1 00:00:00,100 --> 00:00:02,450 The following content is provided under a Creative 2 00:00:02,450 --> 00:00:03,830 Commons license. 3 00:00:03,830 --> 00:00:06,070 Your support will help MIT OpenCourseWare 4 00:00:06,070 --> 00:00:10,170 continue to offer high quality educational resources for free. 5 00:00:10,170 --> 00:00:12,710 To make a donation or to view additional materials 6 00:00:12,710 --> 00:00:16,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,620 --> 00:00:17,278 at ocw.mit.edu. 8 00:00:21,235 --> 00:00:22,110 PROFESSOR: All right. 9 00:00:22,110 --> 00:00:23,190 Thoughts. 10 00:00:23,190 --> 00:00:24,499 Important concepts of the week? 11 00:00:24,499 --> 00:00:25,540 AUDIENCE: Principal axes. 12 00:00:25,540 --> 00:00:26,520 PROFESSOR: Principal axes. 13 00:00:26,520 --> 00:00:27,020 Good one. 14 00:00:35,840 --> 00:00:36,770 Another one. 15 00:00:36,770 --> 00:00:38,706 AUDIENCE: Dynamic versus static balances. 16 00:00:38,706 --> 00:00:39,414 PROFESSOR: Right. 17 00:00:52,860 --> 00:00:53,738 Another one. 18 00:00:53,738 --> 00:00:54,990 AUDIENCE: Symmetry. 19 00:00:54,990 --> 00:00:55,490 Symmetry. 20 00:00:55,490 --> 00:00:56,310 PROFESSOR: Symmetry. 21 00:00:56,310 --> 00:00:56,809 Right. 22 00:01:02,860 --> 00:01:07,100 This is in the context of things like the mass moment of inertia 23 00:01:07,100 --> 00:01:07,730 matrix. 24 00:01:07,730 --> 00:01:08,230 Yes? 25 00:01:08,230 --> 00:01:09,605 AUDIENCE: Parallel axis theorem. 26 00:01:09,605 --> 00:01:10,980 PROFESSOR: Parallel axis theorem. 27 00:01:10,980 --> 00:01:11,530 OK. 28 00:01:11,530 --> 00:01:13,280 We haven't done much with it yet but we're 29 00:01:13,280 --> 00:01:14,360 going to come back to it. 30 00:01:26,301 --> 00:01:26,800 All right. 31 00:01:26,800 --> 00:01:27,205 Pretty good list. 32 00:01:27,205 --> 00:01:28,204 We only had one lecture. 33 00:01:32,320 --> 00:01:33,632 Dynamic and static balancing. 34 00:01:33,632 --> 00:01:35,090 Let's talk about that for a second. 35 00:01:43,580 --> 00:01:45,100 Let's talk about static balancing. 36 00:01:45,100 --> 00:01:48,960 Tell me what are features of static balancing? 37 00:01:48,960 --> 00:01:51,700 How can you-- you have a quiz problem, 38 00:01:51,700 --> 00:01:56,490 it's on the final-- you have an object and an axis about which 39 00:01:56,490 --> 00:01:58,770 its spinning and you're asked, is this statically 40 00:01:58,770 --> 00:02:00,570 balanced or not. 41 00:02:00,570 --> 00:02:01,570 What would you look for? 42 00:02:01,570 --> 00:02:02,611 Christina. 43 00:02:02,611 --> 00:02:04,720 AUDIENCE: [INAUDIBLE]. 44 00:02:04,720 --> 00:02:06,415 PROFESSOR: G needs to be-- 45 00:02:06,415 --> 00:02:07,720 AUDIENCE: [INAUDIBLE] axis. 46 00:02:07,720 --> 00:02:11,920 PROFESSOR: So another way of saying that is the axis must-- 47 00:02:11,920 --> 00:02:14,400 the axis of rotation must pass through-- 48 00:02:14,400 --> 00:02:15,370 AUDIENCE: The center. 49 00:02:15,370 --> 00:02:16,530 PROFESSOR: --G. Is that what you're trying to say? 50 00:02:16,530 --> 00:02:18,140 The center of mass? 51 00:02:18,140 --> 00:02:19,560 Everybody agree about that? 52 00:02:19,560 --> 00:02:20,060 All right. 53 00:02:20,060 --> 00:02:23,650 Now, if I have weird shaped object, 54 00:02:23,650 --> 00:02:25,820 I know where the center of mass, and I 55 00:02:25,820 --> 00:02:32,060 stick the axis of rotation at any angle at all through it, 56 00:02:32,060 --> 00:02:34,280 is it statically balance regardless 57 00:02:34,280 --> 00:02:38,210 of where I have the x-axis of rotation passing 58 00:02:38,210 --> 00:02:39,612 through the center of mass? 59 00:02:39,612 --> 00:02:40,456 AUDIENCE: Yes. 60 00:02:40,456 --> 00:02:41,840 PROFESSOR: I hear a yes. 61 00:02:41,840 --> 00:02:43,860 How about the other people? 62 00:02:43,860 --> 00:02:44,660 AUDIENCE: Yeah. 63 00:02:44,660 --> 00:02:45,160 OK. 64 00:02:45,160 --> 00:02:47,740 So how do you-- I think that's probably right. 65 00:02:47,740 --> 00:02:48,850 But how do you test it? 66 00:02:48,850 --> 00:02:51,870 How do you test to see-- what's a simple test you 67 00:02:51,870 --> 00:02:54,456 can form to see if something is statically balanced? 68 00:02:54,456 --> 00:02:56,330 AUDIENCE: See if it goes through a low point. 69 00:02:56,330 --> 00:02:57,038 PROFESSOR: Right. 70 00:02:57,038 --> 00:03:01,490 So you make the axis that you're rotating about horizontal 71 00:03:01,490 --> 00:03:03,720 and see if the thing seeks a low point. 72 00:03:03,720 --> 00:03:09,430 Because if it does, it's telling you that what 73 00:03:09,430 --> 00:03:10,570 about the center of mass? 74 00:03:10,570 --> 00:03:11,880 AUDIENCE: It's not going through the-- 75 00:03:11,880 --> 00:03:13,430 PROFESSOR: It's not going through the axis. 76 00:03:13,430 --> 00:03:15,471 To go to a low point, it means the center of mass 77 00:03:15,471 --> 00:03:17,470 is somewhere below the axis of spin. 78 00:03:17,470 --> 00:03:18,054 Right? 79 00:03:18,054 --> 00:03:19,970 But any axis at all, as long as its through G, 80 00:03:19,970 --> 00:03:22,630 you are statically balanced. 81 00:03:22,630 --> 00:03:25,520 Dynamic balancing. 82 00:03:25,520 --> 00:03:29,060 What are the-- what would you look for on an object to say 83 00:03:29,060 --> 00:03:30,875 is this dynamically balanced or not? 84 00:03:30,875 --> 00:03:31,875 What kind of properties? 85 00:03:36,706 --> 00:03:38,622 AUDIENCE: There would be no unbalanced torques 86 00:03:38,622 --> 00:03:41,590 if you decided to rotate it. 87 00:03:41,590 --> 00:03:44,990 PROFESSOR: So you're looking for now unbalanced torques. 88 00:03:44,990 --> 00:03:47,766 AUDIENCE: Because I'm like if it-- if the object is rotating 89 00:03:47,766 --> 00:03:49,890 about the axis that you have it on, 90 00:03:49,890 --> 00:03:58,790 you won't need to supply an extra torque to try and keep 91 00:03:58,790 --> 00:04:00,447 it balanced. 