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 --> 00:00:10,160 continue to offer high-quality educational resources for free. 5 00:00:10,160 --> 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 --> 00:00:17,325 at ocw.mit.edu. 8 00:00:21,730 --> 00:00:24,510 PROFESSOR: --get started. 9 00:00:24,510 --> 00:00:27,880 What do you think are some important new concepts 10 00:00:27,880 --> 00:00:32,189 that we've been talking about in the last week? 11 00:00:32,189 --> 00:00:32,800 Make a list. 12 00:00:32,800 --> 00:00:33,526 Betsy. 13 00:00:33,526 --> 00:00:34,984 AUDIENCE: Various ways to calculate 14 00:00:34,984 --> 00:00:37,377 kinetic energy [INAUDIBLE]. 15 00:00:37,377 --> 00:00:37,960 PROFESSOR: OK. 16 00:00:45,896 --> 00:00:47,384 All right? 17 00:00:47,384 --> 00:00:48,561 AUDIENCE: Virtual work. 18 00:00:48,561 --> 00:00:49,560 PROFESSOR: Virtual work. 19 00:01:02,150 --> 00:01:04,309 Something else. 20 00:01:04,309 --> 00:01:05,295 [INAUDIBLE] 21 00:01:05,295 --> 00:01:07,267 AUDIENCE: The Lagrange equation. 22 00:01:07,267 --> 00:01:08,253 PROFESSOR: All right. 23 00:01:18,610 --> 00:01:19,364 [INAUDIBLE] 24 00:01:19,364 --> 00:01:21,280 AUDIENCE: [INAUDIBLE] generalized coordinates. 25 00:01:21,280 --> 00:01:22,738 PROFESSOR: Generalized coordinates. 26 00:01:34,670 --> 00:01:37,286 Anything else on your list? 27 00:01:37,286 --> 00:01:38,494 AUDIENCE: Generalized forces. 28 00:01:43,490 --> 00:01:45,050 PROFESSOR: OK. 29 00:01:45,050 --> 00:01:46,455 Lagrange equations. 30 00:01:46,455 --> 00:01:48,820 That's a pretty good list. 31 00:01:48,820 --> 00:01:51,340 Now, we've talked about quite a few new things lately. 32 00:01:51,340 --> 00:01:53,650 So is there any-- did you come in here today 33 00:01:53,650 --> 00:01:56,400 with any questions about something 34 00:01:56,400 --> 00:02:00,645 that just isn't sitting right on these topics or anything else? 35 00:02:00,645 --> 00:02:02,520 And I think write up a couple extra questions 36 00:02:02,520 --> 00:02:05,540 that we may be able to cover them as we go along. 37 00:02:05,540 --> 00:02:06,788 Steven? 38 00:02:06,788 --> 00:02:08,784 AUDIENCE: So when you solve the entire thing, 39 00:02:08,784 --> 00:02:10,780 everything in general is coordinates, right? 40 00:02:10,780 --> 00:02:12,776 But say you want to find the actual numbers. 41 00:02:12,776 --> 00:02:18,196 Do you calculate them as your [INAUDIBLE], 42 00:02:18,196 --> 00:02:19,320 or do you just [INAUDIBLE]? 43 00:02:22,030 --> 00:02:26,100 PROFESSOR: So this is really how to use the equations you end up 44 00:02:26,100 --> 00:02:28,970 with in practical situations? 45 00:02:28,970 --> 00:02:31,015 OK. 46 00:02:31,015 --> 00:02:33,270 I'm just going to answer this one. 47 00:02:33,270 --> 00:02:36,400 So when you choose your generalized coordinates, 48 00:02:36,400 --> 00:02:39,620 you'll probably choose about the same ones 49 00:02:39,620 --> 00:02:42,470 you'd choose if you did the direct method. 50 00:02:42,470 --> 00:02:48,580 So if you did direct method and got equations of motion, 51 00:02:48,580 --> 00:02:50,960 would you be asking the same question? 52 00:02:50,960 --> 00:02:51,918 AUDIENCE: I don't know. 53 00:02:51,918 --> 00:02:52,700 PROFESSOR: All right? 54 00:02:52,700 --> 00:02:53,200 OK. 55 00:02:53,200 --> 00:02:57,050 So really, you want to choose coordinates 56 00:02:57,050 --> 00:03:02,330 that are going to be practically useful in the end. 57 00:03:02,330 --> 00:03:02,830 OK? 58 00:03:02,830 --> 00:03:04,538 And so it shouldn't matter whether you're 59 00:03:04,538 --> 00:03:06,930 doing it by Lagrange or doing it by the direct method 60 00:03:06,930 --> 00:03:08,336 to get to them, you're going to use them 61 00:03:08,336 --> 00:03:09,480 in the same way in the end. 62 00:03:12,140 --> 00:03:15,580 Now, we have two methods to get equations of motion now. 63 00:03:15,580 --> 00:03:17,910 When you're doing tough problems, complicated problems, 64 00:03:17,910 --> 00:03:22,030 I would always do it one way and use the other way to check it. 65 00:03:22,030 --> 00:03:23,280 I'd end up doing it both ways. 66 00:03:23,280 --> 00:03:26,410 If it was really important, you do it both ways. 67 00:03:26,410 --> 00:03:28,720 And one will give you insight about the other. 68 00:03:28,720 --> 00:03:30,920 Yesterday in lecture, I was talking about, 69 00:03:30,920 --> 00:03:37,570 does it make sense to have a Coriolis term in this equation? 70 00:03:37,570 --> 00:03:39,610 Would you expect it? 71 00:03:39,610 --> 00:03:41,820 And that kind of common sense checking. 72 00:03:41,820 --> 00:03:42,320 All right. 73 00:03:42,320 --> 00:03:43,230 Any other questions? 74 00:03:43,230 --> 00:03:45,770 Good question. 75 00:03:45,770 --> 00:03:48,140 Generalized coordinates or forces or anything? 76 00:03:48,140 --> 00:03:48,640 All right. 77 00:03:48,640 --> 00:03:49,830 Let's get on. 78 00:03:49,830 --> 00:03:53,180 Let's keep working here. 79 00:03:53,180 --> 00:04:02,390 So I have an assignment for you, and it's here. 80 00:04:02,390 --> 00:04:04,040 This is a familiar problem that you've 81 00:04:04,040 --> 00:04:06,180 worked before by other methods. 