WEBVTT 00:00:00.060 --> 00:00:02.430 The following content is provided under a Creative 00:00:02.430 --> 00:00:03.800 Commons license. 00:00:03.800 --> 00:00:06.050 Your support will help MIT OpenCourseWare 00:00:06.050 --> 00:00:10.140 continue to offer high-quality educational resources for free. 00:00:10.140 --> 00:00:12.690 To make a donation, or to view additional materials 00:00:12.690 --> 00:00:16.600 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.600 --> 00:00:17.297 at ocw.mit.edu. 00:00:20.854 --> 00:00:22.645 PROFESSOR: Let's get on with some dynamics. 00:00:26.490 --> 00:00:32.020 So the place I'm going to begin is just 00:00:32.020 --> 00:00:34.820 a comment about mechanical engineering courses. 00:00:34.820 --> 00:00:39.010 The first, and you may have heard this already in classes, 00:00:39.010 --> 00:00:42.440 you'll be taking subject 2001 if you're 00:00:42.440 --> 00:00:45.770 Course 2 majors through 2009, and if you're 00:00:45.770 --> 00:00:48.700 2-A, most of the odd ones. 00:00:48.700 --> 00:00:57.920 But the subjects 2001 through 2005 00:00:57.920 --> 00:01:00.510 are really basically engineering science subjects that are all 00:01:00.510 --> 00:01:02.650 foundational to mechanical engineering, 00:01:02.650 --> 00:01:05.780 and they all have a common or property through them. 00:01:05.780 --> 00:01:10.220 And that is that we make observations of the world, 00:01:10.220 --> 00:01:13.340 and we try to understand them. 00:01:13.340 --> 00:01:14.610 We pose problems. 00:01:14.610 --> 00:01:20.710 Why-- 400 years ago, is the sun in the center 00:01:20.710 --> 00:01:23.620 of the solar system or not? 00:01:23.620 --> 00:01:32.470 And we try to produce models that explain the problem. 00:01:32.470 --> 00:01:37.530 So here's the problem, the question of the day. 00:01:37.530 --> 00:01:40.460 We try to produce models to describe it, 00:01:40.460 --> 00:01:49.370 and we make observations, measurements, to see 00:01:49.370 --> 00:01:52.000 if our models are correct. 00:01:52.000 --> 00:01:56.540 And if we feed that information back into the models, 00:01:56.540 --> 00:02:00.160 we try out the models, we test it against more observations, 00:02:00.160 --> 00:02:02.419 and you go round and round. 00:02:02.419 --> 00:02:03.960 And this is kind of the fundamental-- 00:02:03.960 --> 00:02:08.630 this is the way all of these basic first five subjects use, 00:02:08.630 --> 00:02:14.140 basically, this method of inquiry. 00:02:14.140 --> 00:02:20.220 So in 2003, the way this system works, 00:02:20.220 --> 00:02:29.140 my kind of mental conception of this modeling process, 00:02:29.140 --> 00:02:30.201 is three things. 00:02:30.201 --> 00:02:31.200 And this applies to you. 00:02:31.200 --> 00:02:32.580 You have a homework problem. 00:02:32.580 --> 00:02:34.850 How do you attack a homework problem? 00:02:34.850 --> 00:02:38.690 You're going to need to describe the motion. 00:02:44.680 --> 00:02:48.840 You're going to need to choose the physical laws-- pick, 00:02:48.840 --> 00:02:51.960 I'll call it because it's short-- 00:02:51.960 --> 00:02:59.180 the physical law that you want to apply 00:02:59.180 --> 00:03:02.210 like f equals ma, conservation of energy, 00:03:02.210 --> 00:03:03.875 conservation of momentum. 00:03:03.875 --> 00:03:06.590 You got to know which physical laws to apply. 00:03:06.590 --> 00:03:16.370 And then finally, third you need to apply the correct math. 00:03:16.370 --> 00:03:18.224 And that's really-- most dynamic problems 00:03:18.224 --> 00:03:19.390 can be broken down this way. 00:03:19.390 --> 00:03:21.530 That's the way I like to conceptually break them down. 00:03:21.530 --> 00:03:23.300 You might have another model, but this is 00:03:23.300 --> 00:03:25.450 the way I'm going to teach it. 00:03:25.450 --> 00:03:28.210 Can you describe the motion, pick the correct physical laws 00:03:28.210 --> 00:03:32.630 to apply to the problem, and able to do the correct math, 00:03:32.630 --> 00:03:35.260 solving the equation of motion, for example. 00:03:35.260 --> 00:03:39.550 And all this is what fits in our models box. 00:03:39.550 --> 00:03:42.400 And we test it against observations and measurements 00:03:42.400 --> 00:03:45.970 and improve those things over time. 00:03:45.970 --> 00:03:50.920 So I'm going to give you-- how many of you like history? 00:03:50.920 --> 00:03:52.884 I find history and history technology 00:03:52.884 --> 00:03:54.050 kind of fun and interesting. 00:03:54.050 --> 00:03:55.924 So I'm going to throw a little bit of history 00:03:55.924 --> 00:03:58.320 into giving you a little quick course 00:03:58.320 --> 00:04:05.000 outline of what we're going to do in this subject this term. 00:04:12.300 --> 00:04:15.150 Because the history dynamics and what 00:04:15.150 --> 00:04:18.130 we're going to do in the course actually track one another 00:04:18.130 --> 00:04:20.110 remarkably closely. 00:04:20.110 --> 00:04:23.410 So if I ever gave you a bunch of names like Galileo, Kepler, 00:04:23.410 --> 00:04:29.280 Descartes, Newton, Copernicus, Euler, Lagrange and Brahe, 00:04:29.280 --> 00:04:30.280 which one comes first? 00:04:35.490 --> 00:04:38.384 Take a guess. 00:04:38.384 --> 00:04:40.764 AUDIENCE: Copernicus. 00:04:40.764 --> 00:04:41.940 PROFESSOR: Good. 00:04:41.940 --> 00:04:42.440 Copernicus. 00:04:48.710 --> 00:04:53.260 So Copernicus was Polish, and the story 00:04:53.260 --> 00:04:58.550 starts long before then, but in about 1,500 Copernicus 00:04:58.550 --> 00:05:00.618 said what? 00:05:00.618 --> 00:05:01.550 AUDIENCE: [INAUDIBLE] 00:05:04.346 --> 00:05:06.230 PROFESSOR: The sun's the center? 00:05:06.230 --> 00:05:07.950 AUDIENCE: [INAUDIBLE] 00:05:07.950 --> 00:05:09.625 PROFESSOR: Or the Earth is the center? 00:05:09.625 --> 00:05:10.310 AUDIENCE: [INAUDIBLE] 00:05:10.310 --> 00:05:11.430 PROFESSOR: Which did say? 00:05:11.430 --> 00:05:14.320 Yes, so Ptolemy, back around 130 AD said, 00:05:14.320 --> 00:05:18.050 well the Earth's the center of the solar system. 00:05:18.050 --> 00:05:19.720 Copernicus came along and said, nope 00:05:19.720 --> 00:05:23.080 I think that, in fact, the sun's the center of the solar system. 00:05:23.080 --> 00:05:27.042 And it for the next 100 years-- more than 100 years, couple 00:05:27.042 --> 00:05:29.250 hundred years-- there was a really raging controversy 00:05:29.250 --> 00:05:30.250 about that. 00:05:30.250 --> 00:05:38.520 So Copernicus, Brahe Kepler-- so I'm 00:05:38.520 --> 00:05:40.520 putting them in rough chronological order here. 00:05:40.520 --> 00:05:41.995 Now, I'm going to run out of board. 00:05:41.995 --> 00:05:42.710 Oh well. 00:05:45.370 --> 00:06:23.130 Galileo, Descartes-- I'm gonna cheat-- OK, Descartes, Newton, 00:06:23.130 --> 00:06:26.580 Euler, and Lagrange. 00:06:30.795 --> 00:06:33.420 So we're going to talk and say a little bit about each of them. 00:06:33.420 --> 00:06:34.955 And now that I'm-- like I told you, 00:06:34.955 --> 00:06:37.330 I haven't used this classroom before so I gotta learn how 00:06:37.330 --> 00:06:39.740 to play this game. 00:06:39.740 --> 00:06:41.780 I need to be able to reach this for a minute. 00:06:41.780 --> 00:06:44.900 So Brahe, he was along about 1,600. 00:06:44.900 --> 00:06:47.450 Brahe was the mathematician that wrote-- 00:06:47.450 --> 00:06:51.060 the imperial mathematician to the emperor in Prague. 