00:48:01,060 --> 00:48:05,570 omega 0, which is the oscillation frequency, 668 00:48:05,570 --> 00:48:06,985 is actually an angular frequency. 669 00:48:06,985 --> 00:48:10,930 It's actually the square root of 3g over 2l. 670 00:48:10,930 --> 00:48:17,380 And you can see that the new omega, the oscillation 671 00:48:17,380 --> 00:48:24,650 frequency with drag force, is the square root of this, 672 00:48:24,650 --> 00:48:29,000 omega 0 squared minus gamma squared over 4. 673 00:48:29,000 --> 00:48:32,230 So what this actually tells us is 674 00:48:32,230 --> 00:48:38,400 that this is going to be smaller, 675 00:48:38,400 --> 00:48:40,740 because of the drag force, OK? 676 00:48:40,740 --> 00:48:43,980 So that's a prediction. 677 00:48:43,980 --> 00:48:47,080 Let's do the experiment and see what is going to happen. 678 00:48:47,080 --> 00:48:49,740 So let's take a look at these two systems. 679 00:48:49,740 --> 00:48:54,870 They have the identical mass, which our technical instructor 680 00:48:54,870 --> 00:48:57,330 actually carefully prepared. 681 00:48:57,330 --> 00:49:00,600 They have the same mass, even though one actually 682 00:49:00,600 --> 00:49:01,980 looks a bit funny. 683 00:49:01,980 --> 00:49:04,740 The other one looks normal, OK? 684 00:49:04,740 --> 00:49:10,590 Now, what I'm going to do is to really try and see 685 00:49:10,590 --> 00:49:13,110 which one is actually oscillating faster, OK? 686 00:49:13,110 --> 00:49:14,802 So let's see. 687 00:49:14,802 --> 00:49:17,380 I release them at the same time. 688 00:49:17,380 --> 00:49:19,470 And you can see that originally they 689 00:49:19,470 --> 00:49:22,120 seem to be oscillating at the same frequency. 690 00:49:22,120 --> 00:49:27,570 But you can see very clearly that the one with the hat 691 00:49:27,570 --> 00:49:31,230 is actually oscillating slower, OK? 692 00:49:31,230 --> 00:49:33,810 So you can see that, OK, 17 of you 693 00:49:33,810 --> 00:49:37,440 actually got the correct answer. 694 00:49:37,440 --> 00:49:40,380 And the most important thing is that you 695 00:49:40,380 --> 00:49:42,780 can see that this simple mass actually 696 00:49:42,780 --> 00:49:46,460 describes and predicts what is going to happen 697 00:49:46,460 --> 00:49:48,930 in my little experiment. 698 00:49:48,930 --> 00:49:51,340 So that is actually really cool. 699 00:49:51,340 --> 00:49:54,890 And I think it's time to take a little break. 700 00:49:54,890 --> 00:49:59,250 And then, we will come back and look at other solutions. 701 00:49:59,250 --> 00:50:01,380 And of course, you are welcome to come 702 00:50:01,380 --> 00:50:04,836 to the front to play with those demonstrations. 703 00:50:11,180 --> 00:50:13,510 So there are two small issues which 704 00:50:13,510 --> 00:50:16,970 were raised during the break. 705 00:50:16,970 --> 00:50:22,730 So the first one is that, if you actually calculate the torque 706 00:50:22,730 --> 00:50:25,520 from this equation-- 707 00:50:25,520 --> 00:50:27,440 so I made a mistake. 708 00:50:27,440 --> 00:50:31,490 The R vector should be actually pointing from the nail 709 00:50:31,490 --> 00:50:33,140 to the center of mass, OK? 710 00:50:33,140 --> 00:50:35,660 So I think that's a trivial mistake. 711 00:50:35,660 --> 00:50:38,630 So if you do this, then you can actually 712 00:50:38,630 --> 00:50:43,010 calculate the tau equal to R cross F. 713 00:50:43,010 --> 00:50:46,200 Then, you actually get this minus sign, OK? 714 00:50:46,200 --> 00:50:51,580 So if I make a mistake in pointing towards the nail, 715 00:50:51,580 --> 00:50:54,650 then you will get no minus sign, then 716 00:50:54,650 --> 00:50:56,750 that didn't really work, OK? 717 00:50:56,750 --> 00:51:00,500 So very good, I'm very happy that you are actually 718 00:51:00,500 --> 00:51:04,490 paying very much attention to capture those. 719 00:51:04,490 --> 00:51:07,040 The second issue is that-- 720 00:51:07,040 --> 00:51:09,590 so now, I'm saying that, OK, now I 721 00:51:09,590 --> 00:51:13,180 have the solution in the complex format. 