82 00:04:06,180 --> 00:04:08,820 And you did, in fact, even find the kinetic and potential 83 00:04:08,820 --> 00:04:11,200 energies, I think, last week for this. 84 00:04:11,200 --> 00:04:16,490 So here's a cart with a rod, uniform rod. 85 00:04:16,490 --> 00:04:21,329 No dashpot at the moment, and no external forces. 86 00:04:21,329 --> 00:04:23,810 I'm going to give you coordinates to use. 87 00:04:23,810 --> 00:04:26,930 So here's our inertial system, deflection of the cart 88 00:04:26,930 --> 00:04:30,080 x, rotation of the rod theta. 89 00:04:30,080 --> 00:04:32,720 We're going to break you into groups of about-- we 90 00:04:32,720 --> 00:04:36,625 got a big group here-- 5, 10. 91 00:04:39,190 --> 00:04:45,860 Well, I want you to break into three groups, kind 92 00:04:45,860 --> 00:04:48,121 of like five or six there, five or six there, 93 00:04:48,121 --> 00:04:49,120 and the same thing here. 94 00:04:49,120 --> 00:04:52,670 And one group each-- so this group in the front 95 00:04:52,670 --> 00:04:57,030 left, compute the kinetic energy of this system. 96 00:04:57,030 --> 00:04:59,910 And the group behind them, compute the potential. 97 00:04:59,910 --> 00:05:01,890 And this group over here, come up 98 00:05:01,890 --> 00:05:14,070 with the velocity of G and the G dot VG, the velocity squared. 99 00:05:14,070 --> 00:05:14,919 OK? 100 00:05:14,919 --> 00:05:16,710 And this won't you take long because you've 101 00:05:16,710 --> 00:05:17,880 done the stuff before. 102 00:05:17,880 --> 00:05:20,088 And then when you get done, somebody from your group, 103 00:05:20,088 --> 00:05:22,430 just when you're done, come up and write it down. 104 00:05:22,430 --> 00:05:23,892 All right. 105 00:05:23,892 --> 00:05:25,270 Let's sort these out. 106 00:05:28,850 --> 00:05:31,780 So let's start with the velocity. 107 00:05:31,780 --> 00:05:35,700 We're going to need the velocity to be able to finish out 108 00:05:35,700 --> 00:05:36,525 the kinetic energy. 109 00:05:36,525 --> 00:05:42,940 So the vector, right, and they've 110 00:05:42,940 --> 00:05:44,165 broken it into two pieces. 111 00:05:46,860 --> 00:05:49,040 And since the others, if you haven't worked on it, 112 00:05:49,040 --> 00:05:49,750 are you guys-- 113 00:05:49,750 --> 00:05:52,205 STUDENT: Can you switch the sine and cosine? 114 00:05:52,205 --> 00:05:56,560 PROFESSOR: I was about to ask you about that. 115 00:05:56,560 --> 00:05:58,080 I think that that's better. 116 00:05:58,080 --> 00:05:59,699 So this is sine, right. 117 00:05:59,699 --> 00:06:01,136 STUDENT: No, that's cosine. 118 00:06:01,136 --> 00:06:02,573 PROFESSOR: I just erased it. 119 00:06:02,573 --> 00:06:05,447 That's cosine, sine. 120 00:06:10,240 --> 00:06:14,310 This is your l theta dot piece and you want it this way. 121 00:06:14,310 --> 00:06:18,420 And that's theta, so it should be sine theta j, cosine theta 122 00:06:18,420 --> 00:06:18,920 i. 123 00:06:21,920 --> 00:06:23,710 I think that's good for the velocity. 124 00:06:23,710 --> 00:06:29,254 And then velocity squared, just the square reach of pieces. 125 00:06:52,650 --> 00:06:54,559 So we have v dot b. 126 00:06:54,559 --> 00:06:56,850 We need to put this into the kinetic energy expression, 127 00:06:56,850 --> 00:07:00,480 but one at a time here. 128 00:07:00,480 --> 00:07:03,210 The potential energy expression, 1/2 kx squared. 129 00:07:03,210 --> 00:07:06,160 Everybody good with that? 130 00:07:06,160 --> 00:07:09,530 Looks like the spring potential energy. 131 00:07:09,530 --> 00:07:16,530 And m2g, 1/2 l is a position of the center of mass. 132 00:07:16,530 --> 00:07:19,760 1 minus cosine l that times 1 minus cosine theta 133 00:07:19,760 --> 00:07:23,230 is the amount that it changes height from the reference. 134 00:07:23,230 --> 00:07:25,785 What's the reference position? 135 00:07:25,785 --> 00:07:27,410 What have you assumed for the reference 136 00:07:27,410 --> 00:07:29,520 position in this formulation? 137 00:07:29,520 --> 00:07:30,020 Yeah? 138 00:07:30,020 --> 00:07:31,619 STUDENT: [INAUDIBLE]. 139 00:07:31,619 --> 00:07:33,160 PROFESSOR: So your reference position 140 00:07:33,160 --> 00:07:37,330 is center of mass when it's hanging straight down. 141 00:07:37,330 --> 00:07:37,990 Great. 142 00:07:37,990 --> 00:07:40,804 So there's your potential energy expression. 143 00:07:40,804 --> 00:07:45,010 Kinetic energy expression-- everybody 144 00:07:45,010 --> 00:07:45,990 comfortable with that? 145 00:07:57,370 --> 00:07:59,040 I'm not totally comfortable with it. 146 00:08:02,460 --> 00:08:05,800 Let's talk about a minute about kinetic energy and say, 147 00:08:05,800 --> 00:08:11,390 what can we use, what formulas can we use here for t? 148 00:08:11,390 --> 00:08:15,230 There's totally general and then we can narrow it down. 149 00:08:15,230 --> 00:08:16,390 What do you suggest? 150 00:08:16,390 --> 00:08:22,018 STUDENT: For the [INAUDIBLE] you can-- 151 00:08:22,018 --> 00:08:24,196 you have to take into account that it's not spinning 152 00:08:24,196 --> 00:08:25,406 about its center of mass. 153 00:08:25,406 --> 00:08:28,310 So you use the general mass formula one half omega 154 00:08:28,310 --> 00:08:30,730 dot, its angular momentum [INAUDIBLE]. 