1 00:00:00,080 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,800 Commons license. 3 00:00:03,800 --> 00:00:06,050 Your support will help MIT OpenCourseWare 4 00:00:06,050 --> 00:00:10,140 continue to offer high quality educational resources for free. 5 00:00:10,140 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:16,590 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,590 --> 00:00:17,297 at ocw.mit.edu. 8 00:00:21,260 --> 00:00:23,680 PROFESSOR: Today we're going to talk all about moments 9 00:00:23,680 --> 00:00:30,760 of inertia, and the last time we did [? muddy ?] cards, there 10 00:00:30,760 --> 00:00:32,860 were a lot of questions. 11 00:00:32,860 --> 00:00:37,990 The most common question-- where did all those terms come from? 12 00:00:37,990 --> 00:00:42,040 And I'm going to do a brief kind of intro 13 00:00:42,040 --> 00:00:47,045 around that so as to facilitate the rest of the discussions. 14 00:00:58,060 --> 00:00:59,940 Some basic assumptions here today. 15 00:00:59,940 --> 00:01:13,190 We're going to consider bodies that 16 00:01:13,190 --> 00:01:30,456 rotate about their center of mass, or fixed points. 17 00:01:34,540 --> 00:01:38,210 That's the kind of problems we're doing. 18 00:01:38,210 --> 00:01:42,610 And secondly-- all through this lecture 19 00:01:42,610 --> 00:01:47,160 today, there's four or five significant assumptions 20 00:01:47,160 --> 00:01:50,125 or conditions that are really important 21 00:01:50,125 --> 00:01:51,125 to the whole discussion. 22 00:01:51,125 --> 00:01:52,630 I'm going to really try to highlight them. 23 00:01:52,630 --> 00:01:53,754 And starting with two here. 24 00:01:53,754 --> 00:02:10,036 The other one is we're going to use reference frames attached 25 00:02:10,036 --> 00:02:10,619 to the bodies. 26 00:02:13,850 --> 00:02:16,740 That's why we did all that work in kinematics figuring out 27 00:02:16,740 --> 00:02:21,560 how to do velocities of things that are in translating 28 00:02:21,560 --> 00:02:23,190 and rotating references frame, it's 29 00:02:23,190 --> 00:02:25,220 so that we can do these problems. 30 00:02:28,860 --> 00:02:38,500 So this said, here's a inertial frame. 31 00:02:42,410 --> 00:02:46,000 Here's some rigid body. 32 00:02:46,000 --> 00:02:48,830 Here's a point on it, that's my origin. 33 00:02:48,830 --> 00:02:56,400 And here is a set of coordinates attached to the rigid body-- 34 00:02:56,400 --> 00:03:00,340 an xyz system-- the body rotates and translates, 35 00:03:00,340 --> 00:03:03,360 it goes with it. 36 00:03:03,360 --> 00:03:09,050 And if we consider a little piece of this body, 37 00:03:09,050 --> 00:03:13,020 a little part-- I'll call it little mass element 38 00:03:13,020 --> 00:03:23,110 mi-- from here to here is some distance y with respect to A, 39 00:03:23,110 --> 00:03:30,060 then the angular momentum of that little mass particle 40 00:03:30,060 --> 00:03:37,760 is by definition of angular momentum Ri cross Pi. 41 00:03:40,660 --> 00:03:43,310 And the P is always the linear momentum 42 00:03:43,310 --> 00:03:46,916 defined in an inertial frame. 43 00:03:46,916 --> 00:03:48,665 That's the definition of angular momentum. 44 00:03:59,310 --> 00:04:04,050 So you have seen this little demo before. 45 00:04:04,050 --> 00:04:08,630 So this is now my rigid body that I'm talking about. 46 00:04:08,630 --> 00:04:14,330 In this case, the axis of rotation is this one. 47 00:04:14,330 --> 00:04:17,410 And I'm just going to define my rigid body 48 00:04:17,410 --> 00:04:21,360 as essentially rotating about this fixed point, 49 00:04:21,360 --> 00:04:23,270 this is A now. 50 00:04:23,270 --> 00:04:32,830 And my coordinate system is xyz, so the rotation is omega z. 51 00:04:32,830 --> 00:04:36,200 And this rigid body-- this is massless, 52 00:04:36,200 --> 00:04:39,470 consists of a single point mass. 53 00:04:39,470 --> 00:04:41,200 So we've done this before, but we 54 00:04:41,200 --> 00:04:43,710 tended to do this kind of problem 55 00:04:43,710 --> 00:04:44,910 using polar coordinates. 56 00:04:44,910 --> 00:04:47,790 And now I really wanted to strictly think in terms of XYZ. 57 00:04:47,790 --> 00:04:49,420 The answers come out exactly the same. 58 00:04:49,420 --> 00:04:50,970 Polar coordinates also moved. 59 00:04:50,970 --> 00:04:54,470 You could do this in rhat, thetahat, z. 60 00:04:54,470 --> 00:04:59,390 We're going to do it in xyz attached to the body. 61 00:05:22,450 --> 00:05:26,150 And y is going into the board, and we'll 62 00:05:26,150 --> 00:05:29,180 have no values in the y direction for this problem. 63 00:05:29,180 --> 00:05:33,150 Here's our little mass, we'll call it m1. 64 00:05:33,150 --> 00:05:38,480 It's at a position out here x1, z1 are its coordinates, 65 00:05:38,480 --> 00:05:41,430 and y is zero. 66 00:05:41,430 --> 00:05:44,710 And we want to compute its angular momentum, 67 00:05:44,710 --> 00:05:57,505 h1 with respect to A. That's our R1A cross P1 and 0. 68 00:05:57,505 --> 00:06:02,800 And for this problem that's x1 in the i hat 69 00:06:02,800 --> 00:06:09,080 plus z1 in the k hat, that's the R-- cross 70 00:06:09,080 --> 00:06:15,030 P and P is mass times velocity, m1. 71 00:06:15,030 --> 00:06:19,980 And the velocity is just omega cross R. 72 00:06:19,980 --> 00:06:21,070 We've done this before. 73 00:06:21,070 --> 00:06:25,010 So we need a length on this thing. 74 00:06:33,410 --> 00:06:35,510 We've done this enough times before I'm assuming 75 00:06:35,510 --> 00:06:38,260 you can make this leap with me. 76 00:06:38,260 --> 00:06:46,510 The momentum is into the board, in the y direction. 77 00:06:46,510 --> 00:06:53,620 The linear momentum is R cross with that. 78 00:06:53,620 --> 00:07:02,260 And that will give us-- let me write it-- omega z. 79 00:07:02,260 --> 00:07:04,310 So that's the linear momentum. 80 00:07:04,310 --> 00:07:10,850 It's in the j direction, its velocity is x1 omega z. 81 00:07:10,850 --> 00:07:14,120 The radius times the rotation rate is the velocity, 82 00:07:14,120 --> 00:07:15,820 the direction is that direction. 83 00:07:15,820 --> 00:07:17,470 So this is your momentum. 84 00:07:17,470 --> 00:07:19,360 You multiply out this cross product, 85 00:07:19,360 --> 00:07:34,410 you get two terms-- m1 x1 squared omega z k, minus m1 x1 86 00:07:34,410 --> 00:07:37,400 z1 omega z i. 87 00:07:40,070 --> 00:07:43,030 And these two terms then we would identify. 88 00:07:43,030 --> 00:07:46,094 This is the angular momentum in the k direction, angular 89 00:07:46,094 --> 00:07:47,260 momentum in the i direction. 90 00:07:47,260 --> 00:07:49,343 There doesn't happen to be any in the z direction, 91 00:07:49,343 --> 00:07:50,760 in general there could be. 92 00:07:50,760 --> 00:07:55,035 But this is h for this particle, particle one. 93 00:07:59,370 --> 00:08:01,760 I want to particularly emphasize this one-- 94 00:08:01,760 --> 00:08:04,380 oops, this is-- excuse me-- z. 95 00:08:04,380 --> 00:08:06,780 And this is the piece we call hx. 96 00:08:10,050 --> 00:08:12,664 So there's a complement of the angular momentum 97 00:08:12,664 --> 00:08:14,080 that's in the z direction, there's 98 00:08:14,080 --> 00:08:15,520 a component in the x direction. 99 00:08:21,350 --> 00:08:24,040 We know that if we compute-- just 100 00:08:24,040 --> 00:08:32,720 to tie it back to previous work a little further-- d 101 00:08:32,720 --> 00:08:41,900 by dt of h1A gives us the external torques 102 00:08:41,900 --> 00:08:44,370 in the system, a vector of torques 103 00:08:44,370 --> 00:08:46,490 when we compute that out. 104 00:08:46,490 --> 00:08:49,270 I'm not going to calculate it, I don't need it 105 00:08:49,270 --> 00:08:51,450 for the purposes of the rest of the discussion. 106 00:08:51,450 --> 00:08:54,180 This one gives you omega z dot term. 107 00:08:54,180 --> 00:08:55,911 This one gets two terms because you have 108 00:08:55,911 --> 00:08:57,190 to take the derivative of i. 109 00:08:57,190 --> 00:09:00,430 So you get three terms which are torque 110 00:09:00,430 --> 00:09:02,700 in the x, torque in the y, torque in the z directions. 111 00:09:02,700 --> 00:09:04,410 Two of them are static, because it's 112 00:09:04,410 --> 00:09:08,320 trying to bend this thing back and out, and one of them, 113 00:09:08,320 --> 00:09:11,040 the z direction one, is the one that makes it spin faster. 114 00:09:13,850 --> 00:09:16,910 So we've seen that, that's a review. 115 00:09:16,910 --> 00:09:39,930 And to make the leap from that to rigid bodies-- Capital 116 00:09:39,930 --> 00:09:43,570 H, now, a collection of particles with respect 117 00:09:43,570 --> 00:09:51,870 to its the origin attached to the rigid body 118 00:09:51,870 --> 00:09:54,082 is going to be the summation over all 119 00:09:54,082 --> 00:09:55,165 the little mass particles. 120 00:10:00,740 --> 00:10:07,254 Their position vector crossed with the linear momentum of all 121 00:10:07,254 --> 00:10:08,420 those little mass particles. 