WEBVTT 00:00:00.100 --> 00:00:02.450 The following content is provided under a Creative 00:00:02.450 --> 00:00:03.830 Commons license. 00:00:03.830 --> 00:00:06.070 Your support will help MIT OpenCourseWare 00:00:06.070 --> 00:00:10.170 continue to offer high quality educational resources for free. 00:00:10.170 --> 00:00:12.710 To make a donation or to view additional materials 00:00:12.710 --> 00:00:16.620 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.620 --> 00:00:17.278 at ocw.mit.edu. 00:00:21.235 --> 00:00:22.110 PROFESSOR: All right. 00:00:22.110 --> 00:00:23.190 Thoughts. 00:00:23.190 --> 00:00:24.499 Important concepts of the week? 00:00:24.499 --> 00:00:25.540 AUDIENCE: Principal axes. 00:00:25.540 --> 00:00:26.520 PROFESSOR: Principal axes. 00:00:26.520 --> 00:00:27.020 Good one. 00:00:35.840 --> 00:00:36.770 Another one. 00:00:36.770 --> 00:00:38.706 AUDIENCE: Dynamic versus static balances. 00:00:38.706 --> 00:00:39.414 PROFESSOR: Right. 00:00:52.860 --> 00:00:53.738 Another one. 00:00:53.738 --> 00:00:54.990 AUDIENCE: Symmetry. 00:00:54.990 --> 00:00:55.490 Symmetry. 00:00:55.490 --> 00:00:56.310 PROFESSOR: Symmetry. 00:00:56.310 --> 00:00:56.809 Right. 00:01:02.860 --> 00:01:07.100 This is in the context of things like the mass moment of inertia 00:01:07.100 --> 00:01:07.730 matrix. 00:01:07.730 --> 00:01:08.230 Yes? 00:01:08.230 --> 00:01:09.605 AUDIENCE: Parallel axis theorem. 00:01:09.605 --> 00:01:10.980 PROFESSOR: Parallel axis theorem. 00:01:10.980 --> 00:01:11.530 OK. 00:01:11.530 --> 00:01:13.280 We haven't done much with it yet but we're 00:01:13.280 --> 00:01:14.360 going to come back to it. 00:01:26.301 --> 00:01:26.800 All right. 00:01:26.800 --> 00:01:27.205 Pretty good list. 00:01:27.205 --> 00:01:28.204 We only had one lecture. 00:01:32.320 --> 00:01:33.632 Dynamic and static balancing. 00:01:33.632 --> 00:01:35.090 Let's talk about that for a second. 00:01:43.580 --> 00:01:45.100 Let's talk about static balancing. 00:01:45.100 --> 00:01:48.960 Tell me what are features of static balancing? 00:01:48.960 --> 00:01:51.700 How can you-- you have a quiz problem, 00:01:51.700 --> 00:01:56.490 it's on the final-- you have an object and an axis about which 00:01:56.490 --> 00:01:58.770 its spinning and you're asked, is this statically 00:01:58.770 --> 00:02:00.570 balanced or not. 00:02:00.570 --> 00:02:01.570 What would you look for? 00:02:01.570 --> 00:02:02.611 Christina. 00:02:02.611 --> 00:02:04.720 AUDIENCE: [INAUDIBLE]. 00:02:04.720 --> 00:02:06.415 PROFESSOR: G needs to be-- 00:02:06.415 --> 00:02:07.720 AUDIENCE: [INAUDIBLE] axis. 00:02:07.720 --> 00:02:11.920 PROFESSOR: So another way of saying that is the axis must-- 00:02:11.920 --> 00:02:14.400 the axis of rotation must pass through-- 00:02:14.400 --> 00:02:15.370 AUDIENCE: The center. 00:02:15.370 --> 00:02:16.530 PROFESSOR: --G. Is that what you're trying to say? 00:02:16.530 --> 00:02:18.140 The center of mass? 00:02:18.140 --> 00:02:19.560 Everybody agree about that? 00:02:19.560 --> 00:02:20.060 All right. 00:02:20.060 --> 00:02:23.650 Now, if I have weird shaped object, 00:02:23.650 --> 00:02:25.820 I know where the center of mass, and I 00:02:25.820 --> 00:02:32.060 stick the axis of rotation at any angle at all through it, 00:02:32.060 --> 00:02:34.280 is it statically balance regardless 00:02:34.280 --> 00:02:38.210 of where I have the x-axis of rotation passing 00:02:38.210 --> 00:02:39.612 through the center of mass? 00:02:39.612 --> 00:02:40.456 AUDIENCE: Yes. 00:02:40.456 --> 00:02:41.840 PROFESSOR: I hear a yes. 00:02:41.840 --> 00:02:43.860 How about the other people? 00:02:43.860 --> 00:02:44.660 AUDIENCE: Yeah. 00:02:44.660 --> 00:02:45.160 OK. 00:02:45.160 --> 00:02:47.740 So how do you-- I think that's probably right. 00:02:47.740 --> 00:02:48.850 But how do you test it? 00:02:48.850 --> 00:02:51.870 How do you test to see-- what's a simple test you 00:02:51.870 --> 00:02:54.456 can form to see if something is statically balanced? 00:02:54.456 --> 00:02:56.330 AUDIENCE: See if it goes through a low point. 00:02:56.330 --> 00:02:57.038 PROFESSOR: Right. 00:02:57.038 --> 00:03:01.490 So you make the axis that you're rotating about horizontal 00:03:01.490 --> 00:03:03.720 and see if the thing seeks a low point. 00:03:03.720 --> 00:03:09.430 Because if it does, it's telling you that what 00:03:09.430 --> 00:03:10.570 about the center of mass? 00:03:10.570 --> 00:03:11.880 AUDIENCE: It's not going through the-- 00:03:11.880 --> 00:03:13.430 PROFESSOR: It's not going through the axis. 00:03:13.430 --> 00:03:15.471 To go to a low point, it means the center of mass 00:03:15.471 --> 00:03:17.470 is somewhere below the axis of spin. 00:03:17.470 --> 00:03:18.054 Right? 00:03:18.054 --> 00:03:19.970 But any axis at all, as long as its through G, 00:03:19.970 --> 00:03:22.630 you are statically balanced. 00:03:22.630 --> 00:03:25.520 Dynamic balancing. 00:03:25.520 --> 00:03:29.060 What are the-- what would you look for on an object to say 00:03:29.060 --> 00:03:30.875 is this dynamically balanced or not? 00:03:30.875 --> 00:03:31.875 What kind of properties? 00:03:36.706 --> 00:03:38.622 AUDIENCE: There would be no unbalanced torques 00:03:38.622 --> 00:03:41.590 if you decided to rotate it. 00:03:41.590 --> 00:03:44.990 PROFESSOR: So you're looking for now unbalanced torques. 