92 00:04:00,447 --> 00:04:01,280 [INTERPOSING VOICES] 93 00:04:01,280 --> 00:04:02,620 PROFESSOR: --pretty convincing. 94 00:04:02,620 --> 00:04:04,492 AUDIENCE: --the axis it's rotating on. 95 00:04:04,492 --> 00:04:05,430 PROFESSOR: OK. 96 00:04:05,430 --> 00:04:07,790 If you put in that last caveat then 97 00:04:07,790 --> 00:04:09,610 it's getting pretty correct. 98 00:04:09,610 --> 00:04:12,310 Can there be a torque on it that's not balanced 99 00:04:12,310 --> 00:04:14,646 and have it be dynamically balanced? 100 00:04:14,646 --> 00:04:15,540 AUDIENCE: Well, yeah. 101 00:04:15,540 --> 00:04:16,415 PROFESSOR: Which one? 102 00:04:18,706 --> 00:04:20,519 AUDIENCE: Within the axis of rotation. 103 00:04:20,519 --> 00:04:24,030 PROFESSOR: If you have a torque that's in the axis of spin 104 00:04:24,030 --> 00:04:26,210 and around the axis of rotation and that torque 105 00:04:26,210 --> 00:04:27,755 will cause what to happen? 106 00:04:27,755 --> 00:04:29,030 AUDIENCE: Rotation. 107 00:04:29,030 --> 00:04:30,930 PROFESSOR: Cause the rotation rate to? 108 00:04:30,930 --> 00:04:31,320 AUDIENCE: Change. 109 00:04:31,320 --> 00:04:32,361 PROFESSOR: Change, right? 110 00:04:32,361 --> 00:04:35,330 That puts energy into the system and accelerates it. 111 00:04:35,330 --> 00:04:38,390 And it's a torque that's in the direction of spin that doesn't 112 00:04:38,390 --> 00:04:39,700 cause dynamic imbalances. 113 00:04:39,700 --> 00:04:44,740 But a torque that is other than in the direction of spin, 114 00:04:44,740 --> 00:04:47,460 that's what a dynamic imbalance is. 115 00:04:47,460 --> 00:04:51,470 So what are some other ways you can say this in terms of things 116 00:04:51,470 --> 00:04:58,330 like evidence of-- let's say I give you H 117 00:04:58,330 --> 00:05:00,922 and I give you omega, how can you just 118 00:05:00,922 --> 00:05:02,380 tell by looking at those two thing? 119 00:05:02,380 --> 00:05:04,755 I give you the angle of momentum and I give you the spin, 120 00:05:04,755 --> 00:05:06,590 how can you tell instantly whether or not 121 00:05:06,590 --> 00:05:08,170 it is dynamically balanced? 122 00:05:08,170 --> 00:05:10,078 AUDIENCE: If there-- each of the components 123 00:05:10,078 --> 00:05:11,510 in the same direction. 124 00:05:11,510 --> 00:05:16,928 PROFESSOR: So you're saying if H and omega are in the-- 125 00:05:16,928 --> 00:05:18,040 AUDIENCE: R line. 126 00:05:18,040 --> 00:05:18,790 PROFESSOR: A line. 127 00:05:18,790 --> 00:05:20,800 They just have to be parallel. 128 00:05:20,800 --> 00:05:24,465 If they're aligned, you do not have any dynamic imbalance. 129 00:05:28,062 --> 00:05:28,770 That sound right? 130 00:05:28,770 --> 00:05:29,555 Yeah? 131 00:05:29,555 --> 00:05:31,138 AUDIENCE: It doesn't matter if they're 132 00:05:31,138 --> 00:05:32,617 parallel or antiparallel, does it? 133 00:05:32,617 --> 00:05:34,325 PROFESSOR: You mean just opposite senses? 134 00:05:34,325 --> 00:05:35,927 AUDIENCE: Yeah. 135 00:05:35,927 --> 00:05:37,885 PROFESSOR: I think that'd be really hard to do. 136 00:05:37,885 --> 00:05:40,030 I don't know of any-- I don't think 137 00:05:40,030 --> 00:05:43,100 you can make that system that has angular momentum that 138 00:05:43,100 --> 00:05:46,980 is negative-- it's actually opposite the direction 139 00:05:46,980 --> 00:05:49,800 of the rotation rate. 140 00:05:49,800 --> 00:05:54,340 I don't think it-- I don't think our universe supports 141 00:05:54,340 --> 00:05:55,940 that physics. 142 00:05:55,940 --> 00:05:59,660 But anyway, in any case, they're aligned, you're OK. 143 00:06:02,930 --> 00:06:03,520 All right? 144 00:06:03,520 --> 00:06:04,900 Let's move on. 145 00:06:04,900 --> 00:06:09,987 So today we're going actually just 146 00:06:09,987 --> 00:06:11,320 have you work a couple problems. 147 00:06:11,320 --> 00:06:14,000 And you really are going to work in groups, and when you finish, 148 00:06:14,000 --> 00:06:15,916 one of the groups is going to come up and fill 149 00:06:15,916 --> 00:06:18,370 in the blanks of the answer to the problem 150 00:06:18,370 --> 00:06:19,370 and we'll talk about it. 151 00:06:19,370 --> 00:06:21,300 And we're going to work two problems. 152 00:06:21,300 --> 00:06:25,150 And they have to do with things that spin. 153 00:06:25,150 --> 00:06:31,483 And as a quick review just of information-- 154 00:06:31,483 --> 00:06:32,465 I guess we can do this. 155 00:06:38,370 --> 00:06:42,475 So the problem is a simple one. 156 00:06:42,475 --> 00:06:44,530 It's basically this. 157 00:06:44,530 --> 00:06:49,150 And I decided to lubricate it and it made the wood swell 158 00:06:49,150 --> 00:06:50,966 and so now it won't spin at all. 159 00:06:50,966 --> 00:06:52,727 Actually, unless I let the axis spin. 160 00:06:52,727 --> 00:06:53,810 So it's just this problem. 161 00:06:57,280 --> 00:07:00,830 The axis passes through the center of mass 162 00:07:00,830 --> 00:07:03,970 and it looks just like this. 163 00:07:03,970 --> 00:07:10,070 And I've got an x-axis aligned with the axis of the rod. 164 00:07:10,070 --> 00:07:11,420 The Z is this way. 165 00:07:11,420 --> 00:07:14,000 And the spin is cap omega. 166 00:07:14,000 --> 00:07:19,890 And the mass moment of inertia matrix 167 00:07:19,890 --> 00:07:22,220 for this problem about G-- and our axis 168 00:07:22,220 --> 00:07:26,220 is passing through G-- so mass moment of inertia matrix times 169 00:07:26,220 --> 00:07:30,220 omega gives you H. So here's the mass moment of inertia matrix. 170 00:07:30,220 --> 00:07:33,490 I claim is going to be diagonal. 