1 00:00:00,070 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,810 Commons license. 3 00:00:03,810 --> 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 --> 00:00:16,600 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,600 --> 00:00:17,258 at ocw.mit.edu. 8 00:00:27,130 --> 00:00:28,630 PROFESSOR: So today I'm mostly going 9 00:00:28,630 --> 00:00:31,780 to talk about two formulas that we are 10 00:00:31,780 --> 00:00:34,870 going to make a lot of use of. 11 00:00:34,870 --> 00:00:37,740 One is the time derivative of linear momentum, 12 00:00:37,740 --> 00:00:40,890 which will be used a lot, has got to be equal to, 13 00:00:40,890 --> 00:00:41,716 for a particle. 14 00:00:45,840 --> 00:00:49,160 We'll just do particles today and then rigid bodies soon. 15 00:00:49,160 --> 00:00:52,546 So for a particle, this is just the mass times 16 00:00:52,546 --> 00:00:53,920 the acceleration of the particle. 17 00:00:56,600 --> 00:00:59,830 The rigid body, it's the total mass times acceleration 18 00:00:59,830 --> 00:01:00,930 in the center of gravity. 19 00:01:00,930 --> 00:01:05,650 And the other formula that we've learned, that we've come up to, 20 00:01:05,650 --> 00:01:14,770 is the sum of the external torques with respect to a point 21 00:01:14,770 --> 00:01:22,770 A on a particle is the time derivative of the angular 22 00:01:22,770 --> 00:01:29,670 momentum of the particle at B with respect to A dt. 23 00:01:29,670 --> 00:01:34,160 And then it's got this nuisance term, this velocity of A 24 00:01:34,160 --> 00:01:40,820 with respect to O cross the linear momentum at the point B 25 00:01:40,820 --> 00:01:44,950 with respect to O. So we talked about this last time. 26 00:01:44,950 --> 00:01:51,880 So this is a particle that's located out here 27 00:01:51,880 --> 00:01:59,410 at B. There's some intermediate point A. And we're 28 00:01:59,410 --> 00:02:03,630 computing the angular momentum of this particle, m, 29 00:02:03,630 --> 00:02:06,880 with respect to A. And A can be moving. 30 00:02:06,880 --> 00:02:09,280 And if it is, when you go to compute torques, 31 00:02:09,280 --> 00:02:11,820 you have to use this kind of messy formula. 32 00:02:11,820 --> 00:02:15,220 We often try to simplify the problems 33 00:02:15,220 --> 00:02:19,450 that we do so that we can either make A a fixed 34 00:02:19,450 --> 00:02:21,860 axis of rotation-- make it rotate 35 00:02:21,860 --> 00:02:25,480 about this point, in which case this goes to zero-- 36 00:02:25,480 --> 00:02:29,100 or there's another time when you can make 37 00:02:29,100 --> 00:02:32,940 the velocity of this point parallel 38 00:02:32,940 --> 00:02:34,420 to the velocity of that point. 39 00:02:34,420 --> 00:02:37,560 And the most common one of all is 40 00:02:37,560 --> 00:02:43,020 when you use this formula about the center of mass. 41 00:02:43,020 --> 00:02:46,240 Then the velocity of the center of mass 42 00:02:46,240 --> 00:02:49,037 is the same direction as the momentum. 43 00:02:49,037 --> 00:02:50,620 And therefore, this term goes to zero. 44 00:02:50,620 --> 00:02:54,890 So we have these two cases where this formula simplifies. 45 00:02:54,890 --> 00:02:57,410 So the way we most often use it is 46 00:02:57,410 --> 00:03:00,920 the summation of the torques with respect to A 47 00:03:00,920 --> 00:03:07,370 is just the derivative of the angular momentum dt. 48 00:03:07,370 --> 00:03:08,250 And that's it. 49 00:03:08,250 --> 00:03:26,900 This is when vA is zero, or vA is parallel to P, such 50 00:03:26,900 --> 00:03:36,560 as A is at the center of mass. 51 00:03:36,560 --> 00:03:39,500 So these are the two formulas that we want to use. 52 00:03:56,580 --> 00:03:58,480 Let's just do a real simple case first. 53 00:04:31,130 --> 00:04:38,070 And this is just my mass on a string spinning 54 00:04:38,070 --> 00:04:39,640 around, constant speed. 55 00:04:44,080 --> 00:04:47,199 I'm going to pretend I'm on a frictionless table 56 00:04:47,199 --> 00:04:48,740 so I don't have to deal with gravity. 57 00:04:48,740 --> 00:04:51,420 This thing is horizontal, just spinning around. 58 00:04:51,420 --> 00:04:56,636 And I'll call this point A. This is my point O. This is x, z. 59 00:04:59,920 --> 00:05:01,900 No, y. 60 00:05:01,900 --> 00:05:04,710 It's looking down on it. 61 00:05:04,710 --> 00:05:09,310 So it's going to have a velocity in this theta hat direction. 62 00:05:09,310 --> 00:05:13,977 And you've got your R hat. 63 00:05:13,977 --> 00:05:15,810 We'll just use polar coordinates to do this. 64 00:05:15,810 --> 00:05:19,940 So the velocity of A with respect to O we 65 00:05:19,940 --> 00:05:25,310 know-- and we'll give this just a length magnitude r1. 66 00:05:25,310 --> 00:05:30,923 So this is going to be theta dot equals a constant. 67 00:05:34,880 --> 00:05:36,910 r dot equals 0. 68 00:05:36,910 --> 00:05:38,690 So it's just fixed length. 69 00:05:38,690 --> 00:05:40,710 Fixed length is r1. 70 00:05:40,710 --> 00:05:43,090 So we know the simple formula for the velocity 71 00:05:43,090 --> 00:05:50,350 of that is r1 theta dot in the theta hat direction. 72 00:05:50,350 --> 00:05:53,910 [INAUDIBLE] hat, excuse me. 73 00:05:53,910 --> 00:06:00,180 And the momentum then is just the mass times that. 74 00:06:04,300 --> 00:06:13,365 And the time derivative of the linear momentum 75 00:06:13,365 --> 00:06:16,420 with respect to this fixed reference frame 76 00:06:16,420 --> 00:06:18,385 is the time derivative of this. 77 00:06:18,385 --> 00:06:19,430 This is constant. 78 00:06:19,430 --> 00:06:20,380 This is constant. 79 00:06:20,380 --> 00:06:21,790 This is constant. 80 00:06:21,790 --> 00:06:25,050 The only thing you have to take is the derivative of theta hat. 81 00:06:25,050 --> 00:06:34,989 So this is mr1 theta dot theta hat time derivative. 82 00:06:34,989 --> 00:06:37,030 But we've worked that one out two or three times. 83 00:06:37,030 --> 00:06:45,600 It's theta dot-- minus theta dot r hat. 84 00:06:56,030 --> 00:06:57,170 We'll put that in. 85 00:06:57,170 --> 00:07:06,290 You get minus mr1 theta dot squared r hat. 86 00:07:06,290 --> 00:07:11,240 And this must be the sum of the external forces. 87 00:07:11,240 --> 00:07:13,140 And if we're, looking down on this, 88 00:07:13,140 --> 00:07:20,690 draw a free body diagram of our mass, looking down on it, 89 00:07:20,690 --> 00:07:25,110 here's r hat. 90 00:07:25,110 --> 00:07:28,780 The theta direction like this, z coming out of the board. 91 00:07:28,780 --> 00:07:33,950 So this is x, y, and o. 92 00:07:33,950 --> 00:07:37,540 There is a inward force on it that we 93 00:07:37,540 --> 00:07:39,830 call the tension in the string. 94 00:07:39,830 --> 00:07:45,830 And some of external force is just minus Tr hat. 95 00:07:45,830 --> 00:07:50,540 So T is mr1 theta dot squared, which 96 00:07:50,540 --> 00:07:53,460 is what we talked about just a second ago. 97 00:07:53,460 --> 00:07:56,320 So this is just a demonstration of what 98 00:07:56,320 --> 00:07:59,050 we were talking about with the survey questions, 99 00:07:59,050 --> 00:08:02,030 that when you spin something, you're 100 00:08:02,030 --> 00:08:05,760 changing its linear momentum. 101 00:08:05,760 --> 00:08:07,570 Not in magnitude. 102 00:08:07,570 --> 00:08:08,900 It stays constant. 