1 00:00:00,080 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,800 Commons license. 3 00:00:03,800 --> 00:00:06,050 Your support will help MIT OpenCourseWare 4 00:00:06,050 --> 00:00:10,150 continue to offer high quality educational resources for free. 5 00:00:10,150 --> 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 --> 00:00:17,260 at ocw.mit.edu. 8 00:00:21,540 --> 00:00:23,640 PROFESSOR: All right, you know the drill 9 00:00:23,640 --> 00:00:26,120 from the previous couple of weeks. 10 00:00:26,120 --> 00:00:31,030 Get your piece of paper out, and write down what you think 11 00:00:31,030 --> 00:00:34,700 were important concepts for the last week or so. 12 00:00:34,700 --> 00:00:35,700 OK. 13 00:00:35,700 --> 00:00:40,910 Let's see what you have on your papers. 14 00:00:40,910 --> 00:00:44,970 AUDIENCE: Center of mass can be used to find the total force. 15 00:00:47,590 --> 00:00:51,010 PROFESSOR: So the concept is center of mass and things 16 00:00:51,010 --> 00:00:52,080 you can use it for. 17 00:00:52,080 --> 00:01:07,110 So I'm going to generalize that to "systems of particles." 18 00:01:07,110 --> 00:01:11,180 And what you've just said is, center of mass 19 00:01:11,180 --> 00:01:13,040 is one thing that comes from that. 20 00:01:13,040 --> 00:01:15,724 Anybody, on this subject of systems of particles, 21 00:01:15,724 --> 00:01:16,890 anybody else have something? 22 00:01:16,890 --> 00:01:18,194 AUDIENCE: Angular momentum? 23 00:01:18,194 --> 00:01:19,360 PROFESSOR: Angular momentum. 24 00:01:19,360 --> 00:01:21,900 So what about it? 25 00:01:21,900 --> 00:01:23,280 AUDIENCE: [INAUDIBLE]. 26 00:01:23,280 --> 00:01:29,190 PROFESSOR: Yeah, so this concept of the summation of the hi's, 27 00:01:29,190 --> 00:01:33,100 equal to H at the system, and that you can use that. 28 00:01:33,100 --> 00:01:35,720 Anything else about systems of particles that we learned? 29 00:01:41,504 --> 00:01:43,920 AUDIENCE: Newton's third law between the particles? 30 00:01:43,920 --> 00:01:45,378 PROFESSOR: Yeah, Newton's third law 31 00:01:45,378 --> 00:01:48,900 can help to support things out, and a consequence-- you 32 00:01:48,900 --> 00:01:51,200 found center of mass, that's useful, 33 00:01:51,200 --> 00:01:55,340 but you take the derivative of the center of mass, 34 00:01:55,340 --> 00:01:59,080 what do you find out? 35 00:01:59,080 --> 00:02:08,646 rG here, dot times the total mass is what? 36 00:02:08,646 --> 00:02:10,930 AUDIENCE: Momentum? 37 00:02:10,930 --> 00:02:13,786 PROFESSOR: Yeah, it's momentum, but it's momentum of what? 38 00:02:13,786 --> 00:02:15,407 AUDIENCE: [INAUDIBLE]. 39 00:02:15,407 --> 00:02:17,740 PROFESSOR: Well, it's the momentum of the entire system, 40 00:02:17,740 --> 00:02:18,240 right? 41 00:02:18,240 --> 00:02:21,680 And so this is the summation of the Pi. 42 00:02:21,680 --> 00:02:24,240 That's kind of important. 43 00:02:24,240 --> 00:02:29,840 And finally, take another derivative, 44 00:02:29,840 --> 00:02:32,820 what did you learn about that? 45 00:02:32,820 --> 00:02:33,596 Pardon? 46 00:02:33,596 --> 00:02:34,450 AUDIENCE: Forces? 47 00:02:34,450 --> 00:02:35,825 PROFESSOR: What kind of forces? 48 00:02:35,825 --> 00:02:36,300 AUDIENCE: External. 49 00:02:36,300 --> 00:02:38,508 PROFESSOR: All the external forces are in the system. 50 00:02:38,508 --> 00:02:39,360 All of them. 51 00:02:44,345 --> 00:02:49,087 So powerful things came out of thinking about center of mass. 52 00:02:49,087 --> 00:02:50,545 All right, what's something else we 53 00:02:50,545 --> 00:02:52,878 did in the week that wasn't around systems of particles? 54 00:02:55,600 --> 00:02:56,540 Yes. 55 00:02:56,540 --> 00:03:00,550 Actually, say your name when I call on you, 56 00:03:00,550 --> 00:03:02,600 I'm gonna try to learn more of your names. 57 00:03:02,600 --> 00:03:03,433 AUDIENCE: Christina. 58 00:03:03,433 --> 00:03:05,040 PROFESSOR: Christina, right. 59 00:03:05,040 --> 00:03:07,465 AUDIENCE: So, when you're looking at equations, 60 00:03:07,465 --> 00:03:09,405 for example, like the force external 61 00:03:09,405 --> 00:03:14,255 is equal to the change in angular momentum. 62 00:03:14,255 --> 00:03:15,871 Looking at what pieces go where, so 63 00:03:15,871 --> 00:03:17,412 if you have a defined moment, does it 64 00:03:17,412 --> 00:03:18,640 go on this side or that side? 65 00:03:18,640 --> 00:03:26,720 PROFESSOR: OK, and I'd put that in the category of free body 66 00:03:26,720 --> 00:03:27,970 diagrams, are you thinking of? 67 00:03:27,970 --> 00:03:29,655 So, constructing free body diagrams. 68 00:03:38,690 --> 00:03:41,915 OK, what else? 69 00:03:41,915 --> 00:03:43,030 AUDIENCE: Eddie. 70 00:03:43,030 --> 00:03:44,786 PROFESSOR: Eddie? 71 00:03:44,786 --> 00:03:47,888 AUDIENCE: The two different methods for finding torque. 72 00:03:47,888 --> 00:03:53,905 PROFESSOR: So, let's go a little further with torque. 73 00:03:53,905 --> 00:03:56,690 What do you use the torque for? 74 00:03:56,690 --> 00:03:58,320 What do you want to find torque for? 75 00:04:01,140 --> 00:04:06,350 To get-- let's generalize this to equations of motion, 76 00:04:06,350 --> 00:04:14,570 so both for forces, and moments. 77 00:04:14,570 --> 00:04:19,649 Two different kinds of problems that we 78 00:04:19,649 --> 00:04:23,497 are able to do by being able to find the torques. 79 00:04:23,497 --> 00:04:24,080 Anything else? 80 00:04:27,620 --> 00:04:29,307 It's important this week. 81 00:04:29,307 --> 00:04:30,223 AUDIENCE: [INAUDIBLE]? 82 00:04:34,470 --> 00:04:37,110 PROFESSOR: Yeah, I mean, this isn't a math course. 83 00:04:37,110 --> 00:04:40,190 Yeah, you've got to be able to solve differential equations, 84 00:04:40,190 --> 00:04:40,690 yeah. 85 00:04:43,300 --> 00:04:47,960 Maybe you're actually onto something here, 86 00:04:47,960 --> 00:04:51,010 but let's put it in the context of what kind of equations 87 00:04:51,010 --> 00:04:53,260 we work on a lot on this week. 88 00:04:53,260 --> 00:04:55,810 What kind of equations of motion? 89 00:04:55,810 --> 00:04:57,720 I don't mean like second order linear, 90 00:04:57,720 --> 00:04:59,960 I mean we're doing dynamics. 91 00:04:59,960 --> 00:05:02,710 We're doing dynamics of what kind? 92 00:05:02,710 --> 00:05:04,000 AUDIENCE: Acceleration? 93 00:05:04,000 --> 00:05:09,800 PROFESSOR: Yeah, but also, a lot of examples this week, 94 00:05:09,800 --> 00:05:12,356 were they rotational problems or translational problems? 95 00:05:12,356 --> 00:05:13,230 AUDIENCE: Rotational? 96 00:05:13,230 --> 00:05:15,530 PROFESSOR: Right, kind of a heavy emphasis this week 97 00:05:15,530 --> 00:05:18,080 on angular momentum, right? 98 00:05:18,080 --> 00:05:33,595 So, using-- That's a pretty good list. 99 00:05:36,580 --> 00:05:38,620 My own list had most of that on it. 100 00:05:38,620 --> 00:05:41,350 Let's see where my list was. 101 00:05:41,350 --> 00:05:45,060 Center of mass, r dot, r double dot. 102 00:05:47,850 --> 00:05:49,790 I wrote these as equations. 