171 00:07:33,490 --> 00:07:35,830 And the test of that, can you tell me whether or not 172 00:07:35,830 --> 00:07:39,520 I've chosen axes that, just from symmetry, you know 173 00:07:39,520 --> 00:07:41,060 will be principal axes? 174 00:07:45,425 --> 00:07:46,687 What do you think? 175 00:07:46,687 --> 00:07:49,104 AUDIENCE: Well, the rod is circular now that [INAUDIBLE]. 176 00:07:49,104 --> 00:07:49,770 PROFESSOR: Yeah. 177 00:07:49,770 --> 00:07:53,939 So one axis-- that's an axis of symmetry is down the shaft. 178 00:07:53,939 --> 00:07:55,230 So that's guaranteed to be one. 179 00:07:55,230 --> 00:07:57,210 Have I chosen one like that? 180 00:07:57,210 --> 00:07:59,335 And any other orthogonal pair after that-- 181 00:07:59,335 --> 00:08:01,860 doesn't matter which way I orient them-- will also be. 182 00:08:01,860 --> 00:08:04,070 So I've got one up and one into the board. 183 00:08:04,070 --> 00:08:07,340 So those are principal axes for this object. 184 00:08:07,340 --> 00:08:09,020 So it's going to be diagonal. 185 00:08:09,020 --> 00:08:13,010 And in fact, it looks-- it's MR squared over 2, 186 00:08:13,010 --> 00:08:15,970 ML squared over 12, and ML squared over 12 187 00:08:15,970 --> 00:08:17,710 when you work it out. 188 00:08:17,710 --> 00:08:21,320 And if you look up in the book, most books for slen-- 189 00:08:21,320 --> 00:08:23,640 what they call slender rods, will 190 00:08:23,640 --> 00:08:25,986 say that this first term is 0. 191 00:08:25,986 --> 00:08:27,360 And that's because they're saying 192 00:08:27,360 --> 00:08:31,250 that L is a lot bigger than R-- the radius of this thing. 193 00:08:31,250 --> 00:08:34,360 And so MR squared is a pretty small number. 194 00:08:34,360 --> 00:08:38,450 So it's energy and rotation spinning this way 195 00:08:38,450 --> 00:08:40,860 is not very big for its angular momentum. 196 00:08:40,860 --> 00:08:42,809 But its angular momentum spinning like this 197 00:08:42,809 --> 00:08:48,377 is much, much larger because you have much greater MR squareds. 198 00:08:48,377 --> 00:08:50,460 So you can leave, for the purpose of this problem, 199 00:08:50,460 --> 00:08:52,460 you can't treat this as 0. 200 00:08:52,460 --> 00:08:55,990 And in your groups, I want you to come up with the omega 201 00:08:55,990 --> 00:08:59,740 vector, H and DHDT. 202 00:08:59,740 --> 00:09:01,480 So do get in groups, talk about it. 203 00:09:01,480 --> 00:09:02,980 You've got a few minutes to do this. 204 00:09:02,980 --> 00:09:04,760 This one's pretty straightforward. 205 00:09:04,760 --> 00:09:07,337 And then you can do another one that's harder so warm up 206 00:09:07,337 --> 00:09:08,920 and find a group to work in and you're 207 00:09:08,920 --> 00:09:10,720 going to work a couple of problems this way. 208 00:09:10,720 --> 00:09:12,525 Got a group that feels pretty good about they're answer? 209 00:09:12,525 --> 00:09:13,210 AUDIENCE: Yes. 210 00:09:13,210 --> 00:09:15,460 PROFESSOR: All right. 211 00:09:15,460 --> 00:09:16,930 Write it up. 212 00:09:16,930 --> 00:09:23,830 Come up, fill in the omega, H and DHDT. 213 00:09:49,200 --> 00:09:49,700 OK. 214 00:09:49,700 --> 00:09:51,750 Can everybody see it? 215 00:09:51,750 --> 00:09:58,080 So you have a Z component only for the omega, a Z component k 216 00:09:58,080 --> 00:10:03,180 hat only for H, and its i omega-- iZZ omega. 217 00:10:03,180 --> 00:10:07,600 And DHDT, the only variable here is omega 218 00:10:07,600 --> 00:10:10,470 and you get an omega dot. 219 00:10:10,470 --> 00:10:15,910 And there's no-- DKDT is 0 because it 220 00:10:15,910 --> 00:10:17,980 doesn't change direction. 221 00:10:17,980 --> 00:10:21,620 Any questions about this? 222 00:10:21,620 --> 00:10:22,140 Yes? 223 00:10:22,140 --> 00:10:23,806 AUDIENCE: So I was a little bit confused 224 00:10:23,806 --> 00:10:30,347 in lecture about how we knew the i hat, j hat, k hat in the H 225 00:10:30,347 --> 00:10:30,847 term. 226 00:10:30,847 --> 00:10:32,755 Is it just because it's only i hat 227 00:10:32,755 --> 00:10:35,140 on the top, j hat in the middle, and k hat on the bottom? 228 00:10:35,140 --> 00:10:35,640 Is that-- 229 00:10:35,640 --> 00:10:39,180 PROFESSOR: So the convention-- if I understand your question 230 00:10:39,180 --> 00:10:44,630 correctly-- the convention when you write out the H vector 231 00:10:44,630 --> 00:10:52,220 is it's the result of multiplying the three 232 00:10:52,220 --> 00:10:54,220 components of the spin. 233 00:10:54,220 --> 00:10:56,290 This is the piece in the i direction, 234 00:10:56,290 --> 00:10:58,400 j direction, k direction. 235 00:10:58,400 --> 00:11:04,270 And you multiply these out you get three results. 236 00:11:04,270 --> 00:11:08,846 Vector times-- and this one is Hx and it is in the i. 237 00:11:08,846 --> 00:11:13,900 The second one from this times the middle row gives you Hy 238 00:11:13,900 --> 00:11:15,530 and it's in the j. 239 00:11:15,530 --> 00:11:18,670 And the third one, this vector times these three 240 00:11:18,670 --> 00:11:22,100 in the bottom row, give you Hz in the k. 241 00:11:25,022 --> 00:11:26,970 AUDIENCE: So I thought when-- I don't know, 242 00:11:26,970 --> 00:11:29,405 I could be completely wrong-- but I thought when you did 243 00:11:29,405 --> 00:11:34,275 [INAUDIBLE] you multiplied this, this, this and then this, this, 244 00:11:34,275 --> 00:11:37,320 this and then-- 245 00:11:37,320 --> 00:11:43,102 PROFESSOR: So we have a capital A, capital B, capital C 246 00:11:43,102 --> 00:11:44,060 terms in the first row? 247 00:11:44,060 --> 00:11:45,620 AUDIENCE: So you do, like, this and-- 248 00:11:45,620 --> 00:11:47,630 PROFESSOR: And you have a capital-- little a, 249 00:11:47,630 --> 00:11:50,370 little b, little c terms in this one. 250 00:11:50,370 --> 00:11:53,140 When you multiply this times this, 251 00:11:53,140 --> 00:11:59,275 you get Aa plus Bb plus Cc. 252 00:11:59,275 --> 00:11:59,900 AUDIENCE: Yeah. 253 00:11:59,900 --> 00:12:01,160 So that's my question. 