00:03:44.990 --> 00:03:47.766 AUDIENCE: Because I'm like if it-- if the object is rotating 00:03:47.766 --> 00:03:49.890 about the axis that you have it on, 00:03:49.890 --> 00:03:58.790 you won't need to supply an extra torque to try and keep 00:03:58.790 --> 00:04:00.447 it balanced. 00:04:00.447 --> 00:04:01.280 [INTERPOSING VOICES] 00:04:01.280 --> 00:04:02.620 PROFESSOR: --pretty convincing. 00:04:02.620 --> 00:04:04.492 AUDIENCE: --the axis it's rotating on. 00:04:04.492 --> 00:04:05.430 PROFESSOR: OK. 00:04:05.430 --> 00:04:07.790 If you put in that last caveat then 00:04:07.790 --> 00:04:09.610 it's getting pretty correct. 00:04:09.610 --> 00:04:12.310 Can there be a torque on it that's not balanced 00:04:12.310 --> 00:04:14.646 and have it be dynamically balanced? 00:04:14.646 --> 00:04:15.540 AUDIENCE: Well, yeah. 00:04:15.540 --> 00:04:16.415 PROFESSOR: Which one? 00:04:18.706 --> 00:04:20.519 AUDIENCE: Within the axis of rotation. 00:04:20.519 --> 00:04:24.030 PROFESSOR: If you have a torque that's in the axis of spin 00:04:24.030 --> 00:04:26.210 and around the axis of rotation and that torque 00:04:26.210 --> 00:04:27.755 will cause what to happen? 00:04:27.755 --> 00:04:29.030 AUDIENCE: Rotation. 00:04:29.030 --> 00:04:30.930 PROFESSOR: Cause the rotation rate to? 00:04:30.930 --> 00:04:31.320 AUDIENCE: Change. 00:04:31.320 --> 00:04:32.361 PROFESSOR: Change, right? 00:04:32.361 --> 00:04:35.330 That puts energy into the system and accelerates it. 00:04:35.330 --> 00:04:38.390 And it's a torque that's in the direction of spin that doesn't 00:04:38.390 --> 00:04:39.700 cause dynamic imbalances. 00:04:39.700 --> 00:04:44.740 But a torque that is other than in the direction of spin, 00:04:44.740 --> 00:04:47.460 that's what a dynamic imbalance is. 00:04:47.460 --> 00:04:51.470 So what are some other ways you can say this in terms of things 00:04:51.470 --> 00:04:58.330 like evidence of-- let's say I give you H 00:04:58.330 --> 00:05:00.922 and I give you omega, how can you just 00:05:00.922 --> 00:05:02.380 tell by looking at those two thing? 00:05:02.380 --> 00:05:04.755 I give you the angle of momentum and I give you the spin, 00:05:04.755 --> 00:05:06.590 how can you tell instantly whether or not 00:05:06.590 --> 00:05:08.170 it is dynamically balanced? 00:05:08.170 --> 00:05:10.078 AUDIENCE: If there-- each of the components 00:05:10.078 --> 00:05:11.510 in the same direction. 00:05:11.510 --> 00:05:16.928 PROFESSOR: So you're saying if H and omega are in the-- 00:05:16.928 --> 00:05:18.040 AUDIENCE: R line. 00:05:18.040 --> 00:05:18.790 PROFESSOR: A line. 00:05:18.790 --> 00:05:20.800 They just have to be parallel. 00:05:20.800 --> 00:05:24.465 If they're aligned, you do not have any dynamic imbalance. 00:05:28.062 --> 00:05:28.770 That sound right? 00:05:28.770 --> 00:05:29.555 Yeah? 00:05:29.555 --> 00:05:31.138 AUDIENCE: It doesn't matter if they're 00:05:31.138 --> 00:05:32.617 parallel or antiparallel, does it? 00:05:32.617 --> 00:05:34.325 PROFESSOR: You mean just opposite senses? 00:05:34.325 --> 00:05:35.927 AUDIENCE: Yeah. 00:05:35.927 --> 00:05:37.885 PROFESSOR: I think that'd be really hard to do. 00:05:37.885 --> 00:05:40.030 I don't know of any-- I don't think 00:05:40.030 --> 00:05:43.100 you can make that system that has angular momentum that 00:05:43.100 --> 00:05:46.980 is negative-- it's actually opposite the direction 00:05:46.980 --> 00:05:49.800 of the rotation rate. 00:05:49.800 --> 00:05:54.340 I don't think it-- I don't think our universe supports 00:05:54.340 --> 00:05:55.940 that physics. 00:05:55.940 --> 00:05:59.660 But anyway, in any case, they're aligned, you're OK. 00:06:02.930 --> 00:06:03.520 All right? 00:06:03.520 --> 00:06:04.900 Let's move on. 00:06:04.900 --> 00:06:09.987 So today we're going actually just 00:06:09.987 --> 00:06:11.320 have you work a couple problems. 00:06:11.320 --> 00:06:14.000 And you really are going to work in groups, and when you finish, 00:06:14.000 --> 00:06:15.916 one of the groups is going to come up and fill 00:06:15.916 --> 00:06:18.370 in the blanks of the answer to the problem 00:06:18.370 --> 00:06:19.370 and we'll talk about it. 00:06:19.370 --> 00:06:21.300 And we're going to work two problems. 00:06:21.300 --> 00:06:25.150 And they have to do with things that spin. 00:06:25.150 --> 00:06:31.483 And as a quick review just of information-- 00:06:31.483 --> 00:06:32.465 I guess we can do this. 00:06:38.370 --> 00:06:42.475 So the problem is a simple one. 00:06:42.475 --> 00:06:44.530 It's basically this. 00:06:44.530 --> 00:06:49.150 And I decided to lubricate it and it made the wood swell 00:06:49.150 --> 00:06:50.966 and so now it won't spin at all. 00:06:50.966 --> 00:06:52.727 Actually, unless I let the axis spin. 00:06:52.727 --> 00:06:53.810 So it's just this problem. 00:06:57.280 --> 00:07:00.830 The axis passes through the center of mass 00:07:00.830 --> 00:07:03.970 and it looks just like this. 00:07:03.970 --> 00:07:10.070 And I've got an x-axis aligned with the axis of the rod. 00:07:10.070 --> 00:07:11.420 The Z is this way. 00:07:11.420 --> 00:07:14.000 And the spin is cap omega. 00:07:14.000 --> 00:07:19.890 And the mass moment of inertia matrix 00:07:19.890 --> 00:07:22.220 for this problem about G-- and our axis 00:07:22.220 --> 00:07:26.220 is passing through G-- so mass moment of inertia matrix times 00:07:26.220 --> 00:07:30.220 omega gives you H. So here's the mass moment of inertia matrix. 00:07:30.220 --> 00:07:33.490 I claim is going to be diagonal. 