122 00:10:08,420 --> 00:10:10,490 Sum all that out. 123 00:10:10,490 --> 00:10:12,870 Because it's a rigid body, this is always 124 00:10:12,870 --> 00:10:16,150 something of the form that looks like omega cross R, 125 00:10:16,150 --> 00:10:20,290 m omega cross R. And I can then write this 126 00:10:20,290 --> 00:10:29,310 as the summation of [? oper ?] i of the mi, 127 00:10:29,310 --> 00:10:39,225 RiA cross omega with respect to 0, cross RiA. 128 00:10:42,430 --> 00:10:49,140 That gives you the velocity, m gives you the linear momentum, 129 00:10:49,140 --> 00:10:50,720 crossed with this one again makes it 130 00:10:50,720 --> 00:10:52,690 into an angular momentum. 131 00:10:52,690 --> 00:10:59,780 So the angular momentum of every particle-- the bits of this 132 00:10:59,780 --> 00:11:07,400 come out looking like an R squared omega. 133 00:11:07,400 --> 00:11:13,410 Or at least has dimensions of R squared omega. 134 00:11:13,410 --> 00:11:16,350 These can be like x1 squared, but these can also 135 00:11:16,350 --> 00:11:21,710 end up being terms like x1 z1, those cross product terms. 136 00:11:21,710 --> 00:11:27,240 But in general, these things all add up. 137 00:11:27,240 --> 00:11:30,830 And this is a vector, this is a vector, 138 00:11:30,830 --> 00:11:33,510 so this is a vector-- these are all vectors, 139 00:11:33,510 --> 00:11:34,540 the result's a vector. 140 00:11:39,040 --> 00:11:42,730 And therefore we would break this down, this whole thing. 141 00:11:42,730 --> 00:11:45,750 Once you write all that out and sum it all up, 142 00:11:45,750 --> 00:11:48,210 you're going to get a piece of it 143 00:11:48,210 --> 00:11:52,440 that we would say that is in the i hat direction 144 00:11:52,440 --> 00:11:54,580 and we call that Hx. 145 00:11:54,580 --> 00:11:58,460 Another piece that's in the j hat direction, which 146 00:11:58,460 --> 00:12:03,540 we call Hy and Hz in the k hat. 147 00:12:03,540 --> 00:12:05,920 That's just how that all shakes out. 148 00:12:05,920 --> 00:12:07,290 In general you get three parts. 149 00:12:11,750 --> 00:12:13,664 Yeah? 150 00:12:13,664 --> 00:12:17,648 AUDIENCE: Why do you have mass multiplied by displacement 151 00:12:17,648 --> 00:12:19,650 and the momentum? 152 00:12:19,650 --> 00:12:24,170 PROFESSOR: Why do we have the mass multiplied by a position 153 00:12:24,170 --> 00:12:27,210 vector times the momentum? 154 00:12:27,210 --> 00:12:30,205 Because that is the definition of angular momentum. 155 00:12:34,450 --> 00:12:36,763 It's R cross P, all right? 156 00:12:36,763 --> 00:12:37,596 [INTERPOSING VOICES] 157 00:12:46,310 --> 00:12:48,120 PROFESSOR: Ah, I see what I've done. 158 00:12:48,120 --> 00:12:48,950 Yeah you're right. 159 00:12:48,950 --> 00:12:52,540 This m doesn't belong-- I got ahead of myself-- 160 00:12:52,540 --> 00:12:54,470 it doesn't belong here. 161 00:12:54,470 --> 00:12:57,680 It pops out of this to here. 162 00:12:57,680 --> 00:12:59,752 Thanks for catching that. 163 00:12:59,752 --> 00:13:00,460 Absolutely right. 164 00:13:15,420 --> 00:13:22,530 So every term here is made up of things that 165 00:13:22,530 --> 00:13:30,310 look like x1, z1, m1 omega x. 166 00:13:30,310 --> 00:13:34,150 Or omega y, or omega z, because this is a vector, 167 00:13:34,150 --> 00:13:37,450 and it can have three components-- a piece in the i, 168 00:13:37,450 --> 00:13:39,960 a piece in the j, a piece in the k direction. 169 00:13:39,960 --> 00:13:45,190 So there's a lot of possible terms, 170 00:13:45,190 --> 00:13:49,950 and we're in the practice of writing this out. 171 00:13:53,570 --> 00:14:00,250 This H vector can be written then 172 00:14:00,250 --> 00:14:05,330 as a-- I'm going to run out of room here. 173 00:14:19,220 --> 00:14:22,220 So the Hx term-- the piece that comes out 174 00:14:22,220 --> 00:14:26,410 with all little bits in the i direction-- 175 00:14:26,410 --> 00:14:35,532 when you just break it apart, we would write it Ixx omega 176 00:14:35,532 --> 00:14:46,725 x, plus Ixy omega y, plus Ixz omega z. 177 00:14:49,330 --> 00:14:54,610 All the i terms that float out of this, 178 00:14:54,610 --> 00:14:58,060 we just collect them together-- all the ones that 179 00:14:58,060 --> 00:15:02,150 are multiplied by omega x ends up being some terms that 180 00:15:02,150 --> 00:15:11,570 look like sums of mi-- x, mi z squared terms, 181 00:15:11,570 --> 00:15:13,726 and there's some y squared terms. 182 00:15:13,726 --> 00:15:15,350 But we collect them all together and we 183 00:15:15,350 --> 00:15:18,730 call this constant in front of it Ixx. 184 00:15:18,730 --> 00:15:21,410 Just what floats out of this stuff. 185 00:15:21,410 --> 00:15:25,030 We break it into three pieces, the part of the angular 186 00:15:25,030 --> 00:15:26,865 momentum due to the rotation in the x, 187 00:15:26,865 --> 00:15:28,780 the part due to the rotation in the y, 188 00:15:28,780 --> 00:15:31,400 the part due to the rotation in the z. 189 00:15:31,400 --> 00:15:34,436 And you do the same thing for Hy, 190 00:15:34,436 --> 00:15:43,930 and you get in Iyz omega x plus an Iyy omega 191 00:15:43,930 --> 00:15:53,660 y plus an Iyz omega z, and you're finally get and Hz term. 192 00:15:53,660 --> 00:16:04,935 Izx omega x, Izy omega y, Izz omega z. 193 00:16:07,560 --> 00:16:10,775 That's everything that falls out of just doing this calculation. 194 00:16:19,890 --> 00:16:24,430 And where in the habit of writing that in a matrix 195 00:16:24,430 --> 00:16:32,870 notation as the product of this thing 196 00:16:32,870 --> 00:16:33,985 called the inertia matrix. 197 00:16:41,120 --> 00:16:43,240 And so forth, with this bottom term 198 00:16:43,240 --> 00:16:52,130 being Izz multiplied by omega x, omega y, omega z. 199 00:16:52,130 --> 00:16:57,310 You multiply that times the top row, you get Hx, middle row, 200 00:16:57,310 --> 00:17:01,240 you get Hy, bottom row, you get the Hz. 201 00:17:01,240 --> 00:17:04,700 So where this stuff comes from is just 202 00:17:04,700 --> 00:17:10,770 from carrying out the summation over all the mass bits. 203 00:17:10,770 --> 00:17:13,504 So it basically starts with the definition of angular momentum. 204 00:17:13,504 --> 00:17:15,420 And this is just a convenient way to write it. 205 00:17:19,910 --> 00:17:33,540 So for example, the Ixz term-- the one in the upper right 206 00:17:33,540 --> 00:17:40,090 there that's part of Hx, the Ixz term-- 207 00:17:40,090 --> 00:17:45,850 is minus the summation of all the little mass bits 208 00:17:45,850 --> 00:17:50,170 times their location xi zi. 209 00:17:50,170 --> 00:17:55,331 And this, the xz, always matches xz. 210 00:17:55,331 --> 00:17:57,580 And when you want to do this over a continuous object, 211 00:17:57,580 --> 00:17:59,820 you integrate that. 212 00:17:59,820 --> 00:18:10,410 And for Izz, this is the summation over i, 213 00:18:10,410 --> 00:18:21,920 all the bits of mi xi squared plus yi squared. 214 00:18:21,920 --> 00:18:24,610 So in general this, by summations, that's 215 00:18:24,610 --> 00:18:26,880 where these terms come from. 216 00:18:26,880 --> 00:18:27,530 They just come. 217 00:18:27,530 --> 00:18:29,670 They've started with this calculation. 218 00:18:34,030 --> 00:18:35,905 And in general, from that calculation. 219 00:18:39,040 --> 00:18:43,990 So for our one particle system, for this thing. 220 00:18:43,990 --> 00:18:49,700 So where this is x and z, it's one single particle. 221 00:18:49,700 --> 00:18:51,425 So for our one particle system. 222 00:19:19,310 --> 00:19:22,700 And there's no y1 in here because it's 223 00:19:22,700 --> 00:19:24,870 a zero in this problem. 224 00:19:24,870 --> 00:19:30,410 The coordinate of that little mass particle is x1, 0, y1. 225 00:19:30,410 --> 00:19:33,380 So in general, this is x1 squared plus yi squared, 226 00:19:33,380 --> 00:19:34,030 but that's 0. 227 00:19:34,030 --> 00:19:36,130 So it's just mx1 squared. 228 00:19:47,220 --> 00:19:49,640 So that's where all these things come from. 229 00:19:49,640 --> 00:19:56,900 Now let's kind of look at-- the units of these things 230 00:19:56,900 --> 00:20:01,040 are always mass times length squared. 231 00:20:01,040 --> 00:20:08,500 The diagonal terms the Ixx, Iyy, Izz terms will all 232 00:20:08,500 --> 00:20:12,190 be of the form of something squared. 233 00:20:12,190 --> 00:20:17,080 And what it is is it's always just the distance 234 00:20:17,080 --> 00:20:24,360 of the mass particles from the axis of rotation, call that r. 235 00:20:24,360 --> 00:20:27,300 It's just a summation of this perpendicular distance 236 00:20:27,300 --> 00:20:29,020 from the axis of rotation. 237 00:20:29,020 --> 00:20:32,990 For Izz, the axis of rotation is z. 238 00:20:32,990 --> 00:20:36,190 And this piece here is always the distance 239 00:20:36,190 --> 00:20:41,720 squared that the mass particle is from the axis of rotation. 240 00:20:41,720 --> 00:20:47,050 Now that's the set up for today. 