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 --> 00:00:10,140 continue to offer high quality educational resources for free. 5 00:00:10,140 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:16,590 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,590 --> 00:00:17,297 at ocw.mit.edu. 8 00:00:21,610 --> 00:00:24,360 PROFESSOR: I'm very interested in teaching in ways 9 00:00:24,360 --> 00:00:26,420 that really help you learn. 10 00:00:26,420 --> 00:00:33,640 And one of the most difficult things to learn 11 00:00:33,640 --> 00:00:37,960 is when you have to unlearn something you believe. 12 00:00:37,960 --> 00:00:40,250 So if you have learned a concept wrong 13 00:00:40,250 --> 00:00:43,630 in your previous education and learnings, 14 00:00:43,630 --> 00:00:46,973 your mind is really reluctant to let go of it. 15 00:00:46,973 --> 00:00:47,960 OK? 16 00:00:47,960 --> 00:00:52,140 And I have a wife who is a middle school science teacher. 17 00:00:52,140 --> 00:00:53,860 She also has a PhD in biology. 18 00:00:53,860 --> 00:00:56,750 But she really studies to understand kids 19 00:00:56,750 --> 00:00:58,050 and how they learn. 20 00:00:58,050 --> 00:01:00,500 And MIT students aren't much different from 6th graders, 21 00:01:00,500 --> 00:01:02,541 to tell you the truth, when it comes to learning. 22 00:01:02,541 --> 00:01:04,144 None of us are. 23 00:01:04,144 --> 00:01:05,935 But they're just a little more transparent. 24 00:01:09,558 --> 00:01:13,430 What you can do to help a person unlearn something they've 25 00:01:13,430 --> 00:01:15,960 learned wrong, the best time is when 26 00:01:15,960 --> 00:01:18,980 you run into something called a "discrepant event." 27 00:01:18,980 --> 00:01:20,685 So when you think you know the answer, 28 00:01:20,685 --> 00:01:23,850 and I'm asking you these questions, and you're wrong, 29 00:01:23,850 --> 00:01:25,730 really try to figure out what it is 30 00:01:25,730 --> 00:01:28,250 you believe that led you to that, 31 00:01:28,250 --> 00:01:31,372 and see if you can undo that in getting to the answer. 32 00:01:31,372 --> 00:01:32,830 So some of these concept questions, 33 00:01:32,830 --> 00:01:34,440 if you'd like to talk about them further, 34 00:01:34,440 --> 00:01:36,800 I think they're great for discussions in your recitation 35 00:01:36,800 --> 00:01:37,300 section. 36 00:01:37,300 --> 00:01:38,940 Look at Professor Gossard here. 37 00:01:38,940 --> 00:01:41,620 So talk about these things, and really try 38 00:01:41,620 --> 00:01:44,690 to get to the bottom of why you perhaps 39 00:01:44,690 --> 00:01:49,040 had a concept that wasn't quite up to what you needed. 40 00:01:49,040 --> 00:01:50,080 OK. 41 00:01:50,080 --> 00:01:53,670 We're going to move on. 42 00:01:53,670 --> 00:01:54,910 OK. 43 00:01:54,910 --> 00:01:57,850 Now we're going to get on to talking about velocities 44 00:01:57,850 --> 00:01:59,290 in rotating frames. 45 00:01:59,290 --> 00:02:02,690 And by velocities, I also mean derivatives 46 00:02:02,690 --> 00:02:04,320 of vectors in rotating frames. 47 00:02:22,230 --> 00:02:24,800 Just to give you a quick reminder of where we started 48 00:02:24,800 --> 00:02:27,900 this conversation, this is concept 49 00:02:27,900 --> 00:02:36,320 if you have three points-- A, B, and O in this fixed frame. 50 00:02:36,320 --> 00:02:44,240 This is r A with respect to O, B with respect to A, 51 00:02:44,240 --> 00:02:47,550 and B with respect to O, all vectors. 52 00:02:47,550 --> 00:02:49,900 And we're interested in computing 53 00:02:49,900 --> 00:02:54,150 the velocities of, say, point B. So we 54 00:02:54,150 --> 00:02:57,880 were talking about the derivative with respect 55 00:02:57,880 --> 00:03:01,650 to time of this vector. 56 00:03:01,650 --> 00:03:06,360 And we can make it up as a sum of the derivatives with respect 57 00:03:06,360 --> 00:03:07,930 to time of these other two vectors. 58 00:03:19,420 --> 00:03:21,370 But now there's a really important point. 59 00:03:21,370 --> 00:03:22,870 When you're taking time derivatives, 60 00:03:22,870 --> 00:03:26,450 you have to be explicit about what 61 00:03:26,450 --> 00:03:28,765 it's with respect to-- what frame of reference 62 00:03:28,765 --> 00:03:29,640 you're talking about. 63 00:03:29,640 --> 00:03:33,580 In this case we want to know the velocity with respect 64 00:03:33,580 --> 00:03:35,110 to this reference frame. 65 00:03:35,110 --> 00:03:37,520 So both of these derivatives have 66 00:03:37,520 --> 00:03:40,640 to be taken with respect to that frame. 67 00:03:40,640 --> 00:03:42,780 Well this one's easy. 68 00:03:42,780 --> 00:03:46,000 This is purely translational. 69 00:03:46,000 --> 00:03:48,980 It has nothing to do with rotations, or won't. 70 00:03:48,980 --> 00:03:53,910 We're going to think in terms of having an object out here. 71 00:03:53,910 --> 00:03:58,320 So maybe a rigid body to which we have attached a rotating 72 00:03:58,320 --> 00:04:02,660 frame at A. And I'll call this the rotating reference 73 00:04:02,660 --> 00:04:05,200 frame A x prime y prime. 74 00:04:05,200 --> 00:04:06,260 And it might be rotating. 75 00:04:06,260 --> 00:04:10,000 But this point is just the point with which we describe 76 00:04:10,000 --> 00:04:11,520 the translation of the body. 77 00:04:11,520 --> 00:04:13,150 So this one's pretty easy. 78 00:04:13,150 --> 00:04:16,029 This one's just the velocity of A with respect to O. 79 00:04:16,029 --> 00:04:21,800 But this one has complications. 80 00:04:21,800 --> 00:04:24,210 And I'm going to do it first, the answer to this, just 81 00:04:24,210 --> 00:04:27,110 intuitively-- appeal to your intuition 82 00:04:27,110 --> 00:04:29,710 about why this should be. 83 00:04:29,710 --> 00:04:33,910 So this is my merry-go-round. 84 00:04:33,910 --> 00:04:36,180 And I've set up my little coordinate system. 85 00:04:36,180 --> 00:04:39,200 This is my A, x prime y prime coordinate system. 86 00:04:39,200 --> 00:04:43,000 I've stuck my dog here on the x-axis. 87 00:04:43,000 --> 00:04:49,280 And if he's running some direction, 88 00:04:49,280 --> 00:04:53,550 he has a velocity with respect to this reference frame. 89 00:04:53,550 --> 00:04:56,210 If you're here at A, watch it, and you'll see him moving. 90 00:04:56,210 --> 00:04:59,400 And it will have nothing to do with the rotation. 91 00:04:59,400 --> 00:05:01,410 You will just see the dog moving. 92 00:05:01,410 --> 00:05:05,260 And if this is rotating, and you're sitting there rotating 93 00:05:05,260 --> 00:05:09,650 with it watching the dog, does the motion of the dog change? 94 00:05:09,650 --> 00:05:11,670 You just see it just the same way, right? 95 00:05:11,670 --> 00:05:12,260 OK. 96 00:05:12,260 --> 00:05:16,340 So there's one part of the motion of the dog that's 97 00:05:16,340 --> 00:05:18,340 with respect to this frame that has 98 00:05:18,340 --> 00:05:20,540 nothing to do with rotation. 99 00:05:20,540 --> 00:05:21,250 OK. 100 00:05:21,250 --> 00:05:27,050 So now let's say in an outside, and from an outside point 101 00:05:27,050 --> 00:05:31,800 of view, if we measure the velocity 102 00:05:31,800 --> 00:05:34,620 of the dog with respect to point A-- that's 103 00:05:34,620 --> 00:05:39,250 the one you can see-- and to observer outside says 104 00:05:39,250 --> 00:05:42,580 what's the velocity of dog with respect to A, 105 00:05:42,580 --> 00:05:48,275 will the answer differ if there's rotation? 106 00:05:52,190 --> 00:05:53,360 OK. 107 00:05:53,360 --> 00:05:55,590 Now, what we're going to find out 108 00:05:55,590 --> 00:05:58,580 is that there's actually two contributions. 109 00:05:58,580 --> 00:06:01,840 So the dog running with respect to this merry-go-round 110 00:06:01,840 --> 00:06:04,930 is one contribution to the velocity. 111 00:06:04,930 --> 00:06:07,410 There's another contribution that comes from the fact 112 00:06:07,410 --> 00:06:09,700 that you're sitting out there watching, 113 00:06:09,700 --> 00:06:12,180 and you see definitely the dog move when the thing rotates, 114 00:06:12,180 --> 00:06:12,880 right? 115 00:06:12,880 --> 00:06:13,380 OK. 116 00:06:13,380 --> 00:06:16,880 And if we only do that, if the only rotation-- 117 00:06:16,880 --> 00:06:19,430 and we'll do a case where the dog's not running, just 118 00:06:19,430 --> 00:06:21,280 sitting still-- what's the velocity 119 00:06:21,280 --> 00:06:24,165 of the dog with respect to point A due to the rotation? 120 00:06:27,990 --> 00:06:30,820 I mean you've done lots of problems like this. 121 00:06:30,820 --> 00:06:32,470 This is just pure rotation problem. 122 00:06:32,470 --> 00:06:35,980 You know the angular rate-- theta dot, or omega. 123 00:06:35,980 --> 00:06:39,300 What's the velocity of point B if point A is fixed 124 00:06:39,300 --> 00:06:41,730 and this thing rotates? 125 00:06:41,730 --> 00:06:42,780 r omega. 126 00:06:42,780 --> 00:06:43,900 You all know that. 127 00:06:43,900 --> 00:06:44,840 OK. 128 00:06:44,840 --> 00:06:50,540 And then we'll call the velocity of the dog relative to point 129 00:06:50,540 --> 00:06:54,280 A due to the fact that he is moving in this frame, 130 00:06:54,280 --> 00:06:58,040 we'll call that velocity to the dog. 131 00:06:58,040 --> 00:07:01,520 From the point of view of in the frame is a different quantity. 132 00:07:01,520 --> 00:07:04,622 They're both vectors, and you can add them together. 133 00:07:04,622 --> 00:07:06,205 So in fact the answer-- I'm just going 134 00:07:06,205 --> 00:07:09,720 to give you the answer to this problem-- the answer to this 135 00:07:09,720 --> 00:07:14,860 is the velocity of A with respect to O 136 00:07:14,860 --> 00:07:24,010 plus the derivative of r B/A with respect to time as seen 137 00:07:24,010 --> 00:07:29,660 from within the A xyz prime frame, 138 00:07:29,660 --> 00:07:32,710 plus a term here due to rotation. 139 00:07:35,540 --> 00:07:41,640 And we know that the magnitude of this answer is some r omega. 140 00:07:41,640 --> 00:07:43,045 We know it looks like r omega. 141 00:07:45,890 --> 00:07:48,760 So to interpret this as meaning the derivative 142 00:07:48,760 --> 00:07:55,500 of this vector-- this thing-- as if you were in the frame. 143 00:07:55,500 --> 00:07:57,450 And that accounts for the dog running. 144 00:07:57,450 --> 00:07:58,850 This is just the dog running. 145 00:08:05,350 --> 00:08:09,360 And mathematically, I actually find it easier 146 00:08:09,360 --> 00:08:12,310 to say what this piece means is this 147 00:08:12,310 --> 00:08:15,180 is the velocity of the dog with respect 148 00:08:15,180 --> 00:08:20,960 to A with omega equal to 0. 149 00:08:20,960 --> 00:08:21,870 It's the same thing. 150 00:08:21,870 --> 00:08:25,490 This is what you see if there's no rotation. 151 00:08:25,490 --> 00:08:27,100 That piece. 152 00:08:27,100 --> 00:08:28,980 And this is the piece with rotation. 153 00:08:28,980 --> 00:08:33,350 But I want you to think something through here. 154 00:08:38,500 --> 00:08:39,760 Come on. 155 00:08:39,760 --> 00:08:42,419 I'm still learning how to do this. 156 00:08:42,419 --> 00:08:42,919 OK. 157 00:08:46,830 --> 00:08:49,590 So I have my little reference frames like this. 158 00:08:49,590 --> 00:08:53,620 I'm going to rotate it in this direction, which given xyz 159 00:08:53,620 --> 00:08:55,570 ought to be in that direction. 160 00:08:55,570 --> 00:08:57,680 So the rotation rate, if it's going like this, 161 00:08:57,680 --> 00:09:00,100 I'm going to give it a constant value, 162 00:09:00,100 --> 00:09:05,620 is that the vector omega is some omega in the k hat direction, 163 00:09:05,620 --> 00:09:07,270 right? 