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 --> 00:00:10,150 continue to offer high quality educational resources for free. 5 00:00:10,150 --> 00:00:12,680 To make a donation or to view additional materials 6 00:00:12,680 --> 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 00:00:20,811 --> 00:00:25,140 PROFESSOR: The purpose of these recitations, small group 9 00:00:25,140 --> 00:00:27,790 recitations, is so that we can get out 10 00:00:27,790 --> 00:00:30,140 the key concepts over the week and what 11 00:00:30,140 --> 00:00:33,217 I call the essential understandings-- what 12 00:00:33,217 --> 00:00:35,050 are the really important points for the week 13 00:00:35,050 --> 00:00:37,380 so that when the first quiz comes, 14 00:00:37,380 --> 00:00:39,050 you will know how to deal with it. 15 00:00:39,050 --> 00:00:42,420 So let's start with that. 16 00:00:42,420 --> 00:00:45,420 But you're going to help me think through this. 17 00:00:45,420 --> 00:00:48,420 So take a minute or two, write down 18 00:00:48,420 --> 00:00:50,310 on a piece of paper two or three things 19 00:00:50,310 --> 00:00:53,010 that you think are the most important things that you 20 00:00:53,010 --> 00:00:55,140 heard, saw, read this week about this course. 21 00:01:01,060 --> 00:01:02,400 Let's report out. 22 00:01:02,400 --> 00:01:04,390 I want one from a number of you. 23 00:01:04,390 --> 00:01:06,702 Who wants to volunteer here? 24 00:01:06,702 --> 00:01:08,506 AUDIENCE: Using different reference frames? 25 00:01:08,506 --> 00:01:09,790 PROFESSOR: Say it again? 26 00:01:09,790 --> 00:01:10,560 AUDIENCE: Using different reference frames. 27 00:01:10,560 --> 00:01:12,393 PROFESSOR: Using different reference frames. 28 00:01:15,540 --> 00:01:19,040 I'm going to write that, once I get the chalkboard. 29 00:01:19,040 --> 00:01:29,540 Using-- I'm going to write it as multiple reference frames. 30 00:01:32,050 --> 00:01:34,582 Close enough? 31 00:01:34,582 --> 00:01:35,290 What's your name? 32 00:01:35,290 --> 00:01:36,123 AUDIENCE: Christina. 33 00:01:36,123 --> 00:01:37,640 PROFESSOR: Christina? 34 00:01:37,640 --> 00:01:38,655 What do you have? 35 00:01:38,655 --> 00:01:40,655 AUDIENCE: All points on a rigid, rotating object 36 00:01:40,655 --> 00:01:42,643 have the same rate of rotation. 37 00:01:49,601 --> 00:01:53,110 PROFESSOR: She said, all points on rigid object that's 38 00:01:53,110 --> 00:01:55,910 rotating, all points have the same rotation rate. 39 00:01:55,910 --> 00:02:14,110 So this is rotation and translation of rigid bodies. 40 00:02:14,110 --> 00:02:16,250 I'm going to generalize what you said 41 00:02:16,250 --> 00:02:19,300 a little bit, because somebody else tell me, 42 00:02:19,300 --> 00:02:20,960 what can you say about translation? 43 00:02:20,960 --> 00:02:28,140 So rotation, key point is, all points share the same rotation 44 00:02:28,140 --> 00:02:28,640 rate. 45 00:02:28,640 --> 00:02:30,920 How about translation? 46 00:02:30,920 --> 00:02:32,380 Two different points on an object-- 47 00:02:32,380 --> 00:02:34,252 what can you say about it? 48 00:02:34,252 --> 00:02:36,710 AUDIENCE: They follow the same paths. 49 00:02:36,710 --> 00:02:37,850 PROFESSOR: Parallel paths. 50 00:02:37,850 --> 00:02:41,650 They go through the exactly same parallel paths. 51 00:02:41,650 --> 00:02:43,850 So those are two key things we remember about that. 52 00:02:43,850 --> 00:02:45,150 How about another point? 53 00:02:45,150 --> 00:02:46,140 AUDIENCE: [INAUDIBLE] 54 00:02:52,575 --> 00:02:54,256 PROFESSOR: OK. 55 00:02:54,256 --> 00:02:55,630 This is actually quite important. 56 00:02:58,270 --> 00:03:00,220 I'm going to write it slightly differently. 57 00:03:00,220 --> 00:03:01,860 We need to talk about this. 58 00:03:01,860 --> 00:03:12,750 And that is that rotations-- to be absolutely correct, 59 00:03:12,750 --> 00:03:23,690 finite rotations are not vectors. 60 00:03:27,940 --> 00:03:30,190 I want to come back to that in a minute. 61 00:03:30,190 --> 00:03:32,450 Lots of possible confusion around that. 62 00:03:32,450 --> 00:03:38,823 One more-- or actually, there's more. 63 00:03:38,823 --> 00:03:41,280 AUDIENCE: The MLM strategy? 64 00:03:41,280 --> 00:03:42,330 PROFESSOR: MLM. 65 00:03:42,330 --> 00:03:43,960 So I mentioned that last time. 66 00:03:43,960 --> 00:03:47,300 That's problem solving my way, which 67 00:03:47,300 --> 00:03:51,085 is first M is figure out the motion, describe the motion. 68 00:03:51,085 --> 00:03:51,835 That's kinematics. 69 00:03:55,880 --> 00:03:58,940 The second term is L. What is that? 70 00:03:58,940 --> 00:03:59,610 Laws. 71 00:03:59,610 --> 00:04:00,510 By the physical laws. 72 00:04:00,510 --> 00:04:02,980 And the second M? 73 00:04:02,980 --> 00:04:04,310 Do the math. 74 00:04:04,310 --> 00:04:06,195 So motion, laws, and math. 75 00:04:09,790 --> 00:04:11,460 There's something else here. 76 00:04:11,460 --> 00:04:13,315 Well, you may have just decided it's 77 00:04:13,315 --> 00:04:14,440 going to encompass in that. 78 00:04:14,440 --> 00:04:18,100 But I want to go a little further than that. 79 00:04:18,100 --> 00:04:20,959 What did we talk a lot about yesterday in the lecture? 80 00:04:20,959 --> 00:04:22,855 AUDIENCE: Different types of acceleration. 81 00:04:22,855 --> 00:04:28,560 PROFESSOR: Accelerations and velocities and translating 82 00:04:28,560 --> 00:04:29,990 and rotating frames. 83 00:04:43,900 --> 00:04:49,290 Translating and rotating frames-- running out of room 84 00:04:49,290 --> 00:04:52,340 here but you get the point. 85 00:04:52,340 --> 00:04:54,510 All right, that's a pretty good list. 86 00:04:54,510 --> 00:04:56,377 If I'd been coming up with a list on my own, 87 00:04:56,377 --> 00:04:57,960 what would have thought was important, 88 00:04:57,960 --> 00:05:00,860 that would've captured most of those things. 89 00:05:00,860 --> 00:05:04,370 Certainly this is really important this week. 90 00:05:04,370 --> 00:05:07,470 And we definitely need to learn how to use translating 91 00:05:07,470 --> 00:05:08,860 and rotating frames. 92 00:05:08,860 --> 00:05:12,190 And you're absolutely in trouble if you don't know this. 93 00:05:12,190 --> 00:05:16,160 This is just sort of fundamental to the whole thing. 94 00:05:16,160 --> 00:05:17,880 And then this is a subtle point. 95 00:05:17,880 --> 00:05:19,630 Let's start right there for a second. 96 00:05:22,150 --> 00:05:28,354 Who has a textbook? 97 00:05:28,354 --> 00:05:29,770 It doesn't actually really matter. 98 00:05:29,770 --> 00:05:31,065 Let me borrow your notes. 99 00:05:35,160 --> 00:05:37,790 Rigid body, got the print on the front. 100 00:05:37,790 --> 00:05:40,160 I'm going to rotate it twice. 101 00:05:40,160 --> 00:05:46,950 The x-axis and call this the z-axis. 102 00:05:46,950 --> 00:05:51,160 It comes out top actually pointing at you. 103 00:05:51,160 --> 00:05:52,840 So I did that right. 