1 00:00:00,070 --> 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 --> 00:00:17,305 at ocw.mit.edu. 8 00:00:21,810 --> 00:00:23,960 PROFESSOR: All right, let's get started. 9 00:00:23,960 --> 00:00:28,420 Today is all about Lagrange method. 10 00:00:28,420 --> 00:00:31,200 We will talk a lot about what we really 11 00:00:31,200 --> 00:00:36,100 mean by generalized coordinates and generalized forces 12 00:00:36,100 --> 00:00:41,220 and then do a number of application examples. 13 00:00:41,220 --> 00:00:46,215 There's a set of notes on Stellar on the Lagrange method. 14 00:00:46,215 --> 00:00:54,340 It's about 10 pages long and I highly recommend you read them. 15 00:00:54,340 --> 00:00:56,590 They're not somehow up with the notes 16 00:00:56,590 --> 00:00:59,240 associated with lecture notes or [INAUDIBLE] 17 00:00:59,240 --> 00:01:00,210 way down at the bottom. 18 00:01:00,210 --> 00:01:02,670 So you have to scroll all the way down in the Stellar website 19 00:01:02,670 --> 00:01:03,211 to find them. 20 00:01:06,300 --> 00:01:09,260 Our second quiz is November 8. 21 00:01:09,260 --> 00:01:13,467 That's a week from next Tuesday. 22 00:01:13,467 --> 00:01:14,750 OK. 23 00:01:14,750 --> 00:01:18,960 Pretty much same format as the first one. 24 00:01:18,960 --> 00:01:20,570 OK. 25 00:01:20,570 --> 00:01:23,985 So let's talk about how to use Lagrange equations. 26 00:01:32,600 --> 00:01:37,200 So I defined what's called the Lagrangian last time. 27 00:01:37,200 --> 00:01:41,180 t minus v. The kinetic energy minus the potential energy 28 00:01:41,180 --> 00:01:43,870 of the entire system. 29 00:01:43,870 --> 00:01:48,940 Total kinetic and total potential energy expressions. 30 00:01:48,940 --> 00:01:51,770 Then we have some quantities. 31 00:01:51,770 --> 00:01:53,540 qj's. 32 00:01:53,540 --> 00:01:59,050 These are defined as the generalized forces. 33 00:01:59,050 --> 00:02:01,920 Generalized coordinates, I should say. 34 00:02:07,260 --> 00:02:12,150 And the capital Q sub j's are the generalized forces. 35 00:02:19,620 --> 00:02:24,550 And the Lagrange equation says that d 36 00:02:24,550 --> 00:02:32,130 by dt the time derivative of the partial of l with respect 37 00:02:32,130 --> 00:02:45,990 to the qj dots, the velocities, minus the partial derivative 38 00:02:45,990 --> 00:02:52,910 of l with respect to the generalized displacements 39 00:02:52,910 --> 00:02:56,670 equals the generalized forces. 40 00:02:56,670 --> 00:03:00,820 And for a typical system, you'll have a number of degrees 41 00:03:00,820 --> 00:03:02,812 of freedom, like say three. 42 00:03:02,812 --> 00:03:04,520 And if you have three degrees of freedom, 43 00:03:04,520 --> 00:03:06,700 you need three equations of motion. 44 00:03:06,700 --> 00:03:10,520 And so the j's will go from one to three in that case. 45 00:03:10,520 --> 00:03:13,470 So the j's here refer to an [? index ?] 46 00:03:13,470 --> 00:03:16,470 that gives you the number of equations that you need. 47 00:03:16,470 --> 00:03:19,840 So you do this calculation for coordinate one, 48 00:03:19,840 --> 00:03:22,610 again for coordinate two, again for coordinate three, 49 00:03:22,610 --> 00:03:26,060 and you get then three equations of motion. 50 00:03:26,060 --> 00:03:27,050 OK 51 00:03:27,050 --> 00:03:34,260 So this is a little obscure. 52 00:03:34,260 --> 00:03:35,520 Let's just plug in. 53 00:03:41,610 --> 00:03:45,680 For l equals t minus v. And just put it in here 54 00:03:45,680 --> 00:03:46,630 and see what happens. 55 00:03:51,600 --> 00:04:02,290 You get d by dt of the partial of t with respect 56 00:04:02,290 --> 00:04:11,600 to qj dot minus d by dt of the partial of v 57 00:04:11,600 --> 00:04:27,975 with respect to qj dot plus-- I'll organize it this way. 58 00:04:30,590 --> 00:04:33,770 Minus the partial of t with respect 59 00:04:33,770 --> 00:04:43,320 to qj plus the partial of v with respect to qj 60 00:04:43,320 --> 00:04:46,400 equals capital Qj. 61 00:04:46,400 --> 00:04:52,200 Now when we first talked about potential energy 62 00:04:52,200 --> 00:04:54,840 a few days ago, we said that for mechanical systems, 63 00:04:54,840 --> 00:05:01,070 the potential energy is not a function of time or-- anybody 64 00:05:01,070 --> 00:05:01,730 remember? 65 00:05:01,730 --> 00:05:04,200 Velocity. 66 00:05:04,200 --> 00:05:05,660 So if the potential energy is not 67 00:05:05,660 --> 00:05:10,420 a function of time nor velocity, what will happen to this term? 68 00:05:10,420 --> 00:05:11,830 This goes away. 69 00:05:11,830 --> 00:05:25,370 So this is 0 for mechanical systems. 70 00:05:25,370 --> 00:05:28,190 If you start getting into electrons moving 71 00:05:28,190 --> 00:05:31,550 and magnetic fields, then you start have a potential energies 72 00:05:31,550 --> 00:05:34,290 involving velocities. 73 00:05:34,290 --> 00:05:37,190 But for mechanical systems, this term's 0. 74 00:05:37,190 --> 00:05:40,120 And I think the bookkeeping. 75 00:05:40,120 --> 00:05:41,915 So this is the form of a Lagrange equations 76 00:05:41,915 --> 00:05:44,120 that I write down when I'm doing problems. 77 00:05:44,120 --> 00:05:44,960 I don't write this. 78 00:05:44,960 --> 00:05:49,530 Mathematicians like elegance. 79 00:05:49,530 --> 00:05:52,470 And this comes down to this is beautifully elegant simple 80 00:05:52,470 --> 00:05:53,730 looking formula. 81 00:05:53,730 --> 00:05:56,670 But I'm an engineer and I like it to be 82 00:05:56,670 --> 00:05:59,710 efficient and practical useful. 83 00:05:59,710 --> 00:06:03,360 This is the practical useful form of Lagrange equations. 84 00:06:03,360 --> 00:06:06,310 So you just use what you need. 85 00:06:06,310 --> 00:06:08,592 Kinetic energy here, kinetic energy there, 86 00:06:08,592 --> 00:06:09,550 potential energy there. 87 00:06:09,550 --> 00:06:11,360 And I number these. 88 00:06:11,360 --> 00:06:13,870 There's a lot of bookkeeping in Lagrange. 89 00:06:13,870 --> 00:06:19,045 So I call it term one, term two, term three, and term four. 90 00:06:19,045 --> 00:06:21,420 Because you have to grind through this quite a few times. 91 00:06:21,420 --> 00:06:23,480 And so when you do, basically you 92 00:06:23,480 --> 00:06:29,820 take one of the results of 1 plus 2 plus 3 equals 4. 93 00:06:29,820 --> 00:06:38,968 And you do that j times to get the equations you're after. 94 00:06:38,968 --> 00:06:39,468 OK. 95 00:06:46,230 --> 00:06:48,450 So now we need to talk a little bit 96 00:06:48,450 --> 00:06:55,391 about what we mean by generalized coordinates. 97 00:06:55,391 --> 00:06:55,890 qj. 98 00:07:11,630 --> 00:07:13,270 What's this word generalized mean? 99 00:07:13,270 --> 00:07:15,865 Generalized just means it doesn't have to be Cartesian. 100 00:07:23,810 --> 00:07:28,280 Not necessarily Cartesian as in xyz. 101 00:07:28,280 --> 00:07:30,910 You got a lot of liberty and how you choose coordinates. 102 00:07:30,910 --> 00:07:33,660 Not necessarily Cartesian. 103 00:07:33,660 --> 00:07:43,050 Not even inertial. 104 00:07:51,360 --> 00:07:55,110 They do have to satisfy certain requirements. 105 00:07:55,110 --> 00:08:02,106 The coordinates, they must be what we call independent. 106 00:08:09,830 --> 00:08:11,803 They must be complete. 107 00:08:19,280 --> 00:08:21,800 So it must be independent and complete 108 00:08:21,800 --> 00:08:31,820 and the system must be holonomic. 109 00:08:36,460 --> 00:08:38,600 I'll get to that in a minute. 110 00:08:38,600 --> 00:08:41,330 So you need to understand what it means to be independent, 111 00:08:41,330 --> 00:08:42,350 complete, and holonomic. 112 00:09:02,410 --> 00:09:04,740 So what do we mean by independent? 113 00:09:04,740 --> 00:09:09,150 So if you have a multiple degree of freedom system 114 00:09:09,150 --> 00:09:13,370 and you fix all but one of the coordinates, 115 00:09:13,370 --> 00:09:18,810 say the system can't move in all but one of its coordinates. 116 00:09:18,810 --> 00:09:20,980 That last degree of freedom still 117 00:09:20,980 --> 00:09:23,760 has to have a complete range of motion. 118 00:09:23,760 --> 00:09:27,310 So if you have a double pendulum and you grab the first mask, 119 00:09:27,310 --> 00:09:29,050 the second mask can still move. 120 00:09:29,050 --> 00:09:33,705 It takes two angles to define your double pendulum. 121 00:09:33,705 --> 00:09:34,330 So independent. 122 00:09:44,000 --> 00:09:58,600 When you fix all but one coordinate, 123 00:09:58,600 --> 00:10:17,470 still have a continuous range of movement 124 00:10:17,470 --> 00:10:20,890 essentially in the free coordinate. 125 00:10:26,720 --> 00:10:29,190 And that's independent. 126 00:10:29,190 --> 00:10:35,040 And we'll do this by example mostly. 127 00:10:35,040 --> 00:10:37,240 And complete. 128 00:10:37,240 --> 00:10:41,250 The complete really means it's capable of locating all parts 129 00:10:41,250 --> 00:10:44,385 of the system at all times. 130 00:11:12,020 --> 00:11:14,875 So let's look at a system here. 131 00:11:14,875 --> 00:11:15,840 It's a double pendulum. 132 00:11:28,535 --> 00:11:35,240 It's a simple one just made out of two particles and strings. 133 00:11:35,240 --> 00:11:38,340 I didn't bring one today. 134 00:11:38,340 --> 00:11:43,700 And I need to pick some coordinates to describe this. 135 00:11:43,700 --> 00:11:49,040 And we'll use some Cartesian coordinates. 136 00:11:49,040 --> 00:11:52,950 Here's an x and a y. 137 00:11:52,950 --> 00:11:54,030 And here's particle one. 138 00:11:56,790 --> 00:12:01,640 And I could choose to describe this system xy coordinates. 139 00:12:01,640 --> 00:12:06,900 And I'll specify the location in the system with coordinates x1 140 00:12:06,900 --> 00:12:08,050 and y1. 141 00:12:08,050 --> 00:12:11,290 Two values to specify the location of that. 142 00:12:11,290 --> 00:12:16,350 And down here I'm going to pick two more values, x2 and y2, 143 00:12:16,350 --> 00:12:22,170 just to describe the-- so x1 is a different coordinate from x2. 144 00:12:22,170 --> 00:12:25,380 x1 is the exposition of particle one. 145 00:12:25,380 --> 00:12:31,150 x2 is the x position of particle two. y1 and y2. 146 00:12:31,150 --> 00:12:34,670 So how many coordinates do I have to describe the system? 