164 00:09:07,270 --> 00:09:16,540 And I have on this body-- here's my wheel, 165 00:09:16,540 --> 00:09:23,770 and here's B, and here's A. And this is my little x-axis, 166 00:09:23,770 --> 00:09:25,230 my x-prime axis. 167 00:09:25,230 --> 00:09:31,240 And it has associated with it a lowercase i hat unit vector. 168 00:09:31,240 --> 00:09:32,910 And off in this direction then would 169 00:09:32,910 --> 00:09:37,860 be the y prime with a j hat unit vector. 170 00:09:37,860 --> 00:09:38,700 OK. 171 00:09:38,700 --> 00:09:42,340 So what's the vector r B/A? 172 00:09:42,340 --> 00:09:46,290 How would you write it in vector notation in components here? 173 00:09:49,750 --> 00:09:54,820 So this is r B/A from here to here. 174 00:09:54,820 --> 00:09:57,313 And what's its direction? 175 00:09:57,313 --> 00:09:59,240 AUDIENCE: i hat? 176 00:09:59,240 --> 00:10:00,249 PROFESSOR: i hat. 177 00:10:00,249 --> 00:10:01,540 And is it positive or negative? 178 00:10:06,180 --> 00:10:09,110 So this is the vector. 179 00:10:09,110 --> 00:10:15,210 At some magnitude r B/A in the positive i hat direction, 180 00:10:15,210 --> 00:10:17,860 the arrow goes, when you say r B/A, 181 00:10:17,860 --> 00:10:22,280 this is B with respect to A, the arrow goes with B. 182 00:10:22,280 --> 00:10:25,485 So this, it's positive r B/A i hat. 183 00:10:25,485 --> 00:10:29,690 And what is the velocity that we have discovered up here, 184 00:10:29,690 --> 00:10:32,350 the velocity of B with respect to A as 185 00:10:32,350 --> 00:10:35,980 measured inside of A xyz? 186 00:10:35,980 --> 00:10:39,180 That's this piece here. 187 00:10:39,180 --> 00:10:41,777 In vector terms? 188 00:10:41,777 --> 00:10:43,610 And we've figured out what its velocity was. 189 00:10:43,610 --> 00:10:45,320 It's some r omega. 190 00:10:45,320 --> 00:10:50,320 The r's r B/A-- the magnitude, the length of that r. 191 00:10:50,320 --> 00:10:54,040 Clearly got to be this length right here. 192 00:10:54,040 --> 00:10:57,800 And the omega is given. 193 00:10:57,800 --> 00:10:59,180 So what direction? 194 00:10:59,180 --> 00:11:00,896 You know that the magnitudes are omega. 195 00:11:00,896 --> 00:11:01,770 What's the direction? 196 00:11:07,764 --> 00:11:08,730 AUDIENCE: j hat? 197 00:11:08,730 --> 00:11:09,438 PROFESSOR: j hat. 198 00:11:09,438 --> 00:11:10,470 OK. 199 00:11:10,470 --> 00:11:13,360 So what we're saying is the velocity 200 00:11:13,360 --> 00:11:21,160 of B with respect to A as seen from inside this A frame is 201 00:11:21,160 --> 00:11:31,780 r B/A omega with respect to O in the j hat direction. 202 00:11:31,780 --> 00:11:35,996 Now this has a k associated with it. 203 00:11:35,996 --> 00:11:37,370 That has an i associated with it. 204 00:11:37,370 --> 00:11:39,550 And this has a j associated with it. 205 00:11:39,550 --> 00:11:43,164 So as a product r omega, what kind of product is it? 206 00:11:43,164 --> 00:11:44,330 What you know about vectors? 207 00:11:44,330 --> 00:11:45,630 What does it have to be? 208 00:11:45,630 --> 00:11:49,160 Is it a dot product between omega and r? 209 00:11:49,160 --> 00:11:51,242 Is it a cross product? 210 00:11:51,242 --> 00:11:52,960 I hear somebody saying cross product. 211 00:11:52,960 --> 00:11:53,880 But what's the order? 212 00:11:53,880 --> 00:11:56,380 So I want you to figure out-- just deduce 213 00:11:56,380 --> 00:11:59,230 what the vector notation is that gets 214 00:11:59,230 --> 00:12:01,749 you the right velocity-- the correct velocity-- 215 00:12:01,749 --> 00:12:03,415 correct magnitude and correct direction. 216 00:12:06,760 --> 00:12:11,150 So the answer is positive r omega j. 217 00:12:16,070 --> 00:12:17,550 AUDIENCE: Omega cross r? 218 00:12:17,550 --> 00:12:18,510 PROFESSOR: Omega. 219 00:12:18,510 --> 00:12:19,850 AUDIENCE: Cross to the r? 220 00:12:19,850 --> 00:12:20,891 PROFESSOR: Omega cross r. 221 00:12:20,891 --> 00:12:29,660 So I hear one person, omega with respect to O cross r B/A. 222 00:12:29,660 --> 00:12:32,220 And these are now both vectors. 223 00:12:32,220 --> 00:12:33,390 Anybody else? 224 00:12:33,390 --> 00:12:34,989 So if we just think about unit vector, 225 00:12:34,989 --> 00:12:36,030 actually, let's check it. 226 00:12:36,030 --> 00:12:37,880 This is k hat direction. 227 00:12:37,880 --> 00:12:40,170 This was i hat direction. 228 00:12:40,170 --> 00:12:44,330 And k cross i is positive j. 229 00:12:44,330 --> 00:12:45,015 So that works. 230 00:12:47,800 --> 00:12:51,980 So in general-- this is, in fact, the correct answer 231 00:12:51,980 --> 00:13:01,370 in general-- is that this term here is omega with respect 232 00:13:01,370 --> 00:13:09,260 to O cross r B/A. You're going to use that a lot. 233 00:13:09,260 --> 00:13:20,500 So to summarize, then-- while that's coming down, 234 00:13:20,500 --> 00:13:30,620 the velocity of this dog in a stationary frame where 235 00:13:30,620 --> 00:13:35,150 A is stationary for a moment is entirely due to the rotation, 236 00:13:35,150 --> 00:13:39,320 and it's omega cross r B/A. And now if the dog starts 237 00:13:39,320 --> 00:13:43,470 to run with respect to this reference frame, 238 00:13:43,470 --> 00:13:46,520 he has some velocity, which is the other piece 239 00:13:46,520 --> 00:13:50,400 is velocity of B with respect to A as seen from A-- 240 00:13:50,400 --> 00:13:52,610 or as you would compute, you'd just set momentarily, 241 00:13:52,610 --> 00:13:54,380 omega equal to zero. 242 00:13:54,380 --> 00:13:58,840 And finally then, the total formula for velocity. 243 00:13:58,840 --> 00:14:02,720 So the velocity of B with respect 244 00:14:02,720 --> 00:14:07,280 to O-- some point on a rotating reference frame-- 245 00:14:07,280 --> 00:14:15,696 is the velocity of the frame plus-- 246 00:14:15,696 --> 00:14:18,330 I won't write it as a derivative now. 247 00:14:18,330 --> 00:14:23,610 Plus the velocity of the point, but as 248 00:14:23,610 --> 00:14:29,180 seen from the point of view of somebody in the frame, 249 00:14:29,180 --> 00:14:37,780 plus omega with respect to O cross r B/A. 250 00:14:37,780 --> 00:14:40,000 So this is an important formula. 251 00:14:40,000 --> 00:14:43,370 You're going to use it a lot. 252 00:14:43,370 --> 00:14:45,640 Couple of subtle points in here. 253 00:14:45,640 --> 00:14:49,490 This one-- this thing from as seen from inside the frame. 254 00:14:49,490 --> 00:14:55,870 It's also the partial derivative of r B/A with respect 255 00:14:55,870 --> 00:15:00,570 to t when omega with respect to O equals zero. 256 00:15:00,570 --> 00:15:04,460 That's another way to say this. 257 00:15:04,460 --> 00:15:07,980 And this term is another subtlety. 258 00:15:07,980 --> 00:15:12,907 I was careful to write with respect to O. Yes? 259 00:15:12,907 --> 00:15:17,786 AUDIENCE: To make sure I'm not confused here on this vector 260 00:15:17,786 --> 00:15:18,286 right there. 261 00:15:21,709 --> 00:15:23,665 PROFESSOR: Yes. 262 00:15:23,665 --> 00:15:27,088 AUDIENCE: That leftmost term is the middle one there? 263 00:15:27,088 --> 00:15:28,570 Or the right one? 264 00:15:28,570 --> 00:15:30,810 PROFESSOR: So this was when we were working 265 00:15:30,810 --> 00:15:37,710 with just the velocity due to the motion of the dog 266 00:15:37,710 --> 00:15:40,490 due to rotation plus-- actually this 267 00:15:40,490 --> 00:15:44,066 was rotation-- what did I say? 268 00:15:44,066 --> 00:15:47,482 AUDIENCE: [INAUDIBLE] experiments. 269 00:15:47,482 --> 00:15:50,166 That the term on the left is the motion 270 00:15:50,166 --> 00:15:52,298 that you can see from the frame. 271 00:15:52,298 --> 00:15:53,923 From the right is the rotation that you 272 00:15:53,923 --> 00:15:56,270 don't see in the [INAUDIBLE], but needs to be [INAUDIBLE]. 273 00:15:56,270 --> 00:15:58,080 PROFESSOR: And I can't remember how I got to this point. 274 00:15:58,080 --> 00:15:59,710 But I think I was making a couple of points, 275 00:15:59,710 --> 00:16:00,860 and rammed them together. 276 00:16:00,860 --> 00:16:02,480 So this is the correct. 277 00:16:02,480 --> 00:16:04,690 Don't focus on this. 278 00:16:04,690 --> 00:16:07,410 This is the summary. 279 00:16:07,410 --> 00:16:12,380 This thing is now rotating, and translating, 280 00:16:12,380 --> 00:16:15,220 and the dog's running. 281 00:16:15,220 --> 00:16:17,400 And I want you to be able to write down 282 00:16:17,400 --> 00:16:20,590 the total velocity of the point B 283 00:16:20,590 --> 00:16:27,400 as seen from a fixed reference frame fixed on the ground. 284 00:16:27,400 --> 00:16:29,600 This is with respect to O. 285 00:16:29,600 --> 00:16:32,400 It has three terms-- the velocity 286 00:16:32,400 --> 00:16:40,280 of A, the velocity of the dog relative to the body 287 00:16:40,280 --> 00:16:43,180 it's on here, and finally, the component 288 00:16:43,180 --> 00:16:45,180 of the velocity-- contribution of the velocity 289 00:16:45,180 --> 00:16:49,630 that you see that's due to the rotation. 290 00:16:49,630 --> 00:16:54,360 Now this turns out to be a really powerful formula. 291 00:16:54,360 --> 00:16:55,560 It's a generalization. 292 00:16:55,560 --> 00:17:03,880 It's a special case of a very general formula. 293 00:17:03,880 --> 00:17:07,740 Let me catch up in my notes here. 294 00:17:07,740 --> 00:17:11,099 Because the following statement is true. 295 00:17:11,099 --> 00:17:35,950 So the time derivative of any vector 296 00:17:35,950 --> 00:17:44,080 that's defined in a rotating frame 297 00:17:44,080 --> 00:17:51,100 is given by-- and we'll call it just some vector 298 00:17:51,100 --> 00:17:54,095 A. It might be, for example, angular momentum. 299 00:17:54,095 --> 00:17:56,590 In fact we'll use this formula a lot 300 00:17:56,590 --> 00:17:58,940 when talking about angular momentum. 301 00:17:58,940 --> 00:18:04,825 The time derivative of this vector A-- 302 00:18:04,825 --> 00:18:09,260 whatever it represents-- position, velocity, 303 00:18:09,260 --> 00:18:17,130 angular momentum-- as seen from a reference frame xy-- O xyz, 304 00:18:17,130 --> 00:18:27,850 reference frame O-- is made up of the derivative of A from 305 00:18:27,850 --> 00:18:31,316 inside the frame plus-- 306 00:18:42,460 --> 00:18:45,150 So it's the same thing. 307 00:18:45,150 --> 00:18:49,720 This was the derivative of r B/A. Has two pieces-- the piece 308 00:18:49,720 --> 00:18:51,980 that came from the dog running and the piece that 309 00:18:51,980 --> 00:18:53,690 comes from rotation. 310 00:18:53,690 --> 00:18:56,680 Any vector, this piece is true. 311 00:18:56,680 --> 00:18:58,160 It's the derivative of the vector 312 00:18:58,160 --> 00:19:03,590 inside the frame in which it's defined plus omega with respect 313 00:19:03,590 --> 00:19:10,510 to the reference frame you want the derivative in cross A. Very 314 00:19:10,510 --> 00:19:11,390 powerful formula. 315 00:19:21,310 --> 00:19:25,090 This now is covered in a couple of different places 316 00:19:25,090 --> 00:19:28,140 in the readings, and in the handout that's 317 00:19:28,140 --> 00:19:30,560 posted called "Kinematics." 318 00:19:30,560 --> 00:19:32,570 So you should read by now the portion 319 00:19:32,570 --> 00:19:34,832 of the kinematics thing, at least on velocities. 320 00:19:34,832 --> 00:19:37,060 The second half of that kinematics handout 321 00:19:37,060 --> 00:19:38,160 is about acceleration. 322 00:19:38,160 --> 00:19:39,740 So we're going to use this formula 323 00:19:39,740 --> 00:19:41,560 to go from velocity to acceleration. 324 00:19:41,560 --> 00:19:43,420 I saw a couple hands up. 325 00:19:43,420 --> 00:19:44,383 Yes? 326 00:19:44,383 --> 00:19:47,281 AUDIENCE: So what is the difference between O sub 327 00:19:47,281 --> 00:19:49,213 xyz and A sub xyz? 328 00:19:49,213 --> 00:19:51,050 Where are they [INAUDIBLE]? 