104 00:05:52,840 --> 00:05:56,300 So now I'm going to do the rotation. 105 00:05:56,300 --> 00:05:59,790 Now this one first. 106 00:05:59,790 --> 00:06:02,770 And then what was the other rotation? 107 00:06:02,770 --> 00:06:04,100 AUDIENCE: Backwards. 108 00:06:04,100 --> 00:06:06,420 PROFESSOR: Different answer, right? 109 00:06:06,420 --> 00:06:08,450 Totally different answer. 110 00:06:08,450 --> 00:06:13,070 You can't add angles as actors. 111 00:06:13,070 --> 00:06:14,330 Doesn't work. 112 00:06:14,330 --> 00:06:19,130 And it's just-- the way I think of it, 113 00:06:19,130 --> 00:06:23,990 mathematics is largely done to help 114 00:06:23,990 --> 00:06:26,710 describe the physical world. 115 00:06:26,710 --> 00:06:29,440 Newton and all those people were figuring out-- needed calculus 116 00:06:29,440 --> 00:06:31,920 to describe the motion of the planets. 117 00:06:31,920 --> 00:06:36,620 Vectors were invented to do analytic geometry. 118 00:06:36,620 --> 00:06:39,190 And it doesn't work for angles. 119 00:06:39,190 --> 00:06:42,230 You just can't use them for angles. 120 00:06:42,230 --> 00:06:44,850 It's just the vector math that they figured out 121 00:06:44,850 --> 00:06:47,670 just wasn't quite clever enough to include angles. 122 00:06:47,670 --> 00:06:52,880 However, vectors can be applied to positions, velocities, 123 00:06:52,880 --> 00:06:57,240 accelerations, and angular velocities 124 00:06:57,240 --> 00:07:01,140 and angular accelerations, but not angles themselves. 125 00:07:01,140 --> 00:07:05,550 That's the basic thing you need to learn from that. 126 00:07:05,550 --> 00:07:10,292 Let's use some multiple frames. 127 00:07:10,292 --> 00:07:11,500 We're going to do that today. 128 00:07:11,500 --> 00:07:16,660 We're going to now apply this and this and this today 129 00:07:16,660 --> 00:07:18,925 to do some problems. 130 00:07:18,925 --> 00:07:23,425 And let me see where I want to go first with this. 131 00:07:39,730 --> 00:07:43,036 So I have a problem that I wanted to do. 132 00:07:49,490 --> 00:07:50,840 And it's a circus ride. 133 00:08:01,900 --> 00:08:02,850 There's an arm. 134 00:08:05,560 --> 00:08:07,910 And that arm is rotating. 135 00:08:07,910 --> 00:08:12,035 Attached to the arm is a cross piece. 136 00:08:12,035 --> 00:08:15,230 And a passenger can sit in each one of these things. 137 00:08:15,230 --> 00:08:16,930 And this is basically horizontal. 138 00:08:16,930 --> 00:08:18,350 You're looking down on it. 139 00:08:18,350 --> 00:08:20,600 So you'd be riding around in these cups at the circus 140 00:08:20,600 --> 00:08:22,120 and it's going around and around. 141 00:08:22,120 --> 00:08:25,770 And I want to know the velocity. 142 00:08:25,770 --> 00:08:34,341 What's the velocity of point B in the O frame? 143 00:08:34,341 --> 00:08:36,590 And so this has to do-- one of the things on this list 144 00:08:36,590 --> 00:08:39,750 might be to get the notation down. 145 00:08:39,750 --> 00:08:41,289 So this is the velocity. 146 00:08:41,289 --> 00:08:42,190 This is the point. 147 00:08:45,280 --> 00:08:46,270 And this is the frame. 148 00:08:52,010 --> 00:08:57,040 So what can you write down? 149 00:08:57,040 --> 00:08:58,560 Just take 30 seconds. 150 00:08:58,560 --> 00:08:59,500 See if you remember. 151 00:08:59,500 --> 00:09:02,490 Write down the general velocity formula 152 00:09:02,490 --> 00:09:04,580 that was put up yesterday-- vector velocity 153 00:09:04,580 --> 00:09:08,430 formula for a point in a moving frame that's 154 00:09:08,430 --> 00:09:11,530 moving in a fixed frame. 155 00:09:11,530 --> 00:09:15,054 Came up with a general formula, had two or three terms in it. 156 00:09:15,054 --> 00:09:16,470 And we'll walk our way through it. 157 00:09:56,350 --> 00:09:59,200 I realize I did something maybe slightly out of order. 158 00:09:59,200 --> 00:10:00,700 So hold that thought. 159 00:10:00,700 --> 00:10:04,150 You've written down what you've got. 160 00:10:04,150 --> 00:10:07,340 We have to do something before you can actually write that. 161 00:10:07,340 --> 00:10:13,990 We haven't actually picked our reference frames, have we? 162 00:10:13,990 --> 00:10:15,380 So think about that for a second. 163 00:10:15,380 --> 00:10:17,590 How would you set up this problem? 164 00:10:17,590 --> 00:10:20,930 What would you make translating reference frames, 165 00:10:20,930 --> 00:10:22,580 your rotating, translating frame-- 166 00:10:22,580 --> 00:10:23,730 where would you assign it? 167 00:10:23,730 --> 00:10:24,980 Think about it for 30 seconds. 168 00:10:43,200 --> 00:10:45,500 Who's got to take a shot at it for me? 169 00:10:45,500 --> 00:10:49,711 Where would you pick reference frames for this problem? 170 00:10:54,875 --> 00:10:55,766 What your name? 171 00:10:55,766 --> 00:10:57,104 AUDIENCE: I'm Ben. 172 00:10:57,104 --> 00:10:59,244 PROFESSOR: Ben. 173 00:10:59,244 --> 00:11:03,240 AUDIENCE: O and along the cross? 174 00:11:03,240 --> 00:11:05,700 PROFESSOR: Here, for sure. 175 00:11:05,700 --> 00:11:10,890 This is your inertial frame-- not moving, right? 176 00:11:10,890 --> 00:11:12,320 And? 177 00:11:12,320 --> 00:11:13,715 AUDIENCE: Two axes on the cross? 178 00:11:13,715 --> 00:11:16,360 PROFESSOR: So you would put one up here? 179 00:11:16,360 --> 00:11:16,860 OK. 180 00:11:19,845 --> 00:11:21,345 I'm going to line up with the cross, 181 00:11:21,345 --> 00:11:25,660 and I'm going to stick out here and call it x2. 182 00:11:25,660 --> 00:11:28,390 And then there'd be a y2 here. 183 00:11:28,390 --> 00:11:32,120 And it rotates with the cross? 184 00:11:32,120 --> 00:11:33,380 All right. 185 00:11:33,380 --> 00:11:33,990 That's good. 186 00:11:33,990 --> 00:11:35,830 Now go back to that equation. 187 00:11:35,830 --> 00:11:38,690 Now give me the velocity, the general expression. 188 00:11:38,690 --> 00:11:42,340 I don't want you working out the details, just what set of terms 189 00:11:42,340 --> 00:11:48,110 would you plug things into now to get the velocity of B and O? 190 00:11:48,110 --> 00:11:52,560 Then we'll evaluate the terms and talk about it, using now 191 00:11:52,560 --> 00:11:53,800 what we've decided here. 192 00:12:00,930 --> 00:12:02,780 OK, somebody help me out. 193 00:12:02,780 --> 00:12:06,050 What's on the right hand side of this equation? 194 00:12:06,050 --> 00:12:08,240 First term, Mary. 195 00:12:08,240 --> 00:12:10,540 AUDIENCE: Velocity of-- 196 00:12:10,540 --> 00:12:12,490 PROFESSOR: What's your name? 197 00:12:12,490 --> 00:12:13,910 Steven? 198 00:12:13,910 --> 00:12:15,660 AUDIENCE: Velocity of A with respect to O. 199 00:12:15,660 --> 00:12:18,780 PROFESSOR: Velocity of A with respect to O. All right. 200 00:12:18,780 --> 00:12:23,460 That's the velocity at this point in this frame, right? 