147 00:12:38,240 --> 00:12:40,330 How many have I used? 148 00:12:40,330 --> 00:12:41,530 Four, right? 149 00:12:41,530 --> 00:12:45,680 How many degrees of freedom do you think this problem has? 150 00:12:45,680 --> 00:12:46,297 Two. 151 00:12:46,297 --> 00:12:48,630 So there's something already a little out of whack here. 152 00:12:51,710 --> 00:12:55,930 But the point is these aren't independent, you'll find. 153 00:12:55,930 --> 00:12:56,961 You just do a test. 154 00:12:56,961 --> 00:12:58,710 You'll find that these aren't independent. 155 00:12:58,710 --> 00:13:03,330 If I fix x1 and x2, systems doesn't move. 156 00:13:07,066 --> 00:13:12,110 If I say this is going to be one and this has got to be three, 157 00:13:12,110 --> 00:13:15,330 this system is now frozen. 158 00:13:15,330 --> 00:13:18,530 So this system of core coordinates is not independent. 159 00:13:21,054 --> 00:13:21,720 What did we say? 160 00:13:21,720 --> 00:13:22,420 Independent. 161 00:13:22,420 --> 00:13:25,240 When you fix all but one coordinate, 162 00:13:25,240 --> 00:13:29,280 you still have continuous range of movement of the final one. 163 00:13:29,280 --> 00:13:30,960 I could fix only just two of these 164 00:13:30,960 --> 00:13:32,610 and I've frozen the system. 165 00:13:32,610 --> 00:13:36,390 I don't even have to go to the extent of fixing three. 166 00:13:36,390 --> 00:13:38,760 I'm assuming the strings are of fixed length. 167 00:13:38,760 --> 00:13:41,540 You can't change the string length. 168 00:13:41,540 --> 00:13:43,610 So this is not a very good choice of coordinates. 169 00:13:43,610 --> 00:13:45,510 And we had a hint that it might not be, 170 00:13:45,510 --> 00:13:47,590 because it's more than we ought to use. 171 00:13:47,590 --> 00:13:50,090 We only really need two. 172 00:13:50,090 --> 00:13:58,240 So and then if we choose these angles, v1 and v2, 173 00:13:58,240 --> 00:13:59,720 let's do the test with that. 174 00:13:59,720 --> 00:14:01,929 Are those independent? 175 00:14:01,929 --> 00:14:03,720 So those are the coordinates of the system. 176 00:14:03,720 --> 00:14:07,880 If you fix v1, is there still free and continuous movement 177 00:14:07,880 --> 00:14:09,880 of v2 of the system? 178 00:14:09,880 --> 00:14:10,540 Sure. 179 00:14:10,540 --> 00:14:14,680 And if you fixed v2, it means you can require this angle stay 180 00:14:14,680 --> 00:14:17,600 rigid like that, and move v2, well, the whole system 181 00:14:17,600 --> 00:14:18,820 will still move. 182 00:14:18,820 --> 00:14:24,160 So v1 and v2 are a system which satisfies the independence 183 00:14:24,160 --> 00:14:25,280 requirement. 184 00:14:25,280 --> 00:14:26,270 Complete. 185 00:14:26,270 --> 00:14:28,410 They're both systems that are complete. 186 00:14:28,410 --> 00:14:31,110 They're both capable locating all points at all times. 187 00:14:33,690 --> 00:14:39,400 But only the pair v1 and v2 in this example 188 00:14:39,400 --> 00:14:43,880 are both independent and complete. 189 00:14:43,880 --> 00:14:52,060 Now, the third requirement is a thing called holonomicity. 190 00:14:52,060 --> 00:14:56,360 And what it means to be holonomic 191 00:14:56,360 --> 00:15:01,860 is that the system, the number of degrees of freedom required 192 00:15:01,860 --> 00:15:07,170 is equal to the number of coordinates 193 00:15:07,170 --> 00:15:10,560 required to completely describe the motion. 194 00:15:10,560 --> 00:15:15,320 Now, every example we've ever done so far in this class 195 00:15:15,320 --> 00:15:16,570 satisfies that. 196 00:15:16,570 --> 00:15:21,470 We picked v1 and v2 and that's all the coordinates 197 00:15:21,470 --> 00:15:24,400 that we need to completely to describe the motion. 198 00:15:24,400 --> 00:15:28,495 Let me see if I can figure out a counter example. 199 00:15:34,690 --> 00:15:36,279 I didn't write down this definition. 200 00:15:36,279 --> 00:15:36,820 So holonomic. 201 00:16:14,280 --> 00:16:17,480 And if the answer to this question is no, 202 00:16:17,480 --> 00:16:21,330 you cannot use Lagrange equations. 203 00:16:21,330 --> 00:16:28,410 So let's see if I can show you an example of a system in which 204 00:16:28,410 --> 00:16:33,070 you need more coordinates than you have degrees of freedom. 205 00:16:33,070 --> 00:16:35,960 I've got a ball. 206 00:16:35,960 --> 00:16:38,740 This is an xy plane. 207 00:16:38,740 --> 00:16:44,744 And I'm not going to allow it to translate in z. 208 00:16:44,744 --> 00:16:49,910 And I'm not going to allow it to rotate about the z-axis. 209 00:16:49,910 --> 00:16:53,140 So those are two constraints. 210 00:16:53,140 --> 00:16:55,430 So this is one rigid body. 211 00:16:55,430 --> 00:16:58,720 In general how many degrees of freedom does it have? 212 00:16:58,720 --> 00:17:00,360 Six. 213 00:17:00,360 --> 00:17:03,040 I'm going to constrain it so no z motion. 214 00:17:03,040 --> 00:17:04,250 Five. 215 00:17:04,250 --> 00:17:06,020 No z rotation. 216 00:17:06,020 --> 00:17:08,710 Four. 217 00:17:08,710 --> 00:17:11,800 It's not going to allow it to slip. 218 00:17:11,800 --> 00:17:13,609 This is x and that's y. 219 00:17:13,609 --> 00:17:16,910 I'm not going to allow it to slip in the x. 220 00:17:16,910 --> 00:17:19,140 So now I've got another. 221 00:17:19,140 --> 00:17:21,520 Now I'm down to three. 222 00:17:21,520 --> 00:17:25,500 And I'm not going to allow it to slip in the y. 223 00:17:25,500 --> 00:17:26,520 Two. 224 00:17:26,520 --> 00:17:30,060 So by our calculus of how many degrees of freedom you need, 225 00:17:30,060 --> 00:17:31,600 we're down to two. 226 00:17:31,600 --> 00:17:33,720 We should be able to completely describe 227 00:17:33,720 --> 00:17:36,870 the motion of this system with two coordinates. 228 00:17:36,870 --> 00:17:37,650 OK. 229 00:17:37,650 --> 00:17:41,230 So I've put this piece of tape on the top. 230 00:17:41,230 --> 00:17:45,060 And it's pointing diagonally. 231 00:17:45,060 --> 00:17:46,490 That way. 232 00:17:46,490 --> 00:17:51,900 And I'm going to roll this ball like this 233 00:17:51,900 --> 00:17:53,104 until it shows up again. 234 00:17:53,104 --> 00:17:55,020 So it's right on top, just the way it started. 235 00:17:57,980 --> 00:18:01,540 Now start off same way again. 236 00:18:01,540 --> 00:18:03,580 I'm going to roll first this way. 237 00:18:06,260 --> 00:18:09,900 And then I'm going to roll this way to the same place. 238 00:18:12,470 --> 00:18:14,815 Where's the stripe? 239 00:18:14,815 --> 00:18:15,800 It's in the back. 240 00:18:18,450 --> 00:18:21,760 So I've gone to the same position 241 00:18:21,760 --> 00:18:25,550 but I've ended up with the ball not in the same orientation 242 00:18:25,550 --> 00:18:26,380 as it was. 243 00:18:26,380 --> 00:18:27,770 I went by two different paths. 244 00:18:35,680 --> 00:18:39,400 And the ball comes up over here rather than up there 245 00:18:39,400 --> 00:18:40,351 where it started. 246 00:18:40,351 --> 00:18:40,850 OK 247 00:18:40,850 --> 00:18:46,050 So to actually describe where the ball is at any place 248 00:18:46,050 --> 00:18:49,890 out here, having gotten there by rolling around, 249 00:18:49,890 --> 00:18:56,095 without slipping and without z rotation, how many coordinates 250 00:18:56,095 --> 00:18:59,290 do you think it'll take to actually specify where 251 00:18:59,290 --> 00:19:03,366 that stripe is at any arbitrary place 252 00:19:03,366 --> 00:19:04,740 that it's gotten to on the plane? 253 00:19:09,510 --> 00:19:10,558 Name them. 254 00:19:10,558 --> 00:19:11,474 AUDIENCE: [INAUDIBLE]. 255 00:19:16,460 --> 00:19:18,220 PROFESSOR: So. 256 00:19:18,220 --> 00:19:19,980 In order to actually fully describe it, 257 00:19:19,980 --> 00:19:21,670 you've got to say where it is x and y 258 00:19:21,670 --> 00:19:24,570 and you actually have to say some kind of theta and phi 259 00:19:24,570 --> 00:19:28,140 rotations that it's gone through so that you know where this is. 260 00:19:28,140 --> 00:19:32,750 So this system is not holonomic. 261 00:19:32,750 --> 00:19:38,167 And it has to be holonomic in order 262 00:19:38,167 --> 00:19:39,250 to use Lagrange equations. 263 00:19:43,310 --> 00:19:45,170 So when you go to do Lagrange problems, 264 00:19:45,170 --> 00:19:48,080 you need to test for your coordinates. 265 00:19:48,080 --> 00:19:51,100 Complete, independent, and holonomic. 266 00:19:51,100 --> 00:19:52,350 And you get pretty good at it. 267 00:20:12,330 --> 00:20:14,850 So here's my Lagrange equations. 268 00:20:14,850 --> 00:20:20,150 And I have itemized these four calculations you have to do. 269 00:20:20,150 --> 00:20:23,620 Call them one, two, three, and four. 270 00:20:23,620 --> 00:20:25,930 And what I'm going to write out is just 271 00:20:25,930 --> 00:20:30,540 to get you to adopt a systematic approach to doing Lagrange. 272 00:20:43,380 --> 00:20:44,670 Left hand side. 273 00:20:44,670 --> 00:20:47,780 To the left hand side of your equations of motion 274 00:20:47,780 --> 00:20:50,010 is everything with t and v. The right hand side 275 00:20:50,010 --> 00:20:52,480 has these generalized forces that you have to deal with. 276 00:20:52,480 --> 00:20:55,740 And generalized forces are the non conservative forces 277 00:20:55,740 --> 00:20:57,194 in the system. 278 00:20:57,194 --> 00:20:59,110 So this is going to get a little bit cookbook, 279 00:20:59,110 --> 00:21:02,900 but it's, I think, appropriate for the moment here. 280 00:21:02,900 --> 00:21:03,810 So step one. 281 00:21:10,550 --> 00:21:14,940 Determine the number of degrees of freedom that you need. 282 00:21:19,200 --> 00:21:26,960 And choose your delta j's. 283 00:21:26,960 --> 00:21:28,620 Not deltas, excuse me. 284 00:21:28,620 --> 00:21:30,640 qj's. 285 00:21:30,640 --> 00:21:31,880 Choose your coordinates. 286 00:21:31,880 --> 00:21:34,210 You find the number of degrees of freedom 287 00:21:34,210 --> 00:21:38,050 and choose the coordinates you're going to use, basically. 288 00:21:43,980 --> 00:21:57,010 Verify complete, independent, holonomic. 289 00:22:03,811 --> 00:22:04,310 Three. 290 00:22:08,270 --> 00:22:14,090 Compute t and v for every rigid body in the system. 291 00:22:14,090 --> 00:22:15,965 Compute your kinetic and potential energies. 