329 00:19:51,050 --> 00:19:51,780 PROFESSOR: OK. 330 00:19:51,780 --> 00:19:58,410 So this is a merry-go-round on a train, 331 00:19:58,410 --> 00:19:59,780 and you're looking down on it. 332 00:19:59,780 --> 00:20:02,950 And the train can move. 333 00:20:02,950 --> 00:20:05,550 And we fixed to the ground, not moving, 334 00:20:05,550 --> 00:20:12,030 a reference frame we call O. And I want to know in this case 335 00:20:12,030 --> 00:20:15,280 the velocity of the dog running around 336 00:20:15,280 --> 00:20:20,650 on this merry-go-round with respect to O. 337 00:20:20,650 --> 00:20:24,290 It's the sum of three vector contributions-- 338 00:20:24,290 --> 00:20:28,710 the velocity of the train, the velocity 339 00:20:28,710 --> 00:20:35,990 of the dog with respect to the merry-go-round-- this 340 00:20:35,990 --> 00:20:37,115 is velocity of the train. 341 00:20:40,110 --> 00:20:43,150 This is with respect to the merry-go-round. 342 00:20:43,150 --> 00:20:46,990 And this is the velocity that you see out there. 343 00:20:46,990 --> 00:20:52,620 So you're sitting in O. You are in this fixed frame. 344 00:20:52,620 --> 00:20:56,170 Which brings up another subtle point about velocities 345 00:20:56,170 --> 00:20:59,890 and inertial frames. 346 00:20:59,890 --> 00:21:04,540 If you are at any fixed point in an inertial frame-- 347 00:21:04,540 --> 00:21:06,710 you don't have to be at O; you can be right 348 00:21:06,710 --> 00:21:10,980 where you are-- this term is always the same. 349 00:21:10,980 --> 00:21:14,900 The velocity of this moving object relative to the frame 350 00:21:14,900 --> 00:21:17,540 is the same to any observer in the frame-- 351 00:21:17,540 --> 00:21:20,281 any fixed observer in the frame. 352 00:21:20,281 --> 00:21:20,780 OK? 353 00:21:23,290 --> 00:21:24,395 And finally, the u. 354 00:21:24,395 --> 00:21:27,200 But the u, the part of the contribution 355 00:21:27,200 --> 00:21:29,290 to the velocity due to the rotation of this thing, 356 00:21:29,290 --> 00:21:30,850 you can see it from out there, right? 357 00:21:30,850 --> 00:21:31,810 It moves. 358 00:21:31,810 --> 00:21:35,830 Well, the rotation contribution is this. 359 00:21:35,830 --> 00:21:38,370 The dog running around is this. 360 00:21:38,370 --> 00:21:42,300 And the movement of the train is the first term. 361 00:21:42,300 --> 00:21:44,240 OK? 362 00:21:44,240 --> 00:21:44,780 All right. 363 00:21:44,780 --> 00:21:47,620 And this is the general formula for the derivative 364 00:21:47,620 --> 00:21:54,280 of a vector in a translating, rotating frame. 365 00:21:54,280 --> 00:21:54,810 Yeah? 366 00:21:54,810 --> 00:22:00,605 AUDIENCE: [INAUDIBLE] vectors beyond the [INAUDIBLE] v 367 00:22:00,605 --> 00:22:02,858 of p with respect to the [INAUDIBLE]. 368 00:22:02,858 --> 00:22:04,316 PROFESSOR: I'm not following, here. 369 00:22:04,316 --> 00:22:07,232 AUDIENCE: Are any of the v, v of A, 370 00:22:07,232 --> 00:22:10,148 for example, or v of B with respect to A, are they 371 00:22:10,148 --> 00:22:11,245 all vectors? 372 00:22:11,245 --> 00:22:12,620 PROFESSOR: These are all vectors. 373 00:22:12,620 --> 00:22:15,190 And I've gotten a little careless 374 00:22:15,190 --> 00:22:18,340 about drawing my underlines when I haven't broken them down 375 00:22:18,340 --> 00:22:19,160 into components. 376 00:22:19,160 --> 00:22:20,590 This is a vector. 377 00:22:20,590 --> 00:22:22,400 And we can do this, because we're 378 00:22:22,400 --> 00:22:26,290 relying on formulas for the sums of vectors. 379 00:22:26,290 --> 00:22:28,260 So it's the vector. 380 00:22:28,260 --> 00:22:31,120 There's a vector describing the velocity of the train. 381 00:22:31,120 --> 00:22:34,590 For example, if it's moving in the capital I hat direction, 382 00:22:34,590 --> 00:22:36,350 you have to say that. 383 00:22:36,350 --> 00:22:41,240 And this may actually be the dog. 384 00:22:41,240 --> 00:22:45,180 At the moment you catch him, the frame's like this. 385 00:22:45,180 --> 00:22:51,070 And let's say he's running in the y direction. 386 00:22:51,070 --> 00:22:52,840 His direction at that instant in time 387 00:22:52,840 --> 00:22:56,730 is in the lowercase j hat direction 388 00:22:56,730 --> 00:22:58,420 relative to this frame, right? 389 00:22:58,420 --> 00:23:01,610 And that j hat direction has some angle 390 00:23:01,610 --> 00:23:04,040 with respect to this frame, which 391 00:23:04,040 --> 00:23:09,320 if you want to reduce these velocities down 392 00:23:09,320 --> 00:23:13,130 to the unit vectors in the inertial frame, 393 00:23:13,130 --> 00:23:13,930 you can do that. 394 00:23:13,930 --> 00:23:15,388 But you're going to have to account 395 00:23:15,388 --> 00:23:19,930 for the angles of this reference frame compared to that one. 396 00:23:19,930 --> 00:23:22,900 So cosine thetas and sine thetas and that kind of thing. 397 00:23:25,740 --> 00:23:34,440 One important point about this general expression 398 00:23:34,440 --> 00:23:38,290 for the derivative of a vector in a rotating frame, which 399 00:23:38,290 --> 00:23:41,250 therefore applies to this case, because this is just 400 00:23:41,250 --> 00:23:44,970 the specific example of derivative of a position 401 00:23:44,970 --> 00:23:47,420 vector giving you velocity. 402 00:23:47,420 --> 00:23:54,130 This term and this term-- these account 403 00:23:54,130 --> 00:23:56,190 for the change in length of the vector. 404 00:23:59,910 --> 00:24:03,100 So just to keep it real, in terms of velocities, 405 00:24:03,100 --> 00:24:05,530 you see a velocity that's due to the fact 406 00:24:05,530 --> 00:24:09,245 that the velocity vector itself is changing in length. 407 00:24:12,660 --> 00:24:15,540 Or in this case, the r vector is changing in length, 408 00:24:15,540 --> 00:24:18,670 so the velocity has something to do with the change in length. 409 00:24:18,670 --> 00:24:23,190 This term is due only to the fact that it is rotating. 410 00:24:23,190 --> 00:24:26,280 If the dog's not running, is his position vector 411 00:24:26,280 --> 00:24:28,790 changing in length? 412 00:24:28,790 --> 00:24:30,220 No. 413 00:24:30,220 --> 00:24:32,370 So this term here comes from taking 414 00:24:32,370 --> 00:24:35,140 this derivative of the position vector 415 00:24:35,140 --> 00:24:37,330 within the rotating frame. 416 00:24:37,330 --> 00:24:40,670 So this accounts for the change in length of that position 417 00:24:40,670 --> 00:24:42,570 vector. 418 00:24:42,570 --> 00:24:44,700 This accounts for the rotation. 419 00:24:44,700 --> 00:24:50,915 This term is the same as seen from in any frame. 420 00:24:50,915 --> 00:24:51,790 This is the subtlety. 421 00:24:54,900 --> 00:24:59,030 This term, you get the same answer if you see it from here. 422 00:24:59,030 --> 00:25:00,870 This velocity, this contribution. 423 00:25:00,870 --> 00:25:07,040 The dog's speed with respect to this frame it's in 424 00:25:07,040 --> 00:25:09,370 is the same to you out there in the fixed frame 425 00:25:09,370 --> 00:25:11,240 as it is to somebody sitting here 426 00:25:11,240 --> 00:25:14,460 in the frame moving with him. 427 00:25:14,460 --> 00:25:15,305 Important point. 428 00:25:19,110 --> 00:25:21,640 It'll become more important as you do problems. 429 00:25:21,640 --> 00:25:23,999 You need to remember that doesn't matter 430 00:25:23,999 --> 00:25:25,040 where you're calculating. 431 00:25:25,040 --> 00:25:27,440 You're given the velocity of the dog 432 00:25:27,440 --> 00:25:30,960 in this frame, and little i j k components, 433 00:25:30,960 --> 00:25:34,810 it is in magnitude exactly the same velocity as you would 434 00:25:34,810 --> 00:25:37,784 see it in the fixed frame. 435 00:25:37,784 --> 00:25:39,950 But then you just have to account for the right unit 436 00:25:39,950 --> 00:25:41,190 vectors and so forth. 437 00:25:41,190 --> 00:25:42,062 Yes. 438 00:25:42,062 --> 00:25:44,894 AUDIENCE: [INAUDIBLE] is the same as the frame, 439 00:25:44,894 --> 00:25:47,249 but the direction is changing? 440 00:25:47,249 --> 00:25:49,540 PROFESSOR: The direction is the same in any frame also. 441 00:25:49,540 --> 00:25:52,410 It's just that you have to decide which unit 442 00:25:52,410 --> 00:25:54,772 vectors to represent it in. 443 00:25:54,772 --> 00:25:56,230 And when you're in this frame, it's 444 00:25:56,230 --> 00:26:00,495 easiest to represent it in little i j k, right? 445 00:26:00,495 --> 00:26:01,870 But if you're in the fixed frame, 446 00:26:01,870 --> 00:26:05,450 you may want to actually know the velocity in the fixed 447 00:26:05,450 --> 00:26:08,584 frames unit vectors, and there's just geometric conversions 448 00:26:08,584 --> 00:26:09,500 that'll give you that. 449 00:26:09,500 --> 00:26:10,460 Yes. 450 00:26:10,460 --> 00:26:13,350 AUDIENCE: But if theta is like a function of time [INAUDIBLE] 451 00:26:13,350 --> 00:26:15,270 how would you use those? 452 00:26:15,270 --> 00:26:17,190 Does it just work the same way? 453 00:26:17,190 --> 00:26:20,320 PROFESSOR: So she says if theta is a function of time, 454 00:26:20,320 --> 00:26:24,220 if you were trying to convert-- this dog's running in the y 455 00:26:24,220 --> 00:26:28,430 direction, which is going in the master frame over there, 456 00:26:28,430 --> 00:26:31,410 is going to have a component in the negative capital I hat 457 00:26:31,410 --> 00:26:34,480 and positive capital J hat direction, right? 458 00:26:34,480 --> 00:26:38,020 And you could break down-- let's say his velocity vector is 459 00:26:38,020 --> 00:26:42,140 like this-- you could break it down into a component that's 460 00:26:42,140 --> 00:26:43,840 parallel to the ground, and a component 461 00:26:43,840 --> 00:26:45,470 perpendicular to the ground. 462 00:26:45,470 --> 00:26:47,630 And those then would be in that reference frame. 463 00:26:51,347 --> 00:26:53,430 You've all, I would think, have done some problems 464 00:26:53,430 --> 00:26:55,690 using polar coordinates. 465 00:26:55,690 --> 00:26:59,390 And the conversion between x and y and r and theta 466 00:26:59,390 --> 00:27:01,820 is you used cosine thetas and sine 467 00:27:01,820 --> 00:27:03,400 thetas to compute it, right? 468 00:27:03,400 --> 00:27:08,700 So read that handout on kinematics, 469 00:27:08,700 --> 00:27:16,940 because it proves this formula by doing just that. 470 00:27:16,940 --> 00:27:18,890 It puts everything in terms of thetas 471 00:27:18,890 --> 00:27:21,370 that are functions of time. 472 00:27:21,370 --> 00:27:23,580 And it's kind of a long, painful process, 473 00:27:23,580 --> 00:27:26,100 but you can do that, grind it out, 474 00:27:26,100 --> 00:27:28,520 and you will then, when you assemble the terms, 475 00:27:28,520 --> 00:27:29,980 you end up with this. 476 00:27:29,980 --> 00:27:31,760 And once you know this formula, it's 477 00:27:31,760 --> 00:27:36,720 far, far easier than to calculate 478 00:27:36,720 --> 00:27:38,889 all this in terms of thetas and theta dots, 479 00:27:38,889 --> 00:27:39,805 and sines and cosines. 480 00:27:44,730 --> 00:27:45,230 OK? 481 00:27:47,961 --> 00:27:48,460 All right. 482 00:27:48,460 --> 00:27:49,000 That's good. 483 00:27:49,000 --> 00:27:50,050 That's important stuff. 484 00:27:50,050 --> 00:27:52,997 We're going to make great use of these formulas 485 00:27:52,997 --> 00:27:54,080 over the rest of the term. 486 00:28:02,980 --> 00:28:06,660 So I'm going to just take a break from kinematics. 487 00:28:06,660 --> 00:28:09,480 It's time to talk a little bit about Newton's laws, 488 00:28:09,480 --> 00:28:12,010 because we really want to be able to do dynamics problems. 489 00:28:12,010 --> 00:28:14,780 And so this has been about describing the motion. 