103 00:05:49,790 --> 00:05:53,260 So this thing in particular, the summation 104 00:05:53,260 --> 00:05:57,490 of the external torques with respect to some point, 105 00:05:57,490 --> 00:06:09,850 for a particle, is dh dt, plus vA, cross P. 106 00:06:09,850 --> 00:06:13,980 And we were just beginning to do sums of collections 107 00:06:13,980 --> 00:06:16,550 of particles, rigid bodies, where 108 00:06:16,550 --> 00:06:18,690 you have maybe more than one particle, 109 00:06:18,690 --> 00:06:24,510 the summation of the torques is the H-- capital 110 00:06:24,510 --> 00:06:35,259 H-- for the summation of the h of i's, dt plus vA in o cross, 111 00:06:35,259 --> 00:06:36,800 and here's where we learned something 112 00:06:36,800 --> 00:06:40,750 up here that you can do down here that was really helpful. 113 00:06:40,750 --> 00:06:42,033 What crossed with what? 114 00:06:42,033 --> 00:06:42,820 AUDIENCE: P. 115 00:06:42,820 --> 00:06:44,420 PROFESSOR: Which P? 116 00:06:44,420 --> 00:06:47,060 AUDIENCE: [INAUDIBLE]. 117 00:06:47,060 --> 00:06:48,890 PROFESSOR: This one up here, right? 118 00:06:48,890 --> 00:06:50,780 And that's actually this. 119 00:06:50,780 --> 00:06:57,620 So this is cross P of the center of mass of the system. 120 00:06:57,620 --> 00:07:03,730 OK good, that's a pretty good list. 121 00:07:03,730 --> 00:07:07,005 So now, let's do a problem. 122 00:07:13,970 --> 00:07:17,080 So here's the problem. 123 00:07:17,080 --> 00:07:20,370 It's a rigid rod, with a mass on the end, 124 00:07:20,370 --> 00:07:24,050 and the rod's massless just to make it easy for the moment. 125 00:07:24,050 --> 00:07:27,920 But this is now a rigid body. 126 00:07:27,920 --> 00:07:30,692 There's a torsional spring here. 127 00:07:30,692 --> 00:07:34,240 And the torsional spring resists angular displacement 128 00:07:34,240 --> 00:07:37,000 with a torque that's equal to the spring constant times 129 00:07:37,000 --> 00:07:40,070 the theta that you push it through. 130 00:07:40,070 --> 00:07:44,950 So it generates a moment, a restoring moment. 131 00:07:44,950 --> 00:07:47,650 When you push it away, it tries to push this thing back. 132 00:07:47,650 --> 00:07:52,190 Gravity is also acting, so it's a pendulum as a rigid body 133 00:07:52,190 --> 00:07:54,100 and a torsional spring. 134 00:07:54,100 --> 00:07:57,550 So the first step-- I want you to eventually find 135 00:07:57,550 --> 00:07:59,590 the equation of motion. 136 00:07:59,590 --> 00:08:03,600 And I want you to start by doing a free body diagram. 137 00:08:03,600 --> 00:08:06,270 But before you go to work, I'm going 138 00:08:06,270 --> 00:08:10,320 to give you a little quick mini-lecture on free body 139 00:08:10,320 --> 00:08:12,550 diagrams. 140 00:08:12,550 --> 00:08:16,220 So, the first thing you do is assign coordinates. 141 00:08:16,220 --> 00:08:21,320 Maybe draw is not-- really assign the coordinates 142 00:08:21,320 --> 00:08:24,390 for the problem. 143 00:08:24,390 --> 00:08:26,707 Determine the number of degrees of freedom. 144 00:08:26,707 --> 00:08:28,290 We haven't talked about that much yet. 145 00:08:28,290 --> 00:08:31,850 It will become much more important in the subject. 146 00:08:31,850 --> 00:08:34,460 The number of degrees of freedom-- but I'll say it once, 147 00:08:34,460 --> 00:08:35,890 and will do much more with this-- 148 00:08:35,890 --> 00:08:40,010 the number of degrees of freedom is the same thing 149 00:08:40,010 --> 00:08:43,440 as the number of independent coordinates 150 00:08:43,440 --> 00:08:45,535 you need to completely describe the motion. 151 00:08:48,070 --> 00:08:50,065 And the number of independent coordinates 152 00:08:50,065 --> 00:08:54,650 that you need is equal to the number of equations of motion 153 00:08:54,650 --> 00:08:56,940 that you'll get. 154 00:08:56,940 --> 00:09:00,500 So this problem, how many coordinates does it take-- 155 00:09:00,500 --> 00:09:04,170 and this is planar motion, it's confined to the board. 156 00:09:04,170 --> 00:09:06,160 How many actual coordinates does it 157 00:09:06,160 --> 00:09:09,120 take to completely describe the position of this thing? 158 00:09:11,920 --> 00:09:16,305 I see a one, how about-- anybody else? 159 00:09:16,305 --> 00:09:16,930 What's the one? 160 00:09:16,930 --> 00:09:19,242 What would you choose? 161 00:09:19,242 --> 00:09:21,180 AUDIENCE: I think I'd probably go with polar. 162 00:09:21,180 --> 00:09:23,606 PROFESSOR: Yeah, well what coordinate itself? 163 00:09:23,606 --> 00:09:24,980 What is the single coordinate you 164 00:09:24,980 --> 00:09:28,980 would use to define the motion of the system? 165 00:09:28,980 --> 00:09:30,355 I hear a theta. 166 00:09:30,355 --> 00:09:31,690 Everybody agree? 167 00:09:31,690 --> 00:09:33,640 Theta of t, if you can figure out theta of t, 168 00:09:33,640 --> 00:09:36,050 you know this position of this thing for all time. 169 00:09:36,050 --> 00:09:37,560 That's all you need, is one degree 170 00:09:37,560 --> 00:09:40,070 of freedom, one coordinate, and in terms 171 00:09:40,070 --> 00:09:42,630 of choosing the coordinate system to use, 172 00:09:42,630 --> 00:09:44,790 basically you were saying, use what? 173 00:09:44,790 --> 00:09:45,494 AUDIENCE: Polar 174 00:09:45,494 --> 00:09:47,285 PROFESSOR: Polar coordinates centered here. 175 00:09:50,140 --> 00:09:52,596 But-- there's your inertial system, 176 00:09:52,596 --> 00:09:54,220 but you're going to go with this point. 177 00:09:54,220 --> 00:09:58,200 This is our A point, and is it moving? 178 00:09:58,200 --> 00:10:00,900 So if we use expressions like this, 179 00:10:00,900 --> 00:10:02,682 what will happen to that term? 180 00:10:02,682 --> 00:10:04,120 It goes away. 181 00:10:04,120 --> 00:10:05,856 AUDIENCE: So if you were using Cartesian, 182 00:10:05,856 --> 00:10:07,320 would it be two degrees of freedom? 183 00:10:07,320 --> 00:10:09,910 PROFESSOR: Nope. 184 00:10:09,910 --> 00:10:11,064 And we'll get to that. 185 00:10:11,064 --> 00:10:12,730 If I divert to that, I won't finish what 186 00:10:12,730 --> 00:10:14,010 I've got planned for today. 187 00:10:14,010 --> 00:10:16,295 It's important and we're going to come back to that. 188 00:10:16,295 --> 00:10:17,420 Has to do with constraints. 189 00:10:21,180 --> 00:10:23,660 So you assign the coordinates, determine the number 190 00:10:23,660 --> 00:10:26,940 of degrees of freedom, assign positive values 191 00:10:26,940 --> 00:10:32,230 to all rotations, displacements, velocities, linear velocities, 192 00:10:32,230 --> 00:10:35,840 and rotational velocities, in the problem. 193 00:10:35,840 --> 00:10:37,500 Positive values of those. 194 00:10:37,500 --> 00:10:40,360 And from that, you deduce the direction 195 00:10:40,360 --> 00:10:42,600 of the resulting forces and moments. 