241 00:20:47,050 --> 00:20:49,880 And when we want to get onto the real meat of the discussion, 242 00:20:49,880 --> 00:20:56,040 talking about things like what principal axes are 243 00:20:56,040 --> 00:20:56,980 and so forth. 244 00:21:10,540 --> 00:21:13,060 Now we're going to move on to the part of this conversation 245 00:21:13,060 --> 00:21:14,425 about principal axes. 246 00:21:29,820 --> 00:21:32,900 So a really important point. 247 00:21:36,280 --> 00:21:43,840 For every rigid body, even weird ones. 248 00:21:43,840 --> 00:21:48,320 For every rigid body, there is a coordinate system 249 00:21:48,320 --> 00:21:53,030 that you can attach to this body, an xyz set of orthogonal 250 00:21:53,030 --> 00:21:56,210 coordinates that you can fix in this body such 251 00:21:56,210 --> 00:22:02,455 that you can make this inertia matrix be diagonal. 252 00:22:05,150 --> 00:22:07,410 Any weird body at all, there is a set 253 00:22:07,410 --> 00:22:10,810 of coordinates that if you use that set, 254 00:22:10,810 --> 00:22:13,570 this matrix turns out to be diagonal. 255 00:22:13,570 --> 00:22:20,640 And what that means in dynamics terms is if you then 256 00:22:20,640 --> 00:22:26,790 rotate the object about one of those axes, 257 00:22:26,790 --> 00:22:29,150 it will be dynamically balanced. 258 00:22:31,860 --> 00:22:39,200 And so when you rotate a shaft-- this is an object for which I 259 00:22:39,200 --> 00:22:43,960 know-- it's a circular uniform disk, 260 00:22:43,960 --> 00:22:47,140 one of the axes through the center is a principal axis. 261 00:22:47,140 --> 00:22:51,500 And if I rotate the object about that-- this one the hole's kind 262 00:22:51,500 --> 00:22:53,690 of out around so it wobbles a bit 263 00:22:53,690 --> 00:22:58,450 but if I rotate it about that it will 264 00:22:58,450 --> 00:23:02,600 make no torques about this axis that are because it's 265 00:23:02,600 --> 00:23:05,120 unbalanced. 266 00:23:05,120 --> 00:23:08,590 It's essentially the dynamic meaning of principal axis-- 267 00:23:08,590 --> 00:23:11,500 an axis about which you can rotate the thing and it'll just 268 00:23:11,500 --> 00:23:18,030 be smooth, no away-from-the-axis torques. 269 00:23:18,030 --> 00:23:19,790 So that's so those are pretty important. 270 00:23:19,790 --> 00:23:22,450 We want to know what those are for rigid bodies 271 00:23:22,450 --> 00:23:24,160 and how to find them. 272 00:23:24,160 --> 00:23:26,590 So there's a mathematical way to find them 273 00:23:26,590 --> 00:23:30,630 and you can just read about in the textbook. 274 00:23:30,630 --> 00:23:37,770 It's sort of a-- I'm trying to think of the mathematical term. 275 00:23:37,770 --> 00:23:40,690 I forgot it. 276 00:23:40,690 --> 00:23:42,220 But what I want to teach you today 277 00:23:42,220 --> 00:23:44,890 is for many, many objects, if you just 278 00:23:44,890 --> 00:23:48,000 look at their symmetries, you can figure out by common sense 279 00:23:48,000 --> 00:23:49,452 where their principal axes are. 280 00:23:49,452 --> 00:23:51,035 So that's what we're going to do next. 281 00:24:01,410 --> 00:24:04,660 We started off saying that we're talking about bodies that 282 00:24:04,660 --> 00:24:07,520 are either rotating about their centers of mass 283 00:24:07,520 --> 00:24:11,030 or about some other fixed point. 284 00:24:11,030 --> 00:24:18,470 So we're going to define our principal axes 285 00:24:18,470 --> 00:24:23,990 assuming we are rotating about the center of mass. 286 00:24:23,990 --> 00:24:28,120 There is an easy method known as the parallel axis theorem 287 00:24:28,120 --> 00:24:29,920 to get to any other point. 288 00:24:29,920 --> 00:24:33,075 So why do we care about doing it about the center of mass? 289 00:24:37,950 --> 00:24:40,200 Why is that useful to know about the properties 290 00:24:40,200 --> 00:24:41,930 around the center for dynamics purposes? 291 00:24:41,930 --> 00:24:43,560 What kind of dynamics problems do you 292 00:24:43,560 --> 00:24:46,720 care about the center of mass? 293 00:24:46,720 --> 00:24:48,327 Rotation about the center of mass. 294 00:24:48,327 --> 00:24:49,243 AUDIENCE: [INAUDIBLE]. 295 00:24:52,820 --> 00:24:55,070 PROFESSOR: So when I throw this thing in the air, it's 296 00:24:55,070 --> 00:24:57,320 rotating, it's translating, and when it rotates, 297 00:24:57,320 --> 00:24:59,260 what's it's rotating about? 298 00:24:59,260 --> 00:25:00,302 AUDIENCE: Center of mass. 299 00:25:00,302 --> 00:25:01,385 PROFESSOR: Center of mass. 300 00:25:01,385 --> 00:25:03,185 So there's just lots and lots of problem 301 00:25:03,185 --> 00:25:08,239 in which in fact the rotation is going to occur around 302 00:25:08,239 --> 00:25:09,030 the center of mass. 303 00:25:09,030 --> 00:25:12,830 Any time the thing is off there and there's nothing 304 00:25:12,830 --> 00:25:14,940 constraining its rotational motion, 305 00:25:14,940 --> 00:25:17,980 the rotation will be about the center of mass. 306 00:25:17,980 --> 00:25:24,250 So that's a good enough reason, and a very practical reason 307 00:25:24,250 --> 00:25:26,470 then for computing these things or knowing 308 00:25:26,470 --> 00:25:30,090 how to find these things about the center of mass. 309 00:25:30,090 --> 00:25:34,690 All right so we're now going to do principal axes, 310 00:25:34,690 --> 00:25:40,270 and we're going to teach you some symmetry rules. 311 00:25:45,160 --> 00:25:48,510 Maybe first a little common sense. 312 00:25:48,510 --> 00:25:52,880 This thing-- is this a principal axis? 313 00:25:52,880 --> 00:25:53,980 No way, right? 314 00:25:53,980 --> 00:25:57,210 And we know for a fact that it has off-diagonal terms 315 00:25:57,210 --> 00:25:59,700 because I've defined my coordinates as being 316 00:25:59,700 --> 00:26:03,840 an x, y, z, like this, and this is often at some strange angle. 317 00:26:03,840 --> 00:26:08,970 So as a practical matter, how could I 318 00:26:08,970 --> 00:26:12,880 alter this object so that this is a principal axis? 319 00:26:12,880 --> 00:26:14,689 Essentially, how would you balance this? 320 00:26:14,689 --> 00:26:15,605 AUDIENCE: [INAUDIBLE]. 321 00:26:19,480 --> 00:26:22,200 PROFESSOR: Take the mass off, fix it. 322 00:26:22,200 --> 00:26:23,385 [LAUGHTER] 323 00:26:23,385 --> 00:26:24,766 Or? 324 00:26:24,766 --> 00:26:27,730 All right. 325 00:26:27,730 --> 00:26:29,390 Where do I put it? 326 00:26:29,390 --> 00:26:31,692 Down here? 327 00:26:31,692 --> 00:26:32,775 Ah, you want it like this. 328 00:26:37,117 --> 00:26:40,380 This is steel going into aluminum, 329 00:26:40,380 --> 00:26:42,755 I have to be careful that I don't strip the threads. 330 00:26:50,170 --> 00:26:50,670 There we go. 331 00:26:54,610 --> 00:26:59,070 And now test one, this is axis of rotation, 332 00:26:59,070 --> 00:27:00,840 is it dynamically in balance? 333 00:27:00,840 --> 00:27:03,830 Smooth as silk. 334 00:27:03,830 --> 00:27:07,210 This now is a principal axis. 335 00:27:07,210 --> 00:27:09,110 Its mass distribution is symmetric. 336 00:27:12,790 --> 00:27:16,070 Where is there a plane of symmetry for this problem, 337 00:27:16,070 --> 00:27:17,537 for this object? 338 00:27:17,537 --> 00:27:20,078 That's what we're going to talk about now, planes of symmetry 339 00:27:20,078 --> 00:27:23,430 and axis of symmetry. 340 00:27:23,430 --> 00:27:26,292 Ah, you're saying one like back? 341 00:27:26,292 --> 00:27:28,000 You're right, that's a plane of symmetry. 342 00:27:28,000 --> 00:27:29,416 Is there another one for this one? 343 00:27:32,150 --> 00:27:32,930 Like this. 344 00:27:32,930 --> 00:27:35,650 So there's two planes of symmetry. 345 00:27:35,650 --> 00:27:40,250 So let's say come up with some symmetry rules. 346 00:27:44,380 --> 00:28:16,290 So this first one is there exists-- 347 00:28:16,290 --> 00:28:19,450 by this I mean diagonal, it's a diagonal matrix. 348 00:28:19,450 --> 00:28:21,980 You can find a set of orthogonal axes 349 00:28:21,980 --> 00:28:24,690 such that this matrix is diagonal. 350 00:28:24,690 --> 00:28:27,545 That's what we mean when we go to find principal axes. 351 00:28:30,990 --> 00:28:32,615 And pick a board here-- rules. 352 00:28:50,560 --> 00:28:52,186 And we're going to have three of them, 353 00:28:52,186 --> 00:28:53,560 so just leave room on your paper. 354 00:28:53,560 --> 00:28:54,400 We're going to build these up. 355 00:28:54,400 --> 00:28:56,590 And I'm going to talk about it, and then you add another one 356 00:28:56,590 --> 00:28:57,190 and so forth. 357 00:29:12,540 --> 00:29:13,320 So rule one. 358 00:29:21,375 --> 00:29:24,826 AUDIENCE: [INAUDIBLE] the inertia matrix [INAUDIBLE]. 359 00:29:24,826 --> 00:29:26,798 PROFESSOR: What is the inertia matrix-- 360 00:29:26,798 --> 00:29:28,780 AUDIENCE: No, I mean the second one. 361 00:29:28,780 --> 00:29:30,520 PROFESSOR: This thing with the diagonals? 362 00:29:30,520 --> 00:29:33,500 This is going to have all the off-diagonal terms are zeroes. 