490 00:28:14,780 --> 00:28:19,124 And now we want to talk about the laws on which we depend 491 00:28:19,124 --> 00:28:20,790 in order to be able to calculate things, 492 00:28:20,790 --> 00:28:24,600 and to draw, write equations of motion and so forth. 493 00:28:24,600 --> 00:28:26,780 So any final questions about kinematics? 494 00:28:26,780 --> 00:28:29,520 Because I'm going to change topics. 495 00:28:29,520 --> 00:28:30,454 Yes. 496 00:28:30,454 --> 00:28:34,406 AUDIENCE: So for that equation to the [INAUDIBLE], then 497 00:28:34,406 --> 00:28:37,534 the acceleration of [INAUDIBLE] with respect 498 00:28:37,534 --> 00:28:39,840 to the reference frame, that's [INAUDIBLE] 499 00:28:39,840 --> 00:28:41,779 is just zero, right? 500 00:28:41,779 --> 00:28:43,820 PROFESSOR: You're asking about acceleration, now? 501 00:28:43,820 --> 00:28:44,710 Do you really mean? 502 00:28:44,710 --> 00:28:46,130 Is that what you mean? 503 00:28:46,130 --> 00:28:48,515 Because I haven't talked much about acceleration. 504 00:28:48,515 --> 00:28:51,490 AUDIENCE: I was just wondering, because also [INAUDIBLE] x 505 00:28:51,490 --> 00:28:53,367 to the velocity change. 506 00:28:53,367 --> 00:28:54,200 PROFESSOR: This one? 507 00:28:54,200 --> 00:28:54,825 AUDIENCE: Yeah. 508 00:28:54,825 --> 00:28:59,960 PROFESSOR: So this equation is true of any vector. 509 00:28:59,960 --> 00:29:02,240 This is basically vector calculus. 510 00:29:02,240 --> 00:29:04,910 It's true of any vector in a rotating frame. 511 00:29:04,910 --> 00:29:08,130 It's time derivative with respect to a fixed frame. 512 00:29:08,130 --> 00:29:10,880 It is the derivative you would see in the rotating 513 00:29:10,880 --> 00:29:14,760 frame, which is just the change of length of the vector, 514 00:29:14,760 --> 00:29:17,640 and the part due to the fact that you are making it rotate. 515 00:29:17,640 --> 00:29:21,490 And we'll do examples of this, I promise you, very soon. 516 00:29:21,490 --> 00:29:22,840 So you'll see how it's applied. 517 00:29:22,840 --> 00:29:23,340 OK? 518 00:29:26,540 --> 00:29:27,660 OK. 519 00:29:27,660 --> 00:29:30,700 So let's do a quick review of Newton. 520 00:29:50,940 --> 00:29:51,770 So Newton's Laws. 521 00:29:51,770 --> 00:29:54,270 He had three of them. 522 00:29:54,270 --> 00:29:56,900 And we're all pretty familiar with the second. 523 00:29:56,900 --> 00:29:59,515 The first, though, is called the law of inertia. 524 00:30:05,760 --> 00:30:07,780 And basically, the first law says, 525 00:30:07,780 --> 00:30:11,360 if an object is motionless in the absence of forces, 526 00:30:11,360 --> 00:30:12,030 what happens? 527 00:30:16,480 --> 00:30:17,410 Stays motionless. 528 00:30:17,410 --> 00:30:19,200 Or, if it's at some constant velocity 529 00:30:19,200 --> 00:30:24,340 in the absence of forces it stays at constant velocity, 530 00:30:24,340 --> 00:30:25,370 right? 531 00:30:25,370 --> 00:30:30,500 So these have been stated many ways over the years. 532 00:30:30,500 --> 00:30:33,450 I'm just going to try to come up with short ones for the board. 533 00:30:33,450 --> 00:31:05,000 So in the absence of forces an object-- 534 00:31:05,000 --> 00:31:07,580 a particle-- moves with constant velocity. 535 00:31:07,580 --> 00:31:10,740 In fact, Newton only talked about motions of particles, 536 00:31:10,740 --> 00:31:13,290 not as in little tiny things. 537 00:31:13,290 --> 00:31:17,160 Not about rigid bodies that have finite dimensions. 538 00:31:17,160 --> 00:31:21,810 He thought of the planets, appropriately, as particles. 539 00:31:21,810 --> 00:31:23,030 Second law. 540 00:31:23,030 --> 00:31:24,710 Second law, I won't write it all out. 541 00:31:24,710 --> 00:31:26,890 But basically F equals ma. 542 00:31:26,890 --> 00:31:30,350 Their vectors equals the time derivative 543 00:31:30,350 --> 00:31:31,547 of the linear momentum. 544 00:31:31,547 --> 00:31:33,130 That's what we know as the second law. 545 00:31:33,130 --> 00:31:38,030 The sum of all the external forces 546 00:31:38,030 --> 00:31:40,610 equals the mass times the acceleration. 547 00:31:40,610 --> 00:31:43,460 Another statement of the second law. 548 00:31:43,460 --> 00:31:44,646 OK? 549 00:31:44,646 --> 00:31:45,520 So you know that one. 550 00:31:45,520 --> 00:31:47,410 That's the one you're most familiar with. 551 00:31:47,410 --> 00:31:48,200 Third one. 552 00:31:48,200 --> 00:31:50,006 What's the third one? 553 00:31:50,006 --> 00:31:52,910 AUDIENCE: Every action has an equal and opposite reaction. 554 00:31:52,910 --> 00:31:56,660 PROFESSOR: So every action has an equal and opposite reaction. 555 00:31:56,660 --> 00:32:00,510 And I'm going to draw a picture for this one. 556 00:32:00,510 --> 00:32:02,800 So here's a particle. 557 00:32:02,800 --> 00:32:05,020 Here's another particle. 558 00:32:05,020 --> 00:32:07,070 This is particle 2. 559 00:32:07,070 --> 00:32:09,120 Particle 1. 560 00:32:09,120 --> 00:32:12,070 There's a force on this particle that's 561 00:32:12,070 --> 00:32:16,920 the force on 2 due to the presence of 1. 562 00:32:16,920 --> 00:32:20,000 They've each got a little gravity, little attraction 563 00:32:20,000 --> 00:32:20,740 from one another. 564 00:32:20,740 --> 00:32:27,540 And this one has a force on particle 1 due to particle 2. 565 00:32:27,540 --> 00:32:28,230 OK? 566 00:32:28,230 --> 00:32:29,313 They could be two planets. 567 00:32:32,590 --> 00:32:38,570 And basically what he said is that for every action, 568 00:32:38,570 --> 00:32:41,010 there's an equal and opposite reaction. 569 00:32:41,010 --> 00:32:47,210 That means that f 21 equals minus f 12. 570 00:32:50,120 --> 00:32:53,035 And this is called the strong form 571 00:32:53,035 --> 00:33:01,450 of Newton's third law, which the forces are equal, 572 00:33:01,450 --> 00:33:04,870 opposite, and collinear. 573 00:33:08,060 --> 00:33:12,160 They actually point exactly opposite one another. 574 00:33:12,160 --> 00:33:14,470 And this is true of mechanical systems. 575 00:33:14,470 --> 00:33:19,850 You get some rather interesting subtleties 576 00:33:19,850 --> 00:33:21,970 when you get into electromagnetic fields, 577 00:33:21,970 --> 00:33:23,897 and charged particles, and things like that. 578 00:33:23,897 --> 00:33:25,480 So you have to think really carefully. 579 00:33:25,480 --> 00:33:29,250 But for the mechanical systems, the strong form 580 00:33:29,250 --> 00:33:32,110 will suit us just fine. 581 00:33:32,110 --> 00:33:33,339 OK. 582 00:33:33,339 --> 00:33:34,380 Those are the three laws. 583 00:33:34,380 --> 00:33:41,540 Newton made one major condition for those to be true. 584 00:33:41,540 --> 00:33:45,690 What is the assumption that must be satisfied for these laws 585 00:33:45,690 --> 00:33:48,094 to be true? 586 00:33:48,094 --> 00:33:50,830 AUDIENCE: [INAUDIBLE] an inertial reference frame? 587 00:33:50,830 --> 00:33:55,190 PROFESSOR: I hear "an inertial reference frame," right? 588 00:33:55,190 --> 00:33:57,770 And that's what he assumed. 589 00:33:57,770 --> 00:34:00,990 You have to be in an inertial frame for these statements 590 00:34:00,990 --> 00:34:02,110 to be true. 591 00:34:02,110 --> 00:34:05,510 So then what I want to spend a few minutes talking about 592 00:34:05,510 --> 00:34:09,845 is basically what's an inertial frame. 593 00:34:09,845 --> 00:34:11,219 Because that's going to be really 594 00:34:11,219 --> 00:34:12,480 important in this subject. 595 00:34:12,480 --> 00:34:15,639 When you start getting on things that move and rotate, 596 00:34:15,639 --> 00:34:18,980 things sometimes are not inertial. 597 00:34:18,980 --> 00:34:21,639 So I'm going to ask you a quick question. 598 00:34:21,639 --> 00:34:24,090 So a reference frame. 599 00:34:24,090 --> 00:34:27,771 So we had our fixed frame up there, sitting on the ground, 600 00:34:27,771 --> 00:34:28,520 here on the Earth. 601 00:34:28,520 --> 00:34:32,139 Is the Earth an inertial reference frame, standing 602 00:34:32,139 --> 00:34:33,360 here observing things? 603 00:34:37,520 --> 00:34:38,540 I want a show of hands. 604 00:34:38,540 --> 00:34:39,820 I really want some participation here. 605 00:34:39,820 --> 00:34:41,940 How many of you think Earth's an inertial frame? 606 00:34:41,940 --> 00:34:42,845 Raise your hand. 607 00:34:42,845 --> 00:34:44,550 It's true. 608 00:34:44,550 --> 00:34:47,580 How many think it's not true? 609 00:34:47,580 --> 00:34:50,000 How many think it depends? 610 00:34:50,000 --> 00:34:52,250 On what? 611 00:34:52,250 --> 00:34:55,250 AUDIENCE: What you are focusing on. 612 00:34:55,250 --> 00:34:57,080 PROFESSOR: What the problem is. 613 00:34:57,080 --> 00:34:59,740 He says it depends on what you're focusing on. 614 00:34:59,740 --> 00:35:04,310 So it really depends on the sizes of the forces 615 00:35:04,310 --> 00:35:07,430 and the motion you're interested in. 616 00:35:07,430 --> 00:35:10,890 Can you give me an example in which the Earth cannot be 617 00:35:10,890 --> 00:35:14,730 assumed to be an inertial frame? 618 00:35:14,730 --> 00:35:16,240 Practical example? 619 00:35:16,240 --> 00:35:17,740 AUDIENCE: Oh. 620 00:35:17,740 --> 00:35:18,890 Not practical. 621 00:35:18,890 --> 00:35:20,874 PROFESSOR: Impractical. 622 00:35:20,874 --> 00:35:23,850 AUDIENCE: The rotation of Earth is slowing? 623 00:35:23,850 --> 00:35:26,167 The rotation of the Earth is slowing? 624 00:35:26,167 --> 00:35:28,000 PROFESSOR: Rotation of the earth is slowing. 625 00:35:28,000 --> 00:35:29,850 That's interesting. 626 00:35:29,850 --> 00:35:32,900 To account for that, you would definitely not be able to just 627 00:35:32,900 --> 00:35:34,370 to assume we're inertial. 628 00:35:34,370 --> 00:35:37,710 But what's an everyday example of where, 629 00:35:37,710 --> 00:35:40,954 if you're trying to solve a problem in this field, 630 00:35:40,954 --> 00:35:42,745 you couldn't make this inertial assumption? 631 00:35:45,896 --> 00:35:47,820 AUDIENCE: Rotation of the planets? 632 00:35:47,820 --> 00:35:49,700 PROFESSOR: Well, no, for the most part, 633 00:35:49,700 --> 00:35:50,870 you could get most of it. 634 00:35:50,870 --> 00:35:52,267 But, yeah. 635 00:35:52,267 --> 00:35:53,850 AUDIENCE: I think the weather, really, 636 00:35:53,850 --> 00:35:56,175 because the wind currents, or like the ocean 637 00:35:56,175 --> 00:35:58,970 currents are affected by the fact that the Earth is round. 638 00:35:58,970 --> 00:36:00,350 PROFESSOR: OK. 639 00:36:00,350 --> 00:36:01,800 That's a good one. 640 00:36:01,800 --> 00:36:05,480 In order to account for the circulation 641 00:36:05,480 --> 00:36:07,920 you have to take into account the Earth's motion. 642 00:36:07,920 --> 00:36:10,125 Let me give you an example. 643 00:36:10,125 --> 00:36:10,625 Clocks. 644 00:36:14,720 --> 00:36:18,200 Pendulum clocks. 645 00:36:18,200 --> 00:36:19,850 Does the speed of a clock change? 646 00:36:19,850 --> 00:36:24,550 Is it different at high noon from midnight? 647 00:36:29,691 --> 00:36:30,440 What do you think? 648 00:36:30,440 --> 00:36:31,930 Yes or no? 649 00:36:31,930 --> 00:36:34,080 How many think that the actual speed 650 00:36:34,080 --> 00:36:35,950 of a clock-- the length of a second-- 651 00:36:35,950 --> 00:36:38,220 would be different at noon on the Earth-- 652 00:36:38,220 --> 00:36:40,274 a pendulum clock-- from midnight? 653 00:36:40,274 --> 00:36:41,690 How many think that might be true? 654 00:36:41,690 --> 00:36:42,880 It's different? 655 00:36:42,880 --> 00:36:45,270 How many don't believe that it would be true? 656 00:36:45,270 --> 00:36:47,140 How many just not raising their hands? 