292 00:22:24,680 --> 00:22:40,616 One, two, three for each qj. 293 00:22:40,616 --> 00:22:41,990 So for every coordinate you have, 294 00:22:41,990 --> 00:22:43,550 you have to go through these computations. 295 00:22:43,550 --> 00:22:45,383 One, two, three, four, for every coordinate. 296 00:22:48,080 --> 00:22:49,500 And this is your left hand side. 297 00:22:52,511 --> 00:22:54,760 And if you don't have any external forces and your non 298 00:22:54,760 --> 00:22:59,400 conservative external forces, then 1 plus 2 plus 3 equals 0. 299 00:22:59,400 --> 00:23:01,930 But if you have non conservative forces, 300 00:23:01,930 --> 00:23:04,330 then you have to compute the right hand side. 301 00:23:04,330 --> 00:23:05,480 So the right hand side. 302 00:23:12,470 --> 00:23:21,560 So for each qj, each generalized coordinate, 303 00:23:21,560 --> 00:23:31,920 you need to find the generalized force that 304 00:23:31,920 --> 00:23:33,120 potentially goes with it. 305 00:23:41,580 --> 00:24:00,450 And you do this by computing the virtual work delta w. 306 00:24:00,450 --> 00:24:02,360 I'll put the little nc up here to remind you 307 00:24:02,360 --> 00:24:04,430 these are for the non conservative forces. 308 00:24:04,430 --> 00:24:30,770 The delta w associated with the virtual displacement delta qj. 309 00:24:30,770 --> 00:24:33,710 So for every generalized coordinate you have, 310 00:24:33,710 --> 00:24:39,320 you're going to try out this little delta of motion 311 00:24:39,320 --> 00:24:47,040 in that coordinate and figure out how much virtual 312 00:24:47,040 --> 00:24:49,320 work you've done. 313 00:24:49,320 --> 00:24:56,950 So delta wj is going to be qj delta k. 314 00:24:56,950 --> 00:25:01,390 So this is the thing you're looking for. 315 00:25:01,390 --> 00:25:05,820 And it's going to be a function of all those external non 316 00:25:05,820 --> 00:25:09,940 conservative forces acting through a little virtual 317 00:25:09,940 --> 00:25:12,310 displacement, a little bit of work will be done. 318 00:25:12,310 --> 00:25:15,140 Mostly I'm going to teach you how to do this by example. 319 00:25:36,350 --> 00:25:43,120 So let's quickly do a really simple trivial system. 320 00:25:43,120 --> 00:25:49,880 Our mass spring dashpot system, single agree freedom mkb. 321 00:25:53,380 --> 00:25:55,520 It's going to take one coordinate 322 00:25:55,520 --> 00:25:57,840 to describe the motion. 323 00:25:57,840 --> 00:26:01,590 X happens to be Cartesian. 324 00:26:01,590 --> 00:26:04,580 There'll be one generalized coordinate. 325 00:26:04,580 --> 00:26:13,470 So qj equals q1 equals qx in this case. 326 00:26:13,470 --> 00:26:15,205 It's our x-coordinate. 327 00:26:15,205 --> 00:26:18,030 Actually I should just call it x. 328 00:26:18,030 --> 00:26:21,060 That's our generalized coordinates for this problem. 329 00:26:21,060 --> 00:26:22,190 Is it complete? 330 00:26:22,190 --> 00:26:23,100 Yeah. 331 00:26:23,100 --> 00:26:24,291 Is it independent? 332 00:26:24,291 --> 00:26:24,790 Yes. 333 00:26:24,790 --> 00:26:25,670 Is it holonomic? 334 00:26:25,670 --> 00:26:26,420 No problem. 335 00:26:29,360 --> 00:26:36,010 We need t 1/2 mx dot squared. 336 00:26:36,010 --> 00:26:42,920 We need v. And we have 1/2 kx squared 337 00:26:42,920 --> 00:26:53,490 for the spring minus mgx for the gravitational potential energy. 338 00:26:53,490 --> 00:26:55,880 And now we can start. 339 00:26:55,880 --> 00:26:59,587 And we have some external non conservative forces. 340 00:26:59,587 --> 00:27:00,170 What are they? 341 00:27:02,790 --> 00:27:04,060 fi non conservative. 342 00:27:08,110 --> 00:27:10,370 And I think I'm going to put an excitation up 343 00:27:10,370 --> 00:27:14,180 here too, some f of t. 344 00:27:14,180 --> 00:27:18,340 So what are the non conservative forces? 345 00:27:18,340 --> 00:27:19,380 Pardon? 346 00:27:19,380 --> 00:27:21,630 AUDIENCE: k x dot. 347 00:27:21,630 --> 00:27:22,970 PROFESSOR: It's not k. 348 00:27:27,090 --> 00:27:28,410 My mistake. 349 00:27:28,410 --> 00:27:31,190 You're correct. 350 00:27:31,190 --> 00:27:33,870 My brain is getting ahead of my writing here. 351 00:27:33,870 --> 00:27:35,663 That's normally b and this would be k. 352 00:27:35,663 --> 00:27:37,530 I'm not trying to really mess you up there. 353 00:27:37,530 --> 00:27:41,010 So would be bx dot, right. 354 00:27:41,010 --> 00:27:43,670 And is there anything else? 355 00:27:43,670 --> 00:27:45,900 Are there any other non conservative forces, 356 00:27:45,900 --> 00:27:48,762 things that could put energy into or out of the system? 357 00:27:48,762 --> 00:27:49,678 AUDIENCE: [INAUDIBLE]. 358 00:27:55,140 --> 00:27:58,015 PROFESSOR: So the damper can certainly extract energy. 359 00:27:58,015 --> 00:27:58,952 AUDIENCE: [INAUDIBLE]. 360 00:27:58,952 --> 00:28:00,160 PROFESSOR: Yeah, the force f. 361 00:28:00,160 --> 00:28:02,420 That external, it might be something that's 362 00:28:02,420 --> 00:28:03,670 making it vibrate or whatever. 363 00:28:03,670 --> 00:28:06,520 But it's an external force, and it could do work on the system. 364 00:28:06,520 --> 00:28:10,750 And it's not a potential. 365 00:28:10,750 --> 00:28:15,115 It's not a spring and it's not gravity. 366 00:28:15,115 --> 00:28:16,910 It's coming up and somebody's shaking 367 00:28:16,910 --> 00:28:18,330 it or something like that. 368 00:28:18,330 --> 00:28:19,960 So f is also non conservative. 369 00:28:19,960 --> 00:28:23,290 So the non conservative forces in this thing 370 00:28:23,290 --> 00:28:31,580 are f in the i direction and minus vx dot 371 00:28:31,580 --> 00:28:32,790 in the i direction. 372 00:28:32,790 --> 00:28:35,350 And we could, in our normal approach using Newton, 373 00:28:35,350 --> 00:28:39,080 we draw a free body diagram and we identify 374 00:28:39,080 --> 00:28:43,260 a bx dot on it and an f on it. 375 00:28:43,260 --> 00:28:47,710 But we'd also have our kx on it. 376 00:28:47,710 --> 00:28:50,480 That would be what our free body diagram would look like. 377 00:28:50,480 --> 00:28:51,830 That's a conservative force. 378 00:28:51,830 --> 00:28:55,580 Oops, and we need an mg. 379 00:28:55,580 --> 00:28:59,350 So we have two conservative forces, kx and mg, 380 00:28:59,350 --> 00:29:03,760 and we have two non conservative forces, bx dot and f. 381 00:29:03,760 --> 00:29:08,160 So in this case, some of the non conservative forces 382 00:29:08,160 --> 00:29:11,710 is that f in the i direction minus bx in the x 383 00:29:11,710 --> 00:29:14,990 dot in the i direction. 384 00:29:14,990 --> 00:29:19,940 So let's do our calculus here. 385 00:29:19,940 --> 00:29:25,520 So 1 d by dt of the partial of the t, which 386 00:29:25,520 --> 00:29:33,520 is 1/2 mx dot squared with respect to x dot. 387 00:29:33,520 --> 00:29:36,970 So that gives me the derivative of x dot squared with respect 388 00:29:36,970 --> 00:29:40,320 to x dot gives me 2 mx dot. 389 00:29:40,320 --> 00:29:45,690 So this is d by dt of mx dot. 390 00:29:45,690 --> 00:29:48,250 But that's mx double dot. 391 00:29:48,250 --> 00:29:49,942 And as you might expect when you're 392 00:29:49,942 --> 00:29:51,400 trying to drive equation of motion, 393 00:29:51,400 --> 00:29:53,608 you're probably going to end up with an mx double dot 394 00:29:53,608 --> 00:29:55,480 in the result. And it always comes out 395 00:29:55,480 --> 00:29:58,830 of these d by dt expressions. 396 00:29:58,830 --> 00:30:01,180 OK, so that's term one. 397 00:30:01,180 --> 00:30:04,190 Term two in this problem. 398 00:30:04,190 --> 00:30:09,690 Minus t with respect to x in this case. 399 00:30:09,690 --> 00:30:11,210 Is t a function of x? 400 00:30:18,560 --> 00:30:21,040 It's 1/2 mx dot squared. 401 00:30:21,040 --> 00:30:24,160 So is t a function of displacement x? 402 00:30:24,160 --> 00:30:26,170 It's a function of velocity in the x direction, 403 00:30:26,170 --> 00:30:28,940 but is it a function of displacement? 404 00:30:28,940 --> 00:30:29,490 No. 405 00:30:29,490 --> 00:30:34,850 So this term is 0. 406 00:30:34,850 --> 00:30:35,610 Three. 407 00:30:35,610 --> 00:30:38,090 Our third term. 408 00:30:38,090 --> 00:30:45,200 Partial of v with respect to x. 409 00:30:45,200 --> 00:30:47,640 Well, where's v 1/2 kx squared. 410 00:30:47,640 --> 00:30:51,075 The derivative of this is kx minus mg. 411 00:30:57,640 --> 00:30:58,595 And we sum those. 412 00:31:05,170 --> 00:31:17,620 So we get mx double dot plus kx minus mg equals. 413 00:31:17,620 --> 00:31:21,490 And on the right hand side this is 4. 414 00:31:21,490 --> 00:31:25,600 Now we need to do four for the right hand side. 415 00:31:25,600 --> 00:31:36,110 And four is really the summation of the fi's, 416 00:31:36,110 --> 00:31:43,880 the individual forces, dotted with dr. 417 00:31:43,880 --> 00:31:44,856 These are both vectors. 418 00:31:47,970 --> 00:31:50,226 dr is the movement. 419 00:31:50,226 --> 00:31:52,600 Little bit of work and it's going to be a delta quantity, 420 00:31:52,600 --> 00:31:54,390 like delta x. 421 00:31:54,390 --> 00:31:56,800 And f are the applied forces. 422 00:31:56,800 --> 00:31:59,010 And you need to sum these up. 423 00:31:59,010 --> 00:32:04,540 So this dr in general is going to be a function of the delta 424 00:32:04,540 --> 00:32:05,260 j's. 425 00:32:05,260 --> 00:32:08,780 The virtual displacements in all the possible degrees of freedom 426 00:32:08,780 --> 00:32:10,000 of the system. 427 00:32:10,000 --> 00:32:11,302 We do them one at a time. 428 00:32:11,302 --> 00:32:13,260 This case we'll only have one, so it's trivial. 429 00:32:13,260 --> 00:32:16,720 But this could be delta one, two, three, four. 430 00:32:16,720 --> 00:32:20,710 And each one of them might do some work 431 00:32:20,710 --> 00:32:24,470 when f moves through it. 432 00:32:24,470 --> 00:32:28,200 But work is f dot the displacement. 433 00:32:28,200 --> 00:32:30,770 So it's the component of the force 434 00:32:30,770 --> 00:32:34,360 in the direction of the movement, the dot product, that 435 00:32:34,360 --> 00:32:36,870 gives you this little bit of virtual work. 436 00:32:36,870 --> 00:32:43,740 OK, so in this problem, this is going to be equal to-- we 437 00:32:43,740 --> 00:32:47,570 actually have an f of t of some function of time in the i 438 00:32:47,570 --> 00:32:58,930 direction minus bx dot in the i direction dotted width delta 439 00:32:58,930 --> 00:33:01,070 x, which is our virtual displacement 440 00:33:01,070 --> 00:33:03,420 in our single generalized coordinate. 