WEBVTT 00:00:00.080 --> 00:00:02.430 The following content is provided under a Creative 00:00:02.430 --> 00:00:03.800 Commons license. 00:00:03.800 --> 00:00:06.050 Your support will help MIT OpenCourseWare 00:00:06.050 --> 00:00:10.140 continue to offer high quality educational resources for free. 00:00:10.140 --> 00:00:12.690 To make a donation or to view additional materials 00:00:12.690 --> 00:00:16.590 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.590 --> 00:00:17.297 at ocw.mit.edu. 00:00:21.260 --> 00:00:23.680 PROFESSOR: Today we're going to talk all about moments 00:00:23.680 --> 00:00:30.760 of inertia, and the last time we did [? muddy ?] cards, there 00:00:30.760 --> 00:00:32.860 were a lot of questions. 00:00:32.860 --> 00:00:37.990 The most common question-- where did all those terms come from? 00:00:37.990 --> 00:00:42.040 And I'm going to do a brief kind of intro 00:00:42.040 --> 00:00:47.045 around that so as to facilitate the rest of the discussions. 00:00:58.060 --> 00:00:59.940 Some basic assumptions here today. 00:00:59.940 --> 00:01:13.190 We're going to consider bodies that 00:01:13.190 --> 00:01:30.456 rotate about their center of mass, or fixed points. 00:01:34.540 --> 00:01:38.210 That's the kind of problems we're doing. 00:01:38.210 --> 00:01:42.610 And secondly-- all through this lecture 00:01:42.610 --> 00:01:47.160 today, there's four or five significant assumptions 00:01:47.160 --> 00:01:50.125 or conditions that are really important 00:01:50.125 --> 00:01:51.125 to the whole discussion. 00:01:51.125 --> 00:01:52.630 I'm going to really try to highlight them. 00:01:52.630 --> 00:01:53.754 And starting with two here. 00:01:53.754 --> 00:02:10.036 The other one is we're going to use reference frames attached 00:02:10.036 --> 00:02:10.619 to the bodies. 00:02:13.850 --> 00:02:16.740 That's why we did all that work in kinematics figuring out 00:02:16.740 --> 00:02:21.560 how to do velocities of things that are in translating 00:02:21.560 --> 00:02:23.190 and rotating references frame, it's 00:02:23.190 --> 00:02:25.220 so that we can do these problems. 00:02:28.860 --> 00:02:38.500 So this said, here's a inertial frame. 00:02:42.410 --> 00:02:46.000 Here's some rigid body. 00:02:46.000 --> 00:02:48.830 Here's a point on it, that's my origin. 00:02:48.830 --> 00:02:56.400 And here is a set of coordinates attached to the rigid body-- 00:02:56.400 --> 00:03:00.340 an xyz system-- the body rotates and translates, 00:03:00.340 --> 00:03:03.360 it goes with it. 00:03:03.360 --> 00:03:09.050 And if we consider a little piece of this body, 00:03:09.050 --> 00:03:13.020 a little part-- I'll call it little mass element 00:03:13.020 --> 00:03:23.110 mi-- from here to here is some distance y with respect to A, 00:03:23.110 --> 00:03:30.060 then the angular momentum of that little mass particle 00:03:30.060 --> 00:03:37.760 is by definition of angular momentum Ri cross Pi. 00:03:40.660 --> 00:03:43.310 And the P is always the linear momentum 00:03:43.310 --> 00:03:46.916 defined in an inertial frame. 00:03:46.916 --> 00:03:48.665 That's the definition of angular momentum. 00:03:59.310 --> 00:04:04.050 So you have seen this little demo before. 00:04:04.050 --> 00:04:08.630 So this is now my rigid body that I'm talking about. 00:04:08.630 --> 00:04:14.330 In this case, the axis of rotation is this one. 00:04:14.330 --> 00:04:17.410 And I'm just going to define my rigid body 00:04:17.410 --> 00:04:21.360 as essentially rotating about this fixed point, 00:04:21.360 --> 00:04:23.270 this is A now. 00:04:23.270 --> 00:04:32.830 And my coordinate system is xyz, so the rotation is omega z. 00:04:32.830 --> 00:04:36.200 And this rigid body-- this is massless, 00:04:36.200 --> 00:04:39.470 consists of a single point mass. 00:04:39.470 --> 00:04:41.200 So we've done this before, but we 00:04:41.200 --> 00:04:43.710 tended to do this kind of problem 00:04:43.710 --> 00:04:44.910 using polar coordinates. 00:04:44.910 --> 00:04:47.790 And now I really wanted to strictly think in terms of XYZ. 00:04:47.790 --> 00:04:49.420 The answers come out exactly the same. 00:04:49.420 --> 00:04:50.970 Polar coordinates also moved. 00:04:50.970 --> 00:04:54.470 You could do this in rhat, thetahat, z. 00:04:54.470 --> 00:04:59.390 We're going to do it in xyz attached to the body. 00:05:22.450 --> 00:05:26.150 And y is going into the board, and we'll 00:05:26.150 --> 00:05:29.180 have no values in the y direction for this problem. 00:05:29.180 --> 00:05:33.150 Here's our little mass, we'll call it m1. 00:05:33.150 --> 00:05:38.480 It's at a position out here x1, z1 are its coordinates, 00:05:38.480 --> 00:05:41.430 and y is zero. 00:05:41.430 --> 00:05:44.710 And we want to compute its angular momentum, 00:05:44.710 --> 00:05:57.505 h1 with respect to A. That's our R1A cross P1 and 0. 00:05:57.505 --> 00:06:02.800 And for this problem that's x1 in the i hat 00:06:02.800 --> 00:06:09.080 plus z1 in the k hat, that's the R-- cross 00:06:09.080 --> 00:06:15.030 P and P is mass times velocity, m1. 00:06:15.030 --> 00:06:19.980 And the velocity is just omega cross R. 00:06:19.980 --> 00:06:21.070 We've done this before. 00:06:21.070 --> 00:06:25.010 So we need a length on this thing. 00:06:33.410 --> 00:06:35.510 We've done this enough times before I'm assuming 00:06:35.510 --> 00:06:38.260 you can make this leap with me. 00:06:38.260 --> 00:06:46.510 The momentum is into the board, in the y direction. 00:06:46.510 --> 00:06:53.620 The linear momentum is R cross with that. 00:06:53.620 --> 00:07:02.260 And that will give us-- let me write it-- omega z. 00:07:02.260 --> 00:07:04.310 So that's the linear momentum. 00:07:04.310 --> 00:07:10.850 It's in the j direction, its velocity is x1 omega z. 00:07:10.850 --> 00:07:14.120 The radius times the rotation rate is the velocity, 00:07:14.120 --> 00:07:15.820 the direction is that direction. 00:07:15.820 --> 00:07:17.470 So this is your momentum. 00:07:17.470 --> 00:07:19.360 You multiply out this cross product, 00:07:19.360 --> 00:07:34.410 you get two terms-- m1 x1 squared omega z k, minus m1 x1 00:07:34.410 --> 00:07:37.400 z1 omega z i. 00:07:40.070 --> 00:07:43.030 And these two terms then we would identify. 00:07:43.030 --> 00:07:46.094 This is the angular momentum in the k direction, angular 00:07:46.094 --> 00:07:47.260 momentum in the i direction. 00:07:47.260 --> 00:07:49.343 There doesn't happen to be any in the z direction, 00:07:49.343 --> 00:07:50.760 in general there could be. 00:07:50.760 --> 00:07:55.035 But this is h for this particle, particle one. 00:07:59.370 --> 00:08:01.760 I want to particularly emphasize this one-- 00:08:01.760 --> 00:08:04.380 oops, this is-- excuse me-- z. 00:08:04.380 --> 00:08:06.780 And this is the piece we call hx. 00:08:10.050 --> 00:08:12.664 So there's a complement of the angular momentum 00:08:12.664 --> 00:08:14.080 that's in the z direction, there's 00:08:14.080 --> 00:08:15.520 a component in the x direction. 00:08:21.350 --> 00:08:24.040 We know that if we compute-- just 00:08:24.040 --> 00:08:32.720 to tie it back to previous work a little further-- d 00:08:32.720 --> 00:08:41.900 by dt of h1A gives us the external torques 00:08:41.900 --> 00:08:44.370 in the system, a vector of torques 00:08:44.370 --> 00:08:46.490 when we compute that out. 00:08:46.490 --> 00:08:49.270 I'm not going to calculate it, I don't need it 00:08:49.270 --> 00:08:51.450 for the purposes of the rest of the discussion. 00:08:51.450 --> 00:08:54.180 This one gives you omega z dot term. 00:08:54.180 --> 00:08:55.911 This one gets two terms because you have 00:08:55.911 --> 00:08:57.190 to take the derivative of i. 00:08:57.190 --> 00:09:00.430 So you get three terms which are torque 00:09:00.430 --> 00:09:02.700 in the x, torque in the y, torque in the z directions. 00:09:02.700 --> 00:09:04.410 Two of them are static, because it's 00:09:04.410 --> 00:09:08.320 trying to bend this thing back and out, and one of them, 00:09:08.320 --> 00:09:11.040 the z direction one, is the one that makes it spin faster. 00:09:13.850 --> 00:09:16.910 So we've seen that, that's a review. 00:09:16.910 --> 00:09:39.930 And to make the leap from that to rigid bodies-- Capital 00:09:39.930 --> 00:09:43.570 H, now, a collection of particles with respect 00:09:43.570 --> 00:09:51.870 to its the origin attached to the rigid body 00:09:51.870 --> 00:09:54.082 is going to be the summation over all 00:09:54.082 --> 00:09:55.165 the little mass particles. 