201 00:12:23,460 --> 00:12:25,670 What else do we need? 202 00:12:25,670 --> 00:12:27,420 What's your name? 203 00:12:27,420 --> 00:12:28,410 Andre? 204 00:12:28,410 --> 00:12:29,660 AUDIENCE: Yeah. [INAUDIBLE] 205 00:12:33,724 --> 00:12:39,820 PROFESSOR: I hear a velocity of V with respect to A. 206 00:12:39,820 --> 00:12:43,550 And what is that-- is that influenced by rotation? 207 00:12:43,550 --> 00:12:45,780 Can you describe what you mean by the velocity 208 00:12:45,780 --> 00:12:49,426 of v and A physically? 209 00:12:49,426 --> 00:12:50,300 AUDIENCE: [INAUDIBLE] 210 00:12:55,738 --> 00:13:00,230 PROFESSOR: So it's as if you were sitting on that frame, 211 00:13:00,230 --> 00:13:00,730 right? 212 00:13:00,730 --> 00:13:03,511 Does the rotation have anything to do with what you see? 213 00:13:03,511 --> 00:13:04,010 No. 214 00:13:04,010 --> 00:13:07,130 So I sometimes remind myself right here 215 00:13:07,130 --> 00:13:08,850 this is omega equals 0. 216 00:13:08,850 --> 00:13:11,330 And you can set the omega equal to 0, what you 217 00:13:11,330 --> 00:13:14,420 would see is what this term is. 218 00:13:14,420 --> 00:13:18,060 Do we need anything more? 219 00:13:18,060 --> 00:13:18,692 Name? 220 00:13:18,692 --> 00:13:19,525 AUDIENCE: Christina. 221 00:13:19,525 --> 00:13:21,525 PROFESSOR: Sorry, you gave it to me once before. 222 00:13:21,525 --> 00:13:22,911 It's going to take me awhile. 223 00:13:22,911 --> 00:13:25,366 AUDIENCE: It's the rotational motion 224 00:13:25,366 --> 00:13:29,785 of B spinning around in there. 225 00:13:29,785 --> 00:13:33,222 So it has to do with the omega as seen in the reference 226 00:13:33,222 --> 00:13:45,740 frame, the origin, cross product with r from the in regards 227 00:13:45,740 --> 00:13:50,543 to the x2 xy. 228 00:13:50,543 --> 00:13:53,700 PROFESSOR: And we have the name of that frame to help us out. 229 00:13:53,700 --> 00:13:58,120 This is then frame A, x2, y2, z2. 230 00:13:58,120 --> 00:14:00,340 If you really wanted to write [INAUDIBLE] 231 00:14:00,340 --> 00:14:05,770 We just call it frame A. so this is would be rB as in NA 232 00:14:05,770 --> 00:14:10,370 And these are all vectors and I often forget to underline them. 233 00:14:10,370 --> 00:14:13,480 Do we have it right? 234 00:14:13,480 --> 00:14:16,380 Anybody want to add to that, fix it? 235 00:14:16,380 --> 00:14:19,050 Correct it? 236 00:14:19,050 --> 00:14:19,734 Steven, right? 237 00:14:19,734 --> 00:14:20,608 AUDIENCE: [INAUDIBLE] 238 00:14:23,356 --> 00:14:25,510 PROFESSOR: I left it vague on purpose. 239 00:14:25,510 --> 00:14:26,810 We need to figure that out. 240 00:14:26,810 --> 00:14:29,160 He asks, is it omega 2 or omega 1? 241 00:14:29,160 --> 00:14:32,930 Really important point we want to make today about what omega 242 00:14:32,930 --> 00:14:34,220 this is. 243 00:14:34,220 --> 00:14:34,970 We'll get to that. 244 00:14:34,970 --> 00:14:35,540 Yeah? 245 00:14:35,540 --> 00:14:38,890 AUDIENCE: Well, if they're rotating in the same direction, 246 00:14:38,890 --> 00:14:41,807 wouldn't it be added in both omega 1 or omega 2? 247 00:14:41,807 --> 00:14:42,640 PROFESSOR: Well, OK. 248 00:14:42,640 --> 00:14:43,890 Let's talk about it right now. 249 00:14:43,890 --> 00:14:49,680 Are we agreed that this is the right formula? 250 00:14:49,680 --> 00:14:52,550 Then let's set about figuring it out. 251 00:14:52,550 --> 00:14:55,770 And we can talk about this term first. 252 00:14:55,770 --> 00:15:01,840 So we want to know, this is the rotation rate 253 00:15:01,840 --> 00:15:08,560 of this arm out here in the base frame. 254 00:15:08,560 --> 00:15:11,440 That's what the notation says. 255 00:15:11,440 --> 00:15:14,890 And we know that the rotation rate of this first arm 256 00:15:14,890 --> 00:15:16,860 in the base frame is this. 257 00:15:16,860 --> 00:15:18,580 And we know that the rotation rate 258 00:15:18,580 --> 00:15:28,842 of this thing with to-- now this has 259 00:15:28,842 --> 00:15:30,550 gotten a little complicated, because this 260 00:15:30,550 --> 00:15:33,190 isn't quite exact enough. 261 00:15:33,190 --> 00:15:35,500 This is omega 2 with respect to this arm. 262 00:15:39,360 --> 00:15:40,930 That's what's given in this problem. 263 00:15:40,930 --> 00:15:49,490 So this is omega 2 with respect to the arm OA. 264 00:15:52,581 --> 00:15:53,080 Yeah? 265 00:15:53,080 --> 00:15:55,032 AUDIENCE: So does that mean it's omega 266 00:15:55,032 --> 00:15:58,936 2xz from coordiante system B? 267 00:15:58,936 --> 00:16:03,040 PROFESSOR: No, coordinate system A x2y2 rotates. 268 00:16:03,040 --> 00:16:06,790 And if you're sitting in there, you wouldn't see it. 269 00:16:06,790 --> 00:16:11,440 So this is correct. 270 00:16:11,440 --> 00:16:14,537 It's the rotation rate as seen in O. 271 00:16:14,537 --> 00:16:16,120 So we need to figure out what that is. 272 00:16:16,120 --> 00:16:17,494 And I'm telling you in this case, 273 00:16:17,494 --> 00:16:20,110 you were given-- you might have been 274 00:16:20,110 --> 00:16:22,150 given the rotation rate in O. 275 00:16:22,150 --> 00:16:22,710 You weren't. 276 00:16:22,710 --> 00:16:25,860 You were given the rotation rate relative to here. 277 00:16:25,860 --> 00:16:30,140 So I'll write it as W2 with respect to this arm OA. 278 00:16:30,140 --> 00:16:39,015 So how do you get-- we need omega in O is what? 279 00:16:39,015 --> 00:16:40,440 Help me out here. 280 00:16:40,440 --> 00:16:43,092 AUDIENCE: Is it O-- should there be a small b at the bottom 281 00:16:43,092 --> 00:16:43,592 [INAUDIBLE]? 282 00:16:50,040 --> 00:16:52,280 PROFESSOR: Good. 283 00:16:52,280 --> 00:16:53,090 But what is it? 284 00:16:56,920 --> 00:16:58,520 Let's deduce it. 285 00:16:58,520 --> 00:17:02,820 If my arm here, this is the first arm. 286 00:17:02,820 --> 00:17:05,950 And this is the at AB link. 287 00:17:05,950 --> 00:17:09,400 Now if omega with respect to this arm, this thing 288 00:17:09,400 --> 00:17:12,660 weren't moving, no rotation rate relative to this, 289 00:17:12,660 --> 00:17:14,869 the whole thing would be straight, right? 290 00:17:14,869 --> 00:17:17,210 And it's going around like this. 291 00:17:17,210 --> 00:17:21,160 What's the rotation rate of the link out here? 292 00:17:21,160 --> 00:17:21,960 Omega 1. 293 00:17:24,609 --> 00:17:29,420 And now this arm's not moving, but this is rotating 294 00:17:29,420 --> 00:17:33,570 relative to it at omega 2. 295 00:17:33,570 --> 00:17:36,310 What's the rotation rate of the link out here? 296 00:17:36,310 --> 00:17:37,270 Just omega 2. 297 00:17:37,270 --> 00:17:41,300 If I put the two together, what is the rotation rate 298 00:17:41,300 --> 00:17:43,454 of this arm, this second link? 299 00:17:43,454 --> 00:17:44,370 AUDIENCE: [INAUDIBLE]. 300 00:17:47,520 --> 00:17:53,210 PROFESSOR: Omega 1 in, certainly in O plus omega 2. 