103 00:08:08,900 --> 00:08:12,080 But it's constantly changing in direction. 104 00:08:12,080 --> 00:08:15,850 And the direction change-- you have 105 00:08:15,850 --> 00:08:20,910 to take this derivative-- gives you the [INAUDIBLE]. 106 00:08:20,910 --> 00:08:26,090 Direction changes causes the centripetal acceleration. 107 00:08:26,090 --> 00:08:29,710 To cause that acceleration, you have to put a force on it. 108 00:08:29,710 --> 00:08:33,059 And the force is that inward tension in the string. 109 00:08:33,059 --> 00:08:37,610 OK, so that's the first really simple one. 110 00:08:37,610 --> 00:08:41,479 Now, let's take a [INAUDIBLE]. 111 00:09:01,580 --> 00:09:03,710 Now let's do the same problem, but let it 112 00:09:03,710 --> 00:09:05,440 be a little bit more general. 113 00:09:05,440 --> 00:09:07,900 I've got a tabletop. 114 00:09:07,900 --> 00:09:10,720 This problem's described in the book as an example. 115 00:09:10,720 --> 00:09:12,175 You've got a hole in the table. 116 00:09:15,176 --> 00:09:17,380 You've got your mass out here. 117 00:09:17,380 --> 00:09:19,010 So this is looking down on the table 118 00:09:19,010 --> 00:09:21,070 and down through the hole, coming out 119 00:09:21,070 --> 00:09:23,610 the bottom of the hole, you have this string. 120 00:09:23,610 --> 00:09:25,070 And you pull on it. 121 00:09:31,000 --> 00:09:37,280 So r dot-- so we're looking down on it. 122 00:09:37,280 --> 00:09:40,565 This thing, again, has some theta. 123 00:09:44,060 --> 00:09:48,755 r hat directions like this, theta hat directions like that. 124 00:09:48,755 --> 00:09:55,855 This thing's spinning, going around and around my table top. 125 00:09:55,855 --> 00:09:57,820 But there's a hole in the tabletop, 126 00:09:57,820 --> 00:09:59,880 and I can pull the string and shorten the string. 127 00:10:02,820 --> 00:10:04,040 So r dot's a constant. 128 00:10:08,700 --> 00:10:11,548 Theta dot will not be a constant. 129 00:10:15,030 --> 00:10:16,160 it will change. 130 00:10:22,743 --> 00:10:24,390 This is my vector r. 131 00:10:27,610 --> 00:10:30,745 So the velocity, we'll call this A. This is my origin, 132 00:10:30,745 --> 00:10:35,790 O. The velocity in this case of the particle with respect 133 00:10:35,790 --> 00:10:47,940 to the fixed inertial frame is r dot r hat plus r theta 134 00:10:47,940 --> 00:10:50,005 dot theta hat. 135 00:10:50,005 --> 00:10:53,620 So now it has two possible velocities, 136 00:10:53,620 --> 00:10:55,319 and I'm going to be pulling it in, 137 00:10:55,319 --> 00:10:57,110 and it's going to be going round and round. 138 00:11:10,110 --> 00:11:41,040 So the linear momentum is just mvA with respect to O. 139 00:11:41,040 --> 00:11:43,930 So with this problem, if I take the time 140 00:11:43,930 --> 00:11:48,920 derivative of the linear momentum, you recognize this. 141 00:11:48,920 --> 00:11:52,020 This is just velocity in polar coordinates. 142 00:11:52,020 --> 00:11:55,930 We've done this derivative before to get the acceleration. 143 00:11:55,930 --> 00:11:58,650 So the acceleration, it's the time derivative of A 144 00:11:58,650 --> 00:12:06,160 with respect to O is acceleration 145 00:12:06,160 --> 00:12:11,620 of A with respect to O. And that's a messy formula, 146 00:12:11,620 --> 00:12:14,270 and we've derived it before. 147 00:12:14,270 --> 00:12:22,870 So that looks like-- and it's got four terms. 148 00:12:22,870 --> 00:12:29,520 r double dot minus r theta dot squared 149 00:12:29,520 --> 00:12:39,400 and the r hat plus r theta double dot plus 2r dot theta 150 00:12:39,400 --> 00:12:43,730 dot in the theta hat direction. 151 00:12:43,730 --> 00:12:47,740 And if multiplied by the mass, that's 152 00:12:47,740 --> 00:12:49,830 the mass times the acceleration. 153 00:12:49,830 --> 00:12:54,830 And this would be equal to the sum of all the external forces 154 00:12:54,830 --> 00:12:56,470 acting on that particle. 155 00:12:56,470 --> 00:12:57,720 That's Newton's second law. 156 00:13:02,740 --> 00:13:07,671 But in our problem here, I've said this is constant. 157 00:13:07,671 --> 00:13:10,095 So I can throw out this term that's going to be zero. 158 00:13:13,360 --> 00:13:14,415 So this term goes away. 159 00:13:16,935 --> 00:13:20,760 But this term's certainly not zero. 160 00:13:20,760 --> 00:13:24,070 This term is not necessarily zero. 161 00:13:24,070 --> 00:13:27,820 This term is certainly not zero. 162 00:13:27,820 --> 00:13:31,300 So what can we do with that? 163 00:13:31,300 --> 00:13:34,450 So now's the time you draw some free body diagrams. 164 00:13:34,450 --> 00:13:37,500 And let's look at the side view. 165 00:13:41,200 --> 00:13:44,930 The side view, here's your particle, 166 00:13:44,930 --> 00:13:52,460 mg down, some normal force up, and a tension pulling in. 167 00:13:55,280 --> 00:13:58,490 So the table's supporting it, gravity down, and just 168 00:13:58,490 --> 00:14:01,390 the string force pulling it. 169 00:14:01,390 --> 00:14:08,192 In the top view, you're looking down on it. 170 00:14:08,192 --> 00:14:09,650 I'm not allowing them to-- assuming 171 00:14:09,650 --> 00:14:12,500 it's a frictionless table, so there's no friction. 172 00:14:12,500 --> 00:14:16,090 So the only force seen from the top is the tension. 173 00:14:16,090 --> 00:14:24,220 There are no forces in the plane in the direction theta hat. 174 00:14:24,220 --> 00:14:27,200 So the sum of the external forces in the theta hat 175 00:14:27,200 --> 00:14:30,810 direction are in this direction. 176 00:14:30,810 --> 00:14:39,770 This is-- so the total forces on this thing from your free body 177 00:14:39,770 --> 00:14:41,755 diagram in the theta hat direction are? 178 00:14:41,755 --> 00:14:42,570 AUDIENCE: Zero. 179 00:14:42,570 --> 00:14:43,410 PROFESSOR: Zero. 180 00:14:43,410 --> 00:14:47,180 And that allows us to take this term and set it equal to 0. 181 00:14:57,860 --> 00:14:59,165 This is the x. 182 00:15:03,040 --> 00:15:07,370 So the theta hat piece is equal to 0 183 00:15:07,370 --> 00:15:17,610 is equal to m r theta double dot plus 2r dot theta dot. 184 00:15:17,610 --> 00:15:24,180 [INAUDIBLE] solve this for-- that's equal to 0. 185 00:15:24,180 --> 00:15:31,410 So r theta double dot equals minus 2r dot theta dot. 186 00:15:34,910 --> 00:15:41,470 So in order for this thing to satisfy Newton's law, 187 00:15:41,470 --> 00:15:47,150 it happens to be that as you pull it in, 188 00:15:47,150 --> 00:15:50,480 the product of r dot and theta dot 189 00:15:50,480 --> 00:15:53,230 gives you the angular acceleration. 190 00:15:58,340 --> 00:16:00,830 And the other term, we just went through that. 191 00:16:00,830 --> 00:16:03,260 So what we did a couple minutes ago. 192 00:16:03,260 --> 00:16:08,790 The force in the r hat direction, 193 00:16:08,790 --> 00:16:12,020 sum of the forces-- this is a derivative 194 00:16:12,020 --> 00:16:18,810 of the linear momentum-- is minus mr theta dot 195 00:16:18,810 --> 00:16:21,450 squared r hat. 196 00:16:21,450 --> 00:16:24,040 And we know from the free body diagram 197 00:16:24,040 --> 00:16:29,680 that that better be equal to minus T in the r hat direction. 198 00:16:29,680 --> 00:16:31,140 So just like before, it tells you 199 00:16:31,140 --> 00:16:34,900 that T equals mr theta dot squared. 