1 00:00:00,050 --> 00:00:01,770 The following content is provided 2 00:00:01,770 --> 00:00:04,010 under a Creative Commons license. 3 00:00:04,010 --> 00:00:06,860 Your support will help MIT OpenCourseWare continue 4 00:00:06,860 --> 00:00:10,720 to offer high quality educational resources for free. 5 00:00:10,720 --> 00:00:13,330 To make a donation or view additional materials 6 00:00:13,330 --> 00:00:17,209 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,209 --> 00:00:17,834 at ocw.mit.edu. 8 00:00:21,430 --> 00:00:25,030 PROFESSOR: I'm Wit Busza, professor of physics at MIT. 9 00:00:25,030 --> 00:00:29,120 I'm joining my colleague, Professor Walter Lewin, 10 00:00:29,120 --> 00:00:31,980 to help you understand the physics of waves 11 00:00:31,980 --> 00:00:33,200 and vibrations. 12 00:00:33,200 --> 00:00:37,360 Now you may well ask, why spend so much effort on waves 13 00:00:37,360 --> 00:00:38,460 and vibrations. 14 00:00:38,460 --> 00:00:41,140 And the answer is very simple. 15 00:00:41,140 --> 00:00:48,100 If you take any system, disturb it from equilibrium, 16 00:00:48,100 --> 00:00:52,410 from a stable equilibrium, the resultant motion 17 00:00:52,410 --> 00:00:54,600 is waves and vibrations. 18 00:00:54,600 --> 00:00:57,100 So it's a very common phenomenon. 19 00:00:57,100 --> 00:01:01,320 Not only is it that very common, understanding waves 20 00:01:01,320 --> 00:01:04,220 and vibrations have very important 21 00:01:04,220 --> 00:01:05,670 practical applications. 22 00:01:05,670 --> 00:01:09,060 And furthermore, the fact that they exist, 23 00:01:09,060 --> 00:01:12,390 that this phenomenon exists, has tremendous consequences 24 00:01:12,390 --> 00:01:14,210 on our world. 25 00:01:14,210 --> 00:01:18,680 If waves and vibrations were different or didn't exist, 26 00:01:18,680 --> 00:01:23,450 you wouldn't recognize our universe. 27 00:01:23,450 --> 00:01:27,780 What is the role I am playing in this course? 28 00:01:27,780 --> 00:01:31,650 To answer that question, I have to remind you 29 00:01:31,650 --> 00:01:34,200 what is the scientific method. 30 00:01:34,200 --> 00:01:38,950 In essence, the scientific method has two components. 31 00:01:38,950 --> 00:01:43,230 The first, you look around and you 32 00:01:43,230 --> 00:01:48,120 describe what you see in the one and only language that 33 00:01:48,120 --> 00:01:50,770 can be used, or we find that can be 34 00:01:50,770 --> 00:01:52,770 used for the description of nature. 35 00:01:52,770 --> 00:01:55,920 That this mathematics, in terms of mathematical equations. 36 00:01:59,130 --> 00:02:02,870 The second aspect is since the universe 37 00:02:02,870 --> 00:02:05,930 is describable in terms of mathematical equations, 38 00:02:05,930 --> 00:02:09,160 we can solve those equations. 39 00:02:09,160 --> 00:02:14,510 And that means predict result of situations, 40 00:02:14,510 --> 00:02:19,080 of experiments, which we've never seen before. 41 00:02:19,080 --> 00:02:23,200 Again, this is important for two reasons. 42 00:02:23,200 --> 00:02:27,350 One, practical-- to be able to predict what will happen. 43 00:02:27,350 --> 00:02:30,680 But the other far more important is 44 00:02:30,680 --> 00:02:35,840 that it is the way we have, the objective way we have 45 00:02:35,840 --> 00:02:40,290 of checking whether our understanding of the universe 46 00:02:40,290 --> 00:02:42,160 is correct or not. 47 00:02:42,160 --> 00:02:49,000 If the predictions do not give the right-- do not correspond 48 00:02:49,000 --> 00:02:51,900 to what one actually sees, you know, your theory, 49 00:02:51,900 --> 00:02:55,250 your understanding is wrong. 50 00:02:55,250 --> 00:02:59,740 My role is related to the second part. 51 00:02:59,740 --> 00:03:05,230 In other words, what I would want to help you learn, 52 00:03:05,230 --> 00:03:10,730 take a given situation, convert it into mathematics, 53 00:03:10,730 --> 00:03:15,200 solve it, and predict what will happen. 54 00:03:15,200 --> 00:03:18,670 We call that problem solving. 55 00:03:18,670 --> 00:03:19,460 OK. 56 00:03:19,460 --> 00:03:23,255 Let me immediately start with a concrete example. 57 00:03:30,010 --> 00:03:36,300 What I have here, describing a situation which 58 00:03:36,300 --> 00:03:39,290 we would like to understand. 59 00:03:39,290 --> 00:03:42,450 Imagine you have an ideal spring, 60 00:03:42,450 --> 00:03:46,190 a spring that obeys Hooke's law. 61 00:03:46,190 --> 00:03:50,380 As I've shown here is the spring constant k, length, 62 00:03:50,380 --> 00:03:54,890 natural length l0, and your suspend it from the ceiling. 63 00:03:54,890 --> 00:03:59,410 Imagine that you take a mass, a small mass, 64 00:03:59,410 --> 00:04:03,990 m, and you attach it to that spring. 65 00:04:03,990 --> 00:04:08,870 At some instant of time, and in the proceeds of attaching it, 66 00:04:08,870 --> 00:04:10,170 you may stretch the string. 67 00:04:10,170 --> 00:04:13,200 So the spring may at this instant not be stretched. 68 00:04:13,200 --> 00:04:15,680 But let's assume while you're attaching 69 00:04:15,680 --> 00:04:17,680 and you've stretched the spring a little bit, 70 00:04:17,680 --> 00:04:20,769 you're holding it, all right. 71 00:04:20,769 --> 00:04:24,820 At that instant, it's velocity is 0, stationary. 72 00:04:24,820 --> 00:04:26,170 You let go. 73 00:04:26,170 --> 00:04:28,470 The question is, what will happen? 74 00:04:28,470 --> 00:04:33,052 Can you predict what will be the motion of that particle? 75 00:04:33,052 --> 00:04:34,800 You know, you've seen this often. 76 00:04:34,800 --> 00:04:37,400 But a priori, it's not obvious what will happen. 77 00:04:37,400 --> 00:04:39,560 The spring may pull the mass up. 78 00:04:39,560 --> 00:04:41,920 The mass may pull the spring more down. 79 00:04:41,920 --> 00:04:43,610 It may oscillate. 80 00:04:43,610 --> 00:04:48,070 Everything, until you've understood what's going on, you 81 00:04:48,070 --> 00:04:51,770 cannot predict the outcome. 82 00:04:51,770 --> 00:04:57,840 So let's assume that at some instant of time, 83 00:04:57,840 --> 00:05:03,790 we call that time t, it's as shown on the right. 84 00:05:03,790 --> 00:05:05,690 In order to be able to describe this, 85 00:05:05,690 --> 00:05:09,940 I have to tell you where these masses are 86 00:05:09,940 --> 00:05:11,170 at these various times. 87 00:05:11,170 --> 00:05:14,540 So I will define a coordinate system. 88 00:05:14,540 --> 00:05:16,580 This is a one dimensional situation. 89 00:05:16,580 --> 00:05:17,830 So I only need one coordinate. 90 00:05:17,830 --> 00:05:19,570 And I'll call it the y. 91 00:05:19,570 --> 00:05:21,920 My y will be up. 92 00:05:21,920 --> 00:05:23,000 All right. 93 00:05:23,000 --> 00:05:26,550 Now I also have to measure things from some location. 94 00:05:26,550 --> 00:05:30,101 So I need to define what I mean by y equals 0. 95 00:05:30,101 --> 00:05:34,450 And I will define y equals 0, the position 96 00:05:34,450 --> 00:05:40,560 where if the mass is at that location, the force of gravity 97 00:05:40,560 --> 00:05:44,130 pulling it down and the spring force pulling it up 98 00:05:44,130 --> 00:05:49,610 cancel, so that there is no net force on the particle. 99 00:05:49,610 --> 00:05:53,062 So y equals 0 is the equilibrium position. 100 00:05:53,062 --> 00:05:55,130 And then the position of the spring 101 00:05:55,130 --> 00:06:00,190 when it has no mass attached is the distance y0 from that 102 00:06:00,190 --> 00:06:04,080 at t equals 0, the position I will say is y initial. 103 00:06:04,080 --> 00:06:05,360 That's some number. 104 00:06:05,360 --> 00:06:07,190 So that's a known quantity. 105 00:06:07,190 --> 00:06:07,790 All right. 106 00:06:07,790 --> 00:06:12,890 And that any other instant time, I defined it as y equals t. 107 00:06:12,890 --> 00:06:16,960 That is the physical situation I wish to understand. 108 00:06:16,960 --> 00:06:20,340 I want to know what happened with that spring. 109 00:06:20,340 --> 00:06:25,430 So now I will translate that into mathematics. 110 00:06:25,430 --> 00:06:28,750 I will now try to give you a mathematical description 111 00:06:28,750 --> 00:06:29,503 of that situation. 112 00:06:32,860 --> 00:06:40,040 So we know that we are dealing with forces and masses. 113 00:06:40,040 --> 00:06:45,620 So to describe that, we use Newtonian mechanics. 114 00:06:45,620 --> 00:06:49,910 So here is now my mathematical description. 115 00:06:49,910 --> 00:06:56,470 The mass is a point m, of mass m, on which two forces act. 116 00:06:56,470 --> 00:07:02,330 There is the force fs due to the spring and the force fg 117 00:07:02,330 --> 00:07:06,680 due to the fact that this mass is in the gravitational field, 118 00:07:06,680 --> 00:07:10,530 and therefore there is a gravitational force 119 00:07:10,530 --> 00:07:11,860 on this mass, fg. 120 00:07:11,860 --> 00:07:13,350 OK. 121 00:07:13,350 --> 00:07:15,540 We call this a force diagram. 122 00:07:15,540 --> 00:07:17,930 Or some people call it the free body diagram. 123 00:07:20,500 --> 00:07:26,700 Now this mass, because of the force acting it, 124 00:07:26,700 --> 00:07:28,870 its motion will change. 125 00:07:28,870 --> 00:07:31,960 And it will have an acceleration, 126 00:07:31,960 --> 00:07:34,230 which I will call a of t. 127 00:07:34,230 --> 00:07:39,430 And by the way, that of course is the second derivative 128 00:07:39,430 --> 00:07:42,400 of y with respect to dt. 129 00:07:42,400 --> 00:07:42,990 It's a vector. 130 00:07:42,990 --> 00:07:46,090 It's in the y direction. 