155 00:08:30,730 --> 00:08:33,590 PROFESSOR: OK, so we have some general formulas. 156 00:08:33,590 --> 00:08:37,510 And let's take a step by step approach to this. 157 00:08:37,510 --> 00:08:41,250 We have how many rigid bodies? 158 00:08:41,250 --> 00:08:42,340 Two. 159 00:08:42,340 --> 00:08:45,110 And you compute the kinetic energy 160 00:08:45,110 --> 00:08:48,545 for each one individually. 161 00:08:48,545 --> 00:08:50,280 That's the safest way to go about this. 162 00:08:50,280 --> 00:08:55,290 So I would stay away initially from doing this. 163 00:08:55,290 --> 00:08:58,319 You lumping the two together. 164 00:08:58,319 --> 00:08:59,860 So I would take the two individually. 165 00:08:59,860 --> 00:09:05,110 So if you take the main mass, the block on wheels, 166 00:09:05,110 --> 00:09:06,065 what is its velocity? 167 00:09:11,170 --> 00:09:13,030 Its velocity is this x dot, right? 168 00:09:13,030 --> 00:09:14,897 And its kinetic energy is? 169 00:09:14,897 --> 00:09:15,914 STUDENT: [INAUDIBLE]. 170 00:09:15,914 --> 00:09:17,330 PROFESSOR: So for the first block, 171 00:09:17,330 --> 00:09:20,480 you have m1x dot squared. 172 00:09:20,480 --> 00:09:21,312 And you're done. 173 00:09:21,312 --> 00:09:22,270 That's the first block. 174 00:09:22,270 --> 00:09:24,050 Now you need the second rigid body. 175 00:09:24,050 --> 00:09:30,407 So the second rigid body, you could 176 00:09:30,407 --> 00:09:34,090 do-- the full, general expression 177 00:09:34,090 --> 00:09:50,960 is one half m2 vg in o dot vg in o plus a half omega H, 178 00:09:50,960 --> 00:09:55,200 with respect to g. 179 00:09:55,200 --> 00:09:56,200 Good with that, Vicente? 180 00:09:59,057 --> 00:10:00,015 You need it transposed? 181 00:10:02,580 --> 00:10:05,460 Good. 182 00:10:05,460 --> 00:10:10,750 And can we simplify that at all? 183 00:10:16,290 --> 00:10:19,870 For example, is it that rod rotating about a fixed point? 184 00:10:23,272 --> 00:10:24,150 It's not. 185 00:10:24,150 --> 00:10:26,940 The point it rotates about moves, right? 186 00:10:26,940 --> 00:10:30,180 So you can't say it's just-- you can't use, for example, 187 00:10:30,180 --> 00:10:32,370 parallel axis theorem and just say it's 1/2 188 00:10:32,370 --> 00:10:35,540 I with respect to that point, theta dot squared. 189 00:10:35,540 --> 00:10:37,780 Won't work. 190 00:10:37,780 --> 00:10:39,710 Can't use that one. 191 00:10:39,710 --> 00:10:41,690 You will find, that if you work this out, 192 00:10:41,690 --> 00:10:47,226 you can say 1/2 I with respect to g omega squared, 193 00:10:47,226 --> 00:10:48,600 in this problem it will work out. 194 00:10:48,600 --> 00:10:52,340 It'll come out to that because this is a planar motion problem 195 00:10:52,340 --> 00:10:56,340 and there's only one component of rotation. 196 00:10:56,340 --> 00:11:06,840 So this will work out to be a 1/2 I with respect zz of m2 197 00:11:06,840 --> 00:11:10,360 with respect to g omega z squared, 198 00:11:10,360 --> 00:11:12,339 in this case theta dot squared. 199 00:11:12,339 --> 00:11:13,880 That's what this term will reduce to. 200 00:11:13,880 --> 00:11:16,320 But don't assume it just out of the box. 201 00:11:16,320 --> 00:11:17,730 And then you need this term. 202 00:11:17,730 --> 00:11:18,900 And that's why we need bg. 203 00:11:21,530 --> 00:11:24,630 So you need to do that. 204 00:11:24,630 --> 00:11:26,690 Put that piece in over there. 205 00:11:26,690 --> 00:11:40,370 So we need to-- so what is Izz about g for m2? 206 00:11:45,320 --> 00:11:47,020 For a rod, slender rod. 207 00:11:51,410 --> 00:11:56,090 So m2 l squared over 12. 208 00:11:56,090 --> 00:11:58,340 That's what you need to put in here. 209 00:11:58,340 --> 00:12:02,930 We have an expression for v over there. 210 00:12:02,930 --> 00:12:04,470 We know everything now. 211 00:12:07,380 --> 00:12:10,485 So now let's apply our Lagrange equations. 212 00:12:16,160 --> 00:12:19,179 And I'm going to need to rearrange the board here 213 00:12:19,179 --> 00:12:19,720 a little bit. 214 00:12:22,680 --> 00:12:24,580 I'm going to need that board space. 215 00:12:24,580 --> 00:12:33,760 So our T is 1/2 m1 x1 dot squared, or x dot squared 216 00:12:33,760 --> 00:12:42,110 plus 1/2 m2 vg.vg. 217 00:12:57,160 --> 00:13:00,820 And now I can cover this up. 218 00:13:00,820 --> 00:13:02,020 All right. 219 00:13:02,020 --> 00:13:05,910 So the next task is, let's work out, 220 00:13:05,910 --> 00:13:08,384 do our Lagrange equations work? 221 00:13:08,384 --> 00:13:10,300 So how many generalized coordinates do we have 222 00:13:10,300 --> 00:13:13,130 and what are they? 223 00:13:13,130 --> 00:13:14,990 x and theta, that we've chosen. 224 00:13:14,990 --> 00:13:19,470 Two degrees of freedom-- they're complete, independent, 225 00:13:19,470 --> 00:13:21,000 polynomic. 226 00:13:21,000 --> 00:13:22,920 And we can use Lagrange equations. 227 00:13:22,920 --> 00:13:26,820 We're going to come up with two equations of motion. 