196 00:10:42,600 --> 00:10:44,560 And this will help you get the signs correct, 197 00:10:44,560 --> 00:10:46,410 especially when you have multi-body problems 198 00:10:46,410 --> 00:10:48,900 like two masses with a spring in between them. 199 00:10:48,900 --> 00:10:51,370 Which way do the forces go on each mass? 200 00:10:57,600 --> 00:10:59,560 In this problem, there's only one coordinate, 201 00:10:59,560 --> 00:11:02,170 so what we're really saying is assign a positive theta, 202 00:11:02,170 --> 00:11:04,760 and assume a positive theta dot. 203 00:11:04,760 --> 00:11:08,030 What are the resulting forces that end up in your free body 204 00:11:08,030 --> 00:11:08,840 diagram? 205 00:11:08,840 --> 00:11:12,410 So now, I want you to draw the free body diagram 206 00:11:12,410 --> 00:11:16,370 for this thing, and up here, final statement, 207 00:11:16,370 --> 00:11:19,510 put in all forces and moments. 208 00:11:19,510 --> 00:11:22,240 Don't leave out anything, because you know 209 00:11:22,240 --> 00:11:23,520 they're not going to matter. 210 00:11:23,520 --> 00:11:25,520 Because it'll get you in trouble as the problems 211 00:11:25,520 --> 00:11:28,130 get more complicated. 212 00:11:28,130 --> 00:11:31,790 OK, draw your free body diagrams. 213 00:11:31,790 --> 00:11:34,115 OK, so we're going to get in groups of three or four, 214 00:11:34,115 --> 00:11:35,410 and compare notes here. 215 00:11:35,410 --> 00:11:39,020 You guys are kind of a natural group-- 216 00:11:39,020 --> 00:11:43,015 check each other's stuff and come up with a final. 217 00:11:43,015 --> 00:11:45,265 AUDIENCE: We were having a debate about whether or not 218 00:11:45,265 --> 00:11:47,750 you were asking us for the free body diagram of the system, 219 00:11:47,750 --> 00:11:49,500 or if you were asking us for the free body 220 00:11:49,500 --> 00:11:52,350 diagram of the mass itself. 221 00:11:52,350 --> 00:11:54,460 PROFESSOR: Yeah, so are there moments 222 00:11:54,460 --> 00:11:57,150 and things involved up here? 223 00:11:57,150 --> 00:11:58,275 Like that torsional spring? 224 00:11:58,275 --> 00:12:00,899 AUDIENCE: When you are including the entire system there, yeah. 225 00:12:00,899 --> 00:12:03,030 PROFESSOR: OK, so this is a rigid body. 226 00:12:03,030 --> 00:12:04,840 I kept saying this is a rigid body. 227 00:12:04,840 --> 00:12:07,220 Happens to have a single mass point in it, 228 00:12:07,220 --> 00:12:09,380 but it's a whole, single, rigid body. 229 00:12:09,380 --> 00:12:12,405 And do the free body diagram for the rigid body. 230 00:12:21,400 --> 00:12:23,370 PROFESSOR: Can you show me something? 231 00:12:23,370 --> 00:12:26,917 Where are you at-- what's your best shot at this so far? 232 00:12:26,917 --> 00:12:29,975 AUDIENCE: The free body diagram needs [INAUDIBLE]. 233 00:12:29,975 --> 00:12:33,310 And we say the spring force is producing the moment. 234 00:12:33,310 --> 00:12:36,685 PROFESSOR: So what's your body that you're working with? 235 00:12:36,685 --> 00:12:39,849 What's the definition of a rigid body? 236 00:12:39,849 --> 00:12:41,390 AUDIENCE: Mass on the end of the rod? 237 00:12:41,390 --> 00:12:43,230 PROFESSOR: OK, so you're still thinking about the particle. 238 00:12:43,230 --> 00:12:44,430 But if you don't think about the particle, 239 00:12:44,430 --> 00:12:47,090 it's kind of hard to get that torsional spring involved. 240 00:12:47,090 --> 00:12:50,110 So you need to think of this as a single, whole body. 241 00:12:50,110 --> 00:12:52,370 One whole, rigid body. 242 00:12:52,370 --> 00:12:56,011 And now, just draw a stick in here, and put the forces on it. 243 00:12:56,011 --> 00:12:58,010 Doesn't matter where the mass is for the purpose 244 00:12:58,010 --> 00:12:59,840 of the free body diagram. 245 00:12:59,840 --> 00:13:01,676 That comes into the-- 246 00:13:01,676 --> 00:13:07,070 AUDIENCE: Couldn't you just use the r cross force? 247 00:13:07,070 --> 00:13:10,190 And get the force in the ball from the-- 248 00:13:10,190 --> 00:13:12,176 PROFESSOR: So you're getting into the equation 249 00:13:12,176 --> 00:13:13,050 of motion part of it. 250 00:13:13,050 --> 00:13:17,370 This is only about assuming this thing has some dynamic moment 251 00:13:17,370 --> 00:13:22,100 in time, and draw the external forces and moments that 252 00:13:22,100 --> 00:13:23,240 act on the object. 253 00:13:23,240 --> 00:13:27,194 And the object is this full-length stick. 254 00:13:27,194 --> 00:13:28,610 AUDIENCE: So in that case, we just 255 00:13:28,610 --> 00:13:31,692 have the spring force acting, in opposition 256 00:13:31,692 --> 00:13:36,885 to [INAUDIBLE] the weight? 257 00:13:36,885 --> 00:13:39,030 PROFESSOR: You have weight, for sure. 258 00:13:39,030 --> 00:13:42,630 You have the spring, puts a moment on the system, 259 00:13:42,630 --> 00:13:45,030 and there are yet potential other forces. 260 00:14:09,984 --> 00:14:12,400 PROFESSOR: I'll give you a couple minutes, because there's 261 00:14:12,400 --> 00:14:14,450 a little confusion around whether it's 262 00:14:14,450 --> 00:14:15,980 a particle or a rigid body. 263 00:14:15,980 --> 00:14:17,700 It's a rigid body. 264 00:14:17,700 --> 00:14:19,470 You've got to deal with the whole body. 265 00:14:19,470 --> 00:14:21,330 All right, let's see what you've got. 266 00:14:21,330 --> 00:14:23,240 Where's your free body diagram? 267 00:14:23,240 --> 00:14:25,200 AUDIENCE: This right here. 268 00:14:25,200 --> 00:14:29,020 PROFESSOR: OK, tell me what you have for forces and moments, 269 00:14:29,020 --> 00:14:31,620 just point to each one of 'em and say what it is. 270 00:14:31,620 --> 00:14:35,190 AUDIENCE: So, we have the weight of the this end, 271 00:14:35,190 --> 00:14:37,200 and we assume this is massless. 272 00:14:37,200 --> 00:14:40,320 PROFESSOR: Yup, but it's rigid, so it's all one system now, 273 00:14:40,320 --> 00:14:41,384 you have one rigid body. 274 00:14:41,384 --> 00:14:43,300 AUDIENCE: So the center of mass is right here, 275 00:14:43,300 --> 00:14:49,130 because there's-- [INAUDIBLE], and then we have reaction 276 00:14:49,130 --> 00:14:51,780 forces at the pin, and then we have the moment from the spring 277 00:14:51,780 --> 00:14:52,280 there. 278 00:14:52,280 --> 00:14:52,863 PROFESSOR: OK. 279 00:14:56,040 --> 00:14:57,954 AUDIENCE: We have the balls moving forever-- 280 00:14:57,954 --> 00:14:59,620 PROFESSOR: So, you've gotta convince one 281 00:14:59,620 --> 00:15:01,150 another who's right, here, so-- 282 00:15:01,150 --> 00:15:03,566 AUDIENCE: Well, it depends on which way the ball's moving. 283 00:15:03,566 --> 00:15:06,460 PROFESSOR: So, that's why there is a little system. 284 00:15:06,460 --> 00:15:11,470 There's a rubric that I-- I recommend you just 285 00:15:11,470 --> 00:15:14,150 use the system. 