301 00:17:53,210 --> 00:17:55,050 It's not with respect to A. I'm just 302 00:17:55,050 --> 00:17:59,060 going to call it with respect to the R maybe. 303 00:17:59,060 --> 00:18:02,301 Even this notation is failing a little bit. 304 00:18:02,301 --> 00:18:03,300 But you get what I mean. 305 00:18:03,300 --> 00:18:05,140 It's omega 1 plus omega 2. 306 00:18:05,140 --> 00:18:09,040 And let's just write it as omega 1 plus omega 2. 307 00:18:09,040 --> 00:18:10,550 And what direction is it in? 308 00:18:10,550 --> 00:18:12,940 It's a vector. 309 00:18:12,940 --> 00:18:14,940 So one of the things we have to pay attention to 310 00:18:14,940 --> 00:18:16,611 are unit vectors. 311 00:18:16,611 --> 00:18:17,110 Yeah? 312 00:18:17,110 --> 00:18:17,984 AUDIENCE: [INAUDIBLE] 313 00:18:17,984 --> 00:18:21,535 PROFESSOR: So this is capital I hat here 314 00:18:21,535 --> 00:18:27,420 and capital J hat there and coming out of the board, K hat. 315 00:18:27,420 --> 00:18:31,310 Now, this is certainly K hat, capital K hat. 316 00:18:31,310 --> 00:18:36,090 This one, though, is relative to-- it's the rotation 317 00:18:36,090 --> 00:18:38,350 rate of this thing. 318 00:18:38,350 --> 00:18:40,220 Here is a reference frame. 319 00:18:40,220 --> 00:18:42,590 What's sticking out this way? 320 00:18:42,590 --> 00:18:44,670 A little k2, right? 321 00:18:44,670 --> 00:18:48,720 But is it parallel to capital K? 322 00:18:48,720 --> 00:18:50,720 Always parallel to capital K? 323 00:18:53,530 --> 00:18:55,580 So they're the same thing. 324 00:18:55,580 --> 00:18:57,431 If unit vectors in this are parallel, 325 00:18:57,431 --> 00:18:58,680 they amount to the same thing. 326 00:18:58,680 --> 00:19:00,920 So we can put capital K, lowercase k, anything 327 00:19:00,920 --> 00:19:02,220 we want here and it's correct. 328 00:19:05,530 --> 00:19:08,750 Now we've got an answer for that. 329 00:19:08,750 --> 00:19:14,800 So when you're given-- when the one thing's attached to another 330 00:19:14,800 --> 00:19:17,090 and you're given the-- if out here 331 00:19:17,090 --> 00:19:23,270 you are given the rotation rate in the base frame, you're done. 332 00:19:23,270 --> 00:19:24,930 But if you're given the rotation rate 333 00:19:24,930 --> 00:19:27,950 relative to some other moving part, 334 00:19:27,950 --> 00:19:30,730 then you have to add them up to get the true rotation rate. 335 00:19:30,730 --> 00:19:33,040 That's the bottom line message. 336 00:19:33,040 --> 00:19:33,550 All right. 337 00:19:33,550 --> 00:19:38,800 So we started-- we're trying to figure out 338 00:19:38,800 --> 00:19:40,780 this expression here. 339 00:19:44,960 --> 00:19:49,020 And we started with one of the harder terms. 340 00:19:49,020 --> 00:19:52,450 And we need to figure out-- to finish it, 341 00:19:52,450 --> 00:19:55,480 though, let's do this over here. 342 00:19:55,480 --> 00:20:16,150 We have velocity of A in-- and we have the velocity of B in A 343 00:20:16,150 --> 00:20:18,860 with no rotation. 344 00:20:18,860 --> 00:20:25,040 And we have omega B in O. And let's finish that. 345 00:20:25,040 --> 00:20:27,781 We know what omega is now. 346 00:20:27,781 --> 00:20:28,280 Whoops. 347 00:20:28,280 --> 00:20:29,360 It's not omega. 348 00:20:29,360 --> 00:20:34,502 The third term is omega B and O cross RBA. 349 00:20:38,210 --> 00:20:40,380 So we've gotten the first bit of this. 350 00:20:40,380 --> 00:20:42,770 Let's finish the problem. 351 00:20:42,770 --> 00:20:50,360 This is omega 1 plus omega 2 times k hat cross with what? 352 00:20:54,460 --> 00:20:55,460 We need a length. 353 00:20:55,460 --> 00:20:57,810 I'll call this L. It's L long. 354 00:21:03,600 --> 00:21:07,290 So what is RB respect to A? 355 00:21:07,290 --> 00:21:07,790 Yes? 356 00:21:07,790 --> 00:21:10,120 AUDIENCE: L X 2 hat? 357 00:21:10,120 --> 00:21:11,535 PROFESSOR: L X 2 hat. 358 00:21:11,535 --> 00:21:13,600 And I'll call that Lj2. 359 00:21:16,850 --> 00:21:20,940 The coordinate is x2. 360 00:21:20,940 --> 00:21:26,830 The unit vector would be I2, not a j, an i. 361 00:21:29,570 --> 00:21:32,100 The unit vector is i2. 362 00:21:32,100 --> 00:21:33,180 OK, great. 363 00:21:33,180 --> 00:21:35,350 Now what is k cross i2? 364 00:21:40,510 --> 00:21:41,440 j2. 365 00:21:41,440 --> 00:21:52,700 So we get omega L omega 1 plus omega 2 j2 hat. 366 00:21:52,700 --> 00:21:55,150 That's that term. 367 00:21:55,150 --> 00:22:03,970 And we need to figure out our other two terms. 368 00:22:03,970 --> 00:22:05,580 What's this term? 369 00:22:09,520 --> 00:22:11,470 Remind yourself of the meaning. 370 00:22:11,470 --> 00:22:15,410 This is the velocity of point B with respect to the A frame, 371 00:22:15,410 --> 00:22:16,750 which is attached to it. 372 00:22:16,750 --> 00:22:20,290 It's on a rigid body. 373 00:22:20,290 --> 00:22:23,776 AUDIENCE: [INAUDIBLE] 374 00:22:23,776 --> 00:22:27,730 PROFESSOR: He said omega 2 times L. She says 0. 375 00:22:27,730 --> 00:22:29,740 Any other? 376 00:22:29,740 --> 00:22:31,580 I hear another 0. 377 00:22:31,580 --> 00:22:32,662 Why 0? 378 00:22:32,662 --> 00:22:36,090 AUDIENCE: Because it's rigidly attached into the ride, 379 00:22:36,090 --> 00:22:37,090 if you're moving around. 380 00:22:37,090 --> 00:22:39,070 It's not moving on the ride versus strapped in. 381 00:22:39,070 --> 00:22:39,778 PROFESSOR: Right. 382 00:22:39,778 --> 00:22:42,900 So this term is always from the point of view of a person 383 00:22:42,900 --> 00:22:45,350 riding on the frame. 384 00:22:45,350 --> 00:22:48,670 Riding on that frame-- so you won't ever see rotation 385 00:22:48,670 --> 00:22:49,850 from inside the frame. 386 00:22:49,850 --> 00:22:51,730 You're just moving with it. 387 00:22:51,730 --> 00:22:54,860 So that's called, in the Williams book, 388 00:22:54,860 --> 00:22:59,480 he calls this term the rel. 389 00:22:59,480 --> 00:23:01,480 It's the relative velocity between these two 390 00:23:01,480 --> 00:23:04,222 points and no rotation. 391 00:23:04,222 --> 00:23:05,430 So what is that in this case? 392 00:23:09,020 --> 00:23:10,570 I hear 0. 393 00:23:10,570 --> 00:23:12,270 Everybody agree it's 0? 394 00:23:12,270 --> 00:23:13,660 It's a rigid link. 395 00:23:13,660 --> 00:23:15,070 Two points don't move. 396 00:23:15,070 --> 00:23:18,310 So now we're just left with this one. 397 00:23:18,310 --> 00:23:21,500 And now, one of the points I really 398 00:23:21,500 --> 00:23:27,060 wanted to drive home today is in fact this problem is one 399 00:23:27,060 --> 00:23:29,820 that, depending on how you set it up, 400 00:23:29,820 --> 00:23:33,840 you can think of as actually having multiple rotating 401 00:23:33,840 --> 00:23:35,440 frames. 402 00:23:35,440 --> 00:23:37,880 And if you do that, what's the correct way 403 00:23:37,880 --> 00:23:42,430 to add up the parts so you get to the right answer? 