200 00:16:34,900 --> 00:16:38,210 So the mass times the centripetal acceleration, 201 00:16:38,210 --> 00:16:41,340 in order to make that centripetal motion happen, 202 00:16:41,340 --> 00:16:48,190 you have to pull on it with a force mr theta dot squared. 203 00:16:48,190 --> 00:16:56,933 So what can you say about the angular momentum 204 00:16:56,933 --> 00:17:01,230 of this particle with respect to O? 205 00:17:08,960 --> 00:17:11,700 So let's write it out. 206 00:17:11,700 --> 00:17:17,520 So H of the particle with respect to O 207 00:17:17,520 --> 00:17:45,810 is rAO cross P. This is r r hat cross P. 208 00:17:45,810 --> 00:17:49,760 So r cross r, you get nothing from that term. 209 00:17:49,760 --> 00:17:53,770 r hat cross theta hat gives you a k, a positive k. 210 00:17:53,770 --> 00:18:15,096 So you get r r squared theta dot in the k direction. 211 00:18:30,950 --> 00:18:32,180 And I'm missing something. 212 00:18:36,120 --> 00:18:37,510 There we go. 213 00:18:37,510 --> 00:18:41,000 m r squared theta dot. 214 00:18:41,000 --> 00:18:44,290 mr theta dot is the linear momentum times 215 00:18:44,290 --> 00:18:46,200 another r gives you the angular momentum, 216 00:18:46,200 --> 00:18:47,950 m r squared theta dot. 217 00:18:47,950 --> 00:18:51,830 And because r hat cross theta dot is k, 218 00:18:51,830 --> 00:18:56,450 this angular momentum is directed upward 219 00:18:56,450 --> 00:18:59,840 about the center of rotation. 220 00:18:59,840 --> 00:19:02,730 So that's our expression for our angular momentum. 221 00:19:02,730 --> 00:19:11,780 But if you take the time derivative of it, 222 00:19:11,780 --> 00:19:12,745 what should it tell us? 223 00:19:12,745 --> 00:19:17,450 Let's go back to our formulas we started with. 224 00:19:17,450 --> 00:19:21,340 When the center of rotation's not moving, 225 00:19:21,340 --> 00:19:24,580 when the point with respect to which you're taking the angular 226 00:19:24,580 --> 00:19:27,740 momentum doesn't move, then that's 227 00:19:27,740 --> 00:19:31,520 one of the cases where you can get rid of those extra terms. 228 00:19:31,520 --> 00:19:35,120 So in this case, we're computing the angular momentum 229 00:19:35,120 --> 00:19:37,330 with respect to O, which doesn't move. 230 00:19:37,330 --> 00:19:41,560 The velocity of O here is zero. 231 00:19:41,560 --> 00:19:46,560 So this allows us just to say that the sum of the torques 232 00:19:46,560 --> 00:19:50,090 with respect about O is just equal to the time 233 00:19:50,090 --> 00:19:53,820 rate of change of H with respect to O. 234 00:19:53,820 --> 00:20:03,960 And to do that, this term can change, 235 00:20:03,960 --> 00:20:05,310 and this term can change. 236 00:20:05,310 --> 00:20:07,790 But how about the derivative of k? 237 00:20:07,790 --> 00:20:08,950 It's just constant, right? 238 00:20:08,950 --> 00:20:11,270 So we're going to get two terms out of this. 239 00:20:11,270 --> 00:20:32,170 We're going to get m2r r dot theta dot k plus r 240 00:20:32,170 --> 00:20:36,115 squared theta double dot k. 241 00:20:44,210 --> 00:20:49,134 What are the external torques in this problem? 242 00:20:49,134 --> 00:20:50,800 You're going to have to pretend that I'm 243 00:20:50,800 --> 00:20:53,710 on a frictionless table top here. 244 00:20:53,710 --> 00:20:57,440 When I'm going around like this, what 245 00:20:57,440 --> 00:21:02,877 are the torques about the center point? 246 00:21:02,877 --> 00:21:04,710 So what are the-- first, what are the forces 247 00:21:04,710 --> 00:21:07,590 acting on the mass? 248 00:21:07,590 --> 00:21:09,040 Just the tension. 249 00:21:09,040 --> 00:21:12,190 And the tension cross the moment arm, 250 00:21:12,190 --> 00:21:13,940 the tension's in the r hat direction. 251 00:21:13,940 --> 00:21:16,020 The string is in the r direction. 252 00:21:16,020 --> 00:21:17,720 r hat cross r hat is-- 253 00:21:17,720 --> 00:21:18,370 AUDIENCE: Zero. 254 00:21:18,370 --> 00:21:19,036 PROFESSOR: Zero. 255 00:21:19,036 --> 00:21:20,870 So there's no torques. 256 00:21:20,870 --> 00:21:25,770 So for this problem, this is equal to 0. 257 00:21:28,500 --> 00:22:06,270 And that then allows us to write-- the m cancels out, 258 00:22:06,270 --> 00:22:08,560 obviously, and I can get rid of-- one 259 00:22:08,560 --> 00:22:10,350 of these r's goes away. 260 00:22:10,350 --> 00:22:17,070 And I'm left with an r dot theta dot. 261 00:22:17,070 --> 00:22:18,740 And I move this to the other side 262 00:22:18,740 --> 00:22:21,300 equal to minus r theta double dot. 263 00:22:21,300 --> 00:22:23,700 And that's what we came up with a minute ago 264 00:22:23,700 --> 00:22:28,624 when when we did the time derivative 265 00:22:28,624 --> 00:22:30,290 of the linear momentum, we learned this. 266 00:22:30,290 --> 00:22:31,920 So we haven't learned much more. 267 00:22:31,920 --> 00:22:34,200 It's just telling us that, well, this thing's 268 00:22:34,200 --> 00:22:35,630 going to accelerate. 269 00:22:35,630 --> 00:22:37,650 It should pull it in. 270 00:22:37,650 --> 00:22:40,690 And it'll accelerate at any instant 271 00:22:40,690 --> 00:22:43,250 in time, whatever r dot is, theta dot is, this'll 272 00:22:43,250 --> 00:22:46,465 be the angular acceleration. 273 00:22:51,030 --> 00:23:02,310 So what will happen then? 274 00:23:02,310 --> 00:23:05,275 And let's do it at two points in time. 275 00:23:21,670 --> 00:23:26,704 And we'll let r2 equals r1 divided by 2. 276 00:23:26,704 --> 00:23:28,370 So I'm just going to pull this thing in. 277 00:23:34,580 --> 00:23:35,705 So let's do the experiment. 278 00:23:38,800 --> 00:23:41,770 I'm going to pull it in about half its length. 279 00:23:41,770 --> 00:23:44,765 It can speed up, slow down, stay the same speed. 280 00:23:49,460 --> 00:23:56,402 I'll get it going and then-- and I'll try not to hit you. 281 00:23:56,402 --> 00:23:59,900 I'll move over here so I hit him instead, OK? 282 00:23:59,900 --> 00:24:03,571 Let's try it again. 283 00:24:03,571 --> 00:24:04,070 All right. 284 00:24:04,070 --> 00:24:06,500 But there's no torques on it. 285 00:24:06,500 --> 00:24:09,130 There's no torque being applied. 286 00:24:09,130 --> 00:24:12,820 The angular momentum is constant, 287 00:24:12,820 --> 00:24:14,530 and yet the thing speeds up. 288 00:24:14,530 --> 00:24:18,290 So I want to ask you a question. 289 00:24:18,290 --> 00:24:21,760 Do you think the kinetic energy is staying the same 290 00:24:21,760 --> 00:24:22,840 or changing? 291 00:24:22,840 --> 00:24:28,740 So how many think the kinetic energy as I go from out there 292 00:24:28,740 --> 00:24:32,500 to in here, the kinetic energy stays the same? 293 00:24:32,500 --> 00:24:33,120 OK. 294 00:24:33,120 --> 00:24:35,820 How many think it's different? 295 00:24:35,820 --> 00:24:36,320 All right. 296 00:24:36,320 --> 00:24:38,655 How many are not so sure? 297 00:24:38,655 --> 00:24:39,280 Let's find out. 298 00:24:42,950 --> 00:24:47,030 So we've determined that if this is zero, 299 00:24:47,030 --> 00:24:52,943 then that means that h with respect to O is a constant. 300 00:24:56,950 --> 00:25:06,314 So h-- this is at T1-- is r1. 301 00:25:10,100 --> 00:25:12,970 Be faster if I look at my notes. 