254 00:12:01,160 --> 00:12:04,245 So if you have the omega x is an i hat, 255 00:12:04,245 --> 00:12:06,875 omega y is a j hat, omega z's a k hat, 256 00:12:06,875 --> 00:12:08,510 how, if you're multiplying like this, 257 00:12:08,510 --> 00:12:11,424 do you get only i hats on the top of the thing. 258 00:12:11,424 --> 00:12:12,090 PROFESSOR: Yeah. 259 00:12:12,090 --> 00:12:15,630 Well, it's because this isn't just a matrix. 260 00:12:15,630 --> 00:12:18,400 This is actually a tensor. 261 00:12:18,400 --> 00:12:22,940 And we haven't gone into messy tensor notation. 262 00:12:22,940 --> 00:12:26,880 So you're just being told a convention here. 263 00:12:26,880 --> 00:12:30,140 And that is, because this is a vector being multiplied 264 00:12:30,140 --> 00:12:35,980 by a tensor, the result will-- even though this is i, j, k, 265 00:12:35,980 --> 00:12:40,820 omega xi times this first term in this tensor, 266 00:12:40,820 --> 00:12:43,350 it gives you an i back. 267 00:12:43,350 --> 00:12:47,380 Omega yj times this one gives you an i back. 268 00:12:47,380 --> 00:12:51,242 Omega zk times that gives you an i back. 269 00:12:51,242 --> 00:12:51,783 AUDIENCE: OK. 270 00:12:51,783 --> 00:12:52,490 That's what I was really confused -- 271 00:12:52,490 --> 00:12:53,160 PROFESSOR: Right. 272 00:12:53,160 --> 00:12:55,409 And that's a great question because it wasn't obvious. 273 00:12:55,409 --> 00:12:58,015 We didn't do the full tensor mathematic. 274 00:12:58,015 --> 00:13:01,230 We just gave you a result. By definition, 275 00:13:01,230 --> 00:13:05,100 this row times that is in the i direction, j direction, k 276 00:13:05,100 --> 00:13:05,690 direction. 277 00:13:05,690 --> 00:13:06,550 Great question. 278 00:13:06,550 --> 00:13:09,340 OK. 279 00:13:09,340 --> 00:13:10,180 OK. 280 00:13:10,180 --> 00:13:11,720 So let's go on to harder problem. 281 00:13:11,720 --> 00:13:12,886 So you're great at that one. 282 00:13:16,150 --> 00:13:18,940 We're going to do this one now. 283 00:13:18,940 --> 00:13:20,255 So we just did this problem. 284 00:13:22,699 --> 00:13:23,990 Now we want to do this problem. 285 00:13:30,780 --> 00:13:42,440 And so the same i matrix is used here. 286 00:13:42,440 --> 00:13:45,460 It's this one and you can let that term is 0 287 00:13:45,460 --> 00:13:47,600 because the coordinate system that 288 00:13:47,600 --> 00:13:55,840 has been defined on this object is x, z, y into the board. 289 00:13:55,840 --> 00:13:58,930 But the direction of spin now is like that. 290 00:13:58,930 --> 00:14:01,210 And we'll call it cap omega again. 291 00:14:01,210 --> 00:14:02,440 But it's in that direction. 292 00:14:02,440 --> 00:14:05,060 And these are 45 degree angles for the purposes 293 00:14:05,060 --> 00:14:05,820 of this problem. 294 00:14:05,820 --> 00:14:11,200 So that's sine cosine of 45 root 2 over 2. 295 00:14:11,200 --> 00:14:14,160 So now, what's omega? 296 00:14:14,160 --> 00:14:15,500 What's H? 297 00:14:15,500 --> 00:14:16,870 And what's DHDT? 298 00:14:16,870 --> 00:14:18,910 So work in groups and sort that one out. 299 00:14:22,140 --> 00:14:23,442 You ready to go? 300 00:14:23,442 --> 00:14:24,477 AUDIENCE: We can be. 301 00:14:24,477 --> 00:14:25,060 PROFESSOR: OK. 302 00:14:25,060 --> 00:14:25,810 Go for it. 303 00:14:36,736 --> 00:14:38,236 We'll probably do this occasionally. 304 00:14:41,230 --> 00:14:42,520 I'll say it again. 305 00:14:42,520 --> 00:14:45,419 I have no intention of embarrassing you at the board. 306 00:14:45,419 --> 00:14:47,960 So the practice will be, you put your answer up, you sit down 307 00:14:47,960 --> 00:14:49,330 and then we talk about it. 308 00:14:49,330 --> 00:14:50,170 It's not about you. 309 00:14:50,170 --> 00:14:51,340 It's about what's on the board. 310 00:14:51,340 --> 00:14:51,839 OK? 311 00:15:04,372 --> 00:15:06,080 AUDIENCE: Should I keep going or should-- 312 00:15:06,080 --> 00:15:06,230 PROFESSOR: No. 313 00:15:06,230 --> 00:15:07,188 Put the whole thing up. 314 00:15:07,188 --> 00:15:08,540 Yep. 315 00:15:08,540 --> 00:15:11,579 And if you want to use that big piece of chalk it shows better. 316 00:15:11,579 --> 00:15:12,370 Just to your right. 317 00:15:54,900 --> 00:15:55,400 OK. 318 00:15:55,400 --> 00:15:58,690 So let's talk about omega first. 319 00:15:58,690 --> 00:16:03,550 How do you figure-- we have a minus root 2 over 2 here. 320 00:16:03,550 --> 00:16:04,310 Can't see that. 321 00:16:04,310 --> 00:16:09,330 Minus root 2 over 2 omega i 0 root 2 over 2 omega k. 322 00:16:09,330 --> 00:16:11,350 Are people-- feel good about that? 323 00:16:11,350 --> 00:16:12,470 Any differences? 324 00:16:12,470 --> 00:16:12,970 All right. 325 00:16:12,970 --> 00:16:15,670 So this thing-- this omega-- that spin clearly 326 00:16:15,670 --> 00:16:19,800 has components that are like that and like this. 327 00:16:19,800 --> 00:16:22,650 So this is your z piece and your i piece. 328 00:16:22,650 --> 00:16:25,490 And it is, indeed, in the minus i hat direction. 329 00:16:25,490 --> 00:16:26,990 So that seems OK. 330 00:16:26,990 --> 00:16:34,640 And H would be some i-- the i times omega. 331 00:16:34,640 --> 00:16:38,210 And the first term, if you take that upper left term as 0, 332 00:16:38,210 --> 00:16:39,490 then you don't get anything. 333 00:16:39,490 --> 00:16:41,150 The second term you don't get anything. 334 00:16:41,150 --> 00:16:44,790 So how do you feel about the angular momentum? 335 00:16:47,780 --> 00:16:49,350 Any differences? 