WEBVTT 00:00:00.070 --> 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 --> 00:00:10.150 continue to offer high quality educational resources for free. 00:00:10.150 --> 00:00:12.680 To make a donation or to view additional materials 00:00:12.680 --> 00:00:16.590 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.590 --> 00:00:17.305 at ocw.mit.edu. 00:00:20.811 --> 00:00:25.140 PROFESSOR: The purpose of these recitations, small group 00:00:25.140 --> 00:00:27.790 recitations, is so that we can get out 00:00:27.790 --> 00:00:30.140 the key concepts over the week and what 00:00:30.140 --> 00:00:33.217 I call the essential understandings-- what 00:00:33.217 --> 00:00:35.050 are the really important points for the week 00:00:35.050 --> 00:00:37.380 so that when the first quiz comes, 00:00:37.380 --> 00:00:39.050 you will know how to deal with it. 00:00:39.050 --> 00:00:42.420 So let's start with that. 00:00:42.420 --> 00:00:45.420 But you're going to help me think through this. 00:00:45.420 --> 00:00:48.420 So take a minute or two, write down 00:00:48.420 --> 00:00:50.310 on a piece of paper two or three things 00:00:50.310 --> 00:00:53.010 that you think are the most important things that you 00:00:53.010 --> 00:00:55.140 heard, saw, read this week about this course. 00:01:01.060 --> 00:01:02.400 Let's report out. 00:01:02.400 --> 00:01:04.390 I want one from a number of you. 00:01:04.390 --> 00:01:06.702 Who wants to volunteer here? 00:01:06.702 --> 00:01:08.506 AUDIENCE: Using different reference frames? 00:01:08.506 --> 00:01:09.790 PROFESSOR: Say it again? 00:01:09.790 --> 00:01:10.560 AUDIENCE: Using different reference frames. 00:01:10.560 --> 00:01:12.393 PROFESSOR: Using different reference frames. 00:01:15.540 --> 00:01:19.040 I'm going to write that, once I get the chalkboard. 00:01:19.040 --> 00:01:29.540 Using-- I'm going to write it as multiple reference frames. 00:01:32.050 --> 00:01:34.582 Close enough? 00:01:34.582 --> 00:01:35.290 What's your name? 00:01:35.290 --> 00:01:36.123 AUDIENCE: Christina. 00:01:36.123 --> 00:01:37.640 PROFESSOR: Christina? 00:01:37.640 --> 00:01:38.655 What do you have? 00:01:38.655 --> 00:01:40.655 AUDIENCE: All points on a rigid, rotating object 00:01:40.655 --> 00:01:42.643 have the same rate of rotation. 00:01:49.601 --> 00:01:53.110 PROFESSOR: She said, all points on rigid object that's 00:01:53.110 --> 00:01:55.910 rotating, all points have the same rotation rate. 00:01:55.910 --> 00:02:14.110 So this is rotation and translation of rigid bodies. 00:02:14.110 --> 00:02:16.250 I'm going to generalize what you said 00:02:16.250 --> 00:02:19.300 a little bit, because somebody else tell me, 00:02:19.300 --> 00:02:20.960 what can you say about translation? 00:02:20.960 --> 00:02:28.140 So rotation, key point is, all points share the same rotation 00:02:28.140 --> 00:02:28.640 rate. 00:02:28.640 --> 00:02:30.920 How about translation? 00:02:30.920 --> 00:02:32.380 Two different points on an object-- 00:02:32.380 --> 00:02:34.252 what can you say about it? 00:02:34.252 --> 00:02:36.710 AUDIENCE: They follow the same paths. 00:02:36.710 --> 00:02:37.850 PROFESSOR: Parallel paths. 00:02:37.850 --> 00:02:41.650 They go through the exactly same parallel paths. 00:02:41.650 --> 00:02:43.850 So those are two key things we remember about that. 00:02:43.850 --> 00:02:45.150 How about another point? 00:02:45.150 --> 00:02:46.140 AUDIENCE: [INAUDIBLE] 00:02:52.575 --> 00:02:54.256 PROFESSOR: OK. 00:02:54.256 --> 00:02:55.630 This is actually quite important. 00:02:58.270 --> 00:03:00.220 I'm going to write it slightly differently. 00:03:00.220 --> 00:03:01.860 We need to talk about this. 00:03:01.860 --> 00:03:12.750 And that is that rotations-- to be absolutely correct, 00:03:12.750 --> 00:03:23.690 finite rotations are not vectors. 00:03:27.940 --> 00:03:30.190 I want to come back to that in a minute. 00:03:30.190 --> 00:03:32.450 Lots of possible confusion around that. 00:03:32.450 --> 00:03:38.823 One more-- or actually, there's more. 00:03:38.823 --> 00:03:41.280 AUDIENCE: The MLM strategy? 00:03:41.280 --> 00:03:42.330 PROFESSOR: MLM. 00:03:42.330 --> 00:03:43.960 So I mentioned that last time. 00:03:43.960 --> 00:03:47.300 That's problem solving my way, which 00:03:47.300 --> 00:03:51.085 is first M is figure out the motion, describe the motion. 00:03:51.085 --> 00:03:51.835 That's kinematics. 00:03:55.880 --> 00:03:58.940 The second term is L. What is that? 00:03:58.940 --> 00:03:59.610 Laws. 00:03:59.610 --> 00:04:00.510 By the physical laws. 00:04:00.510 --> 00:04:02.980 And the second M? 00:04:02.980 --> 00:04:04.310 Do the math. 00:04:04.310 --> 00:04:06.195 So motion, laws, and math. 00:04:09.790 --> 00:04:11.460 There's something else here. 00:04:11.460 --> 00:04:13.315 Well, you may have just decided it's 00:04:13.315 --> 00:04:14.440 going to encompass in that. 00:04:14.440 --> 00:04:18.100 But I want to go a little further than that. 00:04:18.100 --> 00:04:20.959 What did we talk a lot about yesterday in the lecture? 00:04:20.959 --> 00:04:22.855 AUDIENCE: Different types of acceleration. 00:04:22.855 --> 00:04:28.560 PROFESSOR: Accelerations and velocities and translating 00:04:28.560 --> 00:04:29.990 and rotating frames. 00:04:43.900 --> 00:04:49.290 Translating and rotating frames-- running out of room 00:04:49.290 --> 00:04:52.340 here but you get the point. 00:04:52.340 --> 00:04:54.510 All right, that's a pretty good list. 00:04:54.510 --> 00:04:56.377 If I'd been coming up with a list on my own, 00:04:56.377 --> 00:04:57.960 what would have thought was important, 00:04:57.960 --> 00:05:00.860 that would've captured most of those things. 00:05:00.860 --> 00:05:04.370 Certainly this is really important this week. 00:05:04.370 --> 00:05:07.470 And we definitely need to learn how to use translating 00:05:07.470 --> 00:05:08.860 and rotating frames. 00:05:08.860 --> 00:05:12.190 And you're absolutely in trouble if you don't know this. 00:05:12.190 --> 00:05:16.160 This is just sort of fundamental to the whole thing. 00:05:16.160 --> 00:05:17.880 And then this is a subtle point. 00:05:17.880 --> 00:05:19.630 Let's start right there for a second. 00:05:22.150 --> 00:05:28.354 Who has a textbook? 00:05:28.354 --> 00:05:29.770 It doesn't actually really matter. 00:05:29.770 --> 00:05:31.065 Let me borrow your notes. 00:05:35.160 --> 00:05:37.790 Rigid body, got the print on the front. 