657 00:36:47,140 --> 00:36:48,190 Come on you guys. 658 00:36:48,190 --> 00:36:49,130 Let's get with it. 659 00:36:49,130 --> 00:36:49,629 OK. 660 00:36:52,600 --> 00:36:55,590 So the effective acceleration of gravity 661 00:36:55,590 --> 00:36:58,800 that that pendulum feels is different at noon 662 00:36:58,800 --> 00:37:01,420 from midnight. 663 00:37:01,420 --> 00:37:07,590 And one reason is because at noon the sun's pulling away 664 00:37:07,590 --> 00:37:09,890 from the surface of the Earth, and at midnight 665 00:37:09,890 --> 00:37:12,664 the sun's gravity is pulling in the same direction 666 00:37:12,664 --> 00:37:14,080 as toward the center of the Earth. 667 00:37:14,080 --> 00:37:15,990 The total effective gravitational 668 00:37:15,990 --> 00:37:18,410 pull that the pendulum field changes 669 00:37:18,410 --> 00:37:21,489 due to the rotation of the Earth, what it feels. 670 00:37:21,489 --> 00:37:23,030 The sum of the forces on the pendulum 671 00:37:23,030 --> 00:37:25,200 includes the Earth's gravity and the sun's gravity. 672 00:37:25,200 --> 00:37:26,790 And the moon also does this. 673 00:37:26,790 --> 00:37:29,750 So there's daily variations in the speed 674 00:37:29,750 --> 00:37:33,160 of pendulum clocks just because of the Earth's 675 00:37:33,160 --> 00:37:35,640 rotation with respect to the sun and the moon. 676 00:37:35,640 --> 00:37:36,471 Yeah. 677 00:37:36,471 --> 00:37:39,560 AUDIENCE: Is that even great enough to be measurable? 678 00:37:39,560 --> 00:37:41,890 PROFESSOR: So she says, "Is that even great enough 679 00:37:41,890 --> 00:37:42,710 to be measurable?" 680 00:37:42,710 --> 00:37:44,160 So I have a friend, a guy named Hugh Hunt. 681 00:37:44,160 --> 00:37:45,950 He's a professor that teaches dynamics 682 00:37:45,950 --> 00:37:47,310 at Cambridge University. 683 00:37:47,310 --> 00:37:52,010 And he is the keeper of the Trinity College clock. 684 00:37:52,010 --> 00:37:53,990 Trinity College is where Newton was. 685 00:37:53,990 --> 00:37:56,620 So he took me to his clock one day up in this tower. 686 00:37:56,620 --> 00:37:58,580 And it's got about a two-meter pendulum on it. 687 00:37:58,580 --> 00:38:00,720 And he has got that clock to run so 688 00:38:00,720 --> 00:38:06,260 that it gains no more than one second per month. 689 00:38:06,260 --> 00:38:07,910 He's really tuned it up carefully. 690 00:38:07,910 --> 00:38:10,630 And the key to being able to do that 691 00:38:10,630 --> 00:38:15,672 is to do things on monthly averages. 692 00:38:15,672 --> 00:38:18,110 So he's got it tuned so that over a month 693 00:38:18,110 --> 00:38:20,540 it just barely gains a tiny bit. 694 00:38:20,540 --> 00:38:22,370 But if you measure it very carefully 695 00:38:22,370 --> 00:38:26,320 over the course of the day, it has amazingly large 696 00:38:26,320 --> 00:38:28,320 fluctuations. 697 00:38:28,320 --> 00:38:30,830 One of them is due to the thing I just described. 698 00:38:30,830 --> 00:38:34,460 So if you're really trying to keep close time, it matters. 699 00:38:34,460 --> 00:38:34,960 OK. 700 00:38:34,960 --> 00:38:37,176 Another one is gunnery. 701 00:38:37,176 --> 00:38:38,670 You're shooting long range. 702 00:38:38,670 --> 00:38:39,920 You're trying to hit a target. 703 00:38:39,920 --> 00:38:41,750 The fact that the earth rotates, you 704 00:38:41,750 --> 00:38:44,640 will not hit the target if you don't 705 00:38:44,640 --> 00:38:48,545 account for the effects that are caused by the Earth's rotation. 706 00:38:48,545 --> 00:38:49,920 And it's one of the first reasons 707 00:38:49,920 --> 00:38:52,997 that people got into understanding 708 00:38:52,997 --> 00:38:55,330 the importance of whether or not it's an inertial frame, 709 00:38:55,330 --> 00:38:59,430 was gunnery in the old days, from naval ships and so forth. 710 00:38:59,430 --> 00:39:00,100 OK. 711 00:39:00,100 --> 00:39:01,724 So we have three laws. 712 00:39:01,724 --> 00:39:04,140 I want to talk a little bit about the first and the third. 713 00:39:04,140 --> 00:39:05,940 We're going to use the second a lot. 714 00:39:05,940 --> 00:39:09,015 So the first law, most people think of the first law 715 00:39:09,015 --> 00:39:12,770 as being a special case of the second, right? 716 00:39:12,770 --> 00:39:15,950 It's just when there's no forces, nothing changes. 717 00:39:15,950 --> 00:39:19,780 But I think the first law is useful in its own right. 718 00:39:19,780 --> 00:39:22,720 And one of the reasons why it's called the law of inertia, 719 00:39:22,720 --> 00:39:26,180 it's the law that allows you to do a test 720 00:39:26,180 --> 00:39:27,930 to discover whether or not you're actually 721 00:39:27,930 --> 00:39:30,220 in an inertial frame. 722 00:39:30,220 --> 00:39:31,770 Useful to be able to do that. 723 00:39:31,770 --> 00:39:36,372 So I'm going to give you an example. 724 00:39:36,372 --> 00:39:38,330 And actually, I did want to ask you a question. 725 00:39:38,330 --> 00:39:43,310 So three possible answers to this question. 726 00:39:43,310 --> 00:39:49,730 An inertial frame can-- how to pose this? 727 00:39:49,730 --> 00:39:54,760 If you're in an inertial frame, can it be accelerating? 728 00:39:54,760 --> 00:39:59,480 If you're in an inertial frame, can it be rotating? 729 00:39:59,480 --> 00:40:01,460 Or if you're in an inertial frame, 730 00:40:01,460 --> 00:40:05,760 it can neither accelerate or rotate. 731 00:40:05,760 --> 00:40:08,290 So which of those three answers is the best answer 732 00:40:08,290 --> 00:40:11,940 for conditions to be in an inertial frame? 733 00:40:11,940 --> 00:40:14,650 Non-accelerating, non-rotating, or both? 734 00:40:14,650 --> 00:40:17,240 How many believe non-accelerating? 735 00:40:20,270 --> 00:40:21,230 Just non-accelerating? 736 00:40:21,230 --> 00:40:23,080 How many believe in just not rotating? 737 00:40:23,080 --> 00:40:25,130 How many believe both? 738 00:40:25,130 --> 00:40:26,310 Both is right. 739 00:40:26,310 --> 00:40:29,260 If you are in a frame which is rotating or accelerating 740 00:40:29,260 --> 00:40:31,270 it's not inertial. 741 00:40:31,270 --> 00:40:31,770 OK. 742 00:40:31,770 --> 00:40:34,420 So just rotation causes it not to be [INAUDIBLE]. 743 00:40:34,420 --> 00:40:35,940 So let's test that. 744 00:40:35,940 --> 00:40:40,070 I'm going to pick two cases quickly. 745 00:40:40,070 --> 00:40:44,820 Let's use this law of inertia to set up 746 00:40:44,820 --> 00:40:48,350 a test to see if a couple different frames are 747 00:40:48,350 --> 00:40:49,320 in fact inertial. 748 00:40:49,320 --> 00:40:51,750 So I've got a cart here. 749 00:40:58,580 --> 00:41:01,430 And I'm sitting here-- or you are. 750 00:41:01,430 --> 00:41:04,220 We're sitting on the cart. 751 00:41:04,220 --> 00:41:13,150 And this cart is accelerating. 752 00:41:17,210 --> 00:41:20,970 Acceleration of A with respect to O 753 00:41:20,970 --> 00:41:26,470 is, I'll call it a naught I hat. 754 00:41:26,470 --> 00:41:29,770 This cart's accelerating in that way, that direction, 755 00:41:29,770 --> 00:41:31,540 the positive I hat direction. 756 00:41:31,540 --> 00:41:32,410 OK. 757 00:41:32,410 --> 00:41:37,070 Now I'm sitting here on the cart-- not very sensitive. 758 00:41:37,070 --> 00:41:41,490 And I want to test whether or not I'm in an inertial frame. 759 00:41:41,490 --> 00:41:44,380 So let's pretend this cart, I've got an air table there, 760 00:41:44,380 --> 00:41:45,325 frictionless table. 761 00:41:45,325 --> 00:41:47,330 And I've got a hockey puck. 762 00:41:47,330 --> 00:41:50,730 And I set it down on the frictionless table 763 00:41:50,730 --> 00:41:52,960 and let it go. 764 00:41:52,960 --> 00:41:53,920 What happens? 765 00:41:53,920 --> 00:41:54,880 What do I observe? 766 00:41:57,327 --> 00:41:58,202 AUDIENCE: [INAUDIBLE] 767 00:42:01,650 --> 00:42:05,660 PROFESSOR: So the puck will accelerate, you're 768 00:42:05,660 --> 00:42:08,640 saying, towards me, right? 769 00:42:08,640 --> 00:42:10,835 AUDIENCE: [INAUDIBLE] 770 00:42:10,835 --> 00:42:11,460 PROFESSOR: Hmm. 771 00:42:11,460 --> 00:42:12,600 It'll move. 772 00:42:12,600 --> 00:42:15,490 So she says the puck will move towards you if you let it go. 773 00:42:15,490 --> 00:42:17,750 So is that an indication of whether or not 774 00:42:17,750 --> 00:42:18,950 you're in an inertial frame? 775 00:42:21,550 --> 00:42:23,360 So the test-- this is first law, now. 776 00:42:23,360 --> 00:42:25,690 So the test-- if you're in this frame, 777 00:42:25,690 --> 00:42:28,940 and you want to know whether or not this is an inertial frame, 778 00:42:28,940 --> 00:42:31,730 and I claim you can set this puck out there, 779 00:42:31,730 --> 00:42:36,024 if you're in an inertial frame, what should it do? 780 00:42:36,024 --> 00:42:36,910 Not move. 781 00:42:36,910 --> 00:42:39,830 If it moves, there's something going on. 782 00:42:39,830 --> 00:42:41,370 Something fishy, right? 783 00:42:41,370 --> 00:42:42,210 OK. 784 00:42:42,210 --> 00:42:46,530 So the acceleration-- and we'll call 785 00:42:46,530 --> 00:42:51,250 the puck at B. So the acceleration of B 786 00:42:51,250 --> 00:42:53,320 with respect to O-- and there's no rotation here. 787 00:42:53,320 --> 00:42:56,320 So the acceleration of B with respect to O, 788 00:42:56,320 --> 00:42:58,710 it can be written as the acceleration of A 789 00:42:58,710 --> 00:43:02,250 with respect to O, plus the acceleration of B 790 00:43:02,250 --> 00:43:11,280 with respect to A. 791 00:43:11,280 --> 00:43:14,990 Now this is O here. 792 00:43:14,990 --> 00:43:16,547 This is an inertial frame. 793 00:43:19,530 --> 00:43:22,390 And if it's an inertial frame, and there 794 00:43:22,390 --> 00:43:26,670 are no forces in the x direction acting on this puck, 795 00:43:26,670 --> 00:43:29,260 then I'm going to say that the sum of the forces on that puck 796 00:43:29,260 --> 00:43:32,100 equal zero. 797 00:43:32,100 --> 00:43:35,000 And therefore, what can I say about the acceleration 798 00:43:35,000 --> 00:43:37,440 of the puck? 799 00:43:37,440 --> 00:43:41,589 What's the acceleration of the puck as seen from O? 800 00:43:41,589 --> 00:43:42,380 It's gotta be zero. 801 00:43:42,380 --> 00:43:47,390 If no forces-- so the summation of the forces with respect 802 00:43:47,390 --> 00:43:51,190 to this O frame in the x direction, 803 00:43:51,190 --> 00:43:54,830 if there's zero, that implies the acceleration of B 804 00:43:54,830 --> 00:43:56,770 with respect to O is got to be zero. 805 00:44:01,730 --> 00:44:06,730 Knowing that, I can now solve for the acceleration of B 806 00:44:06,730 --> 00:44:09,610 with respect to A. And that's going 807 00:44:09,610 --> 00:44:13,315 to be minus the acceleration of A with respect to O. 808 00:44:13,315 --> 00:44:18,425 And that's minus a naught I hat. 809 00:44:23,070 --> 00:44:27,420 So you are correct in saying that it moves. 810 00:44:27,420 --> 00:44:31,390 But it actually accelerates. 811 00:44:31,390 --> 00:44:35,354 From your point of view, what you see sitting there in A, 812 00:44:35,354 --> 00:44:36,895 you were going to see this accelerate 813 00:44:36,895 --> 00:44:38,310 in the opposite direction. 814 00:44:38,310 --> 00:44:40,080 And that's a dead giveaway that you're not 815 00:44:40,080 --> 00:44:41,890 in an inertial frame. 816 00:44:41,890 --> 00:44:42,390 OK. 817 00:44:50,380 --> 00:44:53,910 Come on. 818 00:44:53,910 --> 00:44:55,270 OK. 819 00:44:55,270 --> 00:44:56,460 So a little harder problem. 820 00:45:05,550 --> 00:45:08,863 Now we've got our merry-go-round. 821 00:45:08,863 --> 00:45:10,220 OK. 822 00:45:10,220 --> 00:45:11,220 And it's fixed. 823 00:45:11,220 --> 00:45:12,395 Not on the train. 824 00:45:12,395 --> 00:45:14,220 It's just sitting here. 825 00:45:14,220 --> 00:45:17,160 But it can spin. 826 00:45:17,160 --> 00:45:20,340 And you're sitting at A. So you're 827 00:45:20,340 --> 00:45:24,040 up above this merry-go-round looking down on it. 828 00:45:24,040 --> 00:45:25,937 And it's rotating. 