302 00:25:12,970 --> 00:25:23,240 So the angular momentum at r1 is mr1 squared 303 00:25:23,240 --> 00:25:27,420 theta 1 dot in the k direction. 304 00:25:27,420 --> 00:25:33,580 And that had better be equal to h at r2. 305 00:25:36,990 --> 00:25:46,010 And that'll be mr2 squared theta 2 dot. 306 00:25:46,010 --> 00:25:48,760 And that's also in the k direction. 307 00:25:48,760 --> 00:25:54,350 But we know that r2 is r1 divided by 2. 308 00:25:54,350 --> 00:25:55,735 So we can plug that in. 309 00:26:07,686 --> 00:26:09,890 I'll bring this over here. 310 00:26:09,890 --> 00:26:20,720 So mr1 squared theta 1 dot is mr1/4 theta 2 dot. 311 00:26:20,720 --> 00:26:22,230 So I can solve for theta 2. 312 00:26:31,780 --> 00:26:35,265 So if I shorten the length by a factor of 2, 313 00:26:35,265 --> 00:26:37,770 the angular velocity goes up by a factor of 4. 314 00:26:52,590 --> 00:26:54,230 And let's check the kinetic energy. 315 00:26:54,230 --> 00:27:00,915 Kinetic energy state 1, 1/2mv1 squared. 316 00:27:04,170 --> 00:27:12,786 That's 1/2mv1 is r theta 1 dot quantity squared. 317 00:27:32,204 --> 00:27:34,080 So what do I have? 318 00:27:34,080 --> 00:27:38,000 I have some expressions for this. 319 00:27:38,000 --> 00:27:51,150 r2 is r1/2, and theta 2 is 4 theta 1 dot quantity squared. 320 00:27:51,150 --> 00:28:27,440 And if you multiply that out-- yeah. 321 00:28:27,440 --> 00:28:33,740 So where's the kinetic energy come from? 322 00:28:33,740 --> 00:28:35,700 AUDIENCE: You're adding energy into the system 323 00:28:35,700 --> 00:28:37,087 by pulling down. 324 00:28:37,087 --> 00:28:39,170 PROFESSOR: So she says we add energy to the system 325 00:28:39,170 --> 00:28:39,850 by pulling down. 326 00:28:39,850 --> 00:28:41,530 So we're doing some work, right? 327 00:28:41,530 --> 00:28:46,692 There's tension in that string equal to mr omega squared. 328 00:28:46,692 --> 00:28:49,430 If you pull it down a certain distance-- in fact, 329 00:28:49,430 --> 00:28:52,560 r1/2-- you're going to do work that's 330 00:28:52,560 --> 00:28:54,960 the integral of the tension times 331 00:28:54,960 --> 00:28:57,350 dr. You integrate it, right? 332 00:28:57,350 --> 00:29:01,280 And that work goes into-- there's conservation 333 00:29:01,280 --> 00:29:02,280 of energy in the system. 334 00:29:02,280 --> 00:29:04,970 That goes into speeding up the rotation, 335 00:29:04,970 --> 00:29:07,730 and yet the angular momentum has stayed constant 336 00:29:07,730 --> 00:29:10,740 throughout the action. 337 00:29:10,740 --> 00:29:12,290 So when I first saw this years ago, 338 00:29:12,290 --> 00:29:13,854 I thought, that's really cool. 339 00:29:13,854 --> 00:29:15,020 That's really quite amazing. 340 00:29:19,050 --> 00:29:21,190 So a nice application of conservation 341 00:29:21,190 --> 00:29:23,580 of angular momentum. 342 00:29:23,580 --> 00:29:27,080 An application of using this formula, the time 343 00:29:27,080 --> 00:29:29,140 rate of change of angular momentum 344 00:29:29,140 --> 00:29:32,080 with respect to a point, it tells you 345 00:29:32,080 --> 00:29:34,980 about the torques applied to the system. 346 00:29:34,980 --> 00:29:37,770 And this is in fact a pretty simple case. 347 00:29:37,770 --> 00:29:43,040 So the last-- let's move on though to doing 348 00:29:43,040 --> 00:29:45,900 a little more complicated case. 349 00:29:45,900 --> 00:29:51,700 And this is similar to the last problem in the homework. 350 00:30:01,547 --> 00:30:02,755 So this is like the homework. 351 00:30:14,380 --> 00:30:17,410 The homework, you got this monkey running up the shaft, 352 00:30:17,410 --> 00:30:17,910 right? 353 00:30:17,910 --> 00:30:20,870 So I don't have a monkey, and it's not running up the shaft. 354 00:30:20,870 --> 00:30:31,810 But I do have just this particle on a shaft 355 00:30:31,810 --> 00:30:35,896 rotating about a central axis. 356 00:30:35,896 --> 00:30:37,940 Now, this is a mechanical necessity 357 00:30:37,940 --> 00:30:38,940 to hold it all together. 358 00:30:38,940 --> 00:30:40,865 But let's just ignore the mass of this center 359 00:30:40,865 --> 00:30:41,740 piece for the moment. 360 00:30:41,740 --> 00:30:47,340 Just think of this as a massless arm with a particle on it. 361 00:30:47,340 --> 00:30:50,680 And I want to calculate angular momentum. 362 00:30:50,680 --> 00:30:52,500 I want to calculate forces. 363 00:30:52,500 --> 00:30:55,245 I want to calculate torques and see what happens. 364 00:31:14,590 --> 00:31:25,340 So I'm going to start by putting my O, x, z frame right 365 00:31:25,340 --> 00:31:27,910 on the level with this mass. 366 00:31:31,165 --> 00:31:32,540 What I'm going to show you now is 367 00:31:32,540 --> 00:31:38,200 that where you put the reference point about which you compute 368 00:31:38,200 --> 00:31:41,510 the angular momentum matters. 369 00:31:41,510 --> 00:31:44,530 You get different answers depending on where you put it. 370 00:31:44,530 --> 00:31:48,810 So I'm going to start by putting it here. 371 00:31:48,810 --> 00:31:50,760 And this we'll call [INAUDIBLE] out here's 372 00:31:50,760 --> 00:31:59,275 my point A. Here's O. This is some angle phi here. 373 00:32:02,390 --> 00:32:10,420 And if this has some length l, then this up here is my r 374 00:32:10,420 --> 00:32:16,270 equals l cosine phi. 375 00:32:16,270 --> 00:32:21,186 And this side would be l sine phi. 376 00:32:21,186 --> 00:32:22,810 These are just the two lengths, but I'm 377 00:32:22,810 --> 00:32:24,910 going to use polar coordinates. 378 00:32:24,910 --> 00:32:28,240 So this is going to be my r hat direction. 379 00:32:28,240 --> 00:32:29,580 Theta hat's into the board. 380 00:32:40,070 --> 00:32:43,120 So let's compute-- and it's a particle, 381 00:32:43,120 --> 00:32:49,050 so I'll continue to use lowercase h of A 382 00:32:49,050 --> 00:32:57,840 with respect to O-- it's a vector-- is r of A with respect 383 00:32:57,840 --> 00:33:02,450 to O cross P linear momentum with respect 384 00:33:02,450 --> 00:33:08,840 to O. This is r r hat. 385 00:33:08,840 --> 00:33:11,230 We just call this l cosine theta, just calling it r. 386 00:33:11,230 --> 00:33:13,920 It's in the r hat direction. 387 00:33:13,920 --> 00:33:18,930 Cross with P, and P, we'd done this two or three times now 388 00:33:18,930 --> 00:33:19,670 today. 389 00:33:19,670 --> 00:33:26,290 It's the mass times r times theta dot. 390 00:33:26,290 --> 00:33:28,040 That's its speed. 391 00:33:28,040 --> 00:33:31,325 And it's in the theta hat direction. 392 00:33:38,190 --> 00:33:42,640 So taking the r hat cross theta hat gives me k. 393 00:33:45,220 --> 00:33:53,970 So I get m r squared theta dot k. 394 00:34:00,347 --> 00:34:01,305 Very simple expression. 395 00:34:05,980 --> 00:34:08,560 Now, this is a fixed axis rotation. 396 00:34:08,560 --> 00:34:12,440 So I want to compute the torques. 397 00:34:12,440 --> 00:34:14,260 Look at my formula. 398 00:34:14,260 --> 00:34:19,860 The vA in this case is the velocity of O. 399 00:34:19,860 --> 00:34:23,870 The point of the axis of rotation doesn't move. 400 00:34:23,870 --> 00:34:26,590 So the second terms go away, and I 401 00:34:26,590 --> 00:34:34,790 can say that the torque of my particle at A with respect to O 402 00:34:34,790 --> 00:34:38,730 is just dhA dt. 403 00:34:43,210 --> 00:34:45,530 So m's a constant. 