00:10:00.740 --> 00:10:07.254 Their position vector crossed with the linear momentum of all 00:10:07.254 --> 00:10:08.420 those little mass particles. 00:10:08.420 --> 00:10:10.490 Sum all that out. 00:10:10.490 --> 00:10:12.870 Because it's a rigid body, this is always 00:10:12.870 --> 00:10:16.150 something of the form that looks like omega cross R, 00:10:16.150 --> 00:10:20.290 m omega cross R. And I can then write this 00:10:20.290 --> 00:10:29.310 as the summation of [? oper ?] i of the mi, 00:10:29.310 --> 00:10:39.225 RiA cross omega with respect to 0, cross RiA. 00:10:42.430 --> 00:10:49.140 That gives you the velocity, m gives you the linear momentum, 00:10:49.140 --> 00:10:50.720 crossed with this one again makes it 00:10:50.720 --> 00:10:52.690 into an angular momentum. 00:10:52.690 --> 00:10:59.780 So the angular momentum of every particle-- the bits of this 00:10:59.780 --> 00:11:07.400 come out looking like an R squared omega. 00:11:07.400 --> 00:11:13.410 Or at least has dimensions of R squared omega. 00:11:13.410 --> 00:11:16.350 These can be like x1 squared, but these can also 00:11:16.350 --> 00:11:21.710 end up being terms like x1 z1, those cross product terms. 00:11:21.710 --> 00:11:27.240 But in general, these things all add up. 00:11:27.240 --> 00:11:30.830 And this is a vector, this is a vector, 00:11:30.830 --> 00:11:33.510 so this is a vector-- these are all vectors, 00:11:33.510 --> 00:11:34.540 the result's a vector. 00:11:39.040 --> 00:11:42.730 And therefore we would break this down, this whole thing. 00:11:42.730 --> 00:11:45.750 Once you write all that out and sum it all up, 00:11:45.750 --> 00:11:48.210 you're going to get a piece of it 00:11:48.210 --> 00:11:52.440 that we would say that is in the i hat direction 00:11:52.440 --> 00:11:54.580 and we call that Hx. 00:11:54.580 --> 00:11:58.460 Another piece that's in the j hat direction, which 00:11:58.460 --> 00:12:03.540 we call Hy and Hz in the k hat. 00:12:03.540 --> 00:12:05.920 That's just how that all shakes out. 00:12:05.920 --> 00:12:07.290 In general you get three parts. 00:12:11.750 --> 00:12:13.664 Yeah? 00:12:13.664 --> 00:12:17.648 AUDIENCE: Why do you have mass multiplied by displacement 00:12:17.648 --> 00:12:19.650 and the momentum? 00:12:19.650 --> 00:12:24.170 PROFESSOR: Why do we have the mass multiplied by a position 00:12:24.170 --> 00:12:27.210 vector times the momentum? 00:12:27.210 --> 00:12:30.205 Because that is the definition of angular momentum. 00:12:34.450 --> 00:12:36.763 It's R cross P, all right? 00:12:36.763 --> 00:12:37.596 [INTERPOSING VOICES] 00:12:46.310 --> 00:12:48.120 PROFESSOR: Ah, I see what I've done. 00:12:48.120 --> 00:12:48.950 Yeah you're right. 00:12:48.950 --> 00:12:52.540 This m doesn't belong-- I got ahead of myself-- 00:12:52.540 --> 00:12:54.470 it doesn't belong here. 00:12:54.470 --> 00:12:57.680 It pops out of this to here. 00:12:57.680 --> 00:12:59.752 Thanks for catching that. 00:12:59.752 --> 00:13:00.460 Absolutely right. 00:13:15.420 --> 00:13:22.530 So every term here is made up of things that 00:13:22.530 --> 00:13:30.310 look like x1, z1, m1 omega x. 00:13:30.310 --> 00:13:34.150 Or omega y, or omega z, because this is a vector, 00:13:34.150 --> 00:13:37.450 and it can have three components-- a piece in the i, 00:13:37.450 --> 00:13:39.960 a piece in the j, a piece in the k direction. 00:13:39.960 --> 00:13:45.190 So there's a lot of possible terms, 00:13:45.190 --> 00:13:49.950 and we're in the practice of writing this out. 00:13:53.570 --> 00:14:00.250 This H vector can be written then 00:14:00.250 --> 00:14:05.330 as a-- I'm going to run out of room here. 00:14:19.220 --> 00:14:22.220 So the Hx term-- the piece that comes out 00:14:22.220 --> 00:14:26.410 with all little bits in the i direction-- 00:14:26.410 --> 00:14:35.532 when you just break it apart, we would write it Ixx omega 00:14:35.532 --> 00:14:46.725 x, plus Ixy omega y, plus Ixz omega z. 00:14:49.330 --> 00:14:54.610 All the i terms that float out of this, 00:14:54.610 --> 00:14:58.060 we just collect them together-- all the ones that 00:14:58.060 --> 00:15:02.150 are multiplied by omega x ends up being some terms that 00:15:02.150 --> 00:15:11.570 look like sums of mi-- x, mi z squared terms, 00:15:11.570 --> 00:15:13.726 and there's some y squared terms. 00:15:13.726 --> 00:15:15.350 But we collect them all together and we 00:15:15.350 --> 00:15:18.730 call this constant in front of it Ixx. 00:15:18.730 --> 00:15:21.410 Just what floats out of this stuff. 00:15:21.410 --> 00:15:25.030 We break it into three pieces, the part of the angular 00:15:25.030 --> 00:15:26.865 momentum due to the rotation in the x, 00:15:26.865 --> 00:15:28.780 the part due to the rotation in the y, 00:15:28.780 --> 00:15:31.400 the part due to the rotation in the z. 00:15:31.400 --> 00:15:34.436 And you do the same thing for Hy, 00:15:34.436 --> 00:15:43.930 and you get in Iyz omega x plus an Iyy omega 00:15:43.930 --> 00:15:53.660 y plus an Iyz omega z, and you're finally get and Hz term. 00:15:53.660 --> 00:16:04.935 Izx omega x, Izy omega y, Izz omega z. 00:16:07.560 --> 00:16:10.775 That's everything that falls out of just doing this calculation. 00:16:19.890 --> 00:16:24.430 And where in the habit of writing that in a matrix 00:16:24.430 --> 00:16:32.870 notation as the product of this thing 00:16:32.870 --> 00:16:33.985 called the inertia matrix. 00:16:41.120 --> 00:16:43.240 And so forth, with this bottom term 00:16:43.240 --> 00:16:52.130 being Izz multiplied by omega x, omega y, omega z. 00:16:52.130 --> 00:16:57.310 You multiply that times the top row, you get Hx, middle row, 00:16:57.310 --> 00:17:01.240 you get Hy, bottom row, you get the Hz. 00:17:01.240 --> 00:17:04.700 So where this stuff comes from is just 00:17:04.700 --> 00:17:10.770 from carrying out the summation over all the mass bits. 00:17:10.770 --> 00:17:13.504 So it basically starts with the definition of angular momentum. 00:17:13.504 --> 00:17:15.420 And this is just a convenient way to write it. 00:17:19.910 --> 00:17:33.540 So for example, the Ixz term-- the one in the upper right 00:17:33.540 --> 00:17:40.090 there that's part of Hx, the Ixz term-- 00:17:40.090 --> 00:17:45.850 is minus the summation of all the little mass bits 00:17:45.850 --> 00:17:50.170 times their location xi zi. 00:17:50.170 --> 00:17:55.331 And this, the xz, always matches xz. 00:17:55.331 --> 00:17:57.580 And when you want to do this over a continuous object, 00:17:57.580 --> 00:17:59.820 you integrate that. 00:17:59.820 --> 00:18:10.410 And for Izz, this is the summation over i, 00:18:10.410 --> 00:18:21.920 all the bits of mi xi squared plus yi squared. 00:18:21.920 --> 00:18:24.610 So in general this, by summations, that's 00:18:24.610 --> 00:18:26.880 where these terms come from. 00:18:26.880 --> 00:18:27.530 They just come. 00:18:27.530 --> 00:18:29.670 They've started with this calculation. 00:18:34.030 --> 00:18:35.905 And in general, from that calculation. 00:18:39.040 --> 00:18:43.990 So for our one particle system, for this thing. 00:18:43.990 --> 00:18:49.700 So where this is x and z, it's one single particle. 00:18:49.700 --> 00:18:51.425 So for our one particle system. 00:19:19.310 --> 00:19:22.700 And there's no y1 in here because it's 00:19:22.700 --> 00:19:24.870 a zero in this problem. 00:19:24.870 --> 00:19:30.410 The coordinate of that little mass particle is x1, 0, y1. 00:19:30.410 --> 00:19:33.380 So in general, this is x1 squared plus yi squared, 00:19:33.380 --> 00:19:34.030 but that's 0. 00:19:34.030 --> 00:19:36.130 So it's just mx1 squared. 00:19:47.220 --> 00:19:49.640 So that's where all these things come from. 00:19:49.640 --> 00:19:56.900 Now let's kind of look at-- the units of these things 00:19:56.900 --> 00:20:01.040 are always mass times length squared. 00:20:01.040 --> 00:20:08.500 The diagonal terms the Ixx, Iyy, Izz terms will all 00:20:08.500 --> 00:20:12.190 be of the form of something squared. 00:20:12.190 --> 00:20:17.080 And what it is is it's always just the distance 00:20:17.080 --> 00:20:24.360 of the mass particles from the axis of rotation, call that r. 00:20:24.360 --> 00:20:27.300 It's just a summation of this perpendicular distance 00:20:27.300 --> 00:20:29.020 from the axis of rotation. 00:20:29.020 --> 00:20:32.990 For Izz, the axis of rotation is z. 00:20:32.990 --> 00:20:36.190 And this piece here is always the distance 00:20:36.190 --> 00:20:41.720 squared that the mass particle is from the axis of rotation. 