441 00:33:03,420 --> 00:33:10,130 And this whole thing is going to be equal to Qx delta x. 442 00:33:15,350 --> 00:33:18,160 So you figure out the virtual work that's done. 443 00:33:21,930 --> 00:33:23,390 So if you do this dot product, this 444 00:33:23,390 --> 00:33:25,020 is also in the i hat direction. 445 00:33:25,020 --> 00:33:27,090 So i dot i, i dot i. 446 00:33:27,090 --> 00:33:28,480 You just get ones. 447 00:33:28,480 --> 00:33:30,890 Because the forces are in the same direction 448 00:33:30,890 --> 00:33:34,340 as the displacement. 449 00:33:34,340 --> 00:33:36,500 You're going to get an ft. 450 00:33:36,500 --> 00:33:40,320 f of t delta x is one of the little bits of virtual work. 451 00:33:40,320 --> 00:33:44,870 And you'll get a minus bx dot delta x. 452 00:33:44,870 --> 00:33:48,800 And that together, those two pieces added together, 453 00:33:48,800 --> 00:33:53,800 are the generalized force times delta x. 454 00:33:53,800 --> 00:33:58,950 This total here gives you delta w non 455 00:33:58,950 --> 00:34:05,460 conservative for in this case coordinate x. 456 00:34:05,460 --> 00:34:10,909 So we're trying to solve for what 457 00:34:10,909 --> 00:34:12,219 goes on the right hand side. 458 00:34:12,219 --> 00:34:13,160 We need the qx. 459 00:34:17,400 --> 00:34:20,469 You notice what'll happen, it'll cancel out 460 00:34:20,469 --> 00:34:24,460 the delta x is the result. And in this case, what 461 00:34:24,460 --> 00:34:40,430 you're left with is Qx equals f of t minus bx dot. 462 00:34:40,430 --> 00:34:41,745 So this is number four. 463 00:34:45,489 --> 00:34:46,429 So Qx. 464 00:34:46,429 --> 00:34:50,120 Delta x is the bit of virtual work that's done. 465 00:34:50,120 --> 00:34:55,040 What goes into our equation of motion is the Qx part. 466 00:34:55,040 --> 00:34:58,330 And we got it by computing the virtual work done 467 00:34:58,330 --> 00:35:01,230 by the applied external non conservative forces 468 00:35:01,230 --> 00:35:06,440 as we imagine them going through delta x. 469 00:35:06,440 --> 00:35:07,570 And we're done. 470 00:35:07,570 --> 00:35:11,005 You have the complete equation of motion for a single degree 471 00:35:11,005 --> 00:35:11,630 freedom system. 472 00:35:11,630 --> 00:35:13,300 You could rearrange it a little bit. 473 00:35:13,300 --> 00:35:18,830 mx double dot plus bx dot plus kx equals mg plus f of t 474 00:35:18,830 --> 00:35:20,260 if you well. 475 00:35:20,260 --> 00:35:24,230 So it's the same thing you would have gotten from using Newton. 476 00:35:24,230 --> 00:35:27,950 In a trivial kind of example, but it 477 00:35:27,950 --> 00:35:31,767 helps to find each of the steps, things that we said 478 00:35:31,767 --> 00:35:32,350 were required. 479 00:35:36,470 --> 00:35:43,740 OK, so we're going to go from there to a much harder problem. 480 00:35:43,740 --> 00:35:51,900 So any issues or questions about definitions, procedure? 481 00:35:51,900 --> 00:35:54,520 So we start getting into multiple degrees of freedom. 482 00:35:54,520 --> 00:35:58,570 You need is set up a careful bookkeeping. 483 00:35:58,570 --> 00:36:01,550 So I just do this myself. 484 00:36:01,550 --> 00:36:07,250 The top of the page, I identify my coordinates, write down t, 485 00:36:07,250 --> 00:36:11,840 write down v. Then I say, OK, coordinate one. 486 00:36:11,840 --> 00:36:14,250 One, two, three, four. 487 00:36:14,250 --> 00:36:15,020 Equation. 488 00:36:15,020 --> 00:36:17,020 Then I started the coordinate two. 489 00:36:17,020 --> 00:36:20,840 Calculus for one, two, three, and four and so forth until you 490 00:36:20,840 --> 00:36:22,431 get to the end. 491 00:36:22,431 --> 00:36:23,290 OK, questions? 492 00:36:23,290 --> 00:36:24,962 Yeah. 493 00:36:24,962 --> 00:36:28,834 AUDIENCE: On the [INAUDIBLE] what's 494 00:36:28,834 --> 00:36:30,931 that thing after the [INAUDIBLE] It's 495 00:36:30,931 --> 00:36:33,690 like an open parentheses [INAUDIBLE]. 496 00:36:33,690 --> 00:36:39,130 PROFESSOR: Oh, these are functions of the delta j's. 497 00:36:39,130 --> 00:36:42,790 This dr, where it comes from, the work 498 00:36:42,790 --> 00:36:45,480 that's being done in a virtual displacement 499 00:36:45,480 --> 00:36:50,810 around a dynamic equilibrium position for the system 500 00:36:50,810 --> 00:36:53,000 is a little movement of the system. 501 00:36:53,000 --> 00:36:56,460 dr. And we express it. 502 00:36:56,460 --> 00:37:02,160 It's expressed in terms of a virtual displacement 503 00:37:02,160 --> 00:37:05,770 of the generalized coordinates of the system. 504 00:37:05,770 --> 00:37:08,540 So where the dr comes from is going to be delta. 505 00:37:08,540 --> 00:37:10,160 In this case, it's only delta x. 506 00:37:12,740 --> 00:37:15,020 And in the next problem, we're going 507 00:37:15,020 --> 00:37:17,370 to do the force in the problem is 508 00:37:17,370 --> 00:37:22,970 not in exactly the same direction as the delta x's 509 00:37:22,970 --> 00:37:24,730 and delta theta's and so forth. 510 00:37:24,730 --> 00:37:28,070 So when you do the dot product, only that complement 511 00:37:28,070 --> 00:37:29,930 of the force that's in the direction 512 00:37:29,930 --> 00:37:33,090 of the virtual displacement does work. 513 00:37:33,090 --> 00:37:34,960 And you account for that. 514 00:37:34,960 --> 00:37:41,330 So let's look into a more difficult problem. 515 00:37:41,330 --> 00:37:42,830 So the problem is this. 516 00:37:42,830 --> 00:37:45,460 I tried to fix assessed before I came to class. 517 00:37:45,460 --> 00:37:51,070 I didn't really quite have the parts and pieces I needed. 518 00:37:51,070 --> 00:37:55,720 But this a piece of steel pipe here. 519 00:37:55,720 --> 00:37:59,630 It's a sleeve on the outside of this rod. 520 00:37:59,630 --> 00:38:06,340 And I've got a spring that's on the outside connected 521 00:38:06,340 --> 00:38:07,390 to this piece. 522 00:38:07,390 --> 00:38:08,397 And so it can do this. 523 00:38:12,640 --> 00:38:16,200 And it's also, though, a pendulum. 524 00:38:16,200 --> 00:38:19,430 So the system I really want to look at is this system. 525 00:38:24,330 --> 00:38:27,960 So this swings back and forth, the thing slides up and down. 526 00:38:27,960 --> 00:38:32,960 So this has multiple sources of kinetic energy, multiple forms 527 00:38:32,960 --> 00:38:36,520 of potential energy. 528 00:38:36,520 --> 00:38:39,730 And for the purpose of the problem, 529 00:38:39,730 --> 00:38:44,500 I'm going to say that there's a force that's always horizontal 530 00:38:44,500 --> 00:38:49,640 acting on this mass pushing this system back and forth. 531 00:38:49,640 --> 00:38:53,050 Some f cosine omega t, always horizontal. 532 00:38:53,050 --> 00:38:54,560 And I want to drive the equations 533 00:38:54,560 --> 00:38:55,600 of motion of the system. 534 00:38:58,390 --> 00:39:02,170 So is it a planar motion problem? 535 00:39:06,060 --> 00:39:08,020 How many rigid bodies involved? 536 00:39:13,260 --> 00:39:15,250 There's two rigid bodies. 537 00:39:15,250 --> 00:39:18,440 Each could have possibly six degrees of freedom. 538 00:39:18,440 --> 00:39:21,740 But when you say it's a planar motion, 539 00:39:21,740 --> 00:39:24,460 you're actually immediately confining each rigid body 540 00:39:24,460 --> 00:39:25,830 to three. 541 00:39:25,830 --> 00:39:29,780 Each rigid body can move x and y and rotate in z. 542 00:39:29,780 --> 00:39:33,220 So when you [? spread ?] out and say this is planar motion, 543 00:39:33,220 --> 00:39:35,940 you've just said each rigid body has max three. 544 00:39:35,940 --> 00:39:39,705 So this is a maximum of six possible. 545 00:39:39,705 --> 00:39:41,080 Where the other three disappeared 546 00:39:41,080 --> 00:39:46,330 to is no z deflection and no rotation in the x or y. 547 00:39:46,330 --> 00:39:53,360 OK, so we have a possible maximum six. 548 00:39:53,360 --> 00:39:56,640 How many degrees of freedom does this problem have? 549 00:39:56,640 --> 00:40:01,609 How many coordinates will we need to completely describe 550 00:40:01,609 --> 00:40:02,650 the motion of the system? 551 00:40:02,650 --> 00:40:04,160 So think about that. 552 00:40:04,160 --> 00:40:05,170 Talk to a neighbor. 553 00:40:05,170 --> 00:40:08,730 Decide on the coordinates that we need to use for this system 554 00:40:08,730 --> 00:40:09,900 while I'm drawing it. 555 00:41:27,180 --> 00:41:28,547 OK. 556 00:41:28,547 --> 00:41:29,380 What did you decide? 557 00:41:29,380 --> 00:41:31,090 How many? 558 00:41:31,090 --> 00:41:31,850 Two. 559 00:41:31,850 --> 00:41:33,350 All right, what would you recommend? 560 00:41:37,994 --> 00:41:38,910 What would you choose? 561 00:41:41,742 --> 00:41:42,242 Pardon? 562 00:41:42,242 --> 00:41:44,630 AUDIENCE: The angle and how far down it is. 563 00:41:44,630 --> 00:41:46,480 PROFESSOR: An angle and a deflection 564 00:41:46,480 --> 00:41:49,860 of what I'm calling m2 here. 565 00:41:49,860 --> 00:41:51,740 So this is m2. 566 00:41:51,740 --> 00:41:52,738 The rod is m1. 567 00:41:55,366 --> 00:42:06,950 And he's suggesting an angle theta and a deflection 568 00:42:06,950 --> 00:42:09,280 which I'll call x1. 569 00:42:09,280 --> 00:42:14,450 And I've attached to this bar, the rod I'm calling it, 570 00:42:14,450 --> 00:42:20,660 a rotating coordinate system x1 y1. 571 00:42:20,660 --> 00:42:25,000 About point A. So A x1 y1's my rotating coordinate system 572 00:42:25,000 --> 00:42:28,490 attached to this rod. 573 00:42:28,490 --> 00:42:29,240 OK. 574 00:42:29,240 --> 00:42:31,520 So I'm going to locate the position of this 575 00:42:31,520 --> 00:42:35,510 by some value x1 measured from point A. 576 00:42:35,510 --> 00:42:40,910 And locate the position of the rod itself by an angle theta. 577 00:42:40,910 --> 00:42:42,700 Good. 578 00:42:42,700 --> 00:42:44,330 Is it complete? 579 00:42:44,330 --> 00:42:48,380 So if you freeze one, do you still have-- the complete. 580 00:42:48,380 --> 00:42:52,522 [INAUDIBLE] describe the motion at any possible position. 