1 00:00:00,060 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,800 Commons license. 3 00:00:03,800 --> 00:00:06,050 Your support will help MIT OpenCourseWare 4 00:00:06,050 --> 00:00:10,140 continue to offer high-quality educational resources for free. 5 00:00:10,140 --> 00:00:12,690 To make a donation, or to view additional materials 6 00:00:12,690 --> 00:00:16,600 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,600 --> 00:00:17,297 at ocw.mit.edu. 8 00:00:20,854 --> 00:00:22,645 PROFESSOR: Let's get on with some dynamics. 9 00:00:26,490 --> 00:00:32,020 So the place I'm going to begin is just 10 00:00:32,020 --> 00:00:34,820 a comment about mechanical engineering courses. 11 00:00:34,820 --> 00:00:39,010 The first, and you may have heard this already in classes, 12 00:00:39,010 --> 00:00:42,440 you'll be taking subject 2001 if you're 13 00:00:42,440 --> 00:00:45,770 Course 2 majors through 2009, and if you're 14 00:00:45,770 --> 00:00:48,700 2-A, most of the odd ones. 15 00:00:48,700 --> 00:00:57,920 But the subjects 2001 through 2005 16 00:00:57,920 --> 00:01:00,510 are really basically engineering science subjects that are all 17 00:01:00,510 --> 00:01:02,650 foundational to mechanical engineering, 18 00:01:02,650 --> 00:01:05,780 and they all have a common or property through them. 19 00:01:05,780 --> 00:01:10,220 And that is that we make observations of the world, 20 00:01:10,220 --> 00:01:13,340 and we try to understand them. 21 00:01:13,340 --> 00:01:14,610 We pose problems. 22 00:01:14,610 --> 00:01:20,710 Why-- 400 years ago, is the sun in the center 23 00:01:20,710 --> 00:01:23,620 of the solar system or not? 24 00:01:23,620 --> 00:01:32,470 And we try to produce models that explain the problem. 25 00:01:32,470 --> 00:01:37,530 So here's the problem, the question of the day. 26 00:01:37,530 --> 00:01:40,460 We try to produce models to describe it, 27 00:01:40,460 --> 00:01:49,370 and we make observations, measurements, to see 28 00:01:49,370 --> 00:01:52,000 if our models are correct. 29 00:01:52,000 --> 00:01:56,540 And if we feed that information back into the models, 30 00:01:56,540 --> 00:02:00,160 we try out the models, we test it against more observations, 31 00:02:00,160 --> 00:02:02,419 and you go round and round. 32 00:02:02,419 --> 00:02:03,960 And this is kind of the fundamental-- 33 00:02:03,960 --> 00:02:08,630 this is the way all of these basic first five subjects use, 34 00:02:08,630 --> 00:02:14,140 basically, this method of inquiry. 35 00:02:14,140 --> 00:02:20,220 So in 2003, the way this system works, 36 00:02:20,220 --> 00:02:29,140 my kind of mental conception of this modeling process, 37 00:02:29,140 --> 00:02:30,201 is three things. 38 00:02:30,201 --> 00:02:31,200 And this applies to you. 39 00:02:31,200 --> 00:02:32,580 You have a homework problem. 40 00:02:32,580 --> 00:02:34,850 How do you attack a homework problem? 41 00:02:34,850 --> 00:02:38,690 You're going to need to describe the motion. 42 00:02:44,680 --> 00:02:48,840 You're going to need to choose the physical laws-- pick, 43 00:02:48,840 --> 00:02:51,960 I'll call it because it's short-- 44 00:02:51,960 --> 00:02:59,180 the physical law that you want to apply 45 00:02:59,180 --> 00:03:02,210 like f equals ma, conservation of energy, 46 00:03:02,210 --> 00:03:03,875 conservation of momentum. 47 00:03:03,875 --> 00:03:06,590 You got to know which physical laws to apply. 48 00:03:06,590 --> 00:03:16,370 And then finally, third you need to apply the correct math. 49 00:03:16,370 --> 00:03:18,224 And that's really-- most dynamic problems 50 00:03:18,224 --> 00:03:19,390 can be broken down this way. 51 00:03:19,390 --> 00:03:21,530 That's the way I like to conceptually break them down. 52 00:03:21,530 --> 00:03:23,300 You might have another model, but this is 53 00:03:23,300 --> 00:03:25,450 the way I'm going to teach it. 54 00:03:25,450 --> 00:03:28,210 Can you describe the motion, pick the correct physical laws 55 00:03:28,210 --> 00:03:32,630 to apply to the problem, and able to do the correct math, 56 00:03:32,630 --> 00:03:35,260 solving the equation of motion, for example. 57 00:03:35,260 --> 00:03:39,550 And all this is what fits in our models box. 58 00:03:39,550 --> 00:03:42,400 And we test it against observations and measurements 59 00:03:42,400 --> 00:03:45,970 and improve those things over time. 60 00:03:45,970 --> 00:03:50,920 So I'm going to give you-- how many of you like history? 61 00:03:50,920 --> 00:03:52,884 I find history and history technology 62 00:03:52,884 --> 00:03:54,050 kind of fun and interesting. 63 00:03:54,050 --> 00:03:55,924 So I'm going to throw a little bit of history 64 00:03:55,924 --> 00:03:58,320 into giving you a little quick course 65 00:03:58,320 --> 00:04:05,000 outline of what we're going to do in this subject this term. 66 00:04:12,300 --> 00:04:15,150 Because the history dynamics and what 67 00:04:15,150 --> 00:04:18,130 we're going to do in the course actually track one another 68 00:04:18,130 --> 00:04:20,110 remarkably closely. 69 00:04:20,110 --> 00:04:23,410 So if I ever gave you a bunch of names like Galileo, Kepler, 70 00:04:23,410 --> 00:04:29,280 Descartes, Newton, Copernicus, Euler, Lagrange and Brahe, 71 00:04:29,280 --> 00:04:30,280 which one comes first? 72 00:04:35,490 --> 00:04:38,384 Take a guess. 73 00:04:38,384 --> 00:04:40,764 AUDIENCE: Copernicus. 74 00:04:40,764 --> 00:04:41,940 PROFESSOR: Good. 75 00:04:41,940 --> 00:04:42,440 Copernicus. 76 00:04:48,710 --> 00:04:53,260 So Copernicus was Polish, and the story 77 00:04:53,260 --> 00:04:58,550 starts long before then, but in about 1,500 Copernicus 78 00:04:58,550 --> 00:05:00,618 said what? 79 00:05:00,618 --> 00:05:01,550 AUDIENCE: [INAUDIBLE] 80 00:05:04,346 --> 00:05:06,230 PROFESSOR: The sun's the center? 81 00:05:06,230 --> 00:05:07,950 AUDIENCE: [INAUDIBLE] 82 00:05:07,950 --> 00:05:09,625 PROFESSOR: Or the Earth is the center? 83 00:05:09,625 --> 00:05:10,310 AUDIENCE: [INAUDIBLE] 84 00:05:10,310 --> 00:05:11,430 PROFESSOR: Which did say? 85 00:05:11,430 --> 00:05:14,320 Yes, so Ptolemy, back around 130 AD said, 86 00:05:14,320 --> 00:05:18,050 well the Earth's the center of the solar system. 87 00:05:18,050 --> 00:05:19,720 Copernicus came along and said, nope 88 00:05:19,720 --> 00:05:23,080 I think that, in fact, the sun's the center of the solar system. 89 00:05:23,080 --> 00:05:27,042 And it for the next 100 years-- more than 100 years, couple 90 00:05:27,042 --> 00:05:29,250 hundred years-- there was a really raging controversy 91 00:05:29,250 --> 00:05:30,250 about that. 92 00:05:30,250 --> 00:05:38,520 So Copernicus, Brahe Kepler-- so I'm 93 00:05:38,520 --> 00:05:40,520 putting them in rough chronological order here. 94 00:05:40,520 --> 00:05:41,995 Now, I'm going to run out of board. 95 00:05:41,995 --> 00:05:42,710 Oh well. 96 00:05:45,370 --> 00:06:23,130 Galileo, Descartes-- I'm gonna cheat-- OK, Descartes, Newton, 97 00:06:23,130 --> 00:06:26,580 Euler, and Lagrange. 98 00:06:30,795 --> 00:06:33,420 So we're going to talk and say a little bit about each of them. 99 00:06:33,420 --> 00:06:34,955 And now that I'm-- like I told you, 100 00:06:34,955 --> 00:06:37,330 I haven't used this classroom before so I gotta learn how 101 00:06:37,330 --> 00:06:39,740 to play this game. 102 00:06:39,740 --> 00:06:41,780 I need to be able to reach this for a minute. 103 00:06:41,780 --> 00:06:44,900 So Brahe, he was along about 1,600. 104 00:06:44,900 --> 00:06:47,450 Brahe was the mathematician that wrote-- 105 00:06:47,450 --> 00:06:51,060 the imperial mathematician to the emperor in Prague. 106 00:06:51,060 --> 00:06:54,530 And he did 20 years of observations. 107 00:06:54,530 --> 00:06:57,020 And he was out to prove that the Earth was 108 00:06:57,020 --> 00:06:59,190 the center of the solar system. 