336 00:16:49,350 --> 00:16:50,040 OK. 337 00:16:50,040 --> 00:16:52,530 And the DHDT. 338 00:16:52,530 --> 00:16:55,110 You take the time derivative of this thing-- 339 00:16:55,110 --> 00:16:57,100 and you got an omega dot. 340 00:16:57,100 --> 00:16:58,100 OK. 341 00:16:58,100 --> 00:16:59,562 That looks OK as far as it goes. 342 00:16:59,562 --> 00:17:00,520 But any other thoughts? 343 00:17:04,108 --> 00:17:07,077 AUDIENCE: So k hat dot is not 0. 344 00:17:07,077 --> 00:17:07,660 PROFESSOR: OK. 345 00:17:07,660 --> 00:17:11,220 So you're saying k hat is changing with time. 346 00:17:11,220 --> 00:17:11,957 AUDIENCE: Yeah. 347 00:17:11,957 --> 00:17:13,540 PROFESSOR: And let's think about that. 348 00:17:13,540 --> 00:17:16,290 So it's-- that's this piece here. 349 00:17:16,290 --> 00:17:20,079 This is omega z k hat. 350 00:17:20,079 --> 00:17:25,940 And as this thing goes around that unit vector is doing this. 351 00:17:25,940 --> 00:17:27,260 Right? 352 00:17:27,260 --> 00:17:29,774 So you need a DkDT. 353 00:17:29,774 --> 00:17:31,690 So do you want to give-- somebody else give me 354 00:17:31,690 --> 00:17:34,863 the second term here? 355 00:17:34,863 --> 00:17:35,488 AUDIENCE: Sure. 356 00:17:38,360 --> 00:17:45,627 ML squared times omega squared because you 357 00:17:45,627 --> 00:17:52,298 have the changing directions in your k hat principal axes. 358 00:17:52,298 --> 00:17:56,840 And that's going to be-- well, times root 2. 359 00:17:56,840 --> 00:17:58,495 PROFESSOR: This is root-- 360 00:17:58,495 --> 00:18:00,840 AUDIENCE: Well, omega squared. 361 00:18:00,840 --> 00:18:02,190 PROFESSOR: OK. 362 00:18:02,190 --> 00:18:04,225 And then what about the root 2 over 2? 363 00:18:04,225 --> 00:18:06,180 AUDIENCE: [INAUDIBLE]. 364 00:18:06,180 --> 00:18:07,889 PROFESSOR: Got to square that too, right? 365 00:18:07,889 --> 00:18:09,388 AUDIENCE: That'd be squared as well. 366 00:18:09,388 --> 00:18:11,136 PROFESSOR: That gives you a half. 367 00:18:11,136 --> 00:18:11,635 OK. 368 00:18:11,635 --> 00:18:13,574 And-- 369 00:18:13,574 --> 00:18:15,070 AUDIENCE: That's over 12 as well. 370 00:18:15,070 --> 00:18:16,100 PROFESSOR: 12. 371 00:18:16,100 --> 00:18:16,980 And-- 372 00:18:16,980 --> 00:18:19,180 AUDIENCE: That would be in the j hat direction. 373 00:18:19,180 --> 00:18:19,770 PROFESSOR: Positive-- 374 00:18:19,770 --> 00:18:20,160 AUDIENCE: Positive. 375 00:18:20,160 --> 00:18:20,659 Yes. 376 00:18:20,659 --> 00:18:21,330 In this case. 377 00:18:21,330 --> 00:18:23,876 PROFESSOR: Started out with a negative but-- 378 00:18:23,876 --> 00:18:25,500 AUDIENCE: Negative [INAUDIBLE]. 379 00:18:25,500 --> 00:18:28,680 PROFESSOR: i cross k is minus j times a minus gives you 380 00:18:28,680 --> 00:18:29,500 the plus, right? 381 00:18:29,500 --> 00:18:35,740 So this looks-- this comes out ML squared omega 382 00:18:35,740 --> 00:18:39,930 squared j hat over 24 I think. 383 00:18:39,930 --> 00:18:40,640 Like that. 384 00:18:40,640 --> 00:18:41,960 That term. 385 00:18:41,960 --> 00:18:43,280 Now-- OK. 386 00:18:43,280 --> 00:18:46,190 And people good with that now? 387 00:18:46,190 --> 00:18:53,150 So just to remind you, when I was doing these problems-- 388 00:18:53,150 --> 00:18:55,485 when I first had to teach this course a few years ago, 389 00:18:55,485 --> 00:18:58,110 and I was trying to figure out an easy way to teach this 390 00:18:58,110 --> 00:19:01,320 and how to do these problems, I got myself all confused 391 00:19:01,320 --> 00:19:05,790 trying to figure out which components of the rotation 392 00:19:05,790 --> 00:19:09,540 vector are rotating and what do I have to cross into it 393 00:19:09,540 --> 00:19:10,930 to get the answer. 394 00:19:10,930 --> 00:19:13,450 And in fact, you don't have to make it 395 00:19:13,450 --> 00:19:15,500 anywhere near that hard. 396 00:19:15,500 --> 00:19:18,730 And let's take a quick look at something here. 397 00:19:25,590 --> 00:19:34,710 H is ML squared over 12 root 2 over 2-- 398 00:19:34,710 --> 00:19:37,890 where did H go-- root 2 over 2 omega k hat. 399 00:19:45,030 --> 00:19:49,030 H is a rotating vector. 400 00:19:49,030 --> 00:19:57,170 A derivative of a rotating vector in an inertial frame 401 00:19:57,170 --> 00:20:03,260 is the derivative of that rotating vector in the rotating 402 00:20:03,260 --> 00:20:04,160 frame. 403 00:20:04,160 --> 00:20:05,870 Which is the same thing as saying 404 00:20:05,870 --> 00:20:08,450 that omega is 0-- that's the change in length 405 00:20:08,450 --> 00:20:15,020 of the vector-- plus omega cross H. 406 00:20:15,020 --> 00:20:18,240 Let's test that and see if that gives us the right answer. 407 00:20:18,240 --> 00:20:22,900 This derivative gives you-- all it does is gives you the theta, 408 00:20:22,900 --> 00:20:23,940 the omega dot back. 409 00:20:23,940 --> 00:20:24,440 Right? 410 00:20:24,440 --> 00:20:25,980 That's that first term. 411 00:20:25,980 --> 00:20:28,050 And the second term should just then 412 00:20:28,050 --> 00:20:37,090 look like your omega, which is root 2 minus root 2 413 00:20:37,090 --> 00:20:47,420 over 2 cap omega i plus root 2 over 2 cap omega k 414 00:20:47,420 --> 00:20:54,270 cross a bunch of constants times k. 415 00:20:54,270 --> 00:20:54,770 Right? 416 00:20:57,600 --> 00:21:04,590 So the k cross k terms are? 417 00:21:04,590 --> 00:21:05,200 AUDIENCE: 0. 418 00:21:05,200 --> 00:21:06,150 PROFESSOR: 0. 419 00:21:06,150 --> 00:21:08,564 The i cross k-- 420 00:21:08,564 --> 00:21:09,480 AUDIENCE: Minus j. 421 00:21:09,480 --> 00:21:10,950 PROFESSOR: --minus j. 422 00:21:10,950 --> 00:21:13,670 And so you get root 2 over 2 times root 2 over 2. 423 00:21:13,670 --> 00:21:18,900 You get what's in H here multiplied 424 00:21:18,900 --> 00:21:20,630 by root 2 over 2 omega hat. 425 00:21:20,630 --> 00:21:27,200 And you get back exactly these two terms. 