00:05:37.790 --> 00:05:40.160 I'm going to rotate it twice. 00:05:40.160 --> 00:05:46.950 The x-axis and call this the z-axis. 00:05:46.950 --> 00:05:51.160 It comes out top actually pointing at you. 00:05:51.160 --> 00:05:52.840 So I did that right. 00:05:52.840 --> 00:05:56.300 So now I'm going to do the rotation. 00:05:56.300 --> 00:05:59.790 Now this one first. 00:05:59.790 --> 00:06:02.770 And then what was the other rotation? 00:06:02.770 --> 00:06:04.100 AUDIENCE: Backwards. 00:06:04.100 --> 00:06:06.420 PROFESSOR: Different answer, right? 00:06:06.420 --> 00:06:08.450 Totally different answer. 00:06:08.450 --> 00:06:13.070 You can't add angles as actors. 00:06:13.070 --> 00:06:14.330 Doesn't work. 00:06:14.330 --> 00:06:19.130 And it's just-- the way I think of it, 00:06:19.130 --> 00:06:23.990 mathematics is largely done to help 00:06:23.990 --> 00:06:26.710 describe the physical world. 00:06:26.710 --> 00:06:29.440 Newton and all those people were figuring out-- needed calculus 00:06:29.440 --> 00:06:31.920 to describe the motion of the planets. 00:06:31.920 --> 00:06:36.620 Vectors were invented to do analytic geometry. 00:06:36.620 --> 00:06:39.190 And it doesn't work for angles. 00:06:39.190 --> 00:06:42.230 You just can't use them for angles. 00:06:42.230 --> 00:06:44.850 It's just the vector math that they figured out 00:06:44.850 --> 00:06:47.670 just wasn't quite clever enough to include angles. 00:06:47.670 --> 00:06:52.880 However, vectors can be applied to positions, velocities, 00:06:52.880 --> 00:06:57.240 accelerations, and angular velocities 00:06:57.240 --> 00:07:01.140 and angular accelerations, but not angles themselves. 00:07:01.140 --> 00:07:05.550 That's the basic thing you need to learn from that. 00:07:05.550 --> 00:07:10.292 Let's use some multiple frames. 00:07:10.292 --> 00:07:11.500 We're going to do that today. 00:07:11.500 --> 00:07:16.660 We're going to now apply this and this and this today 00:07:16.660 --> 00:07:18.925 to do some problems. 00:07:18.925 --> 00:07:23.425 And let me see where I want to go first with this. 00:07:39.730 --> 00:07:43.036 So I have a problem that I wanted to do. 00:07:49.490 --> 00:07:50.840 And it's a circus ride. 00:08:01.900 --> 00:08:02.850 There's an arm. 00:08:05.560 --> 00:08:07.910 And that arm is rotating. 00:08:07.910 --> 00:08:12.035 Attached to the arm is a cross piece. 00:08:12.035 --> 00:08:15.230 And a passenger can sit in each one of these things. 00:08:15.230 --> 00:08:16.930 And this is basically horizontal. 00:08:16.930 --> 00:08:18.350 You're looking down on it. 00:08:18.350 --> 00:08:20.600 So you'd be riding around in these cups at the circus 00:08:20.600 --> 00:08:22.120 and it's going around and around. 00:08:22.120 --> 00:08:25.770 And I want to know the velocity. 00:08:25.770 --> 00:08:34.341 What's the velocity of point B in the O frame? 00:08:34.341 --> 00:08:36.590 And so this has to do-- one of the things on this list 00:08:36.590 --> 00:08:39.750 might be to get the notation down. 00:08:39.750 --> 00:08:41.289 So this is the velocity. 00:08:41.289 --> 00:08:42.190 This is the point. 00:08:45.280 --> 00:08:46.270 And this is the frame. 00:08:52.010 --> 00:08:57.040 So what can you write down? 00:08:57.040 --> 00:08:58.560 Just take 30 seconds. 00:08:58.560 --> 00:08:59.500 See if you remember. 00:08:59.500 --> 00:09:02.490 Write down the general velocity formula 00:09:02.490 --> 00:09:04.580 that was put up yesterday-- vector velocity 00:09:04.580 --> 00:09:08.430 formula for a point in a moving frame that's 00:09:08.430 --> 00:09:11.530 moving in a fixed frame. 00:09:11.530 --> 00:09:15.054 Came up with a general formula, had two or three terms in it. 00:09:15.054 --> 00:09:16.470 And we'll walk our way through it. 00:09:56.350 --> 00:09:59.200 I realize I did something maybe slightly out of order. 00:09:59.200 --> 00:10:00.700 So hold that thought. 00:10:00.700 --> 00:10:04.150 You've written down what you've got. 00:10:04.150 --> 00:10:07.340 We have to do something before you can actually write that. 00:10:07.340 --> 00:10:13.990 We haven't actually picked our reference frames, have we? 00:10:13.990 --> 00:10:15.380 So think about that for a second. 00:10:15.380 --> 00:10:17.590 How would you set up this problem? 00:10:17.590 --> 00:10:20.930 What would you make translating reference frames, 00:10:20.930 --> 00:10:22.580 your rotating, translating frame-- 00:10:22.580 --> 00:10:23.730 where would you assign it? 00:10:23.730 --> 00:10:24.980 Think about it for 30 seconds. 00:10:43.200 --> 00:10:45.500 Who's got to take a shot at it for me? 00:10:45.500 --> 00:10:49.711 Where would you pick reference frames for this problem? 00:10:54.875 --> 00:10:55.766 What your name? 00:10:55.766 --> 00:10:57.104 AUDIENCE: I'm Ben. 00:10:57.104 --> 00:10:59.244 PROFESSOR: Ben. 00:10:59.244 --> 00:11:03.240 AUDIENCE: O and along the cross? 00:11:03.240 --> 00:11:05.700 PROFESSOR: Here, for sure. 00:11:05.700 --> 00:11:10.890 This is your inertial frame-- not moving, right? 00:11:10.890 --> 00:11:12.320 And? 00:11:12.320 --> 00:11:13.715 AUDIENCE: Two axes on the cross? 00:11:13.715 --> 00:11:16.360 PROFESSOR: So you would put one up here? 00:11:16.360 --> 00:11:16.860 OK. 00:11:19.845 --> 00:11:21.345 I'm going to line up with the cross, 00:11:21.345 --> 00:11:25.660 and I'm going to stick out here and call it x2. 00:11:25.660 --> 00:11:28.390 And then there'd be a y2 here. 00:11:28.390 --> 00:11:32.120 And it rotates with the cross? 00:11:32.120 --> 00:11:33.380 All right. 00:11:33.380 --> 00:11:33.990 That's good. 00:11:33.990 --> 00:11:35.830 Now go back to that equation. 00:11:35.830 --> 00:11:38.690 Now give me the velocity, the general expression. 00:11:38.690 --> 00:11:42.340 I don't want you working out the details, just what set of terms 00:11:42.340 --> 00:11:48.110 would you plug things into now to get the velocity of B and O? 00:11:48.110 --> 00:11:52.560 Then we'll evaluate the terms and talk about it, using now 00:11:52.560 --> 00:11:53.800 what we've decided here. 00:12:00.930 --> 00:12:02.780 OK, somebody help me out. 00:12:02.780 --> 00:12:06.050 What's on the right hand side of this equation? 00:12:06.050 --> 00:12:08.240 First term, Mary. 00:12:08.240 --> 00:12:10.540 AUDIENCE: Velocity of-- 00:12:10.540 --> 00:12:12.490 PROFESSOR: What's your name? 00:12:12.490 --> 00:12:13.910 Steven? 00:12:13.910 --> 00:12:15.660 AUDIENCE: Velocity of A with respect to O. 00:12:15.660 --> 00:12:18.780 PROFESSOR: Velocity of A with respect to O. All right. 