131 00:07:46,090 --> 00:07:49,030 And in order so I don't have to write things over 132 00:07:49,030 --> 00:07:52,970 and too many things in these equations, 133 00:07:52,970 --> 00:07:56,960 I will define the symbol y with two dots on it 134 00:07:56,960 --> 00:08:01,310 as the second derivative of y with respect to time. 135 00:08:01,310 --> 00:08:04,120 y dot is the first derivative-- in other words, 136 00:08:04,120 --> 00:08:07,120 the velocity, et cetera. 137 00:08:07,120 --> 00:08:11,450 And by the way, you may notice I'm going very slowly here. 138 00:08:11,450 --> 00:08:13,870 I'm doing that intentionally. 139 00:08:13,870 --> 00:08:19,600 I'm going to go here in gory detail every part, you know, 140 00:08:19,600 --> 00:08:23,320 because often I know that when one goes 141 00:08:23,320 --> 00:08:26,900 to a lecture, or studies in a book, et cetera, 142 00:08:26,900 --> 00:08:31,560 you look at some step from one step to another. 143 00:08:31,560 --> 00:08:33,450 And you can't figure it out. 144 00:08:33,450 --> 00:08:37,289 The reason for it often is not that you are not smart enough 145 00:08:37,289 --> 00:08:41,010 to do it, is but the because the teacher 146 00:08:41,010 --> 00:08:43,120 or whoever wrote the book, et cetera, 147 00:08:43,120 --> 00:08:45,760 is so familiar with the material he will 148 00:08:45,760 --> 00:08:49,170 do several steps in his head or her head, 149 00:08:49,170 --> 00:08:51,050 and you don't know about it. 150 00:08:51,050 --> 00:08:53,510 For this first example, I will try 151 00:08:53,510 --> 00:08:55,790 to avoid anything of that kind. 152 00:08:55,790 --> 00:08:57,070 Later on, I'll go faster. 153 00:08:57,070 --> 00:08:59,130 And I'll do the same as everybody else. 154 00:08:59,130 --> 00:09:02,900 But the moment, as I say, I'm going in gory detail. 155 00:09:02,900 --> 00:09:03,590 OK. 156 00:09:03,590 --> 00:09:09,135 So this is the diagram, this free body diagram, 157 00:09:09,135 --> 00:09:11,570 of the situation. 158 00:09:11,570 --> 00:09:15,750 And I know from Newtonian mechanics 159 00:09:15,750 --> 00:09:20,360 that if there are forces acting on that mass, 160 00:09:20,360 --> 00:09:24,010 that mass will have an acceleration, which 161 00:09:24,010 --> 00:09:27,070 will be equal to the net force acting on it, 162 00:09:27,070 --> 00:09:32,540 divided by the mass, the inertia, of that system. 163 00:09:32,540 --> 00:09:35,100 So that is what a will be. 164 00:09:35,100 --> 00:09:38,570 I further know the force is vector, 165 00:09:38,570 --> 00:09:41,830 so the net force acting on this mass 166 00:09:41,830 --> 00:09:44,250 is the sum of those, the vectorial sum, 167 00:09:44,250 --> 00:09:45,890 of those two forces. 168 00:09:45,890 --> 00:09:51,020 So f is the sum of the force due to the spring 169 00:09:51,020 --> 00:09:54,280 and due to gravity. 170 00:09:54,280 --> 00:09:57,230 Next, we also know something about the spring. 171 00:09:57,230 --> 00:09:59,050 I told you at the beginning that I'm 172 00:09:59,050 --> 00:10:01,510 considering an ideal spring. 173 00:10:01,510 --> 00:10:03,750 So for the purpose of this problem, 174 00:10:03,750 --> 00:10:07,370 I'm assuming I have this fictitious thing, a spring 175 00:10:07,370 --> 00:10:10,070 which essentially has no mass, massless, 176 00:10:10,070 --> 00:10:13,240 which obeys exactly Hooke's law. 177 00:10:13,240 --> 00:10:17,230 And here I can't help digress and point out to you 178 00:10:17,230 --> 00:10:19,440 that that's a terrible misnomer. 179 00:10:19,440 --> 00:10:23,130 There is no Hooke's law of nature. 180 00:10:23,130 --> 00:10:27,575 It is an empirical relation which tells you the force 181 00:10:27,575 --> 00:10:31,490 that the spring exerts when you stretch it a certain distance, 182 00:10:31,490 --> 00:10:32,620 all right. 183 00:10:32,620 --> 00:10:34,750 But anyway, it's stuck historically. 184 00:10:34,750 --> 00:10:35,990 It's Hooke's law. 185 00:10:35,990 --> 00:10:38,810 So Hooke's law, from Hooke's law, 186 00:10:38,810 --> 00:10:45,380 I know what will be the force, fs, when the situation is as 187 00:10:45,380 --> 00:10:49,340 shown over there, all right, at time t. 188 00:10:49,340 --> 00:10:53,930 So at this time, this extension of this spring 189 00:10:53,930 --> 00:10:57,480 will be, of course, y0 minus yt. 190 00:10:57,480 --> 00:10:58,170 OK. 191 00:10:58,170 --> 00:11:01,910 And so I get that the force due to the spring 192 00:11:01,910 --> 00:11:05,590 will be the spring constant times its extension 193 00:11:05,590 --> 00:11:07,310 at that instant of time. 194 00:11:07,310 --> 00:11:08,900 It is a vector. 195 00:11:08,900 --> 00:11:18,320 And y0 is a bigger number than yt, this is a positive number. 196 00:11:18,320 --> 00:11:22,540 Therefore, the stretched spring will pull the mass up. 197 00:11:22,540 --> 00:11:24,080 So this is in the y direction. 198 00:11:24,080 --> 00:11:26,900 This is plus. 199 00:11:26,900 --> 00:11:28,890 How about the gravitational force? 200 00:11:28,890 --> 00:11:34,630 Well, that is, of course, the minus mg, 201 00:11:34,630 --> 00:11:37,390 the force of the gravitational field on that. 202 00:11:37,390 --> 00:11:40,390 And it's minus in the y direction 203 00:11:40,390 --> 00:11:45,110 here, because it's pulling this mass down. 204 00:11:45,110 --> 00:11:45,980 OK. 205 00:11:45,980 --> 00:11:49,090 Now what else do we know? 206 00:11:49,090 --> 00:11:53,230 We know that we could get everything done very carefully. 207 00:11:53,230 --> 00:11:56,030 We know that we defined y equals 0 208 00:11:56,030 --> 00:11:58,250 to be the equilibrium position. 209 00:11:58,250 --> 00:12:05,935 Therefore, when y is 0, we know that the second derivative of y 210 00:12:05,935 --> 00:12:06,760 is 0. 211 00:12:06,760 --> 00:12:08,350 It's not accelerating. 212 00:12:08,350 --> 00:12:10,840 So that's a condition we must not forget. 213 00:12:10,840 --> 00:12:14,910 Another thing we know that initially, in other words, 214 00:12:14,910 --> 00:12:20,770 at t equals 0, the position of that mass is y initial. 215 00:12:20,770 --> 00:12:26,720 Finally, I told you that the velocity of that mass 216 00:12:26,720 --> 00:12:30,230 was 0 at t equals 0, stationary. 217 00:12:30,230 --> 00:12:35,040 So this is the beginning of our translating all the information 218 00:12:35,040 --> 00:12:38,940 we gathered here into mathematics. 219 00:12:48,400 --> 00:12:52,150 Let me continue now using this information 220 00:12:52,150 --> 00:12:57,210 and try to reduce it to the minimum set of equations. 221 00:12:57,210 --> 00:13:03,190 From a equals fm, from this, I get that the acceleration 222 00:13:03,190 --> 00:13:06,140 is the total force divided by m. 223 00:13:06,140 --> 00:13:10,380 I can now replace these two forces 224 00:13:10,380 --> 00:13:12,790 from the information I wrote over there. 225 00:13:12,790 --> 00:13:17,210 And so that is equal, 1 over-- this is f over m. 226 00:13:17,210 --> 00:13:19,590 It's 1 of m times the net force, which 227 00:13:19,590 --> 00:13:22,430 is the force due to the spring minus the force 228 00:13:22,430 --> 00:13:24,160 due to the gravity, OK. 229 00:13:24,160 --> 00:13:33,260 So from this, I can now actually write an algebraic equation, 230 00:13:33,260 --> 00:13:34,730 rather than the vector one. 231 00:13:34,730 --> 00:13:37,185 Notice - ha ha. 232 00:13:37,185 --> 00:13:39,940 I have noticed myself even something. 233 00:13:39,940 --> 00:13:45,230 Here this has to be in the direction of y. 234 00:13:45,230 --> 00:13:47,070 OK. 235 00:13:47,070 --> 00:13:51,720 This is a vector equation, but all the parts 236 00:13:51,720 --> 00:13:55,820 are in the same direction, in the y direction. 237 00:13:55,820 --> 00:14:01,700 Therefore, I can rewrite this just the equation for the one 238 00:14:01,700 --> 00:14:07,000 component and not bother to write the y hats throughout. 239 00:14:07,000 --> 00:14:12,470 So this equation I've rewritten now just removing y hat. 240 00:14:12,470 --> 00:14:17,860 So this is how the mass will be accelerating. 241 00:14:17,860 --> 00:14:20,230 Unfortunately, it's a single equation, 242 00:14:20,230 --> 00:14:22,260 but I have more than one unknown in it. 243 00:14:22,260 --> 00:14:24,920 Because I don't know y0. 244 00:14:24,920 --> 00:14:26,600 And I don't know y of t. 245 00:14:26,600 --> 00:14:29,940 Clearly, I won't be able to solve that equation, all right. 246 00:14:29,940 --> 00:14:33,280 But at that stage, I go back to the information 247 00:14:33,280 --> 00:14:35,270 I told you at the beginning. 248 00:14:35,270 --> 00:14:40,420 We defined y equals 0 to be the place where 249 00:14:40,420 --> 00:14:44,500 a y double dot, the second derivative, is 0. 250 00:14:44,500 --> 00:14:50,640 Therefore, I can write that at position. 251 00:14:50,640 --> 00:14:54,510 When y of t is 0, this is 0. 252 00:14:54,510 --> 00:15:00,210 So 0 is equal to 1 over m, into ky minus mg. 253 00:15:00,210 --> 00:15:04,540 And I immediately from this get that ky0 is mg. 254 00:15:04,540 --> 00:15:09,800 Therefore, I have found what y0 is. 255 00:15:09,800 --> 00:15:10,440 OK. 256 00:15:10,440 --> 00:15:11,220 Great. 257 00:15:11,220 --> 00:15:14,270 So there's only one unknown here. 258 00:15:14,270 --> 00:15:18,390 So using this information in here, 259 00:15:18,390 --> 00:15:22,530 I end up immediately with this equation, 260 00:15:22,530 --> 00:15:24,880 that the second derivative of y with respect 261 00:15:24,880 --> 00:15:27,500 to time, the acceleration of the mass 262 00:15:27,500 --> 00:15:34,620 is equal to minus k over m times y of t. 263 00:15:34,620 --> 00:15:41,630 At this stage, I will really find this quantity, k over m. 264 00:15:41,630 --> 00:15:46,410 For the time being, you can look at it as just for convenience, 265 00:15:46,410 --> 00:15:48,330 less to write on the board. 266 00:15:48,330 --> 00:15:54,110 But later, you see this will help us understand 267 00:15:54,110 --> 00:15:56,310 how to deal with different situations. 