228 00:13:26,820 --> 00:13:28,160 And we're going to apply this. 229 00:13:28,160 --> 00:13:29,420 That's the Lagrange equation. 230 00:13:29,420 --> 00:13:30,919 We're going to apply it twice, where 231 00:13:30,919 --> 00:13:34,750 l is defined as T minus v. 232 00:13:34,750 --> 00:13:38,960 If you just plug in l into this expression and just expand it, 233 00:13:38,960 --> 00:13:41,680 you get-- instead of two terms, you get four terms, 234 00:13:41,680 --> 00:13:43,250 because you have these two guys. 235 00:13:43,250 --> 00:13:46,450 And this term-- for mechanical systems, 236 00:13:46,450 --> 00:13:48,270 what can you say about this term generally? 237 00:13:48,270 --> 00:13:52,110 That's the derivative of v with respect 238 00:13:52,110 --> 00:13:55,941 to Q dots to velocities. 239 00:13:55,941 --> 00:13:56,440 Why? 240 00:13:59,940 --> 00:14:01,440 STUDENT: Conservative forces? 241 00:14:01,440 --> 00:14:04,460 PROFESSOR: No, just that you find it for mechanical systems, 242 00:14:04,460 --> 00:14:09,030 springs and gravity, you will never find that the potential 243 00:14:09,030 --> 00:14:12,877 energy as a function of time or velocity just 244 00:14:12,877 --> 00:14:14,835 isn't-- and if it's not a function of velocity, 245 00:14:14,835 --> 00:14:17,085 you take a derivative with respect to velocity you get 246 00:14:17,085 --> 00:14:17,930 0's. 247 00:14:17,930 --> 00:14:20,115 So this goes to 0 for mechanical systems. 248 00:14:23,160 --> 00:14:25,370 An exception for non-mechanical would 249 00:14:25,370 --> 00:14:28,350 be like a charged particle in a magnetic field. 250 00:14:28,350 --> 00:14:31,800 Then the forces get involved with velocities and so forth. 251 00:14:31,800 --> 00:14:32,850 It gets messy. 252 00:14:32,850 --> 00:14:34,160 0 for mechanical systems. 253 00:14:34,160 --> 00:14:38,620 So we really don't have to deal with three terms-- 254 00:14:38,620 --> 00:14:41,650 that one, that one, that one, and then on the right 255 00:14:41,650 --> 00:14:43,600 hand side are generalized forces. 256 00:14:43,600 --> 00:14:48,430 So we can break into four smaller groups. 257 00:14:48,430 --> 00:14:52,270 Two groups are going to do the x equation. 258 00:14:52,270 --> 00:14:59,160 You have to take these Qj's is this problem are Q1 is x 259 00:14:59,160 --> 00:15:02,500 and Q2 is theta. 260 00:15:02,500 --> 00:15:08,660 So for the x equations, we need to do these derivatives. 261 00:15:08,660 --> 00:15:20,450 And for the theta equation, we need to do these computations. 262 00:15:20,450 --> 00:15:27,380 So let's have one group A do these. 263 00:15:27,380 --> 00:15:33,255 Group B do these and group C do these and a D group do those. 264 00:15:33,255 --> 00:15:38,110 So break yourselves into four groups 265 00:15:38,110 --> 00:15:45,360 and we'll do A, B, C here in the center, 266 00:15:45,360 --> 00:15:47,400 group here, four or five, four or five, 267 00:15:47,400 --> 00:15:50,822 or group here the C group, and D group over here. 268 00:15:50,822 --> 00:15:53,030 Do these calculations and let's get our two equations 269 00:15:53,030 --> 00:15:53,880 in motion. 270 00:15:53,880 --> 00:15:56,199 And when you get your stuff done, 271 00:15:56,199 --> 00:15:58,740 so the A group, when you finish, come up here and right here, 272 00:15:58,740 --> 00:16:00,110 write your stuff. 273 00:16:00,110 --> 00:16:02,834 And the B group, write your answer here, and the C group 274 00:16:02,834 --> 00:16:03,500 and the D group. 275 00:16:03,500 --> 00:16:05,666 As soon as you get it done, come up and put it down. 276 00:16:08,020 --> 00:16:16,120 We got this term, this term, this term, and this term. 277 00:16:16,120 --> 00:16:17,560 Did you guys check? 278 00:16:17,560 --> 00:16:20,711 Did you guys get a little time, B group 279 00:16:20,711 --> 00:16:21,710 to check on the A group? 280 00:16:26,700 --> 00:16:27,880 Which one was it? 281 00:16:27,880 --> 00:16:29,446 Who's checking on whom? 282 00:16:29,446 --> 00:16:31,070 You're checking on-- what do you think? 283 00:16:31,070 --> 00:16:32,798 STUDENT: We got the same answer. 284 00:16:32,798 --> 00:16:40,860 PROFESSOR: So main mass acceleration, the second mass, 285 00:16:40,860 --> 00:16:45,390 its total acceleration, these pieces, and there's 286 00:16:45,390 --> 00:16:47,450 an acceleration that's Eulerian and then 287 00:16:47,450 --> 00:16:49,110 there's an acceleration that is-- 288 00:16:49,110 --> 00:16:52,790 what's this term related to? 289 00:16:52,790 --> 00:16:56,690 You expect it to come up? 290 00:16:56,690 --> 00:16:58,690 And the kx term, and all these are 291 00:16:58,690 --> 00:17:00,510 going to equal to the generalized forces 292 00:17:00,510 --> 00:17:02,950 of any non-conservative forces. 293 00:17:02,950 --> 00:17:04,569 So you're OK with this one. 294 00:17:04,569 --> 00:17:07,170 Let's move on to this one, then. 295 00:17:07,170 --> 00:17:10,490 Who's the check group here? 296 00:17:10,490 --> 00:17:11,240 What do you think? 297 00:17:11,240 --> 00:17:14,759 STUDENT: I think they made [INAUDIBLE]. 