722 00:51:13,180 --> 00:51:17,450 So I have a Z plus and I have a Z minus, OK? 723 00:51:17,450 --> 00:51:21,170 And then I would like to go to the real world, right? 724 00:51:21,170 --> 00:51:23,720 Because the imaginary thing is actually 725 00:51:23,720 --> 00:51:27,740 hidden in some kind of motion in the actual dimension, 726 00:51:27,740 --> 00:51:31,680 et cetera, I would like to go back to reality, OK? 727 00:51:31,680 --> 00:51:36,790 And what I said in the class is that I take the real part 728 00:51:36,790 --> 00:51:38,310 of one of the solutions. 729 00:51:38,310 --> 00:51:41,230 And I can also take a real part of i 730 00:51:41,230 --> 00:51:43,440 times one of the solutions. 731 00:51:43,440 --> 00:51:46,400 But of course, you can also do this 732 00:51:46,400 --> 00:51:50,440 by doing a linear combination of the solutions, right? 733 00:51:50,440 --> 00:51:53,100 As we actually discussed last time, 734 00:51:53,100 --> 00:51:55,100 the linear combination of the solutions 735 00:51:55,100 --> 00:51:59,990 is also a solution to the same equation of motion, 736 00:51:59,990 --> 00:52:02,010 since this one is actually linear. 737 00:52:02,010 --> 00:52:06,960 Therefore, what I actually do is actually 738 00:52:06,960 --> 00:52:11,850 to sum the two solutions, Z plus and Z minus and divide it by 2. 739 00:52:11,850 --> 00:52:20,320 Or actually, I can actually do a minus i/2 times Z plus minus Z 740 00:52:20,320 --> 00:52:21,260 minus, OK? 741 00:52:21,260 --> 00:52:24,740 And then I can also extract this sign term here, OK? 742 00:52:24,740 --> 00:52:28,610 So that should be the correct explanation of the two 743 00:52:28,610 --> 00:52:33,620 solutions in the real axis, OK? 744 00:52:33,620 --> 00:52:36,300 Any questions so far? 745 00:52:36,300 --> 00:52:40,200 Thank you very much for capturing those. 746 00:52:40,200 --> 00:52:41,910 Ok, so now, you can see that we have 747 00:52:41,910 --> 00:52:46,530 been discussing the equation of motion of this functional form. 748 00:52:46,530 --> 00:52:49,780 And the one thing which is really, really interesting 749 00:52:49,780 --> 00:52:57,170 is that the solution, when we take a small drag force limit, 750 00:52:57,170 --> 00:53:01,560 actually we arrive at a beautiful solution that 751 00:53:01,560 --> 00:53:06,560 looks like this, A exponential minus gamma over 2 752 00:53:06,560 --> 00:53:09,370 t cosine omega t plus phi. 753 00:53:09,370 --> 00:53:13,080 That actually predicts the oscillation, OK? 754 00:53:13,080 --> 00:53:17,310 At the same time, it also says that the amplitude 755 00:53:17,310 --> 00:53:23,060 is actually going to drop exponentially, but never 0, OK? 756 00:53:23,060 --> 00:53:28,650 Finally, we also know that this solution actually tells us 757 00:53:28,650 --> 00:53:33,840 that, if we have a spring mass system oscillating up and down, 758 00:53:33,840 --> 00:53:40,770 if we have a rod like what we actually solve in a class, 759 00:53:40,770 --> 00:53:46,010 this object is going to pass through 0, the equilibrium 760 00:53:46,010 --> 00:53:48,450 position, an infinite number of times, right? 761 00:53:48,450 --> 00:53:51,630 Because the cosine is always there. 762 00:53:51,630 --> 00:53:53,587 Therefore, although the amplitude 763 00:53:53,587 --> 00:53:55,170 will become very small, but it's still 764 00:53:55,170 --> 00:53:59,910 oscillating forever until the end of the universe, OK? 765 00:53:59,910 --> 00:54:03,560 All right, so that's actually what we have learned. 766 00:54:03,560 --> 00:54:06,660 And also, one thing which we learned last time 767 00:54:06,660 --> 00:54:12,010 is that simple harmonic motion, like this one, which 768 00:54:12,010 --> 00:54:15,310 we were just showing here, or this one, 769 00:54:15,310 --> 00:54:18,330 which is actually a mass oscillating back and forth 770 00:54:18,330 --> 00:54:24,830 on the track, is actually just a projection of a circular motion 771 00:54:24,830 --> 00:54:26,760 in a complex plane, OK? 772 00:54:26,760 --> 00:54:30,770 And what we are really seeing here in front of you 773 00:54:30,770 --> 00:54:34,950 is actually a projection to the real axis, OK? 774 00:54:34,950 --> 00:54:37,850 So that's actually a really remarkable result 775 00:54:37,850 --> 00:54:40,530 and a beautiful picture. 