00:12:18.780 --> 00:12:23.460 That's the velocity at this point in this frame, right? 00:12:23.460 --> 00:12:25.670 What else do we need? 00:12:25.670 --> 00:12:27.420 What's your name? 00:12:27.420 --> 00:12:28.410 Andre? 00:12:28.410 --> 00:12:29.660 AUDIENCE: Yeah. [INAUDIBLE] 00:12:33.724 --> 00:12:39.820 PROFESSOR: I hear a velocity of V with respect to A. 00:12:39.820 --> 00:12:43.550 And what is that-- is that influenced by rotation? 00:12:43.550 --> 00:12:45.780 Can you describe what you mean by the velocity 00:12:45.780 --> 00:12:49.426 of v and A physically? 00:12:49.426 --> 00:12:50.300 AUDIENCE: [INAUDIBLE] 00:12:55.738 --> 00:13:00.230 PROFESSOR: So it's as if you were sitting on that frame, 00:13:00.230 --> 00:13:00.730 right? 00:13:00.730 --> 00:13:03.511 Does the rotation have anything to do with what you see? 00:13:03.511 --> 00:13:04.010 No. 00:13:04.010 --> 00:13:07.130 So I sometimes remind myself right here 00:13:07.130 --> 00:13:08.850 this is omega equals 0. 00:13:08.850 --> 00:13:11.330 And you can set the omega equal to 0, what you 00:13:11.330 --> 00:13:14.420 would see is what this term is. 00:13:14.420 --> 00:13:18.060 Do we need anything more? 00:13:18.060 --> 00:13:18.692 Name? 00:13:18.692 --> 00:13:19.525 AUDIENCE: Christina. 00:13:19.525 --> 00:13:21.525 PROFESSOR: Sorry, you gave it to me once before. 00:13:21.525 --> 00:13:22.911 It's going to take me awhile. 00:13:22.911 --> 00:13:25.366 AUDIENCE: It's the rotational motion 00:13:25.366 --> 00:13:29.785 of B spinning around in there. 00:13:29.785 --> 00:13:33.222 So it has to do with the omega as seen in the reference 00:13:33.222 --> 00:13:45.740 frame, the origin, cross product with r from the in regards 00:13:45.740 --> 00:13:50.543 to the x2 xy. 00:13:50.543 --> 00:13:53.700 PROFESSOR: And we have the name of that frame to help us out. 00:13:53.700 --> 00:13:58.120 This is then frame A, x2, y2, z2. 00:13:58.120 --> 00:14:00.340 If you really wanted to write [INAUDIBLE] 00:14:00.340 --> 00:14:05.770 We just call it frame A. so this is would be rB as in NA 00:14:05.770 --> 00:14:10.370 And these are all vectors and I often forget to underline them. 00:14:10.370 --> 00:14:13.480 Do we have it right? 00:14:13.480 --> 00:14:16.380 Anybody want to add to that, fix it? 00:14:16.380 --> 00:14:19.050 Correct it? 00:14:19.050 --> 00:14:19.734 Steven, right? 00:14:19.734 --> 00:14:20.608 AUDIENCE: [INAUDIBLE] 00:14:23.356 --> 00:14:25.510 PROFESSOR: I left it vague on purpose. 00:14:25.510 --> 00:14:26.810 We need to figure that out. 00:14:26.810 --> 00:14:29.160 He asks, is it omega 2 or omega 1? 00:14:29.160 --> 00:14:32.930 Really important point we want to make today about what omega 00:14:32.930 --> 00:14:34.220 this is. 00:14:34.220 --> 00:14:34.970 We'll get to that. 00:14:34.970 --> 00:14:35.540 Yeah? 00:14:35.540 --> 00:14:38.890 AUDIENCE: Well, if they're rotating in the same direction, 00:14:38.890 --> 00:14:41.807 wouldn't it be added in both omega 1 or omega 2? 00:14:41.807 --> 00:14:42.640 PROFESSOR: Well, OK. 00:14:42.640 --> 00:14:43.890 Let's talk about it right now. 00:14:43.890 --> 00:14:49.680 Are we agreed that this is the right formula? 00:14:49.680 --> 00:14:52.550 Then let's set about figuring it out. 00:14:52.550 --> 00:14:55.770 And we can talk about this term first. 00:14:55.770 --> 00:15:01.840 So we want to know, this is the rotation rate 00:15:01.840 --> 00:15:08.560 of this arm out here in the base frame. 00:15:08.560 --> 00:15:11.440 That's what the notation says. 00:15:11.440 --> 00:15:14.890 And we know that the rotation rate of this first arm 00:15:14.890 --> 00:15:16.860 in the base frame is this. 00:15:16.860 --> 00:15:18.580 And we know that the rotation rate 00:15:18.580 --> 00:15:28.842 of this thing with to-- now this has 00:15:28.842 --> 00:15:30.550 gotten a little complicated, because this 00:15:30.550 --> 00:15:33.190 isn't quite exact enough. 00:15:33.190 --> 00:15:35.500 This is omega 2 with respect to this arm. 00:15:39.360 --> 00:15:40.930 That's what's given in this problem. 00:15:40.930 --> 00:15:49.490 So this is omega 2 with respect to the arm OA. 00:15:52.581 --> 00:15:53.080 Yeah? 00:15:53.080 --> 00:15:55.032 AUDIENCE: So does that mean it's omega 00:15:55.032 --> 00:15:58.936 2xz from coordiante system B? 00:15:58.936 --> 00:16:03.040 PROFESSOR: No, coordinate system A x2y2 rotates. 00:16:03.040 --> 00:16:06.790 And if you're sitting in there, you wouldn't see it. 00:16:06.790 --> 00:16:11.440 So this is correct. 00:16:11.440 --> 00:16:14.537 It's the rotation rate as seen in O. 00:16:14.537 --> 00:16:16.120 So we need to figure out what that is. 00:16:16.120 --> 00:16:17.494 And I'm telling you in this case, 00:16:17.494 --> 00:16:20.110 you were given-- you might have been 00:16:20.110 --> 00:16:22.150 given the rotation rate in O. 00:16:22.150 --> 00:16:22.710 You weren't. 00:16:22.710 --> 00:16:25.860 You were given the rotation rate relative to here. 00:16:25.860 --> 00:16:30.140 So I'll write it as W2 with respect to this arm OA. 00:16:30.140 --> 00:16:39.015 So how do you get-- we need omega in O is what? 00:16:39.015 --> 00:16:40.440 Help me out here. 00:16:40.440 --> 00:16:43.092 AUDIENCE: Is it O-- should there be a small b at the bottom 00:16:43.092 --> 00:16:43.592 [INAUDIBLE]? 00:16:50.040 --> 00:16:52.280 PROFESSOR: Good. 00:16:52.280 --> 00:16:53.090 But what is it? 00:16:56.920 --> 00:16:58.520 Let's deduce it. 00:16:58.520 --> 00:17:02.820 If my arm here, this is the first arm. 00:17:02.820 --> 00:17:05.950 And this is the at AB link. 00:17:05.950 --> 00:17:09.400 Now if omega with respect to this arm, this thing 00:17:09.400 --> 00:17:12.660 weren't moving, no rotation rate relative to this, 00:17:12.660 --> 00:17:14.869 the whole thing would be straight, right? 00:17:14.869 --> 00:17:17.210 And it's going around like this. 00:17:17.210 --> 00:17:21.160 What's the rotation rate of the link out here? 00:17:21.160 --> 00:17:21.960 Omega 1. 00:17:24.609 --> 00:17:29.420 And now this arm's not moving, but this is rotating 00:17:29.420 --> 00:17:33.570 relative to it at omega 2. 00:17:33.570 --> 00:17:36.310 What's the rotation rate of the link out here? 00:17:36.310 --> 00:17:37.270 Just omega 2. 00:17:37.270 --> 00:17:41.300 If I put the two together, what is the rotation rate 00:17:41.300 --> 00:17:43.454 of this arm, this second link? 00:17:43.454 --> 00:17:44.370 AUDIENCE: [INAUDIBLE]. 