109 00:06:59,190 --> 00:07:02,460 And then Kepler actually worked with him as a mathematician, 110 00:07:02,460 --> 00:07:07,250 and then took over as the imperial mathematician. 111 00:07:07,250 --> 00:07:12,160 And he took Brahe's data-- 20 years of astronomical data 112 00:07:12,160 --> 00:07:14,370 without the use of the telescope-- 113 00:07:14,370 --> 00:07:20,720 and used it come up with the three laws of planetary motion. 114 00:07:20,720 --> 00:07:26,130 And so his first and second laws were put out about 1609. 115 00:07:26,130 --> 00:07:28,650 And one of the laws is, like, equal area 116 00:07:28,650 --> 00:07:30,160 swept out in equal time. 117 00:07:30,160 --> 00:07:31,330 Have you hear that one? 118 00:07:31,330 --> 00:07:34,620 That actually turns out to be a statement of conservation 119 00:07:34,620 --> 00:07:37,100 of angular momentum, which we'll talk quite a bit 120 00:07:37,100 --> 00:07:38,960 about the course. 121 00:07:38,960 --> 00:07:43,520 Then came Galileo, and I'm not putting their birth and death 122 00:07:43,520 --> 00:07:44,020 dates here. 123 00:07:44,020 --> 00:07:45,700 I'm kind of putting in a period of time 124 00:07:45,700 --> 00:07:49,070 in which kind of important things happened around him. 125 00:07:49,070 --> 00:07:54,670 So 401 years ago a really important thing happened. 126 00:07:54,670 --> 00:08:00,460 Galileo, in 1609, turned the telescope on Jupiter, 127 00:08:00,460 --> 00:08:02,192 and saw what? 128 00:08:02,192 --> 00:08:03,104 AUDIENCE: [INAUDIBLE] 129 00:08:03,104 --> 00:08:05,420 PROFESSOR: Four moons, right? 130 00:08:05,420 --> 00:08:08,060 And then they really started having some data 131 00:08:08,060 --> 00:08:12,000 with which to really argue against the Ptolymaic view 132 00:08:12,000 --> 00:08:13,840 of the solar system. 133 00:08:16,800 --> 00:08:21,095 Descartes is an important figure to us. 134 00:08:21,095 --> 00:08:26,950 And in the period of about 1630 to 1644-- in that period 135 00:08:26,950 --> 00:08:32,809 Descartes began what is today known as analytic geometry. 136 00:08:32,809 --> 00:08:36,130 He was geometer, he studied Euclid a lot. 137 00:08:36,130 --> 00:08:40,110 But then he came up with a Cartesian coordinate system, 138 00:08:40,110 --> 00:08:43,370 xyz, and the beginnings of analytic geometry, 139 00:08:43,370 --> 00:08:47,790 which is essentially algebra, coordinates, and geometry all 140 00:08:47,790 --> 00:08:48,540 put together. 141 00:08:48,540 --> 00:08:52,170 And we are going to make great use of analytic geometry 142 00:08:52,170 --> 00:08:53,830 in this course. 143 00:08:53,830 --> 00:08:59,433 Then came Newton, kind of in his actual lifespan, 1643. 144 00:08:59,433 --> 00:09:04,070 It's kind of interesting that he spans these people. 145 00:09:04,070 --> 00:09:10,174 And in about 1666 is when he first-- the first statement 146 00:09:10,174 --> 00:09:11,340 of the three laws of motion. 147 00:09:15,370 --> 00:09:30,430 Then Euler, and he's 1707 to 1783, and that's his lifespan. 148 00:09:30,430 --> 00:09:34,780 Euler came up-- Newton never talked about angular momentum. 149 00:09:34,780 --> 00:09:36,260 He mostly talked about particles. 150 00:09:36,260 --> 00:09:40,950 Euler put Newton's three laws into mathematics. 151 00:09:40,950 --> 00:09:44,810 Euler taught us about angular momentum, 152 00:09:44,810 --> 00:09:53,140 and torque being dh dt in most cases. 153 00:09:53,140 --> 00:09:55,630 He's the most prolific mathematician all time, 154 00:09:55,630 --> 00:09:58,230 solved all sorts of important problems. 155 00:09:58,230 --> 00:10:00,410 And then finally, is Lagrange. 156 00:10:00,410 --> 00:10:08,740 And Lagrange, in about 1788, uses an energy method, energy 157 00:10:08,740 --> 00:10:17,140 and the concept of work to give us equations of motion. 158 00:10:17,140 --> 00:10:24,940 So the course, 203, stands on the shoulders 159 00:10:24,940 --> 00:10:26,140 of all these people. 160 00:10:26,140 --> 00:10:29,810 But with Descartes, we start with kinematics, really. 161 00:10:29,810 --> 00:10:33,350 This is analytic geometry. 162 00:10:33,350 --> 00:10:36,200 And that's where we're going to start today is with kinematics. 163 00:10:36,200 --> 00:10:37,840 And very soon thereafter, we're going 164 00:10:37,840 --> 00:10:40,210 to review Newton, the three laws, and what 165 00:10:40,210 --> 00:10:45,400 we call the direct method for finding equations of motion. 166 00:10:45,400 --> 00:10:48,020 Conservation of momentum, fact that 167 00:10:48,020 --> 00:10:50,070 force-- some of the forces on an object 168 00:10:50,070 --> 00:10:53,390 equals mass times acceleration, or it's a time derivative 169 00:10:53,390 --> 00:10:55,730 of its linear momentum. 170 00:10:55,730 --> 00:10:59,350 And we use that to derive equations of motion. 171 00:10:59,350 --> 00:11:05,490 So we're going to go kinematics into doing the direct method 172 00:11:05,490 --> 00:11:09,580 to getting equations of motion. 173 00:11:09,580 --> 00:11:13,750 And we go from there into angular momentum, 174 00:11:13,750 --> 00:11:18,500 and what Euler gave us-- the same thing, torque. 175 00:11:18,500 --> 00:11:21,840 We're going to do quite a lot with angular momentum. 176 00:11:21,840 --> 00:11:25,020 Because I know you know a lot about f equals ma 177 00:11:25,020 --> 00:11:28,380 and you've done lots of problems 801 applying that. 178 00:11:28,380 --> 00:11:30,960 You've done some problems on rigid body rotations. 179 00:11:30,960 --> 00:11:33,084 But I think there's a lot more you 180 00:11:33,084 --> 00:11:34,750 need to understand about this, and we'll 181 00:11:34,750 --> 00:11:36,083 spend quite a bit of time on it. 182 00:11:39,100 --> 00:11:42,280 And then near the last third the course 183 00:11:42,280 --> 00:11:47,800 we shift, because Lagrange said that if you just write down 184 00:11:47,800 --> 00:11:53,030 expressions for energy, kinetic and potential energy, 185 00:11:53,030 --> 00:11:57,340 without any consideration of Newton's laws 186 00:11:57,340 --> 00:12:01,769 and the direct method, you can derive the equations of motion. 187 00:12:01,769 --> 00:12:02,810 That's pretty remarkable. 188 00:12:02,810 --> 00:12:06,200 So there are actually two independent roots to coming up 189 00:12:06,200 --> 00:12:07,650 with equations of motion. 190 00:12:07,650 --> 00:12:10,400 And in this course, about the last third of the course, 191 00:12:10,400 --> 00:12:13,400 we're going to teach you about Lagrange. 192 00:12:13,400 --> 00:12:17,320 And then all these things are going 193 00:12:17,320 --> 00:12:20,522 to be-- one of the applications that are important engineers 194 00:12:20,522 --> 00:12:21,605 is the study of vibration. 195 00:12:25,320 --> 00:12:28,540 So we'll be looking at vibration examples 196 00:12:28,540 --> 00:12:32,250 as we go through the course, and applying 197 00:12:32,250 --> 00:12:36,090 these different methods to first, modeling, 198 00:12:36,090 --> 00:12:38,455 and then solving interesting vibration problems. 199 00:12:40,960 --> 00:12:42,980 Which brings-- ah, I have a question for you. 200 00:12:42,980 --> 00:12:46,270 So how many of you were in this classroom last May 201 00:12:46,270 --> 00:12:49,060 with Professor Haynes Miller, and I showed up one day 202 00:12:49,060 --> 00:12:51,420 and we talked about vibration? 203 00:12:51,420 --> 00:12:52,220 How many remember? 204 00:12:52,220 --> 00:12:54,480 I told you I was going to ask this question, right? 205 00:12:54,480 --> 00:12:55,000 Great. 206 00:12:55,000 --> 00:12:57,670 OK, it's good to see you here again, 207 00:12:57,670 --> 00:13:01,400 and we will talk about vibration in this course. 