00:07:33.490 --> 00:07:35.830 And the test of that, can you tell me whether or not 00:07:35.830 --> 00:07:39.520 I've chosen axes that, just from symmetry, you know 00:07:39.520 --> 00:07:41.060 will be principal axes? 00:07:45.425 --> 00:07:46.687 What do you think? 00:07:46.687 --> 00:07:49.104 AUDIENCE: Well, the rod is circular now that [INAUDIBLE]. 00:07:49.104 --> 00:07:49.770 PROFESSOR: Yeah. 00:07:49.770 --> 00:07:53.939 So one axis-- that's an axis of symmetry is down the shaft. 00:07:53.939 --> 00:07:55.230 So that's guaranteed to be one. 00:07:55.230 --> 00:07:57.210 Have I chosen one like that? 00:07:57.210 --> 00:07:59.335 And any other orthogonal pair after that-- 00:07:59.335 --> 00:08:01.860 doesn't matter which way I orient them-- will also be. 00:08:01.860 --> 00:08:04.070 So I've got one up and one into the board. 00:08:04.070 --> 00:08:07.340 So those are principal axes for this object. 00:08:07.340 --> 00:08:09.020 So it's going to be diagonal. 00:08:09.020 --> 00:08:13.010 And in fact, it looks-- it's MR squared over 2, 00:08:13.010 --> 00:08:15.970 ML squared over 12, and ML squared over 12 00:08:15.970 --> 00:08:17.710 when you work it out. 00:08:17.710 --> 00:08:21.320 And if you look up in the book, most books for slen-- 00:08:21.320 --> 00:08:23.640 what they call slender rods, will 00:08:23.640 --> 00:08:25.986 say that this first term is 0. 00:08:25.986 --> 00:08:27.360 And that's because they're saying 00:08:27.360 --> 00:08:31.250 that L is a lot bigger than R-- the radius of this thing. 00:08:31.250 --> 00:08:34.360 And so MR squared is a pretty small number. 00:08:34.360 --> 00:08:38.450 So it's energy and rotation spinning this way 00:08:38.450 --> 00:08:40.860 is not very big for its angular momentum. 00:08:40.860 --> 00:08:42.809 But its angular momentum spinning like this 00:08:42.809 --> 00:08:48.377 is much, much larger because you have much greater MR squareds. 00:08:48.377 --> 00:08:50.460 So you can leave, for the purpose of this problem, 00:08:50.460 --> 00:08:52.460 you can't treat this as 0. 00:08:52.460 --> 00:08:55.990 And in your groups, I want you to come up with the omega 00:08:55.990 --> 00:08:59.740 vector, H and DHDT. 00:08:59.740 --> 00:09:01.480 So do get in groups, talk about it. 00:09:01.480 --> 00:09:02.980 You've got a few minutes to do this. 00:09:02.980 --> 00:09:04.760 This one's pretty straightforward. 00:09:04.760 --> 00:09:07.337 And then you can do another one that's harder so warm up 00:09:07.337 --> 00:09:08.920 and find a group to work in and you're 00:09:08.920 --> 00:09:10.720 going to work a couple of problems this way. 00:09:10.720 --> 00:09:12.525 Got a group that feels pretty good about they're answer? 00:09:12.525 --> 00:09:13.210 AUDIENCE: Yes. 00:09:13.210 --> 00:09:15.460 PROFESSOR: All right. 00:09:15.460 --> 00:09:16.930 Write it up. 00:09:16.930 --> 00:09:23.830 Come up, fill in the omega, H and DHDT. 00:09:49.200 --> 00:09:49.700 OK. 00:09:49.700 --> 00:09:51.750 Can everybody see it? 00:09:51.750 --> 00:09:58.080 So you have a Z component only for the omega, a Z component k 00:09:58.080 --> 00:10:03.180 hat only for H, and its i omega-- iZZ omega. 00:10:03.180 --> 00:10:07.600 And DHDT, the only variable here is omega 00:10:07.600 --> 00:10:10.470 and you get an omega dot. 00:10:10.470 --> 00:10:15.910 And there's no-- DKDT is 0 because it 00:10:15.910 --> 00:10:17.980 doesn't change direction. 00:10:17.980 --> 00:10:21.620 Any questions about this? 00:10:21.620 --> 00:10:22.140 Yes? 00:10:22.140 --> 00:10:23.806 AUDIENCE: So I was a little bit confused 00:10:23.806 --> 00:10:30.347 in lecture about how we knew the i hat, j hat, k hat in the H 00:10:30.347 --> 00:10:30.847 term. 00:10:30.847 --> 00:10:32.755 Is it just because it's only i hat 00:10:32.755 --> 00:10:35.140 on the top, j hat in the middle, and k hat on the bottom? 00:10:35.140 --> 00:10:35.640 Is that-- 00:10:35.640 --> 00:10:39.180 PROFESSOR: So the convention-- if I understand your question 00:10:39.180 --> 00:10:44.630 correctly-- the convention when you write out the H vector 00:10:44.630 --> 00:10:52.220 is it's the result of multiplying the three 00:10:52.220 --> 00:10:54.220 components of the spin. 00:10:54.220 --> 00:10:56.290 This is the piece in the i direction, 00:10:56.290 --> 00:10:58.400 j direction, k direction. 00:10:58.400 --> 00:11:04.270 And you multiply these out you get three results. 00:11:04.270 --> 00:11:08.846 Vector times-- and this one is Hx and it is in the i. 00:11:08.846 --> 00:11:13.900 The second one from this times the middle row gives you Hy 00:11:13.900 --> 00:11:15.530 and it's in the j. 00:11:15.530 --> 00:11:18.670 And the third one, this vector times these three 00:11:18.670 --> 00:11:22.100 in the bottom row, give you Hz in the k. 00:11:25.022 --> 00:11:26.970 AUDIENCE: So I thought when-- I don't know, 00:11:26.970 --> 00:11:29.405 I could be completely wrong-- but I thought when you did 00:11:29.405 --> 00:11:34.275 [INAUDIBLE] you multiplied this, this, this and then this, this, 00:11:34.275 --> 00:11:37.320 this and then-- 00:11:37.320 --> 00:11:43.102 PROFESSOR: So we have a capital A, capital B, capital C 00:11:43.102 --> 00:11:44.060 terms in the first row? 00:11:44.060 --> 00:11:45.620 AUDIENCE: So you do, like, this and-- 00:11:45.620 --> 00:11:47.630 PROFESSOR: And you have a capital-- little a, 00:11:47.630 --> 00:11:50.370 little b, little c terms in this one. 00:11:50.370 --> 00:11:53.140 When you multiply this times this, 00:11:53.140 --> 00:11:59.275 you get Aa plus Bb plus Cc. 00:11:59.275 --> 00:11:59.900 AUDIENCE: Yeah. 00:11:59.900 --> 00:12:01.160 So that's my question. 