829 00:45:25,937 --> 00:45:27,520 Out there you're in an inertial frame. 830 00:45:27,520 --> 00:45:29,728 But now you come over to here, and you sit right here 831 00:45:29,728 --> 00:45:31,840 at the center. 832 00:45:31,840 --> 00:45:33,390 And if you're right at the center, 833 00:45:33,390 --> 00:45:36,200 you might not actually feel a thing, right? 834 00:45:36,200 --> 00:45:37,820 And there's no windows. 835 00:45:37,820 --> 00:45:43,330 So this could be a spaceship out there, slowly rolling over. 836 00:45:43,330 --> 00:45:46,450 So you're sitting inside of this system, 837 00:45:46,450 --> 00:45:49,200 no windows, right at the center, can't feel a thing. 838 00:45:49,200 --> 00:45:52,350 And I want you to construct a test-- be 839 00:45:52,350 --> 00:45:54,810 using the first law-- that'll tell you whether or not 840 00:45:54,810 --> 00:45:56,350 you're in an inertial frame. 841 00:45:56,350 --> 00:45:57,110 What might you do? 842 00:46:02,185 --> 00:46:04,310 AUDIENCE: Set a ball down on the ground [INAUDIBLE] 843 00:46:04,310 --> 00:46:05,556 roll off to the edge. 844 00:46:05,556 --> 00:46:06,430 PROFESSOR: All right. 845 00:46:06,430 --> 00:46:07,804 He says set a ball on the ground, 846 00:46:07,804 --> 00:46:09,330 and see if it rolls out to the edge. 847 00:46:09,330 --> 00:46:09,830 Right? 848 00:46:09,830 --> 00:46:12,010 See if it moves. 849 00:46:12,010 --> 00:46:14,302 AUDIENCE: Just [INAUDIBLE] is this [INAUDIBLE] 850 00:46:14,302 --> 00:46:16,510 under the effect of gravity, or is there [INAUDIBLE]? 851 00:46:19,350 --> 00:46:24,140 PROFESSOR: Let's really make it a merry-go-round. 852 00:46:24,140 --> 00:46:25,880 So there's gravity. 853 00:46:25,880 --> 00:46:27,770 So I want gravity to be useful here. 854 00:46:27,770 --> 00:46:29,630 It keeps the thing on the surface. 855 00:46:29,630 --> 00:46:31,400 Doesn't just go drifting off. 856 00:46:31,400 --> 00:46:33,760 So yeah, let's say we have gravity. 857 00:46:33,760 --> 00:46:35,260 The axis is vertical. 858 00:46:35,260 --> 00:46:38,430 You're sitting here, but you can't see outside, 859 00:46:38,430 --> 00:46:40,280 and you want to do this test. 860 00:46:40,280 --> 00:46:42,240 So do you agree if you set the ball down, 861 00:46:42,240 --> 00:46:44,030 you might learn something? 862 00:46:44,030 --> 00:46:44,830 OK. 863 00:46:44,830 --> 00:46:48,190 So you set the ball down. 864 00:46:53,990 --> 00:46:55,310 And you're here watching. 865 00:46:55,310 --> 00:46:57,050 And here's the ball. 866 00:46:57,050 --> 00:47:00,030 And let's say you've got a ball, and you actually 867 00:47:00,030 --> 00:47:02,380 have a string on the ball. 868 00:47:02,380 --> 00:47:03,490 Set it out there. 869 00:47:07,480 --> 00:47:09,800 So initially you've got this ball out there. 870 00:47:13,180 --> 00:47:15,710 And you're sitting here at the center. 871 00:47:15,710 --> 00:47:19,610 And this merry-go-round's going round and around. 872 00:47:19,610 --> 00:47:23,200 What can you sense that tells you 873 00:47:23,200 --> 00:47:25,570 that you're not in an inertial frame, 874 00:47:25,570 --> 00:47:27,050 if you're holding onto this string? 875 00:47:27,050 --> 00:47:30,130 Does the ball move, first of all? 876 00:47:32,376 --> 00:47:34,000 Would the ball move if I'm sitting here 877 00:47:34,000 --> 00:47:35,958 hanging onto the string, and I set it out there 878 00:47:35,958 --> 00:47:37,560 and set it down? 879 00:47:37,560 --> 00:47:38,480 No, it won't move. 880 00:47:38,480 --> 00:47:40,920 But what do you feel in the string? 881 00:47:40,920 --> 00:47:42,020 Tension in the string. 882 00:47:42,020 --> 00:47:42,580 OK. 883 00:47:42,580 --> 00:47:45,520 So now you've got indication number one 884 00:47:45,520 --> 00:47:47,811 that there's something fishy. 885 00:47:47,811 --> 00:47:48,310 OK. 886 00:47:48,310 --> 00:47:49,518 Now you let go of the string. 887 00:47:52,460 --> 00:47:55,270 What should happen? 888 00:47:55,270 --> 00:47:56,590 So you [INAUDIBLE] go out. 889 00:47:56,590 --> 00:47:57,420 All right. 890 00:47:57,420 --> 00:48:00,340 But now I'm going to ask you a little harder question. 891 00:48:00,340 --> 00:48:05,104 What direction should the ball travel in once you release it? 892 00:48:08,008 --> 00:48:09,460 AUDIENCE: Radially outwards. 893 00:48:09,460 --> 00:48:10,668 PROFESSOR: Radially outwards. 894 00:48:10,668 --> 00:48:13,790 I have one shot at radially outwards. 895 00:48:13,790 --> 00:48:14,720 Any other thoughts? 896 00:48:14,720 --> 00:48:16,136 AUDIENCE: From your point of view, 897 00:48:16,136 --> 00:48:18,136 it wouldn't seem to go in a straight line. 898 00:48:18,136 --> 00:48:21,270 It would seem to curve off to one side. 899 00:48:21,270 --> 00:48:25,110 PROFESSOR: So I have one postulate 900 00:48:25,110 --> 00:48:28,790 that it will curve-- it will go away and curve off, right? 901 00:48:28,790 --> 00:48:31,160 Which way would it curve? 902 00:48:31,160 --> 00:48:32,790 So you're saying not radially. 903 00:48:32,790 --> 00:48:34,052 You say it's going to curve. 904 00:48:34,052 --> 00:48:35,350 AUDIENCE: Opposite direction that you're spinning. 905 00:48:35,350 --> 00:48:36,683 PROFESSOR: Opposite to the spin. 906 00:48:36,683 --> 00:48:40,239 He says it will curve opposite to the spin. 907 00:48:40,239 --> 00:48:42,030 AUDIENCE: But only from your point of view. 908 00:48:42,030 --> 00:48:43,405 PROFESSOR: This is from the point 909 00:48:43,405 --> 00:48:47,619 of view on the merry-go-round. 910 00:48:47,619 --> 00:48:49,160 Point of view off the merry-go-round, 911 00:48:49,160 --> 00:48:51,790 might be easier to reason this. 912 00:48:51,790 --> 00:48:55,070 So now you're up above the merry-go-round in an inertial 913 00:48:55,070 --> 00:48:57,590 frame, up above this merry-go-round, just 914 00:48:57,590 --> 00:49:01,670 looking down, like sitting up in a tree and looking down on it. 915 00:49:01,670 --> 00:49:02,540 What do you see? 916 00:49:07,930 --> 00:49:11,360 AUDIENCE: [INAUDIBLE], but if you're on the merry-go-round 917 00:49:11,360 --> 00:49:12,405 then you [INAUDIBLE] 918 00:49:12,405 --> 00:49:13,280 PROFESSOR: All right. 919 00:49:13,280 --> 00:49:16,030 So this is an argument for if you're on the merry-go-round, 920 00:49:16,030 --> 00:49:18,070 you'll see radial motion. 921 00:49:18,070 --> 00:49:20,490 If you're in the tree, you'll see the curve. 922 00:49:20,490 --> 00:49:21,550 So we're going to do-- 923 00:49:21,550 --> 00:49:23,508 AUDIENCE: I would like to retract my statement. 924 00:49:26,100 --> 00:49:27,240 PROFESSOR: OK. 925 00:49:27,240 --> 00:49:27,740 All right. 926 00:49:27,740 --> 00:49:29,790 I think we'd better take a vote here. 927 00:49:29,790 --> 00:49:38,860 So the possible answers are from-- how do we frame this? 928 00:49:38,860 --> 00:49:43,840 From an inertial frame, looking down on it, 929 00:49:43,840 --> 00:49:47,660 answer A is it will go in a curved path. 930 00:49:47,660 --> 00:49:52,310 Answer B is it will go in a straight, radial line. 931 00:49:52,310 --> 00:49:57,900 Answer C is it does-- any other guesses? 932 00:49:57,900 --> 00:49:58,950 No other guesses. 933 00:49:58,950 --> 00:50:00,880 So it's the only two choices. 934 00:50:00,880 --> 00:50:03,670 It curves, or it goes in a straight, radial line. 935 00:50:03,670 --> 00:50:06,210 So how many vote for-- and everybody has to participate. 936 00:50:06,210 --> 00:50:09,060 How many vote for it goes a straight radial line? 937 00:50:09,060 --> 00:50:09,940 Let's have it. 938 00:50:09,940 --> 00:50:12,772 Straight out radial line from the point of view 939 00:50:12,772 --> 00:50:13,980 of the fixed reference frame. 940 00:50:13,980 --> 00:50:15,450 OK. 941 00:50:15,450 --> 00:50:16,220 A goodly number. 942 00:50:16,220 --> 00:50:16,780 OK. 943 00:50:16,780 --> 00:50:19,400 How many from the point of view of that fixed frame, looking 944 00:50:19,400 --> 00:50:22,050 down on it, will see it curve? 945 00:50:22,050 --> 00:50:23,900 How many vote for that? 946 00:50:23,900 --> 00:50:26,800 And how many didn't vote? 947 00:50:26,800 --> 00:50:28,460 Those are don't knows, huh? 948 00:50:28,460 --> 00:50:29,670 All right. 949 00:50:29,670 --> 00:50:33,130 So A and B are wrong. 950 00:50:35,640 --> 00:50:37,880 A and B are dead wrong. 951 00:50:37,880 --> 00:50:42,720 And you could have figured out what 952 00:50:42,720 --> 00:50:46,540 the answer is if you went right back to basics, back 953 00:50:46,540 --> 00:50:48,180 to Newton's Laws. 954 00:50:48,180 --> 00:50:56,170 When the string is released, the sum of the forces on the object 955 00:50:56,170 --> 00:50:57,760 are what? 956 00:50:57,760 --> 00:50:59,970 In the direction that it can travel. 957 00:50:59,970 --> 00:51:02,350 Still got gravity pushing on it, but it 958 00:51:02,350 --> 00:51:04,200 can't go in that direction. 959 00:51:04,200 --> 00:51:07,010 So if you did a free body diagram, 960 00:51:07,010 --> 00:51:12,145 what are the forces in that horizontal [? direct ?] plane 961 00:51:12,145 --> 00:51:13,020 that it's sitting on? 962 00:51:16,401 --> 00:51:17,642 I hear centrifugal force. 963 00:51:17,642 --> 00:51:18,475 I hear frictionless. 964 00:51:18,475 --> 00:51:20,260 Let's make the table frictionless 965 00:51:20,260 --> 00:51:22,250 so it can easy to move. 966 00:51:22,250 --> 00:51:23,110 No friction. 967 00:51:23,110 --> 00:51:25,186 I hear centrifugal force. 968 00:51:25,186 --> 00:51:25,940 AUDIENCE: Zero. 969 00:51:25,940 --> 00:51:29,368 PROFESSOR: He says zero. 970 00:51:29,368 --> 00:51:31,604 AUDIENCE: There's the force that you 971 00:51:31,604 --> 00:51:34,835 get from tangent of [? to ?] less perpendicular 972 00:51:34,835 --> 00:51:36,823 to the radial direction. 973 00:51:36,823 --> 00:51:39,308 PROFESSOR: What's perpendicular to the radial direction? 974 00:51:39,308 --> 00:51:42,290 AUDIENCE: [INAUDIBLE] it had to [INAUDIBLE] B traveling 975 00:51:42,290 --> 00:51:44,775 [INAUDIBLE] at one point had to have [INAUDIBLE] force 976 00:51:44,775 --> 00:51:49,506 with tangential to the circle. 977 00:51:49,506 --> 00:51:51,880 PROFESSOR: So you were the one that provided that for us. 978 00:51:51,880 --> 00:51:53,340 You're on the merry-go-round. 979 00:51:53,340 --> 00:51:55,680 You set it out there, and held the string. 980 00:51:55,680 --> 00:51:58,140 You set out there in some radius r. 981 00:51:58,140 --> 00:52:01,870 And so because when you set it down, you're already turning. 982 00:52:01,870 --> 00:52:04,790 So you just set it down with respect to the merry-go-round. 983 00:52:04,790 --> 00:52:05,556 It's not moving. 984 00:52:05,556 --> 00:52:06,430 You just set it down. 985 00:52:06,430 --> 00:52:08,020 OK? 986 00:52:08,020 --> 00:52:12,800 What are the forces-- so just draw the free-body diagram. 987 00:52:12,800 --> 00:52:14,940 Here's this puck. 988 00:52:14,940 --> 00:52:16,760 And you've released the string. 989 00:52:16,760 --> 00:52:21,400 And you got mg downwards, and you have a normal force 990 00:52:21,400 --> 00:52:22,440 upwards. 991 00:52:22,440 --> 00:52:26,270 And you have no friction. 992 00:52:26,270 --> 00:52:28,920 And I hear centrifugal force, but-- 993 00:52:28,920 --> 00:52:30,880 AUDIENCE: [INAUDIBLE] 994 00:52:30,880 --> 00:52:32,460 PROFESSOR: Well, that was a guess. 995 00:52:32,460 --> 00:52:36,430 Somebody said straight out, and the answer is no. 996 00:52:36,430 --> 00:52:38,632 And we're going to figure it out. 997 00:52:38,632 --> 00:52:40,590 First of all, we've got to sort out the forces. 998 00:52:40,590 --> 00:52:43,560 Well, what are the forces on this thing? 