00:06:51.060 --> 00:06:54.530 And he did 20 years of observations. 00:06:54.530 --> 00:06:57.020 And he was out to prove that the Earth was 00:06:57.020 --> 00:06:59.190 the center of the solar system. 00:06:59.190 --> 00:07:02.460 And then Kepler actually worked with him as a mathematician, 00:07:02.460 --> 00:07:07.250 and then took over as the imperial mathematician. 00:07:07.250 --> 00:07:12.160 And he took Brahe's data-- 20 years of astronomical data 00:07:12.160 --> 00:07:14.370 without the use of the telescope-- 00:07:14.370 --> 00:07:20.720 and used it come up with the three laws of planetary motion. 00:07:20.720 --> 00:07:26.130 And so his first and second laws were put out about 1609. 00:07:26.130 --> 00:07:28.650 And one of the laws is, like, equal area 00:07:28.650 --> 00:07:30.160 swept out in equal time. 00:07:30.160 --> 00:07:31.330 Have you hear that one? 00:07:31.330 --> 00:07:34.620 That actually turns out to be a statement of conservation 00:07:34.620 --> 00:07:37.100 of angular momentum, which we'll talk quite a bit 00:07:37.100 --> 00:07:38.960 about the course. 00:07:38.960 --> 00:07:43.520 Then came Galileo, and I'm not putting their birth and death 00:07:43.520 --> 00:07:44.020 dates here. 00:07:44.020 --> 00:07:45.700 I'm kind of putting in a period of time 00:07:45.700 --> 00:07:49.070 in which kind of important things happened around him. 00:07:49.070 --> 00:07:54.670 So 401 years ago a really important thing happened. 00:07:54.670 --> 00:08:00.460 Galileo, in 1609, turned the telescope on Jupiter, 00:08:00.460 --> 00:08:02.192 and saw what? 00:08:02.192 --> 00:08:03.104 AUDIENCE: [INAUDIBLE] 00:08:03.104 --> 00:08:05.420 PROFESSOR: Four moons, right? 00:08:05.420 --> 00:08:08.060 And then they really started having some data 00:08:08.060 --> 00:08:12.000 with which to really argue against the Ptolymaic view 00:08:12.000 --> 00:08:13.840 of the solar system. 00:08:16.800 --> 00:08:21.095 Descartes is an important figure to us. 00:08:21.095 --> 00:08:26.950 And in the period of about 1630 to 1644-- in that period 00:08:26.950 --> 00:08:32.809 Descartes began what is today known as analytic geometry. 00:08:32.809 --> 00:08:36.130 He was geometer, he studied Euclid a lot. 00:08:36.130 --> 00:08:40.110 But then he came up with a Cartesian coordinate system, 00:08:40.110 --> 00:08:43.370 xyz, and the beginnings of analytic geometry, 00:08:43.370 --> 00:08:47.790 which is essentially algebra, coordinates, and geometry all 00:08:47.790 --> 00:08:48.540 put together. 00:08:48.540 --> 00:08:52.170 And we are going to make great use of analytic geometry 00:08:52.170 --> 00:08:53.830 in this course. 00:08:53.830 --> 00:08:59.433 Then came Newton, kind of in his actual lifespan, 1643. 00:08:59.433 --> 00:09:04.070 It's kind of interesting that he spans these people. 00:09:04.070 --> 00:09:10.174 And in about 1666 is when he first-- the first statement 00:09:10.174 --> 00:09:11.340 of the three laws of motion. 00:09:15.370 --> 00:09:30.430 Then Euler, and he's 1707 to 1783, and that's his lifespan. 00:09:30.430 --> 00:09:34.780 Euler came up-- Newton never talked about angular momentum. 00:09:34.780 --> 00:09:36.260 He mostly talked about particles. 00:09:36.260 --> 00:09:40.950 Euler put Newton's three laws into mathematics. 00:09:40.950 --> 00:09:44.810 Euler taught us about angular momentum, 00:09:44.810 --> 00:09:53.140 and torque being dh dt in most cases. 00:09:53.140 --> 00:09:55.630 He's the most prolific mathematician all time, 00:09:55.630 --> 00:09:58.230 solved all sorts of important problems. 00:09:58.230 --> 00:10:00.410 And then finally, is Lagrange. 00:10:00.410 --> 00:10:08.740 And Lagrange, in about 1788, uses an energy method, energy 00:10:08.740 --> 00:10:17.140 and the concept of work to give us equations of motion. 00:10:17.140 --> 00:10:24.940 So the course, 203, stands on the shoulders 00:10:24.940 --> 00:10:26.140 of all these people. 00:10:26.140 --> 00:10:29.810 But with Descartes, we start with kinematics, really. 00:10:29.810 --> 00:10:33.350 This is analytic geometry. 00:10:33.350 --> 00:10:36.200 And that's where we're going to start today is with kinematics. 00:10:36.200 --> 00:10:37.840 And very soon thereafter, we're going 00:10:37.840 --> 00:10:40.210 to review Newton, the three laws, and what 00:10:40.210 --> 00:10:45.400 we call the direct method for finding equations of motion. 00:10:45.400 --> 00:10:48.020 Conservation of momentum, fact that 00:10:48.020 --> 00:10:50.070 force-- some of the forces on an object 00:10:50.070 --> 00:10:53.390 equals mass times acceleration, or it's a time derivative 00:10:53.390 --> 00:10:55.730 of its linear momentum. 00:10:55.730 --> 00:10:59.350 And we use that to derive equations of motion. 00:10:59.350 --> 00:11:05.490 So we're going to go kinematics into doing the direct method 00:11:05.490 --> 00:11:09.580 to getting equations of motion. 00:11:09.580 --> 00:11:13.750 And we go from there into angular momentum, 00:11:13.750 --> 00:11:18.500 and what Euler gave us-- the same thing, torque. 00:11:18.500 --> 00:11:21.840 We're going to do quite a lot with angular momentum. 00:11:21.840 --> 00:11:25.020 Because I know you know a lot about f equals ma 00:11:25.020 --> 00:11:28.380 and you've done lots of problems 801 applying that. 00:11:28.380 --> 00:11:30.960 You've done some problems on rigid body rotations. 00:11:30.960 --> 00:11:33.084 But I think there's a lot more you 00:11:33.084 --> 00:11:34.750 need to understand about this, and we'll 00:11:34.750 --> 00:11:36.083 spend quite a bit of time on it. 00:11:39.100 --> 00:11:42.280 And then near the last third the course 00:11:42.280 --> 00:11:47.800 we shift, because Lagrange said that if you just write down 00:11:47.800 --> 00:11:53.030 expressions for energy, kinetic and potential energy, 00:11:53.030 --> 00:11:57.340 without any consideration of Newton's laws 00:11:57.340 --> 00:12:01.769 and the direct method, you can derive the equations of motion. 00:12:01.769 --> 00:12:02.810 That's pretty remarkable. 00:12:02.810 --> 00:12:06.200 So there are actually two independent roots to coming up 00:12:06.200 --> 00:12:07.650 with equations of motion. 00:12:07.650 --> 00:12:10.400 And in this course, about the last third of the course, 00:12:10.400 --> 00:12:13.400 we're going to teach you about Lagrange. 00:12:13.400 --> 00:12:17.320 And then all these things are going 00:12:17.320 --> 00:12:20.522 to be-- one of the applications that are important engineers 00:12:20.522 --> 00:12:21.605 is the study of vibration. 00:12:25.320 --> 00:12:28.540 So we'll be looking at vibration examples 00:12:28.540 --> 00:12:32.250 as we go through the course, and applying 00:12:32.250 --> 00:12:36.090 these different methods to first, modeling, 00:12:36.090 --> 00:12:38.455 and then solving interesting vibration problems. 00:12:40.960 --> 00:12:42.980 Which brings-- ah, I have a question for you. 00:12:42.980 --> 00:12:46.270 So how many of you were in this classroom last May 00:12:46.270 --> 00:12:49.060 with Professor Haynes Miller, and I showed up one day 00:12:49.060 --> 00:12:51.420 and we talked about vibration? 00:12:51.420 --> 00:12:52.220 How many remember? 00:12:52.220 --> 00:12:54.480 I told you I was going to ask this question, right? 00:12:54.480 --> 00:12:55.000 Great. 00:12:55.000 --> 00:12:57.670 OK, it's good to see you here again, 00:12:57.670 --> 00:13:01.400 and we will talk about vibration in this course. 