776 00:54:40,530 --> 00:54:44,670 And of course, we can actually also plug in the solution 777 00:54:44,670 --> 00:54:47,530 with damping. 778 00:54:47,530 --> 00:54:50,900 So what is actually the picture in this language, 779 00:54:50,900 --> 00:54:54,130 in this exact same language? 780 00:54:54,130 --> 00:54:58,270 If we actually follow the locus, then basically 781 00:54:58,270 --> 00:55:00,980 what you are going to see is that this thing actually 782 00:55:00,980 --> 00:55:02,530 spirals. 783 00:55:02,530 --> 00:55:06,400 And the amplitude is actually getting smaller and smaller 784 00:55:06,400 --> 00:55:10,820 and is sucked into this black hole in the 0, 0, OK? 785 00:55:10,820 --> 00:55:14,440 So you can see that now the picture 786 00:55:14,440 --> 00:55:18,170 looks as if there is something really rotating 787 00:55:18,170 --> 00:55:19,660 in the complex plane. 788 00:55:19,660 --> 00:55:21,790 And it's actually approaching 0. 789 00:55:21,790 --> 00:55:23,770 Because the amplitude is actually 790 00:55:23,770 --> 00:55:25,670 getting smaller and smaller. 791 00:55:25,670 --> 00:55:29,960 But this whole thing is still rotating, OK? 792 00:55:29,960 --> 00:55:33,370 OK, that's really nice. 793 00:55:33,370 --> 00:55:39,360 All right, so now, this is actually a special case. 794 00:55:39,360 --> 00:55:45,000 When we actually assume that gamma is actually pretty small. 795 00:55:45,000 --> 00:55:49,800 So you have very small drag force, OK? 796 00:55:49,800 --> 00:55:54,330 So let's actually check what would happen. 797 00:55:54,330 --> 00:55:58,200 If I now start to increase the drag force, 798 00:55:58,200 --> 00:56:02,700 make this hat larger, larger, and larger, 799 00:56:02,700 --> 00:56:08,660 introducing more and more drag, what is going to happen? 800 00:56:08,660 --> 00:56:13,790 OK, so now, I consider the second situation, 801 00:56:13,790 --> 00:56:22,190 omega 0 squared equal to gamma squared over 4. 802 00:56:22,190 --> 00:56:25,520 OK, so when the gamma is very small, 803 00:56:25,520 --> 00:56:29,150 what we see is that this is actually underdamped, right? 804 00:56:29,150 --> 00:56:32,150 So the damping is really small. 805 00:56:32,150 --> 00:56:38,050 But if I increase the gamma to a critical value, 806 00:56:38,050 --> 00:56:41,840 now omega 0 squared happens to be equal to gamma 807 00:56:41,840 --> 00:56:43,960 squared over 4, OK? 808 00:56:43,960 --> 00:56:56,660 I call this a critically damped oscillator, OK? 809 00:56:56,660 --> 00:56:58,770 So what does that mean? 810 00:56:58,770 --> 00:57:07,070 That means omega is equal to 0, you see? 811 00:57:07,070 --> 00:57:10,190 This is our definition of omega, right? 812 00:57:10,190 --> 00:57:15,080 If omega 0 squared is equal to gamma squared of over 4, 813 00:57:15,080 --> 00:57:19,670 then omega is equal to 0. 814 00:57:19,670 --> 00:57:22,970 That is actually the critical moment 815 00:57:22,970 --> 00:57:28,720 the system stops oscillating, OK? 816 00:57:28,720 --> 00:57:33,050 So it is not oscillating anymore. 817 00:57:33,050 --> 00:57:38,220 So now, I can actually start from the solution 818 00:57:38,220 --> 00:57:41,150 I obtained from 1, OK? 819 00:57:41,150 --> 00:57:43,910 Then, I can actually now make use 820 00:57:43,910 --> 00:57:48,950 of these two solutions, the theta plus and the theta minus. 821 00:57:53,730 --> 00:58:00,250 Theta plus t would be equal to exponential minus gamma 822 00:58:00,250 --> 00:58:05,940 over 2 t cosine omega t. 823 00:58:05,940 --> 00:58:09,240 When omega goes to 0, what is going 824 00:58:09,240 --> 00:58:13,640 to happen is that this is actually becoming, which value? 825 00:58:13,640 --> 00:58:15,600 Anybody know? 826 00:58:15,600 --> 00:58:19,010 If omega is 0, what is going to happen? 827 00:58:19,010 --> 00:58:20,180 1, yeah. 828 00:58:20,180 --> 00:58:21,650 OK, 1, right? 