363 00:29:37,310 --> 00:29:38,470 That's what I mean. 364 00:29:38,470 --> 00:29:43,070 It's a diagonal matrix, the mass moments of inertia and products 365 00:29:43,070 --> 00:29:48,160 of inertia-- you only have Ixx, Iyy, Izz if you properly 366 00:29:48,160 --> 00:29:51,620 pick this set of orthogonal coordinates attached 367 00:29:51,620 --> 00:29:54,060 to the body-- if you pick them right. 368 00:29:54,060 --> 00:29:56,510 What it means if then you spin that object about one 369 00:29:56,510 --> 00:30:00,560 of those axes, it's in balance. 370 00:30:00,560 --> 00:30:41,080 So rule one, if there it is an axis of symmetry-- Remember 371 00:30:41,080 --> 00:30:44,385 the caveat here is we're talking about uniform density objects 372 00:30:44,385 --> 00:30:48,156 or at least objects in which the density is symmetrically 373 00:30:48,156 --> 00:30:48,655 distributed. 374 00:30:51,200 --> 00:30:53,860 So the geometric symmetry and mass symmetry 375 00:30:53,860 --> 00:30:56,690 mean the same thing. 376 00:30:56,690 --> 00:30:57,630 What does that mean? 377 00:30:57,630 --> 00:30:58,840 So axis of symmetry. 378 00:31:03,440 --> 00:31:06,455 Does this have an axis of symmetry? 379 00:31:09,680 --> 00:31:10,689 Where? 380 00:31:10,689 --> 00:31:14,601 AUDIENCE: It has multiple ones down the middle 381 00:31:14,601 --> 00:31:16,527 and it's also symmetric [INAUDIBLE]. 382 00:31:16,527 --> 00:31:18,110 PROFESSOR: That's a plane of symmetry, 383 00:31:18,110 --> 00:31:20,625 the [? other ones you think, ?] axis of symmetry. 384 00:31:20,625 --> 00:31:22,725 AUDIENCE: Well I mean the line right through it. 385 00:31:22,725 --> 00:31:23,600 PROFESSOR: Which way? 386 00:31:23,600 --> 00:31:25,001 AUDIENCE: Well any. 387 00:31:25,001 --> 00:31:25,935 PROFESSOR: Nope. 388 00:31:25,935 --> 00:31:27,650 AUDIENCE: Perpendicular to it. 389 00:31:27,650 --> 00:31:29,530 PROFESSOR: Axis of symmetry means 390 00:31:29,530 --> 00:31:33,800 that there is a mirror image from that point-- 391 00:31:33,800 --> 00:31:36,540 that point mirror image over here is always identical. 392 00:31:36,540 --> 00:31:40,670 But you think any from angle, it doesn't matter where you start, 393 00:31:40,670 --> 00:31:42,250 it's reflected through the axis. 394 00:31:42,250 --> 00:31:44,590 So things that are circularly shaped 395 00:31:44,590 --> 00:31:47,760 tend to have axes of symmetry. 396 00:31:47,760 --> 00:31:51,680 So this has an axis of symmetry. 397 00:31:51,680 --> 00:31:54,695 Everything is reflected exactly just across the axes, 398 00:31:54,695 --> 00:31:58,090 you don't have to pick any particular line. 399 00:31:58,090 --> 00:32:02,510 So what it's saying if it's an axis of symmetry, 400 00:32:02,510 --> 00:32:05,630 then it is a principal axis, and rotation about that 401 00:32:05,630 --> 00:32:08,440 axis-- the things will be nice perfectly in balance. 402 00:32:08,440 --> 00:32:12,340 If you go through all the hairy details of calculating 403 00:32:12,340 --> 00:32:16,490 the hard way, the Ixx, Iyy, et cetera terms-- 404 00:32:16,490 --> 00:32:18,600 all of the off diagonal terms will come out zero. 405 00:32:18,600 --> 00:32:23,230 And the reason is that for every mass particle over here, 406 00:32:23,230 --> 00:32:26,070 let's say there's a little bit right here in the corner, 407 00:32:26,070 --> 00:32:30,130 there is one exactly like it on the other side. 408 00:32:30,130 --> 00:32:33,450 So if this one we're trying to bend this thing up as it 409 00:32:33,450 --> 00:32:37,080 spins around, this one over here is telling it to come back. 410 00:32:37,080 --> 00:32:38,446 Yeah? 411 00:32:38,446 --> 00:32:42,293 AUDIENCE: How come this is not an axis, if you hold it-- 412 00:32:42,293 --> 00:32:42,793 yeah-- 413 00:32:42,793 --> 00:32:43,759 [INTERPOSING VOICES] 414 00:32:43,759 --> 00:32:45,700 PROFESSOR: So if you do this. 415 00:32:45,700 --> 00:32:48,470 So I think it's kind of just the definition we're getting to 416 00:32:48,470 --> 00:32:50,590 of an axis of symmetry. 417 00:32:50,590 --> 00:32:58,490 Because in simple terms, what the object looks 418 00:32:58,490 --> 00:33:01,310 like this way is it's a narrow object, 419 00:33:01,310 --> 00:33:04,170 it is in fact a mirror image reflection. 420 00:33:04,170 --> 00:33:06,770 But it looks different when you go to this angle. 421 00:33:06,770 --> 00:33:09,020 And it looks different when you go to this angle 422 00:33:09,020 --> 00:33:13,370 so it's not an axis of symmetry, that's a plane of symmetry. 423 00:33:13,370 --> 00:33:14,980 We're going talk about that next. 424 00:33:14,980 --> 00:33:17,480 So axes of symmetry-- the thing essentially 425 00:33:17,480 --> 00:33:23,050 looks the same at all orientations about that axis. 426 00:33:23,050 --> 00:33:27,130 But it is certainly dynamically balanced. 427 00:33:27,130 --> 00:33:30,770 Every mass bit here balances one over there, it doesn't wobble. 428 00:33:30,770 --> 00:33:31,640 All right. 429 00:33:31,640 --> 00:33:33,990 So that's for sure true. 430 00:33:33,990 --> 00:33:42,890 Now keep in mind that exists a set of orthogonal axes. 431 00:33:42,890 --> 00:33:44,890 So that means once you find one, you 432 00:33:44,890 --> 00:33:47,970 know that the other two are perpendicular to it 433 00:33:47,970 --> 00:33:50,390 and perpendicular to each other. 434 00:33:50,390 --> 00:33:54,560 So for this system, when I got the x and y here, 435 00:33:54,560 --> 00:33:56,810 there has to be an x and y perpendicular 436 00:33:56,810 --> 00:34:00,690 that that are the other two. 437 00:34:00,690 --> 00:34:02,364 And because it's actually symmetric, 438 00:34:02,364 --> 00:34:04,280 it doesn't matter where you out them, you just 439 00:34:04,280 --> 00:34:07,800 could say it's these two, these two, it doesn't matter. 440 00:34:07,800 --> 00:34:09,300 You just have to decide where you're 441 00:34:09,300 --> 00:34:12,989 going to put them on the object, they're all the same. 442 00:34:12,989 --> 00:34:17,900 So for this axis symmetry, they're just two more. 443 00:34:17,900 --> 00:34:22,610 They're going to be in the body perpendicular to the first. 444 00:34:22,610 --> 00:34:26,699 Now we're going to do about g so the book uses g 445 00:34:26,699 --> 00:34:29,281 to talk about the center of mass, so I'll use g. 446 00:34:29,281 --> 00:34:30,739 So the center of mass of this thing 447 00:34:30,739 --> 00:34:35,469 is right in the middle of this and in the middle here. 448 00:34:35,469 --> 00:34:38,010 So it's inside this body. 449 00:34:38,010 --> 00:34:41,370 Right in the dead center of it. 450 00:34:41,370 --> 00:34:44,585 So the three principal axes are an orthogonal set, one of which 451 00:34:44,585 --> 00:34:45,960 goes through it and the other two 452 00:34:45,960 --> 00:34:47,790 are embedded in it right angles. 453 00:34:47,790 --> 00:34:49,989 So this is the simplest rule. 454 00:34:49,989 --> 00:34:53,480 If you have an axis of symmetry, you know right way just 455 00:34:53,480 --> 00:34:55,980 about everything you need to know about the principal axes 456 00:34:55,980 --> 00:34:56,659 of the body. 457 00:35:09,060 --> 00:35:09,630 Shoot. 458 00:35:09,630 --> 00:35:12,782 AUDIENCE: It seems like then there would only 459 00:35:12,782 --> 00:35:16,470 be axis of symmetry in objects that are either round 460 00:35:16,470 --> 00:35:17,720 or spherical, is that correct? 461 00:35:17,720 --> 00:35:19,470 PROFESSOR: Pretty much. 462 00:35:19,470 --> 00:35:21,610 AUDIENCE: So it's a very limited special case. 463 00:35:21,610 --> 00:35:25,870 PROFESSOR: Yeah but there sure appear an awful lot in machines 464 00:35:25,870 --> 00:35:28,930 because they have this beautiful property of just being balanced 465 00:35:28,930 --> 00:35:31,800 in all directions. 466 00:35:31,800 --> 00:35:34,180 So everything rotating in the world 467 00:35:34,180 --> 00:35:38,595 tends to have an almost perfect axial symmetry. 468 00:35:44,230 --> 00:35:45,650 Just thinking about order here. 469 00:35:45,650 --> 00:35:48,850 Let's do this and then I'm going to go do a couple of examples. 470 00:35:48,850 --> 00:36:09,170 The second rule-- if you have one plane of symmetry, 471 00:36:09,170 --> 00:36:11,070 a plane of symmetry. 472 00:36:11,070 --> 00:36:13,420 So this is kind of the opposite direction, 473 00:36:13,420 --> 00:36:15,670 in this you have the least information. 474 00:36:15,670 --> 00:36:18,867 Here's an object, does it have a plane of symmetry? 475 00:36:18,867 --> 00:36:19,450 AUDIENCE: Yes. 476 00:36:19,450 --> 00:36:20,158 PROFESSOR: Where? 477 00:36:20,158 --> 00:36:21,990 AUDIENCE: Straight through [INAUDIBLE]. 478 00:36:21,990 --> 00:36:23,082 PROFESSOR: Show me. 479 00:36:23,082 --> 00:36:24,151 That cut. 480 00:36:24,151 --> 00:36:24,650 All right. 481 00:36:24,650 --> 00:36:27,620 So I've got these little dotted marks on this thing. 482 00:36:27,620 --> 00:36:32,320 So if I slice through that, I create 483 00:36:32,320 --> 00:36:33,920 two pieces that are identical. 484 00:36:33,920 --> 00:36:35,320 So that's a plane of symmetry. 485 00:36:35,320 --> 00:36:41,710 There is image match across that plane at every point. 