268 00:15:56,310 --> 00:15:58,390 But for the time being, you can just 269 00:15:58,390 --> 00:16:01,830 think of this as a convenience, so I 270 00:16:01,830 --> 00:16:03,650 can write less on the board. 271 00:16:03,650 --> 00:16:08,640 And I end up finally with one equation. 272 00:16:08,640 --> 00:16:11,590 The second derivative of y with respect to time 273 00:16:11,590 --> 00:16:15,650 is equal to minus a constant, that's k over m, right, 274 00:16:15,650 --> 00:16:18,210 times the value of y times t. 275 00:16:18,210 --> 00:16:22,960 This is the equation of motion for this mass. 276 00:16:22,960 --> 00:16:28,420 It tells me in mathematical form how the motion of that mass 277 00:16:28,420 --> 00:16:31,070 changes with time. 278 00:16:31,070 --> 00:16:35,320 I can now actually predict what will 279 00:16:35,320 --> 00:16:39,840 happen in this particular situation. 280 00:16:39,840 --> 00:16:46,570 Because I know what was the motion of it at time 0. 281 00:16:46,570 --> 00:16:52,150 I know that at time 0, the position was y initial. 282 00:16:52,150 --> 00:16:55,800 And the velocity was 0. 283 00:16:55,800 --> 00:17:03,120 These three lines are completely equivalent from the point 284 00:17:03,120 --> 00:17:06,920 of view of understanding the motion of the mass 285 00:17:06,920 --> 00:17:09,950 to our original description. 286 00:17:09,950 --> 00:17:14,010 This is a physical description of the situation. 287 00:17:14,010 --> 00:17:17,920 This is a mathematical description 288 00:17:17,920 --> 00:17:18,904 of the same situation. 289 00:17:22,450 --> 00:17:24,540 So we've achieved step one. 290 00:17:24,540 --> 00:17:27,730 We've translated a physical situation 291 00:17:27,730 --> 00:17:30,310 into a mathematical one. 292 00:17:30,310 --> 00:17:33,260 Let me now try from this, I should 293 00:17:33,260 --> 00:17:35,260 be able to predict what this mass will do. 294 00:17:42,420 --> 00:17:42,920 OK. 295 00:17:47,660 --> 00:17:51,450 I'm now switching into the world the mathematics. 296 00:17:51,450 --> 00:17:54,330 As I just am repeating here, I've 297 00:17:54,330 --> 00:17:57,040 gone away from a physical description 298 00:17:57,040 --> 00:17:59,460 to a mathematical description. 299 00:17:59,460 --> 00:18:02,820 This is pure mathematics. 300 00:18:02,820 --> 00:18:05,280 I have an equation, a mathematical equation, 301 00:18:05,280 --> 00:18:07,470 for y of t. 302 00:18:07,470 --> 00:18:10,130 It's a second order differential equation. 303 00:18:10,130 --> 00:18:15,310 I had the boundary conditions, or initial conditions. 304 00:18:15,310 --> 00:18:19,490 I can solve that using mathematics. 305 00:18:19,490 --> 00:18:20,090 OK. 306 00:18:20,090 --> 00:18:20,970 Let's do that. 307 00:18:20,970 --> 00:18:22,880 So I'm now doing pure mathematics. 308 00:18:22,880 --> 00:18:25,490 I don't want to teach you math. 309 00:18:25,490 --> 00:18:28,540 That's the role of the math department, all right. 310 00:18:28,540 --> 00:18:32,010 So how do I solve that equation? 311 00:18:32,010 --> 00:18:35,000 And let me tell you how I solve it. 312 00:18:35,000 --> 00:18:39,680 I am make use of the so-called uniqueness theorem. 313 00:18:39,680 --> 00:18:43,430 I know, or the mathematicians have told me, 314 00:18:43,430 --> 00:18:58,360 that if I find a solution to that equation which satisfies-- 315 00:18:58,360 --> 00:19:03,390 if I find a solution which satisfies that equation, 316 00:19:03,390 --> 00:19:09,080 and if it has the right number of arbitrary constant, 317 00:19:09,080 --> 00:19:14,770 then I have found the one and only general equation, 318 00:19:14,770 --> 00:19:18,600 which is a solution of that. 319 00:19:18,600 --> 00:19:19,490 Let me be concrete. 320 00:19:23,170 --> 00:19:39,780 y of t equals to a cosine omega t plus phi, where A and phi are 321 00:19:39,780 --> 00:19:42,215 arbitrary, are arbitrary. 322 00:19:47,280 --> 00:19:49,310 They are some arbitrary numbers. 323 00:19:49,310 --> 00:19:50,450 But a number is there. 324 00:19:50,450 --> 00:19:53,470 This can be 7 and this can be 21 degrees, 325 00:19:53,470 --> 00:19:55,998 or whatever, but any number. 326 00:19:55,998 --> 00:20:01,090 This equation satisfies my differential equation. 327 00:20:01,090 --> 00:20:02,830 If you don't believe me, try it. 328 00:20:02,830 --> 00:20:06,960 Differentiate this twice, all right, 329 00:20:06,960 --> 00:20:11,580 for any value of A and phi and you'll satisfy that equation. 330 00:20:11,580 --> 00:20:16,490 So this is a solution which satisfy that equation. 331 00:20:16,490 --> 00:20:19,390 It has the right number of arbitrary constants, 332 00:20:19,390 --> 00:20:23,410 that two arbitrary constants in here. 333 00:20:23,410 --> 00:20:25,830 And therefore, this is the only solution 334 00:20:25,830 --> 00:20:29,130 in the universe of that equation. 335 00:20:29,130 --> 00:20:30,300 OK. 336 00:20:30,300 --> 00:20:33,990 Now, so being a physicist, I don't 337 00:20:33,990 --> 00:20:36,370 care how I got the solution. 338 00:20:36,370 --> 00:20:39,020 Once I had the solution, if I know 339 00:20:39,020 --> 00:20:42,950 it's the only one that exists, I'm home. 340 00:20:42,950 --> 00:20:46,037 Now you can say well, I suppose I didn't guess it. 341 00:20:46,037 --> 00:20:47,120 Well, there are many ways. 342 00:20:47,120 --> 00:20:48,550 You go on the web and find it. 343 00:20:48,550 --> 00:20:49,940 You go in the book and find it. 344 00:20:49,940 --> 00:20:52,680 You ask your friends what it is, all right. 345 00:20:52,680 --> 00:20:54,540 That's mathematics. 346 00:20:54,540 --> 00:20:58,540 And once you've found the solution, we can go on. 347 00:20:58,540 --> 00:21:02,020 All right, so this is the solution of that equation. 348 00:21:02,020 --> 00:21:05,430 Next, if that's y, what is y dot? 349 00:21:05,430 --> 00:21:08,070 What is the rate of change of t? 350 00:21:08,070 --> 00:21:15,136 That's going to equal minus omega 0 A sine omega 0 t 351 00:21:15,136 --> 00:21:17,424 plus phi, OK. 352 00:21:20,910 --> 00:21:23,585 Can I predict what will happen? 353 00:21:23,585 --> 00:21:24,590 All right. 354 00:21:24,590 --> 00:21:28,310 I still need, in order to be able to predict what 355 00:21:28,310 --> 00:21:30,570 will happen, I need to find out what 356 00:21:30,570 --> 00:21:36,090 are the values of A and phi which 357 00:21:36,090 --> 00:21:40,250 satisfy the other information right here. 358 00:21:40,250 --> 00:21:44,830 See, I told you that we reduce that physical situation 359 00:21:44,830 --> 00:21:47,480 to a differential equation, the equation of motion 360 00:21:47,480 --> 00:21:53,660 for this mass, including the information about where it was 361 00:21:53,660 --> 00:21:57,780 at some instant of time, how it was moving, et cetera. 362 00:21:57,780 --> 00:22:05,470 So I need to make sure that this equation satisfies 363 00:22:05,470 --> 00:22:08,010 these boundary conditions. 364 00:22:08,010 --> 00:22:12,100 In other words, it its thees boundary conditions 365 00:22:12,100 --> 00:22:17,130 which will determine what are the A's and phi 366 00:22:17,130 --> 00:22:21,930 for the particular problem that I had there, OK. 367 00:22:21,930 --> 00:22:26,560 And so what I do is-- let's, for example, takes here. 368 00:22:26,560 --> 00:22:28,040 Because I see 0. 369 00:22:28,040 --> 00:22:36,000 Y dot t is 0, all right, at t equals 0. 370 00:22:36,000 --> 00:22:45,100 Well, when t equals 0, this is minus omega 0 A sine phi. 371 00:22:45,100 --> 00:22:46,440 OK. 372 00:22:46,440 --> 00:22:51,535 Therefore, I immediately conclude that phi is 0. 373 00:22:55,020 --> 00:22:56,740 OK. 374 00:22:56,740 --> 00:23:00,840 Next, I know-- so now that I know that phi is 0, 375 00:23:00,840 --> 00:23:03,850 I can go back to this equation. 376 00:23:03,850 --> 00:23:05,510 This is now 0. 377 00:23:05,510 --> 00:23:12,730 And we know that y at t equals 0 is y initial. 378 00:23:12,730 --> 00:23:17,130 But that t equals 0 cosine of 0 is 1. 379 00:23:17,130 --> 00:23:21,010 Therefore, A is y initial. 380 00:23:21,010 --> 00:23:32,530 And so I get finitely y of t is equal to y initial, all right, 381 00:23:32,530 --> 00:23:38,300 times cosine omega 0. 382 00:23:38,300 --> 00:23:41,340 Let me now replace it with the 1-- well, 383 00:23:41,340 --> 00:23:45,570 let me leave it as omega 0 t pluse 0. 384 00:23:45,570 --> 00:23:50,450 That, and I can rewrite this, putting all the numbers 385 00:23:50,450 --> 00:23:58,280 that I have, y initial cosine. 386 00:23:58,280 --> 00:24:02,050 And I'll now even replace omega 0 by k 387 00:24:02,050 --> 00:24:05,385 over n, square root of k over n, times t. 388 00:24:10,420 --> 00:24:17,820 Notice there are no unknown quantities in here. 389 00:24:17,820 --> 00:24:22,210 This tells me two things. 390 00:24:22,210 --> 00:24:28,590 At any instant of time, I can calculate 391 00:24:28,590 --> 00:24:32,340 where this mass will be. 392 00:24:32,340 --> 00:24:35,360 It's given by this equation. 393 00:24:35,360 --> 00:24:39,230 Secondly, I can describe the kind of motion it does. 394 00:24:39,230 --> 00:24:41,160 What is this equation? 395 00:24:41,160 --> 00:24:44,060 As a function of time, this corresponds 396 00:24:44,060 --> 00:24:48,060 to an oscillating position y. 397 00:24:48,060 --> 00:24:51,160 So this mass, when I let go, will oscillate. 398 00:24:55,140 --> 00:24:56,720 What will be the period? 399 00:24:56,720 --> 00:25:01,360 How long will it take before it comes back to where it started? 400 00:25:01,360 --> 00:25:06,320 Well, the period t will be how much time 401 00:25:06,320 --> 00:25:12,360 do I have to add to this t, so that the angle here 402 00:25:12,360 --> 00:25:14,230 changes by 2 pi? 403 00:25:14,230 --> 00:25:21,128 Well, that's obviously 2 pi root of m over k. 404 00:25:21,128 --> 00:25:23,510 OK. 405 00:25:23,510 --> 00:25:26,640 So I've achieved what I wanted to do. 406 00:25:26,640 --> 00:25:29,910 I've taken a physical situation. 