829 00:58:21,650 --> 00:58:27,155 So that will give me exponential minus gamma over 2 t. 830 00:58:30,010 --> 00:58:32,730 Theta minus t-- 831 00:58:32,730 --> 00:58:35,880 OK, I can do the same trick and see what will happen. 832 00:58:35,880 --> 00:58:39,550 So I take theta minus t, which is actually 833 00:58:39,550 --> 00:58:42,110 obtained from the exercise number one 834 00:58:42,110 --> 00:58:44,450 when we discussed the underdamped system. 835 00:58:44,450 --> 00:58:47,250 Then, you actually get exponential minus gamma 836 00:58:47,250 --> 00:58:52,010 over 2 t sine omega t. 837 00:58:52,010 --> 00:59:00,140 When omega goes to 0, actually then I get 0 this time, OK? 838 00:59:00,140 --> 00:59:01,920 So that doesn't really work, right? 839 00:59:01,920 --> 00:59:05,080 Because if I have a solution which is 0, then 840 00:59:05,080 --> 00:59:06,800 it's not describing anything, right? 841 00:59:06,800 --> 00:59:09,410 I can always add 0 to the solution. 842 00:59:09,410 --> 00:59:10,700 But that doesn't help you. 843 00:59:10,700 --> 00:59:17,470 OK, so instead of taking the limit of this function, 844 00:59:17,470 --> 00:59:21,170 actually we choose to actually do 845 00:59:21,170 --> 00:59:25,960 theta minus t divided by omega. 846 00:59:25,960 --> 00:59:31,310 And then, we actually make this omega approaching 0. 847 00:59:31,310 --> 00:59:34,310 Then basically, I get exponential minus gamma 848 00:59:34,310 --> 00:59:42,280 over 2 t sine omega t divided by omega, OK? 849 00:59:42,280 --> 00:59:48,210 If I have this omega approaching to 0, 850 00:59:48,210 --> 00:59:53,130 then this is actually roughly just exponential minus gamma 851 00:59:53,130 --> 00:59:57,930 over 2 t omega t over omega. 852 00:59:57,930 --> 01:00:05,010 And this is actually giving you t times exponential minus 853 01:00:05,010 --> 01:00:06,220 gamma over 2 t. 854 01:00:09,580 --> 01:00:10,904 Any questions so far? 855 01:00:13,870 --> 01:00:15,556 Yes. 856 01:00:15,556 --> 01:00:18,448 AUDIENCE: Completely unrelated, but is 857 01:00:18,448 --> 01:00:22,600 that a negative sign in front of the theta minus negative 1/2? 858 01:00:22,600 --> 01:00:23,750 YEN-JIE LEE: This one? 859 01:00:23,750 --> 01:00:24,690 AUDIENCE: Yeah. 860 01:00:24,690 --> 01:00:25,440 YEN-JIE LEE: Yeah. 861 01:00:25,440 --> 01:00:28,465 So actually, OK, yeah. 862 01:00:28,465 --> 01:00:30,840 AUDIENCE: In front of the 1/2 is that a negative sign? 863 01:00:30,840 --> 01:00:32,590 YEN-JIE LEE: Yes, this is a negative sign. 864 01:00:35,090 --> 01:00:39,990 OK, any other questions? 865 01:00:39,990 --> 01:00:45,840 OK, so you can see that now I arrive at two solutions. 866 01:00:45,840 --> 01:00:50,320 One is actually proportional to exponential minus gamma 867 01:00:50,320 --> 01:00:51,980 over 2 t. 868 01:00:51,980 --> 01:00:54,550 The other one is actually proportional to t 869 01:00:54,550 --> 01:00:59,150 times exponential minus gamma over 2 t, OK? 870 01:00:59,150 --> 01:01:03,680 So you can see that the cosine or sine term disappeared, 871 01:01:03,680 --> 01:01:04,700 right? 872 01:01:04,700 --> 01:01:10,000 So that means you are never oscillating, OK? 873 01:01:10,000 --> 01:01:13,210 So this is actually what we see in this slide, 874 01:01:13,210 --> 01:01:15,730 this so-called critically damped, OK? 875 01:01:15,730 --> 01:01:18,730 When actually, omega 0 squared is 876 01:01:18,730 --> 01:01:23,830 equal to gamma squared over 4. 877 01:01:23,830 --> 01:01:26,620 And you can see that what is going to happen 878 01:01:26,620 --> 01:01:34,540 is that this mass or this rod is going to pass 0 879 01:01:34,540 --> 01:01:39,070 only one time at most, OK? 880 01:01:39,070 --> 01:01:41,980 And it could actually never passed 0, 881 01:01:41,980 --> 01:01:45,210 if you actually set up the initial condition correctly, 882 01:01:45,210 --> 01:01:45,710 OK? 883 01:01:45,710 --> 01:01:50,380 So one thing which I can do is I really shoot this mass really, 884 01:01:50,380 --> 01:01:53,230 really, very forcefully, so that I have 885 01:01:53,230 --> 01:01:56,040 a very large initial velocity. 