00:12:01.160 --> 00:12:04.245 So if you have the omega x is an i hat, 00:12:04.245 --> 00:12:06.875 omega y is a j hat, omega z's a k hat, 00:12:06.875 --> 00:12:08.510 how, if you're multiplying like this, 00:12:08.510 --> 00:12:11.424 do you get only i hats on the top of the thing. 00:12:11.424 --> 00:12:12.090 PROFESSOR: Yeah. 00:12:12.090 --> 00:12:15.630 Well, it's because this isn't just a matrix. 00:12:15.630 --> 00:12:18.400 This is actually a tensor. 00:12:18.400 --> 00:12:22.940 And we haven't gone into messy tensor notation. 00:12:22.940 --> 00:12:26.880 So you're just being told a convention here. 00:12:26.880 --> 00:12:30.140 And that is, because this is a vector being multiplied 00:12:30.140 --> 00:12:35.980 by a tensor, the result will-- even though this is i, j, k, 00:12:35.980 --> 00:12:40.820 omega xi times this first term in this tensor, 00:12:40.820 --> 00:12:43.350 it gives you an i back. 00:12:43.350 --> 00:12:47.380 Omega yj times this one gives you an i back. 00:12:47.380 --> 00:12:51.242 Omega zk times that gives you an i back. 00:12:51.242 --> 00:12:51.783 AUDIENCE: OK. 00:12:51.783 --> 00:12:52.490 That's what I was really confused -- 00:12:52.490 --> 00:12:53.160 PROFESSOR: Right. 00:12:53.160 --> 00:12:55.409 And that's a great question because it wasn't obvious. 00:12:55.409 --> 00:12:58.015 We didn't do the full tensor mathematic. 00:12:58.015 --> 00:13:01.230 We just gave you a result. By definition, 00:13:01.230 --> 00:13:05.100 this row times that is in the i direction, j direction, k 00:13:05.100 --> 00:13:05.690 direction. 00:13:05.690 --> 00:13:06.550 Great question. 00:13:06.550 --> 00:13:09.340 OK. 00:13:09.340 --> 00:13:10.180 OK. 00:13:10.180 --> 00:13:11.720 So let's go on to harder problem. 00:13:11.720 --> 00:13:12.886 So you're great at that one. 00:13:16.150 --> 00:13:18.940 We're going to do this one now. 00:13:18.940 --> 00:13:20.255 So we just did this problem. 00:13:22.699 --> 00:13:23.990 Now we want to do this problem. 00:13:30.780 --> 00:13:42.440 And so the same i matrix is used here. 00:13:42.440 --> 00:13:45.460 It's this one and you can let that term is 0 00:13:45.460 --> 00:13:47.600 because the coordinate system that 00:13:47.600 --> 00:13:55.840 has been defined on this object is x, z, y into the board. 00:13:55.840 --> 00:13:58.930 But the direction of spin now is like that. 00:13:58.930 --> 00:14:01.210 And we'll call it cap omega again. 00:14:01.210 --> 00:14:02.440 But it's in that direction. 00:14:02.440 --> 00:14:05.060 And these are 45 degree angles for the purposes 00:14:05.060 --> 00:14:05.820 of this problem. 00:14:05.820 --> 00:14:11.200 So that's sine cosine of 45 root 2 over 2. 00:14:11.200 --> 00:14:14.160 So now, what's omega? 00:14:14.160 --> 00:14:15.500 What's H? 00:14:15.500 --> 00:14:16.870 And what's DHDT? 00:14:16.870 --> 00:14:18.910 So work in groups and sort that one out. 00:14:22.140 --> 00:14:23.442 You ready to go? 00:14:23.442 --> 00:14:24.477 AUDIENCE: We can be. 00:14:24.477 --> 00:14:25.060 PROFESSOR: OK. 00:14:25.060 --> 00:14:25.810 Go for it. 00:14:36.736 --> 00:14:38.236 We'll probably do this occasionally. 00:14:41.230 --> 00:14:42.520 I'll say it again. 00:14:42.520 --> 00:14:45.419 I have no intention of embarrassing you at the board. 00:14:45.419 --> 00:14:47.960 So the practice will be, you put your answer up, you sit down 00:14:47.960 --> 00:14:49.330 and then we talk about it. 00:14:49.330 --> 00:14:50.170 It's not about you. 00:14:50.170 --> 00:14:51.340 It's about what's on the board. 00:14:51.340 --> 00:14:51.839 OK? 00:15:04.372 --> 00:15:06.080 AUDIENCE: Should I keep going or should-- 00:15:06.080 --> 00:15:06.230 PROFESSOR: No. 00:15:06.230 --> 00:15:07.188 Put the whole thing up. 00:15:07.188 --> 00:15:08.540 Yep. 00:15:08.540 --> 00:15:11.579 And if you want to use that big piece of chalk it shows better. 00:15:11.579 --> 00:15:12.370 Just to your right. 00:15:54.900 --> 00:15:55.400 OK. 00:15:55.400 --> 00:15:58.690 So let's talk about omega first. 00:15:58.690 --> 00:16:03.550 How do you figure-- we have a minus root 2 over 2 here. 00:16:03.550 --> 00:16:04.310 Can't see that. 00:16:04.310 --> 00:16:09.330 Minus root 2 over 2 omega i 0 root 2 over 2 omega k. 00:16:09.330 --> 00:16:11.350 Are people-- feel good about that? 00:16:11.350 --> 00:16:12.470 Any differences? 00:16:12.470 --> 00:16:12.970 All right. 00:16:12.970 --> 00:16:15.670 So this thing-- this omega-- that spin clearly 00:16:15.670 --> 00:16:19.800 has components that are like that and like this. 00:16:19.800 --> 00:16:22.650 So this is your z piece and your i piece. 00:16:22.650 --> 00:16:25.490 And it is, indeed, in the minus i hat direction. 00:16:25.490 --> 00:16:26.990 So that seems OK. 00:16:26.990 --> 00:16:34.640 And H would be some i-- the i times omega. 00:16:34.640 --> 00:16:38.210 And the first term, if you take that upper left term as 0, 00:16:38.210 --> 00:16:39.490 then you don't get anything. 00:16:39.490 --> 00:16:41.150 The second term you don't get anything. 00:16:41.150 --> 00:16:44.790 So how do you feel about the angular momentum? 00:16:47.780 --> 00:16:49.350 Any differences? 