208 00:13:01,400 --> 00:13:03,750 So there's kind of the subject outline built 209 00:13:03,750 --> 00:13:07,350 on the shoulders of these people in history 210 00:13:07,350 --> 00:13:09,400 that made important contributions to dynamics. 211 00:13:16,020 --> 00:13:17,790 Any questions about the history? 212 00:13:17,790 --> 00:13:19,920 If you want to know, one of my TAs 213 00:13:19,920 --> 00:13:22,590 compiled a pretty neat little summary. 214 00:13:22,590 --> 00:13:24,880 Maybe I will see if I go back and find this. 215 00:13:24,880 --> 00:13:26,810 I just printed out and sent it-- how many of 216 00:13:26,810 --> 00:13:29,018 you like to know a little bit more about the history? 217 00:13:29,018 --> 00:13:31,194 These are like two liners on each person. 218 00:13:31,194 --> 00:13:31,860 Anybody want it? 219 00:13:31,860 --> 00:13:33,580 Is it worth my time to send this out? 220 00:13:33,580 --> 00:13:36,450 OK, it's kind of fun. 221 00:13:36,450 --> 00:13:45,980 So let's do an example of this modeling describing the motion, 222 00:13:45,980 --> 00:13:49,760 picking physical laws, applying the math. 223 00:13:49,760 --> 00:13:55,150 And that'll get us launched in the course. 224 00:13:55,150 --> 00:13:58,890 And we'll do it using Newton and the direct method. 225 00:14:19,510 --> 00:14:24,760 So last May, Haynes Miller and I talked about vibration. 226 00:14:24,760 --> 00:14:27,400 So I'm going to start with a vibration problem. 227 00:14:30,180 --> 00:14:31,310 And I brought one. 228 00:14:31,310 --> 00:14:35,370 So here's my couple of lead weights 229 00:14:35,370 --> 00:14:37,262 and a couple of springs. 230 00:14:37,262 --> 00:14:39,720 So really I just want to talk about-- this is the problem I 231 00:14:39,720 --> 00:14:40,960 want to talk about. 232 00:14:40,960 --> 00:14:42,560 Now you've done this problem before. 233 00:14:42,560 --> 00:14:46,480 Haynes Miller and I did it last May. 234 00:14:46,480 --> 00:14:48,870 And you've no doubt it in other classes. 235 00:14:48,870 --> 00:14:51,830 OK, it's a system which has a spring, a mass, 236 00:14:51,830 --> 00:14:54,310 it exhibits something called a natural frequency. 237 00:14:54,310 --> 00:14:58,220 But let's see what it takes to just initially begin 238 00:14:58,220 --> 00:15:03,410 to follow this modeling method to arrive 239 00:15:03,410 --> 00:15:05,160 at an equation of motion for this problem. 240 00:15:14,770 --> 00:15:17,930 So what do I mean by when I say, describe the motion? 241 00:15:17,930 --> 00:15:19,590 Really what that boils down to if you 242 00:15:19,590 --> 00:15:23,240 have to assign a coordinate system 243 00:15:23,240 --> 00:15:26,419 so that you can actually say where the object's moving. 244 00:15:26,419 --> 00:15:27,710 And I'm going to pick one here. 245 00:15:27,710 --> 00:15:30,980 So here's-- coordinate system going to be really important 246 00:15:30,980 --> 00:15:31,660 in this course. 247 00:15:34,480 --> 00:15:41,670 And I'll give us an xyz Cartesian coordinate system. 248 00:15:44,680 --> 00:15:48,250 And I'm going to try to adopt the habit, for the most part, 249 00:15:48,250 --> 00:15:51,960 during the course that this o marks this origin, 250 00:15:51,960 --> 00:15:54,340 but it also names the frame. 251 00:15:54,340 --> 00:15:56,650 So we're going to talk about things 252 00:15:56,650 --> 00:15:58,860 in that are reference frames. 253 00:15:58,860 --> 00:16:01,500 And most important one that we need 254 00:16:01,500 --> 00:16:04,080 to know about in the course is an inertial reference frame, 255 00:16:04,080 --> 00:16:06,250 and when you can use it, and when a system 256 00:16:06,250 --> 00:16:08,210 is inertial and is not. 257 00:16:08,210 --> 00:16:10,691 So I'm gonna say that this is inertial. 258 00:16:10,691 --> 00:16:11,690 It's fixed to the Earth. 259 00:16:11,690 --> 00:16:12,960 It's not moving. 260 00:16:12,960 --> 00:16:17,250 And we're going to use this coordinate x to describe 261 00:16:17,250 --> 00:16:19,110 the motion of this mass. 262 00:16:19,110 --> 00:16:31,560 And the motion is going to be-- this x is from the zero spring 263 00:16:31,560 --> 00:16:33,850 force position. 264 00:16:33,850 --> 00:16:35,440 It's actually quite important that you 265 00:16:35,440 --> 00:16:38,230 pick-- that you have to say what's 266 00:16:38,230 --> 00:16:42,580 the condition in the spring of the system when x is 0 267 00:16:42,580 --> 00:16:44,300 So we're going to say it's, when there's 268 00:16:44,300 --> 00:16:45,841 no force in the spring means it's not 269 00:16:45,841 --> 00:16:48,782 stretch, that's where 0 is. 270 00:16:48,782 --> 00:16:50,490 So we've established a coordinate system. 271 00:16:55,370 --> 00:16:57,630 Second, we need to apply physical laws. 272 00:17:07,300 --> 00:17:11,800 Now, I'm going to do this problem by f equals 273 00:17:11,800 --> 00:17:13,680 ma, Newton's second law. 274 00:17:13,680 --> 00:17:16,190 Sum of the external forces is equal to mass 275 00:17:16,190 --> 00:17:17,740 times the acceleration. 276 00:17:17,740 --> 00:17:21,460 So that's the law I'm going to apply. 277 00:17:21,460 --> 00:17:25,700 Sum of the external forces, it's a vector 278 00:17:25,700 --> 00:17:28,640 but we're just doing the x component only so we don't have 279 00:17:28,640 --> 00:17:32,930 to carry along vector notation, is equal to, in this case, 280 00:17:32,930 --> 00:17:36,350 mass times acceleration. 281 00:17:36,350 --> 00:17:38,520 So that's the law we're going to apply. 282 00:17:38,520 --> 00:17:44,975 And then finally the math to solve the equation of motion 283 00:17:44,975 --> 00:17:46,970 that we find, that'll be the third piece. 284 00:17:46,970 --> 00:17:49,090 But part of applying the physics, in order 285 00:17:49,090 --> 00:17:57,230 to do this now, we need what I call an FBD. 286 00:17:57,230 --> 00:17:58,602 What do you suppose that is? 287 00:17:58,602 --> 00:17:59,810 AUDIENCE: Free body diagrams. 288 00:17:59,810 --> 00:18:01,400 PROFESSOR: Free body diagrams. 289 00:18:01,400 --> 00:18:03,140 You've used these many times before, 290 00:18:03,140 --> 00:18:04,950 so we're going to do those. 291 00:18:04,950 --> 00:18:10,210 And free body diagrams-- 292 00:18:10,210 --> 00:18:12,780 And I'm going to teach you, at least 293 00:18:12,780 --> 00:18:14,960 the way I go about doing free body diagrams, 294 00:18:14,960 --> 00:18:17,150 as things get more and more complicated, 295 00:18:17,150 --> 00:18:20,280 you're going to have to be more sophisticated in the way 296 00:18:20,280 --> 00:18:22,350 that you do these things. 297 00:18:22,350 --> 00:18:25,750 So I just have some simple little rules 298 00:18:25,750 --> 00:18:29,610 to do free body diagrams that keep you from getting hung up 299 00:18:29,610 --> 00:18:31,480 on sign conventions. 300 00:18:31,480 --> 00:18:33,940 I think the thing people make most mistakes about is 301 00:18:33,940 --> 00:18:36,030 they get confused about signs. 302 00:18:36,030 --> 00:18:39,390 So I'll try to show you how I do it. 303 00:18:39,390 --> 00:18:54,540 So first you draw forces that you know, basically 304 00:18:54,540 --> 00:18:56,720 in the direction in which they act. 305 00:18:56,720 --> 00:18:59,150 Seems obvious. 306 00:18:59,150 --> 00:19:07,115 So when you know the direction-- so 307 00:19:07,115 --> 00:19:08,490 this is a really trivial problem, 308 00:19:08,490 --> 00:19:17,590 but the method here is very specific. 309 00:19:17,590 --> 00:19:18,780 So what's an example? 310 00:19:18,780 --> 00:19:20,070 Well, gravity. 311 00:19:20,070 --> 00:19:23,960 So we'll start our free body diagram. 312 00:19:23,960 --> 00:19:26,490 Gravity acts at the center of mass. 313 00:19:26,490 --> 00:19:27,550 It's downward. 314 00:19:27,550 --> 00:19:29,970 This is what I mean by the direction in which it acts. 315 00:19:29,970 --> 00:19:34,120 And it has magnitude, mg. 316 00:19:34,120 --> 00:19:35,740 OK. 317 00:19:35,740 --> 00:19:38,220 Now the other forces aren't so obvious. 