00:13:01.400 --> 00:13:03.750 So there's kind of the subject outline built 00:13:03.750 --> 00:13:07.350 on the shoulders of these people in history 00:13:07.350 --> 00:13:09.400 that made important contributions to dynamics. 00:13:16.020 --> 00:13:17.790 Any questions about the history? 00:13:17.790 --> 00:13:19.920 If you want to know, one of my TAs 00:13:19.920 --> 00:13:22.590 compiled a pretty neat little summary. 00:13:22.590 --> 00:13:24.880 Maybe I will see if I go back and find this. 00:13:24.880 --> 00:13:26.810 I just printed out and sent it-- how many of 00:13:26.810 --> 00:13:29.018 you like to know a little bit more about the history? 00:13:29.018 --> 00:13:31.194 These are like two liners on each person. 00:13:31.194 --> 00:13:31.860 Anybody want it? 00:13:31.860 --> 00:13:33.580 Is it worth my time to send this out? 00:13:33.580 --> 00:13:36.450 OK, it's kind of fun. 00:13:36.450 --> 00:13:45.980 So let's do an example of this modeling describing the motion, 00:13:45.980 --> 00:13:49.760 picking physical laws, applying the math. 00:13:49.760 --> 00:13:55.150 And that'll get us launched in the course. 00:13:55.150 --> 00:13:58.890 And we'll do it using Newton and the direct method. 00:14:19.510 --> 00:14:24.760 So last May, Haynes Miller and I talked about vibration. 00:14:24.760 --> 00:14:27.400 So I'm going to start with a vibration problem. 00:14:30.180 --> 00:14:31.310 And I brought one. 00:14:31.310 --> 00:14:35.370 So here's my couple of lead weights 00:14:35.370 --> 00:14:37.262 and a couple of springs. 00:14:37.262 --> 00:14:39.720 So really I just want to talk about-- this is the problem I 00:14:39.720 --> 00:14:40.960 want to talk about. 00:14:40.960 --> 00:14:42.560 Now you've done this problem before. 00:14:42.560 --> 00:14:46.480 Haynes Miller and I did it last May. 00:14:46.480 --> 00:14:48.870 And you've no doubt it in other classes. 00:14:48.870 --> 00:14:51.830 OK, it's a system which has a spring, a mass, 00:14:51.830 --> 00:14:54.310 it exhibits something called a natural frequency. 00:14:54.310 --> 00:14:58.220 But let's see what it takes to just initially begin 00:14:58.220 --> 00:15:03.410 to follow this modeling method to arrive 00:15:03.410 --> 00:15:05.160 at an equation of motion for this problem. 00:15:14.770 --> 00:15:17.930 So what do I mean by when I say, describe the motion? 00:15:17.930 --> 00:15:19.590 Really what that boils down to if you 00:15:19.590 --> 00:15:23.240 have to assign a coordinate system 00:15:23.240 --> 00:15:26.419 so that you can actually say where the object's moving. 00:15:26.419 --> 00:15:27.710 And I'm going to pick one here. 00:15:27.710 --> 00:15:30.980 So here's-- coordinate system going to be really important 00:15:30.980 --> 00:15:31.660 in this course. 00:15:34.480 --> 00:15:41.670 And I'll give us an xyz Cartesian coordinate system. 00:15:44.680 --> 00:15:48.250 And I'm going to try to adopt the habit, for the most part, 00:15:48.250 --> 00:15:51.960 during the course that this o marks this origin, 00:15:51.960 --> 00:15:54.340 but it also names the frame. 00:15:54.340 --> 00:15:56.650 So we're going to talk about things 00:15:56.650 --> 00:15:58.860 in that are reference frames. 00:15:58.860 --> 00:16:01.500 And most important one that we need 00:16:01.500 --> 00:16:04.080 to know about in the course is an inertial reference frame, 00:16:04.080 --> 00:16:06.250 and when you can use it, and when a system 00:16:06.250 --> 00:16:08.210 is inertial and is not. 00:16:08.210 --> 00:16:10.691 So I'm gonna say that this is inertial. 00:16:10.691 --> 00:16:11.690 It's fixed to the Earth. 00:16:11.690 --> 00:16:12.960 It's not moving. 00:16:12.960 --> 00:16:17.250 And we're going to use this coordinate x to describe 00:16:17.250 --> 00:16:19.110 the motion of this mass. 00:16:19.110 --> 00:16:31.560 And the motion is going to be-- this x is from the zero spring 00:16:31.560 --> 00:16:33.850 force position. 00:16:33.850 --> 00:16:35.440 It's actually quite important that you 00:16:35.440 --> 00:16:38.230 pick-- that you have to say what's 00:16:38.230 --> 00:16:42.580 the condition in the spring of the system when x is 0 00:16:42.580 --> 00:16:44.300 So we're going to say it's, when there's 00:16:44.300 --> 00:16:45.841 no force in the spring means it's not 00:16:45.841 --> 00:16:48.782 stretch, that's where 0 is. 00:16:48.782 --> 00:16:50.490 So we've established a coordinate system. 00:16:55.370 --> 00:16:57.630 Second, we need to apply physical laws. 00:17:07.300 --> 00:17:11.800 Now, I'm going to do this problem by f equals 00:17:11.800 --> 00:17:13.680 ma, Newton's second law. 00:17:13.680 --> 00:17:16.190 Sum of the external forces is equal to mass 00:17:16.190 --> 00:17:17.740 times the acceleration. 00:17:17.740 --> 00:17:21.460 So that's the law I'm going to apply. 00:17:21.460 --> 00:17:25.700 Sum of the external forces, it's a vector 00:17:25.700 --> 00:17:28.640 but we're just doing the x component only so we don't have 00:17:28.640 --> 00:17:32.930 to carry along vector notation, is equal to, in this case, 00:17:32.930 --> 00:17:36.350 mass times acceleration. 00:17:36.350 --> 00:17:38.520 So that's the law we're going to apply. 00:17:38.520 --> 00:17:44.975 And then finally the math to solve the equation of motion 00:17:44.975 --> 00:17:46.970 that we find, that'll be the third piece. 00:17:46.970 --> 00:17:49.090 But part of applying the physics, in order 00:17:49.090 --> 00:17:57.230 to do this now, we need what I call an FBD. 00:17:57.230 --> 00:17:58.602 What do you suppose that is? 00:17:58.602 --> 00:17:59.810 AUDIENCE: Free body diagrams. 00:17:59.810 --> 00:18:01.400 PROFESSOR: Free body diagrams. 00:18:01.400 --> 00:18:03.140 You've used these many times before, 00:18:03.140 --> 00:18:04.950 so we're going to do those. 00:18:04.950 --> 00:18:10.210 And free body diagrams-- 00:18:10.210 --> 00:18:12.780 And I'm going to teach you, at least 00:18:12.780 --> 00:18:14.960 the way I go about doing free body diagrams, 00:18:14.960 --> 00:18:17.150 as things get more and more complicated, 00:18:17.150 --> 00:18:20.280 you're going to have to be more sophisticated in the way 00:18:20.280 --> 00:18:22.350 that you do these things. 00:18:22.350 --> 00:18:25.750 So I just have some simple little rules 00:18:25.750 --> 00:18:29.610 to do free body diagrams that keep you from getting hung up 00:18:29.610 --> 00:18:31.480 on sign conventions. 00:18:31.480 --> 00:18:33.940 I think the thing people make most mistakes about is 00:18:33.940 --> 00:18:36.030 they get confused about signs. 00:18:36.030 --> 00:18:39.390 So I'll try to show you how I do it. 00:18:39.390 --> 00:18:54.540 So first you draw forces that you know, basically 00:18:54.540 --> 00:18:56.720 in the direction in which they act. 00:18:56.720 --> 00:18:59.150 Seems obvious. 00:18:59.150 --> 00:19:07.115 So when you know the direction-- so 00:19:07.115 --> 00:19:08.490 this is a really trivial problem, 00:19:08.490 --> 00:19:17.590 but the method here is very specific. 00:19:17.590 --> 00:19:18.780 So what's an example? 00:19:18.780 --> 00:19:20.070 Well, gravity. 00:19:20.070 --> 00:19:23.960 So we'll start our free body diagram. 00:19:23.960 --> 00:19:26.490 Gravity acts at the center of mass. 00:19:26.490 --> 00:19:27.550 It's downward. 