886 01:01:56,040 --> 01:01:58,690 And what it actually is going to do, 887 01:01:58,690 --> 01:02:00,630 like the right-hand side diagram, 888 01:02:00,630 --> 01:02:03,550 is that, oh, you overshoot the 0 a bit. 889 01:02:03,550 --> 01:02:09,040 Then, it goes back almost exponentially, OK? 890 01:02:09,040 --> 01:02:13,830 So at most, you can only pass through 0 one 891 01:02:13,830 --> 01:02:18,830 time, if you do this kind of initial condition, OK? 892 01:02:18,830 --> 01:02:22,520 So that is actually pretty interesting. 893 01:02:22,520 --> 01:02:29,370 And there are practical applications of this solution, 894 01:02:29,370 --> 01:02:30,030 actually. 895 01:02:30,030 --> 01:02:32,870 So for example, we have the door closed. 896 01:02:32,870 --> 01:02:34,490 So it's also here, right? 897 01:02:34,490 --> 01:02:38,030 The door closed, you would like to have the door go back 898 01:02:38,030 --> 01:02:42,290 to the original closed mold, the position 899 01:02:42,290 --> 01:02:45,740 of equilibrium position actually really fast, OK? 900 01:02:45,740 --> 01:02:50,870 So what you can do is really design this door close 901 01:02:50,870 --> 01:02:54,700 so that it actually matches with the critical dampness 902 01:02:54,700 --> 01:02:58,460 situation, of your condition, so that actually you would go back 903 01:02:58,460 --> 01:03:02,478 to 0 really quick, OK? 904 01:03:02,478 --> 01:03:06,380 Any questions? 905 01:03:06,380 --> 01:03:11,760 OK, so now, what we could do is that, instead 906 01:03:11,760 --> 01:03:21,810 of having a very small drag force, or we'll a slightly 907 01:03:21,810 --> 01:03:23,610 larger drag force, so that actually 908 01:03:23,610 --> 01:03:28,320 reach the critically damped situation, what we could do 909 01:03:28,320 --> 01:03:33,820 is that we put the whole system into water, right? 910 01:03:33,820 --> 01:03:37,450 Then, the drag force will be very big, OK? 911 01:03:37,450 --> 01:03:42,160 And we would like to see what is going to happen, OK? 912 01:03:42,160 --> 01:03:46,690 So in this case is the third situation. 913 01:03:46,690 --> 01:03:50,680 The third situation is that omega 0 squared is actually 914 01:03:50,680 --> 01:03:55,960 smaller than gamma squared over 4. 915 01:03:55,960 --> 01:04:03,220 So you have huge drag force, OK? 916 01:04:06,050 --> 01:04:09,770 So that would give you a situation 917 01:04:09,770 --> 01:04:17,037 which is called overdamped oscillator. 918 01:04:21,870 --> 01:04:27,140 Now, I have, again, alpha is equal to i gamma 919 01:04:27,140 --> 01:04:32,600 over 2 plus/minus square root of omega 0 920 01:04:32,600 --> 01:04:37,180 squared minus gamma squared over 4, right? 921 01:04:37,180 --> 01:04:39,820 I'm just copying from here, OK? 922 01:04:39,820 --> 01:04:42,960 And that will be equal to i-- 923 01:04:42,960 --> 01:04:45,850 I can take out the i, OK?-- 924 01:04:45,850 --> 01:04:50,800 gamma over 2 plus/minus square root 925 01:04:50,800 --> 01:04:56,777 of gamma squared over 4 minus omega 0 squared. 926 01:05:02,140 --> 01:05:10,250 Now, I can actually define gamma plus/minus 927 01:05:10,250 --> 01:05:18,270 equal to gamma over 2 plus/minus square root of gamma 928 01:05:18,270 --> 01:05:25,580 squared over 4 minus omega 0 squared, OK? 929 01:05:25,580 --> 01:05:29,240 Then basically, the solution-- 930 01:05:29,240 --> 01:05:32,340 actually now, I already have the solution. 931 01:05:32,340 --> 01:05:36,720 So basically, the two solutions would be looking like this. 932 01:05:36,720 --> 01:05:41,670 Theta of t would be equal to A plus some kind 933 01:05:41,670 --> 01:05:48,990 of constant exponential minus gamma plus t plus A 934 01:05:48,990 --> 01:05:56,070 minus exponential minus gamma minus t, OK? 935 01:05:56,070 --> 01:05:59,070 Because this is actually becoming already-- 936 01:05:59,070 --> 01:06:04,730 OK, so alpha is actually i times gamma plus/minus. 937 01:06:04,730 --> 01:06:08,360 Therefore, if you put it back into this, 938 01:06:08,360 --> 01:06:09,990 then basically what you are getting 939 01:06:09,990 --> 01:06:18,450 is exponential minus gamma plus t or exponential minus 940 01:06:18,450 --> 01:06:20,700 gamma minus t, OK? 941 01:06:20,700 --> 01:06:22,920 So that's already a real function. 