404 00:23:42,430 --> 00:23:46,020 Because we've left this one for the last. 405 00:23:46,020 --> 00:23:49,890 And I want to make sure you go away 406 00:23:49,890 --> 00:23:56,030 knowing a formula you can always use, and it's going to work. 407 00:23:56,030 --> 00:23:57,580 And the formula we can always use 408 00:23:57,580 --> 00:23:59,820 is the one that's of this form. 409 00:23:59,820 --> 00:24:02,560 Every one of these problems, including multiple links 410 00:24:02,560 --> 00:24:05,160 and things, you can build up by doing 411 00:24:05,160 --> 00:24:09,230 a sequence of this problem again and again and again, 412 00:24:09,230 --> 00:24:11,450 until you get the whole answer. 413 00:24:11,450 --> 00:24:13,580 So we've actually done what I would 414 00:24:13,580 --> 00:24:16,400 call the outer problem first. 415 00:24:16,400 --> 00:24:18,330 We've worked out this thing. 416 00:24:18,330 --> 00:24:20,050 We have to do the inner problem now. 417 00:24:20,050 --> 00:24:21,841 We could have done it in a different order, 418 00:24:21,841 --> 00:24:24,390 but I need to know the velocity of this point. 419 00:24:24,390 --> 00:24:31,105 And just to get you in the habit of using the vector equation, 420 00:24:31,105 --> 00:24:35,280 that we have, I want to know the velocity of A in O. 421 00:24:35,280 --> 00:24:40,370 And I'm going to attach a rotating 422 00:24:40,370 --> 00:24:49,270 frame to this arm, x1, y1. 423 00:24:49,270 --> 00:24:51,610 It rotates with this arm at that rate. 424 00:24:54,310 --> 00:24:56,900 And I want you to use that frame to solve 425 00:24:56,900 --> 00:25:00,600 for the velocity of this point. 426 00:25:00,600 --> 00:25:09,580 And that would be-- velocity of point A in O 427 00:25:09,580 --> 00:25:26,090 would be the-- this frame now is an O little x1, y1, z1. 428 00:25:26,090 --> 00:25:27,970 It's a rotating frame, right? 429 00:25:27,970 --> 00:25:30,450 Because the O's are going to get confusing. 430 00:25:30,450 --> 00:25:32,680 Better not call it O. We'll call this rotating 431 00:25:32,680 --> 00:25:35,000 one-- in Williams, he uses a lowercase o, 432 00:25:35,000 --> 00:25:37,380 but it's hard to do on the board. 433 00:25:37,380 --> 00:25:39,670 Let's call this c. 434 00:25:39,670 --> 00:25:43,030 So this is a frame, C, x1, y1. 435 00:25:45,590 --> 00:25:48,080 So this frame will be my c frame. 436 00:25:48,080 --> 00:25:50,940 So I want to know the velocity of point A. 437 00:25:50,940 --> 00:25:52,830 It's the velocity of what? 438 00:25:58,180 --> 00:26:00,500 If you get stuck, use that top formula up there, 439 00:26:00,500 --> 00:26:02,975 put in the right points. 440 00:26:05,800 --> 00:26:08,686 So what's the first term mean? 441 00:26:08,686 --> 00:26:15,470 It's the velocity of the-- this time the rotating frame, 442 00:26:15,470 --> 00:26:18,000 does the rotating frame translate? 443 00:26:18,000 --> 00:26:19,270 We have a rotating frame. 444 00:26:19,270 --> 00:26:21,370 Does it have any translational velocity? 445 00:26:21,370 --> 00:26:23,265 No, but you still have the right to turn it 446 00:26:23,265 --> 00:26:25,110 down and set it equal to 0. 447 00:26:25,110 --> 00:26:27,410 So what's the right term? 448 00:26:27,410 --> 00:26:28,712 How do you write it? 449 00:26:34,040 --> 00:26:34,910 Right? 450 00:26:34,910 --> 00:26:36,760 It's the velocity of my reference frame. 451 00:26:36,760 --> 00:26:38,759 It's the transitional velocity of that reference 452 00:26:38,759 --> 00:26:40,970 frame in the O frame. 453 00:26:40,970 --> 00:26:42,940 And that's what it is. 454 00:26:42,940 --> 00:26:56,930 And in this case, it's 0 plus velocity 455 00:26:56,930 --> 00:27:01,742 of A with respect to c. 456 00:27:01,742 --> 00:27:03,202 And I'll remind you again. 457 00:27:03,202 --> 00:27:04,910 It's as if you were now rotating with it, 458 00:27:04,910 --> 00:27:09,120 and you're sitting at c, looking at A. What's its speed? 459 00:27:09,120 --> 00:27:11,580 0. 460 00:27:11,580 --> 00:27:20,300 Plus omega-- what omega? 461 00:27:20,300 --> 00:27:22,629 Seen where? 462 00:27:22,629 --> 00:27:26,010 Measured from where? 463 00:27:26,010 --> 00:27:27,900 Measured with respect to what frame? 464 00:27:30,530 --> 00:27:39,170 I hear on O, cross with-- we need length? 465 00:27:39,170 --> 00:27:42,700 We'll make this length capital R, scalar. 466 00:27:46,110 --> 00:27:50,335 So what's the cross product here? 467 00:27:50,335 --> 00:27:51,796 What's the unit vector? 468 00:27:54,718 --> 00:27:57,370 Correct unit vector? 469 00:27:57,370 --> 00:27:59,006 Not x. 470 00:27:59,006 --> 00:28:00,600 x is the coordinate. 471 00:28:00,600 --> 00:28:02,400 The unit vector is-- 472 00:28:02,400 --> 00:28:03,340 AUDIENCE: [INAUDIBLE] 473 00:28:06,971 --> 00:28:08,852 PROFESSOR: Right? 474 00:28:08,852 --> 00:28:09,816 That turns 0. 475 00:28:09,816 --> 00:28:10,780 That turns 0. 476 00:28:10,780 --> 00:28:12,710 This turns 0. 477 00:28:12,710 --> 00:28:14,870 Omega 1, And what's the unit vector associated 478 00:28:14,870 --> 00:28:15,930 with this omega 1? 479 00:28:19,090 --> 00:28:26,440 k cross i 1 j hat. 480 00:28:26,440 --> 00:28:34,560 And so we have R omega 1 j1 hat. 481 00:28:34,560 --> 00:28:38,800 And now, we should be able to write out 482 00:28:38,800 --> 00:28:44,340 the full answer of the velocity of B 483 00:28:44,340 --> 00:28:59,230 in O is the velocity of A, which is R omega 1 j1 hat 484 00:28:59,230 --> 00:29:02,510 plus velocity of B with respect to A, 485 00:29:02,510 --> 00:29:15,950 which was 0 plus this term, which we figured out as L omega 486 00:29:15,950 --> 00:29:21,976 1 plus omega 2 times j2 hat. 487 00:29:30,866 --> 00:29:32,490 So we have two or three sub-- this kind 488 00:29:32,490 --> 00:29:36,260 of a hard problem, actually, for the first time out. 489 00:29:36,260 --> 00:29:41,240 Because it has a number of subtle concepts built into it. 490 00:29:41,240 --> 00:29:42,940 You actually have two rotating bodies. 491 00:29:42,940 --> 00:29:44,023 How do you deal with them? 492 00:29:44,023 --> 00:29:49,150 Well, you do sequential applications of that vector 493 00:29:49,150 --> 00:29:50,176 velocity formula. 494 00:29:50,176 --> 00:29:50,676 Yeah? 495 00:29:50,676 --> 00:29:52,818 AUDIENCE: So I was wondering why we 496 00:29:52,818 --> 00:29:56,388 made another coordinate system that's rotating with the arm 497 00:29:56,388 --> 00:30:01,660 to solve for the velocity of A [INAUDIBLE]. 498 00:30:01,660 --> 00:30:03,395 PROFESSOR: She asked, why did we bother 499 00:30:03,395 --> 00:30:04,520 to make this another frame. 500 00:30:07,520 --> 00:30:11,620 The problems are going to get nastier and nastier. 501 00:30:11,620 --> 00:30:13,880 I could have asked you, when I walked into class, what 502 00:30:13,880 --> 00:30:17,340 is the velocity of point A. And you would have said, well, 503 00:30:17,340 --> 00:30:18,554 obviously R omega. 