286 00:15:14,150 --> 00:15:16,660 This is basically a system for getting equations of motion, 287 00:15:16,660 --> 00:15:18,565 and this is important, that one. 288 00:15:22,150 --> 00:15:23,910 So then if you know it's positive, 289 00:15:23,910 --> 00:15:26,470 then you can figure out which direction the reaction is. 290 00:15:29,300 --> 00:15:31,700 All right, I hate stopping all the fun. 291 00:15:31,700 --> 00:15:36,200 You guys are doing well. 292 00:15:36,200 --> 00:15:41,390 There is great value in talking to one another, 293 00:15:41,390 --> 00:15:44,010 and convincing one another of what 294 00:15:44,010 --> 00:15:46,020 the right way to do something is. 295 00:15:46,020 --> 00:15:48,600 You'll have two different points of view, and talking it out, 296 00:15:48,600 --> 00:15:50,440 It's amazing how fast you can make progress. 297 00:15:50,440 --> 00:15:54,680 So do that, do that when you're home solving problems. 298 00:15:54,680 --> 00:15:57,270 So what was the assignment here? 299 00:15:57,270 --> 00:15:58,460 Draw a free body diagram. 300 00:15:58,460 --> 00:16:00,070 Let's do it. 301 00:16:00,070 --> 00:16:02,400 So it's a rigid body. 302 00:16:02,400 --> 00:16:05,350 All one body, even though the mass is all concentrated down 303 00:16:05,350 --> 00:16:11,380 here, the forces-- what are forces that are on this thing? 304 00:16:11,380 --> 00:16:12,390 Give me one. 305 00:16:12,390 --> 00:16:14,890 Wait, mg, right? 306 00:16:14,890 --> 00:16:18,185 OK, what else? 307 00:16:18,185 --> 00:16:19,351 AUDIENCE: Normal force? 308 00:16:19,351 --> 00:16:20,350 PROFESSOR: Normal force. 309 00:16:20,350 --> 00:16:20,790 Where? 310 00:16:20,790 --> 00:16:21,665 AUDIENCE: On the top. 311 00:16:21,665 --> 00:16:24,910 PROFESSOR: OK, so you're talking about a reaction force up here? 312 00:16:24,910 --> 00:16:26,850 And have you chosen a coordinate system? 313 00:16:26,850 --> 00:16:28,550 We already kind of decided, didn't we, 314 00:16:28,550 --> 00:16:31,010 on polar coordinates for this, right? 315 00:16:31,010 --> 00:16:34,480 So are the reaction forces known? 316 00:16:34,480 --> 00:16:35,690 To start with, no. 317 00:16:35,690 --> 00:16:40,880 So what's the cleverest way to assign them 318 00:16:40,880 --> 00:16:43,716 on your free body diagram? 319 00:16:43,716 --> 00:16:45,690 AUDIENCE: [INAUDIBLE]. 320 00:16:45,690 --> 00:16:50,790 PROFESSOR: Well, and associated with your coordinates, right? 321 00:16:50,790 --> 00:16:55,700 So, I would have two reaction forces up here, one 322 00:16:55,700 --> 00:16:57,810 in each of which directions? 323 00:16:57,810 --> 00:16:58,740 AUDIENCE: Radial. 324 00:16:58,740 --> 00:16:59,475 PROFESSOR: Radial, and? 325 00:16:59,475 --> 00:17:00,350 AUDIENCE: Tangential. 326 00:17:00,350 --> 00:17:05,970 PROFESSOR: OK, and the positive radial, positive tangential, 327 00:17:05,970 --> 00:17:06,920 like that, right? 328 00:17:06,920 --> 00:17:11,230 So I would say, I have an unknown F theta force, 329 00:17:11,230 --> 00:17:15,750 and I have an unknown FR force, and I draw them 330 00:17:15,750 --> 00:17:18,579 in the positive directions because I have no clue. 331 00:17:18,579 --> 00:17:19,900 So just make them positive. 332 00:17:19,900 --> 00:17:21,910 You assume direction for them. 333 00:17:21,910 --> 00:17:26,250 Force is here, g there, what about the tension 334 00:17:26,250 --> 00:17:28,053 in the string? 335 00:17:28,053 --> 00:17:29,530 AUDIENCE: [INAUDIBLE]. 336 00:17:29,530 --> 00:17:31,700 PROFESSOR: Does it belong in this problem? 337 00:17:31,700 --> 00:17:33,520 Why not? 338 00:17:33,520 --> 00:17:36,910 The rigid body-- it's an internal force. 339 00:17:36,910 --> 00:17:40,380 It's the little fij fji thing, it's 340 00:17:40,380 --> 00:17:42,300 those things that cancel internally, right? 341 00:17:42,300 --> 00:17:43,410 So you don't have to deal with it. 342 00:17:43,410 --> 00:17:45,450 It's not an external force to the rigid body. 343 00:17:45,450 --> 00:17:48,090 So this is great. 344 00:17:48,090 --> 00:17:51,160 The next step in this is to find an equation of motion. 345 00:17:51,160 --> 00:17:52,570 Gotta to be speedy about this. 346 00:17:59,580 --> 00:18:00,870 So, what-- 347 00:18:00,870 --> 00:18:02,030 AUDIENCE: [INAUDIBLE]. 348 00:18:02,030 --> 00:18:03,825 PROFESSOR: Pardon? 349 00:18:03,825 --> 00:18:04,600 AUDIENCE: Moment. 350 00:18:04,600 --> 00:18:05,350 PROFESSOR: Moment. 351 00:18:05,350 --> 00:18:06,690 You'd use moment to do it. 352 00:18:06,690 --> 00:18:08,012 About what point? 353 00:18:08,012 --> 00:18:10,322 AUDIENCE: About the point up at the top. 354 00:18:10,322 --> 00:18:12,155 PROFESSOR: All right, so if we call that A, 355 00:18:12,155 --> 00:18:14,502 you'd use angular momentum, and torques, 356 00:18:14,502 --> 00:18:15,460 and that sort of thing? 357 00:18:15,460 --> 00:18:16,110 Is that what you're saying? 358 00:18:16,110 --> 00:18:16,890 Moments about it? 359 00:18:16,890 --> 00:18:18,014 AUDIENCE: Oh, I was just wondering if you 360 00:18:18,014 --> 00:18:19,305 were gonna draw it up on there. 361 00:18:19,305 --> 00:18:21,400 PROFESSOR: No, I'm gonna have you do it yourself. 362 00:18:21,400 --> 00:18:22,440 Do it. 363 00:18:22,440 --> 00:18:23,529 So now-- 364 00:18:23,529 --> 00:18:26,070 AUDIENCE: Well, is she talking about how there's the moment-- 365 00:18:26,070 --> 00:18:27,060 the resistent moment. 366 00:18:27,060 --> 00:18:27,560 [INAUDIBLE] 367 00:18:27,560 --> 00:18:28,925 PROFESSOR: Oh. 368 00:18:28,925 --> 00:18:30,590 Did we forget that one? 369 00:18:30,590 --> 00:18:35,560 So which direction is it? 370 00:18:35,560 --> 00:18:37,408 Clockwise, or counterclockwise? 371 00:18:37,408 --> 00:18:38,900 AUDIENCE: It's clockwise. 372 00:18:38,900 --> 00:18:42,400 PROFESSOR: All right, so there's a moment about this point that 373 00:18:42,400 --> 00:18:44,630 is-- we know what it is. 374 00:18:44,630 --> 00:18:50,110 It's in that direction that has value kt times theta. 375 00:18:50,110 --> 00:18:53,280 Any time you know what it is, don't just make it an unknown, 376 00:18:53,280 --> 00:18:53,980 write it out. 377 00:18:53,980 --> 00:18:56,730 You know it's that, and you know it's in this direction. 378 00:18:56,730 --> 00:18:59,470 And we're going to write-- how many equations of motion 379 00:18:59,470 --> 00:19:00,010 will we get? 380 00:19:00,010 --> 00:19:00,850 AUDIENCE: [INAUDIBLE]. 381 00:19:00,850 --> 00:19:02,766 PROFESSOR: And we've chosen polar coordinates. 382 00:19:05,330 --> 00:19:08,760 You could do this using forces. 383 00:19:08,760 --> 00:19:11,251 Just Newton's second law. 384 00:19:11,251 --> 00:19:12,750 But then you have to solve for what? 385 00:19:15,780 --> 00:19:18,980 If you write force balance, F equals ma, 386 00:19:18,980 --> 00:19:22,480 you have these up here, and you don't know them. 387 00:19:22,480 --> 00:19:24,480 So the reason to use torques in this problem 388 00:19:24,480 --> 00:19:28,210 is because, are these going to appear in your answer? 389 00:19:28,210 --> 00:19:28,840 No. 390 00:19:28,840 --> 00:19:30,511 So that's the reason to use torques. 391 00:19:30,511 --> 00:19:31,010 Yeah. 392 00:19:31,010 --> 00:19:34,230 AUDIENCE: Is that k-- [INAUDIBLE]. 