942 01:06:22,920 --> 01:06:26,610 And the linear combination of these two solutions 943 01:06:26,610 --> 01:06:33,030 is our final, full solution to the equation of motion. 944 01:06:33,030 --> 01:06:36,670 OK, again, what we are going to see 945 01:06:36,670 --> 01:06:41,700 is that actually the drag force is huge. 946 01:06:41,700 --> 01:06:45,090 I just throw the whole system into water. 947 01:06:45,090 --> 01:06:49,840 And the water is really trying to stop the oscillation, really 948 01:06:49,840 --> 01:06:50,760 very much. 949 01:06:50,760 --> 01:06:54,030 Therefore, you can see that, huh, again, 950 01:06:54,030 --> 01:06:57,510 I don't have any oscillation, OK? 951 01:06:57,510 --> 01:07:00,180 If I am very, very strong, I really 952 01:07:00,180 --> 01:07:06,450 start the initial velocity or initial angular 953 01:07:06,450 --> 01:07:12,300 velocity really high, I actually give a huge amount of energy 954 01:07:12,300 --> 01:07:15,960 into the system, then, at most again, I 955 01:07:15,960 --> 01:07:20,310 can actually have the system to pass through the equilibrium 956 01:07:20,310 --> 01:07:22,530 position only one time. 957 01:07:22,530 --> 01:07:28,740 Then, this whole system will slowly recover, 958 01:07:28,740 --> 01:07:36,390 because exponential function we see here. 959 01:07:36,390 --> 01:07:42,571 The amplitude is going to be decaying exponentially, OK? 960 01:07:42,571 --> 01:07:45,560 Any questions? 961 01:07:45,560 --> 01:07:50,180 So let's actually do a quick demonstration here, OK? 962 01:07:50,180 --> 01:07:56,540 So here, this is actually the original little ball here, 963 01:07:56,540 --> 01:07:59,060 a metal one, which actually you can 964 01:07:59,060 --> 01:08:04,790 see that this is really going to go back and forth really 965 01:08:04,790 --> 01:08:05,720 nicely. 966 01:08:05,720 --> 01:08:09,860 And you can see that, because of the friction, 967 01:08:09,860 --> 01:08:12,710 actually the amplitude is becoming 968 01:08:12,710 --> 01:08:15,200 smaller and smaller, OK? 969 01:08:15,200 --> 01:08:19,189 So that actually matches with this situation, right? 970 01:08:19,189 --> 01:08:22,850 So it's actually an underdamped situation. 971 01:08:22,850 --> 01:08:26,529 This ball, in an idealized situation, 972 01:08:26,529 --> 01:08:32,359 is going to go through 0 infinite number of times, OK? 973 01:08:32,359 --> 01:08:35,279 So now, what I am going to do is now 974 01:08:35,279 --> 01:08:39,500 I change this ball to something which is different, OK? 975 01:08:39,500 --> 01:08:43,569 This is actually made of magnets, OK? 976 01:08:43,569 --> 01:08:45,770 And let's see what is going to happen. 977 01:08:45,770 --> 01:08:47,950 So now, you can see that, because this is actually 978 01:08:47,950 --> 01:08:51,460 made of magnets, therefore, the drag force will be colossal, 979 01:08:51,460 --> 01:08:53,396 will be very, very big. 980 01:08:53,396 --> 01:08:56,378 And let's see what will happen. 981 01:08:59,859 --> 01:09:03,460 You see that the drag force is huge. 982 01:09:03,460 --> 01:09:06,899 Therefore, you see I put it here so that it 983 01:09:06,899 --> 01:09:08,699 has big initial velocity. 984 01:09:12,611 --> 01:09:16,800 It only passes through 0 once, right? 985 01:09:16,800 --> 01:09:19,750 Of course, it now is actually approaching the zero really, 986 01:09:19,750 --> 01:09:21,319 really slowly, exponentially. 987 01:09:21,319 --> 01:09:23,460 But it is not 0, OK? 988 01:09:23,460 --> 01:09:27,870 So it only passes through the 0 if you believe the math, 989 01:09:27,870 --> 01:09:30,580 only once, OK? 990 01:09:30,580 --> 01:09:35,470 Just to show that this is a real deal-- 991 01:09:35,470 --> 01:09:36,859 OK, now, whoa, right? 992 01:09:42,770 --> 01:09:48,189 Oh, I'm not trying to destroy the classroom, OK? 993 01:09:48,189 --> 01:09:50,229 So you can actually play with this 994 01:09:50,229 --> 01:09:52,670 after we finish your lecture, OK? 995 01:09:55,200 --> 01:09:58,170 I would like to ask you a question. 