999 00:52:43,560 --> 00:52:45,640 So I heard centrifugal force. 1000 00:52:45,640 --> 00:52:48,654 AUDIENCE: Yeah, I think the only force is the centrifugal force. 1001 00:52:48,654 --> 00:52:52,450 But there was a velocity that's not caused by the force. 1002 00:52:52,450 --> 00:52:55,943 So the velocity that you got was velocity that you told us about 1003 00:52:55,943 --> 00:52:59,295 that [INAUDIBLE] that starts out-- 1004 00:52:59,295 --> 00:53:00,420 PROFESSOR: You set it down. 1005 00:53:00,420 --> 00:53:02,000 And once you set it down, it's there. 1006 00:53:02,000 --> 00:53:02,996 And then? 1007 00:53:02,996 --> 00:53:04,900 AUDIENCE: Yeah, your initial velocity 1008 00:53:04,900 --> 00:53:06,804 from setting it out there. 1009 00:53:06,804 --> 00:53:08,232 PROFESSOR: Yes. 1010 00:53:08,232 --> 00:53:10,945 AUDIENCE: The change in direction-- the force only 1011 00:53:10,945 --> 00:53:13,370 change direction of the ball. 1012 00:53:13,370 --> 00:53:16,037 But once the string cuts, and the direction no longer 1013 00:53:16,037 --> 00:53:18,705 changes, and the ball only goes the direction that 1014 00:53:18,705 --> 00:53:21,620 is tangential to the circle. 1015 00:53:21,620 --> 00:53:23,585 PROFESSOR: All right. 1016 00:53:23,585 --> 00:53:25,040 The man has it right. 1017 00:53:25,040 --> 00:53:26,490 You probably couldn't hear him. 1018 00:53:26,490 --> 00:53:28,250 He says that when you let it go, it 1019 00:53:28,250 --> 00:53:31,430 goes tangential to the circle. 1020 00:53:31,430 --> 00:53:33,210 And that's true. 1021 00:53:33,210 --> 00:53:35,710 The forces on in this direction-- 1022 00:53:35,710 --> 00:53:39,290 in the horizontal direction-- once you release the object, 1023 00:53:39,290 --> 00:53:41,940 there are no forces. 1024 00:53:41,940 --> 00:53:46,530 Centrifugal force is a construction of convenience 1025 00:53:46,530 --> 00:53:48,340 called a fictitious force. 1026 00:53:48,340 --> 00:53:49,750 And we'll talk about that later. 1027 00:53:49,750 --> 00:53:52,270 It is not a real force. 1028 00:53:52,270 --> 00:53:55,380 It as a result of an acceleration. 1029 00:53:55,380 --> 00:53:57,410 And it's the result of the acceleration when you 1030 00:53:57,410 --> 00:53:59,010 are making it go in a circle. 1031 00:53:59,010 --> 00:54:01,250 There is indeed-- that tension is 1032 00:54:01,250 --> 00:54:03,760 what some people call the centrifugal force holding it 1033 00:54:03,760 --> 00:54:04,260 there. 1034 00:54:04,260 --> 00:54:07,980 But once you release it, that's no longer there. 1035 00:54:07,980 --> 00:54:09,900 And if there are no external forces 1036 00:54:09,900 --> 00:54:13,640 acting on the object in the horizontal direction, what's 1037 00:54:13,640 --> 00:54:14,990 the mass times the acceleration? 1038 00:54:14,990 --> 00:54:17,270 AUDIENCE: [INAUDIBLE] 1039 00:54:17,270 --> 00:54:19,580 PROFESSOR: Therefore what's the acceleration? 1040 00:54:19,580 --> 00:54:20,840 Zero. 1041 00:54:20,840 --> 00:54:21,660 And its velocity. 1042 00:54:21,660 --> 00:54:24,508 What's its velocity at that moment in time? 1043 00:54:24,508 --> 00:54:25,436 AUDIENCE: [INAUDIBLE] 1044 00:54:25,436 --> 00:54:28,164 PROFESSOR: Omega. 1045 00:54:28,164 --> 00:54:29,580 R omega. 1046 00:54:29,580 --> 00:54:31,700 So you have an R omega velocity. 1047 00:54:31,700 --> 00:54:34,010 It's tangential to the motion. 1048 00:54:34,010 --> 00:54:42,100 So let's make this my little x, and this my axes 1049 00:54:42,100 --> 00:54:43,980 attached to the merry-go-round. 1050 00:54:43,980 --> 00:54:48,270 This thing is moving in the j-hat direction 1051 00:54:48,270 --> 00:54:54,430 at R B with respect to A in the j-hat direction. 1052 00:54:54,430 --> 00:54:59,750 That is the velocity of B with respect to A 1053 00:54:59,750 --> 00:55:02,778 at the moment you release it. 1054 00:55:02,778 --> 00:55:03,670 OK? 1055 00:55:03,670 --> 00:55:07,800 So it had better run off tangential to the circle 1056 00:55:07,800 --> 00:55:09,110 at the moment of release. 1057 00:55:09,110 --> 00:55:10,670 So let's test it. 1058 00:55:10,670 --> 00:55:14,200 Now I want you to be my quality control person. 1059 00:55:14,200 --> 00:55:16,519 If I smack somebody with this, I'm going to hurt them. 1060 00:55:16,519 --> 00:55:17,060 AUDIENCE: No. 1061 00:55:17,060 --> 00:55:18,780 PROFESSOR: Squeeze it. 1062 00:55:18,780 --> 00:55:20,880 Soft and harmless, right? 1063 00:55:20,880 --> 00:55:22,790 OK. 1064 00:55:22,790 --> 00:55:24,960 So when should I release it if I want 1065 00:55:24,960 --> 00:55:28,320 to hit the MIT sweatshirt sitting up there? 1066 00:55:32,264 --> 00:55:34,110 When it's out here, right? 1067 00:55:34,110 --> 00:55:35,375 I'll see if I can do it. 1068 00:55:43,150 --> 00:55:46,044 All right. 1069 00:55:46,044 --> 00:55:47,210 I'd better not try it again. 1070 00:55:47,210 --> 00:55:50,166 I probably can't do it twice in a row. 1071 00:55:50,166 --> 00:55:51,790 AUDIENCE: But that was a straight line. 1072 00:55:51,790 --> 00:55:53,430 PROFESSOR: It was in a straight line. 1073 00:55:53,430 --> 00:55:54,055 AUDIENCE: Yeah. 1074 00:55:54,055 --> 00:55:55,490 So one of the guesses was-- 1075 00:55:55,490 --> 00:55:59,210 PROFESSOR: Radial straight line was one guess. 1076 00:55:59,210 --> 00:56:00,070 This is? 1077 00:56:00,070 --> 00:56:01,558 AUDIENCE: Tangential straight line. 1078 00:56:01,558 --> 00:56:02,550 PROFESSOR: Got it. 1079 00:56:02,550 --> 00:56:06,703 So this tangential straight line is the right answer. 1080 00:56:06,703 --> 00:56:08,152 AUDIENCE: [INAUDIBLE] the velocity 1081 00:56:08,152 --> 00:56:11,540 is equal to the radius [INAUDIBLE]. 1082 00:56:11,540 --> 00:56:13,375 PROFESSOR: I can't quite hear you. 1083 00:56:13,375 --> 00:56:16,285 AUDIENCE: Here it says that the velocity of [INAUDIBLE] 1084 00:56:16,285 --> 00:56:18,710 PROFESSOR: Ah, I left out the omega, didn't I? 1085 00:56:18,710 --> 00:56:21,040 Sorry about that. 1086 00:56:21,040 --> 00:56:26,820 So it is omega is the result of omega cross r B/A. 1087 00:56:26,820 --> 00:56:31,446 So it's omega with respect to O, r B/A in the j direction. 1088 00:56:31,446 --> 00:56:32,322 Yeah. 1089 00:56:32,322 --> 00:56:35,214 AUDIENCE: [INAUDIBLE] the person on the merry-go-round 1090 00:56:35,214 --> 00:56:36,657 does not see that. 1091 00:56:36,657 --> 00:56:37,240 PROFESSOR: Ah. 1092 00:56:37,240 --> 00:56:39,570 So actually I almost forgot this. 1093 00:56:39,570 --> 00:56:41,290 She is talking about what does the person 1094 00:56:41,290 --> 00:56:42,560 see on the merry-go-round? 1095 00:56:45,442 --> 00:56:47,400 What do you actually see on the merry-go-round? 1096 00:56:50,050 --> 00:56:51,330 Certainly doesn't go radial. 1097 00:56:51,330 --> 00:56:53,800 We've proven that, right? 1098 00:56:53,800 --> 00:56:57,670 But the person on the merry-go-round, 1099 00:56:57,670 --> 00:56:59,890 is you're turning, and this thing is going off 1100 00:56:59,890 --> 00:57:04,510 in a straight line with respect to the fixed frame, 1101 00:57:04,510 --> 00:57:06,480 and you're turning away from it. 1102 00:57:06,480 --> 00:57:10,900 So looking down on it, I release it here. 1103 00:57:10,900 --> 00:57:13,490 It goes off in that direction. 1104 00:57:13,490 --> 00:57:16,520 Your point of observation, you see it here. 1105 00:57:16,520 --> 00:57:22,460 But now a short time later, when you've rotated to this point, 1106 00:57:22,460 --> 00:57:25,200 and you're just keeping your eye on it, 1107 00:57:25,200 --> 00:57:27,640 you see this thing start to move. 1108 00:57:27,640 --> 00:57:30,850 But this is the spot that it was sitting on, 1109 00:57:30,850 --> 00:57:32,930 which is now moved to here. 1110 00:57:32,930 --> 00:57:35,530 But its position is now there. 1111 00:57:35,530 --> 00:57:37,300 You see it moving away from you. 1112 00:57:37,300 --> 00:57:42,580 And as you get up to, say, here, then it 1113 00:57:42,580 --> 00:57:48,069 will have moved out a radius this far. 1114 00:57:48,069 --> 00:57:50,110 Well, actually it's not-- you don't know how fast 1115 00:57:50,110 --> 00:57:52,330 you're-- well, you're actually going exactly the same speed it 1116 00:57:52,330 --> 00:57:52,830 is. 1117 00:57:52,830 --> 00:57:57,610 So you've gone the arc length here. 1118 00:57:57,610 --> 00:58:02,280 It's gone a quarter of a circle out to about there. 1119 00:58:02,280 --> 00:58:03,360 Same point right here. 1120 00:58:03,360 --> 00:58:07,090 But now you see it as being going off like that. 1121 00:58:07,090 --> 00:58:10,040 To you, it's hooking off in the direction opposite 1122 00:58:10,040 --> 00:58:11,800 to the direction of rotation. 1123 00:58:11,800 --> 00:58:14,630 So the young man up there who described that early on 1124 00:58:14,630 --> 00:58:16,910 was exactly right. 1125 00:58:16,910 --> 00:58:21,177 So from your point of view, it goes mrrmm, like that. 1126 00:58:21,177 --> 00:58:22,760 And then if you go all the way around, 1127 00:58:22,760 --> 00:58:23,880 it'll appear to come back. 1128 00:58:23,880 --> 00:58:27,370 It'll be further away, but you'll 1129 00:58:27,370 --> 00:58:29,600 be back down to the point where you see it released. 1130 00:58:29,600 --> 00:58:34,440 So it'll look like it goes from your point of view. 1131 00:58:34,440 --> 00:58:35,970 OK. 1132 00:58:35,970 --> 00:58:36,530 Good one. 1133 00:58:44,410 --> 00:58:47,220 OK, we've got a few minutes left. 1134 00:58:47,220 --> 00:58:50,507 I want to do something really important with the third law. 1135 00:58:50,507 --> 00:58:52,340 So the first law's actually pretty important 1136 00:58:52,340 --> 00:58:54,545 and very handy. 1137 00:58:54,545 --> 00:58:56,670 The second law, we're going to make lots of use of. 1138 00:58:56,670 --> 00:58:59,010 So I want to talk a minute about the third law. 1139 00:58:59,010 --> 00:59:08,090 The third law is responsible for a law that we use, 1140 00:59:08,090 --> 00:59:10,650 or an application that we use all the time. 1141 00:59:10,650 --> 00:59:13,500 And let's see how this works out. 1142 00:59:13,500 --> 00:59:15,070 So Newton's third law is the one that 1143 00:59:15,070 --> 00:59:17,530 says F 21 is equal to F 12. 1144 00:59:21,250 --> 00:59:24,170 So we have these two particles. 1145 00:59:24,170 --> 00:59:31,040 And I called them 2 and 1. 1146 00:59:31,040 --> 00:59:36,780 And I'll give this one mass 1, mass 2. 1147 00:59:36,780 --> 00:59:39,650 And I'm going to say second law tells me 1148 00:59:39,650 --> 00:59:50,250 that the sum of the forces, vectors, external forces, on 2 1149 00:59:50,250 --> 00:59:54,130 is-- and this one, let's say it has things, 1150 00:59:54,130 --> 00:59:55,830 active forces acting on it. 1151 00:59:55,830 --> 00:59:57,940 F i's. 1152 00:59:57,940 --> 01:00:02,120 And it has this little f 21. 1153 01:00:02,120 --> 01:00:09,730 So the forces here are the external forces 1154 01:00:09,730 --> 01:00:21,220 that are vectors, plus f, [? some ?] of the forces on 2. 1155 01:00:21,220 --> 01:00:23,290 So forces on 2 due to 1. 1156 01:00:23,290 --> 01:00:26,146 That's external influence of the other particle. 1157 01:00:26,146 --> 01:00:27,930 OK? 1158 01:00:27,930 --> 01:00:40,520 And this had better be equal to m 2 a 2, I'll call it. 1159 01:00:40,520 --> 01:00:44,100 And with respect to some fixed, some inertial frame. 1160 01:00:44,100 --> 01:00:49,210 So the sum of all these external forces on that particle 2 1161 01:00:49,210 --> 01:00:51,460 had better be equal to the mass times the acceleration 1162 01:00:51,460 --> 01:00:53,120 of particle 2. 1163 01:00:53,120 --> 01:00:56,370 And the same thing can be said about particle 1. 1164 01:00:56,370 --> 01:00:58,260 So I've really confused things here. 1165 01:00:58,260 --> 01:01:00,910 There's 2, and there's 1. 