393 00:19:34,230 --> 00:19:36,710 PROFESSOR: This is k sub t to distinguish it 394 00:19:36,710 --> 00:19:38,640 as a torsional spring. 395 00:19:38,640 --> 00:19:40,390 AUDIENCE: So [INAUDIBLE] multiplied by L-- 396 00:19:40,390 --> 00:19:45,770 PROFESSOR: No, it doesn't have an L, it's right here. 397 00:19:45,770 --> 00:19:50,920 It's a spring who gives a moment resistance to being deflected. 398 00:19:50,920 --> 00:19:53,210 kt theta is units of torques. 399 00:19:53,210 --> 00:19:57,120 And this isn't units of force per unit displacement, 400 00:19:57,120 --> 00:20:01,480 this is units of torque per radian. 401 00:20:01,480 --> 00:20:02,110 OK? 402 00:20:02,110 --> 00:20:02,610 Yeah. 403 00:20:02,610 --> 00:20:03,526 AUDIENCE: [INAUDIBLE]. 404 00:20:06,390 --> 00:20:10,160 PROFESSOR: Well, you have assigned polar coordinates, 405 00:20:10,160 --> 00:20:13,470 and my assumption was that-- I drew it right here. 406 00:20:13,470 --> 00:20:14,762 There is positive theta hat. 407 00:20:17,480 --> 00:20:19,006 Right? 408 00:20:19,006 --> 00:20:21,310 AUDIENCE: So isn't our torque negative? 409 00:20:21,310 --> 00:20:22,280 PROFESSOR: Yeah. 410 00:20:22,280 --> 00:20:24,410 But see, I've drawn-- this negative 411 00:20:24,410 --> 00:20:27,590 is accounted for by the direction of the arrow. 412 00:20:27,590 --> 00:20:30,240 So let's now write the equation of motion, 413 00:20:30,240 --> 00:20:31,870 and to do the equation of motion, 414 00:20:31,870 --> 00:20:34,970 you're going to say that the sum of the what? 415 00:20:34,970 --> 00:20:35,720 AUDIENCE: Torques. 416 00:20:35,720 --> 00:20:37,250 PROFESSOR: External torques. 417 00:20:37,250 --> 00:20:38,650 About what point? 418 00:20:38,650 --> 00:20:39,350 AUDIENCE: A. 419 00:20:39,350 --> 00:20:43,216 PROFESSOR: A is equal to? 420 00:20:43,216 --> 00:20:45,077 AUDIENCE: dH dt? 421 00:20:45,077 --> 00:20:46,660 PROFESSOR: And I'll use the capital H, 422 00:20:46,660 --> 00:20:48,360 because we're now working our way onto rigid bodies. 423 00:20:48,360 --> 00:20:49,776 We could have more than one point, 424 00:20:49,776 --> 00:20:53,230 we could have more than one mass point, dH dt plus? 425 00:20:53,230 --> 00:20:54,880 AUDIENCE: v. 426 00:20:54,880 --> 00:20:58,030 PROFESSOR: vA cross P, and what's that in this problem? 427 00:20:58,030 --> 00:20:58,930 AUDIENCE: 0. 428 00:20:58,930 --> 00:21:00,870 PROFESSOR: That guy conveniently is 0. 429 00:21:00,870 --> 00:21:03,650 So, do it. 430 00:21:03,650 --> 00:21:05,060 You can now do this problem. 431 00:21:05,060 --> 00:21:08,270 Figure out, write down an equation of motion, 432 00:21:08,270 --> 00:21:10,016 using some of external torques and-- so 433 00:21:10,016 --> 00:21:11,640 you're gonna have to figure out angular 434 00:21:11,640 --> 00:21:13,150 momentum and some torques. 435 00:21:19,020 --> 00:21:21,370 I hate pulling you guys away. 436 00:21:21,370 --> 00:21:22,780 Lots of good thinking going on. 437 00:21:25,680 --> 00:21:27,880 All right, drag yourselves away, help me out here. 438 00:21:33,250 --> 00:21:37,094 Which groups figured out the external torques? 439 00:21:37,094 --> 00:21:37,760 What do you got? 440 00:21:37,760 --> 00:21:39,656 Tell me what to do here. 441 00:21:39,656 --> 00:21:41,520 AUDIENCE: We have the one for gravity. 442 00:21:41,520 --> 00:21:42,770 PROFESSOR: Tell me what it is. 443 00:21:42,770 --> 00:21:44,140 I wanna write the terms down. 444 00:21:44,140 --> 00:21:46,720 AUDIENCE: It is negative Lmg sine theta. 445 00:21:46,720 --> 00:21:53,854 PROFESSOR: Minus mgL sine theta, in what direction? 446 00:21:53,854 --> 00:21:56,185 AUDIENCE: k hat? 447 00:21:56,185 --> 00:21:57,810 PROFESSOR: Others agree with that term? 448 00:22:00,607 --> 00:22:01,107 Hmm? 449 00:22:01,107 --> 00:22:02,023 AUDIENCE: [INAUDIBLE]. 450 00:22:05,499 --> 00:22:07,790 PROFESSOR: Wait a second, just one term at a time here. 451 00:22:07,790 --> 00:22:11,105 The gravity term, do we have the sign correct? 452 00:22:11,105 --> 00:22:12,890 And then we have the moment arm correct? 453 00:22:12,890 --> 00:22:16,580 So this is the moment arm, r cross force. 454 00:22:16,580 --> 00:22:18,530 Looks like minus k to me. 455 00:22:18,530 --> 00:22:20,540 mgL sine theta. 456 00:22:20,540 --> 00:22:21,819 Looks pretty good. 457 00:22:21,819 --> 00:22:22,610 What other torques? 458 00:22:25,220 --> 00:22:27,540 AUDIENCE: The moment from the spring. 459 00:22:27,540 --> 00:22:29,580 PROFESSOR: Moment from the spring. 460 00:22:29,580 --> 00:22:31,530 Tell me how to write that. 461 00:22:31,530 --> 00:22:36,810 AUDIENCE: That's in the negative k hat section, so minus m hat. 462 00:22:36,810 --> 00:22:39,690 Well, then [INAUDIBLE] kt theta. 463 00:22:39,690 --> 00:22:43,180 PROFESSOR: kt theta, also in k hat direction. 464 00:22:43,180 --> 00:22:47,960 And our free body digram, the forces produce moments. 465 00:22:50,690 --> 00:22:52,110 Is everybody happy with that? 466 00:22:52,110 --> 00:22:54,360 Are we done? 467 00:22:54,360 --> 00:22:56,600 Any questions about it? 468 00:22:56,600 --> 00:22:58,650 Why it's used the way it is? 469 00:22:58,650 --> 00:22:59,870 OK. 470 00:22:59,870 --> 00:23:02,160 Now we need this piece of it. 471 00:23:02,160 --> 00:23:04,220 So what's the H? 472 00:23:04,220 --> 00:23:05,970 For rigid body, the summation of all the 473 00:23:05,970 --> 00:23:08,670 bits-- how many bits are there? 474 00:23:08,670 --> 00:23:10,050 Just the one. 475 00:23:10,050 --> 00:23:15,290 And that's sum r, cross P. Can somebody 476 00:23:15,290 --> 00:23:17,160 tell me what they got for the p, is 477 00:23:17,160 --> 00:23:20,310 the momentum of our little mass here, 478 00:23:20,310 --> 00:23:24,840 with respect to an inertial frame, which is also 479 00:23:24,840 --> 00:23:26,590 A because it's not moving. 480 00:23:26,590 --> 00:23:37,520 And r is L. So this is L r hat cross m, what? 481 00:23:37,520 --> 00:23:42,110 L, theta dot is its velocity, mass times velocity, 482 00:23:42,110 --> 00:23:44,210 and its direction? 483 00:23:44,210 --> 00:23:45,210 Theta hat. 484 00:23:45,210 --> 00:23:51,450 r cross theta is k, so this looks like mL 485 00:23:51,450 --> 00:23:57,190 squared theta dot k hat. 486 00:23:57,190 --> 00:23:58,900 Are we all in agreement? 487 00:23:58,900 --> 00:23:59,416 dH dt? 488 00:24:02,410 --> 00:24:06,243 This k changed direction, so the only time-dependent thing-- 489 00:24:06,243 --> 00:24:08,360 AUDIENCE: I think you're missing a dot. 490 00:24:08,360 --> 00:24:11,100 PROFESSOR: Oh, thank you, we're going to need that. 491 00:24:11,100 --> 00:24:14,354 All right, the derivative is? 492 00:24:14,354 --> 00:24:15,270 AUDIENCE: [INAUDIBLE]. 493 00:24:18,090 --> 00:24:21,130 PROFESSOR: OK, that's equal to that. 494 00:24:21,130 --> 00:24:24,070 It's all k's, you need just one equation out of this, 495 00:24:24,070 --> 00:24:26,500 you don't have to break it down into subcomponents. 