426 00:21:27,200 --> 00:21:29,800 So it's that easy. 427 00:21:29,800 --> 00:21:31,940 It's just the derivative of a rotating vector. 428 00:21:31,940 --> 00:21:35,560 Just doing omega cross H is the easiest 429 00:21:35,560 --> 00:21:41,190 way to deal with that derivative of the rotating piece. 430 00:21:45,110 --> 00:21:46,650 Any rotating vector, you can take 431 00:21:46,650 --> 00:21:48,620 this time derivative that way. 432 00:21:48,620 --> 00:21:50,570 All right. 433 00:21:50,570 --> 00:21:51,982 We've got a few minutes. 434 00:21:51,982 --> 00:21:53,690 I want to-- actually, any other questions 435 00:21:53,690 --> 00:21:56,920 about this kind of problem? 436 00:21:56,920 --> 00:21:59,910 In physics, most of the problems you've worked before, 437 00:21:59,910 --> 00:22:02,370 that involved rotation, are planar-- what 438 00:22:02,370 --> 00:22:04,090 we call planar motion problems. 439 00:22:04,090 --> 00:22:08,380 The axis of spin was always perpendicular to the plane 440 00:22:08,380 --> 00:22:11,090 and the rest the problem was confined to the plane. 441 00:22:11,090 --> 00:22:15,300 So hockey pucks sliding along and stuff like that. 442 00:22:15,300 --> 00:22:18,730 But it was always assumed that the axis of rotation 443 00:22:18,730 --> 00:22:20,880 was perpendicular to the plane and that the angular 444 00:22:20,880 --> 00:22:23,400 momentum was parallel. 445 00:22:23,400 --> 00:22:26,190 So this is actually-- this is 3D problem. 446 00:22:26,190 --> 00:22:27,740 This is a 3D dynamics problem. 447 00:22:27,740 --> 00:22:30,890 As soon as that H and the omega are different directions, 448 00:22:30,890 --> 00:22:34,470 you can come up with torques in all three directions. 449 00:22:34,470 --> 00:22:37,210 Right? 450 00:22:37,210 --> 00:22:39,830 All right. 451 00:22:39,830 --> 00:22:43,544 So any other questions about this? 452 00:22:43,544 --> 00:22:46,210 And if not, I want to talk about something that was on the quiz. 453 00:22:49,606 --> 00:22:50,106 OK. 454 00:22:54,980 --> 00:23:00,060 So here's-- this is the one problem on the quiz that gave 455 00:23:00,060 --> 00:23:04,880 more conceptual difficulty than any other single problem. 456 00:23:04,880 --> 00:23:06,580 People made mistakes on other problems, 457 00:23:06,580 --> 00:23:10,310 but got into conceptual trouble with this problem. 458 00:23:10,310 --> 00:23:13,580 Remember, you had a vehicle driving up a bridge 459 00:23:13,580 --> 00:23:17,560 and the bridge could be changing at some angular rate 460 00:23:17,560 --> 00:23:21,110 and it has some angular acceleration. 461 00:23:21,110 --> 00:23:23,554 And we said to keep the problem simple, 462 00:23:23,554 --> 00:23:25,220 that you could treat this as a particle. 463 00:23:25,220 --> 00:23:26,761 And that really means you didn't have 464 00:23:26,761 --> 00:23:29,040 to deal with angular momentum. 465 00:23:29,040 --> 00:23:30,700 i omega for the object. 466 00:23:30,700 --> 00:23:33,570 You could just treat it like a particle. 467 00:23:33,570 --> 00:23:38,750 And one of the-- where the confusion came from 468 00:23:38,750 --> 00:23:41,594 was in figuring out free body diagrams. 469 00:23:41,594 --> 00:23:43,760 You're told in the problem the language is something 470 00:23:43,760 --> 00:23:50,620 like, the action of the tires on the road result in a net force 471 00:23:50,620 --> 00:23:55,360 up the incline called t. 472 00:23:55,360 --> 00:23:57,850 We didn't call it friction, we didn't say anything, 473 00:23:57,850 --> 00:24:01,100 but people-- a number of people got confused about what's 474 00:24:01,100 --> 00:24:02,570 that have to do with friction. 475 00:24:02,570 --> 00:24:05,960 How does friction come into this? 476 00:24:05,960 --> 00:24:08,570 So let's do the free body diagram for this thing. 477 00:24:08,570 --> 00:24:11,586 So tell me what's-- here's my car, 478 00:24:11,586 --> 00:24:13,460 tell me what to put on the free body diagram. 479 00:24:17,050 --> 00:24:17,997 AUDIENCE: Weight. 480 00:24:17,997 --> 00:24:18,580 PROFESSOR: OK. 481 00:24:18,580 --> 00:24:19,079 Mg? 482 00:24:19,079 --> 00:24:19,770 AUDIENCE: Yep. 483 00:24:19,770 --> 00:24:20,110 PROFESSOR: All right. 484 00:24:20,110 --> 00:24:20,943 So you've got an Mg. 485 00:24:23,820 --> 00:24:24,320 Next. 486 00:24:24,320 --> 00:24:25,236 AUDIENCE: [INAUDIBLE]. 487 00:24:27,487 --> 00:24:28,570 PROFESSOR: A normal force? 488 00:24:28,570 --> 00:24:29,070 OK. 489 00:24:29,070 --> 00:24:34,047 That's going this way we'll call it. 490 00:24:34,047 --> 00:24:35,980 AUDIENCE: T. 491 00:24:35,980 --> 00:24:39,040 PROFESSOR: T. OK. 492 00:24:39,040 --> 00:24:40,800 Now, what about the tires on the road? 493 00:24:40,800 --> 00:24:42,582 What about friction? 494 00:24:42,582 --> 00:24:43,499 AUDIENCE: [INAUDIBLE]. 495 00:24:43,499 --> 00:24:44,248 PROFESSOR: Pardon? 496 00:24:44,248 --> 00:24:45,380 AUDIENCE: That is T. 497 00:24:45,380 --> 00:24:49,430 PROFESSOR: That is T. So you think of the free body diagram, 498 00:24:49,430 --> 00:24:52,650 you just say, what are all the possible sources 499 00:24:52,650 --> 00:24:54,960 of external force on this thing? 500 00:24:54,960 --> 00:24:59,180 And they come from gravity-- a body force-- and then 501 00:24:59,180 --> 00:25:02,650 other things that are in contact with it. 502 00:25:02,650 --> 00:25:05,180 And the wheels are in contact with the road. 503 00:25:05,180 --> 00:25:07,580 And through the wheels you get the normal force. 504 00:25:07,580 --> 00:25:10,370 And through the wheels you get any active friction. 505 00:25:10,370 --> 00:25:13,310 So that is the-- that's total net friction force. 506 00:25:17,982 --> 00:25:19,440 And then once you got that far, you 507 00:25:19,440 --> 00:25:21,440 were asked to come up with an equation of motion 508 00:25:21,440 --> 00:25:23,360 in this direction. 