404 00:34:45,530 --> 00:34:46,830 r's a constant. 405 00:34:46,830 --> 00:34:48,719 k's a constant. 406 00:34:48,719 --> 00:34:51,534 The only thing that has a time derivative is theta dot, 407 00:34:51,534 --> 00:34:54,136 and it becomes theta double dot. 408 00:34:54,136 --> 00:35:01,690 This is m r squared theta double dot k. 409 00:35:01,690 --> 00:35:07,630 And that is equal-- well, that's equal to the sum 410 00:35:07,630 --> 00:35:10,790 of the external torques. 411 00:35:10,790 --> 00:35:14,140 So what physically does that mean? 412 00:35:14,140 --> 00:35:17,010 What physically is that telling us? 413 00:35:17,010 --> 00:35:20,110 It's telling us theta double dot is the angular 414 00:35:20,110 --> 00:35:22,850 acceleration of this thing speeding up, 415 00:35:22,850 --> 00:35:25,270 going faster and faster. 416 00:35:25,270 --> 00:35:29,590 It takes torque to make that happen. 417 00:35:29,590 --> 00:35:34,310 If it's going at constant rate, what's theta double dot? 418 00:35:34,310 --> 00:35:35,650 Zero. 419 00:35:35,650 --> 00:35:38,390 So at constant rate, the torque required 420 00:35:38,390 --> 00:35:42,070 to make this thing go constant rate is zero. 421 00:35:42,070 --> 00:35:43,080 Makes sense. 422 00:35:43,080 --> 00:35:45,200 But if it were speeding up, if you're 423 00:35:45,200 --> 00:35:47,330 making it go faster and faster and faster, 424 00:35:47,330 --> 00:35:48,890 it requires torque to drive it. 425 00:35:48,890 --> 00:35:50,260 And that's the amount of torque. 426 00:35:50,260 --> 00:35:55,512 And the torque is around the axis of spin. 427 00:35:55,512 --> 00:35:56,470 Pretty straightforward. 428 00:36:02,670 --> 00:36:07,670 So if I did dP [? d, ?] if I took the time 429 00:36:07,670 --> 00:36:11,760 derivative of just the linear momentum for this particle, 430 00:36:11,760 --> 00:36:13,180 we've done it here today. 431 00:36:13,180 --> 00:36:17,810 If I took the time derivative of it, what would I get? 432 00:36:17,810 --> 00:36:18,440 It's a force. 433 00:36:18,440 --> 00:36:19,314 And what's the force? 434 00:36:25,730 --> 00:36:27,650 It's a constant rotation rate. 435 00:36:27,650 --> 00:36:32,930 Take the time derivative of P. dP dt gives me 436 00:36:32,930 --> 00:36:34,340 AUDIENCE: [INAUDIBLE]. 437 00:36:34,340 --> 00:36:35,781 PROFESSOR: Mass times? 438 00:36:35,781 --> 00:36:37,890 AUDIENCE: [INAUDIBLE]. 439 00:36:37,890 --> 00:36:40,720 PROFESSOR: Mv squared over r is mass times acceleration. 440 00:36:40,720 --> 00:36:42,574 The acceleration is which kind? 441 00:36:42,574 --> 00:36:43,490 AUDIENCE: [INAUDIBLE]. 442 00:36:43,490 --> 00:36:44,060 PROFESSOR: Centripetal. 443 00:36:44,060 --> 00:36:46,030 So you just get the same thing back again. 444 00:36:46,030 --> 00:36:51,660 So there's a force acting inwards words this thing 445 00:36:51,660 --> 00:36:56,630 to make it go in a circle that is the mr theta dot squared 446 00:36:56,630 --> 00:37:00,270 term that we've seen so many times. 447 00:37:00,270 --> 00:37:09,420 But now what I want to do is move the point about which I 448 00:37:09,420 --> 00:37:10,940 compute this angular momentum. 449 00:37:16,740 --> 00:37:19,070 And now I'm going to put it here, 450 00:37:19,070 --> 00:37:22,220 the point of attachment of the arm. 451 00:37:22,220 --> 00:37:27,280 So here's O, x, z. 452 00:37:27,280 --> 00:37:28,670 Everything else stays the same. 453 00:37:28,670 --> 00:37:31,460 All I've done is move the point. 454 00:37:31,460 --> 00:37:35,960 And I want to compute the angular momentum 455 00:37:35,960 --> 00:37:43,570 of this A with respect to O. 456 00:37:43,570 --> 00:37:44,650 Well, that's r. 457 00:37:44,650 --> 00:37:54,380 This is now r of A with respect to O, this vector. 458 00:37:54,380 --> 00:37:59,140 This distance here, in polar cylindrical coordinates, is z. 459 00:38:04,210 --> 00:38:05,430 And this is r. 460 00:38:05,430 --> 00:38:06,290 Just as before. 461 00:38:06,290 --> 00:38:08,870 The r hasn't changed. 462 00:38:08,870 --> 00:38:11,960 And there is an r hat in this direction, 463 00:38:11,960 --> 00:38:17,660 theta hat into the board, and a k hat in the z direction. 464 00:38:17,660 --> 00:38:19,750 Those are our unit vectors. 465 00:38:19,750 --> 00:38:34,240 So this is-- rAO is r r hat plus z k hat. 466 00:38:34,240 --> 00:38:38,660 That's the position vector. 467 00:38:38,660 --> 00:38:43,610 And I'm going to cross that with the momentum of A 468 00:38:43,610 --> 00:38:55,030 with respect to O. And the momentum 469 00:38:55,030 --> 00:39:04,180 is mr. Theta dot is the velocity. 470 00:39:04,180 --> 00:39:05,450 And what's its direction? 471 00:39:08,000 --> 00:39:10,410 Theta hat, right? 472 00:39:10,410 --> 00:39:12,820 Mass times velocity, momentum. 473 00:39:12,820 --> 00:39:17,540 And now we need to carry this out. 474 00:39:17,540 --> 00:39:22,230 The r hat term times theta hat gives you a k. 475 00:39:27,195 --> 00:39:35,760 m r squared theta dot k. 476 00:39:35,760 --> 00:39:44,000 And this term, k cross theta hat, gives me a minus r. 477 00:39:44,000 --> 00:39:57,915 Minus mrz theta dot k. 478 00:40:03,616 --> 00:40:05,820 AUDIENCE: [INAUDIBLE]. 479 00:40:05,820 --> 00:40:06,820 PROFESSOR: You're right. 480 00:40:06,820 --> 00:40:08,510 Thank you. 481 00:40:08,510 --> 00:40:13,240 Because I would have a disaster if I let that progress. 482 00:40:13,240 --> 00:40:16,745 This looks like that, right? k cross theta is minus r hat. 483 00:40:19,260 --> 00:40:21,346 r cross theta is a k. 484 00:40:21,346 --> 00:40:23,650 k cross theta is a minus r. 485 00:40:23,650 --> 00:40:26,850 I get two terms. 486 00:40:26,850 --> 00:40:30,505 And this is now an expression for the angular momentum of A 487 00:40:30,505 --> 00:40:34,960 with respect to O. And let's see if I have enough room 488 00:40:34,960 --> 00:40:36,360 to draw it here. 489 00:40:36,360 --> 00:40:41,120 There is a piece of it here. 490 00:40:41,120 --> 00:40:49,180 This is h in the z direction, is this arrow. 491 00:40:49,180 --> 00:40:55,245 And then there's a piece in the r hat direction, like this. 492 00:40:55,245 --> 00:40:59,780 This is h in the r hat direction. 493 00:40:59,780 --> 00:41:05,420 And the sum of those two is that. 494 00:41:05,420 --> 00:41:10,930 So this is hA with respect to O, this guy. 495 00:41:10,930 --> 00:41:15,829 Perpendicular to the shaft. 496 00:41:15,829 --> 00:41:17,620 It will turn out it really is perpendicular 497 00:41:17,620 --> 00:41:20,530 if you work out the numbers. 498 00:41:20,530 --> 00:41:27,860 Totally different result than when I did it here. 499 00:41:27,860 --> 00:41:32,550 So this one, m r squared theta dot k, m r squared theta dot k. 500 00:41:32,550 --> 00:41:34,450 Hey, that term's the same. 501 00:41:34,450 --> 00:41:38,360 So when I did this, I got just the k term. 502 00:41:38,360 --> 00:41:40,900 And now I've moved this thing down, and I get a second term. 503 00:41:44,550 --> 00:41:47,712 So now what we want to know is, what about the torques 504 00:41:47,712 --> 00:41:48,295 in the system? 