00:17:47.520 --> 00:17:53.210 PROFESSOR: Omega 1 in, certainly in O plus omega 2. 00:17:53.210 --> 00:17:55.050 It's not with respect to A. I'm just 00:17:55.050 --> 00:17:59.060 going to call it with respect to the R maybe. 00:17:59.060 --> 00:18:02.301 Even this notation is failing a little bit. 00:18:02.301 --> 00:18:03.300 But you get what I mean. 00:18:03.300 --> 00:18:05.140 It's omega 1 plus omega 2. 00:18:05.140 --> 00:18:09.040 And let's just write it as omega 1 plus omega 2. 00:18:09.040 --> 00:18:10.550 And what direction is it in? 00:18:10.550 --> 00:18:12.940 It's a vector. 00:18:12.940 --> 00:18:14.940 So one of the things we have to pay attention to 00:18:14.940 --> 00:18:16.611 are unit vectors. 00:18:16.611 --> 00:18:17.110 Yeah? 00:18:17.110 --> 00:18:17.984 AUDIENCE: [INAUDIBLE] 00:18:17.984 --> 00:18:21.535 PROFESSOR: So this is capital I hat here 00:18:21.535 --> 00:18:27.420 and capital J hat there and coming out of the board, K hat. 00:18:27.420 --> 00:18:31.310 Now, this is certainly K hat, capital K hat. 00:18:31.310 --> 00:18:36.090 This one, though, is relative to-- it's the rotation 00:18:36.090 --> 00:18:38.350 rate of this thing. 00:18:38.350 --> 00:18:40.220 Here is a reference frame. 00:18:40.220 --> 00:18:42.590 What's sticking out this way? 00:18:42.590 --> 00:18:44.670 A little k2, right? 00:18:44.670 --> 00:18:48.720 But is it parallel to capital K? 00:18:48.720 --> 00:18:50.720 Always parallel to capital K? 00:18:53.530 --> 00:18:55.580 So they're the same thing. 00:18:55.580 --> 00:18:57.431 If unit vectors in this are parallel, 00:18:57.431 --> 00:18:58.680 they amount to the same thing. 00:18:58.680 --> 00:19:00.920 So we can put capital K, lowercase k, anything 00:19:00.920 --> 00:19:02.220 we want here and it's correct. 00:19:05.530 --> 00:19:08.750 Now we've got an answer for that. 00:19:08.750 --> 00:19:14.800 So when you're given-- when the one thing's attached to another 00:19:14.800 --> 00:19:17.090 and you're given the-- if out here 00:19:17.090 --> 00:19:23.270 you are given the rotation rate in the base frame, you're done. 00:19:23.270 --> 00:19:24.930 But if you're given the rotation rate 00:19:24.930 --> 00:19:27.950 relative to some other moving part, 00:19:27.950 --> 00:19:30.730 then you have to add them up to get the true rotation rate. 00:19:30.730 --> 00:19:33.040 That's the bottom line message. 00:19:33.040 --> 00:19:33.550 All right. 00:19:33.550 --> 00:19:38.800 So we started-- we're trying to figure out 00:19:38.800 --> 00:19:40.780 this expression here. 00:19:44.960 --> 00:19:49.020 And we started with one of the harder terms. 00:19:49.020 --> 00:19:52.450 And we need to figure out-- to finish it, 00:19:52.450 --> 00:19:55.480 though, let's do this over here. 00:19:55.480 --> 00:20:16.150 We have velocity of A in-- and we have the velocity of B in A 00:20:16.150 --> 00:20:18.860 with no rotation. 00:20:18.860 --> 00:20:25.040 And we have omega B in O. And let's finish that. 00:20:25.040 --> 00:20:27.781 We know what omega is now. 00:20:27.781 --> 00:20:28.280 Whoops. 00:20:28.280 --> 00:20:29.360 It's not omega. 00:20:29.360 --> 00:20:34.502 The third term is omega B and O cross RBA. 00:20:38.210 --> 00:20:40.380 So we've gotten the first bit of this. 00:20:40.380 --> 00:20:42.770 Let's finish the problem. 00:20:42.770 --> 00:20:50.360 This is omega 1 plus omega 2 times k hat cross with what? 00:20:54.460 --> 00:20:55.460 We need a length. 00:20:55.460 --> 00:20:57.810 I'll call this L. It's L long. 00:21:03.600 --> 00:21:07.290 So what is RB respect to A? 00:21:07.290 --> 00:21:07.790 Yes? 00:21:07.790 --> 00:21:10.120 AUDIENCE: L X 2 hat? 00:21:10.120 --> 00:21:11.535 PROFESSOR: L X 2 hat. 00:21:11.535 --> 00:21:13.600 And I'll call that Lj2. 00:21:16.850 --> 00:21:20.940 The coordinate is x2. 00:21:20.940 --> 00:21:26.830 The unit vector would be I2, not a j, an i. 00:21:29.570 --> 00:21:32.100 The unit vector is i2. 00:21:32.100 --> 00:21:33.180 OK, great. 00:21:33.180 --> 00:21:35.350 Now what is k cross i2? 00:21:40.510 --> 00:21:41.440 j2. 00:21:41.440 --> 00:21:52.700 So we get omega L omega 1 plus omega 2 j2 hat. 00:21:52.700 --> 00:21:55.150 That's that term. 00:21:55.150 --> 00:22:03.970 And we need to figure out our other two terms. 00:22:03.970 --> 00:22:05.580 What's this term? 00:22:09.520 --> 00:22:11.470 Remind yourself of the meaning. 00:22:11.470 --> 00:22:15.410 This is the velocity of point B with respect to the A frame, 00:22:15.410 --> 00:22:16.750 which is attached to it. 00:22:16.750 --> 00:22:20.290 It's on a rigid body. 00:22:20.290 --> 00:22:23.776 AUDIENCE: [INAUDIBLE] 00:22:23.776 --> 00:22:27.730 PROFESSOR: He said omega 2 times L. She says 0. 00:22:27.730 --> 00:22:29.740 Any other? 00:22:29.740 --> 00:22:31.580 I hear another 0. 00:22:31.580 --> 00:22:32.662 Why 0? 00:22:32.662 --> 00:22:36.090 AUDIENCE: Because it's rigidly attached into the ride, 00:22:36.090 --> 00:22:37.090 if you're moving around. 00:22:37.090 --> 00:22:39.070 It's not moving on the ride versus strapped in. 00:22:39.070 --> 00:22:39.778 PROFESSOR: Right. 00:22:39.778 --> 00:22:42.900 So this term is always from the point of view of a person 00:22:42.900 --> 00:22:45.350 riding on the frame. 00:22:45.350 --> 00:22:48.670 Riding on that frame-- so you won't ever see rotation 00:22:48.670 --> 00:22:49.850 from inside the frame. 00:22:49.850 --> 00:22:51.730 You're just moving with it. 00:22:51.730 --> 00:22:54.860 So that's called, in the Williams book, 00:22:54.860 --> 00:22:59.480 he calls this term the rel. 00:22:59.480 --> 00:23:01.480 It's the relative velocity between these two 00:23:01.480 --> 00:23:04.222 points and no rotation. 00:23:04.222 --> 00:23:05.430 So what is that in this case? 00:23:09.020 --> 00:23:10.570 I hear 0. 00:23:10.570 --> 00:23:12.270 Everybody agree it's 0? 00:23:12.270 --> 00:23:13.660 It's a rigid link. 00:23:13.660 --> 00:23:15.070 Two points don't move. 00:23:15.070 --> 00:23:18.310 So now we're just left with this one. 00:23:18.310 --> 00:23:21.500 And now, one of the points I really 00:23:21.500 --> 00:23:27.060 wanted to drive home today is in fact this problem is one 00:23:27.060 --> 00:23:29.820 that, depending on how you set it up, 00:23:29.820 --> 00:23:33.840 you can think of as actually having multiple rotating 00:23:33.840 --> 00:23:35.440 frames. 