00:20:41.720 --> 00:20:47.050 Now that's the set up for today. 00:20:47.050 --> 00:20:49.880 And when we want to get onto the real meat of the discussion, 00:20:49.880 --> 00:20:56.040 talking about things like what principal axes are 00:20:56.040 --> 00:20:56.980 and so forth. 00:21:10.540 --> 00:21:13.060 Now we're going to move on to the part of this conversation 00:21:13.060 --> 00:21:14.425 about principal axes. 00:21:29.820 --> 00:21:32.900 So a really important point. 00:21:36.280 --> 00:21:43.840 For every rigid body, even weird ones. 00:21:43.840 --> 00:21:48.320 For every rigid body, there is a coordinate system 00:21:48.320 --> 00:21:53.030 that you can attach to this body, an xyz set of orthogonal 00:21:53.030 --> 00:21:56.210 coordinates that you can fix in this body such 00:21:56.210 --> 00:22:02.455 that you can make this inertia matrix be diagonal. 00:22:05.150 --> 00:22:07.410 Any weird body at all, there is a set 00:22:07.410 --> 00:22:10.810 of coordinates that if you use that set, 00:22:10.810 --> 00:22:13.570 this matrix turns out to be diagonal. 00:22:13.570 --> 00:22:20.640 And what that means in dynamics terms is if you then 00:22:20.640 --> 00:22:26.790 rotate the object about one of those axes, 00:22:26.790 --> 00:22:29.150 it will be dynamically balanced. 00:22:31.860 --> 00:22:39.200 And so when you rotate a shaft-- this is an object for which I 00:22:39.200 --> 00:22:43.960 know-- it's a circular uniform disk, 00:22:43.960 --> 00:22:47.140 one of the axes through the center is a principal axis. 00:22:47.140 --> 00:22:51.500 And if I rotate the object about that-- this one the hole's kind 00:22:51.500 --> 00:22:53.690 of out around so it wobbles a bit 00:22:53.690 --> 00:22:58.450 but if I rotate it about that it will 00:22:58.450 --> 00:23:02.600 make no torques about this axis that are because it's 00:23:02.600 --> 00:23:05.120 unbalanced. 00:23:05.120 --> 00:23:08.590 It's essentially the dynamic meaning of principal axis-- 00:23:08.590 --> 00:23:11.500 an axis about which you can rotate the thing and it'll just 00:23:11.500 --> 00:23:18.030 be smooth, no away-from-the-axis torques. 00:23:18.030 --> 00:23:19.790 So that's so those are pretty important. 00:23:19.790 --> 00:23:22.450 We want to know what those are for rigid bodies 00:23:22.450 --> 00:23:24.160 and how to find them. 00:23:24.160 --> 00:23:26.590 So there's a mathematical way to find them 00:23:26.590 --> 00:23:30.630 and you can just read about in the textbook. 00:23:30.630 --> 00:23:37.770 It's sort of a-- I'm trying to think of the mathematical term. 00:23:37.770 --> 00:23:40.690 I forgot it. 00:23:40.690 --> 00:23:42.220 But what I want to teach you today 00:23:42.220 --> 00:23:44.890 is for many, many objects, if you just 00:23:44.890 --> 00:23:48.000 look at their symmetries, you can figure out by common sense 00:23:48.000 --> 00:23:49.452 where their principal axes are. 00:23:49.452 --> 00:23:51.035 So that's what we're going to do next. 00:24:01.410 --> 00:24:04.660 We started off saying that we're talking about bodies that 00:24:04.660 --> 00:24:07.520 are either rotating about their centers of mass 00:24:07.520 --> 00:24:11.030 or about some other fixed point. 00:24:11.030 --> 00:24:18.470 So we're going to define our principal axes 00:24:18.470 --> 00:24:23.990 assuming we are rotating about the center of mass. 00:24:23.990 --> 00:24:28.120 There is an easy method known as the parallel axis theorem 00:24:28.120 --> 00:24:29.920 to get to any other point. 00:24:29.920 --> 00:24:33.075 So why do we care about doing it about the center of mass? 00:24:37.950 --> 00:24:40.200 Why is that useful to know about the properties 00:24:40.200 --> 00:24:41.930 around the center for dynamics purposes? 00:24:41.930 --> 00:24:43.560 What kind of dynamics problems do you 00:24:43.560 --> 00:24:46.720 care about the center of mass? 00:24:46.720 --> 00:24:48.327 Rotation about the center of mass. 00:24:48.327 --> 00:24:49.243 AUDIENCE: [INAUDIBLE]. 00:24:52.820 --> 00:24:55.070 PROFESSOR: So when I throw this thing in the air, it's 00:24:55.070 --> 00:24:57.320 rotating, it's translating, and when it rotates, 00:24:57.320 --> 00:24:59.260 what's it's rotating about? 00:24:59.260 --> 00:25:00.302 AUDIENCE: Center of mass. 00:25:00.302 --> 00:25:01.385 PROFESSOR: Center of mass. 00:25:01.385 --> 00:25:03.185 So there's just lots and lots of problem 00:25:03.185 --> 00:25:08.239 in which in fact the rotation is going to occur around 00:25:08.239 --> 00:25:09.030 the center of mass. 00:25:09.030 --> 00:25:12.830 Any time the thing is off there and there's nothing 00:25:12.830 --> 00:25:14.940 constraining its rotational motion, 00:25:14.940 --> 00:25:17.980 the rotation will be about the center of mass. 00:25:17.980 --> 00:25:24.250 So that's a good enough reason, and a very practical reason 00:25:24.250 --> 00:25:26.470 then for computing these things or knowing 00:25:26.470 --> 00:25:30.090 how to find these things about the center of mass. 00:25:30.090 --> 00:25:34.690 All right so we're now going to do principal axes, 00:25:34.690 --> 00:25:40.270 and we're going to teach you some symmetry rules. 00:25:45.160 --> 00:25:48.510 Maybe first a little common sense. 00:25:48.510 --> 00:25:52.880 This thing-- is this a principal axis? 00:25:52.880 --> 00:25:53.980 No way, right? 00:25:53.980 --> 00:25:57.210 And we know for a fact that it has off-diagonal terms 00:25:57.210 --> 00:25:59.700 because I've defined my coordinates as being 00:25:59.700 --> 00:26:03.840 an x, y, z, like this, and this is often at some strange angle. 00:26:03.840 --> 00:26:08.970 So as a practical matter, how could I 00:26:08.970 --> 00:26:12.880 alter this object so that this is a principal axis? 00:26:12.880 --> 00:26:14.689 Essentially, how would you balance this? 00:26:14.689 --> 00:26:15.605 AUDIENCE: [INAUDIBLE]. 00:26:19.480 --> 00:26:22.200 PROFESSOR: Take the mass off, fix it. 00:26:22.200 --> 00:26:23.385 [LAUGHTER] 00:26:23.385 --> 00:26:24.766 Or? 00:26:24.766 --> 00:26:27.730 All right. 00:26:27.730 --> 00:26:29.390 Where do I put it? 00:26:29.390 --> 00:26:31.692 Down here? 00:26:31.692 --> 00:26:32.775 Ah, you want it like this. 00:26:37.117 --> 00:26:40.380 This is steel going into aluminum, 00:26:40.380 --> 00:26:42.755 I have to be careful that I don't strip the threads. 00:26:50.170 --> 00:26:50.670 There we go. 00:26:54.610 --> 00:26:59.070 And now test one, this is axis of rotation, 00:26:59.070 --> 00:27:00.840 is it dynamically in balance? 00:27:00.840 --> 00:27:03.830 Smooth as silk. 00:27:03.830 --> 00:27:07.210 This now is a principal axis. 00:27:07.210 --> 00:27:09.110 Its mass distribution is symmetric. 00:27:12.790 --> 00:27:16.070 Where is there a plane of symmetry for this problem, 00:27:16.070 --> 00:27:17.537 for this object? 00:27:17.537 --> 00:27:20.078 That's what we're going to talk about now, planes of symmetry 00:27:20.078 --> 00:27:23.430 and axis of symmetry. 00:27:23.430 --> 00:27:26.292 Ah, you're saying one like back? 00:27:26.292 --> 00:27:28.000 You're right, that's a plane of symmetry. 00:27:28.000 --> 00:27:29.416 Is there another one for this one? 00:27:32.150 --> 00:27:32.930 Like this. 00:27:32.930 --> 00:27:35.650 So there's two planes of symmetry. 00:27:35.650 --> 00:27:40.250 So let's say come up with some symmetry rules. 00:27:44.380 --> 00:28:16.290 So this first one is there exists-- 00:28:16.290 --> 00:28:19.450 by this I mean diagonal, it's a diagonal matrix. 00:28:19.450 --> 00:28:21.980 You can find a set of orthogonal axes 00:28:21.980 --> 00:28:24.690 such that this matrix is diagonal. 00:28:24.690 --> 00:28:27.545 That's what we mean when we go to find principal axes. 00:28:30.990 --> 00:28:32.615 And pick a board here-- rules. 00:28:50.560 --> 00:28:52.186 And we're going to have three of them, 00:28:52.186 --> 00:28:53.560 so just leave room on your paper. 00:28:53.560 --> 00:28:54.400 We're going to build these up. 00:28:54.400 --> 00:28:56.590 And I'm going to talk about it, and then you add another one 00:28:56.590 --> 00:28:57.190 and so forth. 00:29:12.540 --> 00:29:13.320 So rule one. 00:29:21.375 --> 00:29:24.826 AUDIENCE: [INAUDIBLE] the inertia matrix [INAUDIBLE]. 00:29:24.826 --> 00:29:26.798 PROFESSOR: What is the inertia matrix-- 00:29:26.798 --> 00:29:28.780 AUDIENCE: No, I mean the second one. 00:29:28.780 --> 00:29:30.520 PROFESSOR: This thing with the diagonals? 00:29:30.520 --> 00:29:33.500 This is going to have all the off-diagonal terms are zeroes. 