496 00:24:26,500 --> 00:24:28,820 And you can collect it all on one side. 497 00:24:28,820 --> 00:24:41,540 mL squared theta double dot plus kt theta plus mgL sine theta 498 00:24:41,540 --> 00:24:43,370 equals 0. 499 00:24:43,370 --> 00:24:47,800 And that's a second order nonlinear ordinary differential 500 00:24:47,800 --> 00:24:50,010 equation, which you could linearize for small motion. 501 00:24:50,010 --> 00:24:53,782 It's just a pendulum, OK? 502 00:24:53,782 --> 00:24:54,718 Good. 503 00:24:54,718 --> 00:24:57,820 We got one other one I want to do before the hour's out. 504 00:25:02,570 --> 00:25:05,715 I got a big piece of cement pipe on the back of a truck. 505 00:25:05,715 --> 00:25:08,710 It's not tied down. 506 00:25:08,710 --> 00:25:12,490 The truck's moving at 3 meters per second. 507 00:25:12,490 --> 00:25:20,220 And the rotation rate of this piece of pipe 508 00:25:20,220 --> 00:25:22,550 is 6 radians per second. 509 00:25:22,550 --> 00:25:24,570 And the truck's accelerating, minus 1/2 510 00:25:24,570 --> 00:25:25,750 a meter per second squared. 511 00:25:25,750 --> 00:25:27,880 It's actually breaking. 512 00:25:27,880 --> 00:25:31,550 The guy's in trouble, right? 513 00:25:31,550 --> 00:25:39,500 So the first thing here is-- omega 6 radians per second, 514 00:25:39,500 --> 00:25:46,490 does it matter, with respect to what, the truck or the ground? 515 00:25:46,490 --> 00:25:47,470 I haven't told you. 516 00:25:51,320 --> 00:25:53,860 Talk in your groups, and give me a quick answer to that. 517 00:26:24,530 --> 00:26:26,219 What do you think? 518 00:26:26,219 --> 00:26:27,260 AUDIENCE: Doesn't matter. 519 00:26:27,260 --> 00:26:27,770 PROFESSOR: Say again? 520 00:26:27,770 --> 00:26:29,070 AUDIENCE: Doesn't matter? 521 00:26:29,070 --> 00:26:32,320 PROFESSOR: So Kristen thinks it doesn't matter. 522 00:26:32,320 --> 00:26:36,370 Anybody wanna counter that? 523 00:26:36,370 --> 00:26:36,870 Why not? 524 00:26:39,870 --> 00:26:42,900 So, you're saying that this could be omega with respect 525 00:26:42,900 --> 00:26:43,980 to the truck? 526 00:26:43,980 --> 00:26:46,280 Or it could be omega with respect to this fixed frame, 527 00:26:46,280 --> 00:26:50,510 and the answer will be the same in either case? 528 00:26:50,510 --> 00:26:51,982 Christine, right? 529 00:26:51,982 --> 00:26:52,815 AUDIENCE: Christina. 530 00:26:52,815 --> 00:26:54,880 PROFESSOR: Christina, alright. 531 00:26:54,880 --> 00:26:56,588 AUDIENCE: So earlier, we had that problem 532 00:26:56,588 --> 00:26:58,909 with-- there was an r, [INAUDIBLE] 533 00:26:58,909 --> 00:27:00,950 and then there was another [INAUDIBLE] over here. 534 00:27:00,950 --> 00:27:04,455 You had to know what this omega was with respect to what, 535 00:27:04,455 --> 00:27:06,760 because this arm's rotating also. 536 00:27:06,760 --> 00:27:10,262 And so, with this, you have your one xy-coordinate system, 537 00:27:10,262 --> 00:27:11,720 and while the truck is translating, 538 00:27:11,720 --> 00:27:13,960 the truck's not gonna be in rotation itself, 539 00:27:13,960 --> 00:27:16,395 so it's staying in the same coordinate plane, 540 00:27:16,395 --> 00:27:17,565 and [INAUDIBLE]. 541 00:27:17,565 --> 00:27:19,106 PROFESSOR: So your argument is if you 542 00:27:19,106 --> 00:27:21,314 were were standing on the ground, seeing it rotating, 543 00:27:21,314 --> 00:27:23,840 you'd say-- you would see it rotated at the same rate 544 00:27:23,840 --> 00:27:26,240 as if you were riding on the truck. 545 00:27:26,240 --> 00:27:27,965 Everybody agree with that? 546 00:27:27,965 --> 00:27:29,840 Does it matter that the truck's accelerating? 547 00:27:35,130 --> 00:27:36,050 AUDIENCE: [INAUDIBLE]. 548 00:27:36,050 --> 00:27:38,591 PROFESSOR: No, it's only whether or not the truck's rotating. 549 00:27:38,591 --> 00:27:42,380 If the truck's not rotating, then rotation rate 550 00:27:42,380 --> 00:27:45,130 seen from this frame or just a translating frame 551 00:27:45,130 --> 00:27:46,500 will always seem the same. 552 00:27:46,500 --> 00:27:49,110 Even if you're accelerating. 553 00:27:49,110 --> 00:27:51,960 OK, so we've done that one. 554 00:27:51,960 --> 00:27:55,240 Now, I want you to find the velocity of point G, given 555 00:27:55,240 --> 00:27:57,950 this information. 556 00:27:57,950 --> 00:28:00,704 So, whip it down quickly, how you 557 00:28:00,704 --> 00:28:01,870 would approach this problem. 558 00:28:01,870 --> 00:28:04,360 See if you can solve it. 559 00:28:04,360 --> 00:28:08,520 G is the center of mass of that big pipe. 560 00:28:08,520 --> 00:28:10,310 OK, we're gonna run out of time. 561 00:28:13,396 --> 00:28:14,520 Who's got an answer for me? 562 00:28:14,520 --> 00:28:16,960 I think they've got it sorted out. 563 00:28:16,960 --> 00:28:18,250 Kristen? 564 00:28:18,250 --> 00:28:20,916 AUDIENCE: So, we have velocity of the truck, minus 565 00:28:20,916 --> 00:28:22,930 [INAUDIBLE]. 566 00:28:22,930 --> 00:28:24,805 PROFESSOR: All right, so let's be systematic. 567 00:28:28,480 --> 00:28:30,190 You have to pick some points, and did you 568 00:28:30,190 --> 00:28:31,610 use a rotating frame? 569 00:28:31,610 --> 00:28:35,622 Use an equation that requires a rotating reference frame? 570 00:28:35,622 --> 00:28:39,439 AUDIENCE: No, because the translational [INAUDIBLE]. 571 00:28:39,439 --> 00:28:40,980 PROFESSOR: So I assume you're talking 572 00:28:40,980 --> 00:28:44,710 about-- we're looking at the velocity of G in o, 573 00:28:44,710 --> 00:28:50,210 is equal to, in general, the velocity of some point 574 00:28:50,210 --> 00:28:52,030 A that you're gonna have to pick, 575 00:28:52,030 --> 00:28:56,850 plus the velocity of G with respect to A. 576 00:28:56,850 --> 00:29:01,890 And this thing has two possible components, right? 577 00:29:01,890 --> 00:29:10,130 One which is the derivative of GA, dt, ignoring the rotation, 578 00:29:10,130 --> 00:29:17,330 plus-- OK. 579 00:29:17,330 --> 00:29:19,180 What's this term in this problem? 580 00:29:19,180 --> 00:29:20,930 Well, we haven't picked the point A. Where 581 00:29:20,930 --> 00:29:22,880 are you going to pick point A? 582 00:29:22,880 --> 00:29:26,490 There's dumb points, and there's good points. 583 00:29:26,490 --> 00:29:28,334 What? 584 00:29:28,334 --> 00:29:29,820 AUDIENCE: [INAUDIBLE]. 585 00:29:29,820 --> 00:29:31,070 PROFESSOR: Front of the truck. 586 00:29:31,070 --> 00:29:31,903 Well, it's possible. 587 00:29:35,340 --> 00:29:37,450 Christina, what do you think? 588 00:29:37,450 --> 00:29:40,065 AUDIENCE: I couldn't-- on the bottom of this part 589 00:29:40,065 --> 00:29:40,597 [INAUDIBLE]. 590 00:29:40,597 --> 00:29:41,930 PROFESSOR: The point of contact? 591 00:29:41,930 --> 00:29:43,249 Is that what you're saying? 592 00:29:43,249 --> 00:29:43,790 Anybody else? 593 00:29:43,790 --> 00:29:44,380 What do you vote for? 594 00:29:44,380 --> 00:29:45,781 Where do you want to put A? 595 00:29:45,781 --> 00:29:47,505 AUDIENCE: Point of contact. 