318 00:19:38,220 --> 00:19:41,050 The force that's put on by the stiffness and this damper 319 00:19:41,050 --> 00:19:43,600 in the spring, which way do you draw them? 320 00:19:43,600 --> 00:19:44,370 What's the sign? 321 00:19:44,370 --> 00:19:46,950 What's the sign convention? 322 00:19:46,950 --> 00:19:49,560 So the convention, the way I go about doing these things, 323 00:19:49,560 --> 00:20:08,900 is I assume positive values for the deflections and velocities. 324 00:20:08,900 --> 00:20:11,960 So in this case, x and x dot. 325 00:20:11,960 --> 00:20:16,530 You just require that the deflections that you're going 326 00:20:16,530 --> 00:20:18,920 to work with are positive. 327 00:20:18,920 --> 00:20:20,650 And then from the positive deflection, 328 00:20:20,650 --> 00:20:22,660 you say which way is the resulting force? 329 00:20:22,660 --> 00:20:25,650 So if the deflection in this is downwards, 330 00:20:25,650 --> 00:20:29,540 which direction is the force that the spring applies 331 00:20:29,540 --> 00:20:32,020 to the mass? 332 00:20:32,020 --> 00:20:32,690 Up, right? 333 00:20:32,690 --> 00:20:35,660 What about if the velocity is downwards, 334 00:20:35,660 --> 00:20:41,360 which direction is the force is the damper puts on the mass? 335 00:20:41,360 --> 00:20:42,080 Also up, right? 336 00:20:42,080 --> 00:20:42,580 OK. 337 00:20:42,580 --> 00:20:48,720 So this allows-- this gives us-- so here's f spring and here's 338 00:20:48,720 --> 00:20:50,070 the f damper. 339 00:20:50,070 --> 00:20:53,610 And other any other forces on this mass? 340 00:20:53,610 --> 00:20:57,190 So spring force, damper force, and the gravitational force. 341 00:21:07,940 --> 00:21:18,385 And so third, you deduce the signs basically 342 00:21:18,385 --> 00:21:19,760 from the direction of the arrows. 343 00:21:25,880 --> 00:21:28,470 First we need what's called your constitutive relationship. 344 00:21:28,470 --> 00:21:32,600 So the spring force, fs, well you've 345 00:21:32,600 --> 00:21:35,370 made x positive so it keeps things nice, 346 00:21:35,370 --> 00:21:41,340 the spring constant's a positive number, so fs is kx. 347 00:21:41,340 --> 00:21:45,430 Fd is bx dot. 348 00:21:45,430 --> 00:21:47,440 And now we write the statement that the sum 349 00:21:47,440 --> 00:21:49,755 of forces in the x direction. 350 00:21:52,490 --> 00:21:55,000 We look at up here, we say well that's going 351 00:21:55,000 --> 00:22:01,680 to fs plus fd minus mg. 352 00:22:01,680 --> 00:22:06,360 So that's-- whoops, I wrote it the wrong way around. 353 00:22:06,360 --> 00:22:10,170 Minus, minus, plus. 354 00:22:10,170 --> 00:22:12,170 Because I'm plus downwards, right? 355 00:22:12,170 --> 00:22:18,430 Well, spring minus fs is minus kx minus bx dot 356 00:22:18,430 --> 00:22:24,809 plus mg equals mx double dot. 357 00:22:24,809 --> 00:22:26,600 And I rearranged this to put all the motion 358 00:22:26,600 --> 00:22:29,720 variables on one side. 359 00:22:29,720 --> 00:22:37,700 mx double dot plus bx dot plus kx equals mg. 360 00:22:40,350 --> 00:22:43,670 So there's my equation of motion, but with a method 361 00:22:43,670 --> 00:22:45,710 for doing the free body diagrams, which 362 00:22:45,710 --> 00:22:47,684 will work with multiple bodies. 363 00:22:47,684 --> 00:22:49,850 So you have two bodies with springs in between them. 364 00:22:49,850 --> 00:22:52,740 This is when the confusion really comes up. 365 00:22:52,740 --> 00:22:55,940 Two bodies with a spring trapped between them. 366 00:22:55,940 --> 00:22:58,490 What's the sign convention? 367 00:22:58,490 --> 00:22:59,540 You do the same thing. 368 00:22:59,540 --> 00:23:02,870 Both bodies exhibit positive motions, 369 00:23:02,870 --> 00:23:05,747 the force that results is proportional to the difference, 370 00:23:05,747 --> 00:23:06,580 and you work it out. 371 00:23:06,580 --> 00:23:08,100 And you'll get the signs right. 372 00:23:08,100 --> 00:23:09,940 OK, so here's our equation of motion arrived 373 00:23:09,940 --> 00:23:12,720 at by doing the direct method. 374 00:23:12,720 --> 00:23:21,130 And if we went on to the third step, which we're not 375 00:23:21,130 --> 00:23:24,570 going to do today, and that is apply the math, 376 00:23:24,570 --> 00:23:27,430 it might because I want you now to describe the motion for me, 377 00:23:27,430 --> 00:23:28,520 solve for the motion. 378 00:23:28,520 --> 00:23:30,570 That means solving the differential equation. 379 00:23:30,570 --> 00:23:33,600 And that's what we did last may in Haynes Miller's class. 380 00:23:33,600 --> 00:23:35,560 We'll come back to this later on. 381 00:23:35,560 --> 00:23:40,547 But for today's purposes, we don't need to go there. 382 00:23:40,547 --> 00:23:42,130 Got something else much more important 383 00:23:42,130 --> 00:23:45,270 to get to about kinematics. 384 00:23:45,270 --> 00:23:47,610 But I want to show you one thing, 385 00:23:47,610 --> 00:23:51,700 and that is just a little tiny introductory taste 386 00:23:51,700 --> 00:23:54,480 to this point. 387 00:23:54,480 --> 00:23:57,390 So I've derived the equation of motion of this 388 00:23:57,390 --> 00:24:00,854 by Newton's laws. 389 00:24:00,854 --> 00:24:02,270 But I'm going to ignore Newton now 390 00:24:02,270 --> 00:24:04,144 and saw I'm going to drive equation of motion 391 00:24:04,144 --> 00:24:04,950 by another way. 392 00:24:04,950 --> 00:24:07,470 And it's an energy technique, and that is-- well 393 00:24:07,470 --> 00:24:11,020 let's talk about the total energy of the system. 394 00:24:11,020 --> 00:24:13,750 It's going to be the sum of a kinetic energy 395 00:24:13,750 --> 00:24:14,920 and a potential energy. 396 00:24:18,560 --> 00:24:21,290 And we'll find that even with Lagrange, there's 397 00:24:21,290 --> 00:24:24,400 a problem with forces on systems that 398 00:24:24,400 --> 00:24:26,430 are what we call non-conservative, 399 00:24:26,430 --> 00:24:28,840 things that either take energy out of, or put energy 400 00:24:28,840 --> 00:24:29,710 into the system. 401 00:24:29,710 --> 00:24:31,850 And the dashpot does that. 402 00:24:31,850 --> 00:24:35,469 Dashpot generates heat and takes energy out of the system. 403 00:24:35,469 --> 00:24:37,510 So I'm going to have to ignore it for the moment. 404 00:24:37,510 --> 00:24:39,750 So the sum of the kinetic and the potential energies 405 00:24:39,750 --> 00:24:45,680 in this problem is a 1/2 kx squared 406 00:24:45,680 --> 00:24:53,040 for the potential of the spring, plus a 1/2 mx dot squared 407 00:24:53,040 --> 00:25:00,600 for the kinetic energy of the mass, and minus mgx 408 00:25:00,600 --> 00:25:04,880 for the potential energy that is due to the object moving 409 00:25:04,880 --> 00:25:07,230 in the gravitational field. 410 00:25:07,230 --> 00:25:11,460 And that's the total energy of the system. 411 00:25:11,460 --> 00:25:14,490 Now my problem, I've allowed no forces. 412 00:25:14,490 --> 00:25:15,740 There's no excitation on here. 413 00:25:15,740 --> 00:25:18,085 This is just free vibration only. 414 00:25:18,085 --> 00:25:20,460 That's all we're talking about, make initial displacement 415 00:25:20,460 --> 00:25:22,530 and it vibrates. 416 00:25:22,530 --> 00:25:24,410 If there's no damping, what can you 417 00:25:24,410 --> 00:25:28,884 say about the total energy of the system? 418 00:25:28,884 --> 00:25:30,198 AUDIENCE: [INAUDIBLE] 419 00:25:30,198 --> 00:25:31,464 PROFESSOR: Say it again. 420 00:25:31,464 --> 00:25:32,380 I heard it over there. 421 00:25:32,380 --> 00:25:33,800 It's got to be constant, right? 422 00:25:33,800 --> 00:25:37,480 All right, well, so this must be constant. 423 00:25:37,480 --> 00:25:55,200 Therefore, the time derivative of my system, it better be 0. 