00:16:49.350 --> 00:16:50.040 OK. 00:16:50.040 --> 00:16:52.530 And the DHDT. 00:16:52.530 --> 00:16:55.110 You take the time derivative of this thing-- 00:16:55.110 --> 00:16:57.100 and you got an omega dot. 00:16:57.100 --> 00:16:58.100 OK. 00:16:58.100 --> 00:16:59.562 That looks OK as far as it goes. 00:16:59.562 --> 00:17:00.520 But any other thoughts? 00:17:04.108 --> 00:17:07.077 AUDIENCE: So k hat dot is not 0. 00:17:07.077 --> 00:17:07.660 PROFESSOR: OK. 00:17:07.660 --> 00:17:11.220 So you're saying k hat is changing with time. 00:17:11.220 --> 00:17:11.957 AUDIENCE: Yeah. 00:17:11.957 --> 00:17:13.540 PROFESSOR: And let's think about that. 00:17:13.540 --> 00:17:16.290 So it's-- that's this piece here. 00:17:16.290 --> 00:17:20.079 This is omega z k hat. 00:17:20.079 --> 00:17:25.940 And as this thing goes around that unit vector is doing this. 00:17:25.940 --> 00:17:27.260 Right? 00:17:27.260 --> 00:17:29.774 So you need a DkDT. 00:17:29.774 --> 00:17:31.690 So do you want to give-- somebody else give me 00:17:31.690 --> 00:17:34.863 the second term here? 00:17:34.863 --> 00:17:35.488 AUDIENCE: Sure. 00:17:38.360 --> 00:17:45.627 ML squared times omega squared because you 00:17:45.627 --> 00:17:52.298 have the changing directions in your k hat principal axes. 00:17:52.298 --> 00:17:56.840 And that's going to be-- well, times root 2. 00:17:56.840 --> 00:17:58.495 PROFESSOR: This is root-- 00:17:58.495 --> 00:18:00.840 AUDIENCE: Well, omega squared. 00:18:00.840 --> 00:18:02.190 PROFESSOR: OK. 00:18:02.190 --> 00:18:04.225 And then what about the root 2 over 2? 00:18:04.225 --> 00:18:06.180 AUDIENCE: [INAUDIBLE]. 00:18:06.180 --> 00:18:07.889 PROFESSOR: Got to square that too, right? 00:18:07.889 --> 00:18:09.388 AUDIENCE: That'd be squared as well. 00:18:09.388 --> 00:18:11.136 PROFESSOR: That gives you a half. 00:18:11.136 --> 00:18:11.635 OK. 00:18:11.635 --> 00:18:13.574 And-- 00:18:13.574 --> 00:18:15.070 AUDIENCE: That's over 12 as well. 00:18:15.070 --> 00:18:16.100 PROFESSOR: 12. 00:18:16.100 --> 00:18:16.980 And-- 00:18:16.980 --> 00:18:19.180 AUDIENCE: That would be in the j hat direction. 00:18:19.180 --> 00:18:19.770 PROFESSOR: Positive-- 00:18:19.770 --> 00:18:20.160 AUDIENCE: Positive. 00:18:20.160 --> 00:18:20.659 Yes. 00:18:20.659 --> 00:18:21.330 In this case. 00:18:21.330 --> 00:18:23.876 PROFESSOR: Started out with a negative but-- 00:18:23.876 --> 00:18:25.500 AUDIENCE: Negative [INAUDIBLE]. 00:18:25.500 --> 00:18:28.680 PROFESSOR: i cross k is minus j times a minus gives you 00:18:28.680 --> 00:18:29.500 the plus, right? 00:18:29.500 --> 00:18:35.740 So this looks-- this comes out ML squared omega 00:18:35.740 --> 00:18:39.930 squared j hat over 24 I think. 00:18:39.930 --> 00:18:40.640 Like that. 00:18:40.640 --> 00:18:41.960 That term. 00:18:41.960 --> 00:18:43.280 Now-- OK. 00:18:43.280 --> 00:18:46.190 And people good with that now? 00:18:46.190 --> 00:18:53.150 So just to remind you, when I was doing these problems-- 00:18:53.150 --> 00:18:55.485 when I first had to teach this course a few years ago, 00:18:55.485 --> 00:18:58.110 and I was trying to figure out an easy way to teach this 00:18:58.110 --> 00:19:01.320 and how to do these problems, I got myself all confused 00:19:01.320 --> 00:19:05.790 trying to figure out which components of the rotation 00:19:05.790 --> 00:19:09.540 vector are rotating and what do I have to cross into it 00:19:09.540 --> 00:19:10.930 to get the answer. 00:19:10.930 --> 00:19:13.450 And in fact, you don't have to make it 00:19:13.450 --> 00:19:15.500 anywhere near that hard. 00:19:15.500 --> 00:19:18.730 And let's take a quick look at something here. 00:19:25.590 --> 00:19:34.710 H is ML squared over 12 root 2 over 2-- 00:19:34.710 --> 00:19:37.890 where did H go-- root 2 over 2 omega k hat. 00:19:45.030 --> 00:19:49.030 H is a rotating vector. 00:19:49.030 --> 00:19:57.170 A derivative of a rotating vector in an inertial frame 00:19:57.170 --> 00:20:03.260 is the derivative of that rotating vector in the rotating 00:20:03.260 --> 00:20:04.160 frame. 00:20:04.160 --> 00:20:05.870 Which is the same thing as saying 00:20:05.870 --> 00:20:08.450 that omega is 0-- that's the change in length 00:20:08.450 --> 00:20:15.020 of the vector-- plus omega cross H. 00:20:15.020 --> 00:20:18.240 Let's test that and see if that gives us the right answer. 00:20:18.240 --> 00:20:22.900 This derivative gives you-- all it does is gives you the theta, 00:20:22.900 --> 00:20:23.940 the omega dot back. 00:20:23.940 --> 00:20:24.440 Right? 00:20:24.440 --> 00:20:25.980 That's that first term. 00:20:25.980 --> 00:20:28.050 And the second term should just then 00:20:28.050 --> 00:20:37.090 look like your omega, which is root 2 minus root 2 00:20:37.090 --> 00:20:47.420 over 2 cap omega i plus root 2 over 2 cap omega k 00:20:47.420 --> 00:20:54.270 cross a bunch of constants times k. 00:20:54.270 --> 00:20:54.770 Right? 00:20:57.600 --> 00:21:04.590 So the k cross k terms are? 00:21:04.590 --> 00:21:05.200 AUDIENCE: 0. 00:21:05.200 --> 00:21:06.150 PROFESSOR: 0. 00:21:06.150 --> 00:21:08.564 The i cross k-- 00:21:08.564 --> 00:21:09.480 AUDIENCE: Minus j. 00:21:09.480 --> 00:21:10.950 PROFESSOR: --minus j. 00:21:10.950 --> 00:21:13.670 And so you get root 2 over 2 times root 2 over 2. 00:21:13.670 --> 00:21:18.900 You get what's in H here multiplied 00:21:18.900 --> 00:21:20.630 by root 2 over 2 omega hat. 00:21:20.630 --> 00:21:27.200 And you get back exactly these two terms. 