00:23:35.440 --> 00:23:37.880 And if you do that, what's the correct way 00:23:37.880 --> 00:23:42.430 to add up the parts so you get to the right answer? 00:23:42.430 --> 00:23:46.020 Because we've left this one for the last. 00:23:46.020 --> 00:23:49.890 And I want to make sure you go away 00:23:49.890 --> 00:23:56.030 knowing a formula you can always use, and it's going to work. 00:23:56.030 --> 00:23:57.580 And the formula we can always use 00:23:57.580 --> 00:23:59.820 is the one that's of this form. 00:23:59.820 --> 00:24:02.560 Every one of these problems, including multiple links 00:24:02.560 --> 00:24:05.160 and things, you can build up by doing 00:24:05.160 --> 00:24:09.230 a sequence of this problem again and again and again, 00:24:09.230 --> 00:24:11.450 until you get the whole answer. 00:24:11.450 --> 00:24:13.580 So we've actually done what I would 00:24:13.580 --> 00:24:16.400 call the outer problem first. 00:24:16.400 --> 00:24:18.330 We've worked out this thing. 00:24:18.330 --> 00:24:20.050 We have to do the inner problem now. 00:24:20.050 --> 00:24:21.841 We could have done it in a different order, 00:24:21.841 --> 00:24:24.390 but I need to know the velocity of this point. 00:24:24.390 --> 00:24:31.105 And just to get you in the habit of using the vector equation, 00:24:31.105 --> 00:24:35.280 that we have, I want to know the velocity of A in O. 00:24:35.280 --> 00:24:40.370 And I'm going to attach a rotating 00:24:40.370 --> 00:24:49.270 frame to this arm, x1, y1. 00:24:49.270 --> 00:24:51.610 It rotates with this arm at that rate. 00:24:54.310 --> 00:24:56.900 And I want you to use that frame to solve 00:24:56.900 --> 00:25:00.600 for the velocity of this point. 00:25:00.600 --> 00:25:09.580 And that would be-- velocity of point A in O 00:25:09.580 --> 00:25:26.090 would be the-- this frame now is an O little x1, y1, z1. 00:25:26.090 --> 00:25:27.970 It's a rotating frame, right? 00:25:27.970 --> 00:25:30.450 Because the O's are going to get confusing. 00:25:30.450 --> 00:25:32.680 Better not call it O. We'll call this rotating 00:25:32.680 --> 00:25:35.000 one-- in Williams, he uses a lowercase o, 00:25:35.000 --> 00:25:37.380 but it's hard to do on the board. 00:25:37.380 --> 00:25:39.670 Let's call this c. 00:25:39.670 --> 00:25:43.030 So this is a frame, C, x1, y1. 00:25:45.590 --> 00:25:48.080 So this frame will be my c frame. 00:25:48.080 --> 00:25:50.940 So I want to know the velocity of point A. 00:25:50.940 --> 00:25:52.830 It's the velocity of what? 00:25:58.180 --> 00:26:00.500 If you get stuck, use that top formula up there, 00:26:00.500 --> 00:26:02.975 put in the right points. 00:26:05.800 --> 00:26:08.686 So what's the first term mean? 00:26:08.686 --> 00:26:15.470 It's the velocity of the-- this time the rotating frame, 00:26:15.470 --> 00:26:18.000 does the rotating frame translate? 00:26:18.000 --> 00:26:19.270 We have a rotating frame. 00:26:19.270 --> 00:26:21.370 Does it have any translational velocity? 00:26:21.370 --> 00:26:23.265 No, but you still have the right to turn it 00:26:23.265 --> 00:26:25.110 down and set it equal to 0. 00:26:25.110 --> 00:26:27.410 So what's the right term? 00:26:27.410 --> 00:26:28.712 How do you write it? 00:26:34.040 --> 00:26:34.910 Right? 00:26:34.910 --> 00:26:36.760 It's the velocity of my reference frame. 00:26:36.760 --> 00:26:38.759 It's the transitional velocity of that reference 00:26:38.759 --> 00:26:40.970 frame in the O frame. 00:26:40.970 --> 00:26:42.940 And that's what it is. 00:26:42.940 --> 00:26:56.930 And in this case, it's 0 plus velocity 00:26:56.930 --> 00:27:01.742 of A with respect to c. 00:27:01.742 --> 00:27:03.202 And I'll remind you again. 00:27:03.202 --> 00:27:04.910 It's as if you were now rotating with it, 00:27:04.910 --> 00:27:09.120 and you're sitting at c, looking at A. What's its speed? 00:27:09.120 --> 00:27:11.580 0. 00:27:11.580 --> 00:27:20.300 Plus omega-- what omega? 00:27:20.300 --> 00:27:22.629 Seen where? 00:27:22.629 --> 00:27:26.010 Measured from where? 00:27:26.010 --> 00:27:27.900 Measured with respect to what frame? 00:27:30.530 --> 00:27:39.170 I hear on O, cross with-- we need length? 00:27:39.170 --> 00:27:42.700 We'll make this length capital R, scalar. 00:27:46.110 --> 00:27:50.335 So what's the cross product here? 00:27:50.335 --> 00:27:51.796 What's the unit vector? 00:27:54.718 --> 00:27:57.370 Correct unit vector? 00:27:57.370 --> 00:27:59.006 Not x. 00:27:59.006 --> 00:28:00.600 x is the coordinate. 00:28:00.600 --> 00:28:02.400 The unit vector is-- 00:28:02.400 --> 00:28:03.340 AUDIENCE: [INAUDIBLE] 00:28:06.971 --> 00:28:08.852 PROFESSOR: Right? 00:28:08.852 --> 00:28:09.816 That turns 0. 00:28:09.816 --> 00:28:10.780 That turns 0. 00:28:10.780 --> 00:28:12.710 This turns 0. 00:28:12.710 --> 00:28:14.870 Omega 1, And what's the unit vector associated 00:28:14.870 --> 00:28:15.930 with this omega 1? 00:28:19.090 --> 00:28:26.440 k cross i 1 j hat. 00:28:26.440 --> 00:28:34.560 And so we have R omega 1 j1 hat. 00:28:34.560 --> 00:28:38.800 And now, we should be able to write out 00:28:38.800 --> 00:28:44.340 the full answer of the velocity of B 00:28:44.340 --> 00:28:59.230 in O is the velocity of A, which is R omega 1 j1 hat 00:28:59.230 --> 00:29:02.510 plus velocity of B with respect to A, 00:29:02.510 --> 00:29:15.950 which was 0 plus this term, which we figured out as L omega 00:29:15.950 --> 00:29:21.976 1 plus omega 2 times j2 hat. 00:29:30.866 --> 00:29:32.490 So we have two or three sub-- this kind 00:29:32.490 --> 00:29:36.260 of a hard problem, actually, for the first time out. 00:29:36.260 --> 00:29:41.240 Because it has a number of subtle concepts built into it. 00:29:41.240 --> 00:29:42.940 You actually have two rotating bodies. 00:29:42.940 --> 00:29:44.023 How do you deal with them? 00:29:44.023 --> 00:29:49.150 Well, you do sequential applications of that vector 00:29:49.150 --> 00:29:50.176 velocity formula. 00:29:50.176 --> 00:29:50.676 Yeah? 00:29:50.676 --> 00:29:52.818 AUDIENCE: So I was wondering why we 00:29:52.818 --> 00:29:56.388 made another coordinate system that's rotating with the arm 00:29:56.388 --> 00:30:01.660 to solve for the velocity of A [INAUDIBLE]. 00:30:01.660 --> 00:30:03.395 PROFESSOR: She asked, why did we bother 00:30:03.395 --> 00:30:04.520 to make this another frame. 00:30:07.520 --> 00:30:11.620 The problems are going to get nastier and nastier. 