1166 01:01:00,910 --> 01:01:01,610 Boy. 1167 01:01:01,610 --> 01:01:02,720 OK. 1168 01:01:02,720 --> 01:01:05,210 Guys, help keep me honest here. 1169 01:01:05,210 --> 01:01:08,300 So now that we can say the sum of the forces on 1 1170 01:01:08,300 --> 01:01:16,560 is equal to these external applied forces, plus f 12-- 1171 01:01:16,560 --> 01:01:19,610 the force caused by the other body. 1172 01:01:19,610 --> 01:01:23,387 And those had better equal to m 1 a 1/0. 1173 01:01:23,387 --> 01:01:23,886 OK. 1174 01:01:27,540 --> 01:01:37,970 And these are also equal to the time derivative of m 2 v 2/0. 1175 01:01:37,970 --> 01:01:40,990 That's the momentum, the time derivative of the momentum. 1176 01:01:40,990 --> 01:01:43,930 It's fixed mass, so it's just the derivative velocity 1177 01:01:43,930 --> 01:01:44,900 gives the acceleration. 1178 01:01:44,900 --> 01:01:47,320 So these are clearly the same formula. 1179 01:01:47,320 --> 01:01:53,880 And this is a time derivative of m 1 v 1 with respect to O dt. 1180 01:01:53,880 --> 01:02:00,370 And we call that P 2 dot, P 1 dot. 1181 01:02:00,370 --> 01:02:02,681 So these are statements that are applying second law 1182 01:02:02,681 --> 01:02:03,805 to each of these particles. 1183 01:02:08,280 --> 01:02:19,310 So if I want to compute the total momentum of the system. 1184 01:02:19,310 --> 01:02:23,300 So the linear momentum, the total momentum of the system 1185 01:02:23,300 --> 01:02:27,070 is going to be P 1 plus P 2. 1186 01:02:27,070 --> 01:02:28,990 They're vectors. 1187 01:02:28,990 --> 01:02:32,640 And I want to take the time derivative 1188 01:02:32,640 --> 01:02:34,510 of that total momentum. 1189 01:02:38,810 --> 01:02:40,950 Just the time derivative of a sum 1190 01:02:40,950 --> 01:02:42,460 is the sum of the time derivatives. 1191 01:02:42,460 --> 01:02:46,225 And so I get a P 1 dot plus a P 2 dot. 1192 01:02:49,230 --> 01:02:50,600 But I know what those are. 1193 01:02:50,600 --> 01:02:52,175 I have expressions for them. 1194 01:02:55,920 --> 01:03:04,750 So this is F 1 external forces plus f 12 1195 01:03:04,750 --> 01:03:14,790 plus the F 2 external forces plus f 21. 1196 01:03:14,790 --> 01:03:15,700 These two things. 1197 01:03:15,700 --> 01:03:18,570 This one plus this one, basically, 1198 01:03:18,570 --> 01:03:23,410 is the sum of the two time derivatives of the momentum. 1199 01:03:23,410 --> 01:03:26,380 But what's the sum of this term and that term? 1200 01:03:26,380 --> 01:03:27,054 AUDIENCE: Zero. 1201 01:03:27,054 --> 01:03:27,720 PROFESSOR: Zero. 1202 01:03:31,560 --> 01:03:35,260 And that then allows you to say that this 1203 01:03:35,260 --> 01:03:45,890 is the sum of the external forces on a system is 1204 01:03:45,890 --> 01:03:49,272 equal to the time rate of change of the linear momentum 1205 01:03:49,272 --> 01:03:49,855 of the system. 1206 01:03:55,670 --> 01:03:56,670 OK. 1207 01:03:56,670 --> 01:04:00,180 And that's basically this statement. 1208 01:04:00,180 --> 01:04:03,160 And it has nothing to do with internal forces. 1209 01:04:08,860 --> 01:04:13,050 And this allows you to say that the time 1210 01:04:13,050 --> 01:04:16,930 rate of change of the linear momentum of a rigid body, 1211 01:04:16,930 --> 01:04:19,720 for example. 1212 01:04:19,720 --> 01:04:22,885 So a rigid body is made up of a whole mess 1213 01:04:22,885 --> 01:04:25,430 of different particles. 1214 01:04:25,430 --> 01:04:28,100 You could separate this into a whole bunch of little chunks, 1215 01:04:28,100 --> 01:04:33,560 and treat each one of them as a particle which has connection, 1216 01:04:33,560 --> 01:04:36,210 has forces with particles next to it. 1217 01:04:36,210 --> 01:04:39,140 And all those internal forces-- so these 1218 01:04:39,140 --> 01:04:43,390 are the internal forces-- all of those internal forces 1219 01:04:43,390 --> 01:04:45,830 are equal and opposite and cancel. 1220 01:04:45,830 --> 01:04:47,780 So it's really the third law that 1221 01:04:47,780 --> 01:04:51,300 allows you to say if you have a system of particles 1222 01:04:51,300 --> 01:04:54,690 that the time rate of change of the total momentum 1223 01:04:54,690 --> 01:04:58,050 of the system is zero if there's no external forces acting 1224 01:04:58,050 --> 01:04:58,550 on it. 1225 01:04:58,550 --> 01:05:01,510 And that's where you get the conservation of momentum-- 1226 01:05:01,510 --> 01:05:04,692 the law of conservation of momentum. 1227 01:05:04,692 --> 01:05:05,600 OK. 1228 01:05:05,600 --> 01:05:07,580 That's a consequence of the third law. 1229 01:05:15,460 --> 01:05:15,960 OK. 1230 01:05:51,590 --> 01:05:55,950 So you have a homework problem that has just about this. 1231 01:05:55,950 --> 01:05:58,840 You have a bunch of particles. 1232 01:05:58,840 --> 01:06:01,230 m 1. 1233 01:06:01,230 --> 01:06:03,128 m i. 1234 01:06:03,128 --> 01:06:10,830 r 1 with respect to O. r i with respect to O. 1235 01:06:10,830 --> 01:06:13,779 And you have all these particles out there. 1236 01:06:13,779 --> 01:06:15,320 And I want to find the center of mass 1237 01:06:15,320 --> 01:06:16,445 of this group of particles. 1238 01:06:20,170 --> 01:06:22,340 So the center of mass is a vector. 1239 01:06:22,340 --> 01:06:24,900 And let's just say it's here. 1240 01:06:24,900 --> 01:06:31,000 So I'm looking for r G with respect to O. 1241 01:06:31,000 --> 01:06:32,910 So it's a quantity that I defined-- 1242 01:06:32,910 --> 01:06:40,566 r G with respect to O-- times the summation of the m i's. 1243 01:06:40,566 --> 01:06:44,560 So I'm postulating there's a place out there that if I 1244 01:06:44,560 --> 01:06:49,120 multiply it by the sum of the m i's, I get the same answer 1245 01:06:49,120 --> 01:06:54,520 as if I summed the m i r i/O's. 1246 01:06:57,190 --> 01:06:59,930 And I'm going to define the center of mass 1247 01:06:59,930 --> 01:07:07,450 as r G with respect to O is the summation of each particle 1248 01:07:07,450 --> 01:07:12,140 times its position vector divided 1249 01:07:12,140 --> 01:07:13,920 by the sum of the masses, which is just 1250 01:07:13,920 --> 01:07:16,398 the total mass of the system. 1251 01:07:16,398 --> 01:07:17,210 OK? 1252 01:07:17,210 --> 01:07:19,910 And that's a definition of center of mass. 1253 01:07:19,910 --> 01:07:21,490 And this is a vector. 1254 01:07:21,490 --> 01:07:23,470 And these are vectors. 1255 01:07:23,470 --> 01:07:24,020 OK? 1256 01:07:24,020 --> 01:07:26,380 So that's all there is to the center of mass. 1257 01:07:26,380 --> 01:07:45,240 And if I take a time derivative of r G/O, 1258 01:07:45,240 --> 01:07:52,025 then it's equal to the summation of the m i times the r i/O's. 1259 01:07:52,025 --> 01:07:53,650 But they're timed derivatives-- and I'm 1260 01:07:53,650 --> 01:07:55,608 going to put a dot right there, so I don't have 1261 01:07:55,608 --> 01:07:59,010 to write out d by dt-- over MT. 1262 01:07:59,010 --> 01:08:05,800 But that's just the summation of the individual momenta of each 1263 01:08:05,800 --> 01:08:07,470 of the particles over MT. 1264 01:08:11,760 --> 01:08:14,235 Very handy little formula. 1265 01:08:16,810 --> 01:08:20,600 And if I take another time derivative, 1266 01:08:20,600 --> 01:08:25,729 so I get an r G/O double dot, then it's 1267 01:08:25,729 --> 01:08:32,029 just the summation of the time derivatives 1268 01:08:32,029 --> 01:08:35,160 of the individual momenta again over this. 1269 01:08:35,160 --> 01:08:36,189 So this is a statement. 1270 01:08:36,189 --> 01:08:37,569 We move this to the other side. 1271 01:08:37,569 --> 01:08:44,720 M T r G/O double dot equals the summation 1272 01:08:44,720 --> 01:08:49,010 of the timed derivatives of the individual momenta. 1273 01:08:49,010 --> 01:08:53,520 And that's the total momentum of the system 1274 01:08:53,520 --> 01:08:55,189 times its timed derivative. 1275 01:08:55,189 --> 01:09:02,420 So just from third law, you can come up 1276 01:09:02,420 --> 01:09:06,740 with all of the linear momentum formulas. 1277 01:09:06,740 --> 01:09:09,229 This is a statement for rigid body. 1278 01:09:09,229 --> 01:09:13,399 The mass times the acceleration of the center of mass. 1279 01:09:13,399 --> 01:09:16,899 The total mass of a rigid body times the acceleration 1280 01:09:16,899 --> 01:09:20,069 of the center of mass is equal to the time 1281 01:09:20,069 --> 01:09:23,481 rate of change of the total momentum of that object. 1282 01:09:23,481 --> 01:09:24,439 Very important formula. 1283 01:09:24,439 --> 01:09:26,020 You've used it a lot, right? 1284 01:09:26,020 --> 01:09:28,990 So I'll tell you a quick story, and then we'll knock off. 1285 01:09:28,990 --> 01:09:33,270 So two, three years ago, we have doctoral exams 1286 01:09:33,270 --> 01:09:34,899 in mechanical engineering. 1287 01:09:34,899 --> 01:09:37,479 And in the dynamics oral exam, we had eight students 1288 01:09:37,479 --> 01:09:39,229 a few years ago. 1289 01:09:39,229 --> 01:09:42,120 And they were asked to find the center of mass of an object. 1290 01:09:42,120 --> 01:09:44,910 It was just a step in a harder problem. 1291 01:09:44,910 --> 01:09:46,810 And seven out of the eight students 1292 01:09:46,810 --> 01:09:49,260 could not remember the definition 1293 01:09:49,260 --> 01:09:50,840 of the center of mass. 1294 01:09:50,840 --> 01:09:54,037 Now they could do Lagrange equations and nasty dynamics 1295 01:09:54,037 --> 01:09:55,620 problems, but they'd kind of forgotten 1296 01:09:55,620 --> 01:09:58,290 some of these really, really basic things. 1297 01:09:58,290 --> 01:10:01,910 So I have one demo to show you, just 1298 01:10:01,910 --> 01:10:03,520 to illustrate center of mass. 1299 01:10:03,520 --> 01:10:08,490 So center of mass, we know how to calculate it. 1300 01:10:08,490 --> 01:10:10,570 And this is a rod. 1301 01:10:10,570 --> 01:10:13,657 How can I simply find the center of mass of this thing? 1302 01:10:13,657 --> 01:10:14,790 AUDIENCE: Balance it. 1303 01:10:14,790 --> 01:10:16,861 PROFESSOR: Balance it. 1304 01:10:16,861 --> 01:10:17,360 OK. 1305 01:10:17,360 --> 01:10:19,980 Young lady here says move my fingers. 1306 01:10:19,980 --> 01:10:22,210 Where do you think the center of mass is? 1307 01:10:22,210 --> 01:10:23,130 Right in the middle. 1308 01:10:23,130 --> 01:10:23,410 OK. 1309 01:10:23,410 --> 01:10:23,993 Let's find it. 1310 01:10:29,780 --> 01:10:31,330 Should this work, by the way? 1311 01:10:31,330 --> 01:10:32,330 You ever done this? 1312 01:10:35,762 --> 01:10:37,345 All right, the center of mass, I ought 1313 01:10:37,345 --> 01:10:39,435 to be able to just about balance this thing there. 1314 01:10:39,435 --> 01:10:41,060 So the center of mass is in the middle. 1315 01:10:41,060 --> 01:10:41,960 That's because I cheated. 1316 01:10:41,960 --> 01:10:43,418 I put a piece of steel in this end. 1317 01:10:43,418 --> 01:10:44,709 OK? 1318 01:10:44,709 --> 01:10:46,010 All right. 1319 01:10:46,010 --> 01:10:49,729 But indeed, you might figure out why is it that I can actually 1320 01:10:49,729 --> 01:10:50,770 do this and make it work. 1321 01:10:50,770 --> 01:10:51,530 You can do it with a broomstick. 1322 01:10:51,530 --> 01:10:53,340 You can do it with any object. 1323 01:10:53,340 --> 01:10:56,330 The key is it'll work so long as the friction coefficients are 1324 01:10:56,330 --> 01:10:58,550 the same on both fingers. 1325 01:10:58,550 --> 01:11:01,050 It's really easy to find center of mass that way. 1326 01:11:01,050 --> 01:11:03,230 So this got a little steel weight in the end. 1327 01:11:03,230 --> 01:11:03,730 OK. 1328 01:11:03,730 --> 01:11:05,220 We're done for today. 1329 01:11:05,220 --> 01:11:09,150 And see you on Thursday.