424 00:25:55,200 --> 00:25:56,890 The energy is constant. 425 00:25:56,890 --> 00:25:58,860 Take it's time derivative, it's got to be 0. 426 00:25:58,860 --> 00:26:00,610 Apply that to the right-hand side of this, 427 00:26:00,610 --> 00:26:13,460 I get kxx dot plus mx dot x double dot minus mgx dot 428 00:26:13,460 --> 00:26:15,530 equals zero. 429 00:26:15,530 --> 00:26:23,690 And I now cancel out the common x dot terms go away. 430 00:26:23,690 --> 00:26:37,580 And I'm left with-- and I've essentially 431 00:26:37,580 --> 00:26:40,600 solved for the equation of motion of this system 432 00:26:40,600 --> 00:26:44,010 without ever looking at conservational momentum, 433 00:26:44,010 --> 00:26:46,695 Newton's laws, only by energy considerations. 434 00:26:50,300 --> 00:26:52,570 OK, so that's a very simple example 435 00:26:52,570 --> 00:26:55,350 of that you can use energy to derive equations of motions. 436 00:26:55,350 --> 00:27:00,840 But you then have to go back and fix it to account for the loss 437 00:27:00,840 --> 00:27:01,977 term, the damping term. 438 00:27:01,977 --> 00:27:04,060 And that you still have to consider it as a force, 439 00:27:04,060 --> 00:27:04,800 we'll find out. 440 00:27:04,800 --> 00:27:09,920 Even was Lagrange you have to go back and consider the work done 441 00:27:09,920 --> 00:27:13,050 by external forces. 442 00:27:13,050 --> 00:27:15,320 OK. 443 00:27:15,320 --> 00:27:18,010 So you've just kind of seen the whole course. 444 00:27:18,010 --> 00:27:19,390 We've described the motion, we've 445 00:27:19,390 --> 00:27:24,090 applied to Newton's laws, the physics to the direct method 446 00:27:24,090 --> 00:27:26,360 to derive the equations of motion, 447 00:27:26,360 --> 00:27:31,540 we have gone to a direct method, and have derived the equations 448 00:27:31,540 --> 00:27:32,660 of motion that way. 449 00:27:32,660 --> 00:27:35,290 And that's basically what you're going to do in the course. 450 00:27:35,290 --> 00:27:37,630 But now you're going to do it with much more 451 00:27:37,630 --> 00:27:39,880 sophisticated tools. 452 00:27:39,880 --> 00:27:42,160 You'll have multiple degree of freedom systems. 453 00:27:42,160 --> 00:27:44,520 The description describing the motion, 454 00:27:44,520 --> 00:27:46,230 is maybe going to be for some of you, 455 00:27:46,230 --> 00:27:48,410 the most challenging part of the course. 456 00:27:48,410 --> 00:27:50,615 And this is a topic we call kinematics. 457 00:27:53,960 --> 00:27:59,380 And that's what we'll turn to next. 458 00:28:13,330 --> 00:28:15,420 So reference frames and vectors. 459 00:28:15,420 --> 00:28:16,840 That's the topic. 460 00:28:16,840 --> 00:28:20,110 This is now that we're talking about kinematics, 461 00:28:20,110 --> 00:28:24,610 and this is all about describing the motion. 462 00:28:24,610 --> 00:28:29,330 So Descartes gave us the Cartesian coordinate system, 463 00:28:29,330 --> 00:28:30,340 and we'll start there. 464 00:28:30,340 --> 00:28:34,080 So imagine this is a fixed frame-- 465 00:28:34,080 --> 00:28:36,600 we'll talk about what makes an inertial frame 466 00:28:36,600 --> 00:28:38,300 the next lecture. 467 00:28:38,300 --> 00:28:39,840 But here we have an inertial frame. 468 00:28:42,860 --> 00:28:51,620 And it's the frame we'll call O-xyz or O for short. 469 00:28:51,620 --> 00:29:05,720 And in this frame, maybe this is me, and up here is a dog, 470 00:29:05,720 --> 00:29:12,840 and I'm going to call this point A and this point B. 471 00:29:12,840 --> 00:29:18,190 And I'm going to describe the positions of these two points 472 00:29:18,190 --> 00:29:19,890 by vectors. 473 00:29:19,890 --> 00:29:24,050 This one will be R, and the notation that I'm going to use 474 00:29:24,050 --> 00:29:30,330 is point and it's measurement with respect something. 475 00:29:30,330 --> 00:29:34,980 Well, it's with respect to this point O in this inertial frame. 476 00:29:34,980 --> 00:29:38,910 So this is A with respect O is the way to read this. 477 00:29:38,910 --> 00:29:41,310 There's another vector here. 478 00:29:41,310 --> 00:29:47,440 This is RB respect to A And finally, 479 00:29:47,440 --> 00:29:53,190 R of B with respect to O They're all vectors on the board. 480 00:29:53,190 --> 00:29:55,660 I'll try to remember to underline them 481 00:29:55,660 --> 00:29:56,860 in the textbooks and things. 482 00:29:56,860 --> 00:30:00,420 They're usually-- vectors are noted with bold letters. 483 00:30:03,970 --> 00:30:07,930 And vectors allow us to say the following. 484 00:30:07,930 --> 00:30:13,180 That R, the position of the dog and the reference 485 00:30:13,180 --> 00:30:16,690 with respect to O, is the sum of these other two vectors. 486 00:30:16,690 --> 00:30:26,390 R of A with respect to O plus R R of B with respect to A. 487 00:30:26,390 --> 00:30:29,200 And mostly to do dynamics we're really 488 00:30:29,200 --> 00:30:32,040 interested in things like velocities and accelerations. 489 00:30:32,040 --> 00:30:34,010 So to get the velocities and accelerations, 490 00:30:34,010 --> 00:30:39,095 we have to take a time derivative of our RBO dt. 491 00:30:41,730 --> 00:30:45,840 And that's going to give us what we'll call the velocity, 492 00:30:45,840 --> 00:30:48,040 obviously you write it as V. And it would 493 00:30:48,040 --> 00:30:53,060 be the velocity of point B with respect to O. And no surprise, 494 00:30:53,060 --> 00:30:58,970 it'll be the velocity of point A plus the velocity of B 495 00:30:58,970 --> 00:31:04,330 with respect to A. 496 00:31:04,330 --> 00:31:13,200 And finally, if we take two derivatives, dt squared, 497 00:31:13,200 --> 00:31:17,690 we'll get the acceleration of B with respect to O. 498 00:31:17,690 --> 00:31:22,730 And that'll be the sum of A-- the acceleration of A 499 00:31:22,730 --> 00:31:28,880 with respect to O plus the acceleration of B 500 00:31:28,880 --> 00:31:32,420 with respect to A. All, again, vectors. 501 00:31:38,490 --> 00:31:41,502 Now, just to look ahead-- this seems all really trivial. 502 00:31:41,502 --> 00:31:43,210 You guys are going to sleep on me, right? 503 00:31:46,620 --> 00:31:52,460 If these are rigid bodies, this is a rigid body that is moving 504 00:31:52,460 --> 00:31:55,360 and maybe rotating. 505 00:31:55,360 --> 00:32:01,370 And B is on it, and A is on it, and O isn't on it. 506 00:32:01,370 --> 00:32:05,000 It starts getting a little tricky. 507 00:32:05,000 --> 00:32:12,000 And this, the derivative of a vector that's attached 508 00:32:12,000 --> 00:32:19,220 to the body somehow has to account for the fact that 509 00:32:19,220 --> 00:32:23,530 if I'm-- the observer's on the body, 510 00:32:23,530 --> 00:32:25,010 this other point's on the body. 511 00:32:25,010 --> 00:32:28,550 Say it's, I'm on this asteroid, and I've got a dog out there, 512 00:32:28,550 --> 00:32:31,000 and the dog's run away from me. 513 00:32:31,000 --> 00:32:36,460 The speed of the dog with respect to me, I can measure. 514 00:32:36,460 --> 00:32:37,949 But if I'm down here looking at it, 515 00:32:37,949 --> 00:32:39,740 it'll look different because it's rotating. 516 00:32:39,740 --> 00:32:41,198 So how do you account for all that? 517 00:32:41,198 --> 00:32:44,810 So taking these derivatives of vectors in moving frames 518 00:32:44,810 --> 00:32:48,590 is where the devil's in the details. 519 00:32:48,590 --> 00:32:54,190 And that's part of what I'm going to be teaching you. 520 00:32:54,190 --> 00:32:55,620 OK. 521 00:32:55,620 --> 00:32:59,065 I'm still learning how to optimize my board use. 522 00:32:59,065 --> 00:33:00,940 I haven't got it perfect yet, but because I'm 523 00:33:00,940 --> 00:33:02,898 having to move around a lot here and improvise. 524 00:33:02,898 --> 00:33:04,670 But we'll persevere. 