509 00:25:23,360 --> 00:25:26,379 In the direction up the bridge. 510 00:25:26,379 --> 00:25:28,920 And how do you-- so when you go to get an equation of motion, 511 00:25:28,920 --> 00:25:33,664 you say, the sum of the external forces is equal to? 512 00:25:33,664 --> 00:25:35,183 AUDIENCE: [INAUDIBLE]. 513 00:25:35,183 --> 00:25:36,099 AUDIENCE: [INAUDIBLE]. 514 00:25:39,800 --> 00:25:43,000 PROFESSOR: And we'll make these in the x direction here. 515 00:25:43,000 --> 00:25:45,230 Is the mass times the acceleration 516 00:25:45,230 --> 00:25:46,660 in the x direction. 517 00:25:46,660 --> 00:25:50,447 And how would-- what terms appear? 518 00:25:50,447 --> 00:25:51,820 AUDIENCE: [INAUDIBLE]. 519 00:25:51,820 --> 00:25:52,570 PROFESSOR: Pardon? 520 00:25:52,570 --> 00:25:53,537 AUDIENCE: T. 521 00:25:53,537 --> 00:25:55,120 PROFESSOR: Well, those are the forces. 522 00:25:55,120 --> 00:25:56,220 They're on the other side. 523 00:25:56,220 --> 00:25:58,140 I want you to come up with the accelerations. 524 00:25:58,140 --> 00:26:00,500 You need to come up with the acceleration terms. 525 00:26:00,500 --> 00:26:01,820 AUDIENCE: [INAUDIBLE]. 526 00:26:01,820 --> 00:26:02,874 PROFESSOR: Pardon? 527 00:26:02,874 --> 00:26:03,790 AUDIENCE: [INAUDIBLE]. 528 00:26:07,710 --> 00:26:10,230 PROFESSOR: So this problem most easily to think through, 529 00:26:10,230 --> 00:26:12,200 I think, in terms of r and theta and polar 530 00:26:12,200 --> 00:26:15,090 coordinates because you've worked dozens of times 531 00:26:15,090 --> 00:26:20,030 with a complete acceleration of something in a rotating system. 532 00:26:20,030 --> 00:26:23,320 And this is a rotating system. 533 00:26:23,320 --> 00:26:26,790 Polar coordinates is a pretty good way to do this problem. 534 00:26:26,790 --> 00:26:30,665 So what are the accelerations in this direction? 535 00:26:35,930 --> 00:26:38,732 AUDIENCE: [INAUDIBLE] r and theta? 536 00:26:38,732 --> 00:26:40,565 PROFESSOR: Well, I do it in terms of-- well, 537 00:26:40,565 --> 00:26:42,210 I do it in terms of r and theta. 538 00:26:42,210 --> 00:26:47,160 Given a theta dot and a theta double dot and x and r 539 00:26:47,160 --> 00:26:50,100 mount up to the same thing. 540 00:26:50,100 --> 00:26:53,210 But I-- if you call it r, you'll recognize the terms 541 00:26:53,210 --> 00:26:55,140 in your acceleration equation. 542 00:26:55,140 --> 00:27:00,838 So what are the accelerations in the direction up to bridge? 543 00:27:00,838 --> 00:27:02,332 AUDIENCE: r double dot? 544 00:27:02,332 --> 00:27:05,320 AUDIENCE: r double dot. 545 00:27:05,320 --> 00:27:08,308 AUDIENCE: Minus r theta dot. 546 00:27:11,296 --> 00:27:13,290 PROFESSOR: And those are all in the-- you 547 00:27:13,290 --> 00:27:14,910 get all of that in the i direction. 548 00:27:14,910 --> 00:27:17,160 And then if you looked at that, obviously, 549 00:27:17,160 --> 00:27:19,990 you could replace r double dot with x double dot. 550 00:27:19,990 --> 00:27:25,212 r with x if you make your origin here. 551 00:27:25,212 --> 00:27:26,280 Which you probably would. 552 00:27:26,280 --> 00:27:27,488 That's the point of rotation. 553 00:27:27,488 --> 00:27:31,160 So that is your-- that's the mass times the acceleration 554 00:27:31,160 --> 00:27:33,190 in the direction of travel. 555 00:27:33,190 --> 00:27:36,810 And it's got to be equal to T and minus 556 00:27:36,810 --> 00:27:41,260 Mg sine theta probably. 557 00:27:41,260 --> 00:27:42,150 Right? 558 00:27:42,150 --> 00:27:43,580 And that's one equation of motion. 559 00:27:43,580 --> 00:27:45,621 And you're going to do another equation of motion 560 00:27:45,621 --> 00:27:47,880 in the theta dot-- theta hat direction. 561 00:27:47,880 --> 00:27:50,150 Because what other-- what accelerations 562 00:27:50,150 --> 00:27:54,060 are present in this direction? 563 00:27:54,060 --> 00:27:57,218 In the y j hat? 564 00:27:57,218 --> 00:27:58,694 AUDIENCE: Theta double dot? 565 00:27:58,694 --> 00:27:59,860 PROFESSOR: Theta double dot. 566 00:27:59,860 --> 00:28:01,210 You're [INAUDIBLE] one. 567 00:28:01,210 --> 00:28:03,770 What else? 568 00:28:03,770 --> 00:28:10,890 So the sum of the forces in the y-- 569 00:28:10,890 --> 00:28:15,562 so they are, in terms of polar coordinate terms? 570 00:28:15,562 --> 00:28:18,734 AUDIENCE: 2 omega-- 2 omega dot. 571 00:28:18,734 --> 00:28:20,400 PROFESSOR: Somebody said this one first. 572 00:28:20,400 --> 00:28:25,985 Plus 2 theta dot is your-- 573 00:28:25,985 --> 00:28:27,702 AUDIENCE: It's [INAUDIBLE]. 574 00:28:27,702 --> 00:28:28,410 PROFESSOR: Right. 575 00:28:28,410 --> 00:28:29,396 2 theta dot-- 576 00:28:29,396 --> 00:28:30,062 AUDIENCE: r dot. 577 00:28:30,062 --> 00:28:31,420 PROFESSOR: r dot. 578 00:28:31,420 --> 00:28:33,720 And that's in the j direction. 579 00:28:33,720 --> 00:28:35,860 And now you've got-- now you can work the problem. 580 00:28:38,380 --> 00:28:44,720 So how would you go about doing this problem where 581 00:28:44,720 --> 00:28:46,950 you can't treat it as a particle any longer? 582 00:28:51,410 --> 00:28:55,624 We've graduated to that because we've 583 00:28:55,624 --> 00:28:57,540 been doing-- we've been doing angular momentum 584 00:28:57,540 --> 00:29:00,450 stuff with mass moment of inertia matrices. 585 00:29:00,450 --> 00:29:06,110 So now you want to do this more as a full fledged dynamics 586 00:29:06,110 --> 00:29:07,910 problem, taking into consideration that. 587 00:29:07,910 --> 00:29:10,230 So how do you-- what is the mass-- what 588 00:29:10,230 --> 00:29:16,320 do you have to modify-- do to modify our free body diagram? 589 00:29:21,042 --> 00:29:22,250 Here's my simplified vehicle. 