486 00:36:41,710 --> 00:36:45,350 So if I have a plane of symmetry, 487 00:36:45,350 --> 00:36:48,430 what do you think you can say about a principal axis? 488 00:36:48,430 --> 00:36:50,085 One of the principal axis? 489 00:36:50,085 --> 00:36:51,540 AUDIENCE: It'll be on that plane. 490 00:36:51,540 --> 00:36:52,980 PROFESSOR: It'll be on that plane. 491 00:36:52,980 --> 00:36:55,229 That turns out to be true, but that's the second point 492 00:36:55,229 --> 00:36:56,268 I want to make. 493 00:36:56,268 --> 00:36:58,042 AUDIENCE: It might be perpendicular. 494 00:36:58,042 --> 00:37:00,500 PROFESSOR: She says there might be one perpendicular to it. 495 00:37:00,500 --> 00:37:02,125 And that's the one I was searching for, 496 00:37:02,125 --> 00:37:03,680 you're right too. 497 00:37:03,680 --> 00:37:07,000 There's going to be a principal axis 498 00:37:07,000 --> 00:37:09,380 that's perpendicular to that plane of symmetry, 499 00:37:09,380 --> 00:37:12,470 and since we want to define our moments of inertia 500 00:37:12,470 --> 00:37:14,230 for this to get started with with respect 501 00:37:14,230 --> 00:37:15,817 to g, where will it pass through? 502 00:37:15,817 --> 00:37:16,650 AUDIENCE: Through g. 503 00:37:16,650 --> 00:37:18,910 PROFESSOR: G, kind of by definition. 504 00:37:18,910 --> 00:37:20,800 So we're going to have it pass through g. 505 00:37:20,800 --> 00:37:33,530 If there's a plane of symmetry, then there 506 00:37:33,530 --> 00:37:47,820 is a principal axis perpendicular to it. 507 00:37:53,550 --> 00:37:58,030 And we'll0 define it, for the purposes of our discussion, 508 00:37:58,030 --> 00:38:00,080 we'll just let it pass through g. 509 00:38:03,540 --> 00:38:06,190 It doesn't have to, but that's how we're 510 00:38:06,190 --> 00:38:07,820 going about this discussion. 511 00:38:07,820 --> 00:38:14,720 Let it pass through the center of mass, this point we call g. 512 00:38:14,720 --> 00:38:18,440 All right, so that means that there's 513 00:38:18,440 --> 00:38:21,690 a center of mass in this thing. 514 00:38:21,690 --> 00:38:24,990 And I'm just guessing roughly where it is. 515 00:38:24,990 --> 00:38:28,040 But in fact if I hung this thing up here like this, 516 00:38:28,040 --> 00:38:34,140 and let gravity find it's natural hanging angle, 517 00:38:34,140 --> 00:38:36,100 and I drew a plumb line down here, 518 00:38:36,100 --> 00:38:40,320 just drew a line that the string would take with a plumb bob. 519 00:38:40,320 --> 00:38:43,310 Then I went to some other point and did it again 520 00:38:43,310 --> 00:38:46,140 and hung a plumb bob on it drew the line-- where they intersect 521 00:38:46,140 --> 00:38:48,487 is the center of mass. 522 00:38:48,487 --> 00:38:50,570 And then you know since it's got symmetry this way 523 00:38:50,570 --> 00:38:51,990 that's in the middle. 524 00:38:51,990 --> 00:38:54,980 So I guesses that that's about where it is. 525 00:38:54,980 --> 00:38:58,750 And there then is a principal axis 526 00:38:58,750 --> 00:39:02,960 of this object that's perpendicular to it, 527 00:39:02,960 --> 00:39:06,060 passing through the plane-- perpendicular to the plane. 528 00:39:06,060 --> 00:39:08,180 And that means now there's two more. 529 00:39:08,180 --> 00:39:12,540 But this gets a little more difficult, and I have no clue. 530 00:39:12,540 --> 00:39:15,390 There are two more, because the principal axes always 531 00:39:15,390 --> 00:39:16,840 come in an orthogonal set. 532 00:39:16,840 --> 00:39:18,830 So now that I know one, I know that there's 533 00:39:18,830 --> 00:39:24,330 two more somewhere around oriented 534 00:39:24,330 --> 00:39:25,880 this way, someplace in this plane. 535 00:39:25,880 --> 00:39:28,210 Maybe one like that, maybe one like this, 536 00:39:28,210 --> 00:39:31,670 such that if you spun it about one of those axes, 537 00:39:31,670 --> 00:39:33,809 it'd be in balance. 538 00:39:33,809 --> 00:39:35,350 You can see that gets a little messy, 539 00:39:35,350 --> 00:39:37,535 I can't guess where it is. 540 00:39:37,535 --> 00:39:39,410 And so there are ways to find it, one of them 541 00:39:39,410 --> 00:39:41,570 would be doing an experiment, seeing which 542 00:39:41,570 --> 00:39:43,570 axis it spins nicely around. 543 00:39:43,570 --> 00:39:45,154 So that's the second rule. 544 00:39:47,820 --> 00:39:50,290 But even just with that one plane of symmetry, 545 00:39:50,290 --> 00:39:53,600 you get some pretty good insight. 546 00:39:53,600 --> 00:39:58,744 Now what if there are two planes of symmetry? 547 00:39:58,744 --> 00:39:59,505 Yeah? 548 00:39:59,505 --> 00:40:01,630 AUDIENCE: You said that the principal axis does not 549 00:40:01,630 --> 00:40:05,290 have to pass through the center of mass? 550 00:40:05,290 --> 00:40:09,606 PROFESSOR: No, I'm saying it doesn't have to pass 551 00:40:09,606 --> 00:40:10,730 through the center of mass. 552 00:40:10,730 --> 00:40:12,590 The question is? 553 00:40:12,590 --> 00:40:15,285 AUDIENCE: How could it not because it 554 00:40:15,285 --> 00:40:17,990 becomes stable [INAUDIBLE]. 555 00:40:17,990 --> 00:40:20,634 PROFESSOR: So she's saying it'd be unstable if it's not 556 00:40:20,634 --> 00:40:21,550 passing through there. 557 00:40:21,550 --> 00:40:23,965 Like if I put an axle through this wheel, 558 00:40:23,965 --> 00:40:27,799 and I spun it about some point out here, 559 00:40:27,799 --> 00:40:29,840 you don't know because I had that shaker in here, 560 00:40:29,840 --> 00:40:32,520 that thing is going to shake like crazy. 561 00:40:32,520 --> 00:40:36,540 But does it produce unbalanced torques 562 00:40:36,540 --> 00:40:39,070 about my rotation point? 563 00:40:41,850 --> 00:40:44,770 It produces centrifugal force that you'll feel like crazy, 564 00:40:44,770 --> 00:40:48,030 but does it produce a torque that you'd 565 00:40:48,030 --> 00:40:51,380 have to resist with some static torque? 566 00:40:51,380 --> 00:40:52,130 What do you think? 567 00:40:55,030 --> 00:40:56,270 It won't. 568 00:40:56,270 --> 00:40:58,870 So there's a nuance to this unbalanced thing, 569 00:40:58,870 --> 00:41:02,060 and I was going to get to it, probably next lecture, 570 00:41:02,060 --> 00:41:07,140 but that is the when we say an object is dynamically balanced, 571 00:41:07,140 --> 00:41:12,170 we mean that it doesn't have any unbalanced torques. 572 00:41:12,170 --> 00:41:18,900 If it is statically balanced, it's 573 00:41:18,900 --> 00:41:22,250 rotating about its center of mass. 574 00:41:22,250 --> 00:41:25,600 But you can be statically unbalanced 575 00:41:25,600 --> 00:41:29,580 and let it go around this axis, but it'll produce no torques. 576 00:41:29,580 --> 00:41:31,210 It's still dynamically balanced. 577 00:41:31,210 --> 00:41:42,430 And the angular momentum of an object which is rotating-- 578 00:41:42,430 --> 00:41:44,490 And this we know has a principal axis here, 579 00:41:44,490 --> 00:41:46,550 I just moved it off to the side. 580 00:41:46,550 --> 00:41:50,290 It's rotating about this. 581 00:41:50,290 --> 00:41:55,440 It is dynamically balanced, and if you computed 582 00:41:55,440 --> 00:42:01,440 about this point now the Ixy, Ixz off-diagonal entries 583 00:42:01,440 --> 00:42:05,360 in that moment of inertia matrix, they're all 0. 584 00:42:05,360 --> 00:42:08,640 You'd get no unbalanced torques, but you 585 00:42:08,640 --> 00:42:10,860 do have an unbalanced centrifugal force 586 00:42:10,860 --> 00:42:12,090 as this thing goes around. 587 00:42:12,090 --> 00:42:13,631 And we'll talk a bit more about that. 588 00:42:13,631 --> 00:42:15,595 AUDIENCE: Is that sort of analogous 589 00:42:15,595 --> 00:42:19,032 to the homework problem we had a few weeks back 590 00:42:19,032 --> 00:42:22,085 with the motorcycle wheel and you just had mass on one side 591 00:42:22,085 --> 00:42:22,960 and not on the other? 592 00:42:22,960 --> 00:42:23,720 PROFESSOR: Right. 593 00:42:23,720 --> 00:42:25,510 So she's asking about the motorcycle wheel problem 594 00:42:25,510 --> 00:42:27,500 where we had that little mass that got there. 595 00:42:27,500 --> 00:42:30,510 I think in the next lecture I'm going to come back 596 00:42:30,510 --> 00:42:32,790 to that problem just so we could tie 597 00:42:32,790 --> 00:42:37,500 a bow around this whole thing and understand why it's 598 00:42:37,500 --> 00:42:39,030 unbalanced, how you can balance it, 599 00:42:39,030 --> 00:42:41,170 and the difference between static unbalance 600 00:42:41,170 --> 00:42:42,880 and dynamic unbalance. 601 00:42:42,880 --> 00:42:45,510 But today, we're talking about symmetry rules. 602 00:42:48,124 --> 00:42:50,290 Finally, so I was saying, let's talk about something 603 00:42:50,290 --> 00:42:51,890 that has two planes of symmetry. 604 00:42:51,890 --> 00:42:53,810 This actually has three planes of symmetry, 605 00:42:53,810 --> 00:42:55,060 but we'll settle for two. 606 00:42:59,820 --> 00:43:02,700 Pick a plane of symmetry for this object. 607 00:43:02,700 --> 00:43:07,444 If I pick-- OK, so she picked the one-- slice it this way. 608 00:43:07,444 --> 00:43:08,360 What about the second? 