00:19:27.550 --> 00:19:29.970 This is what I mean by the direction in which it acts. 00:19:29.970 --> 00:19:34.120 And it has magnitude, mg. 00:19:34.120 --> 00:19:35.740 OK. 00:19:35.740 --> 00:19:38.220 Now the other forces aren't so obvious. 00:19:38.220 --> 00:19:41.050 The force that's put on by the stiffness and this damper 00:19:41.050 --> 00:19:43.600 in the spring, which way do you draw them? 00:19:43.600 --> 00:19:44.370 What's the sign? 00:19:44.370 --> 00:19:46.950 What's the sign convention? 00:19:46.950 --> 00:19:49.560 So the convention, the way I go about doing these things, 00:19:49.560 --> 00:20:08.900 is I assume positive values for the deflections and velocities. 00:20:08.900 --> 00:20:11.960 So in this case, x and x dot. 00:20:11.960 --> 00:20:16.530 You just require that the deflections that you're going 00:20:16.530 --> 00:20:18.920 to work with are positive. 00:20:18.920 --> 00:20:20.650 And then from the positive deflection, 00:20:20.650 --> 00:20:22.660 you say which way is the resulting force? 00:20:22.660 --> 00:20:25.650 So if the deflection in this is downwards, 00:20:25.650 --> 00:20:29.540 which direction is the force that the spring applies 00:20:29.540 --> 00:20:32.020 to the mass? 00:20:32.020 --> 00:20:32.690 Up, right? 00:20:32.690 --> 00:20:35.660 What about if the velocity is downwards, 00:20:35.660 --> 00:20:41.360 which direction is the force is the damper puts on the mass? 00:20:41.360 --> 00:20:42.080 Also up, right? 00:20:42.080 --> 00:20:42.580 OK. 00:20:42.580 --> 00:20:48.720 So this allows-- this gives us-- so here's f spring and here's 00:20:48.720 --> 00:20:50.070 the f damper. 00:20:50.070 --> 00:20:53.610 And other any other forces on this mass? 00:20:53.610 --> 00:20:57.190 So spring force, damper force, and the gravitational force. 00:21:07.940 --> 00:21:18.385 And so third, you deduce the signs basically 00:21:18.385 --> 00:21:19.760 from the direction of the arrows. 00:21:25.880 --> 00:21:28.470 First we need what's called your constitutive relationship. 00:21:28.470 --> 00:21:32.600 So the spring force, fs, well you've 00:21:32.600 --> 00:21:35.370 made x positive so it keeps things nice, 00:21:35.370 --> 00:21:41.340 the spring constant's a positive number, so fs is kx. 00:21:41.340 --> 00:21:45.430 Fd is bx dot. 00:21:45.430 --> 00:21:47.440 And now we write the statement that the sum 00:21:47.440 --> 00:21:49.755 of forces in the x direction. 00:21:52.490 --> 00:21:55.000 We look at up here, we say well that's going 00:21:55.000 --> 00:22:01.680 to fs plus fd minus mg. 00:22:01.680 --> 00:22:06.360 So that's-- whoops, I wrote it the wrong way around. 00:22:06.360 --> 00:22:10.170 Minus, minus, plus. 00:22:10.170 --> 00:22:12.170 Because I'm plus downwards, right? 00:22:12.170 --> 00:22:18.430 Well, spring minus fs is minus kx minus bx dot 00:22:18.430 --> 00:22:24.809 plus mg equals mx double dot. 00:22:24.809 --> 00:22:26.600 And I rearranged this to put all the motion 00:22:26.600 --> 00:22:29.720 variables on one side. 00:22:29.720 --> 00:22:37.700 mx double dot plus bx dot plus kx equals mg. 00:22:40.350 --> 00:22:43.670 So there's my equation of motion, but with a method 00:22:43.670 --> 00:22:45.710 for doing the free body diagrams, which 00:22:45.710 --> 00:22:47.684 will work with multiple bodies. 00:22:47.684 --> 00:22:49.850 So you have two bodies with springs in between them. 00:22:49.850 --> 00:22:52.740 This is when the confusion really comes up. 00:22:52.740 --> 00:22:55.940 Two bodies with a spring trapped between them. 00:22:55.940 --> 00:22:58.490 What's the sign convention? 00:22:58.490 --> 00:22:59.540 You do the same thing. 00:22:59.540 --> 00:23:02.870 Both bodies exhibit positive motions, 00:23:02.870 --> 00:23:05.747 the force that results is proportional to the difference, 00:23:05.747 --> 00:23:06.580 and you work it out. 00:23:06.580 --> 00:23:08.100 And you'll get the signs right. 00:23:08.100 --> 00:23:09.940 OK, so here's our equation of motion arrived 00:23:09.940 --> 00:23:12.720 at by doing the direct method. 00:23:12.720 --> 00:23:21.130 And if we went on to the third step, which we're not 00:23:21.130 --> 00:23:24.570 going to do today, and that is apply the math, 00:23:24.570 --> 00:23:27.430 it might because I want you now to describe the motion for me, 00:23:27.430 --> 00:23:28.520 solve for the motion. 00:23:28.520 --> 00:23:30.570 That means solving the differential equation. 00:23:30.570 --> 00:23:33.600 And that's what we did last may in Haynes Miller's class. 00:23:33.600 --> 00:23:35.560 We'll come back to this later on. 00:23:35.560 --> 00:23:40.547 But for today's purposes, we don't need to go there. 00:23:40.547 --> 00:23:42.130 Got something else much more important 00:23:42.130 --> 00:23:45.270 to get to about kinematics. 00:23:45.270 --> 00:23:47.610 But I want to show you one thing, 00:23:47.610 --> 00:23:51.700 and that is just a little tiny introductory taste 00:23:51.700 --> 00:23:54.480 to this point. 00:23:54.480 --> 00:23:57.390 So I've derived the equation of motion of this 00:23:57.390 --> 00:24:00.854 by Newton's laws. 00:24:00.854 --> 00:24:02.270 But I'm going to ignore Newton now 00:24:02.270 --> 00:24:04.144 and saw I'm going to drive equation of motion 00:24:04.144 --> 00:24:04.950 by another way. 00:24:04.950 --> 00:24:07.470 And it's an energy technique, and that is-- well 00:24:07.470 --> 00:24:11.020 let's talk about the total energy of the system. 00:24:11.020 --> 00:24:13.750 It's going to be the sum of a kinetic energy 00:24:13.750 --> 00:24:14.920 and a potential energy. 00:24:18.560 --> 00:24:21.290 And we'll find that even with Lagrange, there's 00:24:21.290 --> 00:24:24.400 a problem with forces on systems that 00:24:24.400 --> 00:24:26.430 are what we call non-conservative, 00:24:26.430 --> 00:24:28.840 things that either take energy out of, or put energy 00:24:28.840 --> 00:24:29.710 into the system. 00:24:29.710 --> 00:24:31.850 And the dashpot does that. 00:24:31.850 --> 00:24:35.469 Dashpot generates heat and takes energy out of the system. 00:24:35.469 --> 00:24:37.510 So I'm going to have to ignore it for the moment. 00:24:37.510 --> 00:24:39.750 So the sum of the kinetic and the potential energies 00:24:39.750 --> 00:24:45.680 in this problem is a 1/2 kx squared 00:24:45.680 --> 00:24:53.040 for the potential of the spring, plus a 1/2 mx dot squared 00:24:53.040 --> 00:25:00.600 for the kinetic energy of the mass, and minus mgx 00:25:00.600 --> 00:25:04.880 for the potential energy that is due to the object moving 00:25:04.880 --> 00:25:07.230 in the gravitational field. 00:25:07.230 --> 00:25:11.460 And that's the total energy of the system. 00:25:11.460 --> 00:25:14.490 Now my problem, I've allowed no forces. 00:25:14.490 --> 00:25:15.740 There's no excitation on here. 00:25:15.740 --> 00:25:18.085 This is just free vibration only. 00:25:18.085 --> 00:25:20.460 That's all we're talking about, make initial displacement 00:25:20.460 --> 00:25:22.530 and it vibrates. 00:25:22.530 --> 00:25:24.410 If there's no damping, what can you 00:25:24.410 --> 00:25:28.884 say about the total energy of the system? 00:25:28.884 --> 00:25:30.198 AUDIENCE: [INAUDIBLE] 00:25:30.198 --> 00:25:31.464 PROFESSOR: Say it again. 00:25:31.464 --> 00:25:32.380 I heard it over there. 00:25:32.380 --> 00:25:33.800 It's got to be constant, right? 00:25:33.800 --> 00:25:37.480 All right, well, so this must be constant. 