581 00:42:52,522 --> 00:42:53,230 Those two things. 582 00:42:53,230 --> 00:42:53,729 Yes. 583 00:42:53,729 --> 00:42:55,750 Is it independent? 584 00:42:55,750 --> 00:42:59,510 If you freeze x, can theta still move? 585 00:42:59,510 --> 00:43:02,310 If you freeze theta, can the x still move? 586 00:43:02,310 --> 00:43:05,360 OK, is it holonomic? 587 00:43:05,360 --> 00:43:06,550 Right. 588 00:43:06,550 --> 00:43:08,240 You need two, we got two and they're 589 00:43:08,240 --> 00:43:09,620 independent and complete. 590 00:43:09,620 --> 00:43:10,680 Good. 591 00:43:10,680 --> 00:43:13,990 Now the harder work starts. 592 00:43:13,990 --> 00:43:17,790 So I'm going to give us the mass of the rod, the mass moment 593 00:43:17,790 --> 00:43:22,920 of inertia the rod about the z-axis but with respect to A. 594 00:43:22,920 --> 00:43:25,440 The length of the rod is l1. 595 00:43:25,440 --> 00:43:31,840 The sleeve mass m2 izz with respect to its g. 596 00:43:31,840 --> 00:43:32,860 So it has a g. 597 00:43:32,860 --> 00:43:36,370 There's also and I'd better call it g2. 598 00:43:36,370 --> 00:43:38,180 That's the g of the sleeve. 599 00:43:38,180 --> 00:43:41,810 There's also a g1. 600 00:43:41,810 --> 00:43:45,000 A center of mass for the rod and a center 601 00:43:45,000 --> 00:43:47,260 of mass for the sleeve. 602 00:43:47,260 --> 00:43:48,910 Those are properties we'll need to know 603 00:43:48,910 --> 00:43:52,000 and I'll give them to you. 604 00:43:52,000 --> 00:43:53,480 OK. 605 00:43:53,480 --> 00:43:57,910 So we need to come up with expressions 606 00:43:57,910 --> 00:44:03,000 for potential energy and kinetic energy. 607 00:44:03,000 --> 00:44:06,920 So this problem, the potential energy's a little messy. 608 00:44:06,920 --> 00:44:09,610 Because you have to pick references. 609 00:44:09,610 --> 00:44:12,100 You have to account for the unstretched length 610 00:44:12,100 --> 00:44:12,695 of the spring. 611 00:44:15,930 --> 00:44:31,360 So call l 0 is the unstretched spring length. 612 00:44:35,240 --> 00:44:37,490 We know that also. 613 00:44:37,490 --> 00:44:42,880 So I propose that the potential energy look like 1/2, 614 00:44:42,880 --> 00:44:45,940 for the spring, anyway, 1/2 the amount that it 615 00:44:45,940 --> 00:44:50,200 stretches in a movement x1. 616 00:44:50,200 --> 00:44:51,610 The amount that it stretches then 617 00:44:51,610 --> 00:44:55,260 should be whatever that x1 position is. 618 00:44:55,260 --> 00:45:03,880 And that x1 position, and I drew it slightly incorrectly. 619 00:45:03,880 --> 00:45:07,390 I'm going to use x1 to locate the center of mass, which 620 00:45:07,390 --> 00:45:08,570 is always a good practice. 621 00:45:08,570 --> 00:45:10,000 So here's the center of mass. 622 00:45:10,000 --> 00:45:15,190 So my x1 goes to the center of mass. 623 00:45:15,190 --> 00:45:16,960 That's x1. 624 00:45:16,960 --> 00:45:20,050 So that's the total distance. 625 00:45:20,050 --> 00:45:23,000 And from that, we need to subtract 626 00:45:23,000 --> 00:45:25,940 l0, the unstretched length of the spring. 627 00:45:25,940 --> 00:45:31,240 And we need to subtract 1/2 the length of the body 628 00:45:31,240 --> 00:45:35,820 because that's that extra bit here. 629 00:45:35,820 --> 00:45:42,990 So this is the amount that the string is actually stretched 630 00:45:42,990 --> 00:45:44,759 when the coordinate is x1. 631 00:45:44,759 --> 00:45:45,800 And you've got to square. 632 00:45:45,800 --> 00:45:51,320 And that'll be the potential energy stored in the spring. 633 00:45:51,320 --> 00:45:55,980 Then we got to do the same thing for the potential energy. 634 00:45:55,980 --> 00:45:58,755 We have two sources of potential energy due to gravity. 635 00:45:58,755 --> 00:45:59,516 And they are? 636 00:46:03,410 --> 00:46:05,350 Two objects, right? 637 00:46:05,350 --> 00:46:06,430 Two potential energy. 638 00:46:06,430 --> 00:46:08,230 So why don't you take a minute and tell me 639 00:46:08,230 --> 00:46:10,210 the potential energy associated with the rod. 640 00:46:13,800 --> 00:46:16,180 So the rod has a center of mass. 641 00:46:16,180 --> 00:46:17,490 It's a pendulum basically. 642 00:46:17,490 --> 00:46:19,320 So it's the same as all the pendulum 643 00:46:19,320 --> 00:46:20,403 problems you've ever seen. 644 00:46:22,620 --> 00:46:28,230 And I would recommend that we use as our reference position 645 00:46:28,230 --> 00:46:32,290 its equilibrium position hanging straight down. 646 00:46:32,290 --> 00:46:34,260 And I'll tell you in advance, I'm 647 00:46:34,260 --> 00:46:37,170 going to use the unstretched spring position this time. 648 00:46:37,170 --> 00:46:38,110 Just stay with that. 649 00:46:38,110 --> 00:46:39,860 That's where it's going to start from. 650 00:46:39,860 --> 00:46:41,870 That's my reference for potential energy. 651 00:46:41,870 --> 00:46:46,470 But does the unstretched spring position 652 00:46:46,470 --> 00:46:50,130 have anything to do with the potential energy of the rod? 653 00:46:50,130 --> 00:46:50,650 No. 654 00:46:50,650 --> 00:46:51,150 OK. 655 00:46:51,150 --> 00:46:55,540 So its reference position is just hanging straight down. 656 00:46:55,540 --> 00:46:56,300 So figure it out. 657 00:46:56,300 --> 00:46:58,280 What's the potential energy expression 658 00:46:58,280 --> 00:47:01,572 for just the rod part? 659 00:47:01,572 --> 00:47:02,280 Think about that. 660 00:47:11,070 --> 00:47:12,920 So I'm going to remind you about something 661 00:47:12,920 --> 00:47:14,510 about potential energy. 662 00:47:14,510 --> 00:47:16,720 Potential energy, one of the requirements about it 663 00:47:16,720 --> 00:47:19,820 is the change in potential energy from one position 664 00:47:19,820 --> 00:47:23,940 to another is path independent. 665 00:47:23,940 --> 00:47:28,950 So you don't actually ever have to do the integral of minus mg 666 00:47:28,950 --> 00:47:30,140 dot dr. 667 00:47:30,140 --> 00:47:31,670 You don't have to do the integral. 668 00:47:31,670 --> 00:47:33,294 You just have to account for the change 669 00:47:33,294 --> 00:47:35,590 in height between its starting position 670 00:47:35,590 --> 00:47:38,220 and its some other position. 671 00:47:42,250 --> 00:47:43,940 Spend a minute or two, think about that. 672 00:47:43,940 --> 00:47:44,440 Work it out. 673 00:47:44,440 --> 00:47:45,935 You got a question? 674 00:47:45,935 --> 00:47:46,435 OK. 675 00:48:14,210 --> 00:48:14,850 Can you talk? 676 00:48:14,850 --> 00:48:17,105 Talk to a neighbor, check your ideas. 677 00:48:54,650 --> 00:48:56,950 So you have a suggestion for me? 678 00:48:56,950 --> 00:48:58,484 Ladies? 679 00:48:58,484 --> 00:48:59,400 AUDIENCE: [INAUDIBLE]. 680 00:49:06,825 --> 00:49:09,300 l1/2 [INAUDIBLE]. 681 00:49:16,127 --> 00:49:16,710 PROFESSOR: OK. 682 00:49:19,440 --> 00:49:21,600 Anybody want to make an improvement on that? 683 00:49:21,600 --> 00:49:23,010 Or they like it? 684 00:49:23,010 --> 00:49:24,212 Improvement? 685 00:49:24,212 --> 00:49:26,810 AUDIENCE: [INAUDIBLE]. 686 00:49:26,810 --> 00:49:29,280 PROFESSOR: 1 minus cosine theta. 687 00:49:29,280 --> 00:49:35,550 So let's put that up and let's figure out if we need that. 688 00:49:35,550 --> 00:49:43,140 We have a bid for cosine theta and 1 minus cosine theta. 689 00:49:43,140 --> 00:49:46,870 So you need to have a potential energy at the reference 690 00:49:46,870 --> 00:49:49,680 and you need to have a potential energy at the final point. 691 00:49:49,680 --> 00:49:51,330 And the difference between the two 692 00:49:51,330 --> 00:49:54,430 is a change in potential energy here. 693 00:49:54,430 --> 00:49:57,450 So what's the reference potential energy 694 00:49:57,450 --> 00:50:01,850 is mg l1 over 2 when it hangs straight down. 695 00:50:01,850 --> 00:50:04,610 And then when it moves up to this other position, 696 00:50:04,610 --> 00:50:10,000 this is the l1 over 2 times this is a delta h. 697 00:50:10,000 --> 00:50:13,920 This is the change in height that it goes through. 698 00:50:13,920 --> 00:50:14,935 So you need the 1 minus. 699 00:50:22,030 --> 00:50:24,585 Do we have the sines right? 700 00:50:24,585 --> 00:50:25,085 Yeah. 701 00:50:30,030 --> 00:50:30,530 OK. 702 00:50:37,970 --> 00:50:39,940 So now we need another term. 703 00:50:39,940 --> 00:50:42,240 And I'll write this one down. 704 00:50:42,240 --> 00:50:43,970 This one's a little messier. 705 00:50:43,970 --> 00:50:49,360 We need a potential energy term due to gravity for the sleeve. 706 00:50:49,360 --> 00:50:51,590 And that's going to mimic this. 707 00:50:51,590 --> 00:50:54,495 You're going to have a term here plus m2g. 708 00:50:58,190 --> 00:50:59,720 And it's reference, I'm just going 709 00:50:59,720 --> 00:51:02,950 to do it as a reference amount minus the final amount. 710 00:51:02,950 --> 00:51:07,610 The reference will be at the initial location 711 00:51:07,610 --> 00:51:20,740 of its center of mass, which is l0 plus l2 over 2 minus m2 g 712 00:51:20,740 --> 00:51:24,520 x1 cosine theta. 713 00:51:27,970 --> 00:51:29,420 Because this one is a little messy 714 00:51:29,420 --> 00:51:30,670 because you've got this thing. 715 00:51:30,670 --> 00:51:34,200 It can move up and down the sleeve. 716 00:51:34,200 --> 00:51:37,010 And if that moves, you've lost your reference. 717 00:51:37,010 --> 00:51:40,070 So you can't do this as a concise little term like this. 718 00:51:40,070 --> 00:51:43,220 You have to separate out the reference 719 00:51:43,220 --> 00:51:46,220 and then this is the final. 720 00:51:46,220 --> 00:51:49,790 And the l0 plus l2, this quantity 721 00:51:49,790 --> 00:51:52,170 here is the starting height. 722 00:51:52,170 --> 00:51:56,074 This x1 cosine theta is the finishing height. 723 00:51:56,074 --> 00:51:57,490 And the difference between the two 724 00:51:57,490 --> 00:51:59,760 gives you the change in the potential energy. 725 00:51:59,760 --> 00:52:02,170 So this is your potential energy expression. 726 00:52:02,170 --> 00:52:04,965 This plus this plus these. 727 00:52:04,965 --> 00:52:05,465 All right. 728 00:52:11,910 --> 00:52:13,180 So what about t? 729 00:52:13,180 --> 00:52:14,880 We got to be able write it. 730 00:52:14,880 --> 00:52:16,877 Kinetic energy is generally easier. 