407 00:25:29,910 --> 00:25:35,160 And I have predicted if I let go what will happen. 408 00:25:35,160 --> 00:25:40,710 This is the motion it will experience. 409 00:25:40,710 --> 00:25:42,750 This is the period. 410 00:25:42,750 --> 00:25:44,830 I can predict the time, et cetera. 411 00:25:44,830 --> 00:25:48,270 At this stage, let's stop for a second 412 00:25:48,270 --> 00:25:52,390 and consider what we've done. 413 00:25:52,390 --> 00:25:56,050 Because it's the essence of-- this 414 00:25:56,050 --> 00:25:59,010 is a good example of the essence of the scientific method. 415 00:26:04,070 --> 00:26:05,855 We have taken a physical situation. 416 00:26:08,870 --> 00:26:15,425 We've described it in terms of mathematics. 417 00:26:20,040 --> 00:26:26,820 Then we made an act of faith that if I 418 00:26:26,820 --> 00:26:33,410 take the mathematical equations and I solve them, 419 00:26:33,410 --> 00:26:37,340 that the resultant answer will actually 420 00:26:37,340 --> 00:26:41,890 correspond to what nature will do. 421 00:26:41,890 --> 00:26:45,540 If you stop to think about that, it's amazing. 422 00:26:45,540 --> 00:26:48,400 Nobody understands that fact. 423 00:26:48,400 --> 00:26:49,140 Why that's true. 424 00:26:49,140 --> 00:26:51,980 Why it happened. 425 00:26:51,980 --> 00:26:53,670 In other words, nobody understands 426 00:26:53,670 --> 00:26:59,230 why nature can be described in terms of mathematics. 427 00:26:59,230 --> 00:27:00,200 OK. 428 00:27:00,200 --> 00:27:06,990 But it is that fact which makes the scientific method possible. 429 00:27:09,880 --> 00:27:15,325 Finally on this note, let me give a quotation from Einstein 430 00:27:15,325 --> 00:27:20,110 which beautifully summarizes what I've just said. 431 00:27:20,110 --> 00:27:24,850 And that is the following, "The most incomprehensible thing 432 00:27:24,850 --> 00:27:29,780 about the universe is that it is comprehensible." 433 00:27:29,780 --> 00:27:34,930 The fact that we can follow this procedure is amazing. 434 00:27:38,000 --> 00:27:39,040 OK. 435 00:27:39,040 --> 00:27:44,180 Let me at this stage go and take another example, all right. 436 00:27:54,820 --> 00:27:56,635 So let's take another example. 437 00:28:05,220 --> 00:28:08,340 Consider the following situation. 438 00:28:08,340 --> 00:28:14,500 I take something like a ruler, a uniform rod. 439 00:28:14,500 --> 00:28:19,165 And I put a nail through it, some kind of a pivot. 440 00:28:19,165 --> 00:28:21,290 There is some pivot. 441 00:28:21,290 --> 00:28:24,340 I pivot the ruler on it. 442 00:28:24,340 --> 00:28:27,880 And it's hanging like this. 443 00:28:27,880 --> 00:28:28,760 OK. 444 00:28:28,760 --> 00:28:34,630 Let's assume the mass is m of the ruler. 445 00:28:34,630 --> 00:28:35,785 The length is l. 446 00:28:41,830 --> 00:28:46,160 It's a uniform ruler, a rod of some kind. 447 00:28:46,160 --> 00:28:54,630 And at t equals 0, I give it an impulse. 448 00:28:54,630 --> 00:29:01,905 I give it a little impulse, so we are now at t equals 0. 449 00:29:05,100 --> 00:29:07,200 We give it an impulse. 450 00:29:07,200 --> 00:29:10,665 At that instant, the ruler is still hanging vertically. 451 00:29:15,350 --> 00:29:18,800 Let me, just so that when you look on the board, 452 00:29:18,800 --> 00:29:21,720 you may be confused in which plane I am. 453 00:29:21,720 --> 00:29:23,460 This is the vertical plane. 454 00:29:23,460 --> 00:29:24,600 So this is up. 455 00:29:29,900 --> 00:29:31,740 So I give it an impulse. 456 00:29:31,740 --> 00:29:38,480 So at that instant of time, this ruler 457 00:29:38,480 --> 00:29:48,260 will have an angular velocity which I will call theta dot. 458 00:29:53,740 --> 00:29:56,550 This is at time equals 0. 459 00:29:56,550 --> 00:29:58,910 And it has some number as a result, 460 00:29:58,910 --> 00:30:01,670 depends how big an impulse I gave it. 461 00:30:01,670 --> 00:30:04,460 And so that you remember what I'm talking about, 462 00:30:04,460 --> 00:30:07,380 I like to give this, instead of using a symbol, 463 00:30:07,380 --> 00:30:16,320 I'll call this angular velocity at t equals 0. 464 00:30:16,320 --> 00:30:20,190 So this is some number, so many radians per second. 465 00:30:20,190 --> 00:30:22,360 That's at t equals 0. 466 00:30:22,360 --> 00:30:25,260 And I'm now going to follow this method again. 467 00:30:25,260 --> 00:30:29,430 I want to know what will be the motion of this. 468 00:30:29,430 --> 00:30:31,240 What's going to happen to this ruler. 469 00:30:31,240 --> 00:30:36,170 Is it going to start spinning around this, like this forever? 470 00:30:36,170 --> 00:30:38,260 What will happen? 471 00:30:38,260 --> 00:30:43,740 So I will try to translate this problem into mathematics. 472 00:30:47,750 --> 00:30:50,530 Because of the mechanical constraint, 473 00:30:50,530 --> 00:30:55,520 at some instant of time, the ruler may be doing this. 474 00:30:55,520 --> 00:30:57,950 Let's call this the time t. 475 00:30:57,950 --> 00:31:01,605 This is time t. 476 00:31:01,605 --> 00:31:04,310 And time t is like this. 477 00:31:04,310 --> 00:31:07,860 And I've got to define some coordinate system. 478 00:31:07,860 --> 00:31:11,930 So I'll take this angle from the vertical. 479 00:31:11,930 --> 00:31:14,621 And I call that theta at time t. 480 00:31:17,870 --> 00:31:19,540 That's why I call this theta dot. 481 00:31:19,540 --> 00:31:22,220 This is the rate of change of that. 482 00:31:22,220 --> 00:31:27,260 So at some instance of time, it will be at this position, 483 00:31:27,260 --> 00:31:28,430 all right. 484 00:31:28,430 --> 00:31:31,320 At that instant of time, it'll have 485 00:31:31,320 --> 00:31:35,110 a velocity in this direction. 486 00:31:35,110 --> 00:31:37,820 And we'll have an acceleration in that direction. 487 00:31:37,820 --> 00:31:40,000 So for example, the acceleration will 488 00:31:40,000 --> 00:31:43,400 be theta double dot at time t. 489 00:31:49,370 --> 00:31:54,680 And just so that at this stage, I will still to remind you 490 00:31:54,680 --> 00:31:58,398 that's alpha, alpha time t. 491 00:32:01,266 --> 00:32:04,090 Because often alpha is used as the acceleration. 492 00:32:04,090 --> 00:32:06,120 So at the moment, I just want you 493 00:32:06,120 --> 00:32:09,220 so you can easy for you to see what I'm talking about. 494 00:32:09,220 --> 00:32:14,880 So at some instance of time, that is the physical situation. 495 00:32:14,880 --> 00:32:21,600 I would like to now convert this into mathematics. 496 00:32:21,600 --> 00:32:23,350 Follow the same procedure as before. 497 00:32:28,980 --> 00:32:33,920 I need to write the equation of motion for this. 498 00:32:33,920 --> 00:32:39,180 And I need to write down the initial conditions. 499 00:32:39,180 --> 00:32:42,820 So how do I do with that? 500 00:32:42,820 --> 00:32:47,250 So now I start off by the free body diagram. 501 00:32:47,250 --> 00:32:50,420 Here is the pivot. 502 00:32:50,420 --> 00:32:51,930 That's the route. 503 00:32:55,800 --> 00:32:58,492 This angle is theta t. 504 00:33:05,120 --> 00:33:08,480 There will be a force acting. 505 00:33:08,480 --> 00:33:11,680 We're now dealing with rigid body motion. 506 00:33:11,680 --> 00:33:18,460 So today we did Newtonian mechanics for masses and forces 507 00:33:18,460 --> 00:33:20,610 through a single point mass and forces. 508 00:33:20,610 --> 00:33:26,320 Now we are doing a Newtonian dynamics for rigid body motion. 509 00:33:26,320 --> 00:33:30,675 You know that if a rigid body is in the gravitational field, 510 00:33:30,675 --> 00:33:36,230 the gravity acts force fg. 511 00:33:36,230 --> 00:33:38,830 We can analyze it, as if there was a force fg 512 00:33:38,830 --> 00:33:44,520 g acting through the center of mass here of the body. 513 00:33:44,520 --> 00:33:46,705 So this length now is l over 2. 514 00:33:52,250 --> 00:33:55,060 So there will be a force fg acting. 515 00:33:55,060 --> 00:34:03,620 And as a result, there will be torques about this point. 516 00:34:03,620 --> 00:34:05,180 Now let me say the following. 517 00:34:05,180 --> 00:34:13,710 We are dealing here with motion, rotations, in a single plane. 518 00:34:13,710 --> 00:34:17,040 And so we are dealing about rotations, 519 00:34:17,040 --> 00:34:22,650 about an axis through this point p. 520 00:34:22,650 --> 00:34:26,560 We're not dealing with three dimensional rotations, 521 00:34:26,560 --> 00:34:30,710 but simple situation where all the motion 522 00:34:30,710 --> 00:34:37,940 is about a single axis, which is perpendicular to this point, p. 523 00:34:40,730 --> 00:34:43,580 There will be a torque about p because 524 00:34:43,580 --> 00:34:46,550 of the gravitational force. 525 00:34:46,550 --> 00:34:50,389 And as a result, there's going to be the acceleration, which 526 00:34:50,389 --> 00:34:55,636 as we've said over there, is theta double dot of t. 527 00:35:00,180 --> 00:35:10,865 Now, we know that torques gives rise to angular acceleration. 528 00:35:16,560 --> 00:35:23,710 Let me define that we will take clockwise motion, clockwise 529 00:35:23,710 --> 00:35:34,860 motion, clockwise rotations to be positive. 530 00:35:42,350 --> 00:35:48,600 So any rotation, this angle, for example, I am sorry. 531 00:35:48,600 --> 00:35:50,160 I meant anti-clockwise. 532 00:35:53,630 --> 00:35:55,830 Anti-clockwise is positive. 533 00:35:55,830 --> 00:35:56,760 Look at this. 534 00:35:56,760 --> 00:36:01,715 If this rotates like that to this angle, this 535 00:36:01,715 --> 00:36:03,900 I take to be a positive number, it's 536 00:36:03,900 --> 00:36:06,200 an anti-clockwise rotation. 