00:29:37.310 --> 00:29:38.470 That's what I mean. 00:29:38.470 --> 00:29:43.070 It's a diagonal matrix, the mass moments of inertia and products 00:29:43.070 --> 00:29:48.160 of inertia-- you only have Ixx, Iyy, Izz if you properly 00:29:48.160 --> 00:29:51.620 pick this set of orthogonal coordinates attached 00:29:51.620 --> 00:29:54.060 to the body-- if you pick them right. 00:29:54.060 --> 00:29:56.510 What it means if then you spin that object about one 00:29:56.510 --> 00:30:00.560 of those axes, it's in balance. 00:30:00.560 --> 00:30:41.080 So rule one, if there it is an axis of symmetry-- Remember 00:30:41.080 --> 00:30:44.385 the caveat here is we're talking about uniform density objects 00:30:44.385 --> 00:30:48.156 or at least objects in which the density is symmetrically 00:30:48.156 --> 00:30:48.655 distributed. 00:30:51.200 --> 00:30:53.860 So the geometric symmetry and mass symmetry 00:30:53.860 --> 00:30:56.690 mean the same thing. 00:30:56.690 --> 00:30:57.630 What does that mean? 00:30:57.630 --> 00:30:58.840 So axis of symmetry. 00:31:03.440 --> 00:31:06.455 Does this have an axis of symmetry? 00:31:09.680 --> 00:31:10.689 Where? 00:31:10.689 --> 00:31:14.601 AUDIENCE: It has multiple ones down the middle 00:31:14.601 --> 00:31:16.527 and it's also symmetric [INAUDIBLE]. 00:31:16.527 --> 00:31:18.110 PROFESSOR: That's a plane of symmetry, 00:31:18.110 --> 00:31:20.625 the [? other ones you think, ?] axis of symmetry. 00:31:20.625 --> 00:31:22.725 AUDIENCE: Well I mean the line right through it. 00:31:22.725 --> 00:31:23.600 PROFESSOR: Which way? 00:31:23.600 --> 00:31:25.001 AUDIENCE: Well any. 00:31:25.001 --> 00:31:25.935 PROFESSOR: Nope. 00:31:25.935 --> 00:31:27.650 AUDIENCE: Perpendicular to it. 00:31:27.650 --> 00:31:29.530 PROFESSOR: Axis of symmetry means 00:31:29.530 --> 00:31:33.800 that there is a mirror image from that point-- 00:31:33.800 --> 00:31:36.540 that point mirror image over here is always identical. 00:31:36.540 --> 00:31:40.670 But you think any from angle, it doesn't matter where you start, 00:31:40.670 --> 00:31:42.250 it's reflected through the axis. 00:31:42.250 --> 00:31:44.590 So things that are circularly shaped 00:31:44.590 --> 00:31:47.760 tend to have axes of symmetry. 00:31:47.760 --> 00:31:51.680 So this has an axis of symmetry. 00:31:51.680 --> 00:31:54.695 Everything is reflected exactly just across the axes, 00:31:54.695 --> 00:31:58.090 you don't have to pick any particular line. 00:31:58.090 --> 00:32:02.510 So what it's saying if it's an axis of symmetry, 00:32:02.510 --> 00:32:05.630 then it is a principal axis, and rotation about that 00:32:05.630 --> 00:32:08.440 axis-- the things will be nice perfectly in balance. 00:32:08.440 --> 00:32:12.340 If you go through all the hairy details of calculating 00:32:12.340 --> 00:32:16.490 the hard way, the Ixx, Iyy, et cetera terms-- 00:32:16.490 --> 00:32:18.600 all of the off diagonal terms will come out zero. 00:32:18.600 --> 00:32:23.230 And the reason is that for every mass particle over here, 00:32:23.230 --> 00:32:26.070 let's say there's a little bit right here in the corner, 00:32:26.070 --> 00:32:30.130 there is one exactly like it on the other side. 00:32:30.130 --> 00:32:33.450 So if this one we're trying to bend this thing up as it 00:32:33.450 --> 00:32:37.080 spins around, this one over here is telling it to come back. 00:32:37.080 --> 00:32:38.446 Yeah? 00:32:38.446 --> 00:32:42.293 AUDIENCE: How come this is not an axis, if you hold it-- 00:32:42.293 --> 00:32:42.793 yeah-- 00:32:42.793 --> 00:32:43.759 [INTERPOSING VOICES] 00:32:43.759 --> 00:32:45.700 PROFESSOR: So if you do this. 00:32:45.700 --> 00:32:48.470 So I think it's kind of just the definition we're getting to 00:32:48.470 --> 00:32:50.590 of an axis of symmetry. 00:32:50.590 --> 00:32:58.490 Because in simple terms, what the object looks 00:32:58.490 --> 00:33:01.310 like this way is it's a narrow object, 00:33:01.310 --> 00:33:04.170 it is in fact a mirror image reflection. 00:33:04.170 --> 00:33:06.770 But it looks different when you go to this angle. 00:33:06.770 --> 00:33:09.020 And it looks different when you go to this angle 00:33:09.020 --> 00:33:13.370 so it's not an axis of symmetry, that's a plane of symmetry. 00:33:13.370 --> 00:33:14.980 We're going talk about that next. 00:33:14.980 --> 00:33:17.480 So axes of symmetry-- the thing essentially 00:33:17.480 --> 00:33:23.050 looks the same at all orientations about that axis. 00:33:23.050 --> 00:33:27.130 But it is certainly dynamically balanced. 00:33:27.130 --> 00:33:30.770 Every mass bit here balances one over there, it doesn't wobble. 00:33:30.770 --> 00:33:31.640 All right. 00:33:31.640 --> 00:33:33.990 So that's for sure true. 00:33:33.990 --> 00:33:42.890 Now keep in mind that exists a set of orthogonal axes. 00:33:42.890 --> 00:33:44.890 So that means once you find one, you 00:33:44.890 --> 00:33:47.970 know that the other two are perpendicular to it 00:33:47.970 --> 00:33:50.390 and perpendicular to each other. 00:33:50.390 --> 00:33:54.560 So for this system, when I got the x and y here, 00:33:54.560 --> 00:33:56.810 there has to be an x and y perpendicular 00:33:56.810 --> 00:34:00.690 that that are the other two. 00:34:00.690 --> 00:34:02.364 And because it's actually symmetric, 00:34:02.364 --> 00:34:04.280 it doesn't matter where you out them, you just 00:34:04.280 --> 00:34:07.800 could say it's these two, these two, it doesn't matter. 00:34:07.800 --> 00:34:09.300 You just have to decide where you're 00:34:09.300 --> 00:34:12.989 going to put them on the object, they're all the same. 00:34:12.989 --> 00:34:17.900 So for this axis symmetry, they're just two more. 00:34:17.900 --> 00:34:22.610 They're going to be in the body perpendicular to the first. 00:34:22.610 --> 00:34:26.699 Now we're going to do about g so the book uses g 00:34:26.699 --> 00:34:29.281 to talk about the center of mass, so I'll use g. 00:34:29.281 --> 00:34:30.739 So the center of mass of this thing 00:34:30.739 --> 00:34:35.469 is right in the middle of this and in the middle here. 00:34:35.469 --> 00:34:38.010 So it's inside this body. 00:34:38.010 --> 00:34:41.370 Right in the dead center of it. 00:34:41.370 --> 00:34:44.585 So the three principal axes are an orthogonal set, one of which 00:34:44.585 --> 00:34:45.960 goes through it and the other two 00:34:45.960 --> 00:34:47.790 are embedded in it right angles. 00:34:47.790 --> 00:34:49.989 So this is the simplest rule. 00:34:49.989 --> 00:34:53.480 If you have an axis of symmetry, you know right way just 00:34:53.480 --> 00:34:55.980 about everything you need to know about the principal axes 00:34:55.980 --> 00:34:56.659 of the body. 00:35:09.060 --> 00:35:09.630 Shoot. 00:35:09.630 --> 00:35:12.782 AUDIENCE: It seems like then there would only 00:35:12.782 --> 00:35:16.470 be axis of symmetry in objects that are either round 00:35:16.470 --> 00:35:17.720 or spherical, is that correct? 00:35:17.720 --> 00:35:19.470 PROFESSOR: Pretty much. 00:35:19.470 --> 00:35:21.610 AUDIENCE: So it's a very limited special case. 00:35:21.610 --> 00:35:25.870 PROFESSOR: Yeah but there sure appear an awful lot in machines 00:35:25.870 --> 00:35:28.930 because they have this beautiful property of just being balanced 00:35:28.930 --> 00:35:31.800 in all directions. 00:35:31.800 --> 00:35:34.180 So everything rotating in the world 00:35:34.180 --> 00:35:38.595 tends to have an almost perfect axial symmetry. 00:35:44.230 --> 00:35:45.650 Just thinking about order here. 00:35:45.650 --> 00:35:48.850 Let's do this and then I'm going to go do a couple of examples. 00:35:48.850 --> 00:36:09.170 The second rule-- if you have one plane of symmetry, 00:36:09.170 --> 00:36:11.070 a plane of symmetry. 00:36:11.070 --> 00:36:13.420 So this is kind of the opposite direction, 00:36:13.420 --> 00:36:15.670 in this you have the least information. 00:36:15.670 --> 00:36:18.867 Here's an object, does it have a plane of symmetry? 00:36:18.867 --> 00:36:19.450 AUDIENCE: Yes. 00:36:19.450 --> 00:36:20.158 PROFESSOR: Where? 00:36:20.158 --> 00:36:21.990 AUDIENCE: Straight through [INAUDIBLE]. 00:36:21.990 --> 00:36:23.082 PROFESSOR: Show me. 00:36:23.082 --> 00:36:24.151 That cut. 00:36:24.151 --> 00:36:24.650 All right. 00:36:24.650 --> 00:36:27.620 So I've got these little dotted marks on this thing. 00:36:27.620 --> 00:36:32.320 So if I slice through that, I create 00:36:32.320 --> 00:36:33.920 two pieces that are identical. 00:36:33.920 --> 00:36:35.320 So that's a plane of symmetry. 