505 00:41:58,110 --> 00:42:01,000 So I want to take the time derivative 506 00:42:01,000 --> 00:42:04,980 of this guy with respect to-- the time 507 00:42:04,980 --> 00:42:08,290 derivative of the angular momentum, which 508 00:42:08,290 --> 00:42:11,160 is going to be equal to the summation 509 00:42:11,160 --> 00:42:14,870 of the external torques with respect to O, 510 00:42:14,870 --> 00:42:18,110 but O is now in a different place. 511 00:42:18,110 --> 00:42:21,100 And so I have to carry out these derivatives. 512 00:42:21,100 --> 00:42:23,130 And this one I did before. 513 00:42:23,130 --> 00:42:32,045 This one just gives me my m r squared theta double dot k hat. 514 00:42:32,045 --> 00:42:36,530 Now, this term, m is a constant. 515 00:42:36,530 --> 00:42:38,800 r is a constant. 516 00:42:38,800 --> 00:42:42,260 z is a constant. 517 00:42:42,260 --> 00:42:49,036 Theta dot is not necessarily a constant. 518 00:42:49,036 --> 00:42:51,600 We're going to let that be a variable. 519 00:42:51,600 --> 00:42:55,540 And r hat is certainly changing direction. 520 00:42:55,540 --> 00:42:57,320 So when I take the derivative of this, 521 00:42:57,320 --> 00:42:59,700 I'm going to get two terms. 522 00:42:59,700 --> 00:43:10,700 So the first one is minus mrz theta double dot r hat. 523 00:43:10,700 --> 00:43:13,505 And that's taking the derivative of this multiplied by that. 524 00:43:13,505 --> 00:43:15,520 And the second term is the derivative 525 00:43:15,520 --> 00:43:17,810 of this multiplied by that. 526 00:43:17,810 --> 00:43:22,200 And so the derivative of r hat is? 527 00:43:26,480 --> 00:43:27,960 AUDIENCE: [INAUDIBLE]. 528 00:43:27,960 --> 00:43:29,410 PROFESSOR: Theta dot theta hat. 529 00:43:29,410 --> 00:43:30,080 Right. 530 00:43:30,080 --> 00:43:30,870 OK. 531 00:43:30,870 --> 00:43:45,210 So minus mrz theta dot theta dot theta hat. 532 00:43:47,720 --> 00:43:50,390 So in other words, this is squared. 533 00:43:50,390 --> 00:43:54,210 Now I have three terms to mess with. 534 00:43:54,210 --> 00:43:58,100 We know what the first term means. 535 00:43:58,100 --> 00:44:00,140 We talked about that. 536 00:44:00,140 --> 00:44:01,220 This is the torque. 537 00:44:01,220 --> 00:44:02,910 These are all torques. 538 00:44:02,910 --> 00:44:06,060 So this is the torque required to do what, the lead term? 539 00:44:09,030 --> 00:44:11,792 AUDIENCE: [INAUDIBLE]. 540 00:44:11,792 --> 00:44:13,000 AUDIENCE: [INAUDIBLE] circle. 541 00:44:13,000 --> 00:44:15,064 PROFESSOR: To make it? 542 00:44:15,064 --> 00:44:16,030 AUDIENCE: [INAUDIBLE]. 543 00:44:16,030 --> 00:44:16,904 PROFESSOR: Go faster. 544 00:44:16,904 --> 00:44:18,990 Change its angular speed, right? 545 00:44:18,990 --> 00:44:23,860 It's just building up the angular momentum in that spin. 546 00:44:23,860 --> 00:44:29,190 So this is the angular spin up. 547 00:44:29,190 --> 00:44:33,405 These other two terms, these are strange things. 548 00:44:33,405 --> 00:44:39,100 Well first, let's take a look at this one. r theta dot squared. 549 00:44:39,100 --> 00:44:40,528 What's that remind you of? 550 00:44:43,280 --> 00:44:49,200 What kind of-- torque is usually some force times a moment arm, 551 00:44:49,200 --> 00:44:51,920 crossed with a moment arm, right? 552 00:44:51,920 --> 00:44:56,040 So we know that there's some forces acting in this system. 553 00:44:56,040 --> 00:44:58,080 It's spinning. 554 00:44:58,080 --> 00:45:03,160 We know that there is a-- in order 555 00:45:03,160 --> 00:45:07,030 to make this thing go round and around-- it 556 00:45:07,030 --> 00:45:09,060 has centripetal acceleration. 557 00:45:09,060 --> 00:45:13,720 Therefore, there must be a force being applied by this shaft 558 00:45:13,720 --> 00:45:17,840 inward that's equal to the mass times 559 00:45:17,840 --> 00:45:22,035 the centripetal acceleration, mr theta dot squared. 560 00:45:26,480 --> 00:45:33,786 So here this guy is mr theta dot squared. 561 00:45:37,060 --> 00:45:39,650 That's the force. 562 00:45:39,650 --> 00:45:42,110 And let's do these. 563 00:45:42,110 --> 00:45:44,770 Let's call this A. Let's Call. 564 00:45:44,770 --> 00:45:55,140 This term here B, this term C. So the C term is-- torque C, 565 00:45:55,140 --> 00:46:01,780 I'll call it-- is some r cross some F. 566 00:46:01,780 --> 00:46:10,110 And the F, I'm telling you, is the centripetal acceleration 567 00:46:10,110 --> 00:46:11,260 times the mass. 568 00:46:11,260 --> 00:46:18,110 And that'll probably be like a minus mr 569 00:46:18,110 --> 00:46:24,200 theta dot squared r hat, right? 570 00:46:24,200 --> 00:46:30,650 And what about the moment arm that that acts about? 571 00:46:30,650 --> 00:46:32,740 What moment arm is perpendicular-- so 572 00:46:32,740 --> 00:46:35,450 that's a force that's acting in. 573 00:46:35,450 --> 00:46:37,244 What moment arm is perpendicular to that? 574 00:46:37,244 --> 00:46:39,160 Because the only thing that's perpendicular to 575 00:46:39,160 --> 00:46:42,187 it lead to torques. 576 00:46:42,187 --> 00:46:42,770 PROFESSOR: Hm? 577 00:46:42,770 --> 00:46:43,550 AUDIENCE: [INAUDIBLE]. 578 00:46:43,550 --> 00:46:44,091 PROFESSOR: z. 579 00:46:52,670 --> 00:46:58,525 k cross r should give me a theta. 580 00:47:02,100 --> 00:47:07,280 Sure enough, there's a minus, sure enough. 581 00:47:07,280 --> 00:47:09,600 And there's the z. 582 00:47:09,600 --> 00:47:12,690 You multiply this out, you get this term. 583 00:47:12,690 --> 00:47:14,820 So this is a strange term. 584 00:47:14,820 --> 00:47:19,880 It's in the theta hat direction. 585 00:47:19,880 --> 00:47:21,100 What is that? 586 00:47:21,100 --> 00:47:25,080 So it's spinning, and it's lined up like this. 587 00:47:25,080 --> 00:47:27,900 Theta hat's in that direction. 588 00:47:27,900 --> 00:47:33,370 Positive k, positive theta, positive r hat. 589 00:47:33,370 --> 00:47:34,510 k's in that direction. 590 00:47:34,510 --> 00:47:36,670 It's telling you there's a torque 591 00:47:36,670 --> 00:47:42,684 being applied about this point in the minus theta direction. 592 00:47:42,684 --> 00:47:44,670 Does that make sense? 593 00:47:44,670 --> 00:47:49,850 You have a-- there's a centripetal acceleration 594 00:47:49,850 --> 00:47:50,660 times a mass. 595 00:47:50,660 --> 00:47:53,860 There's a force times a moment arm. 596 00:47:53,860 --> 00:47:59,317 This force is trying to bend this thing out. 597 00:47:59,317 --> 00:48:01,900 If this thing had a hinge down here, and I started to spin it, 598 00:48:01,900 --> 00:48:02,608 what would it do? 599 00:48:02,608 --> 00:48:04,960 It would just flop out, right? 600 00:48:04,960 --> 00:48:08,670 There's got to be a torque keeping that from happening. 601 00:48:08,670 --> 00:48:10,190 And that's a torque. 602 00:48:10,190 --> 00:48:11,870 If it wants to go that way, there's 603 00:48:11,870 --> 00:48:15,360 got to be a torque going the other way keeping it in place. 604 00:48:15,360 --> 00:48:18,330 And that's what that term is. 605 00:48:18,330 --> 00:48:18,990 So now we know. 606 00:48:18,990 --> 00:48:20,710 So this is the Euler. 607 00:48:20,710 --> 00:48:22,630 This is the spin up. 608 00:48:22,630 --> 00:48:26,410 This is keeping this thing from flopping out. 609 00:48:26,410 --> 00:48:27,990 What's this guy? 610 00:48:27,990 --> 00:48:33,640 This is yet another torque, and it's in the r, minus r, 611 00:48:33,640 --> 00:48:34,140 direction. 