525 00:33:04,670 --> 00:33:09,160 You need to remember a couple things about vectors, 526 00:33:09,160 --> 00:33:13,916 how to add them, dot products. 527 00:33:13,916 --> 00:33:15,290 If you've forgotten these things, 528 00:33:15,290 --> 00:33:18,469 you need to go back and review them really quickly. 529 00:33:18,469 --> 00:33:20,510 There's usually a little review section the book, 530 00:33:20,510 --> 00:33:22,301 so you need to practice that sort of thing. 531 00:33:24,700 --> 00:33:26,710 Couple other little facts you need to remember. 532 00:33:26,710 --> 00:33:32,687 So the derivative of the sum of two vectors 533 00:33:32,687 --> 00:33:34,145 is just the sum of the derivatives. 534 00:33:42,130 --> 00:33:45,180 And quite importantly, we're going 535 00:33:45,180 --> 00:33:48,170 to make use of this one a lot, is the derivative 536 00:33:48,170 --> 00:33:49,760 of a product of two things. 537 00:33:49,760 --> 00:33:53,340 One of them be in a vector, some function maybe of time 538 00:33:53,340 --> 00:34:01,840 and a here is derivative of f with respect to t times a, 539 00:34:01,840 --> 00:34:11,139 plus the derivative of a with respect to t times f. 540 00:34:11,139 --> 00:34:12,750 That we'll make a lot use of. 541 00:34:12,750 --> 00:34:16,350 So just your basic calculus. 542 00:34:16,350 --> 00:34:20,070 So now, I want to take up-- let's talk about the simplest 543 00:34:20,070 --> 00:34:22,690 form of being able to do these derivatives 544 00:34:22,690 --> 00:34:27,429 and calculate these velocities, when everything's 545 00:34:27,429 --> 00:34:33,199 described in terms of Cartesian coordinates. 546 00:34:33,199 --> 00:34:36,946 Now I'm going to give you a little look ahead because I'm 547 00:34:36,946 --> 00:34:42,620 going to try to avoid confusion as much as possible here. 548 00:34:42,620 --> 00:34:47,204 The hardest problem is when you have a rigid body, 549 00:34:47,204 --> 00:34:49,570 you got the dog on it, you've got the observer on it, 550 00:34:49,570 --> 00:34:52,300 it's rotating, and translating. 551 00:34:52,300 --> 00:34:56,260 And to take this derivative, you end up with a number of terms. 552 00:34:56,260 --> 00:34:58,470 The simplest problem is just something 553 00:34:58,470 --> 00:35:00,410 in a fixed Cartesian coordinate system. 554 00:35:00,410 --> 00:35:02,240 So we're going to start with a simple one, 555 00:35:02,240 --> 00:35:04,239 and build our way up to the complicated one, OK? 556 00:35:07,030 --> 00:35:10,650 But let's now, we're going to do the really, the simplest one. 557 00:35:10,650 --> 00:35:15,050 We're going to do velocity and acceleration 558 00:35:15,050 --> 00:35:16,980 in Cartesian coordinates. 559 00:35:23,750 --> 00:35:29,750 And basically I should say fixed Cartesian coordinates, 560 00:35:29,750 --> 00:35:30,410 not moving. 561 00:35:33,160 --> 00:35:37,900 All right, so now let's consider the dog out here, 562 00:35:37,900 --> 00:35:42,840 and his position in the Cartesian coordinate system. 563 00:35:42,840 --> 00:35:45,620 And I could write that and you'll, 564 00:35:45,620 --> 00:35:48,250 without any loss of generality here, you'll 565 00:35:48,250 --> 00:35:52,427 know what I mean if I say RBx component. 566 00:35:52,427 --> 00:35:54,260 And I'm going to stop writing the slash O's, 567 00:35:54,260 --> 00:35:57,320 because this is now all in this fixed reference frame. 568 00:35:57,320 --> 00:36:01,080 And it's in I-hat direction. 569 00:36:01,080 --> 00:36:10,590 And I've got another component, RBy in the J-hat, and an RBz 570 00:36:10,590 --> 00:36:13,344 in the K-hat. 571 00:36:13,344 --> 00:36:15,010 And I want to take the time derivative-- 572 00:36:15,010 --> 00:36:16,301 I was looking for the velocity. 573 00:36:16,301 --> 00:36:18,680 I want to calculate the velocity. 574 00:36:18,680 --> 00:36:25,175 So the velocity here of BNO is d by dt of RBO. 575 00:36:25,175 --> 00:36:28,070 . 576 00:36:28,070 --> 00:36:31,200 And now this is now the product of two things, 577 00:36:31,200 --> 00:36:33,950 so I've got to use that formula over here. 578 00:36:33,950 --> 00:36:35,960 Product one turn times the other, and so forth. 579 00:36:35,960 --> 00:36:39,330 So I go to these, and I say OK, so this 580 00:36:39,330 --> 00:36:53,690 is R dot Bx times I plus R dot By times J plus R dot 581 00:36:53,690 --> 00:37:00,610 Bz times K. And then the other-- the flip side of that is I 582 00:37:00,610 --> 00:37:06,650 have to take the derivatives of I times RBx, the derivative J 583 00:37:06,650 --> 00:37:07,810 and so forth. 584 00:37:07,810 --> 00:37:10,940 But what's the derivative of, let's say, I? 585 00:37:10,940 --> 00:37:15,380 Capital I is my unit vector in the fixed reference 586 00:37:15,380 --> 00:37:17,795 frame, my O-xyz frame. 587 00:37:17,795 --> 00:37:19,045 0 So it's a constant. 588 00:37:19,045 --> 00:37:23,390 It is unit length, and it points in a direction that it's fixed. 589 00:37:23,390 --> 00:37:25,240 So what's its derivative? 590 00:37:25,240 --> 00:37:26,990 It's going to have a 0 derivative. 591 00:37:26,990 --> 00:37:30,770 So the second part of this-- second bits of that is zero. 592 00:37:30,770 --> 00:37:36,480 So that's the velocity in Cartesian coordinates of my dog 593 00:37:36,480 --> 00:37:40,750 out there running around. 594 00:37:40,750 --> 00:37:43,477 And the acceleration, in a similar way, 595 00:37:43,477 --> 00:37:45,810 now to get the acceleration, you take another derivative 596 00:37:45,810 --> 00:37:46,309 of this. 597 00:37:46,309 --> 00:37:48,920 And again, you'll have to take derivatives of I, J, and K, 598 00:37:48,920 --> 00:37:50,620 and again they're going to be 0. 599 00:37:50,620 --> 00:37:57,300 So you will find that the acceleration then, is just R 600 00:37:57,300 --> 00:38:02,910 double dot x term in the plus R double 601 00:38:02,910 --> 00:38:10,485 dot By in the J plus r double dot Bz in the K. 602 00:38:10,485 --> 00:38:12,610 That would be our acceleration term, and it's easy. 603 00:38:20,219 --> 00:38:22,760 Now imagine that we are doing this in polar coordinates, unit 604 00:38:22,760 --> 00:38:25,390 vectors in polar coordinates. 605 00:38:25,390 --> 00:38:27,350 Let me check, last year the students told me 606 00:38:27,350 --> 00:38:30,110 that in your physics courses, you 607 00:38:30,110 --> 00:38:34,380 use unit vectors R-hat, theta-hat, and K. Is 608 00:38:34,380 --> 00:38:35,450 that right? 609 00:38:35,450 --> 00:38:38,580 So I'll use those unit vectors so they look familiar, 610 00:38:38,580 --> 00:38:40,090 because in polar coordinates people 611 00:38:40,090 --> 00:38:41,760 use lots of different things. 612 00:38:41,760 --> 00:38:45,710 But think about it, in polar coordinates, theta-- 613 00:38:45,710 --> 00:38:48,460 it's a fixed, maybe, coordinate system, 614 00:38:48,460 --> 00:38:54,020 but now theta goes like this and R moves with theta, right? 615 00:38:54,020 --> 00:38:55,550 So the unit vector is pointing here, 616 00:38:55,550 --> 00:38:58,070 but over time it might move down to here. 617 00:38:58,070 --> 00:39:01,710 And unit vector has changed direction, 618 00:39:01,710 --> 00:39:04,210 and its derivative in time is no longer 0. 619 00:39:04,210 --> 00:39:06,810 So it starts getting messy as soon as the unit vectors 620 00:39:06,810 --> 00:39:07,680 change in time. 621 00:39:07,680 --> 00:39:10,690 And so that's one of our objectives here 622 00:39:10,690 --> 00:39:12,420 is to get to that point and describe 623 00:39:12,420 --> 00:39:14,130 how you handle those cases. 624 00:39:46,450 --> 00:39:51,460 So a quick point about velocity. 625 00:39:51,460 --> 00:39:54,250 You need to really understand what we mean by velocity. 626 00:39:54,250 --> 00:40:00,010 So here's our Cartesian system. 