00:36:35.320 --> 00:36:41.710 There is image match across that plane at every point. 00:36:41.710 --> 00:36:45.350 So if I have a plane of symmetry, 00:36:45.350 --> 00:36:48.430 what do you think you can say about a principal axis? 00:36:48.430 --> 00:36:50.085 One of the principal axis? 00:36:50.085 --> 00:36:51.540 AUDIENCE: It'll be on that plane. 00:36:51.540 --> 00:36:52.980 PROFESSOR: It'll be on that plane. 00:36:52.980 --> 00:36:55.229 That turns out to be true, but that's the second point 00:36:55.229 --> 00:36:56.268 I want to make. 00:36:56.268 --> 00:36:58.042 AUDIENCE: It might be perpendicular. 00:36:58.042 --> 00:37:00.500 PROFESSOR: She says there might be one perpendicular to it. 00:37:00.500 --> 00:37:02.125 And that's the one I was searching for, 00:37:02.125 --> 00:37:03.680 you're right too. 00:37:03.680 --> 00:37:07.000 There's going to be a principal axis 00:37:07.000 --> 00:37:09.380 that's perpendicular to that plane of symmetry, 00:37:09.380 --> 00:37:12.470 and since we want to define our moments of inertia 00:37:12.470 --> 00:37:14.230 for this to get started with with respect 00:37:14.230 --> 00:37:15.817 to g, where will it pass through? 00:37:15.817 --> 00:37:16.650 AUDIENCE: Through g. 00:37:16.650 --> 00:37:18.910 PROFESSOR: G, kind of by definition. 00:37:18.910 --> 00:37:20.800 So we're going to have it pass through g. 00:37:20.800 --> 00:37:33.530 If there's a plane of symmetry, then there 00:37:33.530 --> 00:37:47.820 is a principal axis perpendicular to it. 00:37:53.550 --> 00:37:58.030 And we'll0 define it, for the purposes of our discussion, 00:37:58.030 --> 00:38:00.080 we'll just let it pass through g. 00:38:03.540 --> 00:38:06.190 It doesn't have to, but that's how we're 00:38:06.190 --> 00:38:07.820 going about this discussion. 00:38:07.820 --> 00:38:14.720 Let it pass through the center of mass, this point we call g. 00:38:14.720 --> 00:38:18.440 All right, so that means that there's 00:38:18.440 --> 00:38:21.690 a center of mass in this thing. 00:38:21.690 --> 00:38:24.990 And I'm just guessing roughly where it is. 00:38:24.990 --> 00:38:28.040 But in fact if I hung this thing up here like this, 00:38:28.040 --> 00:38:34.140 and let gravity find it's natural hanging angle, 00:38:34.140 --> 00:38:36.100 and I drew a plumb line down here, 00:38:36.100 --> 00:38:40.320 just drew a line that the string would take with a plumb bob. 00:38:40.320 --> 00:38:43.310 Then I went to some other point and did it again 00:38:43.310 --> 00:38:46.140 and hung a plumb bob on it drew the line-- where they intersect 00:38:46.140 --> 00:38:48.487 is the center of mass. 00:38:48.487 --> 00:38:50.570 And then you know since it's got symmetry this way 00:38:50.570 --> 00:38:51.990 that's in the middle. 00:38:51.990 --> 00:38:54.980 So I guesses that that's about where it is. 00:38:54.980 --> 00:38:58.750 And there then is a principal axis 00:38:58.750 --> 00:39:02.960 of this object that's perpendicular to it, 00:39:02.960 --> 00:39:06.060 passing through the plane-- perpendicular to the plane. 00:39:06.060 --> 00:39:08.180 And that means now there's two more. 00:39:08.180 --> 00:39:12.540 But this gets a little more difficult, and I have no clue. 00:39:12.540 --> 00:39:15.390 There are two more, because the principal axes always 00:39:15.390 --> 00:39:16.840 come in an orthogonal set. 00:39:16.840 --> 00:39:18.830 So now that I know one, I know that there's 00:39:18.830 --> 00:39:24.330 two more somewhere around oriented 00:39:24.330 --> 00:39:25.880 this way, someplace in this plane. 00:39:25.880 --> 00:39:28.210 Maybe one like that, maybe one like this, 00:39:28.210 --> 00:39:31.670 such that if you spun it about one of those axes, 00:39:31.670 --> 00:39:33.809 it'd be in balance. 00:39:33.809 --> 00:39:35.350 You can see that gets a little messy, 00:39:35.350 --> 00:39:37.535 I can't guess where it is. 00:39:37.535 --> 00:39:39.410 And so there are ways to find it, one of them 00:39:39.410 --> 00:39:41.570 would be doing an experiment, seeing which 00:39:41.570 --> 00:39:43.570 axis it spins nicely around. 00:39:43.570 --> 00:39:45.154 So that's the second rule. 00:39:47.820 --> 00:39:50.290 But even just with that one plane of symmetry, 00:39:50.290 --> 00:39:53.600 you get some pretty good insight. 00:39:53.600 --> 00:39:58.744 Now what if there are two planes of symmetry? 00:39:58.744 --> 00:39:59.505 Yeah? 00:39:59.505 --> 00:40:01.630 AUDIENCE: You said that the principal axis does not 00:40:01.630 --> 00:40:05.290 have to pass through the center of mass? 00:40:05.290 --> 00:40:09.606 PROFESSOR: No, I'm saying it doesn't have to pass 00:40:09.606 --> 00:40:10.730 through the center of mass. 00:40:10.730 --> 00:40:12.590 The question is? 00:40:12.590 --> 00:40:15.285 AUDIENCE: How could it not because it 00:40:15.285 --> 00:40:17.990 becomes stable [INAUDIBLE]. 00:40:17.990 --> 00:40:20.634 PROFESSOR: So she's saying it'd be unstable if it's not 00:40:20.634 --> 00:40:21.550 passing through there. 00:40:21.550 --> 00:40:23.965 Like if I put an axle through this wheel, 00:40:23.965 --> 00:40:27.799 and I spun it about some point out here, 00:40:27.799 --> 00:40:29.840 you don't know because I had that shaker in here, 00:40:29.840 --> 00:40:32.520 that thing is going to shake like crazy. 00:40:32.520 --> 00:40:36.540 But does it produce unbalanced torques 00:40:36.540 --> 00:40:39.070 about my rotation point? 00:40:41.850 --> 00:40:44.770 It produces centrifugal force that you'll feel like crazy, 00:40:44.770 --> 00:40:48.030 but does it produce a torque that you'd 00:40:48.030 --> 00:40:51.380 have to resist with some static torque? 00:40:51.380 --> 00:40:52.130 What do you think? 00:40:55.030 --> 00:40:56.270 It won't. 00:40:56.270 --> 00:40:58.870 So there's a nuance to this unbalanced thing, 00:40:58.870 --> 00:41:02.060 and I was going to get to it, probably next lecture, 00:41:02.060 --> 00:41:07.140 but that is the when we say an object is dynamically balanced, 00:41:07.140 --> 00:41:12.170 we mean that it doesn't have any unbalanced torques. 00:41:12.170 --> 00:41:18.900 If it is statically balanced, it's 00:41:18.900 --> 00:41:22.250 rotating about its center of mass. 00:41:22.250 --> 00:41:25.600 But you can be statically unbalanced 00:41:25.600 --> 00:41:29.580 and let it go around this axis, but it'll produce no torques. 00:41:29.580 --> 00:41:31.210 It's still dynamically balanced. 00:41:31.210 --> 00:41:42.430 And the angular momentum of an object which is rotating-- 00:41:42.430 --> 00:41:44.490 And this we know has a principal axis here, 00:41:44.490 --> 00:41:46.550 I just moved it off to the side. 00:41:46.550 --> 00:41:50.290 It's rotating about this. 00:41:50.290 --> 00:41:55.440 It is dynamically balanced, and if you computed 00:41:55.440 --> 00:42:01.440 about this point now the Ixy, Ixz off-diagonal entries 00:42:01.440 --> 00:42:05.360 in that moment of inertia matrix, they're all 0. 00:42:05.360 --> 00:42:08.640 You'd get no unbalanced torques, but you 00:42:08.640 --> 00:42:10.860 do have an unbalanced centrifugal force 00:42:10.860 --> 00:42:12.090 as this thing goes around. 00:42:12.090 --> 00:42:13.631 And we'll talk a bit more about that. 00:42:13.631 --> 00:42:15.595 AUDIENCE: Is that sort of analogous 00:42:15.595 --> 00:42:19.032 to the homework problem we had a few weeks back 00:42:19.032 --> 00:42:22.085 with the motorcycle wheel and you just had mass on one side 00:42:22.085 --> 00:42:22.960 and not on the other? 00:42:22.960 --> 00:42:23.720 PROFESSOR: Right. 00:42:23.720 --> 00:42:25.510 So she's asking about the motorcycle wheel problem 00:42:25.510 --> 00:42:27.500 where we had that little mass that got there. 00:42:27.500 --> 00:42:30.510 I think in the next lecture I'm going to come back 00:42:30.510 --> 00:42:32.790 to that problem just so we could tie 00:42:32.790 --> 00:42:37.500 a bow around this whole thing and understand why it's 00:42:37.500 --> 00:42:39.030 unbalanced, how you can balance it, 00:42:39.030 --> 00:42:41.170 and the difference between static unbalance 00:42:41.170 --> 00:42:42.880 and dynamic unbalance. 00:42:42.880 --> 00:42:45.510 But today, we're talking about symmetry rules. 00:42:48.124 --> 00:42:50.290 Finally, so I was saying, let's talk about something 00:42:50.290 --> 00:42:51.890 that has two planes of symmetry. 00:42:51.890 --> 00:42:53.810 This actually has three planes of symmetry, 00:42:53.810 --> 00:42:55.060 but we'll settle for two. 00:42:59.820 --> 00:43:02.700 Pick a plane of symmetry for this object. 