537 00:36:06,200 --> 00:36:09,710 Similarly, if this acceleration is a positive number, 538 00:36:09,710 --> 00:36:13,140 it's accelerating in this direction. 539 00:36:13,140 --> 00:36:18,430 Since we are dealing with rotations about a single axis, 540 00:36:18,430 --> 00:36:22,800 we don't have to go to the vector formulation. 541 00:36:22,800 --> 00:36:27,500 We can consider it just the magnitude. 542 00:36:27,500 --> 00:36:36,380 And we know that the acceleration 543 00:36:36,380 --> 00:36:43,660 is equal to the torque divided by the moment of inertia. 544 00:36:43,660 --> 00:36:45,845 Or you may have seen it the other way. 545 00:36:45,845 --> 00:36:48,580 Torque Equals I alpha. 546 00:36:56,620 --> 00:36:58,090 I prefer it this way. 547 00:36:58,090 --> 00:36:59,640 For me, it's more logical. 548 00:36:59,640 --> 00:37:03,040 The angular acceleration is a consequence of the torque. 549 00:37:03,040 --> 00:37:05,940 So I write it like that. 550 00:37:05,940 --> 00:37:12,460 So this is the dynamic equation, which tells you 551 00:37:12,460 --> 00:37:17,530 how the motion of this mass changes with time. 552 00:37:17,530 --> 00:37:18,860 All right. 553 00:37:18,860 --> 00:37:26,460 So alpha is theta double dot of t. 554 00:37:26,460 --> 00:37:28,240 OK. 555 00:37:28,240 --> 00:37:32,120 What is the torque at that instant of time? 556 00:37:32,120 --> 00:37:42,860 Well, you know general torque is r cross f, all right. 557 00:37:42,860 --> 00:37:44,730 That's true in three dimension. 558 00:37:44,730 --> 00:37:46,590 So it will apply here. 559 00:37:46,590 --> 00:37:58,960 So the torque is going to be this force times this distance. 560 00:37:58,960 --> 00:38:00,000 OK. 561 00:38:00,000 --> 00:38:01,870 So it's going to be-- let's write it. 562 00:38:01,870 --> 00:38:15,030 The force is mg right, times l over 2 sine theta, 563 00:38:15,030 --> 00:38:16,160 theta of time t. 564 00:38:16,160 --> 00:38:18,800 OK. 565 00:38:18,800 --> 00:38:27,300 That's the torque about this axis p on this rod, all right. 566 00:38:27,300 --> 00:38:35,010 And it's divided by I, where I is the moment of inertia 567 00:38:35,010 --> 00:38:39,100 of this rod about an axis through p 568 00:38:39,100 --> 00:38:41,670 perpendicular to the board. 569 00:38:41,670 --> 00:38:43,020 OK. 570 00:38:43,020 --> 00:38:45,990 Now we need to calculate the moment, 571 00:38:45,990 --> 00:38:49,650 in order to continue further, we need to calculate I. 572 00:38:49,650 --> 00:38:52,500 Since we know this mass of the rod. 573 00:38:52,500 --> 00:38:54,560 And we know it's a uniform rod. 574 00:38:54,560 --> 00:38:59,280 And we know it's length l, we can calculate it. 575 00:38:59,280 --> 00:39:00,490 You know how to do it. 576 00:39:00,490 --> 00:39:04,440 If you don't, you can look it up in the book on mechanics, 577 00:39:04,440 --> 00:39:05,150 all right. 578 00:39:05,150 --> 00:39:07,950 Or just look up the moments of inertia. 579 00:39:07,950 --> 00:39:16,060 And you will find that the moment of inertia, 580 00:39:16,060 --> 00:39:22,600 you will find that the moment of inertia I for a rod like that 581 00:39:22,600 --> 00:39:29,700 is 1/3 the mass times the length squared. 582 00:39:29,700 --> 00:39:30,930 OK. 583 00:39:30,930 --> 00:39:32,330 So now I have to continue. 584 00:39:32,330 --> 00:39:35,190 But I've run out of board space. 585 00:39:35,190 --> 00:39:38,860 So I'm going to erase the board at the far end. 586 00:39:38,860 --> 00:39:42,730 And we'll continue from there. 587 00:39:42,730 --> 00:39:45,440 So I erased the board. 588 00:39:45,440 --> 00:39:48,930 And then so that you don't have to look backwards and forwards, 589 00:39:48,930 --> 00:39:53,470 I've started rewriting it and I realized that I actually 590 00:39:53,470 --> 00:39:55,047 missed the negative sign. 591 00:39:55,047 --> 00:39:56,380 So I'm going to correct it here. 592 00:39:56,380 --> 00:39:58,710 So that's why it's completely written out. 593 00:39:58,710 --> 00:40:01,650 So let me just remind you. 594 00:40:01,650 --> 00:40:05,040 The situation we have is this rod, 595 00:40:05,040 --> 00:40:07,930 which at time t, we define this angle 596 00:40:07,930 --> 00:40:10,410 to be theta t, the rotation of the rod. 597 00:40:10,410 --> 00:40:13,230 It has an acceleration, theta double dot t. 598 00:40:13,230 --> 00:40:18,090 And we are considering rotations about an axis perpendicular 599 00:40:18,090 --> 00:40:20,540 to the board through this point here. 600 00:40:20,540 --> 00:40:21,450 OK. 601 00:40:21,450 --> 00:40:24,540 We know, that was the last thing we did, 602 00:40:24,540 --> 00:40:29,780 that the acceleration is given by the torque divided 603 00:40:29,780 --> 00:40:31,430 by the moment of inertia. 604 00:40:31,430 --> 00:40:33,330 All right. 605 00:40:33,330 --> 00:40:38,450 The torque is mg, l over 2 sine theta, I derived it 606 00:40:38,450 --> 00:40:40,790 for you before, divided by I. 607 00:40:40,790 --> 00:40:44,430 But what I neglected to put a negative sign. 608 00:40:44,430 --> 00:40:46,760 And that you could do in your head, right. 609 00:40:46,760 --> 00:40:53,040 Consider we've taken all the rotations to be positive 610 00:40:53,040 --> 00:40:55,050 if they're anti-clockwise. 611 00:40:55,050 --> 00:40:58,210 So this angle is a positive rotation. 612 00:40:58,210 --> 00:41:02,420 This would be, this direction would be a positive rotation. 613 00:41:02,420 --> 00:41:05,980 But the torque if you look at this, 614 00:41:05,980 --> 00:41:08,270 there is a force acting down on this. 615 00:41:08,270 --> 00:41:11,970 So about this point, it's trying to rotate 616 00:41:11,970 --> 00:41:14,380 this in the clockwise direction. 617 00:41:14,380 --> 00:41:15,580 And so it's minus. 618 00:41:15,580 --> 00:41:19,020 And I didn't-- it would have naturally come out if I did 619 00:41:19,020 --> 00:41:25,912 the full vector calculation, the torque is R times F. 620 00:41:25,912 --> 00:41:28,120 It would have come out, the sign would have come out. 621 00:41:28,120 --> 00:41:30,450 So that's where this minus sign comes in. 622 00:41:30,450 --> 00:41:33,480 OK, so this is where we got on the board over there. 623 00:41:33,480 --> 00:41:35,720 And now let's continue. 624 00:41:35,720 --> 00:41:38,810 We can replace I from here. 625 00:41:38,810 --> 00:41:45,530 And we get that theta double dot of t 626 00:41:45,530 --> 00:42:00,380 is equal to minus, all right, 3 halves, 3 l, 3 halves, not l. 627 00:42:00,380 --> 00:42:01,700 It's divided by 2. 628 00:42:01,700 --> 00:42:02,800 l is at the bottom. 629 00:42:02,800 --> 00:42:04,060 Sorry. 630 00:42:04,060 --> 00:42:13,936 3 halves g over l times sine theta, theta of t. 631 00:42:13,936 --> 00:42:14,436 OK. 632 00:42:17,540 --> 00:42:20,770 I'm sorry. 633 00:42:20,770 --> 00:42:25,770 Sine theta of t. 634 00:42:25,770 --> 00:42:33,160 OK, as before, to simplify it, I will write omega, I'll define. 635 00:42:33,160 --> 00:42:47,880 Let's define omega 0 squared to be equal to 3 halves g over l. 636 00:42:47,880 --> 00:42:53,760 With this definition, we get that theta double 637 00:42:53,760 --> 00:43:05,750 dot t is equal to minus omega 0 squared sine theta of t. 638 00:43:05,750 --> 00:43:06,770 OK. 639 00:43:06,770 --> 00:43:13,680 So this is our equation of motion for this problem. 640 00:43:13,680 --> 00:43:16,350 That's the equation of motion. 641 00:43:16,350 --> 00:43:20,140 And these are the boundary conditions. 642 00:43:20,140 --> 00:43:25,670 So these three equations are a translation 643 00:43:25,670 --> 00:43:28,770 of this problem in the language of mathematics. 644 00:43:33,270 --> 00:43:37,910 If we now want to predict what will happen 645 00:43:37,910 --> 00:43:41,250 to this rod at some other time, we 646 00:43:41,250 --> 00:43:43,380 have to solve these equations. 647 00:43:43,380 --> 00:43:47,710 And admit now, I have a problem. 648 00:43:47,710 --> 00:43:52,140 If you remember, when we did it for the spring, 649 00:43:52,140 --> 00:43:58,310 the equation of motion was one where I guessed the answer. 650 00:43:58,310 --> 00:44:04,800 I don't know what the answer is of this. 651 00:44:04,800 --> 00:44:07,520 If you go into books, you will find 652 00:44:07,520 --> 00:44:11,590 that this is not one of the differential equations 653 00:44:11,590 --> 00:44:13,800 which you can analytically solve. 654 00:44:13,800 --> 00:44:16,650 It's, in fact, a second order differential equations 655 00:44:16,650 --> 00:44:19,700 with transcendental functions in it. 656 00:44:19,700 --> 00:44:23,870 So this is not something we know the answer to. 657 00:44:23,870 --> 00:44:28,530 So the only thing if I want to now predict what will happen, 658 00:44:28,530 --> 00:44:31,660 I have to numerically solve this. 659 00:44:31,660 --> 00:44:35,160 And then I can-- I have enough information. 660 00:44:35,160 --> 00:44:39,390 I can numerically solve this equation with these boundary 661 00:44:39,390 --> 00:44:41,980 conditions and predict what will happen. 662 00:44:41,980 --> 00:44:44,840 That's not very instructive for the purpose of course 663 00:44:44,840 --> 00:44:46,120 at the moment. 664 00:44:46,120 --> 00:44:48,300 So let me do something else. 665 00:44:48,300 --> 00:44:50,930 OK, let me modify the problem. 666 00:44:50,930 --> 00:44:55,280 Rather than take the problem we took, let me say, 667 00:44:55,280 --> 00:44:59,420 how about if I took this rod and gave it 668 00:44:59,420 --> 00:45:01,720 only a very tiny impulse. 669 00:45:01,720 --> 00:45:04,460 So this angle is small. 