298 00:17:14,759 --> 00:17:17,353 PROFESSOR: Do you think there's a problem here? 299 00:17:17,353 --> 00:17:19,889 STUDENT: There might be. 300 00:17:19,889 --> 00:17:21,951 PROFESSOR: Can you give me an alternative? 301 00:17:21,951 --> 00:17:26,720 STUDENT: It may be that their dt d theta there should 302 00:17:26,720 --> 00:17:28,640 be in the time derivative. 303 00:17:33,220 --> 00:17:34,430 PROFESSOR: All right. 304 00:17:34,430 --> 00:17:38,775 So we need d, the derivative of this with respect to theta dot. 305 00:17:41,960 --> 00:17:45,420 The first term doesn't give me anything, the mx dot squared. 306 00:17:45,420 --> 00:17:49,730 The third term gives you-- should give you 307 00:17:49,730 --> 00:17:56,200 an Izz g theta double dot, eventually, right? 308 00:17:56,200 --> 00:18:05,370 So for sure, this d by dt, the partial of T 309 00:18:05,370 --> 00:18:08,120 with respect to theta dot. 310 00:18:08,120 --> 00:18:09,570 So we just run through it. 311 00:18:09,570 --> 00:18:11,510 The first term gives us nothing. 312 00:18:11,510 --> 00:18:18,090 The third piece gives us, with respect to theta dot Izz, 313 00:18:18,090 --> 00:18:19,030 theta dot. 314 00:18:19,030 --> 00:18:20,774 And no one wipes out the 1/2. 315 00:18:20,774 --> 00:18:22,690 The time derivative makes it theta double dot. 316 00:18:22,690 --> 00:18:28,600 So the third term's going to be an Izz g theta double dot. 317 00:18:28,600 --> 00:18:32,490 And it's the second term that needs a derivative of this 318 00:18:32,490 --> 00:18:34,090 with respect to theta dot. 319 00:18:37,690 --> 00:18:45,470 So both terms are going to yield some stuff, right? 320 00:18:45,470 --> 00:18:47,650 A lot of stuff. 321 00:18:47,650 --> 00:18:49,710 All right, I'm going to write down 322 00:18:49,710 --> 00:18:54,070 how this should work it out, rather than try to grind it out 323 00:18:54,070 --> 00:18:54,860 real time here. 324 00:19:06,750 --> 00:19:39,501 Izz 325 00:19:39,501 --> 00:19:40,000 All right. 326 00:19:40,000 --> 00:19:42,270 These are the terms that should appear. 327 00:19:44,880 --> 00:19:47,260 This is the piece about g. 328 00:19:47,260 --> 00:19:52,470 This is the-- no, that's not quite right. 329 00:19:55,470 --> 00:19:56,670 That's the piece about g. 330 00:19:56,670 --> 00:20:04,605 This should be about-- and then this term. 331 00:20:08,000 --> 00:20:10,080 That's how it actually should check out. 332 00:20:10,080 --> 00:20:11,720 Do you agree with me? 333 00:20:11,720 --> 00:20:12,740 Looks OK? 334 00:20:12,740 --> 00:20:22,660 And then if we add to that the m2 g l over 2 sine theta, 335 00:20:22,660 --> 00:20:25,350 which is our gravitational potential energy, all of that 336 00:20:25,350 --> 00:20:30,510 added together ought to be equal to q theta. 337 00:20:30,510 --> 00:20:34,400 So let's move on to looking at the generalized forces 338 00:20:34,400 --> 00:20:36,040 for this problem. 339 00:20:36,040 --> 00:20:43,020 So don't know where you guys went wrong on this. 340 00:20:43,020 --> 00:20:46,535 But if you have any questions-- we 341 00:20:46,535 --> 00:20:47,910 can talk about this for a minute. 342 00:20:52,370 --> 00:20:54,180 STUDENT: [INAUDIBLE]. 343 00:20:54,180 --> 00:20:57,830 PROFESSOR: d t d theta? 344 00:20:57,830 --> 00:21:05,240 So you got to take the partial derivative of this expression 345 00:21:05,240 --> 00:21:07,446 with respect to theta dot. 346 00:21:07,446 --> 00:21:10,180 This piece gives you a contribution 347 00:21:10,180 --> 00:21:13,136 that will be Izz theta dot. 348 00:21:13,136 --> 00:21:14,510 And you take its time derivative. 349 00:21:14,510 --> 00:21:19,840 So that's pretty obvious why it gives you the first piece. 350 00:21:19,840 --> 00:21:21,714 STUDENT: I think the problem is understanding 351 00:21:21,714 --> 00:21:25,286 in the first place that they did [INAUDIBLE]. 352 00:21:25,286 --> 00:21:26,760 PROFESSOR: Oh. 353 00:21:26,760 --> 00:21:27,672 STUDENT: [INAUDIBLE]. 354 00:21:31,320 --> 00:21:33,180 PROFESSOR: Then the second one, when 355 00:21:33,180 --> 00:21:36,740 you take the derivative of this expression with respect 356 00:21:36,740 --> 00:21:41,180 to-- this is T with respect to theta 357 00:21:41,180 --> 00:21:44,190 dot of this part, this expression, 358 00:21:44,190 --> 00:21:49,370 you get 2 times what's inside times 359 00:21:49,370 --> 00:21:56,160 the derivative of the inside with respect to theta dot. 360 00:21:56,160 --> 00:22:03,270 And that'll give you another L over 2 cosine theta. 361 00:22:03,270 --> 00:22:05,150 And I think you're done. 362 00:22:05,150 --> 00:22:08,690 This times this stuff, right? 363 00:22:08,690 --> 00:22:10,680 2 times this times the derivative 364 00:22:10,680 --> 00:22:12,976 of the inside, which is-- the derivative 365 00:22:12,976 --> 00:22:15,070 of the inside with respect to theta dot 366 00:22:15,070 --> 00:22:17,790 should be L over 2 cosine theta. 367 00:22:17,790 --> 00:22:22,360 So you get an x dot plus L over 2 theta 368 00:22:22,360 --> 00:22:29,490 dot cosine theta 2 times that times 369 00:22:29,490 --> 00:22:32,925 the derivative of the inside with respect to theta dot. 370 00:22:32,925 --> 00:22:35,840 This is the only term that contributes is that. 371 00:22:35,840 --> 00:22:39,210 And then we've already done the derivative of this with respect 372 00:22:39,210 --> 00:22:41,787 to theta dot. 373 00:22:41,787 --> 00:22:42,370 Wait a minute. 