00:25:37.480 --> 00:25:55.200 Therefore, the time derivative of my system, it better be 0. 00:25:55.200 --> 00:25:56.890 The energy is constant. 00:25:56.890 --> 00:25:58.860 Take it's time derivative, it's got to be 0. 00:25:58.860 --> 00:26:00.610 Apply that to the right-hand side of this, 00:26:00.610 --> 00:26:13.460 I get kxx dot plus mx dot x double dot minus mgx dot 00:26:13.460 --> 00:26:15.530 equals zero. 00:26:15.530 --> 00:26:23.690 And I now cancel out the common x dot terms go away. 00:26:23.690 --> 00:26:37.580 And I'm left with-- and I've essentially 00:26:37.580 --> 00:26:40.600 solved for the equation of motion of this system 00:26:40.600 --> 00:26:44.010 without ever looking at conservational momentum, 00:26:44.010 --> 00:26:46.695 Newton's laws, only by energy considerations. 00:26:50.300 --> 00:26:52.570 OK, so that's a very simple example 00:26:52.570 --> 00:26:55.350 of that you can use energy to derive equations of motions. 00:26:55.350 --> 00:27:00.840 But you then have to go back and fix it to account for the loss 00:27:00.840 --> 00:27:01.977 term, the damping term. 00:27:01.977 --> 00:27:04.060 And that you still have to consider it as a force, 00:27:04.060 --> 00:27:04.800 we'll find out. 00:27:04.800 --> 00:27:09.920 Even was Lagrange you have to go back and consider the work done 00:27:09.920 --> 00:27:13.050 by external forces. 00:27:13.050 --> 00:27:15.320 OK. 00:27:15.320 --> 00:27:18.010 So you've just kind of seen the whole course. 00:27:18.010 --> 00:27:19.390 We've described the motion, we've 00:27:19.390 --> 00:27:24.090 applied to Newton's laws, the physics to the direct method 00:27:24.090 --> 00:27:26.360 to derive the equations of motion, 00:27:26.360 --> 00:27:31.540 we have gone to a direct method, and have derived the equations 00:27:31.540 --> 00:27:32.660 of motion that way. 00:27:32.660 --> 00:27:35.290 And that's basically what you're going to do in the course. 00:27:35.290 --> 00:27:37.630 But now you're going to do it with much more 00:27:37.630 --> 00:27:39.880 sophisticated tools. 00:27:39.880 --> 00:27:42.160 You'll have multiple degree of freedom systems. 00:27:42.160 --> 00:27:44.520 The description describing the motion, 00:27:44.520 --> 00:27:46.230 is maybe going to be for some of you, 00:27:46.230 --> 00:27:48.410 the most challenging part of the course. 00:27:48.410 --> 00:27:50.615 And this is a topic we call kinematics. 00:27:53.960 --> 00:27:59.380 And that's what we'll turn to next. 00:28:13.330 --> 00:28:15.420 So reference frames and vectors. 00:28:15.420 --> 00:28:16.840 That's the topic. 00:28:16.840 --> 00:28:20.110 This is now that we're talking about kinematics, 00:28:20.110 --> 00:28:24.610 and this is all about describing the motion. 00:28:24.610 --> 00:28:29.330 So Descartes gave us the Cartesian coordinate system, 00:28:29.330 --> 00:28:30.340 and we'll start there. 00:28:30.340 --> 00:28:34.080 So imagine this is a fixed frame-- 00:28:34.080 --> 00:28:36.600 we'll talk about what makes an inertial frame 00:28:36.600 --> 00:28:38.300 the next lecture. 00:28:38.300 --> 00:28:39.840 But here we have an inertial frame. 00:28:42.860 --> 00:28:51.620 And it's the frame we'll call O-xyz or O for short. 00:28:51.620 --> 00:29:05.720 And in this frame, maybe this is me, and up here is a dog, 00:29:05.720 --> 00:29:12.840 and I'm going to call this point A and this point B. 00:29:12.840 --> 00:29:18.190 And I'm going to describe the positions of these two points 00:29:18.190 --> 00:29:19.890 by vectors. 00:29:19.890 --> 00:29:24.050 This one will be R, and the notation that I'm going to use 00:29:24.050 --> 00:29:30.330 is point and it's measurement with respect something. 00:29:30.330 --> 00:29:34.980 Well, it's with respect to this point O in this inertial frame. 00:29:34.980 --> 00:29:38.910 So this is A with respect O is the way to read this. 00:29:38.910 --> 00:29:41.310 There's another vector here. 00:29:41.310 --> 00:29:47.440 This is RB respect to A And finally, 00:29:47.440 --> 00:29:53.190 R of B with respect to O They're all vectors on the board. 00:29:53.190 --> 00:29:55.660 I'll try to remember to underline them 00:29:55.660 --> 00:29:56.860 in the textbooks and things. 00:29:56.860 --> 00:30:00.420 They're usually-- vectors are noted with bold letters. 00:30:03.970 --> 00:30:07.930 And vectors allow us to say the following. 00:30:07.930 --> 00:30:13.180 That R, the position of the dog and the reference 00:30:13.180 --> 00:30:16.690 with respect to O, is the sum of these other two vectors. 00:30:16.690 --> 00:30:26.390 R of A with respect to O plus R R of B with respect to A. 00:30:26.390 --> 00:30:29.200 And mostly to do dynamics we're really 00:30:29.200 --> 00:30:32.040 interested in things like velocities and accelerations. 00:30:32.040 --> 00:30:34.010 So to get the velocities and accelerations, 00:30:34.010 --> 00:30:39.095 we have to take a time derivative of our RBO dt. 00:30:41.730 --> 00:30:45.840 And that's going to give us what we'll call the velocity, 00:30:45.840 --> 00:30:48.040 obviously you write it as V. And it would 00:30:48.040 --> 00:30:53.060 be the velocity of point B with respect to O. And no surprise, 00:30:53.060 --> 00:30:58.970 it'll be the velocity of point A plus the velocity of B 00:30:58.970 --> 00:31:04.330 with respect to A. 00:31:04.330 --> 00:31:13.200 And finally, if we take two derivatives, dt squared, 00:31:13.200 --> 00:31:17.690 we'll get the acceleration of B with respect to O. 00:31:17.690 --> 00:31:22.730 And that'll be the sum of A-- the acceleration of A 00:31:22.730 --> 00:31:28.880 with respect to O plus the acceleration of B 00:31:28.880 --> 00:31:32.420 with respect to A. All, again, vectors. 00:31:38.490 --> 00:31:41.502 Now, just to look ahead-- this seems all really trivial. 00:31:41.502 --> 00:31:43.210 You guys are going to sleep on me, right? 00:31:46.620 --> 00:31:52.460 If these are rigid bodies, this is a rigid body that is moving 00:31:52.460 --> 00:31:55.360 and maybe rotating. 00:31:55.360 --> 00:32:01.370 And B is on it, and A is on it, and O isn't on it. 00:32:01.370 --> 00:32:05.000 It starts getting a little tricky. 00:32:05.000 --> 00:32:12.000 And this, the derivative of a vector that's attached 00:32:12.000 --> 00:32:19.220 to the body somehow has to account for the fact that 00:32:19.220 --> 00:32:23.530 if I'm-- the observer's on the body, 00:32:23.530 --> 00:32:25.010 this other point's on the body. 00:32:25.010 --> 00:32:28.550 Say it's, I'm on this asteroid, and I've got a dog out there, 00:32:28.550 --> 00:32:31.000 and the dog's run away from me. 00:32:31.000 --> 00:32:36.460 The speed of the dog with respect to me, I can measure. 00:32:36.460 --> 00:32:37.949 But if I'm down here looking at it, 00:32:37.949 --> 00:32:39.740 it'll look different because it's rotating. 00:32:39.740 --> 00:32:41.198 So how do you account for all that? 00:32:41.198 --> 00:32:44.810 So taking these derivatives of vectors in moving frames 00:32:44.810 --> 00:32:48.590 is where the devil's in the details. 00:32:48.590 --> 00:32:54.190 And that's part of what I'm going to be teaching you. 00:32:54.190 --> 00:32:55.620 OK. 00:32:55.620 --> 00:32:59.065 I'm still learning how to optimize my board use. 00:32:59.065 --> 00:33:00.940 I haven't got it perfect yet, but because I'm 00:33:00.940 --> 00:33:02.898 having to move around a lot here and improvise. 