609 00:43:11,135 --> 00:43:13,057 Like that. 610 00:43:13,057 --> 00:43:13,890 Then a third, right? 611 00:43:13,890 --> 00:43:15,014 There's even one like that. 612 00:43:15,014 --> 00:43:16,980 So this has three planes of symmetry. 613 00:43:16,980 --> 00:43:25,510 But if you have two planes of symmetry that intersect, 614 00:43:25,510 --> 00:43:29,820 that are orthogonal to one another, 615 00:43:29,820 --> 00:43:33,000 what do you think you can say about that line 616 00:43:33,000 --> 00:43:34,894 of intersection? 617 00:43:34,894 --> 00:43:36,310 AUDIENCE: It's the principal axis. 618 00:43:36,310 --> 00:43:39,100 PROFESSOR: It sure is. 619 00:43:39,100 --> 00:43:41,670 And it's probably right through g. 620 00:43:41,670 --> 00:43:57,490 So if you have two planes of symmetry-- 621 00:43:57,490 --> 00:43:59,352 Now make them orthogonal. 622 00:43:59,352 --> 00:44:01,060 You can make all sorts of symmetry rules, 623 00:44:01,060 --> 00:44:04,230 and I'm just picking these three to help you out. 624 00:44:04,230 --> 00:44:07,340 This just to help you see principal axes. 625 00:44:07,340 --> 00:44:12,990 If you have two orthogonal planes 626 00:44:12,990 --> 00:44:46,050 of symmetry their intersection-- and once you know that, 627 00:44:46,050 --> 00:44:48,110 then you go back to rule two. 628 00:44:48,110 --> 00:44:50,236 And it tells you everything else you need to know. 629 00:44:50,236 --> 00:44:51,860 Because you have one plane of symmetry, 630 00:44:51,860 --> 00:44:57,080 you know there is a principal axis perpendicular to it. 631 00:44:57,080 --> 00:44:59,720 Well if you have two planes of symmetry, the rule still holds. 632 00:44:59,720 --> 00:45:01,710 There's one perpendicular to each one. 633 00:45:01,710 --> 00:45:05,010 The intersection, let's say, of this plane of symmetry 634 00:45:05,010 --> 00:45:07,910 and this plane of symmetry is a line 635 00:45:07,910 --> 00:45:11,200 which goes right through the center of this thing that way. 636 00:45:11,200 --> 00:45:13,710 So there's a principal axis this way. 637 00:45:13,710 --> 00:45:16,200 But since there is a plane of symmetry here, 638 00:45:16,200 --> 00:45:19,350 there must also be a principal axis perpendicular to it. 639 00:45:19,350 --> 00:45:23,730 So sure enough, three principal axes for this thing 640 00:45:23,730 --> 00:45:28,420 are through the center, perpendicular this way, 641 00:45:28,420 --> 00:45:29,550 perpendicular that way. 642 00:45:29,550 --> 00:45:30,450 You instantly know. 643 00:45:30,450 --> 00:45:32,210 Two planes of symmetry-- you instantly 644 00:45:32,210 --> 00:45:37,400 know where the three orthogonal principal axes are that past 645 00:45:37,400 --> 00:45:38,783 through the center of mass. 646 00:45:38,783 --> 00:45:39,769 Yeah? 647 00:45:39,769 --> 00:45:40,269 [? 648 00:45:40,269 --> 00:45:42,094 AUDIENCE: Does this all apply ?] just like a constant mass 649 00:45:42,094 --> 00:45:43,040 throughout? 650 00:45:43,040 --> 00:45:48,150 PROFESSOR: Not constant, symmetrically 651 00:45:48,150 --> 00:45:51,150 distributed density. 652 00:45:51,150 --> 00:45:55,280 Right so I'm choosing my words carefully so 653 00:45:55,280 --> 00:45:57,970 that I succeed in the following-- 654 00:45:57,970 --> 00:46:01,130 that the planes defining mass symmetry 655 00:46:01,130 --> 00:46:05,479 will be the same as the planes defining geometric similarity. 656 00:46:05,479 --> 00:46:07,770 But you actually don't have to have a constant density, 657 00:46:07,770 --> 00:46:11,420 it just has to be distributed so that what I just said is true. 658 00:46:11,420 --> 00:46:13,290 So that the geometric symmetries are 659 00:46:13,290 --> 00:46:17,527 the same as the mass distribution symmetries. 660 00:46:17,527 --> 00:46:19,610 All right so those are my three rules of symmetry. 661 00:46:19,610 --> 00:46:20,651 You could make up others. 662 00:46:20,651 --> 00:46:22,630 Those are the three that I've made up 663 00:46:22,630 --> 00:46:24,315 to help you see objects. 664 00:46:31,230 --> 00:46:35,110 That object, it's a circular disk 665 00:46:35,110 --> 00:46:42,430 put on top of another object such that their centers 666 00:46:42,430 --> 00:46:45,290 of mass line up. 667 00:46:45,290 --> 00:46:50,151 Where are the principal axes of this object using those rules? 668 00:46:55,930 --> 00:46:58,180 If you think you know one, tell me. 669 00:47:05,124 --> 00:47:06,674 AUDIENCE: Through the middle. 670 00:47:06,674 --> 00:47:08,590 PROFESSOR: Through the middle of both of them. 671 00:47:08,590 --> 00:47:11,498 Probably, good guess. 672 00:47:11,498 --> 00:47:12,426 How about another one? 673 00:47:16,610 --> 00:47:20,280 Where does this thing have planes of symmetry? 674 00:47:20,280 --> 00:47:23,640 AUDIENCE: So there's a plane of symmetry if you cut it in half. 675 00:47:23,640 --> 00:47:26,520 [INAUDIBLE] cut it in half, [INAUDIBLE]. 676 00:47:26,520 --> 00:47:27,960 PROFESSOR: OK, and? 677 00:47:27,960 --> 00:47:30,360 AUDIENCE: The other way. 678 00:47:30,360 --> 00:47:32,965 PROFESSOR: One like that, we've got all three. 679 00:47:32,965 --> 00:47:35,590 And if we're going to want it to go through the center of mass, 680 00:47:35,590 --> 00:47:37,964 then we're going to have to find where the center of mass 681 00:47:37,964 --> 00:47:40,470 is this way, but it's about there. 682 00:47:40,470 --> 00:47:42,500 So just using the symmetry ideas, 683 00:47:42,500 --> 00:47:45,800 you can right away figure out where these principal axes be. 684 00:47:45,800 --> 00:47:48,680 And that means from a dynamic point of view, 685 00:47:48,680 --> 00:47:50,960 if you spin it about one of those axes, 686 00:47:50,960 --> 00:47:52,530 it's nice and dynamically balanced. 687 00:47:52,530 --> 00:47:56,000 If you spin it off in some other weird direction 688 00:47:56,000 --> 00:47:59,830 is it necessarily dynamically balanced 689 00:47:59,830 --> 00:48:01,750 about that axis of spin? 690 00:48:08,730 --> 00:48:10,330 So let me restate that question. 691 00:48:16,620 --> 00:48:21,120 We know that this thing has an axis of symmetry principal 692 00:48:21,120 --> 00:48:25,690 axis through the center, and another one this way, 693 00:48:25,690 --> 00:48:26,890 and another one this way. 694 00:48:26,890 --> 00:48:28,650 And if I spin it about any one of those, 695 00:48:28,650 --> 00:48:30,240 it's dynamically balanced. 696 00:48:30,240 --> 00:48:35,940 But if I pick some other strange direction for the spin, 697 00:48:35,940 --> 00:48:40,180 and I spin it about that axis, will I feel unbalanced torques 698 00:48:40,180 --> 00:48:42,040 on this axle, on the bearings having 699 00:48:42,040 --> 00:48:44,080 to hold this thing in place? 700 00:48:44,080 --> 00:48:46,900 Yeah, you better believe it, this thing wobbles like crazy. 701 00:48:46,900 --> 00:48:56,030 So the principal axes are a property of the object, 702 00:48:56,030 --> 00:48:59,430 they're not a property of the angular momentum. 703 00:48:59,430 --> 00:49:04,567 The angular momentum comes then from multiplying 704 00:49:04,567 --> 00:49:06,150 the mass moment of inertia that you've 705 00:49:06,150 --> 00:49:11,930 determined times the actual rotation vector. 706 00:49:11,930 --> 00:49:15,590 And you'll find out then you get angular components of angular 707 00:49:15,590 --> 00:49:18,220 momentum that are not in the direction of spin, 708 00:49:18,220 --> 00:49:21,160 and as soon as that happens, you have unbalanced terms. 709 00:49:25,600 --> 00:49:28,235 I've got to get on to something else to help you do homework. 710 00:50:09,730 --> 00:50:18,960 So for my disk, with z coming out of the board, the Izz-- 711 00:50:18,960 --> 00:50:24,680 so let's say here's x, y, z coming out of the board-- Izz, 712 00:50:24,680 --> 00:50:28,530 the mass moment of inertia about this z axis 713 00:50:28,530 --> 00:50:35,070 is, from the basic definition, the summation of the mis, 714 00:50:35,070 --> 00:50:41,510 xi squared plus yi squared. 715 00:50:41,510 --> 00:50:43,900 It's just that for every little mass particle 716 00:50:43,900 --> 00:50:48,790 it's the radius squared away from the center of rotation. 717 00:50:48,790 --> 00:50:51,320 That's what the x squared plus y squared is. 718 00:50:51,320 --> 00:50:54,370 And we can turn this into an integral. 719 00:50:54,370 --> 00:50:57,890 It's the integral of r squared, that distance, 720 00:50:57,890 --> 00:51:01,570 times the little mass bit that's there. 721 00:51:01,570 --> 00:51:04,660 And that's the same. 722 00:51:04,660 --> 00:51:09,740 If you wanted to do the integral as x squared plus y squared dm. 723 00:51:12,880 --> 00:51:19,360 But to do this integral for a nice circular, symmetric disk, 724 00:51:19,360 --> 00:51:30,506 you can pick a little mass bit that has thickness dr and width 725 00:51:30,506 --> 00:51:32,510 rd theta. 726 00:51:32,510 --> 00:51:35,380 And this angle here is d theta. 