504 00:30:22,470 --> 00:30:24,540 Why are we going to all this trouble, when 505 00:30:24,540 --> 00:30:28,960 everybody knows from high school physics that it's just R omega? 506 00:30:28,960 --> 00:30:30,930 And the answer is is because we're 507 00:30:30,930 --> 00:30:33,060 going to get to doing really nasty problems. 508 00:30:33,060 --> 00:30:36,890 And I want to make sure you understand all the subtleties 509 00:30:36,890 --> 00:30:41,120 about how we get these. 510 00:30:41,120 --> 00:30:44,380 So we started simple, but I did it the long, hard way. 511 00:30:44,380 --> 00:30:48,410 Because later on, if I'd walked in at the beginning 512 00:30:48,410 --> 00:30:51,460 and just asked you right off the bat, what's 513 00:30:51,460 --> 00:30:54,669 the velocity of this point-- go for it-- 514 00:30:54,669 --> 00:30:56,210 you guys would have failed miserably. 515 00:30:58,760 --> 00:31:01,400 It's not much harder, but it takes 516 00:31:01,400 --> 00:31:03,920 two sequential applications of what you think 517 00:31:03,920 --> 00:31:06,050 is obvious when you walked into class. 518 00:31:06,050 --> 00:31:07,095 So that's why. 519 00:31:07,095 --> 00:31:08,970 We're just doing it the hard way, so that you 520 00:31:08,970 --> 00:31:11,610 get all the little nuances. 521 00:31:11,610 --> 00:31:12,397 Yeah? 522 00:31:12,397 --> 00:31:14,882 AUDIENCE: So why [INAUDIBLE] there for omega 2, 523 00:31:14,882 --> 00:31:18,609 we have it with respect to arm AB [INAUDIBLE] with respect 524 00:31:18,609 --> 00:31:20,846 to arm OA? 525 00:31:20,846 --> 00:31:23,228 PROFESSOR: Why is it that way? 526 00:31:23,228 --> 00:31:24,644 AUDIENCE: [INAUDIBLE] with respect 527 00:31:24,644 --> 00:31:28,045 to arm AB and when you wrote [INAUDIBLE] with respect 528 00:31:28,045 --> 00:31:29,511 to arm OA? 529 00:31:29,511 --> 00:31:32,100 PROFESSOR: When I wrote the equation for-- 530 00:31:32,100 --> 00:31:34,710 AUDIENCE: For the omega 2. 531 00:31:34,710 --> 00:31:36,800 There you're saying it's with respect to OA, 532 00:31:36,800 --> 00:31:39,356 and there you say its respect to AB. 533 00:31:39,356 --> 00:31:42,550 PROFESSOR: Oh, I see. 534 00:31:42,550 --> 00:31:44,190 Because this is wrong. 535 00:31:44,190 --> 00:31:45,180 AUDIENCE: [INAUDIBLE] 536 00:31:51,060 --> 00:31:57,307 AUDIENCE: My question deals with j1 and j2-- are they the same? 537 00:31:57,307 --> 00:31:58,640 PROFESSOR: So he has a question. 538 00:31:58,640 --> 00:32:04,170 And that's the final, subtle point I want to get to today. 539 00:32:04,170 --> 00:32:04,840 Good question. 540 00:32:04,840 --> 00:32:06,173 He's saying, are these the same? 541 00:32:06,173 --> 00:32:08,390 Are they different? 542 00:32:08,390 --> 00:32:09,850 How do we deal with it? 543 00:32:09,850 --> 00:32:11,960 So a question for you. 544 00:32:11,960 --> 00:32:14,280 In general, if you're asked or given 545 00:32:14,280 --> 00:32:17,520 a problem like we just did, and you arrive at a solution, 546 00:32:17,520 --> 00:32:23,380 is it OK to give an answer where you had unit 547 00:32:23,380 --> 00:32:24,740 vectors in multiple frames? 548 00:32:24,740 --> 00:32:28,520 And neither of these unit vectors are in the base frame. 549 00:32:28,520 --> 00:32:30,790 And yet, the answer we're claiming 550 00:32:30,790 --> 00:32:34,270 is that this is the velocity of B in O. 551 00:32:34,270 --> 00:32:38,130 And here we've got unit vectors that are not in the base frame. 552 00:32:38,130 --> 00:32:39,825 Is it a legit equation or not? 553 00:32:44,640 --> 00:32:45,584 What do you think? 554 00:32:48,250 --> 00:32:49,501 See a lot of no's out there. 555 00:32:53,547 --> 00:32:54,880 I think we better figure it out. 556 00:33:37,090 --> 00:33:40,530 So we have unit on the arm. 557 00:33:40,530 --> 00:33:43,430 This is my c and O here. 558 00:33:43,430 --> 00:33:46,040 On the arm, I have frame that rotates 559 00:33:46,040 --> 00:33:50,070 with it that has unit vectors in the direction of the arm of i1 560 00:33:50,070 --> 00:33:52,830 and j1. 561 00:33:52,830 --> 00:33:55,410 So here's i1. 562 00:33:55,410 --> 00:33:57,050 It's unit long. 563 00:33:57,050 --> 00:33:59,730 Here is the angle theta. 564 00:33:59,730 --> 00:34:06,000 Here's j1 and the angles. 565 00:34:06,000 --> 00:34:12,360 And I want to know-- this is i1 and this is j1-- 566 00:34:12,360 --> 00:34:15,780 can I express i1 and j1 in terms of capital 567 00:34:15,780 --> 00:34:18,400 I, capital J, the unit vectors in the base frame? 568 00:34:24,960 --> 00:34:27,780 I want to express them in terms of unit vectors 569 00:34:27,780 --> 00:34:31,500 that are in this rigid, non-moving, non-rotating, 570 00:34:31,500 --> 00:34:34,230 inertial frame. 571 00:34:34,230 --> 00:34:39,449 So down here, this is the i hat direction 572 00:34:39,449 --> 00:34:45,280 and this is the j hat direction, right-- not moving. 573 00:34:45,280 --> 00:34:48,139 So this is just a unit thing, unit long. 574 00:34:48,139 --> 00:34:51,760 Can I project it onto its i component and capital J 575 00:34:51,760 --> 00:34:52,449 component? 576 00:34:52,449 --> 00:34:53,170 All right. 577 00:34:53,170 --> 00:34:59,630 So i1, it looks to me like cosine theta I 578 00:34:59,630 --> 00:35:05,715 hat plus sine theta J hat. 579 00:35:05,715 --> 00:35:06,370 Do you agree? 580 00:35:10,524 --> 00:35:13,650 Just standard trick, right? 581 00:35:13,650 --> 00:35:16,665 And this one, takes me a minute to figure this out. 582 00:35:19,960 --> 00:35:23,200 Which is the theta here? 583 00:35:23,200 --> 00:35:24,280 This is theta. 584 00:35:24,280 --> 00:35:25,910 That's 90 minus theta. 585 00:35:25,910 --> 00:35:27,195 So this must be theta, right? 586 00:35:30,116 --> 00:35:32,110 There's a theta here. 587 00:35:32,110 --> 00:35:35,430 And if this is unit long, what's that? 588 00:35:35,430 --> 00:35:43,350 That projection there is-- so j1 has two components, 589 00:35:43,350 --> 00:35:54,540 minus sine theta I plus cosine theta J. 590 00:35:54,540 --> 00:35:57,611 And I highly recommend you write that one down. 591 00:35:57,611 --> 00:35:59,110 Make sure you can drive it yourself. 592 00:35:59,110 --> 00:36:01,610 You're going to need it again and again and again and again. 593 00:36:06,450 --> 00:36:09,890 Now, could we do the same thing for-- could we 594 00:36:09,890 --> 00:36:17,230 convert J2 to the base frame? 595 00:36:24,030 --> 00:36:28,520 And it rotates, so this x2 can be at any arbitrary position. 596 00:36:28,520 --> 00:36:32,360 But in order to do the problem, you have to pick a position. 597 00:36:32,360 --> 00:36:35,930 And then you'd have to do draw an angle. 598 00:36:35,930 --> 00:36:40,570 And then you'd have to apply this formula. 