00:33:02.898 --> 00:33:04.670 But we'll persevere. 00:33:04.670 --> 00:33:09.160 You need to remember a couple things about vectors, 00:33:09.160 --> 00:33:13.916 how to add them, dot products. 00:33:13.916 --> 00:33:15.290 If you've forgotten these things, 00:33:15.290 --> 00:33:18.469 you need to go back and review them really quickly. 00:33:18.469 --> 00:33:20.510 There's usually a little review section the book, 00:33:20.510 --> 00:33:22.301 so you need to practice that sort of thing. 00:33:24.700 --> 00:33:26.710 Couple other little facts you need to remember. 00:33:26.710 --> 00:33:32.687 So the derivative of the sum of two vectors 00:33:32.687 --> 00:33:34.145 is just the sum of the derivatives. 00:33:42.130 --> 00:33:45.180 And quite importantly, we're going 00:33:45.180 --> 00:33:48.170 to make use of this one a lot, is the derivative 00:33:48.170 --> 00:33:49.760 of a product of two things. 00:33:49.760 --> 00:33:53.340 One of them be in a vector, some function maybe of time 00:33:53.340 --> 00:34:01.840 and a here is derivative of f with respect to t times a, 00:34:01.840 --> 00:34:11.139 plus the derivative of a with respect to t times f. 00:34:11.139 --> 00:34:12.750 That we'll make a lot use of. 00:34:12.750 --> 00:34:16.350 So just your basic calculus. 00:34:16.350 --> 00:34:20.070 So now, I want to take up-- let's talk about the simplest 00:34:20.070 --> 00:34:22.690 form of being able to do these derivatives 00:34:22.690 --> 00:34:27.429 and calculate these velocities, when everything's 00:34:27.429 --> 00:34:33.199 described in terms of Cartesian coordinates. 00:34:33.199 --> 00:34:36.946 Now I'm going to give you a little look ahead because I'm 00:34:36.946 --> 00:34:42.620 going to try to avoid confusion as much as possible here. 00:34:42.620 --> 00:34:47.204 The hardest problem is when you have a rigid body, 00:34:47.204 --> 00:34:49.570 you got the dog on it, you've got the observer on it, 00:34:49.570 --> 00:34:52.300 it's rotating, and translating. 00:34:52.300 --> 00:34:56.260 And to take this derivative, you end up with a number of terms. 00:34:56.260 --> 00:34:58.470 The simplest problem is just something 00:34:58.470 --> 00:35:00.410 in a fixed Cartesian coordinate system. 00:35:00.410 --> 00:35:02.240 So we're going to start with a simple one, 00:35:02.240 --> 00:35:04.239 and build our way up to the complicated one, OK? 00:35:07.030 --> 00:35:10.650 But let's now, we're going to do the really, the simplest one. 00:35:10.650 --> 00:35:15.050 We're going to do velocity and acceleration 00:35:15.050 --> 00:35:16.980 in Cartesian coordinates. 00:35:23.750 --> 00:35:29.750 And basically I should say fixed Cartesian coordinates, 00:35:29.750 --> 00:35:30.410 not moving. 00:35:33.160 --> 00:35:37.900 All right, so now let's consider the dog out here, 00:35:37.900 --> 00:35:42.840 and his position in the Cartesian coordinate system. 00:35:42.840 --> 00:35:45.620 And I could write that and you'll, 00:35:45.620 --> 00:35:48.250 without any loss of generality here, you'll 00:35:48.250 --> 00:35:52.427 know what I mean if I say RBx component. 00:35:52.427 --> 00:35:54.260 And I'm going to stop writing the slash O's, 00:35:54.260 --> 00:35:57.320 because this is now all in this fixed reference frame. 00:35:57.320 --> 00:36:01.080 And it's in I-hat direction. 00:36:01.080 --> 00:36:10.590 And I've got another component, RBy in the J-hat, and an RBz 00:36:10.590 --> 00:36:13.344 in the K-hat. 00:36:13.344 --> 00:36:15.010 And I want to take the time derivative-- 00:36:15.010 --> 00:36:16.301 I was looking for the velocity. 00:36:16.301 --> 00:36:18.680 I want to calculate the velocity. 00:36:18.680 --> 00:36:25.175 So the velocity here of BNO is d by dt of RBO. 00:36:25.175 --> 00:36:28.070 . 00:36:28.070 --> 00:36:31.200 And now this is now the product of two things, 00:36:31.200 --> 00:36:33.950 so I've got to use that formula over here. 00:36:33.950 --> 00:36:35.960 Product one turn times the other, and so forth. 00:36:35.960 --> 00:36:39.330 So I go to these, and I say OK, so this 00:36:39.330 --> 00:36:53.690 is R dot Bx times I plus R dot By times J plus R dot 00:36:53.690 --> 00:37:00.610 Bz times K. And then the other-- the flip side of that is I 00:37:00.610 --> 00:37:06.650 have to take the derivatives of I times RBx, the derivative J 00:37:06.650 --> 00:37:07.810 and so forth. 00:37:07.810 --> 00:37:10.940 But what's the derivative of, let's say, I? 00:37:10.940 --> 00:37:15.380 Capital I is my unit vector in the fixed reference 00:37:15.380 --> 00:37:17.795 frame, my O-xyz frame. 00:37:17.795 --> 00:37:19.045 0 So it's a constant. 00:37:19.045 --> 00:37:23.390 It is unit length, and it points in a direction that it's fixed. 00:37:23.390 --> 00:37:25.240 So what's its derivative? 00:37:25.240 --> 00:37:26.990 It's going to have a 0 derivative. 00:37:26.990 --> 00:37:30.770 So the second part of this-- second bits of that is zero. 00:37:30.770 --> 00:37:36.480 So that's the velocity in Cartesian coordinates of my dog 00:37:36.480 --> 00:37:40.750 out there running around. 00:37:40.750 --> 00:37:43.477 And the acceleration, in a similar way, 00:37:43.477 --> 00:37:45.810 now to get the acceleration, you take another derivative 00:37:45.810 --> 00:37:46.309 of this. 00:37:46.309 --> 00:37:48.920 And again, you'll have to take derivatives of I, J, and K, 00:37:48.920 --> 00:37:50.620 and again they're going to be 0. 00:37:50.620 --> 00:37:57.300 So you will find that the acceleration then, is just R 00:37:57.300 --> 00:38:02.910 double dot x term in the plus R double 00:38:02.910 --> 00:38:10.485 dot By in the J plus r double dot Bz in the K. 00:38:10.485 --> 00:38:12.610 That would be our acceleration term, and it's easy. 00:38:20.219 --> 00:38:22.760 Now imagine that we are doing this in polar coordinates, unit 00:38:22.760 --> 00:38:25.390 vectors in polar coordinates. 00:38:25.390 --> 00:38:27.350 Let me check, last year the students told me 00:38:27.350 --> 00:38:30.110 that in your physics courses, you 00:38:30.110 --> 00:38:34.380 use unit vectors R-hat, theta-hat, and K. Is 00:38:34.380 --> 00:38:35.450 that right? 00:38:35.450 --> 00:38:38.580 So I'll use those unit vectors so they look familiar, 00:38:38.580 --> 00:38:40.090 because in polar coordinates people 00:38:40.090 --> 00:38:41.760 use lots of different things. 00:38:41.760 --> 00:38:45.710 But think about it, in polar coordinates, theta-- 00:38:45.710 --> 00:38:48.460 it's a fixed, maybe, coordinate system, 00:38:48.460 --> 00:38:54.020 but now theta goes like this and R moves with theta, right? 00:38:54.020 --> 00:38:55.550 So the unit vector is pointing here, 00:38:55.550 --> 00:38:58.070 but over time it might move down to here. 00:38:58.070 --> 00:39:01.710 And unit vector has changed direction, 00:39:01.710 --> 00:39:04.210 and its derivative in time is no longer 0. 00:39:04.210 --> 00:39:06.810 So it starts getting messy as soon as the unit vectors 00:39:06.810 --> 00:39:07.680 change in time. 00:39:07.680 --> 00:39:10.690 And so that's one of our objectives here 00:39:10.690 --> 00:39:12.420 is to get to that point and describe 00:39:12.420 --> 00:39:14.130 how you handle those cases. 00:39:46.450 --> 00:39:51.460 So a quick point about velocity. 00:39:51.460 --> 00:39:54.250 You need to really understand what we mean by velocity. 00:39:54.250 --> 00:40:00.010 So here's our Cartesian system. 