00:43:02.700 --> 00:43:07.444 If I pick-- OK, so she picked the one-- slice it this way. 00:43:07.444 --> 00:43:08.360 What about the second? 00:43:11.135 --> 00:43:13.057 Like that. 00:43:13.057 --> 00:43:13.890 Then a third, right? 00:43:13.890 --> 00:43:15.014 There's even one like that. 00:43:15.014 --> 00:43:16.980 So this has three planes of symmetry. 00:43:16.980 --> 00:43:25.510 But if you have two planes of symmetry that intersect, 00:43:25.510 --> 00:43:29.820 that are orthogonal to one another, 00:43:29.820 --> 00:43:33.000 what do you think you can say about that line 00:43:33.000 --> 00:43:34.894 of intersection? 00:43:34.894 --> 00:43:36.310 AUDIENCE: It's the principal axis. 00:43:36.310 --> 00:43:39.100 PROFESSOR: It sure is. 00:43:39.100 --> 00:43:41.670 And it's probably right through g. 00:43:41.670 --> 00:43:57.490 So if you have two planes of symmetry-- 00:43:57.490 --> 00:43:59.352 Now make them orthogonal. 00:43:59.352 --> 00:44:01.060 You can make all sorts of symmetry rules, 00:44:01.060 --> 00:44:04.230 and I'm just picking these three to help you out. 00:44:04.230 --> 00:44:07.340 This just to help you see principal axes. 00:44:07.340 --> 00:44:12.990 If you have two orthogonal planes 00:44:12.990 --> 00:44:46.050 of symmetry their intersection-- and once you know that, 00:44:46.050 --> 00:44:48.110 then you go back to rule two. 00:44:48.110 --> 00:44:50.236 And it tells you everything else you need to know. 00:44:50.236 --> 00:44:51.860 Because you have one plane of symmetry, 00:44:51.860 --> 00:44:57.080 you know there is a principal axis perpendicular to it. 00:44:57.080 --> 00:44:59.720 Well if you have two planes of symmetry, the rule still holds. 00:44:59.720 --> 00:45:01.710 There's one perpendicular to each one. 00:45:01.710 --> 00:45:05.010 The intersection, let's say, of this plane of symmetry 00:45:05.010 --> 00:45:07.910 and this plane of symmetry is a line 00:45:07.910 --> 00:45:11.200 which goes right through the center of this thing that way. 00:45:11.200 --> 00:45:13.710 So there's a principal axis this way. 00:45:13.710 --> 00:45:16.200 But since there is a plane of symmetry here, 00:45:16.200 --> 00:45:19.350 there must also be a principal axis perpendicular to it. 00:45:19.350 --> 00:45:23.730 So sure enough, three principal axes for this thing 00:45:23.730 --> 00:45:28.420 are through the center, perpendicular this way, 00:45:28.420 --> 00:45:29.550 perpendicular that way. 00:45:29.550 --> 00:45:30.450 You instantly know. 00:45:30.450 --> 00:45:32.210 Two planes of symmetry-- you instantly 00:45:32.210 --> 00:45:37.400 know where the three orthogonal principal axes are that past 00:45:37.400 --> 00:45:38.783 through the center of mass. 00:45:38.783 --> 00:45:39.769 Yeah? 00:45:39.769 --> 00:45:40.269 [? 00:45:40.269 --> 00:45:42.094 AUDIENCE: Does this all apply ?] just like a constant mass 00:45:42.094 --> 00:45:43.040 throughout? 00:45:43.040 --> 00:45:48.150 PROFESSOR: Not constant, symmetrically 00:45:48.150 --> 00:45:51.150 distributed density. 00:45:51.150 --> 00:45:55.280 Right so I'm choosing my words carefully so 00:45:55.280 --> 00:45:57.970 that I succeed in the following-- 00:45:57.970 --> 00:46:01.130 that the planes defining mass symmetry 00:46:01.130 --> 00:46:05.479 will be the same as the planes defining geometric similarity. 00:46:05.479 --> 00:46:07.770 But you actually don't have to have a constant density, 00:46:07.770 --> 00:46:11.420 it just has to be distributed so that what I just said is true. 00:46:11.420 --> 00:46:13.290 So that the geometric symmetries are 00:46:13.290 --> 00:46:17.527 the same as the mass distribution symmetries. 00:46:17.527 --> 00:46:19.610 All right so those are my three rules of symmetry. 00:46:19.610 --> 00:46:20.651 You could make up others. 00:46:20.651 --> 00:46:22.630 Those are the three that I've made up 00:46:22.630 --> 00:46:24.315 to help you see objects. 00:46:31.230 --> 00:46:35.110 That object, it's a circular disk 00:46:35.110 --> 00:46:42.430 put on top of another object such that their centers 00:46:42.430 --> 00:46:45.290 of mass line up. 00:46:45.290 --> 00:46:50.151 Where are the principal axes of this object using those rules? 00:46:55.930 --> 00:46:58.180 If you think you know one, tell me. 00:47:05.124 --> 00:47:06.674 AUDIENCE: Through the middle. 00:47:06.674 --> 00:47:08.590 PROFESSOR: Through the middle of both of them. 00:47:08.590 --> 00:47:11.498 Probably, good guess. 00:47:11.498 --> 00:47:12.426 How about another one? 00:47:16.610 --> 00:47:20.280 Where does this thing have planes of symmetry? 00:47:20.280 --> 00:47:23.640 AUDIENCE: So there's a plane of symmetry if you cut it in half. 00:47:23.640 --> 00:47:26.520 [INAUDIBLE] cut it in half, [INAUDIBLE]. 00:47:26.520 --> 00:47:27.960 PROFESSOR: OK, and? 00:47:27.960 --> 00:47:30.360 AUDIENCE: The other way. 00:47:30.360 --> 00:47:32.965 PROFESSOR: One like that, we've got all three. 00:47:32.965 --> 00:47:35.590 And if we're going to want it to go through the center of mass, 00:47:35.590 --> 00:47:37.964 then we're going to have to find where the center of mass 00:47:37.964 --> 00:47:40.470 is this way, but it's about there. 00:47:40.470 --> 00:47:42.500 So just using the symmetry ideas, 00:47:42.500 --> 00:47:45.800 you can right away figure out where these principal axes be. 00:47:45.800 --> 00:47:48.680 And that means from a dynamic point of view, 00:47:48.680 --> 00:47:50.960 if you spin it about one of those axes, 00:47:50.960 --> 00:47:52.530 it's nice and dynamically balanced. 00:47:52.530 --> 00:47:56.000 If you spin it off in some other weird direction 00:47:56.000 --> 00:47:59.830 is it necessarily dynamically balanced 00:47:59.830 --> 00:48:01.750 about that axis of spin? 00:48:08.730 --> 00:48:10.330 So let me restate that question. 00:48:16.620 --> 00:48:21.120 We know that this thing has an axis of symmetry principal 00:48:21.120 --> 00:48:25.690 axis through the center, and another one this way, 00:48:25.690 --> 00:48:26.890 and another one this way. 00:48:26.890 --> 00:48:28.650 And if I spin it about any one of those, 00:48:28.650 --> 00:48:30.240 it's dynamically balanced. 00:48:30.240 --> 00:48:35.940 But if I pick some other strange direction for the spin, 00:48:35.940 --> 00:48:40.180 and I spin it about that axis, will I feel unbalanced torques 00:48:40.180 --> 00:48:42.040 on this axle, on the bearings having 00:48:42.040 --> 00:48:44.080 to hold this thing in place? 00:48:44.080 --> 00:48:46.900 Yeah, you better believe it, this thing wobbles like crazy. 00:48:46.900 --> 00:48:56.030 So the principal axes are a property of the object, 00:48:56.030 --> 00:48:59.430 they're not a property of the angular momentum. 00:48:59.430 --> 00:49:04.567 The angular momentum comes then from multiplying 00:49:04.567 --> 00:49:06.150 the mass moment of inertia that you've 00:49:06.150 --> 00:49:11.930 determined times the actual rotation vector. 00:49:11.930 --> 00:49:15.590 And you'll find out then you get angular components of angular 00:49:15.590 --> 00:49:18.220 momentum that are not in the direction of spin, 00:49:18.220 --> 00:49:21.160 and as soon as that happens, you have unbalanced terms. 00:49:25.600 --> 00:49:28.235 I've got to get on to something else to help you do homework. 00:50:09.730 --> 00:50:18.960 So for my disk, with z coming out of the board, the Izz-- 00:50:18.960 --> 00:50:24.680 so let's say here's x, y, z coming out of the board-- Izz, 00:50:24.680 --> 00:50:28.530 the mass moment of inertia about this z axis 00:50:28.530 --> 00:50:35.070 is, from the basic definition, the summation of the mis, 00:50:35.070 --> 00:50:41.510 xi squared plus yi squared. 00:50:41.510 --> 00:50:43.900 It's just that for every little mass particle 00:50:43.900 --> 00:50:48.790 it's the radius squared away from the center of rotation. 00:50:48.790 --> 00:50:51.320 That's what the x squared plus y squared is. 00:50:51.320 --> 00:50:54.370 And we can turn this into an integral. 00:50:54.370 --> 00:50:57.890 It's the integral of r squared, that distance, 00:50:57.890 --> 00:51:01.570 times the little mass bit that's there. 00:51:01.570 --> 00:51:04.660 And that's the same. 00:51:04.660 --> 00:51:09.740 If you wanted to do the integral as x squared plus y squared dm. 00:51:12.880 --> 00:51:19.360 But to do this integral for a nice circular, symmetric disk, 00:51:19.360 --> 00:51:30.506 you can pick a little mass bit that has thickness dr and width 00:51:30.506 --> 00:51:32.510 rd theta. 00:51:32.510 --> 00:51:35.380 And this angle here is d theta. 