612 00:48:36,960 --> 00:48:38,780 So let's see if we can intuitively 613 00:48:38,780 --> 00:48:40,600 figure this one out. 614 00:48:40,600 --> 00:48:44,560 Well, there's an r theta double dot. 615 00:48:44,560 --> 00:48:46,260 That should look familiar. 616 00:48:46,260 --> 00:48:49,350 r theta double dot. 617 00:48:49,350 --> 00:48:53,060 If this thing is accelerating, angular acceleration, 618 00:48:53,060 --> 00:48:59,650 speeding up, out here that mass says, I'm here right now. 619 00:48:59,650 --> 00:49:01,765 In order for me to go a little bit faster, 620 00:49:01,765 --> 00:49:04,320 I'm accelerating in that direction. 621 00:49:04,320 --> 00:49:06,070 There must be a force being applied 622 00:49:06,070 --> 00:49:09,260 by this rod in that direction. 623 00:49:09,260 --> 00:49:15,390 And that force times a moment arm perpendicular to it, z, 624 00:49:15,390 --> 00:49:20,870 is a moment in the r direction. 625 00:49:20,870 --> 00:49:23,770 So as this thing spins up, if this thing could, 626 00:49:23,770 --> 00:49:25,740 it would fall back. 627 00:49:25,740 --> 00:49:27,820 But this rod is stiff and won't let it do that. 628 00:49:27,820 --> 00:49:30,980 So this is the torque down here that 629 00:49:30,980 --> 00:49:33,660 is required to keep this thing moving. 630 00:49:36,200 --> 00:49:38,690 So this is really quite amazing. 631 00:49:41,200 --> 00:49:46,550 Do either of these torques, these second ones, this one 632 00:49:46,550 --> 00:49:50,880 and this one, do they contribute to-- do they do any work? 633 00:49:53,460 --> 00:49:56,330 Do they add energy to the system? 634 00:49:56,330 --> 00:49:59,720 See, work means force through a distance. 635 00:49:59,720 --> 00:50:02,020 They don't do actually any work. 636 00:50:02,020 --> 00:50:04,260 They are static torques just required 637 00:50:04,260 --> 00:50:06,370 to hold the system together. 638 00:50:06,370 --> 00:50:07,590 This one does some work. 639 00:50:07,590 --> 00:50:10,715 It actually makes-- this one leads to energy accumulating, 640 00:50:10,715 --> 00:50:13,000 just going faster and faster faster. 641 00:50:13,000 --> 00:50:15,060 These are just holding the thing together. 642 00:50:15,060 --> 00:50:18,720 But the amazing thing is that you 643 00:50:18,720 --> 00:50:21,930 can use angular momentum to calculate things 644 00:50:21,930 --> 00:50:24,040 like these torques. 645 00:50:24,040 --> 00:50:30,520 So if you were designing this, these forces 646 00:50:30,520 --> 00:50:35,490 acting on this lump out here are producing torques 647 00:50:35,490 --> 00:50:39,950 about this point, which are the same thing as-- in 2.001 you'll 648 00:50:39,950 --> 00:50:42,200 be doing bending moments. 649 00:50:42,200 --> 00:50:45,124 It creates a bending moment in the shaft. 650 00:50:45,124 --> 00:50:47,040 And if you don't make the shaft strong enough, 651 00:50:47,040 --> 00:50:49,270 it'll break it off. 652 00:50:49,270 --> 00:50:54,250 So the torque about this point is the bending moment 653 00:50:54,250 --> 00:50:58,230 in the r direction and in the theta direction. 654 00:50:58,230 --> 00:51:00,680 It's trying to be bent in two different ways. 655 00:51:00,680 --> 00:51:03,480 And you can calculate the stresses down here caused 656 00:51:03,480 --> 00:51:05,020 by those moments. 657 00:51:05,020 --> 00:51:08,760 And that would help you design the thing. 658 00:51:08,760 --> 00:51:11,590 So you not only get dynamics information 659 00:51:11,590 --> 00:51:13,770 out of taking things like the time rate of change 660 00:51:13,770 --> 00:51:15,250 of angular momentum. 661 00:51:15,250 --> 00:51:20,050 You get some of the static information as well. 662 00:51:20,050 --> 00:51:24,200 The other thing to remember, the really important point 663 00:51:24,200 --> 00:51:29,720 of the lecture, is that angular momentum 664 00:51:29,720 --> 00:51:34,910 changes depending on where you pick a reference point. 665 00:51:34,910 --> 00:51:38,450 So when we picked the reference point just opposite it, 666 00:51:38,450 --> 00:51:41,500 we got none of the information about the torques down here. 667 00:51:41,500 --> 00:51:43,720 Because with respect to this point, 668 00:51:43,720 --> 00:51:47,890 there are no torques except the one speeded up. 669 00:51:47,890 --> 00:51:51,940 The centripetal force here doesn't cause torques. 670 00:51:51,940 --> 00:51:57,179 The force out here-- there are no other torques, just 671 00:51:57,179 --> 00:51:58,470 the one to make it spin faster. 672 00:51:58,470 --> 00:52:00,220 But as soon as I move it down here, 673 00:52:00,220 --> 00:52:04,350 I learn something of considerable value. 674 00:52:04,350 --> 00:52:07,404 So the homework problem has some of the same things 675 00:52:07,404 --> 00:52:08,820 on it, except the monkey's moving. 676 00:52:08,820 --> 00:52:11,780 So you get even a little bit more 677 00:52:11,780 --> 00:52:14,760 interesting information out of it. 678 00:52:14,760 --> 00:52:15,260 All right. 679 00:52:28,090 --> 00:52:32,050 So that went faster than I thought. 680 00:52:32,050 --> 00:52:34,040 So that gives some time for some questions. 681 00:52:34,040 --> 00:52:35,230 I could see several. 682 00:52:35,230 --> 00:52:38,048 So we'll start there and then go here. 683 00:52:38,048 --> 00:52:39,920 AUDIENCE: What exactly [INAUDIBLE] 684 00:52:39,920 --> 00:52:42,730 torque B is balancing out? 685 00:52:42,730 --> 00:52:44,704 PROFESSOR: So say again? 686 00:52:44,704 --> 00:52:48,600 AUDIENCE: Torque B [INAUDIBLE], what 687 00:52:48,600 --> 00:52:51,515 exactly is that balancing out? 688 00:52:51,515 --> 00:52:52,390 PROFESSOR: This term? 689 00:52:52,390 --> 00:52:54,017 You're asking about this term? 690 00:52:54,017 --> 00:52:55,267 AUDIENCE: [INAUDIBLE]. 691 00:52:55,267 --> 00:52:55,850 PROFESSOR: OK. 692 00:52:55,850 --> 00:52:57,915 You want me to explain again what this one means? 693 00:53:02,850 --> 00:53:09,130 This is a term associated with increasing the angular speed. 694 00:53:09,130 --> 00:53:21,910 So let's see if we can't-- so this B term, 695 00:53:21,910 --> 00:53:27,510 I'll call it the torque associated with B, 696 00:53:27,510 --> 00:53:34,120 minus mrz theta double dot r hat. 697 00:53:34,120 --> 00:53:42,260 Now, that is going to be some r cross some F. 698 00:53:42,260 --> 00:53:44,310 And if we can get some physical insight, 699 00:53:44,310 --> 00:53:46,820 if we could figure out what they are. 700 00:53:46,820 --> 00:53:55,740 So the mass, the force, is the mass times an acceleration. 701 00:53:55,740 --> 00:53:58,450 There's an acceleration r theta double dot, which 702 00:53:58,450 --> 00:54:03,690 is the speed of this thing is increasing speed. 703 00:54:03,690 --> 00:54:07,490 And the r is going to be z. 704 00:54:07,490 --> 00:54:11,860 z k hat cross-- and I'm guessing that it's a force that 705 00:54:11,860 --> 00:54:18,820 looks like mr theta double dot. 706 00:54:18,820 --> 00:54:26,415 And this is in the theta hat direction. 707 00:54:29,550 --> 00:54:33,440 k cross theta hat gives me minus r hat. 708 00:54:33,440 --> 00:54:34,964 r hat and the minus. 