00:40:00.010 --> 00:40:05.300 Here's this point out here B. And now, this 00:40:05.300 --> 00:40:10.070 is the dog running around, and the path of the dog 00:40:10.070 --> 00:40:14.440 might have been like this. 00:40:14.440 --> 00:40:16.885 And right in here he's going this direction. 00:40:19.460 --> 00:40:30.800 And in a little time, in delta t, 00:40:30.800 --> 00:40:38.610 he moves by an amount delta RB with respect to O. 00:40:38.610 --> 00:40:41.100 And that's what this is. 00:40:41.100 --> 00:40:44.015 He's moved this little bit in time delta t. 00:40:44.015 --> 00:40:46.320 And he happens to be going off in that direction. 00:40:46.320 --> 00:40:49.450 So this then is R prime, I'll call it, 00:40:49.450 --> 00:40:53.790 of B with respect to O, and this is our original RB with respect 00:40:53.790 --> 00:40:59.780 to O. So we can say that his new position, RB 00:40:59.780 --> 00:41:09.030 with respect to prime is RBO plus delta R. 00:41:09.030 --> 00:41:10.550 And these are all vectors. 00:41:14.800 --> 00:41:24.332 And the velocity of B with respect to O 00:41:24.332 --> 00:41:32.800 is just equal to this limit of delta RBO over delta t 00:41:32.800 --> 00:41:37.740 as t goes to 0. 00:41:37.740 --> 00:41:39.520 So what direction is the velocity? 00:41:43.858 --> 00:41:47.410 The velocity is in the direction of the change, not 00:41:47.410 --> 00:41:50.600 the original vector, it was in the direction of the change. 00:41:50.600 --> 00:41:52.580 And in fact, if the path of the dog 00:41:52.580 --> 00:41:58.310 is like this, at the instant you compute the velocity, 00:41:58.310 --> 00:42:02.710 you're computing the tangent to the path of the dog. 00:42:02.710 --> 00:42:04.950 So that's what velocity is at any instant time 00:42:04.950 --> 00:42:06.990 is a tangent to the path. 00:42:06.990 --> 00:42:08.650 And that's a good concept to remember. 00:42:49.660 --> 00:42:53.400 So we're still in this fixed Cartesian space, 00:42:53.400 --> 00:42:55.190 and I have of couple of points. 00:42:55.190 --> 00:42:57.250 I'll make it really trivial here. 00:42:57.250 --> 00:43:04.820 Here's B, and here's A, and the velocity of B-- 00:43:04.820 --> 00:43:05.590 where's my number? 00:43:09.950 --> 00:43:13.920 We'll make this 10 feet per second. 00:43:13.920 --> 00:43:19.300 And it's in the J-hat direction. 00:43:19.300 --> 00:43:25.610 And A, this is the velocity of BNO. 00:43:25.610 --> 00:43:33.190 The velocity of ANO, we'll say is 4 feet per second, 00:43:33.190 --> 00:43:35.270 also in the J direction. 00:43:35.270 --> 00:43:38.590 And I want to know what's the velocity of B with respect 00:43:38.590 --> 00:43:44.005 to A. So now I'm chasing the dog, he's running at 10, 00:43:44.005 --> 00:43:47.460 I'm running at 4. 00:43:47.460 --> 00:43:50.060 How do I perceive the speed of the dog? 00:43:50.060 --> 00:43:52.170 Well, to do this in vectors, which 00:43:52.170 --> 00:43:54.420 is the point of the exercise here, 00:43:54.420 --> 00:43:58.000 is we have the expressions we started with over there. 00:43:58.000 --> 00:44:00.370 And we're going to use these a lot in the course. 00:44:00.370 --> 00:44:04.120 So the velocity of B with respect to O 00:44:04.120 --> 00:44:05.730 is the velocity of A with respect 00:44:05.730 --> 00:44:09.520 to O plus the velocity of B with respect to A. 00:44:09.520 --> 00:44:12.140 And if I want to know velocity of B with respect to A, 00:44:12.140 --> 00:44:12.990 I just solve this. 00:44:18.140 --> 00:44:22.540 So velocity of B with respect to O minus the velocity of A 00:44:22.540 --> 00:44:26.340 with respect to O, and in this case that's 10 minus 4 00:44:26.340 --> 00:44:32.380 is 6 in the J. 00:44:32.380 --> 00:44:35.470 Point of the exercise is to manipulate the vector 00:44:35.470 --> 00:44:36.930 expressions like this. 00:44:36.930 --> 00:44:39.220 So take whatever known quantities you have 00:44:39.220 --> 00:44:40.470 and solve for the unknown one. 00:44:40.470 --> 00:44:42.553 In this case, I want to know the relative velocity 00:44:42.553 --> 00:44:44.490 between the two, and it's this. 00:44:49.970 --> 00:44:53.620 If I'm here, and I'm watching the dog, 00:44:53.620 --> 00:44:57.110 that's how I perceive the speed of the dog relative to me, 00:44:57.110 --> 00:44:58.200 right? 00:44:58.200 --> 00:45:00.180 6 feet per second in the J direction. 00:45:00.180 --> 00:45:05.510 What's the speed of the dog from the point of view of over here? 00:45:05.510 --> 00:45:09.570 The speed of the dog relative to me. 00:45:13.450 --> 00:45:17.140 So it's again the velocity of B with respect to A, 00:45:17.140 --> 00:45:22.300 but from a different position in this fixed reference frame. 00:45:29.930 --> 00:45:31.740 Really important point, actually. 00:45:31.740 --> 00:45:34.085 This is a really important conceptual point. 00:45:37.300 --> 00:45:38.570 Somebody be bold. 00:45:38.570 --> 00:45:40.490 What's the speed with respect to O? 00:45:40.490 --> 00:45:44.570 The velocity of B with respect to A seen from O, 00:45:44.570 --> 00:45:48.074 as computed from O, measured from O. Got radar down there, 00:45:48.074 --> 00:45:49.115 and you're tracking them. 00:45:54.065 --> 00:45:56.045 AUDIENCE: [INAUDIBLE] 00:45:56.045 --> 00:45:59.470 PROFESSOR: In what direction? 00:45:59.470 --> 00:46:00.430 AUDIENCE: [INAUDIBLE] 00:46:03.310 --> 00:46:04.760 PROFESSOR: Yeah. 00:46:04.760 --> 00:46:06.110 It's the same. 00:46:06.110 --> 00:46:09.160 The point is it's the same. 00:46:09.160 --> 00:46:15.740 If you're in a fixed reference frame, a vector of velocity 00:46:15.740 --> 00:46:20.760 is the same as seen from any point in the frame. 00:46:20.760 --> 00:46:23.680 Any fixed point in the frame of velocity is always the same. 00:46:23.680 --> 00:46:27.140 And in fact, in this case, the velocity-- this is a moving 00:46:27.140 --> 00:46:30.820 point and the velocity of him with respect to me 00:46:30.820 --> 00:46:33.590 this is different six feet per second. 00:46:33.590 --> 00:46:35.570 And I, from here, say the velocity 00:46:35.570 --> 00:46:39.140 of that guy with respect to this guy is still 6 feet per second. 00:46:39.140 --> 00:46:43.930 Any place in that frame or even any point 00:46:43.930 --> 00:46:45.470 moving at constant velocity, you're 00:46:45.470 --> 00:46:47.955 going to see the same answer. 00:46:47.955 --> 00:46:50.740 So it doesn't matter where you are 00:46:50.740 --> 00:46:54.510 to compute the velocity of B with respect to A. 00:46:54.510 --> 00:46:57.011 That's the important point. 00:46:57.011 --> 00:46:57.510 OK. 00:47:12.010 --> 00:47:15.550 OK, we got to pick up with, and I may not quite finish, 00:47:15.550 --> 00:47:22.795 but I am going to introduce the next complexity. 00:47:40.960 --> 00:47:41.460 OK. 00:48:03.340 --> 00:48:06.120 So what we just arrived at a minute ago 00:48:06.120 --> 00:48:08.590 is that the velocity as seen from O 00:48:08.590 --> 00:48:13.220 is the same as the velocity as seen from A. And A is me, 00:48:13.220 --> 00:48:15.140 and I'm moving, and I'm chasing the dog. 00:48:15.140 --> 00:48:18.295 So I'm a moving reference frame, I'm 00:48:18.295 --> 00:48:20.590 what's called a translating reference frame. 