00:30:11.620 --> 00:30:13.880 I could have asked you, when I walked into class, what 00:30:13.880 --> 00:30:17.340 is the velocity of point A. And you would have said, well, 00:30:17.340 --> 00:30:18.554 obviously R omega. 00:30:22.470 --> 00:30:24.540 Why are we going to all this trouble, when 00:30:24.540 --> 00:30:28.960 everybody knows from high school physics that it's just R omega? 00:30:28.960 --> 00:30:30.930 And the answer is is because we're 00:30:30.930 --> 00:30:33.060 going to get to doing really nasty problems. 00:30:33.060 --> 00:30:36.890 And I want to make sure you understand all the subtleties 00:30:36.890 --> 00:30:41.120 about how we get these. 00:30:41.120 --> 00:30:44.380 So we started simple, but I did it the long, hard way. 00:30:44.380 --> 00:30:48.410 Because later on, if I'd walked in at the beginning 00:30:48.410 --> 00:30:51.460 and just asked you right off the bat, what's 00:30:51.460 --> 00:30:54.669 the velocity of this point-- go for it-- 00:30:54.669 --> 00:30:56.210 you guys would have failed miserably. 00:30:58.760 --> 00:31:01.400 It's not much harder, but it takes 00:31:01.400 --> 00:31:03.920 two sequential applications of what you think 00:31:03.920 --> 00:31:06.050 is obvious when you walked into class. 00:31:06.050 --> 00:31:07.095 So that's why. 00:31:07.095 --> 00:31:08.970 We're just doing it the hard way, so that you 00:31:08.970 --> 00:31:11.610 get all the little nuances. 00:31:11.610 --> 00:31:12.397 Yeah? 00:31:12.397 --> 00:31:14.882 AUDIENCE: So why [INAUDIBLE] there for omega 2, 00:31:14.882 --> 00:31:18.609 we have it with respect to arm AB [INAUDIBLE] with respect 00:31:18.609 --> 00:31:20.846 to arm OA? 00:31:20.846 --> 00:31:23.228 PROFESSOR: Why is it that way? 00:31:23.228 --> 00:31:24.644 AUDIENCE: [INAUDIBLE] with respect 00:31:24.644 --> 00:31:28.045 to arm AB and when you wrote [INAUDIBLE] with respect 00:31:28.045 --> 00:31:29.511 to arm OA? 00:31:29.511 --> 00:31:32.100 PROFESSOR: When I wrote the equation for-- 00:31:32.100 --> 00:31:34.710 AUDIENCE: For the omega 2. 00:31:34.710 --> 00:31:36.800 There you're saying it's with respect to OA, 00:31:36.800 --> 00:31:39.356 and there you say its respect to AB. 00:31:39.356 --> 00:31:42.550 PROFESSOR: Oh, I see. 00:31:42.550 --> 00:31:44.190 Because this is wrong. 00:31:44.190 --> 00:31:45.180 AUDIENCE: [INAUDIBLE] 00:31:51.060 --> 00:31:57.307 AUDIENCE: My question deals with j1 and j2-- are they the same? 00:31:57.307 --> 00:31:58.640 PROFESSOR: So he has a question. 00:31:58.640 --> 00:32:04.170 And that's the final, subtle point I want to get to today. 00:32:04.170 --> 00:32:04.840 Good question. 00:32:04.840 --> 00:32:06.173 He's saying, are these the same? 00:32:06.173 --> 00:32:08.390 Are they different? 00:32:08.390 --> 00:32:09.850 How do we deal with it? 00:32:09.850 --> 00:32:11.960 So a question for you. 00:32:11.960 --> 00:32:14.280 In general, if you're asked or given 00:32:14.280 --> 00:32:17.520 a problem like we just did, and you arrive at a solution, 00:32:17.520 --> 00:32:23.380 is it OK to give an answer where you had unit 00:32:23.380 --> 00:32:24.740 vectors in multiple frames? 00:32:24.740 --> 00:32:28.520 And neither of these unit vectors are in the base frame. 00:32:28.520 --> 00:32:30.790 And yet, the answer we're claiming 00:32:30.790 --> 00:32:34.270 is that this is the velocity of B in O. 00:32:34.270 --> 00:32:38.130 And here we've got unit vectors that are not in the base frame. 00:32:38.130 --> 00:32:39.825 Is it a legit equation or not? 00:32:44.640 --> 00:32:45.584 What do you think? 00:32:48.250 --> 00:32:49.501 See a lot of no's out there. 00:32:53.547 --> 00:32:54.880 I think we better figure it out. 00:33:37.090 --> 00:33:40.530 So we have unit on the arm. 00:33:40.530 --> 00:33:43.430 This is my c and O here. 00:33:43.430 --> 00:33:46.040 On the arm, I have frame that rotates 00:33:46.040 --> 00:33:50.070 with it that has unit vectors in the direction of the arm of i1 00:33:50.070 --> 00:33:52.830 and j1. 00:33:52.830 --> 00:33:55.410 So here's i1. 00:33:55.410 --> 00:33:57.050 It's unit long. 00:33:57.050 --> 00:33:59.730 Here is the angle theta. 00:33:59.730 --> 00:34:06.000 Here's j1 and the angles. 00:34:06.000 --> 00:34:12.360 And I want to know-- this is i1 and this is j1-- 00:34:12.360 --> 00:34:15.780 can I express i1 and j1 in terms of capital 00:34:15.780 --> 00:34:18.400 I, capital J, the unit vectors in the base frame? 00:34:24.960 --> 00:34:27.780 I want to express them in terms of unit vectors 00:34:27.780 --> 00:34:31.500 that are in this rigid, non-moving, non-rotating, 00:34:31.500 --> 00:34:34.230 inertial frame. 00:34:34.230 --> 00:34:39.449 So down here, this is the i hat direction 00:34:39.449 --> 00:34:45.280 and this is the j hat direction, right-- not moving. 00:34:45.280 --> 00:34:48.139 So this is just a unit thing, unit long. 00:34:48.139 --> 00:34:51.760 Can I project it onto its i component and capital J 00:34:51.760 --> 00:34:52.449 component? 00:34:52.449 --> 00:34:53.170 All right. 00:34:53.170 --> 00:34:59.630 So i1, it looks to me like cosine theta I 00:34:59.630 --> 00:35:05.715 hat plus sine theta J hat. 00:35:05.715 --> 00:35:06.370 Do you agree? 00:35:10.524 --> 00:35:13.650 Just standard trick, right? 00:35:13.650 --> 00:35:16.665 And this one, takes me a minute to figure this out. 00:35:19.960 --> 00:35:23.200 Which is the theta here? 00:35:23.200 --> 00:35:24.280 This is theta. 00:35:24.280 --> 00:35:25.910 That's 90 minus theta. 00:35:25.910 --> 00:35:27.195 So this must be theta, right? 00:35:30.116 --> 00:35:32.110 There's a theta here. 00:35:32.110 --> 00:35:35.430 And if this is unit long, what's that? 00:35:35.430 --> 00:35:43.350 That projection there is-- so j1 has two components, 00:35:43.350 --> 00:35:54.540 minus sine theta I plus cosine theta J. 00:35:54.540 --> 00:35:57.611 And I highly recommend you write that one down. 00:35:57.611 --> 00:35:59.110 Make sure you can drive it yourself. 00:35:59.110 --> 00:36:01.610 You're going to need it again and again and again and again. 00:36:06.450 --> 00:36:09.890 Now, could we do the same thing for-- could we 00:36:09.890 --> 00:36:17.230 convert J2 to the base frame? 00:36:24.030 --> 00:36:28.520 And it rotates, so this x2 can be at any arbitrary position. 00:36:28.520 --> 00:36:32.360 But in order to do the problem, you have to pick a position. 00:36:32.360 --> 00:36:35.930 And then you'd have to do draw an angle. 00:36:35.930 --> 00:36:40.570 And then you'd have to apply this formula. 