727 00:51:38,390 --> 00:51:40,360 And that's a little bit of area. 728 00:51:40,360 --> 00:51:44,973 That's a little dA which has area r dr d theta. 729 00:51:48,400 --> 00:51:50,520 It's just length times width. 730 00:51:50,520 --> 00:51:53,280 When it's small enough, it's a little rectangle, 731 00:51:53,280 --> 00:51:56,280 and it has that area. 732 00:51:56,280 --> 00:52:02,500 And it has a volume, dV-- the volume of that thing 733 00:52:02,500 --> 00:52:05,960 is just the area times the thickness of it. 734 00:52:05,960 --> 00:52:10,500 So here's our disk here, but it has some thickness, 735 00:52:10,500 --> 00:52:12,840 and I'll call that h. 736 00:52:12,840 --> 00:52:20,500 So the volume is just h r dr d theta. 737 00:52:20,500 --> 00:52:25,355 And the mass, dm, is a density times dV. 738 00:52:31,330 --> 00:52:34,260 So I want to integrate this, all I 739 00:52:34,260 --> 00:52:37,450 have to integrate the integral then of r 740 00:52:37,450 --> 00:52:50,700 squared dm is the integral from 0 to 2pi, 0 to r of rho dV. 741 00:52:50,700 --> 00:52:57,890 Rho h-- oh, and I need an r squared-- r squared 742 00:52:57,890 --> 00:53:04,830 dV is rho h r dr d theta. 743 00:53:04,830 --> 00:53:08,170 So this is 1802 integrals, right? 744 00:53:08,170 --> 00:53:12,790 So is any of this a function of theta? 745 00:53:12,790 --> 00:53:14,334 No, so it's a trivial integral, you 746 00:53:14,334 --> 00:53:16,250 integrate that over theta, you just get theta, 747 00:53:16,250 --> 00:53:18,140 evaluate it 0 to 2pi. 748 00:53:18,140 --> 00:53:25,980 So this is 2pi rho h, can all come to the outside, integral 0 749 00:53:25,980 --> 00:53:33,550 to R of r cubed dr. And that ends up-- 750 00:53:33,550 --> 00:53:37,090 the r cubed goes to r to the 4th over 4. 751 00:53:37,090 --> 00:53:48,490 And the final result of this one is 2pi r to the 4th over 4 752 00:53:48,490 --> 00:53:54,300 rho h, and when new account for h times pi r squared 753 00:53:54,300 --> 00:53:57,030 is the volume times rho is the mass. 754 00:53:57,030 --> 00:54:03,260 This all works out to be m r squared over 2. 755 00:54:03,260 --> 00:54:08,800 So Izz-- so I needed to do this once for you. 756 00:54:08,800 --> 00:54:13,230 For simple things integrate, Izz in this case, 757 00:54:13,230 --> 00:54:15,650 you just integrate it out, account 758 00:54:15,650 --> 00:54:18,650 for all little mass bits, that is the mass moment of inertia 759 00:54:18,650 --> 00:54:25,390 with respect to the axis passing through the center like this. 760 00:54:25,390 --> 00:54:26,662 Pardon? 761 00:54:26,662 --> 00:54:28,150 AUDIENCE: [INAUDIBLE] 762 00:54:28,150 --> 00:54:30,358 PROFESSOR: It's this, this is what I'm talking about. 763 00:54:42,610 --> 00:54:44,220 Moving on to the last bit. 764 00:54:53,010 --> 00:54:57,720 So we need to know how to be able rotate things 765 00:54:57,720 --> 00:55:01,160 about places other than their centers of mass. 766 00:55:01,160 --> 00:55:05,210 So this is a stick, I can rotate about the center of mass, 767 00:55:05,210 --> 00:55:06,890 but it's more interesting if I rotate it 768 00:55:06,890 --> 00:55:08,520 about some other point. 769 00:55:08,520 --> 00:55:10,926 It makes it a pendulum when I do it around here. 770 00:55:10,926 --> 00:55:12,300 So I need to be able to calculate 771 00:55:12,300 --> 00:55:14,710 mass moments of inertia about a point that's 772 00:55:14,710 --> 00:55:16,730 not through the center of mass. 773 00:55:16,730 --> 00:55:21,300 I know you've seen this before in [? 8.01, ?] 774 00:55:21,300 --> 00:55:23,435 so this is going to be a quick reminder. 775 00:55:26,500 --> 00:55:29,090 But I'll show you where it comes from. 776 00:55:29,090 --> 00:55:30,070 So here's my stick. 777 00:55:39,980 --> 00:55:48,190 And it has a center of mass, and that's where G is located here. 778 00:55:48,190 --> 00:55:50,155 It has a total length l. 779 00:55:53,260 --> 00:56:04,220 I'm going to give it a thickness b, a width a. 780 00:56:04,220 --> 00:56:07,780 So it's a stick. 781 00:56:07,780 --> 00:56:12,120 a wide, b thick, l long. 782 00:56:12,120 --> 00:56:15,720 Uniform has a center gravity right in the middle. 783 00:56:15,720 --> 00:56:23,130 And I'm going to attach to this stick-- 784 00:56:23,130 --> 00:56:25,080 and this point is kind of hard to draw. 785 00:56:25,080 --> 00:56:28,140 This point is at the center of the stick, OK? 786 00:56:28,140 --> 00:56:30,750 I'm going to put my coordinate system attached 787 00:56:30,750 --> 00:56:33,730 at the center of gravity, center of mass, 788 00:56:33,730 --> 00:56:40,060 and I'm going to make it the-- that's x prime downward, 789 00:56:40,060 --> 00:56:45,680 z prime, and y prime is then going off that way. 790 00:56:45,680 --> 00:56:47,960 So this is a body set of coordinates 791 00:56:47,960 --> 00:56:49,790 at the center of mass. 792 00:56:49,790 --> 00:56:51,700 x prime, y prime, z prime. 793 00:56:51,700 --> 00:56:53,890 And x happens to be down. 794 00:56:53,890 --> 00:56:58,270 And I want to calculate my mass moment of inertia 795 00:56:58,270 --> 00:57:06,050 with respect to a point up here that is d, this distance. 796 00:57:06,050 --> 00:57:10,750 I've moved up the x-axis an amount d. 797 00:57:10,750 --> 00:57:14,120 I'm going to set a new coordinate system up here. 798 00:57:14,120 --> 00:57:18,010 So if this was z prime, my new z is here. 799 00:57:18,010 --> 00:57:19,370 It's getting a little messy. 800 00:57:19,370 --> 00:57:24,220 Maybe I'll do just a face view. 801 00:57:27,660 --> 00:57:33,400 If my previously y prime and x prime 802 00:57:33,400 --> 00:57:36,120 were like that, z coming out of the board, 803 00:57:36,120 --> 00:57:39,420 now I have a new system that is y 804 00:57:39,420 --> 00:57:42,960 and x like this, z still coming out of the board. 805 00:57:47,350 --> 00:57:48,856 Now the coordinate. 806 00:57:51,500 --> 00:57:56,500 So how do I calculate mass moment of inertia? 807 00:57:56,500 --> 00:58:02,670 Well I want Izz. 808 00:58:06,400 --> 00:58:08,770 I probably know Iz'z'. 809 00:58:08,770 --> 00:58:13,480 Iz'z' is the mass moment of inertia about this point. 810 00:58:13,480 --> 00:58:15,790 I know it's a principal axis from all the things we 811 00:58:15,790 --> 00:58:18,290 just-- that square block is the same as this. 812 00:58:18,290 --> 00:58:20,590 That's a principal axis in the z prime direction. 813 00:58:20,590 --> 00:58:24,715 I know the Izz' with respect to G, 814 00:58:24,715 --> 00:58:27,960 I want to know what with respect to this point. 815 00:58:27,960 --> 00:58:33,510 So well Izz, which is my new location 816 00:58:33,510 --> 00:58:40,700 up here, and we'll call it A. So Izz here with respect to A 817 00:58:40,700 --> 00:58:44,609 is the integral of r squared dm. 818 00:58:44,609 --> 00:58:46,192 We've got to do the same integral now. 819 00:58:48,980 --> 00:58:57,515 But that's the integral of x squared plus y squared dm. 820 00:59:01,300 --> 00:59:04,420 Now I can look at this and I can say oh, well, these 821 00:59:04,420 --> 00:59:07,440 are d-- this is separated by d. 822 00:59:07,440 --> 00:59:08,740 I only moved it in the x. 823 00:59:08,740 --> 00:59:11,810 The ys haven't moved and the zs didn't change. 824 00:59:11,810 --> 00:59:15,360 I just moved my point only in the x direction. 825 00:59:15,360 --> 00:59:20,820 So I can now say that in terms of my new coordinate, 826 00:59:20,820 --> 00:59:26,670 it's the same as x prime plus d, the distance from here 827 00:59:26,670 --> 00:59:32,750 to a point down her, some arbitrary mass point 828 00:59:32,750 --> 00:59:38,500 xi is going to be xi' plus d. 829 00:59:38,500 --> 00:59:43,390 So to do this integral in the new coordinates, 830 00:59:43,390 --> 00:59:51,070 this is going to be the integral of x prime plus d squared 831 00:59:51,070 --> 00:59:53,560 plus y. 832 00:59:53,560 --> 01:00:01,714 Now y prime equals y and z prime equals z. 833 01:00:01,714 --> 01:00:02,630 Those haven't changed. 834 01:00:02,630 --> 01:00:05,720 I didn't move my new coordinate system 835 01:00:05,720 --> 01:00:07,940 in the y direction or the z direction. 836 01:00:07,940 --> 01:00:11,200 So the coordinate in the new system in y 837 01:00:11,200 --> 01:00:12,390 is the same as before. 838 01:00:12,390 --> 01:00:15,030 So this is just y prime squared. 839 01:00:15,030 --> 01:00:18,910 And this whole thing times d, integrated times 840 01:00:18,910 --> 01:00:21,835 every little mass bit. 841 01:00:21,835 --> 01:00:26,690 If I square this, I get x prime squared, 2x'd, d squared, 842 01:00:26,690 --> 01:00:27,960 plus y squared. 843 01:00:27,960 --> 01:00:31,870 So this integral, Izz with respect 844 01:00:31,870 --> 01:00:35,500 to A, when you rearrange it, looks 845 01:00:35,500 --> 01:00:43,780 like x prime squared plus y prime squared dm 846 01:00:43,780 --> 01:00:47,690 and the integral of a sum is the sum of the integrals. 