590 00:29:22,250 --> 00:29:24,730 What are the forces on it that you need to deal with? 591 00:29:32,050 --> 00:29:34,304 AUDIENCE: Well, there are two normal forces. 592 00:29:34,304 --> 00:29:35,470 There are two normal forces. 593 00:29:35,470 --> 00:29:35,806 PROFESSOR: Yeah. 594 00:29:35,806 --> 00:29:37,140 So you can add-- need two because you've 595 00:29:37,140 --> 00:29:38,590 got two wheels on this thing. 596 00:29:38,590 --> 00:29:43,320 And so there's axle one here- so you get one here and one here. 597 00:29:43,320 --> 00:29:46,260 I call this one N1, N2. 598 00:29:46,260 --> 00:29:48,472 What else? 599 00:29:48,472 --> 00:29:50,892 AUDIENCE: [INAUDIBLE]. 600 00:29:50,892 --> 00:29:52,350 The same T and Mg. 601 00:29:52,350 --> 00:29:54,040 PROFESSOR: So you've got a g somewhere. 602 00:29:54,040 --> 00:29:57,740 And so you still have an Mg term. 603 00:29:57,740 --> 00:30:00,098 What other external forces? 604 00:30:00,098 --> 00:30:01,430 AUDIENCE: T? 605 00:30:01,430 --> 00:30:04,100 PROFESSOR: T. But now T's the problem. 606 00:30:04,100 --> 00:30:06,880 So as I gave you before, T is the total net force. 607 00:30:06,880 --> 00:30:09,786 But that's not adequate in this problem any longer. 608 00:30:09,786 --> 00:30:11,384 AUDIENCE: [INAUDIBLE] 609 00:30:11,384 --> 00:30:12,050 PROFESSOR: Yeah. 610 00:30:12,050 --> 00:30:17,440 So we're going to end up having a friction force that's 611 00:30:17,440 --> 00:30:25,105 supplied here-- f1-- and another one applied here-- f2. 612 00:30:25,105 --> 00:30:31,250 And f1 plus f2 would be T. OK. 613 00:30:31,250 --> 00:30:34,968 So how many unknowns do we have? 614 00:30:40,584 --> 00:30:41,520 AUDIENCE: Four? 615 00:30:41,520 --> 00:30:43,395 PROFESSOR: Well, it looks like four are here. 616 00:30:43,395 --> 00:30:46,600 Plus any motion stuff that you have to solve for. 617 00:30:46,600 --> 00:30:50,260 So how many equations of motion can you write? 618 00:30:50,260 --> 00:30:53,830 Remember, you get one equation for every vector component 619 00:30:53,830 --> 00:30:54,330 direction. 620 00:30:54,330 --> 00:30:57,360 So x, y, z gives you three components 621 00:30:57,360 --> 00:30:59,690 and you could conceivably write three equations. 622 00:30:59,690 --> 00:31:02,060 So tell me the relevant equations of motion 623 00:31:02,060 --> 00:31:03,930 that we could come up with for this problem. 624 00:31:09,738 --> 00:31:12,650 AUDIENCE: Sum forces in x, sum forces in y. 625 00:31:12,650 --> 00:31:13,390 PROFESSOR: OK. 626 00:31:13,390 --> 00:31:15,550 I argue-- sum of the forces in x. 627 00:31:20,360 --> 00:31:21,110 That's still true. 628 00:31:21,110 --> 00:31:22,250 Yep. 629 00:31:22,250 --> 00:31:23,970 Sum of the forces in y you said. 630 00:31:27,610 --> 00:31:29,591 Is this 0? 631 00:31:29,591 --> 00:31:30,090 No. 632 00:31:30,090 --> 00:31:32,170 Because that bridge is moving. 633 00:31:32,170 --> 00:31:33,010 OK. 634 00:31:33,010 --> 00:31:33,978 What else? 635 00:31:33,978 --> 00:31:36,120 AUDIENCE: Torque. 636 00:31:36,120 --> 00:31:37,330 PROFESSOR: OK. 637 00:31:37,330 --> 00:31:39,640 And how do you do that? 638 00:31:39,640 --> 00:31:42,746 Sum of the torques in what direction? 639 00:31:42,746 --> 00:31:44,520 AUDIENCE: Theta hat. 640 00:31:44,520 --> 00:31:47,400 PROFESSOR: Well, no. 641 00:31:47,400 --> 00:31:47,900 Yeah. 642 00:31:47,900 --> 00:31:48,470 Well, you could. 643 00:31:48,470 --> 00:31:49,000 Excuse me. 644 00:31:49,000 --> 00:31:51,759 I shouldn't say that. 645 00:31:51,759 --> 00:31:53,300 Do you have more than one torque term 646 00:31:53,300 --> 00:31:54,100 that you can work with here? 647 00:31:54,100 --> 00:31:55,350 More than one torque equation? 648 00:32:04,637 --> 00:32:05,220 AUDIENCE: Yes? 649 00:32:05,220 --> 00:32:05,955 AUDIENCE: Yes. 650 00:32:05,955 --> 00:32:08,172 PROFESSOR: How's that? 651 00:32:08,172 --> 00:32:09,880 So let's just think through these things. 652 00:32:09,880 --> 00:32:13,340 You could have a torque-- you could have a torque in, 653 00:32:13,340 --> 00:32:16,560 this is x, y, z out of the board. 654 00:32:16,560 --> 00:32:19,150 You could have a z torque making this thing trying 655 00:32:19,150 --> 00:32:22,110 to pitch up and down. 656 00:32:22,110 --> 00:32:24,575 You could have an x torque making it trying to rollover. 657 00:32:24,575 --> 00:32:26,700 But we don't have any information in that direction 658 00:32:26,700 --> 00:32:28,150 and its constrained. 659 00:32:28,150 --> 00:32:32,890 And we could have a y torque making it try to go like that. 660 00:32:32,890 --> 00:32:34,800 And we don't have much information there. 661 00:32:34,800 --> 00:32:38,580 So it seems like we've got a z torque 662 00:32:38,580 --> 00:32:39,940 equation that we could write. 663 00:32:45,920 --> 00:32:49,450 And that's one-- seems like we got three equations. 664 00:32:49,450 --> 00:32:51,420 Sum of the forces x, sum of the forces y, 665 00:32:51,420 --> 00:32:53,210 and a torque in the z. 666 00:32:53,210 --> 00:32:59,162 And doesn't seem like quite enough. 667 00:32:59,162 --> 00:33:00,870 And that's as far as I've gotten with it. 668 00:33:00,870 --> 00:33:04,190 I just thought of this on the way over here. 669 00:33:04,190 --> 00:33:06,950 So I'd sit down and start working this out and trying 670 00:33:06,950 --> 00:33:08,950 to figure out, where is my fourth equation going 671 00:33:08,950 --> 00:33:09,491 to come from? 672 00:33:13,960 --> 00:33:17,884 And I'm not exactly sure quite yet. 673 00:33:17,884 --> 00:33:20,457 But, generally, I'd go about setting it up like that 674 00:33:20,457 --> 00:33:21,915 and start working those things out. 675 00:33:26,470 --> 00:33:27,170 OK. 676 00:33:27,170 --> 00:33:29,210 Any other questions or questions about the quiz? 677 00:33:40,260 --> 00:33:42,210 We're done early.