00:48:20.590 --> 00:48:22.490 So now we're going to take the next step. 00:48:22.490 --> 00:48:24.890 We had a fixed reference frame before purely, 00:48:24.890 --> 00:48:26.390 and now I want to talk about having 00:48:26.390 --> 00:48:28.680 the idea, the concept of having a moving reference 00:48:28.680 --> 00:48:32.200 frame within a fixed one. 00:48:32.200 --> 00:48:35.571 So this is the reference frame O capital XYZ. 00:48:35.571 --> 00:48:37.820 And this little reference frame now is attached to me, 00:48:37.820 --> 00:48:41.050 and it's A, and I call it x-prime y-prime. 00:48:41.050 --> 00:48:43.360 So just so you can-- it's going to be 00:48:43.360 --> 00:48:46.910 hard to tell this X from this X if I don't do something 00:48:46.910 --> 00:48:49.860 like a prime. 00:48:49.860 --> 00:48:52.990 So that this is the concept of a translating coordinate system 00:48:52.990 --> 00:48:56.410 attached to a body, like a rigid body, for example. 00:48:56.410 --> 00:48:59.180 We're going to do lots of rigid body dynamics here. 00:48:59.180 --> 00:49:01.720 And within this coordinate system, 00:49:01.720 --> 00:49:05.759 I can compute the velocity of B with respect to A, 00:49:05.759 --> 00:49:07.300 and I'll get exactly the same answer. 00:49:07.300 --> 00:49:10.930 I'll get that 6 feet per second in the J direction. 00:49:10.930 --> 00:49:14.340 So it's as if-- so this concept of being 00:49:14.340 --> 00:49:17.110 able to have a reference frame attached to a body 00:49:17.110 --> 00:49:20.220 and translating with it, you can measure things within it, 00:49:20.220 --> 00:49:24.160 get the answer, and then convert that answer to here 00:49:24.160 --> 00:49:27.780 if you're using a different coordinate. 00:49:27.780 --> 00:49:31.070 You could use polar coordinates here and rectangular here, 00:49:31.070 --> 00:49:33.610 but they still can be related to one another. 00:49:33.610 --> 00:49:34.830 We'll do problems like that. 00:49:43.410 --> 00:49:45.820 OK. 00:49:45.820 --> 00:49:51.340 So now what I'm doing is I told you like in the readings, 00:49:51.340 --> 00:49:56.740 the end game is to be able to talk about translating 00:49:56.740 --> 00:49:59.510 and rotating bodies, and do dynamics 00:49:59.510 --> 00:50:01.950 in three dimensions with translating and rotating 00:50:01.950 --> 00:50:03.090 objects. 00:50:03.090 --> 00:50:07.740 And we're going to get there somewhat step by step. 00:50:07.740 --> 00:50:10.390 But I want you to understand the end game 00:50:10.390 --> 00:50:11.950 so you know where we're going. 00:50:11.950 --> 00:50:14.110 And you need to have a couple of concepts in mind. 00:50:17.820 --> 00:50:23.950 So the first concept is that this is a rigid body now. 00:50:23.950 --> 00:50:27.800 And you can describe the motion of rigid bodies 00:50:27.800 --> 00:50:34.360 by the summation, the combination of a translation 00:50:34.360 --> 00:50:36.410 and a rotation. 00:50:36.410 --> 00:50:39.680 And of the rigid body, if you can describe its translation, 00:50:39.680 --> 00:50:41.510 and you can describe its rotation, 00:50:41.510 --> 00:50:43.430 you have the complete motion. 00:50:43.430 --> 00:50:47.470 So you got to understand what do we mean by what's really 00:50:47.470 --> 00:50:49.700 the definition of translation. 00:50:49.700 --> 00:50:53.096 So translation-- so I've got this-- 00:50:53.096 --> 00:50:55.450 I'll call it a merry-go-round. 00:50:55.450 --> 00:50:58.510 We'll use a merry-go-round example in a minute. 00:50:58.510 --> 00:51:01.845 And you're observers in a fixed inertial frame 00:51:01.845 --> 00:51:05.140 up above this merry-go-round looking down. 00:51:05.140 --> 00:51:06.640 OK. 00:51:06.640 --> 00:51:09.890 But so you can see it, I got to turn it on its side. 00:51:09.890 --> 00:51:11.680 So here's my merry-go-round. 00:51:11.680 --> 00:51:13.840 And if it's not rotating, but let's 00:51:13.840 --> 00:51:20.320 say it's sitting on a train, on a flat bed and moving along. 00:51:20.320 --> 00:51:22.360 It's translating. 00:51:22.360 --> 00:51:24.620 And when you say a body translates, 00:51:24.620 --> 00:51:31.510 any two points on the body move in parallel paths. 00:51:31.510 --> 00:51:33.844 So two points, my thumb and my finger-- 00:51:33.844 --> 00:51:35.260 if I'm just going along with this, 00:51:35.260 --> 00:51:37.510 those two paths are traveling parallel to one another. 00:51:40.780 --> 00:51:50.180 If I got Y pointing up, the body does this, 00:51:50.180 --> 00:51:52.605 is it rotating and translating? 00:51:52.605 --> 00:51:53.480 AUDIENCE: [INAUDIBLE] 00:51:53.480 --> 00:51:59.095 PROFESSOR: Are any two points on a moving in parallel paths? 00:51:59.095 --> 00:52:00.050 Right? 00:52:00.050 --> 00:52:01.200 OK. 00:52:01.200 --> 00:52:02.800 When it goes through curved things, 00:52:02.800 --> 00:52:05.090 it's called curvilinear translation. 00:52:05.090 --> 00:52:07.220 But it's still just translation. 00:52:07.220 --> 00:52:09.200 OK, so I'll stop and hold steady. 00:52:09.200 --> 00:52:13.620 The train stopped, and the thing-- let it rotate. 00:52:13.620 --> 00:52:17.200 So that's pure rotation. 00:52:17.200 --> 00:52:20.340 And the thing to remember about pure rotation 00:52:20.340 --> 00:52:26.372 is that anywhere on the body rotates at the same rate. 00:52:26.372 --> 00:52:29.800 If this is going around once a second, 00:52:29.800 --> 00:52:31.880 the rotation rate is one rotation 00:52:31.880 --> 00:52:35.450 per second, 360 degrees, 2 pi radians per second 00:52:35.450 --> 00:52:37.180 is its rotation rate. 00:52:37.180 --> 00:52:39.710 Every point on the body experiences the same rotation 00:52:39.710 --> 00:52:40.950 rate. 00:52:40.950 --> 00:52:42.760 That's a really important one to remember. 00:52:46.240 --> 00:52:48.640 If I'm holding still, merry-go-round's 00:52:48.640 --> 00:52:53.300 going round and round, it has a fixed axis of rotation, right? 00:52:53.300 --> 00:52:58.590 But do rotating bodies have to have fixed axes of rotation? 00:52:58.590 --> 00:53:07.605 So if I throw that up in the air, not hanging onto it, 00:53:07.605 --> 00:53:11.200 it's got gravity acting on it, it's rotating. 00:53:11.200 --> 00:53:12.675 What's a rotate about? 00:53:12.675 --> 00:53:13.550 AUDIENCE: [INAUDIBLE] 00:53:13.550 --> 00:53:15.180 PROFESSOR: Center of mass, OK. 00:53:15.180 --> 00:53:18.570 Is the center of mass moving? 00:53:18.570 --> 00:53:21.700 So this is clearly-- this is an example of rotation 00:53:21.700 --> 00:53:22.580 plus translation. 00:53:25.430 --> 00:53:28.660 It rotates about an axis but the axis can move. 00:53:28.660 --> 00:53:31.099 That's another important concept that we 00:53:31.099 --> 00:53:33.390 have to allow in order to be able to do these problems. 00:53:33.390 --> 00:53:35.410 But this is now general motion, it's 00:53:35.410 --> 00:53:38.050 a combination of translation and rotation, 00:53:38.050 --> 00:53:40.329 and we figure out each of those two pieces, 00:53:40.329 --> 00:53:42.620 then we can describe the complete motion of the system. 00:53:47.140 --> 00:53:57.660 All right, where we'll pick up next time is then doing that. 00:53:57.660 --> 00:53:59.076 And it would help actually, if you 00:53:59.076 --> 00:54:03.570 go read that reading, especially up to chapter 16, 00:54:03.570 --> 00:54:07.700 we have to get into to taking derivatives of vectors which 00:54:07.700 --> 00:54:10.700 are rotating, and come up with a general formula 00:54:10.700 --> 00:54:13.680 allows us to do velocities and accelerations 00:54:13.680 --> 00:54:14.660 under those conditions. 00:54:14.660 --> 00:54:18.940 See you on Tuesday next.