00:21:27.200 --> 00:21:29.800 So it's that easy. 00:21:29.800 --> 00:21:31.940 It's just the derivative of a rotating vector. 00:21:31.940 --> 00:21:35.560 Just doing omega cross H is the easiest 00:21:35.560 --> 00:21:41.190 way to deal with that derivative of the rotating piece. 00:21:45.110 --> 00:21:46.650 Any rotating vector, you can take 00:21:46.650 --> 00:21:48.620 this time derivative that way. 00:21:48.620 --> 00:21:50.570 All right. 00:21:50.570 --> 00:21:51.982 We've got a few minutes. 00:21:51.982 --> 00:21:53.690 I want to-- actually, any other questions 00:21:53.690 --> 00:21:56.920 about this kind of problem? 00:21:56.920 --> 00:21:59.910 In physics, most of the problems you've worked before, 00:21:59.910 --> 00:22:02.370 that involved rotation, are planar-- what 00:22:02.370 --> 00:22:04.090 we call planar motion problems. 00:22:04.090 --> 00:22:08.380 The axis of spin was always perpendicular to the plane 00:22:08.380 --> 00:22:11.090 and the rest the problem was confined to the plane. 00:22:11.090 --> 00:22:15.300 So hockey pucks sliding along and stuff like that. 00:22:15.300 --> 00:22:18.730 But it was always assumed that the axis of rotation 00:22:18.730 --> 00:22:20.880 was perpendicular to the plane and that the angular 00:22:20.880 --> 00:22:23.400 momentum was parallel. 00:22:23.400 --> 00:22:26.190 So this is actually-- this is 3D problem. 00:22:26.190 --> 00:22:27.740 This is a 3D dynamics problem. 00:22:27.740 --> 00:22:30.890 As soon as that H and the omega are different directions, 00:22:30.890 --> 00:22:34.470 you can come up with torques in all three directions. 00:22:34.470 --> 00:22:37.210 Right? 00:22:37.210 --> 00:22:39.830 All right. 00:22:39.830 --> 00:22:43.544 So any other questions about this? 00:22:43.544 --> 00:22:46.210 And if not, I want to talk about something that was on the quiz. 00:22:49.606 --> 00:22:50.106 OK. 00:22:54.980 --> 00:23:00.060 So here's-- this is the one problem on the quiz that gave 00:23:00.060 --> 00:23:04.880 more conceptual difficulty than any other single problem. 00:23:04.880 --> 00:23:06.580 People made mistakes on other problems, 00:23:06.580 --> 00:23:10.310 but got into conceptual trouble with this problem. 00:23:10.310 --> 00:23:13.580 Remember, you had a vehicle driving up a bridge 00:23:13.580 --> 00:23:17.560 and the bridge could be changing at some angular rate 00:23:17.560 --> 00:23:21.110 and it has some angular acceleration. 00:23:21.110 --> 00:23:23.554 And we said to keep the problem simple, 00:23:23.554 --> 00:23:25.220 that you could treat this as a particle. 00:23:25.220 --> 00:23:26.761 And that really means you didn't have 00:23:26.761 --> 00:23:29.040 to deal with angular momentum. 00:23:29.040 --> 00:23:30.700 i omega for the object. 00:23:30.700 --> 00:23:33.570 You could just treat it like a particle. 00:23:33.570 --> 00:23:38.750 And one of the-- where the confusion came from 00:23:38.750 --> 00:23:41.594 was in figuring out free body diagrams. 00:23:41.594 --> 00:23:43.760 You're told in the problem the language is something 00:23:43.760 --> 00:23:50.620 like, the action of the tires on the road result in a net force 00:23:50.620 --> 00:23:55.360 up the incline called t. 00:23:55.360 --> 00:23:57.850 We didn't call it friction, we didn't say anything, 00:23:57.850 --> 00:24:01.100 but people-- a number of people got confused about what's 00:24:01.100 --> 00:24:02.570 that have to do with friction. 00:24:02.570 --> 00:24:05.960 How does friction come into this? 00:24:05.960 --> 00:24:08.570 So let's do the free body diagram for this thing. 00:24:08.570 --> 00:24:11.586 So tell me what's-- here's my car, 00:24:11.586 --> 00:24:13.460 tell me what to put on the free body diagram. 00:24:17.050 --> 00:24:17.997 AUDIENCE: Weight. 00:24:17.997 --> 00:24:18.580 PROFESSOR: OK. 00:24:18.580 --> 00:24:19.079 Mg? 00:24:19.079 --> 00:24:19.770 AUDIENCE: Yep. 00:24:19.770 --> 00:24:20.110 PROFESSOR: All right. 00:24:20.110 --> 00:24:20.943 So you've got an Mg. 00:24:23.820 --> 00:24:24.320 Next. 00:24:24.320 --> 00:24:25.236 AUDIENCE: [INAUDIBLE]. 00:24:27.487 --> 00:24:28.570 PROFESSOR: A normal force? 00:24:28.570 --> 00:24:29.070 OK. 00:24:29.070 --> 00:24:34.047 That's going this way we'll call it. 00:24:34.047 --> 00:24:35.980 AUDIENCE: T. 00:24:35.980 --> 00:24:39.040 PROFESSOR: T. OK. 00:24:39.040 --> 00:24:40.800 Now, what about the tires on the road? 00:24:40.800 --> 00:24:42.582 What about friction? 00:24:42.582 --> 00:24:43.499 AUDIENCE: [INAUDIBLE]. 00:24:43.499 --> 00:24:44.248 PROFESSOR: Pardon? 00:24:44.248 --> 00:24:45.380 AUDIENCE: That is T. 00:24:45.380 --> 00:24:49.430 PROFESSOR: That is T. So you think of the free body diagram, 00:24:49.430 --> 00:24:52.650 you just say, what are all the possible sources 00:24:52.650 --> 00:24:54.960 of external force on this thing? 00:24:54.960 --> 00:24:59.180 And they come from gravity-- a body force-- and then 00:24:59.180 --> 00:25:02.650 other things that are in contact with it. 00:25:02.650 --> 00:25:05.180 And the wheels are in contact with the road. 00:25:05.180 --> 00:25:07.580 And through the wheels you get the normal force. 00:25:07.580 --> 00:25:10.370 And through the wheels you get any active friction. 00:25:10.370 --> 00:25:13.310 So that is the-- that's total net friction force. 00:25:17.982 --> 00:25:19.440 And then once you got that far, you 00:25:19.440 --> 00:25:21.440 were asked to come up with an equation of motion 00:25:21.440 --> 00:25:23.360 in this direction. 