627 00:40:00,010 --> 00:40:05,300 Here's this point out here B. And now, this 628 00:40:05,300 --> 00:40:10,070 is the dog running around, and the path of the dog 629 00:40:10,070 --> 00:40:14,440 might have been like this. 630 00:40:14,440 --> 00:40:16,885 And right in here he's going this direction. 631 00:40:19,460 --> 00:40:30,800 And in a little time, in delta t, 632 00:40:30,800 --> 00:40:38,610 he moves by an amount delta RB with respect to O. 633 00:40:38,610 --> 00:40:41,100 And that's what this is. 634 00:40:41,100 --> 00:40:44,015 He's moved this little bit in time delta t. 635 00:40:44,015 --> 00:40:46,320 And he happens to be going off in that direction. 636 00:40:46,320 --> 00:40:49,450 So this then is R prime, I'll call it, 637 00:40:49,450 --> 00:40:53,790 of B with respect to O, and this is our original RB with respect 638 00:40:53,790 --> 00:40:59,780 to O. So we can say that his new position, RB 639 00:40:59,780 --> 00:41:09,030 with respect to prime is RBO plus delta R. 640 00:41:09,030 --> 00:41:10,550 And these are all vectors. 641 00:41:14,800 --> 00:41:24,332 And the velocity of B with respect to O 642 00:41:24,332 --> 00:41:32,800 is just equal to this limit of delta RBO over delta t 643 00:41:32,800 --> 00:41:37,740 as t goes to 0. 644 00:41:37,740 --> 00:41:39,520 So what direction is the velocity? 645 00:41:43,858 --> 00:41:47,410 The velocity is in the direction of the change, not 646 00:41:47,410 --> 00:41:50,600 the original vector, it was in the direction of the change. 647 00:41:50,600 --> 00:41:52,580 And in fact, if the path of the dog 648 00:41:52,580 --> 00:41:58,310 is like this, at the instant you compute the velocity, 649 00:41:58,310 --> 00:42:02,710 you're computing the tangent to the path of the dog. 650 00:42:02,710 --> 00:42:04,950 So that's what velocity is at any instant time 651 00:42:04,950 --> 00:42:06,990 is a tangent to the path. 652 00:42:06,990 --> 00:42:08,650 And that's a good concept to remember. 653 00:42:49,660 --> 00:42:53,400 So we're still in this fixed Cartesian space, 654 00:42:53,400 --> 00:42:55,190 and I have of couple of points. 655 00:42:55,190 --> 00:42:57,250 I'll make it really trivial here. 656 00:42:57,250 --> 00:43:04,820 Here's B, and here's A, and the velocity of B-- 657 00:43:04,820 --> 00:43:05,590 where's my number? 658 00:43:09,950 --> 00:43:13,920 We'll make this 10 feet per second. 659 00:43:13,920 --> 00:43:19,300 And it's in the J-hat direction. 660 00:43:19,300 --> 00:43:25,610 And A, this is the velocity of BNO. 661 00:43:25,610 --> 00:43:33,190 The velocity of ANO, we'll say is 4 feet per second, 662 00:43:33,190 --> 00:43:35,270 also in the J direction. 663 00:43:35,270 --> 00:43:38,590 And I want to know what's the velocity of B with respect 664 00:43:38,590 --> 00:43:44,005 to A. So now I'm chasing the dog, he's running at 10, 665 00:43:44,005 --> 00:43:47,460 I'm running at 4. 666 00:43:47,460 --> 00:43:50,060 How do I perceive the speed of the dog? 667 00:43:50,060 --> 00:43:52,170 Well, to do this in vectors, which 668 00:43:52,170 --> 00:43:54,420 is the point of the exercise here, 669 00:43:54,420 --> 00:43:58,000 is we have the expressions we started with over there. 670 00:43:58,000 --> 00:44:00,370 And we're going to use these a lot in the course. 671 00:44:00,370 --> 00:44:04,120 So the velocity of B with respect to O 672 00:44:04,120 --> 00:44:05,730 is the velocity of A with respect 673 00:44:05,730 --> 00:44:09,520 to O plus the velocity of B with respect to A. 674 00:44:09,520 --> 00:44:12,140 And if I want to know velocity of B with respect to A, 675 00:44:12,140 --> 00:44:12,990 I just solve this. 676 00:44:18,140 --> 00:44:22,540 So velocity of B with respect to O minus the velocity of A 677 00:44:22,540 --> 00:44:26,340 with respect to O, and in this case that's 10 minus 4 678 00:44:26,340 --> 00:44:32,380 is 6 in the J. 679 00:44:32,380 --> 00:44:35,470 Point of the exercise is to manipulate the vector 680 00:44:35,470 --> 00:44:36,930 expressions like this. 681 00:44:36,930 --> 00:44:39,220 So take whatever known quantities you have 682 00:44:39,220 --> 00:44:40,470 and solve for the unknown one. 683 00:44:40,470 --> 00:44:42,553 In this case, I want to know the relative velocity 684 00:44:42,553 --> 00:44:44,490 between the two, and it's this. 685 00:44:49,970 --> 00:44:53,620 If I'm here, and I'm watching the dog, 686 00:44:53,620 --> 00:44:57,110 that's how I perceive the speed of the dog relative to me, 687 00:44:57,110 --> 00:44:58,200 right? 688 00:44:58,200 --> 00:45:00,180 6 feet per second in the J direction. 689 00:45:00,180 --> 00:45:05,510 What's the speed of the dog from the point of view of over here? 690 00:45:05,510 --> 00:45:09,570 The speed of the dog relative to me. 691 00:45:13,450 --> 00:45:17,140 So it's again the velocity of B with respect to A, 692 00:45:17,140 --> 00:45:22,300 but from a different position in this fixed reference frame. 693 00:45:29,930 --> 00:45:31,740 Really important point, actually. 694 00:45:31,740 --> 00:45:34,085 This is a really important conceptual point. 695 00:45:37,300 --> 00:45:38,570 Somebody be bold. 696 00:45:38,570 --> 00:45:40,490 What's the speed with respect to O? 697 00:45:40,490 --> 00:45:44,570 The velocity of B with respect to A seen from O, 698 00:45:44,570 --> 00:45:48,074 as computed from O, measured from O. Got radar down there, 699 00:45:48,074 --> 00:45:49,115 and you're tracking them. 700 00:45:54,065 --> 00:45:56,045 AUDIENCE: [INAUDIBLE] 701 00:45:56,045 --> 00:45:59,470 PROFESSOR: In what direction? 702 00:45:59,470 --> 00:46:00,430 AUDIENCE: [INAUDIBLE] 703 00:46:03,310 --> 00:46:04,760 PROFESSOR: Yeah. 704 00:46:04,760 --> 00:46:06,110 It's the same. 705 00:46:06,110 --> 00:46:09,160 The point is it's the same. 706 00:46:09,160 --> 00:46:15,740 If you're in a fixed reference frame, a vector of velocity 707 00:46:15,740 --> 00:46:20,760 is the same as seen from any point in the frame. 708 00:46:20,760 --> 00:46:23,680 Any fixed point in the frame of velocity is always the same. 709 00:46:23,680 --> 00:46:27,140 And in fact, in this case, the velocity-- this is a moving 710 00:46:27,140 --> 00:46:30,820 point and the velocity of him with respect to me 711 00:46:30,820 --> 00:46:33,590 this is different six feet per second. 712 00:46:33,590 --> 00:46:35,570 And I, from here, say the velocity 713 00:46:35,570 --> 00:46:39,140 of that guy with respect to this guy is still 6 feet per second. 714 00:46:39,140 --> 00:46:43,930 Any place in that frame or even any point 715 00:46:43,930 --> 00:46:45,470 moving at constant velocity, you're 716 00:46:45,470 --> 00:46:47,955 going to see the same answer. 717 00:46:47,955 --> 00:46:50,740 So it doesn't matter where you are 718 00:46:50,740 --> 00:46:54,510 to compute the velocity of B with respect to A. 719 00:46:54,510 --> 00:46:57,011 That's the important point. 720 00:46:57,011 --> 00:46:57,510 OK. 721 00:47:12,010 --> 00:47:15,550 OK, we got to pick up with, and I may not quite finish, 722 00:47:15,550 --> 00:47:22,795 but I am going to introduce the next complexity. 723 00:47:40,960 --> 00:47:41,460 OK. 724 00:48:03,340 --> 00:48:06,120 So what we just arrived at a minute ago 725 00:48:06,120 --> 00:48:08,590 is that the velocity as seen from O 726 00:48:08,590 --> 00:48:13,220 is the same as the velocity as seen from A. And A is me, 727 00:48:13,220 --> 00:48:15,140 and I'm moving, and I'm chasing the dog. 728 00:48:15,140 --> 00:48:18,295 So I'm a moving reference frame, I'm 729 00:48:18,295 --> 00:48:20,590 what's called a translating reference frame. 