00:36:40.570 --> 00:36:44.790 And so you're going to end up with an i2 00:36:44.790 --> 00:36:53.630 and some cosine phi capital I plus sine phi capital 00:36:53.630 --> 00:37:04.450 J. And the same thing, j2 is minus sine phi i plus cosine 00:37:04.450 --> 00:37:07.280 theta J. 00:37:07.280 --> 00:37:12.870 So we'll do a trivial example, solve a trivial case. 00:37:12.870 --> 00:37:15.540 What is the instantaneous velocity 00:37:15.540 --> 00:37:20.950 at the moment that the coordinate system is lined up 00:37:20.950 --> 00:37:23.580 as we see, and B is sitting right here? 00:37:33.430 --> 00:37:39.070 So we've got to go look at our answer. 00:37:39.070 --> 00:37:41.300 Where was our final answer? 00:37:41.300 --> 00:37:47.075 Velocity of this guy here, right? 00:37:50.520 --> 00:37:54.335 What would be the contribution of this term? 00:37:57.060 --> 00:37:58.800 We have to take each term and convert it 00:37:58.800 --> 00:38:03.910 to the base system and capital IJ terms, right? 00:38:03.910 --> 00:38:06.640 You do it one term at a time and add up the components. 00:38:06.640 --> 00:38:11.570 So how do you break this one down and put it into capital I, 00:38:11.570 --> 00:38:14.484 capital J components? 00:38:14.484 --> 00:38:15.858 AUDIENCE: Substitute? 00:38:15.858 --> 00:38:17.740 PROFESSOR: Yeah, what's the answer? 00:38:23.180 --> 00:38:25.310 So j, if it's lined up like this, 00:38:25.310 --> 00:38:27.744 j2 is importing in what direction? 00:38:27.744 --> 00:38:28.550 Up. 00:38:28.550 --> 00:38:31.240 And what is that in this system? 00:38:31.240 --> 00:38:34.470 Just capital J. 00:38:34.470 --> 00:38:36.030 At this instant in time, that's just 00:38:36.030 --> 00:38:42.080 capital J. Trivial calculation, because this angle 00:38:42.080 --> 00:38:43.950 is 90 degrees. 00:38:43.950 --> 00:38:48.130 Plug in 90 degrees, this term goes to 0, this term goes to 1. 00:38:48.130 --> 00:38:52.310 J2 is capital J. 00:38:52.310 --> 00:38:57.620 And what about the other term, J1? 00:38:57.620 --> 00:39:00.982 You just got to-- it's J1, right? 00:39:00.982 --> 00:39:02.440 So you just gotta go with the flow. 00:39:02.440 --> 00:39:03.220 It is is. 00:39:03.220 --> 00:39:07.530 You'd substitute this in for J1 right here, 00:39:07.530 --> 00:39:10.680 and you'd have R1 omega 1 cosine theta 00:39:10.680 --> 00:39:16.100 sine theta and j and k terms, plus this thing, capital J. 00:39:16.100 --> 00:39:18.790 And you have just converted the answer, 00:39:18.790 --> 00:39:22.230 which was in terms of unit vectors in rotating 00:39:22.230 --> 00:39:24.530 to different rotating frames. 00:39:24.530 --> 00:39:26.730 You've converted it all down to the base frame. 00:39:31.730 --> 00:39:32.640 AUDIENCE: [INAUDIBLE] 00:39:38.120 --> 00:39:39.350 PROFESSOR: Oops, I'm sorry. 00:39:39.350 --> 00:39:40.530 I just made a mistake. 00:39:40.530 --> 00:39:42.352 You guys got to get better catching me. 00:39:45.390 --> 00:39:47.780 That now make sense? 00:39:47.780 --> 00:39:51.710 So phi is the angle that the j2 unit vector makes 00:39:51.710 --> 00:39:55.190 with the inertial frame, right? 00:39:55.190 --> 00:40:00.182 And theta is the angle that the j1 or i1 00:40:00.182 --> 00:40:01.390 make with the inertial frame. 00:40:01.390 --> 00:40:01.890 Yes? 00:40:01.890 --> 00:40:05.150 AUDIENCE: Phi is 0, though, right? 00:40:05.150 --> 00:40:12.410 PROFESSOR: In this case, phi is 0. 00:40:12.410 --> 00:40:13.950 Does that still work out over there? 00:40:13.950 --> 00:40:18.812 Sine of phi is 0 and cosine of 0 is one and you get j. 00:40:18.812 --> 00:40:23.250 AUDIENCE: Then why did we didn't plug in anything for j? 00:40:23.250 --> 00:40:24.340 PROFESSOR: We did. 00:40:27.440 --> 00:40:30.040 There isn't a simple answer for it. 00:40:30.040 --> 00:40:32.070 And so you have to use the full expression. 00:40:32.070 --> 00:40:35.810 I just got lazy and didn't want to write it out. 00:40:35.810 --> 00:40:36.880 The answer is this. 00:40:40.010 --> 00:40:42.770 Stick in 30 degrees if you want, and then you'll get numbers. 00:40:47.130 --> 00:40:50.150 So real important point that we discovered 00:40:50.150 --> 00:40:56.040 is that the answers are correct, expressed in rotating unit 00:40:56.040 --> 00:40:59.860 vectors, expressed in different unit vectors, 00:40:59.860 --> 00:41:01.210 different rotating ones. 00:41:01.210 --> 00:41:04.240 This is correct, because you can take this 00:41:04.240 --> 00:41:08.090 and you can reduce it down to the base frame. 00:41:08.090 --> 00:41:12.060 So you will be-- usually in problems that you're given, 00:41:12.060 --> 00:41:15.290 you'll be asked to express the answer in terms of unit 00:41:15.290 --> 00:41:16.434 vectors in the base frame. 00:41:16.434 --> 00:41:17.850 Or you'll be told you can leave it 00:41:17.850 --> 00:41:22.300 in whatever is your comfortable set of unit vectors. 00:41:22.300 --> 00:41:25.610 Most of the time, the first ones you'll arrive at 00:41:25.610 --> 00:41:27.885 are the ones in terms of the rotating coordinates that 00:41:27.885 --> 00:41:29.530 are easier to use. 00:41:29.530 --> 00:41:31.990 The more natural answer falls out in terms of these. 00:41:35.220 --> 00:41:36.330 Good. 00:41:36.330 --> 00:41:40.560 All right, and we've got three, four minutes left. 00:41:40.560 --> 00:41:43.730 What have I confused you with here? 00:41:43.730 --> 00:41:51.920 So key concepts-- what have we-- what 00:41:51.920 --> 00:41:55.820 hasn't been clear or maybe we didn't cover it 00:41:55.820 --> 00:41:58.502 yet-- another point. 00:41:58.502 --> 00:42:00.090 AUDIENCE: So the reason we chose those 00:42:00.090 --> 00:42:04.462 as the starting reference frames instead of I hats and theta 00:42:04.462 --> 00:42:05.127 hats? 00:42:05.127 --> 00:42:07.210 PROFESSOR: Only because at the beginning of class, 00:42:07.210 --> 00:42:09.410 we talked about it-- which frames do we want to use, 00:42:09.410 --> 00:42:11.290 and then we chose those. 00:42:11.290 --> 00:42:15.630 Could we have used a polar coordinate system 00:42:15.630 --> 00:42:18.390 to do this problem? 00:42:18.390 --> 00:42:19.790 Sure. 00:42:19.790 --> 00:42:23.705 Twice You do it once in each-- 00:42:23.705 --> 00:42:25.425 AUDIENCE: Is there a way to know up 00:42:25.425 --> 00:42:32.855 front which one would simplify down to the inertial i hats 00:42:32.855 --> 00:42:34.316 and j hats more simply? 00:42:34.316 --> 00:42:35.795 PROFESSOR: The easiest way? 00:42:35.795 --> 00:42:37.261 Is there a way to know upfront? 