374 00:22:42,370 --> 00:22:42,869 We haven't. 375 00:22:42,869 --> 00:22:45,330 This one, now we got another term here. 376 00:22:45,330 --> 00:22:52,030 So this one gives you 2 times the expression times 377 00:22:52,030 --> 00:23:04,954 the-- this would give you L theta dot sine theta. 378 00:23:04,954 --> 00:23:07,120 But now you have to take the derivative with respect 379 00:23:07,120 --> 00:23:12,300 to theta dot, which gives you what? 380 00:23:12,300 --> 00:23:14,940 STUDENT: [INAUDIBLE]. 381 00:23:14,940 --> 00:23:16,920 PROFESSOR: Another L over 2 sine theta? 382 00:23:22,100 --> 00:23:24,540 Something like that. 383 00:23:24,540 --> 00:23:30,015 So you end up with the theta dot L, L squared over 2, 384 00:23:30,015 --> 00:23:32,440 theta dot L squared sine squared. 385 00:23:32,440 --> 00:23:35,220 And you probably get a theta dot L squared 386 00:23:35,220 --> 00:23:36,520 cosine squared over here. 387 00:23:36,520 --> 00:23:38,250 And those two add together to give you 388 00:23:38,250 --> 00:23:47,060 a theta dot L squared over-- theta dot L squared, I guess. 389 00:23:47,060 --> 00:23:50,170 Those collapse together. 390 00:23:50,170 --> 00:23:55,310 Those come together to give you the other piece of this. 391 00:23:55,310 --> 00:23:59,094 STUDENT: [INAUDIBLE] one of the coefficients. 392 00:23:59,094 --> 00:24:02,775 But where is the derivative [INAUDIBLE]. 393 00:24:02,775 --> 00:24:04,900 PROFESSOR: No, the derivative with respect to theta 394 00:24:04,900 --> 00:24:08,254 only comes in in the potential energy term. 395 00:24:08,254 --> 00:24:12,238 STUDENT: So what's number two? 396 00:24:12,238 --> 00:24:14,360 PROFESSOR: OK. 397 00:24:14,360 --> 00:24:15,267 All right, yep. 398 00:24:15,267 --> 00:24:15,850 You need that. 399 00:24:15,850 --> 00:24:19,130 And so T with respect to theta, and you 400 00:24:19,130 --> 00:24:20,690 do have theta pieces in there. 401 00:24:20,690 --> 00:24:22,140 And it does kick out more pieces. 402 00:24:22,140 --> 00:24:23,701 STUDENT: [INAUDIBLE]. 403 00:24:23,701 --> 00:24:24,700 PROFESSOR: No, I didn't. 404 00:24:24,700 --> 00:24:27,270 Haven't even done that piece yet. 405 00:24:27,270 --> 00:24:31,740 So you do that piece, a couple things cancel. 406 00:24:31,740 --> 00:24:37,030 And you end up with-- so I don't have time to work it out, 407 00:24:37,030 --> 00:24:38,440 to write it all up on the board. 408 00:24:38,440 --> 00:24:43,940 But the complete solution for this is posted. 409 00:24:43,940 --> 00:24:47,600 So Professor Gossard, who teaches the other three 410 00:24:47,600 --> 00:24:52,010 recitation sections, writes these up and posts the answers. 411 00:24:52,010 --> 00:24:53,160 And so they're on Stellar. 412 00:24:53,160 --> 00:24:57,940 So you get the gory details of each of these pieces. 413 00:24:57,940 --> 00:25:04,444 Let's go on to talk about generalized forces, 414 00:25:04,444 --> 00:25:05,610 while we have a few minutes. 415 00:25:08,410 --> 00:25:10,200 The way it was set up, were there-- 416 00:25:10,200 --> 00:25:11,450 what are the right hand sides? 417 00:25:11,450 --> 00:25:14,000 Are there any generalized external 418 00:25:14,000 --> 00:25:18,500 non-conservative forces, the way the problem was first posed? 419 00:25:18,500 --> 00:25:19,140 None. 420 00:25:19,140 --> 00:25:20,190 So let's put in a couple. 421 00:25:20,190 --> 00:25:26,535 Let's add dashpot, b here, and an external force here. 422 00:25:26,535 --> 00:25:29,240 Call it F1 of T. 423 00:25:29,240 --> 00:25:33,560 So now, what's Qx and Q theta? 424 00:25:39,830 --> 00:25:43,260 That's an exercise I think you can all go through, but just 425 00:25:43,260 --> 00:25:46,300 check with your groups. 426 00:25:46,300 --> 00:25:50,060 Figure out the generalized forces. 427 00:25:50,060 --> 00:25:54,420 And do it by imagining-- for this one, for the x equation, 428 00:25:54,420 --> 00:25:58,100 say OK, you have a small, virtual displacement delta x. 429 00:25:58,100 --> 00:26:00,255 What's the virtual work that's done? 430 00:26:00,255 --> 00:26:05,330 Your delta w, then, will be thing you're looking for, 431 00:26:05,330 --> 00:26:06,150 delta x. 432 00:26:06,150 --> 00:26:07,250 And the same thing. 433 00:26:07,250 --> 00:26:09,005 This is the x1. 434 00:26:09,005 --> 00:26:12,480 And you get a similar x thing when you do they one for theta. 435 00:26:12,480 --> 00:26:14,370 It would be Q theta, delta theta. 436 00:26:14,370 --> 00:26:16,790 Figure out the work done and that'll tell you 437 00:26:16,790 --> 00:26:20,650 what Qx and Q theta are. 438 00:26:20,650 --> 00:26:24,960 So the total work of the non-conservative forces 439 00:26:24,960 --> 00:26:26,460 through these virtual displacements, 440 00:26:26,460 --> 00:26:28,880 you can just add them up. 441 00:26:28,880 --> 00:26:31,090 So there's a contribution that comes from a delta x. 442 00:26:31,090 --> 00:26:32,460 And there will be another contribution 443 00:26:32,460 --> 00:26:33,350 from the delta theta. 