00:51:38.390 --> 00:51:40.360 And that's a little bit of area. 00:51:40.360 --> 00:51:44.973 That's a little dA which has area r dr d theta. 00:51:48.400 --> 00:51:50.520 It's just length times width. 00:51:50.520 --> 00:51:53.280 When it's small enough, it's a little rectangle, 00:51:53.280 --> 00:51:56.280 and it has that area. 00:51:56.280 --> 00:52:02.500 And it has a volume, dV-- the volume of that thing 00:52:02.500 --> 00:52:05.960 is just the area times the thickness of it. 00:52:05.960 --> 00:52:10.500 So here's our disk here, but it has some thickness, 00:52:10.500 --> 00:52:12.840 and I'll call that h. 00:52:12.840 --> 00:52:20.500 So the volume is just h r dr d theta. 00:52:20.500 --> 00:52:25.355 And the mass, dm, is a density times dV. 00:52:31.330 --> 00:52:34.260 So I want to integrate this, all I 00:52:34.260 --> 00:52:37.450 have to integrate the integral then of r 00:52:37.450 --> 00:52:50.700 squared dm is the integral from 0 to 2pi, 0 to r of rho dV. 00:52:50.700 --> 00:52:57.890 Rho h-- oh, and I need an r squared-- r squared 00:52:57.890 --> 00:53:04.830 dV is rho h r dr d theta. 00:53:04.830 --> 00:53:08.170 So this is 1802 integrals, right? 00:53:08.170 --> 00:53:12.790 So is any of this a function of theta? 00:53:12.790 --> 00:53:14.334 No, so it's a trivial integral, you 00:53:14.334 --> 00:53:16.250 integrate that over theta, you just get theta, 00:53:16.250 --> 00:53:18.140 evaluate it 0 to 2pi. 00:53:18.140 --> 00:53:25.980 So this is 2pi rho h, can all come to the outside, integral 0 00:53:25.980 --> 00:53:33.550 to R of r cubed dr. And that ends up-- 00:53:33.550 --> 00:53:37.090 the r cubed goes to r to the 4th over 4. 00:53:37.090 --> 00:53:48.490 And the final result of this one is 2pi r to the 4th over 4 00:53:48.490 --> 00:53:54.300 rho h, and when new account for h times pi r squared 00:53:54.300 --> 00:53:57.030 is the volume times rho is the mass. 00:53:57.030 --> 00:54:03.260 This all works out to be m r squared over 2. 00:54:03.260 --> 00:54:08.800 So Izz-- so I needed to do this once for you. 00:54:08.800 --> 00:54:13.230 For simple things integrate, Izz in this case, 00:54:13.230 --> 00:54:15.650 you just integrate it out, account 00:54:15.650 --> 00:54:18.650 for all little mass bits, that is the mass moment of inertia 00:54:18.650 --> 00:54:25.390 with respect to the axis passing through the center like this. 00:54:25.390 --> 00:54:26.662 Pardon? 00:54:26.662 --> 00:54:28.150 AUDIENCE: [INAUDIBLE] 00:54:28.150 --> 00:54:30.358 PROFESSOR: It's this, this is what I'm talking about. 00:54:42.610 --> 00:54:44.220 Moving on to the last bit. 00:54:53.010 --> 00:54:57.720 So we need to know how to be able rotate things 00:54:57.720 --> 00:55:01.160 about places other than their centers of mass. 00:55:01.160 --> 00:55:05.210 So this is a stick, I can rotate about the center of mass, 00:55:05.210 --> 00:55:06.890 but it's more interesting if I rotate it 00:55:06.890 --> 00:55:08.520 about some other point. 00:55:08.520 --> 00:55:10.926 It makes it a pendulum when I do it around here. 00:55:10.926 --> 00:55:12.300 So I need to be able to calculate 00:55:12.300 --> 00:55:14.710 mass moments of inertia about a point that's 00:55:14.710 --> 00:55:16.730 not through the center of mass. 00:55:16.730 --> 00:55:21.300 I know you've seen this before in [? 8.01, ?] 00:55:21.300 --> 00:55:23.435 so this is going to be a quick reminder. 00:55:26.500 --> 00:55:29.090 But I'll show you where it comes from. 00:55:29.090 --> 00:55:30.070 So here's my stick. 00:55:39.980 --> 00:55:48.190 And it has a center of mass, and that's where G is located here. 00:55:48.190 --> 00:55:50.155 It has a total length l. 00:55:53.260 --> 00:56:04.220 I'm going to give it a thickness b, a width a. 00:56:04.220 --> 00:56:07.780 So it's a stick. 00:56:07.780 --> 00:56:12.120 a wide, b thick, l long. 00:56:12.120 --> 00:56:15.720 Uniform has a center gravity right in the middle. 00:56:15.720 --> 00:56:23.130 And I'm going to attach to this stick-- 00:56:23.130 --> 00:56:25.080 and this point is kind of hard to draw. 00:56:25.080 --> 00:56:28.140 This point is at the center of the stick, OK? 00:56:28.140 --> 00:56:30.750 I'm going to put my coordinate system attached 00:56:30.750 --> 00:56:33.730 at the center of gravity, center of mass, 00:56:33.730 --> 00:56:40.060 and I'm going to make it the-- that's x prime downward, 00:56:40.060 --> 00:56:45.680 z prime, and y prime is then going off that way. 00:56:45.680 --> 00:56:47.960 So this is a body set of coordinates 00:56:47.960 --> 00:56:49.790 at the center of mass. 00:56:49.790 --> 00:56:51.700 x prime, y prime, z prime. 00:56:51.700 --> 00:56:53.890 And x happens to be down. 00:56:53.890 --> 00:56:58.270 And I want to calculate my mass moment of inertia 00:56:58.270 --> 00:57:06.050 with respect to a point up here that is d, this distance. 00:57:06.050 --> 00:57:10.750 I've moved up the x-axis an amount d. 00:57:10.750 --> 00:57:14.120 I'm going to set a new coordinate system up here. 00:57:14.120 --> 00:57:18.010 So if this was z prime, my new z is here. 00:57:18.010 --> 00:57:19.370 It's getting a little messy. 00:57:19.370 --> 00:57:24.220 Maybe I'll do just a face view. 00:57:27.660 --> 00:57:33.400 If my previously y prime and x prime 00:57:33.400 --> 00:57:36.120 were like that, z coming out of the board, 00:57:36.120 --> 00:57:39.420 now I have a new system that is y 00:57:39.420 --> 00:57:42.960 and x like this, z still coming out of the board. 00:57:47.350 --> 00:57:48.856 Now the coordinate. 00:57:51.500 --> 00:57:56.500 So how do I calculate mass moment of inertia? 00:57:56.500 --> 00:58:02.670 Well I want Izz. 00:58:06.400 --> 00:58:08.770 I probably know Iz'z'. 00:58:08.770 --> 00:58:13.480 Iz'z' is the mass moment of inertia about this point. 00:58:13.480 --> 00:58:15.790 I know it's a principal axis from all the things we 00:58:15.790 --> 00:58:18.290 just-- that square block is the same as this. 00:58:18.290 --> 00:58:20.590 That's a principal axis in the z prime direction. 00:58:20.590 --> 00:58:24.715 I know the Izz' with respect to G, 00:58:24.715 --> 00:58:27.960 I want to know what with respect to this point. 00:58:27.960 --> 00:58:33.510 So well Izz, which is my new location 00:58:33.510 --> 00:58:40.700 up here, and we'll call it A. So Izz here with respect to A 00:58:40.700 --> 00:58:44.609 is the integral of r squared dm. 00:58:44.609 --> 00:58:46.192 We've got to do the same integral now. 00:58:48.980 --> 00:58:57.515 But that's the integral of x squared plus y squared dm. 00:59:01.300 --> 00:59:04.420 Now I can look at this and I can say oh, well, these 00:59:04.420 --> 00:59:07.440 are d-- this is separated by d. 00:59:07.440 --> 00:59:08.740 I only moved it in the x. 00:59:08.740 --> 00:59:11.810 The ys haven't moved and the zs didn't change. 00:59:11.810 --> 00:59:15.360 I just moved my point only in the x direction. 00:59:15.360 --> 00:59:20.820 So I can now say that in terms of my new coordinate, 00:59:20.820 --> 00:59:26.670 it's the same as x prime plus d, the distance from here 00:59:26.670 --> 00:59:32.750 to a point down her, some arbitrary mass point 00:59:32.750 --> 00:59:38.500 xi is going to be xi' plus d. 00:59:38.500 --> 00:59:43.390 So to do this integral in the new coordinates, 00:59:43.390 --> 00:59:51.070 this is going to be the integral of x prime plus d squared 00:59:51.070 --> 00:59:53.560 plus y. 00:59:53.560 --> 01:00:01.714 Now y prime equals y and z prime equals z. 01:00:01.714 --> 01:00:02.630 Those haven't changed. 01:00:02.630 --> 01:00:05.720 I didn't move my new coordinate system 01:00:05.720 --> 01:00:07.940 in the y direction or the z direction. 01:00:07.940 --> 01:00:11.200 So the coordinate in the new system in y 01:00:11.200 --> 01:00:12.390 is the same as before. 01:00:12.390 --> 01:00:15.030 So this is just y prime squared. 01:00:15.030 --> 01:00:18.910 And this whole thing times d, integrated times 01:00:18.910 --> 01:00:21.835 every little mass bit. 01:00:21.835 --> 01:00:26.690 If I square this, I get x prime squared, 2x'd, d squared, 01:00:26.690 --> 01:00:27.960 plus y squared. 01:00:27.960 --> 01:00:31.870 So this integral, Izz with respect 01:00:31.870 --> 01:00:35.500 to A, when you rearrange it, looks 01:00:35.500 --> 01:00:43.780 like x prime squared plus y prime squared dm 01:00:43.780 --> 01:00:47.690 and the integral of a sum is the sum of the integrals. 