00:25:23.360 --> 00:25:26.379 In the direction up the bridge. 00:25:26.379 --> 00:25:28.920 And how do you-- so when you go to get an equation of motion, 00:25:28.920 --> 00:25:33.664 you say, the sum of the external forces is equal to? 00:25:33.664 --> 00:25:35.183 AUDIENCE: [INAUDIBLE]. 00:25:35.183 --> 00:25:36.099 AUDIENCE: [INAUDIBLE]. 00:25:39.800 --> 00:25:43.000 PROFESSOR: And we'll make these in the x direction here. 00:25:43.000 --> 00:25:45.230 Is the mass times the acceleration 00:25:45.230 --> 00:25:46.660 in the x direction. 00:25:46.660 --> 00:25:50.447 And how would-- what terms appear? 00:25:50.447 --> 00:25:51.820 AUDIENCE: [INAUDIBLE]. 00:25:51.820 --> 00:25:52.570 PROFESSOR: Pardon? 00:25:52.570 --> 00:25:53.537 AUDIENCE: T. 00:25:53.537 --> 00:25:55.120 PROFESSOR: Well, those are the forces. 00:25:55.120 --> 00:25:56.220 They're on the other side. 00:25:56.220 --> 00:25:58.140 I want you to come up with the accelerations. 00:25:58.140 --> 00:26:00.500 You need to come up with the acceleration terms. 00:26:00.500 --> 00:26:01.820 AUDIENCE: [INAUDIBLE]. 00:26:01.820 --> 00:26:02.874 PROFESSOR: Pardon? 00:26:02.874 --> 00:26:03.790 AUDIENCE: [INAUDIBLE]. 00:26:07.710 --> 00:26:10.230 PROFESSOR: So this problem most easily to think through, 00:26:10.230 --> 00:26:12.200 I think, in terms of r and theta and polar 00:26:12.200 --> 00:26:15.090 coordinates because you've worked dozens of times 00:26:15.090 --> 00:26:20.030 with a complete acceleration of something in a rotating system. 00:26:20.030 --> 00:26:23.320 And this is a rotating system. 00:26:23.320 --> 00:26:26.790 Polar coordinates is a pretty good way to do this problem. 00:26:26.790 --> 00:26:30.665 So what are the accelerations in this direction? 00:26:35.930 --> 00:26:38.732 AUDIENCE: [INAUDIBLE] r and theta? 00:26:38.732 --> 00:26:40.565 PROFESSOR: Well, I do it in terms of-- well, 00:26:40.565 --> 00:26:42.210 I do it in terms of r and theta. 00:26:42.210 --> 00:26:47.160 Given a theta dot and a theta double dot and x and r 00:26:47.160 --> 00:26:50.100 mount up to the same thing. 00:26:50.100 --> 00:26:53.210 But I-- if you call it r, you'll recognize the terms 00:26:53.210 --> 00:26:55.140 in your acceleration equation. 00:26:55.140 --> 00:27:00.838 So what are the accelerations in the direction up to bridge? 00:27:00.838 --> 00:27:02.332 AUDIENCE: r double dot? 00:27:02.332 --> 00:27:05.320 AUDIENCE: r double dot. 00:27:05.320 --> 00:27:08.308 AUDIENCE: Minus r theta dot. 00:27:11.296 --> 00:27:13.290 PROFESSOR: And those are all in the-- you 00:27:13.290 --> 00:27:14.910 get all of that in the i direction. 00:27:14.910 --> 00:27:17.160 And then if you looked at that, obviously, 00:27:17.160 --> 00:27:19.990 you could replace r double dot with x double dot. 00:27:19.990 --> 00:27:25.212 r with x if you make your origin here. 00:27:25.212 --> 00:27:26.280 Which you probably would. 00:27:26.280 --> 00:27:27.488 That's the point of rotation. 00:27:27.488 --> 00:27:31.160 So that is your-- that's the mass times the acceleration 00:27:31.160 --> 00:27:33.190 in the direction of travel. 00:27:33.190 --> 00:27:36.810 And it's got to be equal to T and minus 00:27:36.810 --> 00:27:41.260 Mg sine theta probably. 00:27:41.260 --> 00:27:42.150 Right? 00:27:42.150 --> 00:27:43.580 And that's one equation of motion. 00:27:43.580 --> 00:27:45.621 And you're going to do another equation of motion 00:27:45.621 --> 00:27:47.880 in the theta dot-- theta hat direction. 00:27:47.880 --> 00:27:50.150 Because what other-- what accelerations 00:27:50.150 --> 00:27:54.060 are present in this direction? 00:27:54.060 --> 00:27:57.218 In the y j hat? 00:27:57.218 --> 00:27:58.694 AUDIENCE: Theta double dot? 00:27:58.694 --> 00:27:59.860 PROFESSOR: Theta double dot. 00:27:59.860 --> 00:28:01.210 You're [INAUDIBLE] one. 00:28:01.210 --> 00:28:03.770 What else? 00:28:03.770 --> 00:28:10.890 So the sum of the forces in the y-- 00:28:10.890 --> 00:28:15.562 so they are, in terms of polar coordinate terms? 00:28:15.562 --> 00:28:18.734 AUDIENCE: 2 omega-- 2 omega dot. 00:28:18.734 --> 00:28:20.400 PROFESSOR: Somebody said this one first. 00:28:20.400 --> 00:28:25.985 Plus 2 theta dot is your-- 00:28:25.985 --> 00:28:27.702 AUDIENCE: It's [INAUDIBLE]. 00:28:27.702 --> 00:28:28.410 PROFESSOR: Right. 00:28:28.410 --> 00:28:29.396 2 theta dot-- 00:28:29.396 --> 00:28:30.062 AUDIENCE: r dot. 00:28:30.062 --> 00:28:31.420 PROFESSOR: r dot. 00:28:31.420 --> 00:28:33.720 And that's in the j direction. 00:28:33.720 --> 00:28:35.860 And now you've got-- now you can work the problem. 00:28:38.380 --> 00:28:44.720 So how would you go about doing this problem where 00:28:44.720 --> 00:28:46.950 you can't treat it as a particle any longer? 00:28:51.410 --> 00:28:55.624 We've graduated to that because we've 00:28:55.624 --> 00:28:57.540 been doing-- we've been doing angular momentum 00:28:57.540 --> 00:29:00.450 stuff with mass moment of inertia matrices. 00:29:00.450 --> 00:29:06.110 So now you want to do this more as a full fledged dynamics 00:29:06.110 --> 00:29:07.910 problem, taking into consideration that. 00:29:07.910 --> 00:29:10.230 So how do you-- what is the mass-- what 00:29:10.230 --> 00:29:16.320 do you have to modify-- do to modify our free body diagram? 00:29:21.042 --> 00:29:22.250 Here's my simplified vehicle. 