596 00:29:47,505 --> 00:29:49,050 PROFESSOR: Point of contact, OK. 597 00:29:49,050 --> 00:29:53,600 If you're using this equation, and you're using these pieces, 598 00:29:53,600 --> 00:29:56,720 you are using the concept of a rotating frame. 599 00:29:56,720 --> 00:29:58,260 If there's any omega in the problem, 600 00:29:58,260 --> 00:30:01,227 you'd better have a rotating frame somewhere. 601 00:30:01,227 --> 00:30:02,810 Unless you're using polar coordinates, 602 00:30:02,810 --> 00:30:05,890 which is sort of a degenerate rotating frame. 603 00:30:05,890 --> 00:30:09,840 So at this point, we're going to make A here. 604 00:30:09,840 --> 00:30:17,160 And you have a coordinate system, little x1 y1, 605 00:30:17,160 --> 00:30:21,730 and attach the A. So you have an A x1 y1, here z1. 606 00:30:21,730 --> 00:30:22,650 And is it rotating? 607 00:30:27,940 --> 00:30:29,720 Is it a rotating coordinate system? 608 00:30:32,340 --> 00:30:36,986 Well, if it's not, then you can't do this. 609 00:30:41,430 --> 00:30:46,740 This assumes-- this term here is the velocity 610 00:30:46,740 --> 00:30:48,250 at which this point and this point 611 00:30:48,250 --> 00:30:50,420 are moving relative to one another, 612 00:30:50,420 --> 00:30:52,825 as if you were sitting on the pipe. 613 00:30:55,720 --> 00:30:59,160 Ignoring the-- you don't see the rotation, 614 00:30:59,160 --> 00:31:01,880 you're moving with the pipe. 615 00:31:01,880 --> 00:31:05,220 It's assuming you have then put a reference frame on the pipe 616 00:31:05,220 --> 00:31:06,330 that moves with the pipe. 617 00:31:06,330 --> 00:31:09,800 So does the little reference frame, reference frame Axyz 618 00:31:09,800 --> 00:31:11,890 move with the pipe? 619 00:31:11,890 --> 00:31:15,970 It better, or you can't use the equation. 620 00:31:15,970 --> 00:31:17,880 So this is attached to the pipe. 621 00:31:17,880 --> 00:31:20,580 And as it rolls, you have to allow that xyz frame 622 00:31:20,580 --> 00:31:22,640 to move with it. 623 00:31:22,640 --> 00:31:23,870 OK. 624 00:31:23,870 --> 00:31:27,260 So in that frame, what's this term? 625 00:31:27,260 --> 00:31:31,340 This is 0 because the two points are fixed in the pipe. 626 00:31:31,340 --> 00:31:35,164 And what's this velocity? 627 00:31:35,164 --> 00:31:36,580 AUDIENCE: 3. 628 00:31:36,580 --> 00:31:40,040 PROFESSOR: 3 meters per second in what direction? 629 00:31:43,250 --> 00:31:46,310 We'll do it in an inertial frame if we want. 630 00:31:46,310 --> 00:31:49,870 And then what's this term? 631 00:31:49,870 --> 00:31:52,730 So it's 6, and we said it doesn't 632 00:31:52,730 --> 00:31:55,070 matter what frame it is, it's all the same rotation 633 00:31:55,070 --> 00:31:56,820 rate, radians per second. 634 00:31:56,820 --> 00:31:58,700 In what direction? 635 00:31:58,700 --> 00:32:03,250 k hat cross, the radius is capital R, 636 00:32:03,250 --> 00:32:05,120 what's this direction? 637 00:32:05,120 --> 00:32:07,040 And now this is at an instant in time. 638 00:32:07,040 --> 00:32:09,110 At this particular instant in time, 639 00:32:09,110 --> 00:32:13,320 you've cleverly drawn the xyz system so it lines up with them 640 00:32:13,320 --> 00:32:17,000 after so it makes the problem easy. 641 00:32:17,000 --> 00:32:19,320 So at the moment, it's either in the big J hat 642 00:32:19,320 --> 00:32:21,840 or in the little j hat. 643 00:32:21,840 --> 00:32:24,190 It's easiest if you do it in big-- call it big 644 00:32:24,190 --> 00:32:26,520 J, because then you're done. 645 00:32:26,520 --> 00:32:33,280 So, k cross J, I think negative I. 646 00:32:33,280 --> 00:32:37,020 And so this should work out to be 647 00:32:37,020 --> 00:32:44,830 3 meters per second positive I. 1 and 1/2 times 6 is 648 00:32:44,830 --> 00:32:47,370 9 and a minus. 649 00:32:47,370 --> 00:32:54,340 9 meters per second in the I direction. 650 00:32:54,340 --> 00:32:59,200 So you are at minus 6 meters per second. 651 00:32:59,200 --> 00:33:03,110 AUDIENCE: Isn't the radius from point G? 652 00:33:03,110 --> 00:33:03,610 [INAUDIBLE] 653 00:33:06,719 --> 00:33:08,010 PROFESSOR: Wouldn't it be what? 654 00:33:08,010 --> 00:33:10,490 AUDIENCE: Does it matter if it's negative? 655 00:33:10,490 --> 00:33:12,800 PROFESSOR: Ah, it matters a lot. 656 00:33:12,800 --> 00:33:20,920 You have to exercise great care when you pick the R vector. 657 00:33:20,920 --> 00:33:25,310 And the R vector goes from your origin 658 00:33:25,310 --> 00:33:27,800 to the point you're talking about. 659 00:33:27,800 --> 00:33:30,950 So the origin, that's your frame, and that's the point. 660 00:33:30,950 --> 00:33:33,230 And when you draw the arrow, the R vector, 661 00:33:33,230 --> 00:33:37,660 it goes from A to G. That's its direction. 662 00:33:37,660 --> 00:33:42,230 And it has-- its length is capital R long. 663 00:33:42,230 --> 00:33:45,110 Yeah, that's a really, really important point, 664 00:33:45,110 --> 00:33:47,680 which way the arrows go. 665 00:33:47,680 --> 00:33:50,850 OK, and so now we're done with that. 666 00:33:50,850 --> 00:33:56,130 Let's see, we've got a minute or two. 667 00:33:56,130 --> 00:33:59,690 Let's talk about the acceleration. 668 00:33:59,690 --> 00:34:02,560 Let's do the acceleration of this problem, acceleration 669 00:34:02,560 --> 00:34:05,330 of G in o, and this is just an exercise 670 00:34:05,330 --> 00:34:07,360 in remembering the terms. 671 00:34:07,360 --> 00:34:10,889 Full 3D vector equation for accelerations. 672 00:34:10,889 --> 00:34:13,934 What's the first term here? 673 00:34:13,934 --> 00:34:14,850 AUDIENCE: [INAUDIBLE]. 674 00:34:17,530 --> 00:34:19,780 PROFESSOR: How about before we even get into-- we're 675 00:34:19,780 --> 00:34:21,759 not in polar coordinates in this. 676 00:34:21,759 --> 00:34:23,300 AUDIENCE: The acceleration of A in o? 677 00:34:23,300 --> 00:34:25,610 PROFESSOR: All right, let's just work it out. 678 00:34:25,610 --> 00:34:28,340 I start with this one myself, acceleration of A 679 00:34:28,340 --> 00:34:30,150 with respect to o. 680 00:34:30,150 --> 00:34:31,270 What's another term? 681 00:34:31,270 --> 00:34:33,389 AUDIENCE: Acceleration of G with respect to A. 682 00:34:33,389 --> 00:34:35,250 PROFESSOR: So the acceleration of G 683 00:34:35,250 --> 00:34:40,050 with respect to A, but no rotation, right? 684 00:34:40,050 --> 00:34:41,650 That term. 685 00:34:41,650 --> 00:34:44,889 Plus, how many more terms do we have to go? 686 00:34:44,889 --> 00:34:46,510 Three, OK. 687 00:34:46,510 --> 00:34:47,360 Give me one. 688 00:34:47,360 --> 00:34:49,316 Somebody else? 689 00:34:49,316 --> 00:34:50,839 AUDIENCE: [INAUDIBLE]. 690 00:34:50,839 --> 00:34:52,005 PROFESSOR: What's your name? 691 00:34:52,005 --> 00:34:53,190 AUDIENCE: Stephen. 692 00:34:53,190 --> 00:34:54,600 PROFESSOR: Stephen. 