444 00:26:33,350 --> 00:26:35,110 And we can figure out each piece. 445 00:26:35,110 --> 00:26:38,580 And you assign each piece to the equation it goes with. 446 00:26:38,580 --> 00:26:44,830 So if you do a small virtual deflection in the x direction, 447 00:26:44,830 --> 00:26:47,662 how much work is done? 448 00:26:47,662 --> 00:26:48,870 Somebody give me a term here. 449 00:26:52,710 --> 00:26:58,390 So work, remember, is F dot d dot dr. 450 00:26:58,390 --> 00:27:00,550 And this dr is a function of our delta 451 00:27:00,550 --> 00:27:04,792 x's delta theta's and so forth. 452 00:27:04,792 --> 00:27:07,780 STUDENT: F of T dx? 453 00:27:07,780 --> 00:27:09,140 PROFESSOR: Delta x? 454 00:27:09,140 --> 00:27:10,830 That'll be some work done. 455 00:27:10,830 --> 00:27:15,970 So that force, external force moves through the full delta x. 456 00:27:15,970 --> 00:27:17,372 And what else? 457 00:27:17,372 --> 00:27:20,144 STUDENT: Minus dx [INAUDIBLE]. 458 00:27:20,144 --> 00:27:22,865 PROFESSOR: So that suggests then we have here, 459 00:27:22,865 --> 00:27:29,305 this is F1 of T in positive I direction minus dx 460 00:27:29,305 --> 00:27:32,590 dot in the opposite direction times delta 461 00:27:32,590 --> 00:27:36,780 x is a virtual work done by-- as you do that. 462 00:27:36,780 --> 00:27:38,030 Now how about the delta theta? 463 00:27:43,460 --> 00:27:46,030 Somebody else-- how much virtual work 464 00:27:46,030 --> 00:27:49,540 is done by these forces F and minus bx? 465 00:27:49,540 --> 00:27:52,740 They're the only non-conservative 466 00:27:52,740 --> 00:27:56,570 external forces in the problem are the dashpot and the F, 467 00:27:56,570 --> 00:27:58,300 whatever it is. 468 00:27:58,300 --> 00:28:00,750 How much work is done by those forces 469 00:28:00,750 --> 00:28:05,980 in a virtual displacement delta theta? 470 00:28:05,980 --> 00:28:08,030 Hand up back there? 471 00:28:08,030 --> 00:28:10,740 I hear a bid for 0. 472 00:28:10,740 --> 00:28:13,120 What do other people think? 473 00:28:13,120 --> 00:28:17,310 Do those forces move at all because you make motion delta 474 00:28:17,310 --> 00:28:18,800 theta? 475 00:28:18,800 --> 00:28:22,560 Is there any dr here that results 476 00:28:22,560 --> 00:28:24,830 because of delta theta in the direction 477 00:28:24,830 --> 00:28:26,570 of any of these applied forces? 478 00:28:26,570 --> 00:28:30,280 At the point of application of these forces, do they move? 479 00:28:30,280 --> 00:28:33,040 So this force is right here. 480 00:28:33,040 --> 00:28:34,910 Does it move because you do a delta theta? 481 00:28:34,910 --> 00:28:35,630 Nope. 482 00:28:35,630 --> 00:28:38,390 And this force, applied right here, 483 00:28:38,390 --> 00:28:40,940 does that point move because of delta theta? 484 00:28:40,940 --> 00:28:46,790 No, so in this problem then this piece here is 0 delta theta. 485 00:28:46,790 --> 00:28:48,430 And total virtual work done is that. 486 00:28:48,430 --> 00:28:51,410 And you assign each piece to its appropriate equation. 487 00:28:51,410 --> 00:28:56,720 So Qx, this is Qx right here, delta x. 488 00:28:56,720 --> 00:29:00,450 Qx belongs up here. 489 00:29:00,450 --> 00:29:03,080 The sum of these things equals Qx. 490 00:29:03,080 --> 00:29:08,690 And in the second case, for the beta equation, it's equal to 0. 491 00:29:08,690 --> 00:29:12,060 So let's make the problems a little bit harder for a second. 492 00:29:12,060 --> 00:29:19,550 Let's put a force, apply a horizontal force here. 493 00:29:19,550 --> 00:29:20,366 We'll call this F2. 494 00:29:24,130 --> 00:29:33,340 So now, what is the work done in this system? 495 00:29:33,340 --> 00:29:36,320 We now have an additional force. 496 00:29:36,320 --> 00:29:39,170 Is there any work done because of delta x? 497 00:29:42,700 --> 00:29:48,489 This is the real-- you understand this piece, 498 00:29:48,489 --> 00:29:50,280 then you really begin to understand how you 499 00:29:50,280 --> 00:29:53,000 do these generalized forces. 500 00:29:53,000 --> 00:30:00,440 Qx delta x-- is the generalized force associated with delta x, 501 00:30:00,440 --> 00:30:02,880 is it affected by this new force? 502 00:30:02,880 --> 00:30:04,940 Is any work done? 503 00:30:04,940 --> 00:30:09,190 So I now cause this little delta x to happen. 504 00:30:09,190 --> 00:30:15,840 Is that force doing work, assuming no other deflections 505 00:30:15,840 --> 00:30:18,501 aren't happening right now? 506 00:30:18,501 --> 00:30:19,000 Why? 507 00:30:23,230 --> 00:30:26,110 So is there-- it's not allowed to move. 508 00:30:26,110 --> 00:30:28,080 Whatever instantaneous position it's in, 509 00:30:28,080 --> 00:30:31,500 it's frozen there in its coordinate. 