731 00:52:16,877 --> 00:52:18,710 Got to account for all the parts and pieces. 732 00:52:18,710 --> 00:52:20,190 So we have to chunks. 733 00:52:20,190 --> 00:52:25,090 And we're going to have rotational kinetic energy 734 00:52:25,090 --> 00:52:28,750 associated with the rod, rotational kinetic energy 735 00:52:28,750 --> 00:52:33,190 associated with the sleeve. 736 00:52:33,190 --> 00:52:35,450 But also some translational kinetic energy 737 00:52:35,450 --> 00:52:37,292 associated with the sleeve. 738 00:52:37,292 --> 00:52:38,625 And I'll write these terms down. 739 00:52:41,860 --> 00:52:43,845 Make the problem go a little faster here. 740 00:52:48,390 --> 00:52:54,655 1/2 izz about A. That's the rod. 741 00:52:57,930 --> 00:53:08,055 Plus 1/2 izz for the sleeve about g. 742 00:53:10,620 --> 00:53:13,920 We'll discuss why the difference here. 743 00:53:13,920 --> 00:53:18,190 And that's theta dot squared. 744 00:53:20,960 --> 00:53:30,870 Now for the kinetic energy that comes from translation 745 00:53:30,870 --> 00:53:33,580 of the center of mass. 746 00:53:33,580 --> 00:53:34,990 Because I'm broken up. 747 00:53:38,930 --> 00:53:40,690 Let me start over. 748 00:53:40,690 --> 00:53:47,680 This system is pinned about A. And the rod is just 749 00:53:47,680 --> 00:53:52,370 simply pinned at A. And the last lecture 750 00:53:52,370 --> 00:53:55,450 I put up these different conditions and simplifications. 751 00:53:55,450 --> 00:54:00,470 You can account for a something about a fixed pin 752 00:54:00,470 --> 00:54:04,405 by computing maximum inertia about A. 753 00:54:04,405 --> 00:54:06,950 It's basically a parallel axis theorem argument. 754 00:54:06,950 --> 00:54:09,700 Times 1/2 times that times theta dot squared. 755 00:54:09,700 --> 00:54:12,140 So this gives you all the kinetic energy in one 756 00:54:12,140 --> 00:54:14,460 go with the rod. 757 00:54:14,460 --> 00:54:19,000 But for the sliding mass, because its position 758 00:54:19,000 --> 00:54:21,580 is changing, you can't do that. 759 00:54:21,580 --> 00:54:26,080 You have to account for the two components of kinetic energy 760 00:54:26,080 --> 00:54:27,060 separately. 761 00:54:27,060 --> 00:54:29,990 This accounts for rotation about g. 762 00:54:29,990 --> 00:54:32,380 Even though g is moving. 763 00:54:32,380 --> 00:54:33,715 That accounts for that energy. 764 00:54:36,140 --> 00:54:37,890 Because it's only a function of theta dot. 765 00:54:37,890 --> 00:54:40,860 It's not a function of that position x. 766 00:54:40,860 --> 00:54:44,060 This term is going to account for the kinetic energy 767 00:54:44,060 --> 00:54:46,890 associated with the movement of the center of mass. 768 00:54:46,890 --> 00:54:53,405 So we need a vg2 in the inertial frame dot vg2. 769 00:54:58,300 --> 00:54:59,620 These be in vectors. 770 00:54:59,620 --> 00:55:02,580 And does that get everything? 771 00:55:02,580 --> 00:55:04,270 I think that does. 772 00:55:04,270 --> 00:55:16,350 So vgo is it certainly has a component that 773 00:55:16,350 --> 00:55:18,800 is its speed sliding up and down the rod, right? 774 00:55:18,800 --> 00:55:20,950 And that's in the i hat direction. 775 00:55:23,760 --> 00:55:28,080 But it has another component due to what? 776 00:55:28,080 --> 00:55:30,370 Can you tell me what it is? 777 00:55:30,370 --> 00:55:34,435 Its contribution to its speed due to its rotation. 778 00:55:44,113 --> 00:55:46,617 AUDIENCE: [INAUDIBLE]. 779 00:55:46,617 --> 00:55:47,950 PROFESSOR: It's got a theta dot. 780 00:55:47,950 --> 00:55:49,500 Yep. 781 00:55:49,500 --> 00:55:51,485 It needs an r, right? 782 00:55:51,485 --> 00:55:52,401 AUDIENCE: [INAUDIBLE]. 783 00:55:58,714 --> 00:55:59,380 PROFESSOR: Yeah. 784 00:55:59,380 --> 00:56:01,246 So this would be an x1 plus. 785 00:56:04,710 --> 00:56:12,210 No actually, I made x1 go right to the-- so just x1 theta 786 00:56:12,210 --> 00:56:15,308 dot in one direction. 787 00:56:15,308 --> 00:56:18,400 Yeah, so j hat here. 788 00:56:18,400 --> 00:56:20,980 Actually that's the moving coordinate system unit 789 00:56:20,980 --> 00:56:22,690 vector in the y direction. 790 00:56:22,690 --> 00:56:26,310 And so we do the dot product. 791 00:56:26,310 --> 00:56:32,810 You get this times itself. i dot i and j dot j. 792 00:56:32,810 --> 00:56:42,190 This quantity here is 1/2 m2 x dot squared 793 00:56:42,190 --> 00:56:46,820 plus x, this next one I guess. 794 00:56:46,820 --> 00:56:51,880 x1 squared of theta dot squared. 795 00:56:51,880 --> 00:56:55,180 That's the kinetic energy of accounting for the velocity 796 00:56:55,180 --> 00:56:56,204 of the center of mass. 797 00:56:56,204 --> 00:56:58,370 So now we have our entire kinetic energy expression. 798 00:57:08,230 --> 00:57:12,330 So now we have how many coordinates? 799 00:57:12,330 --> 00:57:12,949 Two, right? 800 00:57:12,949 --> 00:57:14,740 How many times do we have to turn the crank 801 00:57:14,740 --> 00:57:17,390 and go through the Lagrangian? 802 00:57:17,390 --> 00:57:19,160 Got to go through it twice. 803 00:57:19,160 --> 00:57:25,640 So let's apply Lagrange here. 804 00:57:28,720 --> 00:57:30,980 And we'll just do number one first. 805 00:57:30,980 --> 00:57:32,720 So and let's see. 806 00:57:32,720 --> 00:57:34,860 Which one do I have on my paper first? 807 00:57:34,860 --> 00:57:38,120 I guess we'll do the x1 equation. 808 00:57:38,120 --> 00:57:40,260 This is delta x1. 809 00:57:40,260 --> 00:57:42,920 So this generalized coordinate x1. 810 00:57:42,920 --> 00:57:45,520 And we need to do term one, which 811 00:57:45,520 --> 00:58:00,210 is in d by dt of partial of t with respect to x1 dot. 812 00:58:00,210 --> 00:58:01,830 OK. 813 00:58:01,830 --> 00:58:06,500 So we look at this and say, well, is this a function? 814 00:58:06,500 --> 00:58:09,415 Is this term a function of x? 815 00:58:09,415 --> 00:58:10,010 Nothing. 816 00:58:10,010 --> 00:58:12,180 You get nothing from there. 817 00:58:12,180 --> 00:58:14,370 Is this term a function of x? 818 00:58:14,370 --> 00:58:16,840 Yeah, it's down here. 819 00:58:16,840 --> 00:58:19,300 We only have to take the derivative of this. 820 00:58:19,300 --> 00:58:22,100 We have to do that job. 821 00:58:22,100 --> 00:58:25,200 So the derivative of this with respect to x 822 00:58:25,200 --> 00:58:32,152 dot, you get a 2x dot here. 823 00:58:32,152 --> 00:58:34,610 Do you get anything from here when you do this with respect 824 00:58:34,610 --> 00:58:35,197 to x dot? 825 00:58:35,197 --> 00:58:36,780 You only get a contribution from here. 826 00:58:36,780 --> 00:58:38,580 The two cancels that. 827 00:58:38,580 --> 00:58:48,930 And so this should look like m2 x1 dot but d by dt. 828 00:58:48,930 --> 00:58:53,400 Do this once in two steps here so you see what happens. 829 00:58:53,400 --> 00:58:57,610 You get an m2 x1 double dot out of that. 830 00:59:08,200 --> 00:59:10,300 So we've gotten the first piece of this. 831 00:59:10,300 --> 00:59:11,690 We've got a couple to go. 832 00:59:11,690 --> 00:59:15,184 But you know a lot about Newton's laws 833 00:59:15,184 --> 00:59:17,100 and you know a lot about calculating equations 834 00:59:17,100 --> 00:59:21,250 of motion now using sum of torques and all that stuff, 835 00:59:21,250 --> 00:59:22,070 right? 836 00:59:22,070 --> 00:59:27,500 So this is just something moving, has a circular motion, 837 00:59:27,500 --> 00:59:29,090 has translational motion. 838 00:59:29,090 --> 00:59:32,720 What other accelerations had better 839 00:59:32,720 --> 00:59:35,860 appear in this equation of motion? 840 00:59:35,860 --> 00:59:39,320 And which equation are we getting? 841 00:59:39,320 --> 00:59:40,695 There's going to be two equations 842 00:59:40,695 --> 00:59:42,790 and it has physical significance to it. 843 00:59:42,790 --> 00:59:46,830 What equation does this begin to look like? 844 00:59:46,830 --> 00:59:50,800 Just physically, what movement is being accounted for here? 845 00:59:50,800 --> 00:59:52,970 Looks like translation in the x direction. 846 00:59:52,970 --> 00:59:54,800 It's this thing sliding. 847 00:59:54,800 --> 00:59:57,200 It's this part of the motion sliding up and down. 848 00:59:57,200 --> 00:59:59,100 You're writing an equation of motion and mx 849 00:59:59,100 --> 01:00:00,405 double dot has units of what? 850 01:00:04,930 --> 01:00:06,540 Torque? 851 01:00:06,540 --> 01:00:07,400 Force. 852 01:00:07,400 --> 01:00:08,490 So it's a force equation. 853 01:00:08,490 --> 01:00:11,869 This is just f equals ma is what this is going to show us. 854 01:00:11,869 --> 01:00:13,660 Remember, the direct method has to give you 855 01:00:13,660 --> 01:00:16,920 the same answer as Lagrange. 856 01:00:16,920 --> 01:00:20,810 So we're getting a force equation. 857 01:00:20,810 --> 01:00:23,460 It's describing mx double dot. 858 01:00:23,460 --> 01:00:26,040 What other acceleration terms do you 859 01:00:26,040 --> 01:00:27,900 expect to appear in this from what you know? 860 01:00:27,900 --> 01:00:28,400 Yeah. 861 01:00:28,400 --> 01:00:28,895 AUDIENCE: [INAUDIBLE]. 862 01:00:28,895 --> 01:00:30,340 PROFESSOR: A centripetal term. 863 01:00:30,340 --> 01:00:32,464 Do you believe there ought to be a centripetal term 864 01:00:32,464 --> 01:00:34,150 in this answer? 865 01:00:34,150 --> 01:00:37,030 Why? 866 01:00:37,030 --> 01:00:39,550 Because it's got circular motion involved. 867 01:00:39,550 --> 01:00:41,380 For sure. 868 01:00:41,380 --> 01:00:42,040 Any others? 869 01:00:42,040 --> 01:00:44,780 Is there any Coriolis in this? 870 01:00:44,780 --> 01:00:45,890 In this direction. 871 01:00:45,890 --> 01:00:49,690 Which direction are we working in? 872 01:00:49,690 --> 01:00:53,680 Is there Coriolis acceleration in the x direction? 873 01:00:53,680 --> 01:00:58,530 By the way, these equations, do we have any ijk's in here? 874 01:00:58,530 --> 01:01:00,440 These are pure scalar equations. 875 01:01:03,090 --> 01:01:05,170 No unit vectors involved. 876 01:01:05,170 --> 01:01:07,810 This equation only described motion in the x. 877 01:01:11,730 --> 01:01:15,270 So will there be a Coriolis force in this acceleration 878 01:01:15,270 --> 01:01:16,220 in this problem? 