599 00:36:40,570 --> 00:36:44,790 And so you're going to end up with an i2 600 00:36:44,790 --> 00:36:53,630 and some cosine phi capital I plus sine phi capital 601 00:36:53,630 --> 00:37:04,450 J. And the same thing, j2 is minus sine phi i plus cosine 602 00:37:04,450 --> 00:37:07,280 theta J. 603 00:37:07,280 --> 00:37:12,870 So we'll do a trivial example, solve a trivial case. 604 00:37:12,870 --> 00:37:15,540 What is the instantaneous velocity 605 00:37:15,540 --> 00:37:20,950 at the moment that the coordinate system is lined up 606 00:37:20,950 --> 00:37:23,580 as we see, and B is sitting right here? 607 00:37:33,430 --> 00:37:39,070 So we've got to go look at our answer. 608 00:37:39,070 --> 00:37:41,300 Where was our final answer? 609 00:37:41,300 --> 00:37:47,075 Velocity of this guy here, right? 610 00:37:50,520 --> 00:37:54,335 What would be the contribution of this term? 611 00:37:57,060 --> 00:37:58,800 We have to take each term and convert it 612 00:37:58,800 --> 00:38:03,910 to the base system and capital IJ terms, right? 613 00:38:03,910 --> 00:38:06,640 You do it one term at a time and add up the components. 614 00:38:06,640 --> 00:38:11,570 So how do you break this one down and put it into capital I, 615 00:38:11,570 --> 00:38:14,484 capital J components? 616 00:38:14,484 --> 00:38:15,858 AUDIENCE: Substitute? 617 00:38:15,858 --> 00:38:17,740 PROFESSOR: Yeah, what's the answer? 618 00:38:23,180 --> 00:38:25,310 So j, if it's lined up like this, 619 00:38:25,310 --> 00:38:27,744 j2 is importing in what direction? 620 00:38:27,744 --> 00:38:28,550 Up. 621 00:38:28,550 --> 00:38:31,240 And what is that in this system? 622 00:38:31,240 --> 00:38:34,470 Just capital J. 623 00:38:34,470 --> 00:38:36,030 At this instant in time, that's just 624 00:38:36,030 --> 00:38:42,080 capital J. Trivial calculation, because this angle 625 00:38:42,080 --> 00:38:43,950 is 90 degrees. 626 00:38:43,950 --> 00:38:48,130 Plug in 90 degrees, this term goes to 0, this term goes to 1. 627 00:38:48,130 --> 00:38:52,310 J2 is capital J. 628 00:38:52,310 --> 00:38:57,620 And what about the other term, J1? 629 00:38:57,620 --> 00:39:00,982 You just got to-- it's J1, right? 630 00:39:00,982 --> 00:39:02,440 So you just gotta go with the flow. 631 00:39:02,440 --> 00:39:03,220 It is is. 632 00:39:03,220 --> 00:39:07,530 You'd substitute this in for J1 right here, 633 00:39:07,530 --> 00:39:10,680 and you'd have R1 omega 1 cosine theta 634 00:39:10,680 --> 00:39:16,100 sine theta and j and k terms, plus this thing, capital J. 635 00:39:16,100 --> 00:39:18,790 And you have just converted the answer, 636 00:39:18,790 --> 00:39:22,230 which was in terms of unit vectors in rotating 637 00:39:22,230 --> 00:39:24,530 to different rotating frames. 638 00:39:24,530 --> 00:39:26,730 You've converted it all down to the base frame. 639 00:39:31,730 --> 00:39:32,640 AUDIENCE: [INAUDIBLE] 640 00:39:38,120 --> 00:39:39,350 PROFESSOR: Oops, I'm sorry. 641 00:39:39,350 --> 00:39:40,530 I just made a mistake. 642 00:39:40,530 --> 00:39:42,352 You guys got to get better catching me. 643 00:39:45,390 --> 00:39:47,780 That now make sense? 644 00:39:47,780 --> 00:39:51,710 So phi is the angle that the j2 unit vector makes 645 00:39:51,710 --> 00:39:55,190 with the inertial frame, right? 646 00:39:55,190 --> 00:40:00,182 And theta is the angle that the j1 or i1 647 00:40:00,182 --> 00:40:01,390 make with the inertial frame. 648 00:40:01,390 --> 00:40:01,890 Yes? 649 00:40:01,890 --> 00:40:05,150 AUDIENCE: Phi is 0, though, right? 650 00:40:05,150 --> 00:40:12,410 PROFESSOR: In this case, phi is 0. 651 00:40:12,410 --> 00:40:13,950 Does that still work out over there? 652 00:40:13,950 --> 00:40:18,812 Sine of phi is 0 and cosine of 0 is one and you get j. 653 00:40:18,812 --> 00:40:23,250 AUDIENCE: Then why did we didn't plug in anything for j? 654 00:40:23,250 --> 00:40:24,340 PROFESSOR: We did. 655 00:40:27,440 --> 00:40:30,040 There isn't a simple answer for it. 656 00:40:30,040 --> 00:40:32,070 And so you have to use the full expression. 657 00:40:32,070 --> 00:40:35,810 I just got lazy and didn't want to write it out. 658 00:40:35,810 --> 00:40:36,880 The answer is this. 659 00:40:40,010 --> 00:40:42,770 Stick in 30 degrees if you want, and then you'll get numbers. 660 00:40:47,130 --> 00:40:50,150 So real important point that we discovered 661 00:40:50,150 --> 00:40:56,040 is that the answers are correct, expressed in rotating unit 662 00:40:56,040 --> 00:40:59,860 vectors, expressed in different unit vectors, 663 00:40:59,860 --> 00:41:01,210 different rotating ones. 664 00:41:01,210 --> 00:41:04,240 This is correct, because you can take this 665 00:41:04,240 --> 00:41:08,090 and you can reduce it down to the base frame. 666 00:41:08,090 --> 00:41:12,060 So you will be-- usually in problems that you're given, 667 00:41:12,060 --> 00:41:15,290 you'll be asked to express the answer in terms of unit 668 00:41:15,290 --> 00:41:16,434 vectors in the base frame. 669 00:41:16,434 --> 00:41:17,850 Or you'll be told you can leave it 670 00:41:17,850 --> 00:41:22,300 in whatever is your comfortable set of unit vectors. 671 00:41:22,300 --> 00:41:25,610 Most of the time, the first ones you'll arrive at 672 00:41:25,610 --> 00:41:27,885 are the ones in terms of the rotating coordinates that 673 00:41:27,885 --> 00:41:29,530 are easier to use. 674 00:41:29,530 --> 00:41:31,990 The more natural answer falls out in terms of these. 675 00:41:35,220 --> 00:41:36,330 Good. 676 00:41:36,330 --> 00:41:40,560 All right, and we've got three, four minutes left. 677 00:41:40,560 --> 00:41:43,730 What have I confused you with here? 678 00:41:43,730 --> 00:41:51,920 So key concepts-- what have we-- what 679 00:41:51,920 --> 00:41:55,820 hasn't been clear or maybe we didn't cover it 680 00:41:55,820 --> 00:41:58,502 yet-- another point. 681 00:41:58,502 --> 00:42:00,090 AUDIENCE: So the reason we chose those 682 00:42:00,090 --> 00:42:04,462 as the starting reference frames instead of I hats and theta 683 00:42:04,462 --> 00:42:05,127 hats? 684 00:42:05,127 --> 00:42:07,210 PROFESSOR: Only because at the beginning of class, 685 00:42:07,210 --> 00:42:09,410 we talked about it-- which frames do we want to use, 686 00:42:09,410 --> 00:42:11,290 and then we chose those. 687 00:42:11,290 --> 00:42:15,630 Could we have used a polar coordinate system 688 00:42:15,630 --> 00:42:18,390 to do this problem? 689 00:42:18,390 --> 00:42:19,790 Sure. 690 00:42:19,790 --> 00:42:23,705 Twice You do it once in each-- 691 00:42:23,705 --> 00:42:25,425 AUDIENCE: Is there a way to know up 692 00:42:25,425 --> 00:42:32,855 front which one would simplify down to the inertial i hats 693 00:42:32,855 --> 00:42:34,316 and j hats more simply? 694 00:42:34,316 --> 00:42:35,795 PROFESSOR: The easiest way? 695 00:42:35,795 --> 00:42:37,261 Is there a way to know upfront? 