00:42:37.261 --> 00:42:37.760 No. 00:42:37.760 --> 00:42:39.850 That's just experience. 00:42:39.850 --> 00:42:43.790 Work lots of problems, and you get good at picking frame. 00:42:43.790 --> 00:42:45.945 We can probably, with time as we meet and talk 00:42:45.945 --> 00:42:47.320 about these things, we'll come up 00:42:47.320 --> 00:42:51.330 with some sort of general insights about how to do that. 00:42:51.330 --> 00:42:52.325 Yes? 00:42:52.325 --> 00:42:54.750 AUDIENCE: Is this picture up in parentheses 00:42:54.750 --> 00:42:58.145 supposed to be those coordinate systems? 00:42:58.145 --> 00:43:01.990 PROFESSOR: This picture is the coordinate system 00:43:01.990 --> 00:43:03.110 of that first arm. 00:43:03.110 --> 00:43:05.860 AUDIENCE: OK, so is that supposed to be phi up there? 00:43:05.860 --> 00:43:06.623 PROFESSOR: Yeah. 00:43:09.780 --> 00:43:11.640 Wait a minute. 00:43:11.640 --> 00:43:12.490 No. 00:43:12.490 --> 00:43:14.050 This is the first arm. 00:43:14.050 --> 00:43:17.750 That is theta and these are ones, right? 00:43:20.920 --> 00:43:22.705 The 2 system would be phis 00:43:22.705 --> 00:43:23.463 AUDIENCE: OK. 00:43:23.463 --> 00:43:26.004 I was just wondering if that was the ride or if that was not. 00:43:26.004 --> 00:43:26.920 PROFESSOR: No. 00:43:26.920 --> 00:43:31.711 This is point A, if you will. 00:43:31.711 --> 00:43:32.460 Well, it could be. 00:43:32.460 --> 00:43:36.050 It's lined up with point A. This is A. 00:43:36.050 --> 00:43:39.050 AUDIENCE: Because I thought we decided that the phi was-- 00:43:39.050 --> 00:43:40.800 PROFESSOR: This is point A, right? 00:43:40.800 --> 00:43:44.590 That is point A. And this is arm CA. 00:43:44.590 --> 00:43:48.852 AUDIENCE: So is this one here the origin of this one? 00:43:48.852 --> 00:43:51.540 PROFESSOR: Well, look at whatever the unit vector is. 00:43:51.540 --> 00:43:56.200 The unit vector in this system is lined up with that arm. 00:43:56.200 --> 00:44:00.300 So this is just a breakdown of these unit vectors 00:44:00.300 --> 00:44:02.110 so I could draw the angles and figure out 00:44:02.110 --> 00:44:03.570 the sines and cosines. 00:44:03.570 --> 00:44:07.730 You could draw a similar picture for i2 j2's. 00:44:07.730 --> 00:44:10.600 And then it would be phi's. 00:44:10.600 --> 00:44:13.110 Good question. 00:44:13.110 --> 00:44:13.610 Yes? 00:44:13.610 --> 00:44:15.150 AUDIENCE: So, since you can choose 00:44:15.150 --> 00:44:17.010 between Cartesian and polar coordinates, 00:44:17.010 --> 00:44:20.400 could you set one in Cartesian, one in polar, 00:44:20.400 --> 00:44:25.350 you can mix and match it or-- is that beneficial 00:44:25.350 --> 00:44:26.835 in some problems? 00:44:26.835 --> 00:44:34.450 PROFESSOR: Polar is-- I don't have time to show you today. 00:44:34.450 --> 00:44:39.680 But for planar motion problems, which 00:44:39.680 --> 00:44:44.170 are things confined to a plane, they rotate, axis of rotation's 00:44:44.170 --> 00:44:46.770 always in the k direction, which is all the problems that you 00:44:46.770 --> 00:44:48.115 ever did in 801 Physics. 00:44:48.115 --> 00:44:50.990 You didn't do general things actually. 00:44:50.990 --> 00:44:56.340 But for planar motion problems, cylindrical coordinates, 00:44:56.340 --> 00:45:00.250 actually, you still need the k to describe the rotation, 00:45:00.250 --> 00:45:01.300 right? 00:45:01.300 --> 00:45:03.150 Polar coordinates, cylindrical coordinates 00:45:03.150 --> 00:45:05.480 are oftentimes really convenient. 00:45:05.480 --> 00:45:08.290 And they're easy to use because you've learned them 00:45:08.290 --> 00:45:09.450 a long, long time ago. 00:45:09.450 --> 00:45:11.890 And you know the relations. 00:45:11.890 --> 00:45:17.439 But you can make it a rotating x1, y1, z1 rotating system 00:45:17.439 --> 00:45:18.480 and it will all work out. 00:45:21.700 --> 00:45:24.440 We came up with this little formula here, right? 00:45:24.440 --> 00:45:27.565 This could just as easily have been r hat. 00:45:30.360 --> 00:45:35.600 And this could have just as easily been no difference 00:45:35.600 --> 00:45:38.860 whatsoever in a planar motion problem, when 00:45:38.860 --> 00:45:42.190 you attach an xy system that rotates with it, 00:45:42.190 --> 00:45:43.970 or I call it r and theta. 00:45:43.970 --> 00:45:45.430 These are the same direction. 00:45:45.430 --> 00:45:46.820 R is in the direction of i1. 00:45:46.820 --> 00:45:48.390 Theta is in the direction of j1. 00:45:51.270 --> 00:45:53.540 So use it when it's convenient, and it's convenient 00:45:53.540 --> 00:45:57.620 a lot of times, especially that nasty acceleration formula. 00:45:57.620 --> 00:45:59.570 In polar coordinates, it reduces down just 00:45:59.570 --> 00:46:01.620 to the set of five terms. 00:46:01.620 --> 00:46:06.330 Memorize it and just tick them off-- Coriolis, centripetal. 00:46:06.330 --> 00:46:07.440 You see them right away. 00:46:07.440 --> 00:46:08.570 You know what they are. 00:46:08.570 --> 00:46:13.180 But there are certain problems, even in planar motion problems 00:46:13.180 --> 00:46:15.610 that polar coordinates don't work for-- doesn't work for. 00:46:15.610 --> 00:46:16.110 [INAUDIBLE] 00:46:20.150 --> 00:46:21.810 And think about that. 00:46:21.810 --> 00:46:23.280 It's actually a simple problem. 00:46:23.280 --> 00:46:24.640 Put a dog on a marry-go-round. 00:46:24.640 --> 00:46:26.420 The dog's running in a random direction 00:46:26.420 --> 00:46:27.372 on the merry-go-round. 00:46:27.372 --> 00:46:29.330 And the merry-go-round is turning at some rate. 00:46:29.330 --> 00:46:32.460 And you only want one rotating coordinate system, r and theta. 00:46:32.460 --> 00:46:35.090 You can't do the problem with polar coordinates. 00:46:35.090 --> 00:46:35.930 Think about it. 00:46:35.930 --> 00:46:38.560 Go away, think about why not. 00:46:38.560 --> 00:46:40.410 I'll tell you the answer in words. 00:46:40.410 --> 00:46:41.320 You go figure it out. 00:46:41.320 --> 00:46:45.080 You can't describe the velocity of the dog 00:46:45.080 --> 00:46:46.203 in polar coordinates. 00:46:51.420 --> 00:46:52.660 The dog is running around. 00:46:52.660 --> 00:46:54.930 If the dog's fixed on the rotating thing, than polar 00:46:54.930 --> 00:46:55.650 coordinates work. 00:46:55.650 --> 00:46:57.691 If the dog's running, you can't do that velocity. 00:46:57.691 --> 00:47:02.155 So you need a more sophisticated coordinate system.