879 01:01:16,220 --> 01:01:18,050 Will there be an Eulerian acceleration 880 01:01:18,050 --> 01:01:19,360 in this equation of motion? 881 01:01:19,360 --> 01:01:21,151 The reason I'm going through this with you, 882 01:01:21,151 --> 01:01:23,590 I want you to start developing your own intuition about 883 01:01:23,590 --> 01:01:25,950 whether or not when you get it at the end 884 01:01:25,950 --> 01:01:27,450 it's got everything it ought to have 885 01:01:27,450 --> 01:01:30,590 and doesn't have stuff it shouldn't have. 886 01:01:30,590 --> 01:01:31,150 OK. 887 01:01:31,150 --> 01:01:35,190 So your forecasting, then we better get a centripetal term. 888 01:01:35,190 --> 01:01:36,550 Well, let's see what happens. 889 01:01:36,550 --> 01:01:39,030 So that was number one. 890 01:01:39,030 --> 01:01:56,010 Number two here is our dt by minus the derivative 891 01:01:56,010 --> 01:01:58,970 with respect to x, in this case. 892 01:01:58,970 --> 01:02:00,460 So we go here. 893 01:02:00,460 --> 01:02:02,770 x1 we've been calling it. 894 01:02:02,770 --> 01:02:04,500 Is this a function of x? 895 01:02:04,500 --> 01:02:05,000 This piece? 896 01:02:05,000 --> 01:02:06,290 Nope, it's x dot. 897 01:02:06,290 --> 01:02:07,391 How about this one? 898 01:02:07,391 --> 01:02:07,890 Right. 899 01:02:07,890 --> 01:02:10,550 Take this derivative, you get 2x. 900 01:02:10,550 --> 01:02:12,580 So this fellow is going to give us 901 01:02:12,580 --> 01:02:22,535 minus m2 x1 theta dot squared. 902 01:02:25,090 --> 01:02:27,620 What's that look like? 903 01:02:27,620 --> 01:02:28,590 There it is. 904 01:02:28,590 --> 01:02:30,980 There's your centripetal term you're expecting to get. 905 01:02:30,980 --> 01:02:31,790 OK. 906 01:02:31,790 --> 01:02:39,900 And step three is plus partial of v with respect to x. 907 01:02:42,450 --> 01:02:44,280 In this case with respect to x. 908 01:02:44,280 --> 01:02:46,900 And where's our potential energy expression? 909 01:02:46,900 --> 01:02:48,530 Well, it's up here. 910 01:02:48,530 --> 01:02:55,910 And where the x dependency is in it. 911 01:02:55,910 --> 01:03:00,480 There is no x in that term and no x in that term. 912 01:03:00,480 --> 01:03:04,220 But we have x's in both of these other terms. 913 01:03:04,220 --> 01:03:07,840 So when we run through this, I'll 914 01:03:07,840 --> 01:03:11,190 write down what we come up with. 915 01:03:11,190 --> 01:03:13,670 We get certainly a spring term. 916 01:03:13,670 --> 01:03:22,950 k x1 minus l0 minus l2 over 2. 917 01:03:22,950 --> 01:03:29,420 So that's the spring piece when you take the derivative. 918 01:03:29,420 --> 01:03:32,010 The two cancels the 1/2 and the derivative of parts 919 01:03:32,010 --> 01:03:33,670 inside just gives you 1. 920 01:03:33,670 --> 01:03:36,790 So that's the first piece of the potential energy expression. 921 01:03:36,790 --> 01:03:41,940 And the second piece is only going to come from here. 922 01:03:41,940 --> 01:03:43,780 The derivative of this with respect to x1 923 01:03:43,780 --> 01:03:47,915 is just m2g cosine theta minus. 924 01:03:56,280 --> 01:03:58,920 And you add those bits together, you 925 01:03:58,920 --> 01:04:11,230 end up with m2 x1 double dot minus m2 x1 theta 926 01:04:11,230 --> 01:04:21,200 dot squared plus k x1 minus l0 minus l2 927 01:04:21,200 --> 01:04:29,440 over 2 minus m2g cosine theta. 928 01:04:29,440 --> 01:04:31,800 So those are the three terms, 1 plus 2 929 01:04:31,800 --> 01:04:34,160 plus 3, that go on the left hand side. 930 01:04:34,160 --> 01:04:39,940 And they're going to equal my qx that I find. 931 01:04:39,940 --> 01:04:42,570 I still have to find what the generalized force is 932 01:04:42,570 --> 01:04:43,386 in the x direction. 933 01:04:55,070 --> 01:04:57,190 So all that's left to do for this problem 934 01:04:57,190 --> 01:05:01,440 is to find q sub x, the generalized force that 935 01:05:01,440 --> 01:05:03,540 goes on the right hand side. 936 01:05:03,540 --> 01:05:10,520 So now let's draw a little diagram here of my system. 937 01:05:10,520 --> 01:05:13,330 And at the end of the sleeve. 938 01:05:13,330 --> 01:05:17,490 So here's my sleeve. 939 01:05:17,490 --> 01:05:18,960 I've applied this force. 940 01:05:21,710 --> 01:05:25,110 This is f of t. 941 01:05:25,110 --> 01:05:30,030 And maybe it's some f not cosine omega t. 942 01:05:30,030 --> 01:05:33,290 It's an oscillatory force, external force. 943 01:05:33,290 --> 01:05:34,190 Make it vibrate. 944 01:05:37,730 --> 01:05:42,550 And I need to know the virtual work done 945 01:05:42,550 --> 01:05:47,405 making that force go through a displacement in what direction? 946 01:05:54,200 --> 01:05:57,340 So this equation is the x1 equation, right? 947 01:05:57,340 --> 01:06:02,430 And so the virtual displacement I'm talking about is delta x1. 948 01:06:02,430 --> 01:06:04,660 And the amount of work that it does 949 01:06:04,660 --> 01:06:09,340 is delta x1 times the component of this force that's 950 01:06:09,340 --> 01:06:11,420 in its direction. 951 01:06:11,420 --> 01:06:19,380 So I'm going to take this force and break it up 952 01:06:19,380 --> 01:06:22,690 into two components. 953 01:06:22,690 --> 01:06:29,228 And if this is my theta, this is also theta. 954 01:06:37,700 --> 01:06:41,505 So this will be f0 product. 955 01:06:41,505 --> 01:06:46,255 And I'll leave out the cosine omega t here. 956 01:06:46,255 --> 01:06:47,690 It's a function of time. 957 01:06:47,690 --> 01:06:51,770 But this side then is cosine theta i. 958 01:06:54,500 --> 01:06:56,460 No, hey, I got this wrong. 959 01:06:56,460 --> 01:06:57,700 I drew this wrong, I'm sorry. 960 01:06:57,700 --> 01:06:59,120 This is theta. 961 01:06:59,120 --> 01:07:00,165 This is going to be sine. 962 01:07:02,700 --> 01:07:08,500 This side is sine theta in the i direction. 963 01:07:08,500 --> 01:07:18,025 And this piece is f0 of t cosine theta in the j. 964 01:07:18,025 --> 01:07:21,260 So I break it up in two parts. 965 01:07:21,260 --> 01:07:26,440 And the virtual work associated with x1 966 01:07:26,440 --> 01:07:33,810 is the thing I'm looking for, qx, dotted with delta x1. 967 01:07:33,810 --> 01:07:40,910 And that is f of t here, the vector 968 01:07:40,910 --> 01:07:45,990 dotted with dr, my little displacement. 969 01:07:45,990 --> 01:07:48,540 But in this case, this then all works out 970 01:07:48,540 --> 01:08:00,080 to be f0 cosine omega t. 971 01:08:00,080 --> 01:08:12,750 And it has sine theta i plus cos theta j components 972 01:08:12,750 --> 01:08:17,890 dotted width delta x in the i. 973 01:08:17,890 --> 01:08:21,620 So you're only going to get i dot j gives you 0, i dot i 974 01:08:21,620 --> 01:08:22,920 gets you 1. 975 01:08:22,920 --> 01:08:26,149 So you're going to get one piece out of this. 976 01:08:26,149 --> 01:08:41,190 This says in the qx equals f0 cosine omega t sine theta. 977 01:08:45,109 --> 01:08:50,250 And start with you have a delta x here and a delta x here 978 01:08:50,250 --> 01:08:52,560 and that gives you the delta virtual work. 979 01:08:56,581 --> 01:08:58,080 Personally when I do these problems, 980 01:08:58,080 --> 01:09:01,390 I have to think in terms of that little virtual deflection. 981 01:09:01,390 --> 01:09:04,670 I've actually figure out what's the virtual work done. 982 01:09:04,670 --> 01:09:07,770 And then at the end I take this out 983 01:09:07,770 --> 01:09:11,660 and this is the qx that I'm looking for. 984 01:09:11,660 --> 01:09:15,100 So my final equation of motion says, 985 01:09:15,100 --> 01:09:26,520 this equals f0 cosine omega t sine theta. 986 01:09:26,520 --> 01:09:30,054 And that's your equation of motion in the x1 direction. 987 01:09:35,680 --> 01:09:38,490 So when you finish one of these, you need to ask yourself, 988 01:09:38,490 --> 01:09:39,479 does this make sense? 989 01:09:39,479 --> 01:09:41,859 Does this jive with my understanding 990 01:09:41,859 --> 01:09:43,115 of Newtonian physics? 991 01:09:46,540 --> 01:09:48,189 Better have a linear acceleration term, 992 01:09:48,189 --> 01:09:50,180 because that's what it's doing. 993 01:09:50,180 --> 01:09:52,960 You have another acceleration term in the same direction 994 01:09:52,960 --> 01:09:54,810 due to centripetal. 995 01:09:54,810 --> 01:09:57,060 A spring force for sure. 996 01:09:57,060 --> 01:10:01,470 And a component of gravity in the direction of motion, 997 01:10:01,470 --> 01:10:04,700 up and down the slide, equal to any external forces 998 01:10:04,700 --> 01:10:05,840 in that direction. 999 01:10:05,840 --> 01:10:08,090 So it makes pretty good sense. 1000 01:10:08,090 --> 01:10:09,220 OK. 1001 01:10:09,220 --> 01:10:12,850 Now, also another test you can do 1002 01:10:12,850 --> 01:10:17,690 is does it satisfy the laws of statics? 1003 01:10:17,690 --> 01:10:19,350 That's another check you could perform. 1004 01:10:19,350 --> 01:10:22,840 Does this thing at static equilibrium tell the truth? 1005 01:10:22,840 --> 01:10:25,230 A static equilibrium all time derivative is 0. 1006 01:10:25,230 --> 01:10:27,430 So this would be 0, this would be 0. 1007 01:10:27,430 --> 01:10:29,620 You know its static equilibrium hangs down, 1008 01:10:29,620 --> 01:10:31,800 so cosine theta is 1. 1009 01:10:31,800 --> 01:10:34,320 Static you don't have any time dependent forces. 1010 01:10:34,320 --> 01:10:35,610 That's 0. 1011 01:10:35,610 --> 01:10:50,630 So the static part of this says that k x1 minus l0 minus l2 1012 01:10:50,630 --> 01:10:57,690 over 2 equals m2g cosine. 1013 01:10:57,690 --> 01:11:02,000 And that's cosine theta is 1, so it's m2g. 1014 01:11:02,000 --> 01:11:06,680 And you could figure out then this must be k times something. 1015 01:11:06,680 --> 01:11:08,480 This is the x. 1016 01:11:08,480 --> 01:11:12,180 This is the amount of the spring stretches, the static stretch 1017 01:11:12,180 --> 01:11:16,730 of the spring, so the spring has an equal and opposite force 1018 01:11:16,730 --> 01:11:18,690 to the weight of the thing m2g. 1019 01:11:18,690 --> 01:11:23,220 So that's another check you can do when doing the problems. 