696 00:42:37,261 --> 00:42:37,760 No. 697 00:42:37,760 --> 00:42:39,850 That's just experience. 698 00:42:39,850 --> 00:42:43,790 Work lots of problems, and you get good at picking frame. 699 00:42:43,790 --> 00:42:45,945 We can probably, with time as we meet and talk 700 00:42:45,945 --> 00:42:47,320 about these things, we'll come up 701 00:42:47,320 --> 00:42:51,330 with some sort of general insights about how to do that. 702 00:42:51,330 --> 00:42:52,325 Yes? 703 00:42:52,325 --> 00:42:54,750 AUDIENCE: Is this picture up in parentheses 704 00:42:54,750 --> 00:42:58,145 supposed to be those coordinate systems? 705 00:42:58,145 --> 00:43:01,990 PROFESSOR: This picture is the coordinate system 706 00:43:01,990 --> 00:43:03,110 of that first arm. 707 00:43:03,110 --> 00:43:05,860 AUDIENCE: OK, so is that supposed to be phi up there? 708 00:43:05,860 --> 00:43:06,623 PROFESSOR: Yeah. 709 00:43:09,780 --> 00:43:11,640 Wait a minute. 710 00:43:11,640 --> 00:43:12,490 No. 711 00:43:12,490 --> 00:43:14,050 This is the first arm. 712 00:43:14,050 --> 00:43:17,750 That is theta and these are ones, right? 713 00:43:20,920 --> 00:43:22,705 The 2 system would be phis 714 00:43:22,705 --> 00:43:23,463 AUDIENCE: OK. 715 00:43:23,463 --> 00:43:26,004 I was just wondering if that was the ride or if that was not. 716 00:43:26,004 --> 00:43:26,920 PROFESSOR: No. 717 00:43:26,920 --> 00:43:31,711 This is point A, if you will. 718 00:43:31,711 --> 00:43:32,460 Well, it could be. 719 00:43:32,460 --> 00:43:36,050 It's lined up with point A. This is A. 720 00:43:36,050 --> 00:43:39,050 AUDIENCE: Because I thought we decided that the phi was-- 721 00:43:39,050 --> 00:43:40,800 PROFESSOR: This is point A, right? 722 00:43:40,800 --> 00:43:44,590 That is point A. And this is arm CA. 723 00:43:44,590 --> 00:43:48,852 AUDIENCE: So is this one here the origin of this one? 724 00:43:48,852 --> 00:43:51,540 PROFESSOR: Well, look at whatever the unit vector is. 725 00:43:51,540 --> 00:43:56,200 The unit vector in this system is lined up with that arm. 726 00:43:56,200 --> 00:44:00,300 So this is just a breakdown of these unit vectors 727 00:44:00,300 --> 00:44:02,110 so I could draw the angles and figure out 728 00:44:02,110 --> 00:44:03,570 the sines and cosines. 729 00:44:03,570 --> 00:44:07,730 You could draw a similar picture for i2 j2's. 730 00:44:07,730 --> 00:44:10,600 And then it would be phi's. 731 00:44:10,600 --> 00:44:13,110 Good question. 732 00:44:13,110 --> 00:44:13,610 Yes? 733 00:44:13,610 --> 00:44:15,150 AUDIENCE: So, since you can choose 734 00:44:15,150 --> 00:44:17,010 between Cartesian and polar coordinates, 735 00:44:17,010 --> 00:44:20,400 could you set one in Cartesian, one in polar, 736 00:44:20,400 --> 00:44:25,350 you can mix and match it or-- is that beneficial 737 00:44:25,350 --> 00:44:26,835 in some problems? 738 00:44:26,835 --> 00:44:34,450 PROFESSOR: Polar is-- I don't have time to show you today. 739 00:44:34,450 --> 00:44:39,680 But for planar motion problems, which 740 00:44:39,680 --> 00:44:44,170 are things confined to a plane, they rotate, axis of rotation's 741 00:44:44,170 --> 00:44:46,770 always in the k direction, which is all the problems that you 742 00:44:46,770 --> 00:44:48,115 ever did in 801 Physics. 743 00:44:48,115 --> 00:44:50,990 You didn't do general things actually. 744 00:44:50,990 --> 00:44:56,340 But for planar motion problems, cylindrical coordinates, 745 00:44:56,340 --> 00:45:00,250 actually, you still need the k to describe the rotation, 746 00:45:00,250 --> 00:45:01,300 right? 747 00:45:01,300 --> 00:45:03,150 Polar coordinates, cylindrical coordinates 748 00:45:03,150 --> 00:45:05,480 are oftentimes really convenient. 749 00:45:05,480 --> 00:45:08,290 And they're easy to use because you've learned them 750 00:45:08,290 --> 00:45:09,450 a long, long time ago. 751 00:45:09,450 --> 00:45:11,890 And you know the relations. 752 00:45:11,890 --> 00:45:17,439 But you can make it a rotating x1, y1, z1 rotating system 753 00:45:17,439 --> 00:45:18,480 and it will all work out. 754 00:45:21,700 --> 00:45:24,440 We came up with this little formula here, right? 755 00:45:24,440 --> 00:45:27,565 This could just as easily have been r hat. 756 00:45:30,360 --> 00:45:35,600 And this could have just as easily been no difference 757 00:45:35,600 --> 00:45:38,860 whatsoever in a planar motion problem, when 758 00:45:38,860 --> 00:45:42,190 you attach an xy system that rotates with it, 759 00:45:42,190 --> 00:45:43,970 or I call it r and theta. 760 00:45:43,970 --> 00:45:45,430 These are the same direction. 761 00:45:45,430 --> 00:45:46,820 R is in the direction of i1. 762 00:45:46,820 --> 00:45:48,390 Theta is in the direction of j1. 763 00:45:51,270 --> 00:45:53,540 So use it when it's convenient, and it's convenient 764 00:45:53,540 --> 00:45:57,620 a lot of times, especially that nasty acceleration formula. 765 00:45:57,620 --> 00:45:59,570 In polar coordinates, it reduces down just 766 00:45:59,570 --> 00:46:01,620 to the set of five terms. 767 00:46:01,620 --> 00:46:06,330 Memorize it and just tick them off-- Coriolis, centripetal. 768 00:46:06,330 --> 00:46:07,440 You see them right away. 769 00:46:07,440 --> 00:46:08,570 You know what they are. 770 00:46:08,570 --> 00:46:13,180 But there are certain problems, even in planar motion problems 771 00:46:13,180 --> 00:46:15,610 that polar coordinates don't work for-- doesn't work for. 772 00:46:15,610 --> 00:46:16,110 [INAUDIBLE] 773 00:46:20,150 --> 00:46:21,810 And think about that. 774 00:46:21,810 --> 00:46:23,280 It's actually a simple problem. 775 00:46:23,280 --> 00:46:24,640 Put a dog on a marry-go-round. 776 00:46:24,640 --> 00:46:26,420 The dog's running in a random direction 777 00:46:26,420 --> 00:46:27,372 on the merry-go-round. 778 00:46:27,372 --> 00:46:29,330 And the merry-go-round is turning at some rate. 779 00:46:29,330 --> 00:46:32,460 And you only want one rotating coordinate system, r and theta. 780 00:46:32,460 --> 00:46:35,090 You can't do the problem with polar coordinates. 781 00:46:35,090 --> 00:46:35,930 Think about it. 782 00:46:35,930 --> 00:46:38,560 Go away, think about why not. 783 00:46:38,560 --> 00:46:40,410 I'll tell you the answer in words. 784 00:46:40,410 --> 00:46:41,320 You go figure it out. 785 00:46:41,320 --> 00:46:45,080 You can't describe the velocity of the dog 786 00:46:45,080 --> 00:46:46,203 in polar coordinates. 787 00:46:51,420 --> 00:46:52,660 The dog is running around. 788 00:46:52,660 --> 00:46:54,930 If the dog's fixed on the rotating thing, than polar 789 00:46:54,930 --> 00:46:55,650 coordinates work. 790 00:46:55,650 --> 00:46:57,691 If the dog's running, you can't do that velocity. 791 00:46:57,691 --> 00:47:02,155 So you need a more sophisticated coordinate system.