01:00:47.690 --> 01:00:49.730 So I break it into bits here, there's 01:00:49.730 --> 01:00:53.340 a d squared, which is a constant, dm. 01:00:53.340 --> 01:01:02.252 And then the last term is plus 2d and it's x prime dm. 01:01:05.410 --> 01:01:08.550 Just multiply this out, rewrite it, break it apart. 01:01:11.932 --> 01:01:12.890 Well let's do this one. 01:01:12.890 --> 01:01:16.360 Integral of x prime squared plus y prime squared dm. 01:01:16.360 --> 01:01:20.620 That's something that we already have a name for. 01:01:20.620 --> 01:01:22.175 This is IGzz. 01:01:27.640 --> 01:01:30.450 It's the original mass moment of inertia 01:01:30.450 --> 01:01:34.980 with respect to the original coordinate system at G 01:01:34.980 --> 01:01:35.950 in the z direction. 01:01:35.950 --> 01:01:39.810 So it's [? IzzG, ?] we already know that. 01:01:39.810 --> 01:01:41.840 That's given for the object. 01:01:41.840 --> 01:01:45.180 Plus, this is the integral of dm over the whole extent 01:01:45.180 --> 01:01:47.654 of the object? 01:01:47.654 --> 01:01:48.820 Just the mass of the object. 01:01:53.320 --> 01:01:58.330 This integral, this is the integral 01:01:58.330 --> 01:02:00.990 in terms of the x-coordinate. 01:02:00.990 --> 01:02:07.210 And every mass bit from here, if I go out here and find one, 01:02:07.210 --> 01:02:09.470 there's an equal and opposite one up here. 01:02:09.470 --> 01:02:11.840 This is the definition of the center of mass. 01:02:11.840 --> 01:02:14.540 This integral, if I'm at the center of mass 01:02:14.540 --> 01:02:18.890 integrating out from it, this is zero because of the definition 01:02:18.890 --> 01:02:21.220 the center of mass. 01:02:21.220 --> 01:02:26.540 And I've just proven the parallel axis theorem. 01:02:26.540 --> 01:02:32.740 Izz about this new point is I about G plus Md Squared, 01:02:32.740 --> 01:02:35.710 where d squared is the distance I've 01:02:35.710 --> 01:02:39.160 moved this z-axis to a new place parallel to it. 01:03:41.040 --> 01:03:48.710 So Izz with respect to G, the original mass moment 01:03:48.710 --> 01:04:01.630 of inertia for Izz is m L squared plus a squared over 12. 01:04:04.450 --> 01:04:20.570 And Ixx m L squared plus b squared over 12. 01:04:20.570 --> 01:04:29.420 And Izz-- oh, we already know that one. 01:04:32.980 --> 01:04:34.460 Wait a minute. 01:04:34.460 --> 01:04:36.440 I haven't told you what that is. 01:04:36.440 --> 01:04:38.510 That's Izz, Ixx, Iyy. 01:04:38.510 --> 01:04:42.180 Just a little messy here. 01:04:42.180 --> 01:04:56.990 Iyy for this problem is-- I have made a mistake. 01:04:56.990 --> 01:04:59.750 Ixx is a squared plus b squared. 01:04:59.750 --> 01:05:08.660 Iyy is m L squared plus b squared. 01:05:08.660 --> 01:05:10.940 So those are the three-- all with respect 01:05:10.940 --> 01:05:13.880 to G-- for this stick stick. 01:05:13.880 --> 01:05:14.785 And I'm going to-- 01:05:17.550 --> 01:05:19.014 AUDIENCE: [INAUDIBLE]? 01:05:19.014 --> 01:05:20.788 Is that also divided by 12? 01:05:20.788 --> 01:05:21.454 PROFESSOR: Yeah. 01:05:28.800 --> 01:05:33.190 That kind of sets us up where I can pick up next time. 01:05:33.190 --> 01:05:42.800 So let's finish by asking ourselves the question, what 01:05:42.800 --> 01:05:45.770 do we think about-- if I've moved to this new put 01:05:45.770 --> 01:05:52.320 new position, and I'm not rotating about the center, 01:05:52.320 --> 01:05:58.000 is this new axis-- this one around the center before, 01:05:58.000 --> 01:06:01.840 that one we know is a principal axis. 01:06:01.840 --> 01:06:03.990 If I rotate about this new place which 01:06:03.990 --> 01:06:08.650 I've defined the mass moment of inertia about that place? 01:06:08.650 --> 01:06:10.454 Is this a principal axis? 01:06:10.454 --> 01:06:11.342 AUDIENCE: Yes. 01:06:11.342 --> 01:06:12.230 AUDIENCE: Yes. 01:06:12.230 --> 01:06:15.320 PROFESSOR: How many think yes? 01:06:15.320 --> 01:06:18.330 How many think no? 01:06:18.330 --> 01:06:22.410 OK, so lots of people not sure. 01:06:22.410 --> 01:06:27.930 So my dynamic definition of principal axis 01:06:27.930 --> 01:06:31.580 is if you can rotate the object about that axis 01:06:31.580 --> 01:06:36.280 and produce no unbalanced torques, it's a principal axis. 01:06:36.280 --> 01:06:39.180 And I can do that and this thing will just spin all day long. 01:06:39.180 --> 01:06:43.390 Now there is a force, you could think of a fictitious force. 01:06:43.390 --> 01:06:45.840 There's a center of mass out here. 01:06:45.840 --> 01:06:49.610 As it spins around, there's a centripetal acceleration 01:06:49.610 --> 01:06:53.270 making it go in the circle, that means that fictitious force is 01:06:53.270 --> 01:06:56.790 like there is a centrifugal force pulling out on it. 01:06:56.790 --> 01:06:57.960 Do I feel that? 01:06:57.960 --> 01:07:00.176 Do I have to this resist that force 01:07:00.176 --> 01:07:01.300 as it goes round and round? 01:07:01.300 --> 01:07:02.530 Yes. 01:07:02.530 --> 01:07:05.960 So that is an unbalance of a kind 01:07:05.960 --> 01:07:09.600 we know as a static imbalance, but it doesn't produce torques 01:07:09.600 --> 01:07:17.118 about my axis right lined up on the center. 01:07:17.118 --> 01:07:19.986 AUDIENCE: Doesn't gravity pull on the center of mass 01:07:19.986 --> 01:07:21.541 [INAUDIBLE]. 01:07:21.541 --> 01:07:23.415 PROFESSOR: Sure, gravity does, but that's now 01:07:23.415 --> 01:07:24.248 a different problem. 01:07:24.248 --> 01:07:28.240 That's what makes this thing act like an oscillator. 01:07:28.240 --> 01:07:30.040 The torques of the kind I'm talking about 01:07:30.040 --> 01:07:34.390 is if I compute the angular momentum of this thing 01:07:34.390 --> 01:07:38.590 and compute dh/dt-- the time rate of change of the angular 01:07:38.590 --> 01:07:41.430 momentum is a torque on the system, right? 01:07:41.430 --> 01:07:44.680 I will get the term that makes it spin faster, 01:07:44.680 --> 01:07:46.640 and I will get, if they exist, terms 01:07:46.640 --> 01:07:51.640 that make it want to bend this way or bend back. 01:07:51.640 --> 01:07:58.650 It only happens if-- if I hold this thing over here, and spin 01:07:58.650 --> 01:08:02.790 it, get it spinning, and I compute the angle momentum 01:08:02.790 --> 01:08:06.030 with respect to this point, will I get torques? 01:08:06.030 --> 01:08:12.360 Yeah, but that's not how I-- that's a different problem. 01:08:12.360 --> 01:08:18.569 The IG is as if I were computing the angular momentum. 01:08:18.569 --> 01:08:22.660 Remember I started defining mass moment of inertia matrix 01:08:22.660 --> 01:08:26.520 based on an angular momentum computation at G. 01:08:26.520 --> 01:08:29.029 So it's right there in the center of this object, 01:08:29.029 --> 01:08:31.560 there's no moment arm that is causing 01:08:31.560 --> 01:08:35.830 torques that's trying to twist this thing about that point. 01:08:35.830 --> 01:08:39.952 So the answer to the question is this is a principal axis. 01:08:39.952 --> 01:08:40.859 Yeah? 01:08:40.859 --> 01:08:44.446 AUDIENCE: So if you take the derivative of the angular 01:08:44.446 --> 01:08:49.370 momentum would you get torques that are not in that direction? 01:08:49.370 --> 01:08:52.450 PROFESSOR: If you get torques that aren't in that direction, 01:08:52.450 --> 01:08:54.979 either you made a mistake doing the math, 01:08:54.979 --> 01:08:59.140 or you were in error in identifying 01:08:59.140 --> 01:09:02.055 the mass moment of inertia matrix to begin with. 01:09:02.055 --> 01:09:03.430 Because if you get torques, there 01:09:03.430 --> 01:09:07.270 must be non-zero off-diagonal terms and in mass moment 01:09:07.270 --> 01:09:08.109 of inertia matrix. 01:09:08.109 --> 01:09:12.140 They are what account for the torques. 01:09:12.140 --> 01:09:15.420 So by this parallel axis theorem, any other axis 01:09:15.420 --> 01:09:18.569 you go to-- if you started at a principal axis, 01:09:18.569 --> 01:09:23.149 any other axis you create is also a principal axis. 01:09:23.149 --> 01:09:26.870 That's the movement of just one axis. 01:09:26.870 --> 01:09:30.490 If you do two, if you move this way and this way, 01:09:30.490 --> 01:09:31.220 all bets are off. 01:09:31.220 --> 01:09:32.386 You get a difference answer. 01:09:32.386 --> 01:09:34.870 And if you're interested in that more complicated problem, 01:09:34.870 --> 01:09:37.250 read that Williams thing because he 01:09:37.250 --> 01:09:40.359 does the complete parallel axis, parallel planes 01:09:40.359 --> 01:09:43.410 and comes up with a super compact little way 01:09:43.410 --> 01:09:45.740 of calculating them. 01:09:45.740 --> 01:09:47.690 See you on Thursday.