1020 01:11:23,220 --> 01:11:31,070 OK, I'll write up the final one. 1021 01:11:31,070 --> 01:11:32,530 We have one more question to go. 1022 01:11:32,530 --> 01:11:36,090 Got to do all the derivatives with respect to theta. 1023 01:11:36,090 --> 01:11:43,390 So you take a minute to decide how many acceleration terms 1024 01:11:43,390 --> 01:11:45,040 and what acceleration terms do you 1025 01:11:45,040 --> 01:11:48,670 expect to see come out of this second equation of motion. 1026 01:11:48,670 --> 01:11:51,870 Because now we're talking about which motion? 1027 01:11:51,870 --> 01:11:53,870 Swinging motion. 1028 01:11:53,870 --> 01:11:55,550 And what's its direction? 1029 01:11:55,550 --> 01:11:58,250 In Newtonian sense, it would have a vector direction. 1030 01:11:58,250 --> 01:12:01,530 It's in what we call j here. 1031 01:12:01,530 --> 01:12:05,680 OK, so you're about to get the j equation. 1032 01:12:05,680 --> 01:12:08,042 What terms do you expect to find in it? 1033 01:12:11,020 --> 01:12:12,730 Talk to your neighbors and sort this out. 1034 01:12:12,730 --> 01:12:15,550 And basically tell me what the answer's going to be. 1035 01:13:02,765 --> 01:13:03,690 What do you think? 1036 01:13:09,660 --> 01:13:12,836 What are we going to get? 1037 01:13:12,836 --> 01:13:15,306 AUDIENCE: We were debating about whether or not 1038 01:13:15,306 --> 01:13:22,240 it was going to be like speeding up in the theta [INAUDIBLE]. 1039 01:13:22,240 --> 01:13:25,330 PROFESSOR: So it is a pendulum, just a weird pendulum. 1040 01:13:25,330 --> 01:13:33,940 So does the theta change speed? 1041 01:13:33,940 --> 01:13:34,460 Sure. 1042 01:13:34,460 --> 01:13:36,680 When he gets up the top of the swing at 0. 1043 01:13:36,680 --> 01:13:38,620 All the way down, it's maximum speed. 1044 01:13:38,620 --> 01:13:41,900 So what term does that imply that you're going to get? 1045 01:13:41,900 --> 01:13:43,960 AUDIENCE: [INAUDIBLE]. 1046 01:13:43,960 --> 01:13:46,110 PROFESSOR: Well, maybe, maybe not. 1047 01:13:46,110 --> 01:13:46,610 Yeah? 1048 01:13:46,610 --> 01:13:47,527 AUDIENCE: [INAUDIBLE]. 1049 01:13:47,527 --> 01:13:49,818 PROFESSOR: Going to get an Eulerian, which means you've 1050 01:13:49,818 --> 01:13:51,140 got a theta double dot term. 1051 01:13:51,140 --> 01:13:53,640 You're expecting a theta double dot term to show up. 1052 01:13:53,640 --> 01:13:54,140 OK. 1053 01:13:54,140 --> 01:13:55,120 What else? 1054 01:13:59,040 --> 01:14:02,860 Will you get a Coriolis term? 1055 01:14:02,860 --> 01:14:04,580 Do you expect a Coriolis term? 1056 01:14:04,580 --> 01:14:07,556 Something that looks like x dot theta dot. 1057 01:14:07,556 --> 01:14:10,074 AUDIENCE: [INAUDIBLE]. 1058 01:14:10,074 --> 01:14:10,740 PROFESSOR: Yeah. 1059 01:14:10,740 --> 01:14:12,573 The thing is sliding up and down the sleeve. 1060 01:14:12,573 --> 01:14:15,640 It has a non 0 value of x dot. 1061 01:14:15,640 --> 01:14:19,650 Any time you got things moving radially 1062 01:14:19,650 --> 01:14:24,020 while something is swinging in a circle, 1063 01:14:24,020 --> 01:14:25,605 you will get Coriolis forces. 1064 01:14:25,605 --> 01:14:28,220 It means the angular momentum of that thing is changing 1065 01:14:28,220 --> 01:14:30,280 and it takes forces to make that happen. 1066 01:14:30,280 --> 01:14:34,515 So here's what this answer looks like. 1067 01:14:52,340 --> 01:14:55,850 That's the one term. 1068 01:14:55,850 --> 01:15:00,530 The two piece gives you 0. 1069 01:15:03,435 --> 01:15:09,210 It's not a function of x in the three piece. 1070 01:15:09,210 --> 01:15:19,290 The potential energy pace gives you m2g x1 sine theta 1071 01:15:19,290 --> 01:15:27,980 plus m1 g l1 over 2 sine theta. 1072 01:15:27,980 --> 01:15:32,930 And the fourth piece, the q theta, 1073 01:15:32,930 --> 01:15:35,536 well, that's just going to be the virtual work done. 1074 01:15:38,660 --> 01:15:40,260 There's a tricky bit to this one. 1075 01:15:40,260 --> 01:15:43,240 Now there's virtual work, but which direction? 1076 01:15:43,240 --> 01:15:49,800 So we have an f dot dr. The only f we have is this. 1077 01:15:49,800 --> 01:15:51,519 What's the dr? 1078 01:15:51,519 --> 01:15:52,517 What direction is it? 1079 01:15:58,010 --> 01:15:59,310 This is the theta coordinate. 1080 01:15:59,310 --> 01:16:01,810 What direction does that give you displacements? 1081 01:16:01,810 --> 01:16:05,616 f dot dr's a displacement, not an angle. 1082 01:16:05,616 --> 01:16:07,490 To get the work done, you got to move a force 1083 01:16:07,490 --> 01:16:08,975 through a distance. 1084 01:16:08,975 --> 01:16:11,100 So the distance, first of all, is in what direction 1085 01:16:11,100 --> 01:16:14,256 when theta moves? 1086 01:16:14,256 --> 01:16:14,756 j. 1087 01:16:14,756 --> 01:16:16,910 Little j hat, right? 1088 01:16:16,910 --> 01:16:21,050 And now if you get a virtual deflection 1089 01:16:21,050 --> 01:16:25,670 of delta theta, what's the virtual displacement? 1090 01:16:25,670 --> 01:16:28,310 You had a virtual change in [? angle ?] delta to put theta. 1091 01:16:28,310 --> 01:16:29,880 But is that the virtual displacement? 1092 01:16:32,970 --> 01:16:39,970 What's the displacement of this point here in the j direction, 1093 01:16:39,970 --> 01:16:42,576 given a virtual displacement delta theta? 1094 01:16:42,576 --> 01:16:43,568 Think that out. 1095 01:16:53,984 --> 01:16:56,067 AUDIENCE: [INAUDIBLE]. 1096 01:16:56,067 --> 01:16:57,400 PROFESSOR: Can't quite hear you. 1097 01:16:57,400 --> 01:16:59,240 AUDIENCE: [INAUDIBLE]. 1098 01:16:59,240 --> 01:17:01,340 PROFESSOR: x1 delta theta will give you 1099 01:17:01,340 --> 01:17:04,710 the motion to displacement at the center of mass 1100 01:17:04,710 --> 01:17:07,560 in that direction. 1101 01:17:07,560 --> 01:17:09,700 x1 comes from here to here. 1102 01:17:09,700 --> 01:17:12,220 So x1 delta theta will give you a little displacement 1103 01:17:12,220 --> 01:17:13,340 in that direction. 1104 01:17:13,340 --> 01:17:17,140 But is that the displacement we care about? 1105 01:17:17,140 --> 01:17:18,600 We need the displacement here. 1106 01:17:18,600 --> 01:17:19,540 So you're close. 1107 01:17:24,240 --> 01:17:32,670 So we're going to get some force dot a displacement dr. 1108 01:17:32,670 --> 01:17:36,370 And that's going to be our force, this guy, 1109 01:17:36,370 --> 01:17:38,530 with its i and j components. 1110 01:17:38,530 --> 01:17:39,780 i and j terms. 1111 01:17:39,780 --> 01:17:47,270 But this term out here is x1 plus l2 over 2 1112 01:17:47,270 --> 01:17:48,950 to get to the end. 1113 01:17:48,950 --> 01:17:52,100 And it's in the j direction. 1114 01:17:52,100 --> 01:17:55,700 So it's a length times A. And you need the delta. 1115 01:17:59,710 --> 01:18:01,640 This quantity. 1116 01:18:01,640 --> 01:18:05,230 And you need a delta theta. 1117 01:18:05,230 --> 01:18:07,490 Delta theta. 1118 01:18:07,490 --> 01:18:10,150 This is the term. 1119 01:18:10,150 --> 01:18:14,590 This is the dr for the system. 1120 01:18:14,590 --> 01:18:18,020 An angle, a virtual deflection in angle times the moment arm 1121 01:18:18,020 --> 01:18:19,650 gives you a distance. 1122 01:18:19,650 --> 01:18:27,350 It's in the j hat direction dotted with the same force 1123 01:18:27,350 --> 01:18:30,950 breaking the force up into its i and j components. 1124 01:18:30,950 --> 01:18:33,680 It had a sine theta i cos theta j. 1125 01:18:33,680 --> 01:18:36,840 So this is going to give me a f cosine omega t 1126 01:18:36,840 --> 01:18:38,960 cos theta j dot j. 1127 01:18:45,912 --> 01:19:02,230 f0 cosine omega t cos theta x1 plus l2 over 2 delta theta 1128 01:19:02,230 --> 01:19:03,930 is the delta w. 1129 01:19:03,930 --> 01:19:07,160 That's the work and the virtual. 1130 01:19:10,030 --> 01:19:15,820 The generalized force q theta is this part of it. 1131 01:19:15,820 --> 01:19:22,580 So this plus this plus this equals 1132 01:19:22,580 --> 01:19:24,800 that on the right hand side. 1133 01:19:24,800 --> 01:19:27,940 So this is part four. 1134 01:19:27,940 --> 01:19:30,750 And look at it. 1135 01:19:30,750 --> 01:19:31,620 Yeah? 1136 01:19:31,620 --> 01:19:32,536 AUDIENCE: [INAUDIBLE]. 1137 01:19:37,690 --> 01:19:41,230 PROFESSOR: So f of t, I didn't want to write it all out. 1138 01:19:41,230 --> 01:19:43,760 This thing breaks into an i and a j piece, 1139 01:19:43,760 --> 01:19:45,310 which is written over there. 1140 01:19:45,310 --> 01:19:48,230 This is the sine theta i cos theta j term. 1141 01:19:51,140 --> 01:19:54,040 Which I brought back from over there. 1142 01:19:54,040 --> 01:19:57,190 And we dot it with the dr that we care about, 1143 01:19:57,190 --> 01:20:02,620 which is this length times that angle in the j direction. 1144 01:20:02,620 --> 01:20:03,650 So j dot. 1145 01:20:03,650 --> 01:20:06,620 We only pick up the j piece of this. 1146 01:20:06,620 --> 01:20:10,740 And that gives us this cosine theta term. 1147 01:20:10,740 --> 01:20:13,070 OK. 1148 01:20:13,070 --> 01:20:14,640 Let's look quickly. 1149 01:20:14,640 --> 01:20:17,730 This is a rotational thing. 1150 01:20:17,730 --> 01:20:19,515 It has units of is it force? 1151 01:20:19,515 --> 01:20:21,125 Is this a force equation? 1152 01:20:24,050 --> 01:20:26,780 i theta double dot has units of what? 1153 01:20:26,780 --> 01:20:27,280 Torque. 1154 01:20:27,280 --> 01:20:28,900 This is a torque equation. 1155 01:20:28,900 --> 01:20:36,080 This is the total mass moment of inertia izz 1156 01:20:36,080 --> 01:20:40,190 with respect to A for this system such that the Eulerian 1157 01:20:40,190 --> 01:20:41,030 acceleration. 1158 01:20:41,030 --> 01:20:44,030 The torque it takes to make that happen 1159 01:20:44,030 --> 01:20:48,280 is the sum of the mass moment of inertia of the rod 1160 01:20:48,280 --> 01:20:51,910 plus the mass moment of inertia of g plus m2 x1 1161 01:20:51,910 --> 01:20:55,340 squared, which looks a lot like the parallel axis theorem. 1162 01:20:55,340 --> 01:20:59,550 This is izz A for the moving mass. 1163 01:20:59,550 --> 01:21:00,710 There's your Coriolis term. 1164 01:21:03,730 --> 01:21:06,120 And here's your potential terms and there's 1165 01:21:06,120 --> 01:21:08,371 your external force. 1166 01:21:08,371 --> 01:21:08,870 OK. 1167 01:21:12,570 --> 01:21:15,560 Talk more about these things in recitation.