996 01:09:58,170 --> 01:10:01,930 After we learned this from this lecture, 997 01:10:01,930 --> 01:10:08,920 there are three situations, underdamped, critically damped, 998 01:10:08,920 --> 01:10:11,230 and overdamped, OK? 999 01:10:11,230 --> 01:10:13,550 I would like to ask you two questions. 1000 01:10:13,550 --> 01:10:18,130 The first one is through this demonstration, OK? 1001 01:10:18,130 --> 01:10:25,190 So, now I have a system which is nicely constructed. 1002 01:10:25,190 --> 01:10:27,990 I hope you can see it, OK? 1003 01:10:27,990 --> 01:10:29,560 You can see it. 1004 01:10:29,560 --> 01:10:36,870 And this system is made of a torsional spring. 1005 01:10:36,870 --> 01:10:39,930 And also, there's a pad here, which 1006 01:10:39,930 --> 01:10:42,170 is connected to the spring, OK? 1007 01:10:42,170 --> 01:10:44,790 If I actually perturb this thing, 1008 01:10:44,790 --> 01:10:49,770 it's going to be oscillating back and forth before I turn 1009 01:10:49,770 --> 01:10:56,070 on the power, so that the lower part is actually 1010 01:10:56,070 --> 01:10:58,930 you have a magnet, OK? 1011 01:10:58,930 --> 01:11:01,140 It's not turned on yet, OK? 1012 01:11:01,140 --> 01:11:04,260 And this magnet is going to provide 1013 01:11:04,260 --> 01:11:10,590 a drag force to actually change the behavior of the system, OK? 1014 01:11:10,590 --> 01:11:15,480 So you can see that, before I turn on the magnetic field, 1015 01:11:15,480 --> 01:11:18,120 the whole system is actually oscillating back and forth 1016 01:11:18,120 --> 01:11:19,170 really nicely. 1017 01:11:19,170 --> 01:11:23,450 As we predicted, small amplitude vibration 1018 01:11:23,450 --> 01:11:26,190 is harmonic oscillation, OK? 1019 01:11:26,190 --> 01:11:27,760 So that's very nice. 1020 01:11:27,760 --> 01:11:32,700 So now, what am I going to do is to turn on the power 1021 01:11:32,700 --> 01:11:35,190 and see what is going to happen. 1022 01:11:35,190 --> 01:11:42,450 After I turn on the power, there's an electric field, OK? 1023 01:11:42,450 --> 01:11:45,730 And this is actually going to be-- 1024 01:11:45,730 --> 01:11:49,870 OK, so the magnetic field is actually turned down. 1025 01:11:49,870 --> 01:11:53,370 Therefore, it is actually acting like a drag force 1026 01:11:53,370 --> 01:11:55,060 to this system, OK? 1027 01:11:55,060 --> 01:11:57,910 So let's actually see what is going to happen. 1028 01:11:57,910 --> 01:12:01,540 Now, I release this. 1029 01:12:01,540 --> 01:12:05,060 The behavior of the system looks like this. 1030 01:12:05,060 --> 01:12:09,730 It first oscillates, and then it stops. 1031 01:12:09,730 --> 01:12:14,940 So the question is, is this a critically damped, underdamped, 1032 01:12:14,940 --> 01:12:16,530 or overdamped system? 1033 01:12:16,530 --> 01:12:19,360 Anybody knows? 1034 01:12:19,360 --> 01:12:19,860 Yeah? 1035 01:12:19,860 --> 01:12:21,330 AUDIENCE: Underdamped. 1036 01:12:21,330 --> 01:12:23,120 YEN-JIE LEE: Yes, this is underdamped. 1037 01:12:23,120 --> 01:12:24,670 How do I see that? 1038 01:12:24,670 --> 01:12:28,360 That is because, when I do this experiment, 1039 01:12:28,360 --> 01:12:33,870 you would pass through 0s multiple times. 1040 01:12:33,870 --> 01:12:37,280 Therefore, there are oscillations coming into play. 1041 01:12:37,280 --> 01:12:40,700 Therefore, I can conclude that the drag force is not 1042 01:12:40,700 --> 01:12:41,750 large enough. 1043 01:12:41,750 --> 01:12:44,900 So that is actually an underdamped situation, OK? 1044 01:12:44,900 --> 01:12:47,990 And the next time, we are going to drag this system. 1045 01:12:47,990 --> 01:12:51,050 I have a second question for you. 1046 01:12:51,050 --> 01:12:55,850 So now, your friends know that you took 8.03. 1047 01:12:55,850 --> 01:12:58,190 Therefore, they will wonder if you can actually 1048 01:12:58,190 --> 01:13:04,880 design a car suspension system, to see if you can actually 1049 01:13:04,880 --> 01:13:06,860 make this design for them. 1050 01:13:06,860 --> 01:13:12,500 When you design this car, which condition 1051 01:13:12,500 --> 01:13:16,610 will you consider to set up the car? 1052 01:13:16,610 --> 01:13:22,700 Will you set it up as underdamped, critically damped, 1053 01:13:22,700 --> 01:13:24,710 or overdamped? 