00:29:22.250 --> 00:29:24.730 What are the forces on it that you need to deal with? 00:29:32.050 --> 00:29:34.304 AUDIENCE: Well, there are two normal forces. 00:29:34.304 --> 00:29:35.470 There are two normal forces. 00:29:35.470 --> 00:29:35.806 PROFESSOR: Yeah. 00:29:35.806 --> 00:29:37.140 So you can add-- need two because you've 00:29:37.140 --> 00:29:38.590 got two wheels on this thing. 00:29:38.590 --> 00:29:43.320 And so there's axle one here- so you get one here and one here. 00:29:43.320 --> 00:29:46.260 I call this one N1, N2. 00:29:46.260 --> 00:29:48.472 What else? 00:29:48.472 --> 00:29:50.892 AUDIENCE: [INAUDIBLE]. 00:29:50.892 --> 00:29:52.350 The same T and Mg. 00:29:52.350 --> 00:29:54.040 PROFESSOR: So you've got a g somewhere. 00:29:54.040 --> 00:29:57.740 And so you still have an Mg term. 00:29:57.740 --> 00:30:00.098 What other external forces? 00:30:00.098 --> 00:30:01.430 AUDIENCE: T? 00:30:01.430 --> 00:30:04.100 PROFESSOR: T. But now T's the problem. 00:30:04.100 --> 00:30:06.880 So as I gave you before, T is the total net force. 00:30:06.880 --> 00:30:09.786 But that's not adequate in this problem any longer. 00:30:09.786 --> 00:30:11.384 AUDIENCE: [INAUDIBLE] 00:30:11.384 --> 00:30:12.050 PROFESSOR: Yeah. 00:30:12.050 --> 00:30:17.440 So we're going to end up having a friction force that's 00:30:17.440 --> 00:30:25.105 supplied here-- f1-- and another one applied here-- f2. 00:30:25.105 --> 00:30:31.250 And f1 plus f2 would be T. OK. 00:30:31.250 --> 00:30:34.968 So how many unknowns do we have? 00:30:40.584 --> 00:30:41.520 AUDIENCE: Four? 00:30:41.520 --> 00:30:43.395 PROFESSOR: Well, it looks like four are here. 00:30:43.395 --> 00:30:46.600 Plus any motion stuff that you have to solve for. 00:30:46.600 --> 00:30:50.260 So how many equations of motion can you write? 00:30:50.260 --> 00:30:53.830 Remember, you get one equation for every vector component 00:30:53.830 --> 00:30:54.330 direction. 00:30:54.330 --> 00:30:57.360 So x, y, z gives you three components 00:30:57.360 --> 00:30:59.690 and you could conceivably write three equations. 00:30:59.690 --> 00:31:02.060 So tell me the relevant equations of motion 00:31:02.060 --> 00:31:03.930 that we could come up with for this problem. 00:31:09.738 --> 00:31:12.650 AUDIENCE: Sum forces in x, sum forces in y. 00:31:12.650 --> 00:31:13.390 PROFESSOR: OK. 00:31:13.390 --> 00:31:15.550 I argue-- sum of the forces in x. 00:31:20.360 --> 00:31:21.110 That's still true. 00:31:21.110 --> 00:31:22.250 Yep. 00:31:22.250 --> 00:31:23.970 Sum of the forces in y you said. 00:31:27.610 --> 00:31:29.591 Is this 0? 00:31:29.591 --> 00:31:30.090 No. 00:31:30.090 --> 00:31:32.170 Because that bridge is moving. 00:31:32.170 --> 00:31:33.010 OK. 00:31:33.010 --> 00:31:33.978 What else? 00:31:33.978 --> 00:31:36.120 AUDIENCE: Torque. 00:31:36.120 --> 00:31:37.330 PROFESSOR: OK. 00:31:37.330 --> 00:31:39.640 And how do you do that? 00:31:39.640 --> 00:31:42.746 Sum of the torques in what direction? 00:31:42.746 --> 00:31:44.520 AUDIENCE: Theta hat. 00:31:44.520 --> 00:31:47.400 PROFESSOR: Well, no. 00:31:47.400 --> 00:31:47.900 Yeah. 00:31:47.900 --> 00:31:48.470 Well, you could. 00:31:48.470 --> 00:31:49.000 Excuse me. 00:31:49.000 --> 00:31:51.759 I shouldn't say that. 00:31:51.759 --> 00:31:53.300 Do you have more than one torque term 00:31:53.300 --> 00:31:54.100 that you can work with here? 00:31:54.100 --> 00:31:55.350 More than one torque equation? 00:32:04.637 --> 00:32:05.220 AUDIENCE: Yes? 00:32:05.220 --> 00:32:05.955 AUDIENCE: Yes. 00:32:05.955 --> 00:32:08.172 PROFESSOR: How's that? 00:32:08.172 --> 00:32:09.880 So let's just think through these things. 00:32:09.880 --> 00:32:13.340 You could have a torque-- you could have a torque in, 00:32:13.340 --> 00:32:16.560 this is x, y, z out of the board. 00:32:16.560 --> 00:32:19.150 You could have a z torque making this thing trying 00:32:19.150 --> 00:32:22.110 to pitch up and down. 00:32:22.110 --> 00:32:24.575 You could have an x torque making it trying to rollover. 00:32:24.575 --> 00:32:26.700 But we don't have any information in that direction 00:32:26.700 --> 00:32:28.150 and its constrained. 00:32:28.150 --> 00:32:32.890 And we could have a y torque making it try to go like that. 00:32:32.890 --> 00:32:34.800 And we don't have much information there. 00:32:34.800 --> 00:32:38.580 So it seems like we've got a z torque 00:32:38.580 --> 00:32:39.940 equation that we could write. 00:32:45.920 --> 00:32:49.450 And that's one-- seems like we got three equations. 00:32:49.450 --> 00:32:51.420 Sum of the forces x, sum of the forces y, 00:32:51.420 --> 00:32:53.210 and a torque in the z. 00:32:53.210 --> 00:32:59.162 And doesn't seem like quite enough. 00:32:59.162 --> 00:33:00.870 And that's as far as I've gotten with it. 00:33:00.870 --> 00:33:04.190 I just thought of this on the way over here. 00:33:04.190 --> 00:33:06.950 So I'd sit down and start working this out and trying 00:33:06.950 --> 00:33:08.950 to figure out, where is my fourth equation going 00:33:08.950 --> 00:33:09.491 to come from? 00:33:13.960 --> 00:33:17.884 And I'm not exactly sure quite yet. 00:33:17.884 --> 00:33:20.457 But, generally, I'd go about setting it up like that 00:33:20.457 --> 00:33:21.915 and start working those things out. 00:33:26.470 --> 00:33:27.170 OK. 00:33:27.170 --> 00:33:29.210 Any other questions or questions about the quiz? 00:33:40.260 --> 00:33:42.210 We're done early.