730 00:48:20,590 --> 00:48:22,490 So now we're going to take the next step. 731 00:48:22,490 --> 00:48:24,890 We had a fixed reference frame before purely, 732 00:48:24,890 --> 00:48:26,390 and now I want to talk about having 733 00:48:26,390 --> 00:48:28,680 the idea, the concept of having a moving reference 734 00:48:28,680 --> 00:48:32,200 frame within a fixed one. 735 00:48:32,200 --> 00:48:35,571 So this is the reference frame O capital XYZ. 736 00:48:35,571 --> 00:48:37,820 And this little reference frame now is attached to me, 737 00:48:37,820 --> 00:48:41,050 and it's A, and I call it x-prime y-prime. 738 00:48:41,050 --> 00:48:43,360 So just so you can-- it's going to be 739 00:48:43,360 --> 00:48:46,910 hard to tell this X from this X if I don't do something 740 00:48:46,910 --> 00:48:49,860 like a prime. 741 00:48:49,860 --> 00:48:52,990 So that this is the concept of a translating coordinate system 742 00:48:52,990 --> 00:48:56,410 attached to a body, like a rigid body, for example. 743 00:48:56,410 --> 00:48:59,180 We're going to do lots of rigid body dynamics here. 744 00:48:59,180 --> 00:49:01,720 And within this coordinate system, 745 00:49:01,720 --> 00:49:05,759 I can compute the velocity of B with respect to A, 746 00:49:05,759 --> 00:49:07,300 and I'll get exactly the same answer. 747 00:49:07,300 --> 00:49:10,930 I'll get that 6 feet per second in the J direction. 748 00:49:10,930 --> 00:49:14,340 So it's as if-- so this concept of being 749 00:49:14,340 --> 00:49:17,110 able to have a reference frame attached to a body 750 00:49:17,110 --> 00:49:20,220 and translating with it, you can measure things within it, 751 00:49:20,220 --> 00:49:24,160 get the answer, and then convert that answer to here 752 00:49:24,160 --> 00:49:27,780 if you're using a different coordinate. 753 00:49:27,780 --> 00:49:31,070 You could use polar coordinates here and rectangular here, 754 00:49:31,070 --> 00:49:33,610 but they still can be related to one another. 755 00:49:33,610 --> 00:49:34,830 We'll do problems like that. 756 00:49:43,410 --> 00:49:45,820 OK. 757 00:49:45,820 --> 00:49:51,340 So now what I'm doing is I told you like in the readings, 758 00:49:51,340 --> 00:49:56,740 the end game is to be able to talk about translating 759 00:49:56,740 --> 00:49:59,510 and rotating bodies, and do dynamics 760 00:49:59,510 --> 00:50:01,950 in three dimensions with translating and rotating 761 00:50:01,950 --> 00:50:03,090 objects. 762 00:50:03,090 --> 00:50:07,740 And we're going to get there somewhat step by step. 763 00:50:07,740 --> 00:50:10,390 But I want you to understand the end game 764 00:50:10,390 --> 00:50:11,950 so you know where we're going. 765 00:50:11,950 --> 00:50:14,110 And you need to have a couple of concepts in mind. 766 00:50:17,820 --> 00:50:23,950 So the first concept is that this is a rigid body now. 767 00:50:23,950 --> 00:50:27,800 And you can describe the motion of rigid bodies 768 00:50:27,800 --> 00:50:34,360 by the summation, the combination of a translation 769 00:50:34,360 --> 00:50:36,410 and a rotation. 770 00:50:36,410 --> 00:50:39,680 And of the rigid body, if you can describe its translation, 771 00:50:39,680 --> 00:50:41,510 and you can describe its rotation, 772 00:50:41,510 --> 00:50:43,430 you have the complete motion. 773 00:50:43,430 --> 00:50:47,470 So you got to understand what do we mean by what's really 774 00:50:47,470 --> 00:50:49,700 the definition of translation. 775 00:50:49,700 --> 00:50:53,096 So translation-- so I've got this-- 776 00:50:53,096 --> 00:50:55,450 I'll call it a merry-go-round. 777 00:50:55,450 --> 00:50:58,510 We'll use a merry-go-round example in a minute. 778 00:50:58,510 --> 00:51:01,845 And you're observers in a fixed inertial frame 779 00:51:01,845 --> 00:51:05,140 up above this merry-go-round looking down. 780 00:51:05,140 --> 00:51:06,640 OK. 781 00:51:06,640 --> 00:51:09,890 But so you can see it, I got to turn it on its side. 782 00:51:09,890 --> 00:51:11,680 So here's my merry-go-round. 783 00:51:11,680 --> 00:51:13,840 And if it's not rotating, but let's 784 00:51:13,840 --> 00:51:20,320 say it's sitting on a train, on a flat bed and moving along. 785 00:51:20,320 --> 00:51:22,360 It's translating. 786 00:51:22,360 --> 00:51:24,620 And when you say a body translates, 787 00:51:24,620 --> 00:51:31,510 any two points on the body move in parallel paths. 788 00:51:31,510 --> 00:51:33,844 So two points, my thumb and my finger-- 789 00:51:33,844 --> 00:51:35,260 if I'm just going along with this, 790 00:51:35,260 --> 00:51:37,510 those two paths are traveling parallel to one another. 791 00:51:40,780 --> 00:51:50,180 If I got Y pointing up, the body does this, 792 00:51:50,180 --> 00:51:52,605 is it rotating and translating? 793 00:51:52,605 --> 00:51:53,480 AUDIENCE: [INAUDIBLE] 794 00:51:53,480 --> 00:51:59,095 PROFESSOR: Are any two points on a moving in parallel paths? 795 00:51:59,095 --> 00:52:00,050 Right? 796 00:52:00,050 --> 00:52:01,200 OK. 797 00:52:01,200 --> 00:52:02,800 When it goes through curved things, 798 00:52:02,800 --> 00:52:05,090 it's called curvilinear translation. 799 00:52:05,090 --> 00:52:07,220 But it's still just translation. 800 00:52:07,220 --> 00:52:09,200 OK, so I'll stop and hold steady. 801 00:52:09,200 --> 00:52:13,620 The train stopped, and the thing-- let it rotate. 802 00:52:13,620 --> 00:52:17,200 So that's pure rotation. 803 00:52:17,200 --> 00:52:20,340 And the thing to remember about pure rotation 804 00:52:20,340 --> 00:52:26,372 is that anywhere on the body rotates at the same rate. 805 00:52:26,372 --> 00:52:29,800 If this is going around once a second, 806 00:52:29,800 --> 00:52:31,880 the rotation rate is one rotation 807 00:52:31,880 --> 00:52:35,450 per second, 360 degrees, 2 pi radians per second 808 00:52:35,450 --> 00:52:37,180 is its rotation rate. 809 00:52:37,180 --> 00:52:39,710 Every point on the body experiences the same rotation 810 00:52:39,710 --> 00:52:40,950 rate. 811 00:52:40,950 --> 00:52:42,760 That's a really important one to remember. 812 00:52:46,240 --> 00:52:48,640 If I'm holding still, merry-go-round's 813 00:52:48,640 --> 00:52:53,300 going round and round, it has a fixed axis of rotation, right? 814 00:52:53,300 --> 00:52:58,590 But do rotating bodies have to have fixed axes of rotation? 815 00:52:58,590 --> 00:53:07,605 So if I throw that up in the air, not hanging onto it, 816 00:53:07,605 --> 00:53:11,200 it's got gravity acting on it, it's rotating. 817 00:53:11,200 --> 00:53:12,675 What's a rotate about? 818 00:53:12,675 --> 00:53:13,550 AUDIENCE: [INAUDIBLE] 819 00:53:13,550 --> 00:53:15,180 PROFESSOR: Center of mass, OK. 820 00:53:15,180 --> 00:53:18,570 Is the center of mass moving? 821 00:53:18,570 --> 00:53:21,700 So this is clearly-- this is an example of rotation 822 00:53:21,700 --> 00:53:22,580 plus translation. 823 00:53:25,430 --> 00:53:28,660 It rotates about an axis but the axis can move. 824 00:53:28,660 --> 00:53:31,099 That's another important concept that we 825 00:53:31,099 --> 00:53:33,390 have to allow in order to be able to do these problems. 826 00:53:33,390 --> 00:53:35,410 But this is now general motion, it's 827 00:53:35,410 --> 00:53:38,050 a combination of translation and rotation, 828 00:53:38,050 --> 00:53:40,329 and we figure out each of those two pieces, 829 00:53:40,329 --> 00:53:42,620 then we can describe the complete motion of the system. 830 00:53:47,140 --> 00:53:57,660 All right, where we'll pick up next time is then doing that. 831 00:53:57,660 --> 00:53:59,076 And it would help actually, if you 832 00:53:59,076 --> 00:54:03,570 go read that reading, especially up to chapter 16, 833 00:54:03,570 --> 00:54:07,700 we have to get into to taking derivatives of vectors which 834 00:54:07,700 --> 00:54:10,700 are rotating, and come up with a general formula 835 00:54:10,700 --> 00:54:13,680 allows us to do velocities and accelerations 836 00:54:13,680 --> 00:54:14,660 under those conditions. 837 00:54:14,660 --> 00:54:18,940 See you on Tuesday next.