693 00:34:54,600 --> 00:35:00,400 You've got a 2 omega, and omega is-- let's 694 00:35:00,400 --> 00:35:03,099 get this unit vector in here. 695 00:35:03,099 --> 00:35:04,140 Well, actually let's not. 696 00:35:04,140 --> 00:35:06,280 2 omega cross r, let's just do that. 697 00:35:06,280 --> 00:35:08,465 OK? 698 00:35:08,465 --> 00:35:09,340 Give me another term. 699 00:35:12,610 --> 00:35:13,701 Somebody else? 700 00:35:13,701 --> 00:35:16,410 AUDIENCE: Omega cross [INAUDIBLE]. 701 00:35:16,410 --> 00:35:20,870 PROFESSOR: Omega cross omega cross r. 702 00:35:20,870 --> 00:35:22,655 And got another one. 703 00:35:22,655 --> 00:35:23,571 AUDIENCE: [INAUDIBLE]. 704 00:35:31,370 --> 00:35:33,250 PROFESSOR: Yeah, and in fact, we'd 705 00:35:33,250 --> 00:35:35,210 better be a little more careful than that. 706 00:35:35,210 --> 00:35:44,550 So it's really the velocity of G with respect to A. 707 00:35:44,550 --> 00:35:46,721 And does it have rotation in it or not? 708 00:35:52,980 --> 00:35:54,090 This is the no rotation. 709 00:35:54,090 --> 00:35:56,710 This is the speed at which the points are moving apart 710 00:35:56,710 --> 00:36:00,000 from one another, right? 711 00:36:00,000 --> 00:36:03,110 Omega cross omega cross r, we've got-- one, two-- 712 00:36:03,110 --> 00:36:06,630 I think we need another term. 713 00:36:06,630 --> 00:36:09,960 Omega dot cross r. 714 00:36:09,960 --> 00:36:17,030 And the r is rGA, this r is GA. 715 00:36:17,030 --> 00:36:20,554 OK, so what's the acceleration of this? 716 00:36:20,554 --> 00:36:21,470 AUDIENCE: [INAUDIBLE]. 717 00:36:21,470 --> 00:36:22,255 PROFESSOR: Nope. 718 00:36:22,255 --> 00:36:23,607 AUDIENCE: Negative half. 719 00:36:23,607 --> 00:36:25,065 PROFESSOR: So this one's minus 1/2. 720 00:36:28,844 --> 00:36:29,510 What's this one? 721 00:36:32,410 --> 00:36:34,800 So you're in the frame of the pipe now, 722 00:36:34,800 --> 00:36:37,180 and this is the speed at which the acceleration of G 723 00:36:37,180 --> 00:36:43,830 with respect to A. It doesn't move, it's fixed length, right? 724 00:36:43,830 --> 00:36:46,760 This one, what's the velocity of G with respect to A, 725 00:36:46,760 --> 00:36:49,175 without rotation involved? 726 00:36:54,070 --> 00:37:02,690 OK, and this one is omega k cross, and this is our, again, 727 00:37:02,690 --> 00:37:04,640 omega k cross. 728 00:37:04,640 --> 00:37:10,360 And this RG, it's R in what direction at this instant? 729 00:37:10,360 --> 00:37:11,260 J, right? 730 00:37:11,260 --> 00:37:12,610 So we could work that out. 731 00:37:12,610 --> 00:37:15,120 And what about this one? 732 00:37:15,120 --> 00:37:17,210 What direction is omega dot in? 733 00:37:20,150 --> 00:37:26,810 k cross R, at this instant, J again. 734 00:37:26,810 --> 00:37:28,370 Do we know that value? 735 00:37:28,370 --> 00:37:29,570 No. 736 00:37:29,570 --> 00:37:31,860 That might be something we have to figure out. 737 00:37:31,860 --> 00:37:35,902 So you could now crank this out. 738 00:37:35,902 --> 00:37:37,610 That'd be the acceleration of that point, 739 00:37:37,610 --> 00:37:40,180 but you'd find out that we don't have that. 740 00:37:40,180 --> 00:37:45,978 Now, how would you go about finding out what omega dot is? 741 00:37:45,978 --> 00:37:48,338 AUDIENCE: Equations of motion? 742 00:37:48,338 --> 00:37:50,870 PROFESSOR: You get the gold star. 743 00:37:50,870 --> 00:37:52,090 Equations of motion. 744 00:37:52,090 --> 00:37:54,620 You need to come up with equations of motion. 745 00:37:54,620 --> 00:38:06,270 And how many equations of motion would we end up writing? 746 00:38:06,270 --> 00:38:08,750 This is equivalent to asking how many degrees of freedom 747 00:38:08,750 --> 00:38:09,320 are there. 748 00:38:12,190 --> 00:38:15,220 I hear 2, I hear a 1. 749 00:38:18,820 --> 00:38:24,930 Now, there's a lot of nuances to this discussion. 750 00:38:24,930 --> 00:38:30,100 Anytime you're given a fixed value for something, 751 00:38:30,100 --> 00:38:34,170 that coordinate is constrained, that parameter's constrained. 752 00:38:36,900 --> 00:38:41,100 We're given the motion of the truck. 753 00:38:41,100 --> 00:38:42,310 It's completely specified. 754 00:38:47,481 --> 00:38:49,730 You could just substitute numbers in for those things. 755 00:38:49,730 --> 00:38:51,650 I mean, like this, it pops up as 1/2. 756 00:38:54,850 --> 00:39:00,210 I think this thing will boil down to one equation. 757 00:39:00,210 --> 00:39:06,110 And you need to pick some set of coordinates to describe it. 758 00:39:06,110 --> 00:39:09,530 So if I'm looking for omega dot, I'd 759 00:39:09,530 --> 00:39:11,630 probably work with torque about that point. 760 00:39:14,140 --> 00:39:16,860 Isolated to just this object, and this 761 00:39:16,860 --> 00:39:19,560 is some of the torque around this point, 762 00:39:19,560 --> 00:39:22,320 and see what happens. 763 00:39:22,320 --> 00:39:29,460 Some of the external torques gotta be equal to what? 764 00:39:29,460 --> 00:39:33,420 DH dt, and then you have to sort out this term. 765 00:39:33,420 --> 00:39:35,300 So in this problem, you know-- what's 766 00:39:35,300 --> 00:39:37,290 the direction of this velocity? 767 00:39:40,150 --> 00:39:44,690 I. What is the unit vector associated 768 00:39:44,690 --> 00:39:48,930 with the linear momentum of that pipe? 769 00:39:48,930 --> 00:39:51,090 What direction is the linear momentum of the pipe? 770 00:39:55,540 --> 00:39:56,600 What's the velocity? 771 00:39:56,600 --> 00:39:59,150 What is this answer? 772 00:39:59,150 --> 00:40:02,760 What's the momentum of that pipe? 773 00:40:02,760 --> 00:40:06,980 Mass times that velocity, right? 774 00:40:06,980 --> 00:40:09,160 Remember, the velocity of the system, the pipe 775 00:40:09,160 --> 00:40:11,850 may have all its mass around the rim, 776 00:40:11,850 --> 00:40:16,370 but the velocity of the momentum of the pipe 777 00:40:16,370 --> 00:40:21,080 is the total mass of the pipe times the velocity 778 00:40:21,080 --> 00:40:22,144 of its center of mass. 779 00:40:22,144 --> 00:40:24,060 And that's the velocity of the center of mass. 780 00:40:24,060 --> 00:40:29,284 So its linear momentum has what unit vector assigned with it? 781 00:40:29,284 --> 00:40:33,930 I. And what is the unit vector was associated with vA? 782 00:40:33,930 --> 00:40:35,625 I. I cross I is? 783 00:40:35,625 --> 00:40:36,650 AUDIENCE: 0. 784 00:40:36,650 --> 00:40:38,566 PROFESSOR: So this is one of those cases where 785 00:40:38,566 --> 00:40:41,080 they're parallel paths, and that term drops out. 786 00:40:41,080 --> 00:40:44,060 So you could come up with it very quickly and easily. 787 00:40:44,060 --> 00:40:46,430 Equation of motion. 788 00:40:46,430 --> 00:40:47,100 OK. 789 00:40:47,100 --> 00:40:49,290 Very good. 790 00:40:49,290 --> 00:40:51,460 See you on Tuesday. 791 00:40:51,460 --> 00:40:54,111 Next Tuesday lecture time will be review. 792 00:40:54,111 --> 00:40:56,360 I think it's gonna be more like one of these sessions, 793 00:40:56,360 --> 00:40:58,940 and then quiz Tuesday night.