670 00:45:04,460 --> 00:45:11,200 Let me make the angles sufficiently small, such 671 00:45:11,200 --> 00:45:18,850 that sine theta of t is always approximately 672 00:45:18,850 --> 00:45:24,200 equal to theta of t. 673 00:45:24,200 --> 00:45:27,690 Depends how well you want to approximate this. 674 00:45:27,690 --> 00:45:31,580 But typically, if you use your calculator or computer, 675 00:45:31,580 --> 00:45:36,000 up to about 10 degrees, that approximation is pretty good. 676 00:45:36,000 --> 00:45:40,490 So I will now change my problem. 677 00:45:40,490 --> 00:45:44,400 And I said OK, let's see whether we can predict analytically 678 00:45:44,400 --> 00:45:47,990 the motion of the rod where I give the impulse, which 679 00:45:47,990 --> 00:45:52,280 is sufficiently small, that this angle is always small. 680 00:45:52,280 --> 00:45:58,830 Under those conditions, note that my equation of motion 681 00:45:58,830 --> 00:46:09,430 becomes theta double dot of t is equal to minus omega squared 682 00:46:09,430 --> 00:46:12,970 times theta at t. 683 00:46:12,970 --> 00:46:17,875 Because sine theta t is always approximately equal to theta t 684 00:46:17,875 --> 00:46:20,070 if I take the angle small enough. 685 00:46:26,800 --> 00:46:30,180 And eureka, I can solve that one. 686 00:46:30,180 --> 00:46:32,705 Because that's exactly the same equation we solved before. 687 00:46:43,880 --> 00:46:46,910 OK, so we get the solution to that equation, 688 00:46:46,910 --> 00:46:58,320 is theta of t is some constant cosine omega 0 t plus phi. 689 00:47:02,530 --> 00:47:07,580 As before, A and phi are some arbitrary constants. 690 00:47:07,580 --> 00:47:11,360 And clearly, if it worked over there for that same equation, 691 00:47:11,360 --> 00:47:12,040 it works here. 692 00:47:12,040 --> 00:47:16,760 The only difference here is we have theta of t instead y of t. 693 00:47:16,760 --> 00:47:19,270 That's just different symbols, but the solution 694 00:47:19,270 --> 00:47:21,960 is exactly the same. 695 00:47:21,960 --> 00:47:25,070 So we know that's the solution of this equation. 696 00:47:25,070 --> 00:47:27,159 We know the boundary conditions. 697 00:47:27,159 --> 00:47:28,950 Therefore, we can predict what will happen. 698 00:47:28,950 --> 00:47:30,700 Let's continue and do that. 699 00:47:30,700 --> 00:47:38,630 So from here, you get theta dot of t is equal minus omega 0 700 00:47:38,630 --> 00:47:46,890 A sine omega 0 t plus phase. 701 00:47:46,890 --> 00:47:48,500 OK. 702 00:47:48,500 --> 00:47:52,610 And we have to put in the boundary conditions. 703 00:47:52,610 --> 00:47:54,280 OK. 704 00:47:54,280 --> 00:48:06,790 Now at t equals 0, OK, we get that this is at t equals 0. 705 00:48:06,790 --> 00:48:13,000 So at t equals 0, we get theta of t 0 706 00:48:13,000 --> 00:48:22,130 is equal to A cosine phi. 707 00:48:22,130 --> 00:48:22,860 OK. 708 00:48:22,860 --> 00:48:30,810 Therefore, phi is pi over 2. 709 00:48:30,810 --> 00:48:32,460 That's a possible value of phi. 710 00:48:35,160 --> 00:48:43,640 Now that gives me that if I is pi over 2 here, 711 00:48:43,640 --> 00:48:55,050 we get to that theta dot of t, which is equal to the angular, 712 00:48:55,050 --> 00:49:04,400 angular velocity at t equal to 0, all right, 713 00:49:04,400 --> 00:49:23,500 will be equal to minus omega 0 A sine omega ) t plus pi over 2. 714 00:49:26,990 --> 00:49:31,810 OK, from which I can get that A is 715 00:49:31,810 --> 00:49:33,860 angular velocity over t plus 0. 716 00:49:33,860 --> 00:49:39,400 And so my final solution is that theta of t 717 00:49:39,400 --> 00:49:52,210 is equal to angular velocity at t equal 0 718 00:49:52,210 --> 00:50:00,230 divided omega 0 sine, sine omega O t. 719 00:50:03,080 --> 00:50:03,580 OK. 720 00:50:03,580 --> 00:50:06,300 And I want to make sure I'm not making a sign mistake again. 721 00:50:06,300 --> 00:50:07,510 I'm not. 722 00:50:07,510 --> 00:50:08,380 All right. 723 00:50:08,380 --> 00:50:12,990 And so, and omega 0 we know, and so the 724 00:50:12,990 --> 00:50:16,640 in terms of knowing quantities, the answer 725 00:50:16,640 --> 00:50:25,340 is angular velocity of t equals 0 over. 726 00:50:25,340 --> 00:50:30,210 And omega 0, we have found defined to be that, 727 00:50:30,210 --> 00:50:39,590 so the square root of 3 g over 2 l times 728 00:50:39,590 --> 00:50:50,090 sine square root 3g over 2l times t. 729 00:50:50,090 --> 00:50:54,150 Now this is theta t. 730 00:50:54,150 --> 00:50:56,780 So we have completely solved the problem. 731 00:50:56,780 --> 00:50:58,550 And we have predicted the motion. 732 00:50:58,550 --> 00:51:04,040 So as before, following this process 733 00:51:04,040 --> 00:51:07,300 of taking the physical situation, 734 00:51:07,300 --> 00:51:10,010 describing it in terms of mathematics, 735 00:51:10,010 --> 00:51:12,600 solving the mathematical equations, 736 00:51:12,600 --> 00:51:14,380 including all the information we have 737 00:51:14,380 --> 00:51:16,470 about the problem, the boundary-- 738 00:51:16,470 --> 00:51:19,590 initial conditions or boundary conditions, 739 00:51:19,590 --> 00:51:24,500 we can predict what will happen to this angle 740 00:51:24,500 --> 00:51:28,670 as a function of time, and also the kind of motion 741 00:51:28,670 --> 00:51:31,150 this is an accelerating motion. 742 00:51:31,150 --> 00:51:35,140 I can also predict, as before, that the period of this 743 00:51:35,140 --> 00:51:43,570 will be 2 pi, 2 pi square root of 2l over 3g, et cetera. 744 00:51:46,720 --> 00:51:49,350 OK. 745 00:51:49,350 --> 00:52:01,750 Now, one of the things you'll notice, that in some ways, 746 00:52:01,750 --> 00:52:03,650 it seems I'm repeating myself. 747 00:52:06,760 --> 00:52:09,520 We took completely different situations, 748 00:52:09,520 --> 00:52:13,490 and yet the result, the equations of motion, 749 00:52:13,490 --> 00:52:17,520 and the results, have good very similar form. 750 00:52:17,520 --> 00:52:22,470 Now this is part of the beauty of the scientific method. 751 00:52:22,470 --> 00:52:29,080 Because it turns out that very many 752 00:52:29,080 --> 00:52:32,940 different physical situations can 753 00:52:32,940 --> 00:52:37,180 be described by the same mathematical equations. 754 00:52:37,180 --> 00:52:43,470 So once you've solved the problem for one 755 00:52:43,470 --> 00:52:48,620 physical situation, you have automatically have solved it 756 00:52:48,620 --> 00:52:53,340 for an almost infinite number of other situations which 757 00:52:53,340 --> 00:52:57,510 are described by this same mathematics. 758 00:52:57,510 --> 00:53:01,790 Finally, let me do just more as a question of practice, 759 00:53:01,790 --> 00:53:06,050 one more problem of this kind that apparently 760 00:53:06,050 --> 00:53:07,630 seems to be completely different. 761 00:53:07,630 --> 00:53:10,485 I'll take a problem from electricity and magnetism. 762 00:53:14,960 --> 00:53:17,170 Let me consider the following situation. 763 00:53:25,120 --> 00:53:29,620 So now we're going to a different problem. 764 00:53:29,620 --> 00:53:34,540 The physical situation is suppose I have two plates, two 765 00:53:34,540 --> 00:53:38,830 metal plates, and I connect them with a wire. 766 00:53:38,830 --> 00:53:46,660 Schematically, it consists of a capacitor C 767 00:53:46,660 --> 00:53:49,507 connected to an inductor. 768 00:53:53,620 --> 00:53:56,320 This is a schematic representation 769 00:53:56,320 --> 00:54:00,090 of two parallel plates connected, short 770 00:54:00,090 --> 00:54:03,860 circuited by a wire. 771 00:54:03,860 --> 00:54:10,150 I will assume for simplicity here that these wires have 772 00:54:10,150 --> 00:54:14,830 no resistance, superconducting, all right. 773 00:54:14,830 --> 00:54:21,370 Any loop like has an inductant L. 774 00:54:21,370 --> 00:54:24,870 And the capacity between these is C. 775 00:54:24,870 --> 00:54:27,600 So this is an L C circuit. 776 00:54:27,600 --> 00:54:32,110 And I'm going to assume that at time equal 0, so this is now 777 00:54:32,110 --> 00:54:41,520 time equals 0, I have a charge here, minus Q0 plus Q0 778 00:54:41,520 --> 00:54:44,580 here, OK. 779 00:54:44,580 --> 00:54:47,970 And let's assume that at time there's even a current flowing, 780 00:54:47,970 --> 00:54:50,500 so I is 0 here. 781 00:54:54,420 --> 00:54:59,780 So this is a system which is disturbed from equilibrium. 782 00:54:59,780 --> 00:55:02,907 And what will happen is a function of time. 783 00:55:05,590 --> 00:55:10,990 I will do the same and almost boring you to tears, 784 00:55:10,990 --> 00:55:13,230 I'm going to, you'll see I'm essentially 785 00:55:13,230 --> 00:55:15,790 doing the same problem again. 786 00:55:15,790 --> 00:55:23,520 I will now consider this circuit at some arbitrary time t, 787 00:55:23,520 --> 00:55:29,420 derive the equation of motion for the charges in the current, 788 00:55:29,420 --> 00:55:32,640 therefore translate this physical situation, 789 00:55:32,640 --> 00:55:34,430 or describe this physical situation 790 00:55:34,430 --> 00:55:37,560 in terms of mathematics, deriving 791 00:55:37,560 --> 00:55:40,960 mathematical equations, solve them, 792 00:55:40,960 --> 00:55:42,680 and predict what will happen. 793 00:55:42,680 --> 00:55:46,370 So I just follow what I just did a second ago. 794 00:55:46,370 --> 00:55:55,590 So at some instant of time, that same circuit L 795 00:55:55,590 --> 00:56:00,530 will have some current I of t to the charge 796 00:56:00,530 --> 00:56:06,840 minus Q of t plus Q of t, all right. 797 00:56:06,840 --> 00:56:07,960 This at time t. 798 00:56:12,830 --> 00:56:19,170 So from this, I can derive the equation of motion. 799 00:56:22,500 --> 00:56:32,340 Let me remind you about Faraday's law. 800 00:56:32,340 --> 00:56:41,290 You know that if you have current coil in the loop, 801 00:56:41,290 --> 00:56:48,780 it produces magnetic flux in that loop. 802 00:56:48,780 --> 00:56:54,000 The changing flux gives rise to an EMF around that loop. 803 00:56:54,000 --> 00:56:57,130 To be specific, Faraday's law I can write. 