510 00:30:31,500 --> 00:30:35,110 But if there's an x component, it moves in x. 511 00:30:35,110 --> 00:30:38,485 So it's angled like this off the cart. 512 00:30:38,485 --> 00:30:40,630 But now delta x does this to it? 513 00:30:40,630 --> 00:30:42,590 Is that force F doing work? 514 00:30:42,590 --> 00:30:43,100 How much? 515 00:30:47,560 --> 00:30:52,600 So then, when you add that one to it, we end up with a-- 516 00:30:52,600 --> 00:30:54,905 and is it in plus direction, and it's exactly 517 00:30:54,905 --> 00:30:56,820 in the same direction as delta x. 518 00:30:56,820 --> 00:30:59,470 There's no components. 519 00:30:59,470 --> 00:31:04,050 The dot product of F2 is in the I direction dot delta 520 00:31:04,050 --> 00:31:06,060 x, which is also in the I. 521 00:31:06,060 --> 00:31:09,940 So you get an additional contribution of F2 delta x. 522 00:31:09,940 --> 00:31:14,540 And so now this generalized force has an F2 in it. 523 00:31:14,540 --> 00:31:16,520 How about the other direction, though? 524 00:31:16,520 --> 00:31:22,110 How about this now Q theta delta theta? 525 00:31:22,110 --> 00:31:23,280 What does it give you? 526 00:31:28,370 --> 00:31:36,090 Now, if you now freeze x, and allow a slight angular 527 00:31:36,090 --> 00:31:41,570 variation delta theta, does this guy do any work? 528 00:31:41,570 --> 00:31:44,250 How much, and is it F2 delta theta? 529 00:31:47,310 --> 00:31:49,010 That's wrong in two ways. 530 00:31:49,010 --> 00:31:51,610 Dimensions are wrong. 531 00:31:51,610 --> 00:31:52,610 Something else is wrong. 532 00:31:52,610 --> 00:31:54,320 So let's draw it. 533 00:31:54,320 --> 00:31:56,080 Here is this thing. 534 00:31:56,080 --> 00:31:58,150 Here's F2. 535 00:31:58,150 --> 00:32:03,352 Delta theta causes the motion of this point in what direction? 536 00:32:03,352 --> 00:32:04,560 Perpendicular to this, right? 537 00:32:04,560 --> 00:32:05,934 So you can think of it like this. 538 00:32:05,934 --> 00:32:10,190 And this would be L delta theta is the actual distance 539 00:32:10,190 --> 00:32:10,865 that it moves. 540 00:32:13,440 --> 00:32:16,050 And so can either say the component 541 00:32:16,050 --> 00:32:19,240 of that motion in the direction of the force, 542 00:32:19,240 --> 00:32:21,160 or you can say the component of the force 543 00:32:21,160 --> 00:32:24,980 in the direction of that motion, dot product between this 544 00:32:24,980 --> 00:32:25,540 and that. 545 00:32:25,540 --> 00:32:27,900 And this is theta. 546 00:32:27,900 --> 00:32:30,390 So what is the component of L delta theta 547 00:32:30,390 --> 00:32:33,510 in the direction of the force? 548 00:32:33,510 --> 00:32:34,570 That's this right here. 549 00:32:44,390 --> 00:32:51,360 And that's now in the I hat dot F2 I. 550 00:32:51,360 --> 00:32:56,510 So our delta w in the theta direction 551 00:32:56,510 --> 00:33:09,900 is Q theta delta theta equals my F2 L cosine theta delta theta. 552 00:33:09,900 --> 00:33:14,070 And to solve for Q theta, now these can disappear. 553 00:33:14,070 --> 00:33:16,860 And Q theta is F2 L cosine theta. 554 00:33:16,860 --> 00:33:18,230 Are the units right? 555 00:33:18,230 --> 00:33:22,370 What are the units of this generalized force, meaning you 556 00:33:22,370 --> 00:33:24,705 got to think about that equation. 557 00:33:24,705 --> 00:33:26,440 The theta equation-- we talked about this 558 00:33:26,440 --> 00:33:27,840 before-- has units of what? 559 00:33:41,610 --> 00:33:46,379 What are the dimensions of term that look like that? 560 00:33:46,379 --> 00:33:48,920 This comes from, when you do it the direct way, the summation 561 00:33:48,920 --> 00:33:50,990 of external torques. 562 00:33:50,990 --> 00:33:52,830 This had better have units of torque, right, 563 00:33:52,830 --> 00:33:55,570 which is units of force times distance. 564 00:33:55,570 --> 00:33:58,610 So force times distance had better 565 00:33:58,610 --> 00:34:01,270 be the units of this equation. 566 00:34:01,270 --> 00:34:04,470 And therefore that had better be a torque. 567 00:34:04,470 --> 00:34:08,360 Is force times length a torque, of units a torque? 568 00:34:08,360 --> 00:34:08,880 Sure. 569 00:34:08,880 --> 00:34:11,949 So that's the correct unit. 570 00:34:11,949 --> 00:34:14,920 And over here, for the x equation, 571 00:34:14,920 --> 00:34:17,480 do we come out with the correct units? 572 00:34:17,480 --> 00:34:19,760 Yeah, come out force. 573 00:34:19,760 --> 00:34:21,989 This naturally works, because the delta theta 574 00:34:21,989 --> 00:34:23,000 is dimensionless. 575 00:34:23,000 --> 00:34:27,190 So the length dr, the length piece that comes in here 576 00:34:27,190 --> 00:34:29,179 stays with this to give you a torque. 577 00:34:29,179 --> 00:34:31,560 Over here, delta x has length in it. 578 00:34:31,560 --> 00:34:33,199 And you're just left with the force. 579 00:34:33,199 --> 00:34:36,739 So you get a force-- this generalized force 580 00:34:36,739 --> 00:34:39,090 for the x direction is a force. 581 00:34:39,090 --> 00:34:41,889 The generalized force in the theta direction 582 00:34:41,889 --> 00:34:44,630 is a moment, a torque. 583 00:34:44,630 --> 00:34:45,610 All right? 584 00:34:45,610 --> 00:34:47,909 Good. 585 00:34:47,909 --> 00:34:50,610 So we're officially done. 586 00:34:50,610 --> 00:34:53,389 But if you have any last questions on this, 587 00:34:53,389 --> 00:34:56,439 I'll stick around and we can chat about it.