1054 01:13:24,710 --> 01:13:27,998 How many of you actually think it should be underdamped? 1055 01:13:31,910 --> 01:13:33,650 No, nobody? 1056 01:13:33,650 --> 01:13:39,390 How many of you actually think it should be overdamped? 1057 01:13:39,390 --> 01:13:45,110 1, 2, 3, 4, OK. 1058 01:13:45,110 --> 01:13:49,250 How many of you actually think it should be critically damped? 1059 01:13:49,250 --> 01:13:51,360 OK, the majority of you think that should 1060 01:13:51,360 --> 01:13:52,930 be the correct design. 1061 01:13:52,930 --> 01:13:59,680 So if you have the car designed as an underdamped situation, 1062 01:13:59,680 --> 01:14:01,470 then, when you drive the car, you 1063 01:14:01,470 --> 01:14:03,280 are going to have very funny style. 1064 01:14:03,280 --> 01:14:05,760 You are going to have this. 1065 01:14:05,760 --> 01:14:07,470 This is the style. 1066 01:14:07,470 --> 01:14:11,580 So the car is going to be oscillating all the time, OK? 1067 01:14:11,580 --> 01:14:14,290 Because it's going to be there. 1068 01:14:14,290 --> 01:14:18,270 And it's really damping really slowly, OK? 1069 01:14:18,270 --> 01:14:22,860 If you design it to be overdamped, 1070 01:14:22,860 --> 01:14:24,930 it would become very bumpy, right? 1071 01:14:24,930 --> 01:14:29,040 So let's take a limit of infinitely large drag 1072 01:14:29,040 --> 01:14:30,600 force constant, OK? 1073 01:14:30,600 --> 01:14:34,590 Then, it's like, when you hit some bump, you go woo! 1074 01:14:34,590 --> 01:14:36,360 Wow! 1075 01:14:36,360 --> 01:14:40,710 It doesn't really help you to reduce the amplitude, OK? 1076 01:14:40,710 --> 01:14:44,440 So the correct answer is you would 1077 01:14:44,440 --> 01:14:50,010 give the advice that you would do it critically damped, OK? 1078 01:14:50,010 --> 01:14:54,420 So before we end the section today, 1079 01:14:54,420 --> 01:14:57,820 I would like to pose a question to you. 1080 01:14:57,820 --> 01:15:02,500 The thing which we have learned from simple harmonic motion 1081 01:15:02,500 --> 01:15:07,090 is that the energy is conserved in a simple harmonic motion, 1082 01:15:07,090 --> 01:15:07,590 OK? 1083 01:15:07,590 --> 01:15:13,500 I have the Fs, the spring force, proportional to minus k times 1084 01:15:13,500 --> 01:15:15,000 x. 1085 01:15:15,000 --> 01:15:18,360 And the energy is conserved, OK? 1086 01:15:18,360 --> 01:15:23,430 But if I add a drag force in the form or minus b times v, 1087 01:15:23,430 --> 01:15:25,170 energy is not conserved, right? 1088 01:15:25,170 --> 01:15:28,070 So you can see that it was actually oscillating. 1089 01:15:28,070 --> 01:15:31,090 Now, it's not oscillating, right? 1090 01:15:31,090 --> 01:15:34,200 This thing has stopped oscillating, OK? 1091 01:15:34,200 --> 01:15:38,400 Why is that the case mathematically? 1092 01:15:38,400 --> 01:15:42,600 OK, we know what is happening physically 1093 01:15:42,600 --> 01:15:43,980 in this physical system. 1094 01:15:43,980 --> 01:15:50,250 Because OK, this Mexican hat is trying to push the air away. 1095 01:15:50,250 --> 01:15:53,280 So what is going to happen is that it's transferring 1096 01:15:53,280 --> 01:15:59,040 the energy from this system to the molecules of the air, OK? 1097 01:15:59,040 --> 01:16:00,780 So it's accelerating the air. 1098 01:16:00,780 --> 01:16:02,700 So the energy goes away. 1099 01:16:02,700 --> 01:16:06,660 But why the mathematical form looks so similar 1100 01:16:06,660 --> 01:16:09,120 and it does different things? 1101 01:16:09,120 --> 01:16:10,820 And think about it. 1102 01:16:10,820 --> 01:16:14,890 And I'm not going to talk about the answer today. 1103 01:16:14,890 --> 01:16:17,910 And thank you very much. 1104 01:16:17,910 --> 01:16:20,890 And we will continue next time to see 1105 01:16:20,890 --> 01:16:24,770 what we can learn if I start to drive the oscillator. 1106 01:16:24,770 --> 01:16:26,490 Bye-bye.