804 00:56:57,130 --> 00:57:06,050 If I take this circuit of the wire, the integral of E 805 00:57:06,050 --> 00:57:16,310 dot dl around a closed loop, that is equal to minus du phi. 806 00:57:16,310 --> 00:57:17,540 5. 807 00:57:17,540 --> 00:57:18,980 Now watch out. 808 00:57:18,980 --> 00:57:21,830 The Greek alphabet has a limited number of letters 809 00:57:21,830 --> 00:57:27,080 so you'll find one constant is reusing the letters. 810 00:57:27,080 --> 00:57:29,020 But at the moment not to confuse you, 811 00:57:29,020 --> 00:57:32,730 I'm going to put here magnetic flux, 812 00:57:32,730 --> 00:57:36,425 total magnetic, total magnetic. 813 00:57:41,050 --> 00:57:44,970 So phi is the total magnetic flux 814 00:57:44,970 --> 00:57:49,416 linking this circuit, all right, dt. 815 00:58:00,750 --> 00:58:05,285 So Faraday's law tells us that the integral of ED 816 00:58:05,285 --> 00:58:09,480 all around this loop will be equal two 817 00:58:09,480 --> 00:58:12,320 minus the rate of change magnetic flux. 818 00:58:12,320 --> 00:58:17,100 This is the dynamic equation which 819 00:58:17,100 --> 00:58:18,930 tells you how this behaves. 820 00:58:18,930 --> 00:58:24,130 It is the analogous to Newton's law f equals ma 821 00:58:24,130 --> 00:58:30,910 in the case of our mass, or 2 torque is I 822 00:58:30,910 --> 00:58:33,290 alpha in the case of rotations, et cetera. 823 00:58:33,290 --> 00:58:35,570 This is the non dynamic equations. 824 00:58:35,570 --> 00:58:38,220 So let me calculate this. 825 00:58:38,220 --> 00:58:41,940 And now I'm going through the-- around this. 826 00:58:41,940 --> 00:58:46,800 And you find since this wire I'm assuming is superconducting, 827 00:58:46,800 --> 00:58:49,600 there can be no electric field inside it. 828 00:58:49,600 --> 00:58:52,880 So the contribution to this line integral 829 00:58:52,880 --> 00:58:56,040 is 0 when I go through the wire. 830 00:58:56,040 --> 00:58:59,780 So the only place where this line integral is non-zero 831 00:58:59,780 --> 00:59:02,390 is between the plates, all right. 832 00:59:02,390 --> 00:59:04,910 And that is simply the potential difference 833 00:59:04,910 --> 00:59:09,560 between those, which is q over c. 834 00:59:09,560 --> 00:59:10,170 OK. 835 00:59:10,170 --> 00:59:17,700 Q at time t over c, where c is the capacitance is. 836 00:59:17,700 --> 00:59:22,360 That is the integral of EDL around that loop. 837 00:59:22,360 --> 00:59:39,090 And that's going to be equal to minus D dI at time t dt. 838 00:59:39,090 --> 00:59:39,960 All right. 839 00:59:39,960 --> 00:59:44,950 Because the magnetic flux, this is 840 00:59:44,950 --> 00:59:50,510 by definition of the inductance or first inductance 841 00:59:50,510 --> 00:59:55,230 is that the total flux linking the circuit when 842 00:59:55,230 --> 00:59:59,660 the current flowing in it is I of I, the total flux 843 00:59:59,660 --> 01:00:04,870 is L times I. Here, I have the rate of change of that flux, 844 01:00:04,870 --> 01:00:07,140 so it's equal to this. 845 01:00:07,140 --> 01:00:08,310 OK. 846 01:00:08,310 --> 01:00:13,330 Now we know by charge conservation, 847 01:00:13,330 --> 01:00:15,538 that the current I of t. 848 01:00:19,130 --> 01:00:20,040 What is the current? 849 01:00:20,040 --> 01:00:22,130 It's the charge is flowing per second 850 01:00:22,130 --> 01:00:28,010 will be equal to the number of charges per second that arrive 851 01:00:28,010 --> 01:00:39,160 at this plate here or the part from there is equal to dQ dt. 852 01:00:39,160 --> 01:00:39,800 OK. 853 01:00:39,800 --> 01:00:42,480 Or in other words, Q dot. 854 01:00:46,290 --> 01:00:47,415 OK, let me continue. 855 01:00:57,120 --> 01:01:04,390 So from these two equations, right, the dIdt therefore, 856 01:01:04,390 --> 01:01:07,970 this is the second thing, so I end up from there 857 01:01:07,970 --> 01:01:13,770 that Q double dot, second derivative of t, 858 01:01:13,770 --> 01:01:21,895 is equal to minus right 1 over LC times Q of t. 859 01:01:24,750 --> 01:01:25,900 Eureka. 860 01:01:25,900 --> 01:01:28,510 We have once again the same equation. 861 01:01:28,510 --> 01:01:32,410 This I can define as before, omega O squared. 862 01:01:32,410 --> 01:01:37,900 If I define that as one over LC, all right, 863 01:01:37,900 --> 01:01:45,730 then what I have here is Q of t double dot is equal to minus 864 01:01:45,730 --> 01:01:48,720 omega ) squared Q of t. 865 01:01:52,060 --> 01:01:55,830 Again, we have come to the same equation. 866 01:01:55,830 --> 01:01:59,440 This is the same equation of motion 867 01:01:59,440 --> 01:02:02,570 as we came in the other two situations. 868 01:02:02,570 --> 01:02:05,360 So the answer will be the same. 869 01:02:05,360 --> 01:02:07,930 The variables will be different here. 870 01:02:07,930 --> 01:02:12,770 It'll be the charge that will be changing with time, 871 01:02:12,770 --> 01:02:15,250 while there in the one case was the angle. 872 01:02:15,250 --> 01:02:21,000 In the other case was the position of the mass. 873 01:02:21,000 --> 01:02:24,250 OK, and the solution to this problem, 874 01:02:24,250 --> 01:02:29,100 I can now write immediately, is Q of t 875 01:02:29,100 --> 01:02:36,040 is A cosine omega 0 t plus fe. 876 01:02:36,040 --> 01:02:40,110 Note that this is the fe is nothing to do with that phi. 877 01:02:40,110 --> 01:02:40,880 OK. 878 01:02:40,880 --> 01:02:50,710 And Q dot of t, which by the way is I of t, 879 01:02:50,710 --> 01:03:02,330 is equal to minus omega o A sine omega o t plus phi. 880 01:03:02,330 --> 01:03:03,550 OK. 881 01:03:03,550 --> 01:03:09,220 Now as before, what actually happens 882 01:03:09,220 --> 01:03:14,180 depends on the initial conditions. 883 01:03:14,180 --> 01:03:19,640 And we look at that picture on the top board. 884 01:03:19,640 --> 01:03:23,980 We know that initially Q is Q 0. 885 01:03:23,980 --> 01:03:28,570 And we know that initially Q dot t is I0. 886 01:03:28,570 --> 01:03:30,800 To save time, I'll just immediately write. 887 01:03:30,800 --> 01:03:36,040 You can do that in your head. 888 01:03:36,040 --> 01:03:41,150 And if you the write that out, you 889 01:03:41,150 --> 01:03:48,120 find that if you've used those two conditions, 890 01:03:48,120 --> 01:03:59,660 you find that time phi this time is equal to minus IO over 891 01:03:59,660 --> 01:04:08,660 Q0 omega 0 and A is equal to Q0 over cosine phi. 892 01:04:11,470 --> 01:04:15,090 I saved time without just solving algebraic equations. 893 01:04:15,090 --> 01:04:17,680 Take these two equations. 894 01:04:17,680 --> 01:04:22,590 Consider t equals 0, the values of those quantities, 895 01:04:22,590 --> 01:04:25,810 and just solve for the two unknowns and you get this. 896 01:04:25,810 --> 01:04:30,790 And so once again, we have predicted what will happen. 897 01:04:30,790 --> 01:04:35,420 And what I would like to just at this stage 898 01:04:35,420 --> 01:04:40,360 emphasize that although we have taken three 899 01:04:40,360 --> 01:04:45,430 different physical situations, in each case, 900 01:04:45,430 --> 01:04:52,590 we took the system, displaced it from equilibrium, let go 901 01:04:52,590 --> 01:04:56,110 and we wanted to see what will happen. 902 01:04:56,110 --> 01:04:59,770 In all the cases, it turned out that 903 01:04:59,770 --> 01:05:03,450 the mathematical description, the mathematical equations 904 01:05:03,450 --> 01:05:05,730 are identical in form. 905 01:05:05,730 --> 01:05:08,890 And so they gave, not surprisingly, 906 01:05:08,890 --> 01:05:12,040 the same kind of motion. 907 01:05:12,040 --> 01:05:15,550 This motion that we see in all those cases, 908 01:05:15,550 --> 01:05:17,980 we called simple harmonic motion. 909 01:05:21,260 --> 01:05:24,230 It has the characteristic that if you displace the system 910 01:05:24,230 --> 01:05:29,830 from equilibrium, it oscillates with harmonic motion, 911 01:05:29,830 --> 01:05:34,180 meaning it oscillates as sine or a cosine 912 01:05:34,180 --> 01:05:37,110 of different phases, et cetera. 913 01:05:37,110 --> 01:05:40,700 If you tell me any one of these systems 914 01:05:40,700 --> 01:05:43,060 where it was at any instant of time, 915 01:05:43,060 --> 01:05:45,880 I can predict it forever in the future. 916 01:05:48,910 --> 01:05:51,545 Now finally, the last few minutes. 917 01:05:55,810 --> 01:06:00,020 Some of you may have noticed that in each of these-- 918 01:06:00,020 --> 01:06:01,940 or I told you at the beginning that I 919 01:06:01,940 --> 01:06:08,530 can take a physical situation and describe it 920 01:06:08,530 --> 01:06:13,730 in terms of mathematics, and thereby predict the future. 921 01:06:13,730 --> 01:06:18,630 But in each case, I in some ways almost cheated. 922 01:06:18,630 --> 01:06:21,880 I said let's consider an ideal spring. 923 01:06:21,880 --> 01:06:25,680 We'll assume it has no mass, that it exactly 924 01:06:25,680 --> 01:06:27,880 obeys Hooke's law. 925 01:06:27,880 --> 01:06:31,810 Or when it came to that rod, I assumed 926 01:06:31,810 --> 01:06:36,800 that it's only displaced by small amounts, 927 01:06:36,800 --> 01:06:44,930 so that's sine theta equals-- I can approximate with theta. 928 01:06:44,930 --> 01:06:48,650 In the case of the electrical circuit, 929 01:06:48,650 --> 01:06:51,820 it maybe not so obvious what I assumed, 930 01:06:51,820 --> 01:06:56,610 but I certainly made assumptions about that the wire is 931 01:06:56,610 --> 01:06:58,930 perfectly conducting, et cetera. 932 01:06:58,930 --> 01:07:02,750 And I didn't discuss in detail what 933 01:07:02,750 --> 01:07:04,800 happens in between the plates or the capacitor 934 01:07:04,800 --> 01:07:06,204 where the fields are, et cetera. 