847 01:00:47,690 --> 01:00:49,730 So I break it into bits here, there's 848 01:00:49,730 --> 01:00:53,340 a d squared, which is a constant, dm. 849 01:00:53,340 --> 01:01:02,252 And then the last term is plus 2d and it's x prime dm. 850 01:01:05,410 --> 01:01:08,550 Just multiply this out, rewrite it, break it apart. 851 01:01:11,932 --> 01:01:12,890 Well let's do this one. 852 01:01:12,890 --> 01:01:16,360 Integral of x prime squared plus y prime squared dm. 853 01:01:16,360 --> 01:01:20,620 That's something that we already have a name for. 854 01:01:20,620 --> 01:01:22,175 This is IGzz. 855 01:01:27,640 --> 01:01:30,450 It's the original mass moment of inertia 856 01:01:30,450 --> 01:01:34,980 with respect to the original coordinate system at G 857 01:01:34,980 --> 01:01:35,950 in the z direction. 858 01:01:35,950 --> 01:01:39,810 So it's [? IzzG, ?] we already know that. 859 01:01:39,810 --> 01:01:41,840 That's given for the object. 860 01:01:41,840 --> 01:01:45,180 Plus, this is the integral of dm over the whole extent 861 01:01:45,180 --> 01:01:47,654 of the object? 862 01:01:47,654 --> 01:01:48,820 Just the mass of the object. 863 01:01:53,320 --> 01:01:58,330 This integral, this is the integral 864 01:01:58,330 --> 01:02:00,990 in terms of the x-coordinate. 865 01:02:00,990 --> 01:02:07,210 And every mass bit from here, if I go out here and find one, 866 01:02:07,210 --> 01:02:09,470 there's an equal and opposite one up here. 867 01:02:09,470 --> 01:02:11,840 This is the definition of the center of mass. 868 01:02:11,840 --> 01:02:14,540 This integral, if I'm at the center of mass 869 01:02:14,540 --> 01:02:18,890 integrating out from it, this is zero because of the definition 870 01:02:18,890 --> 01:02:21,220 the center of mass. 871 01:02:21,220 --> 01:02:26,540 And I've just proven the parallel axis theorem. 872 01:02:26,540 --> 01:02:32,740 Izz about this new point is I about G plus Md Squared, 873 01:02:32,740 --> 01:02:35,710 where d squared is the distance I've 874 01:02:35,710 --> 01:02:39,160 moved this z-axis to a new place parallel to it. 875 01:03:41,040 --> 01:03:48,710 So Izz with respect to G, the original mass moment 876 01:03:48,710 --> 01:04:01,630 of inertia for Izz is m L squared plus a squared over 12. 877 01:04:04,450 --> 01:04:20,570 And Ixx m L squared plus b squared over 12. 878 01:04:20,570 --> 01:04:29,420 And Izz-- oh, we already know that one. 879 01:04:32,980 --> 01:04:34,460 Wait a minute. 880 01:04:34,460 --> 01:04:36,440 I haven't told you what that is. 881 01:04:36,440 --> 01:04:38,510 That's Izz, Ixx, Iyy. 882 01:04:38,510 --> 01:04:42,180 Just a little messy here. 883 01:04:42,180 --> 01:04:56,990 Iyy for this problem is-- I have made a mistake. 884 01:04:56,990 --> 01:04:59,750 Ixx is a squared plus b squared. 885 01:04:59,750 --> 01:05:08,660 Iyy is m L squared plus b squared. 886 01:05:08,660 --> 01:05:10,940 So those are the three-- all with respect 887 01:05:10,940 --> 01:05:13,880 to G-- for this stick stick. 888 01:05:13,880 --> 01:05:14,785 And I'm going to-- 889 01:05:17,550 --> 01:05:19,014 AUDIENCE: [INAUDIBLE]? 890 01:05:19,014 --> 01:05:20,788 Is that also divided by 12? 891 01:05:20,788 --> 01:05:21,454 PROFESSOR: Yeah. 892 01:05:28,800 --> 01:05:33,190 That kind of sets us up where I can pick up next time. 893 01:05:33,190 --> 01:05:42,800 So let's finish by asking ourselves the question, what 894 01:05:42,800 --> 01:05:45,770 do we think about-- if I've moved to this new put 895 01:05:45,770 --> 01:05:52,320 new position, and I'm not rotating about the center, 896 01:05:52,320 --> 01:05:58,000 is this new axis-- this one around the center before, 897 01:05:58,000 --> 01:06:01,840 that one we know is a principal axis. 898 01:06:01,840 --> 01:06:03,990 If I rotate about this new place which 899 01:06:03,990 --> 01:06:08,650 I've defined the mass moment of inertia about that place? 900 01:06:08,650 --> 01:06:10,454 Is this a principal axis? 901 01:06:10,454 --> 01:06:11,342 AUDIENCE: Yes. 902 01:06:11,342 --> 01:06:12,230 AUDIENCE: Yes. 903 01:06:12,230 --> 01:06:15,320 PROFESSOR: How many think yes? 904 01:06:15,320 --> 01:06:18,330 How many think no? 905 01:06:18,330 --> 01:06:22,410 OK, so lots of people not sure. 906 01:06:22,410 --> 01:06:27,930 So my dynamic definition of principal axis 907 01:06:27,930 --> 01:06:31,580 is if you can rotate the object about that axis 908 01:06:31,580 --> 01:06:36,280 and produce no unbalanced torques, it's a principal axis. 909 01:06:36,280 --> 01:06:39,180 And I can do that and this thing will just spin all day long. 910 01:06:39,180 --> 01:06:43,390 Now there is a force, you could think of a fictitious force. 911 01:06:43,390 --> 01:06:45,840 There's a center of mass out here. 912 01:06:45,840 --> 01:06:49,610 As it spins around, there's a centripetal acceleration 913 01:06:49,610 --> 01:06:53,270 making it go in the circle, that means that fictitious force is 914 01:06:53,270 --> 01:06:56,790 like there is a centrifugal force pulling out on it. 915 01:06:56,790 --> 01:06:57,960 Do I feel that? 916 01:06:57,960 --> 01:07:00,176 Do I have to this resist that force 917 01:07:00,176 --> 01:07:01,300 as it goes round and round? 918 01:07:01,300 --> 01:07:02,530 Yes. 919 01:07:02,530 --> 01:07:05,960 So that is an unbalance of a kind 920 01:07:05,960 --> 01:07:09,600 we know as a static imbalance, but it doesn't produce torques 921 01:07:09,600 --> 01:07:17,118 about my axis right lined up on the center. 922 01:07:17,118 --> 01:07:19,986 AUDIENCE: Doesn't gravity pull on the center of mass 923 01:07:19,986 --> 01:07:21,541 [INAUDIBLE]. 924 01:07:21,541 --> 01:07:23,415 PROFESSOR: Sure, gravity does, but that's now 925 01:07:23,415 --> 01:07:24,248 a different problem. 926 01:07:24,248 --> 01:07:28,240 That's what makes this thing act like an oscillator. 927 01:07:28,240 --> 01:07:30,040 The torques of the kind I'm talking about 928 01:07:30,040 --> 01:07:34,390 is if I compute the angular momentum of this thing 929 01:07:34,390 --> 01:07:38,590 and compute dh/dt-- the time rate of change of the angular 930 01:07:38,590 --> 01:07:41,430 momentum is a torque on the system, right? 931 01:07:41,430 --> 01:07:44,680 I will get the term that makes it spin faster, 932 01:07:44,680 --> 01:07:46,640 and I will get, if they exist, terms 933 01:07:46,640 --> 01:07:51,640 that make it want to bend this way or bend back. 934 01:07:51,640 --> 01:07:58,650 It only happens if-- if I hold this thing over here, and spin 935 01:07:58,650 --> 01:08:02,790 it, get it spinning, and I compute the angle momentum 936 01:08:02,790 --> 01:08:06,030 with respect to this point, will I get torques? 937 01:08:06,030 --> 01:08:12,360 Yeah, but that's not how I-- that's a different problem. 938 01:08:12,360 --> 01:08:18,569 The IG is as if I were computing the angular momentum. 939 01:08:18,569 --> 01:08:22,660 Remember I started defining mass moment of inertia matrix 940 01:08:22,660 --> 01:08:26,520 based on an angular momentum computation at G. 941 01:08:26,520 --> 01:08:29,029 So it's right there in the center of this object, 942 01:08:29,029 --> 01:08:31,560 there's no moment arm that is causing 943 01:08:31,560 --> 01:08:35,830 torques that's trying to twist this thing about that point. 944 01:08:35,830 --> 01:08:39,952 So the answer to the question is this is a principal axis. 945 01:08:39,952 --> 01:08:40,859 Yeah? 946 01:08:40,859 --> 01:08:44,446 AUDIENCE: So if you take the derivative of the angular 947 01:08:44,446 --> 01:08:49,370 momentum would you get torques that are not in that direction? 948 01:08:49,370 --> 01:08:52,450 PROFESSOR: If you get torques that aren't in that direction, 949 01:08:52,450 --> 01:08:54,979 either you made a mistake doing the math, 950 01:08:54,979 --> 01:08:59,140 or you were in error in identifying 951 01:08:59,140 --> 01:09:02,055 the mass moment of inertia matrix to begin with. 952 01:09:02,055 --> 01:09:03,430 Because if you get torques, there 953 01:09:03,430 --> 01:09:07,270 must be non-zero off-diagonal terms and in mass moment 954 01:09:07,270 --> 01:09:08,109 of inertia matrix. 955 01:09:08,109 --> 01:09:12,140 They are what account for the torques. 956 01:09:12,140 --> 01:09:15,420 So by this parallel axis theorem, any other axis 957 01:09:15,420 --> 01:09:18,569 you go to-- if you started at a principal axis, 958 01:09:18,569 --> 01:09:23,149 any other axis you create is also a principal axis. 959 01:09:23,149 --> 01:09:26,870 That's the movement of just one axis. 960 01:09:26,870 --> 01:09:30,490 If you do two, if you move this way and this way, 961 01:09:30,490 --> 01:09:31,220 all bets are off. 962 01:09:31,220 --> 01:09:32,386 You get a difference answer. 963 01:09:32,386 --> 01:09:34,870 And if you're interested in that more complicated problem, 964 01:09:34,870 --> 01:09:37,250 read that Williams thing because he 965 01:09:37,250 --> 01:09:40,359 does the complete parallel axis, parallel planes 966 01:09:40,359 --> 01:09:43,410 and comes up with a super compact little way 967 01:09:43,410 --> 01:09:45,740 of calculating them. 968 01:09:45,740 --> 01:09:47,690 See you on Thursday.