935 01:07:06,204 --> 01:07:07,370 One is doing approximations. 936 01:07:10,190 --> 01:07:14,510 In reality, if you look at any physical situation 937 01:07:14,510 --> 01:07:19,150 in the world, it's always incredibly complicated. 938 01:07:23,340 --> 01:07:27,990 It's never that you have an idealized situation like this. 939 01:07:27,990 --> 01:07:32,570 So to what extent does what we have just done 940 01:07:32,570 --> 01:07:35,660 correspond-- is it useful at all. 941 01:07:35,660 --> 01:07:40,930 And the way I'll answer it is by another example. 942 01:07:40,930 --> 01:07:45,040 This is the last thing I'll do on the top of a simple harmonic 943 01:07:45,040 --> 01:07:47,520 motion, last problem. 944 01:07:47,520 --> 01:07:51,290 Suppose I'm looking out of the window. 945 01:07:51,290 --> 01:08:00,110 And I see there is a tree and a branch and a bird lands on it. 946 01:08:00,110 --> 01:08:04,740 Do I understand what will happen? 947 01:08:04,740 --> 01:08:07,576 It's clearly extremely complicated. 948 01:08:07,576 --> 01:08:10,800 The mechanics of the branch is complicated. 949 01:08:10,800 --> 01:08:13,240 There is air friction. 950 01:08:13,240 --> 01:08:15,460 Nothing is simple about it. 951 01:08:15,460 --> 01:08:19,685 And yet you and I can predict what will happen. 952 01:08:19,685 --> 01:08:21,850 You know what will happen. 953 01:08:21,850 --> 01:08:27,520 As the bird lands, it'll start oscillating and finally 954 01:08:27,520 --> 01:08:34,830 come to rest, very much like harmonic motion. 955 01:08:34,830 --> 01:08:41,120 I claim I can use the word, I understand what's going on. 956 01:08:41,120 --> 01:08:44,149 And the reason why I claim that is the following. 957 01:08:44,149 --> 01:08:49,439 That to understand something, all 958 01:08:49,439 --> 01:08:54,160 I would like to understand the general features of what's 959 01:08:54,160 --> 01:08:54,680 going on. 960 01:08:54,680 --> 01:08:58,350 I don't need to know what every atom in the branch 961 01:08:58,350 --> 01:09:02,020 is going on in the process of trying 962 01:09:02,020 --> 01:09:04,410 to understand what the bird is doing. 963 01:09:04,410 --> 01:09:06,990 If I want to understand what atoms are doing, 964 01:09:06,990 --> 01:09:09,979 that's a different story. 965 01:09:09,979 --> 01:09:20,330 And so one of the important abilities we have to develop 966 01:09:20,330 --> 01:09:26,890 is to be able to, when you see some situation, 967 01:09:26,890 --> 01:09:33,120 model it in terms of the most important aspects 968 01:09:33,120 --> 01:09:35,779 of the situation. 969 01:09:35,779 --> 01:09:38,000 And let me be concrete. 970 01:09:38,000 --> 01:09:45,080 In this case, I can say look, I can model this approximately 971 01:09:45,080 --> 01:09:49,460 as the branch I'll treat as a spring, 972 01:09:49,460 --> 01:09:51,500 of some spring constant k. 973 01:09:51,500 --> 01:09:54,430 The bird I'm going to treat as a mass m. 974 01:09:54,430 --> 01:09:57,940 And I'm going to consider this situation 975 01:09:57,940 --> 01:10:07,210 to be modeled by a mass being placed on a spring and let go. 976 01:10:07,210 --> 01:10:09,580 Now is that going to be exactly this? 977 01:10:09,580 --> 01:10:11,330 No. 978 01:10:11,330 --> 01:10:15,200 But from the point of view of understanding 979 01:10:15,200 --> 01:10:18,470 the general features of this, it will 980 01:10:18,470 --> 01:10:23,310 be a reasonable approximation. 981 01:10:23,310 --> 01:10:28,080 Now how can I check, this is the scientific method, 982 01:10:28,080 --> 01:10:33,350 that this is a good approximation is the following. 983 01:10:33,350 --> 01:10:35,950 Make a prediction. 984 01:10:35,950 --> 01:10:41,160 Suppose when I see the bird landing, 985 01:10:41,160 --> 01:10:52,760 it makes five oscillations, five oscillations in ten seconds. 986 01:10:55,380 --> 01:10:57,650 OK. 987 01:10:57,650 --> 01:11:03,190 I can predict approximately once the oscillations have 988 01:11:03,190 --> 01:11:10,030 died out how much the bird has compressed, distorted, 989 01:11:10,030 --> 01:11:11,230 this branch. 990 01:11:11,230 --> 01:11:15,230 In other words, from the moment it landed, 991 01:11:15,230 --> 01:11:19,550 what distance will the branch, its position, 992 01:11:19,550 --> 01:11:23,682 be lowered when it comes to rest. 993 01:11:26,990 --> 01:11:28,590 So let's model it. 994 01:11:28,590 --> 01:11:32,830 I'll model it first of all, as I did here, 995 01:11:32,830 --> 01:11:38,260 as a mass and a spring problem using-- this is now 996 01:11:38,260 --> 01:11:40,930 the same problem we did at the beginning that is still 997 01:11:40,930 --> 01:11:42,630 on the board here. 998 01:11:42,630 --> 01:11:46,890 I can calculate that for this idealized situation, 999 01:11:46,890 --> 01:11:50,240 the period will be equal to 2 pi square root m 1000 01:11:50,240 --> 01:11:54,110 over k, for this idealized model. 1001 01:11:54,110 --> 01:11:57,760 In reality, there's friction. 1002 01:11:57,760 --> 01:12:02,850 So this oscillation will be dumped out. 1003 01:12:02,850 --> 01:12:06,580 And you would have learned from Professor Walter Lewin 1004 01:12:06,580 --> 01:12:12,120 that if you have a damped oscillator, 1005 01:12:12,120 --> 01:12:20,420 the frequency of oscillations does not depend significantly 1006 01:12:20,420 --> 01:12:24,830 on how much damping it is, provided it is weak damping. 1007 01:12:24,830 --> 01:12:30,200 So I would make the assumption that the period 1008 01:12:30,200 --> 01:12:33,200 will be given by that. 1009 01:12:33,200 --> 01:12:38,750 I also know that at the time when the motion has been damped 1010 01:12:38,750 --> 01:12:44,620 out and the bird has come to rest, at that instant, 1011 01:12:44,620 --> 01:12:46,810 there's no net force on the bird. 1012 01:12:46,810 --> 01:12:49,290 And so the force of gravity on it 1013 01:12:49,290 --> 01:12:54,560 will be equal to the restoring force due to the spring. 1014 01:12:54,560 --> 01:12:57,400 So mg is equal to kl. 1015 01:12:57,400 --> 01:13:03,170 From this, I get that m over k is l over g. 1016 01:13:03,170 --> 01:13:04,680 All right. 1017 01:13:04,680 --> 01:13:07,860 But we know what the period is. 1018 01:13:07,860 --> 01:13:10,580 We said five oscillations in 10 seconds. 1019 01:13:10,580 --> 01:13:14,250 So the period is two seconds. 1020 01:13:14,250 --> 01:13:17,830 So two seconds will equal 2 pi divided 1021 01:13:17,830 --> 01:13:25,820 by but this, which is l over g, the square root of l over g. 1022 01:13:25,820 --> 01:13:29,540 Square this and calculate the one unknown l 1023 01:13:29,540 --> 01:13:33,210 and you get one meter. 1024 01:13:33,210 --> 01:13:36,050 So my prediction is that this bird 1025 01:13:36,050 --> 01:13:44,514 will, after it's settled down, roughly be lower by one meter. 1026 01:13:44,514 --> 01:13:46,430 It's certainly not going to be one millimeter. 1027 01:13:46,430 --> 01:13:48,410 It's not going to be one centimeter. 1028 01:13:48,410 --> 01:13:50,740 It's not going be 10 meters. 1029 01:13:50,740 --> 01:13:53,620 And if you go and you measure it, 1030 01:13:53,620 --> 01:13:56,920 you find this is approximately correct. 1031 01:13:56,920 --> 01:14:01,860 The fact that I can predict it is for me 1032 01:14:01,860 --> 01:14:05,550 the same as saying I understand what's going on. 1033 01:14:05,550 --> 01:14:08,270 I realize it's not exact. 1034 01:14:08,270 --> 01:14:13,070 But with the approximations that I've made, 1035 01:14:13,070 --> 01:14:17,030 I get an answer which is consistent with what 1036 01:14:17,030 --> 01:14:19,250 is observed. 1037 01:14:19,250 --> 01:14:25,690 Today I have tried to tell you what my role in this course 1038 01:14:25,690 --> 01:14:32,210 is, what I'm trying to help you learn. 1039 01:14:32,210 --> 01:14:35,910 I intentionally went very slowly. 1040 01:14:35,910 --> 01:14:39,270 I used the word gory detail. 1041 01:14:39,270 --> 01:14:42,200 I tried, in particularly in the first problem, 1042 01:14:42,200 --> 01:14:46,140 not to miss any steps. 1043 01:14:46,140 --> 01:14:51,870 And what we covered today is the phenomenon 1044 01:14:51,870 --> 01:14:54,330 of simple harmonic motion. 1045 01:14:54,330 --> 01:15:01,870 It occurs whenever you have any system which 1046 01:15:01,870 --> 01:15:07,460 is displaced from equilibrium where the restoring force is 1047 01:15:07,460 --> 01:15:11,060 proportional to the displacement. 1048 01:15:11,060 --> 01:15:13,460 And it illustrated that you could 1049 01:15:13,460 --> 01:15:18,010 have very, very different physical situations which, 1050 01:15:18,010 --> 01:15:20,990 when translated into mathematics, 1051 01:15:20,990 --> 01:15:24,840 give essentially the same problem. 1052 01:15:24,840 --> 01:15:32,760 So it's a beautiful example of the scientific method where 1053 01:15:32,760 --> 01:15:40,310 we utilize this same-- well, once we've learned it 1054 01:15:40,310 --> 01:15:46,780 for one system, we can apply the results to another system. 1055 01:15:46,780 --> 01:15:50,160 So as I said, today I did simple harmonic motion. 1056 01:15:50,160 --> 01:15:53,730 Next time, we would be considering problems 1057 01:15:53,730 --> 01:15:56,130 to do with simple harmonic motion, 1058 01:15:56,130 --> 01:15:59,560 but which includes friction, damping. 1059 01:15:59,560 --> 01:16:04,700 We'll then go on to talk about harmonic oscillators which 1060 01:16:04,700 --> 01:16:05,630 are driven. 1061 01:16:05,630 --> 01:16:07,690 So we have driven harmonic motion. 1062 01:16:07,690 --> 01:16:09,460 And gradually in the course, we'll 1063 01:16:09,460 --> 01:16:13,900 go to more and more decrease of freedom, waves, et cetera.