709 00:54:38,130 --> 00:54:40,940 So this looks like a plausible explanation for where 710 00:54:40,940 --> 00:54:42,080 this might have come from. 711 00:54:42,080 --> 00:54:48,360 So this is the force speeding up, attempting 712 00:54:48,360 --> 00:54:49,615 to speed up this mass. 713 00:54:49,615 --> 00:54:53,450 There's a force pushing on it that's given to it by this rod. 714 00:54:53,450 --> 00:54:56,950 This rod is pushing on it to make it go faster. 715 00:54:56,950 --> 00:55:01,600 Mass times acceleration would be mr theta double dot. 716 00:55:01,600 --> 00:55:04,110 And the moment arm is this distance 717 00:55:04,110 --> 00:55:06,440 from here down to the point at which 718 00:55:06,440 --> 00:55:12,240 I've been computing my reference point from here to here at z. 719 00:55:12,240 --> 00:55:16,600 So force times z, r cross F, puts it 720 00:55:16,600 --> 00:55:19,985 in the-- ends up in the minus r direction. 721 00:55:19,985 --> 00:55:23,636 It's got to be this direction, k cross theta hat. 722 00:55:23,636 --> 00:55:25,070 The force is this way. 723 00:55:31,542 --> 00:55:32,250 Think about this. 724 00:55:32,250 --> 00:55:34,040 Force is that way. 725 00:55:34,040 --> 00:55:37,240 The r is this way. 726 00:55:37,240 --> 00:55:41,840 The r cross F is this way. 727 00:55:41,840 --> 00:55:43,920 And that's where you get the minus sign. 728 00:55:43,920 --> 00:55:47,130 It's in the minus r hat direction. 729 00:55:47,130 --> 00:55:51,890 But it comes from trying to speed up this mass. 730 00:55:51,890 --> 00:55:58,260 And if this was a floppy, weak link, as it tries to speed up, 731 00:55:58,260 --> 00:56:00,300 it would try to bend back. 732 00:56:00,300 --> 00:56:04,070 It would flop back as this thing tries to make it go faster. 733 00:56:04,070 --> 00:56:05,600 It will say, no, I don't want to go. 734 00:56:05,600 --> 00:56:06,710 Lay back on me. 735 00:56:06,710 --> 00:56:11,320 And that would we going in the-- this way, 736 00:56:11,320 --> 00:56:12,710 to keep from doing that, you have 737 00:56:12,710 --> 00:56:16,420 to put a torque on it this way. 738 00:56:16,420 --> 00:56:17,460 Just trying to speed up. 739 00:56:17,460 --> 00:56:19,220 It's trying to lay back, and you're saying nope, 740 00:56:19,220 --> 00:56:19,940 can't do that. 741 00:56:19,940 --> 00:56:21,985 Go like this. 742 00:56:21,985 --> 00:56:22,860 So that's the B term. 743 00:56:26,020 --> 00:56:30,194 AUDIENCE: If you do the problem with a situation like that, 744 00:56:30,194 --> 00:56:32,150 how do you know where to set it? 745 00:56:32,150 --> 00:56:33,180 PROFESSOR: How do you know where to set it? 746 00:56:33,180 --> 00:56:34,959 Well, that's a good question. 747 00:56:34,959 --> 00:56:37,250 He's saying, when you're doing a problem like this, how 748 00:56:37,250 --> 00:56:39,570 do you know where to pick the reference frame? 749 00:56:39,570 --> 00:56:42,856 Well, ask yourself what it is you want to know. 750 00:56:42,856 --> 00:56:46,980 And in fact, now that you know that from angular momentum 751 00:56:46,980 --> 00:56:50,470 of mechanical things, you can actually 752 00:56:50,470 --> 00:56:53,940 get static torques on the system, 753 00:56:53,940 --> 00:56:56,640 ask yourself where you want to know those torques. 754 00:56:56,640 --> 00:56:58,480 In this case, if you're designing this, 755 00:56:58,480 --> 00:57:01,200 you want to know whether you're going to break this thing off. 756 00:57:01,200 --> 00:57:04,760 And it's probably going to break right down at the bottom 757 00:57:04,760 --> 00:57:07,140 where the moment arms are the greatest. 758 00:57:07,140 --> 00:57:08,650 So that's why you pick that point. 759 00:57:11,910 --> 00:57:18,460 It comes a lot with experience will help you choose. 760 00:57:18,460 --> 00:57:20,990 But the amazing thing is this information's 761 00:57:20,990 --> 00:57:23,200 all stored in the angular momentum 762 00:57:23,200 --> 00:57:25,120 if you pick it in the right place. 763 00:57:25,120 --> 00:57:25,700 Another hand. 764 00:57:31,100 --> 00:57:32,240 OK. 765 00:57:32,240 --> 00:57:34,390 Yes, Phillip. 766 00:57:34,390 --> 00:57:37,785 AUDIENCE: I had a question about the direction for r hat. 767 00:57:37,785 --> 00:57:40,210 I thought it had to come out of the origin. 768 00:57:40,210 --> 00:57:42,635 But you have it going in the x direction. 769 00:57:47,510 --> 00:57:49,350 PROFESSOR: So I've been using polar-- he's 770 00:57:49,350 --> 00:57:51,230 asking the direction of r hat. 771 00:57:51,230 --> 00:57:53,590 So I've just been using polar coordinates. 772 00:57:53,590 --> 00:57:58,710 And polar coordinates is cylindrical, technically. 773 00:57:58,710 --> 00:58:05,380 This problem has a z direction upwards, r direction 774 00:58:05,380 --> 00:58:09,960 radially outwards, r hat, and theta as drawn here 775 00:58:09,960 --> 00:58:14,110 would be into the board, given the position of the mass. 776 00:58:14,110 --> 00:58:20,530 Looking down on it, here's my O. And looking down, 777 00:58:20,530 --> 00:58:23,930 this would be my x and my y. 778 00:58:23,930 --> 00:58:27,810 And this is some random arbitrary position here. 779 00:58:27,810 --> 00:58:31,260 And this is theta. 780 00:58:31,260 --> 00:58:33,770 Looking down on it, in this plane 781 00:58:33,770 --> 00:58:41,690 is r hat theta hat, k hat coming out of the board. 782 00:58:41,690 --> 00:58:53,130 Side view, x, z, and my system is like this. 783 00:58:53,130 --> 00:59:02,130 Now the theta is into the board, and the r direction 784 00:59:02,130 --> 00:59:02,950 is this way. 785 00:59:02,950 --> 00:59:03,880 That's r hat. 786 00:59:03,880 --> 00:59:05,850 And this is z. 787 00:59:05,850 --> 00:59:09,840 This is the z-coordinate upwards. 788 00:59:09,840 --> 00:59:15,260 So the position vector, the thing we call rA with respect 789 00:59:15,260 --> 00:59:21,410 to O, is indeed the length of this whole thing. 790 00:59:21,410 --> 00:59:29,740 But it is made up of a component in the z direction 791 00:59:29,740 --> 00:59:39,688 plus a component here that we call r in the r hat direction. 792 00:59:39,688 --> 00:59:42,592 AUDIENCE: [INAUDIBLE] example where we calculated from 793 00:59:42,592 --> 00:59:45,980 the bottom rather than the top circle, 794 00:59:45,980 --> 00:59:50,094 then we got a value for the angular momentum that 795 00:59:50,094 --> 00:59:51,788 doesn't have a theta hat component, 796 00:59:51,788 --> 00:59:53,040 but as the thing spins-- 797 00:59:53,040 --> 00:59:54,300 PROFESSOR: Which are you referring to? 798 00:59:54,300 --> 00:59:54,883 Are you talk-- 799 00:59:54,883 --> 00:59:58,660 AUDIENCE: h about the point O down-- 800 00:59:58,660 --> 01:00:00,350 PROFESSOR: Which example, this guy or-- 801 01:00:00,350 --> 01:00:01,516 AUDIENCE: Yeah, [INAUDIBLE]. 802 01:00:01,516 --> 01:00:03,000 PROFESSOR: OK, so point. 803 01:00:03,000 --> 01:00:04,240 Tell me what you mean here. 804 01:00:04,240 --> 01:00:05,656 AUDIENCE: Right here. [INAUDIBLE]. 805 01:00:05,656 --> 01:00:08,326 PROFESSOR: When we computed, not here, but down here. 806 01:00:08,326 --> 01:00:11,302 AUDIENCE: We got that there was no theta component, 807 01:00:11,302 --> 01:00:14,774 but as this spins around, theta is changing. 808 01:00:14,774 --> 01:00:16,975 And if it's always opposite, shouldn't there 809 01:00:16,975 --> 01:00:17,850 be a theta component? 810 01:00:17,850 --> 01:00:18,860 PROFESSOR: A theta component of what? 811 01:00:18,860 --> 01:00:20,280 AUDIENCE: Angular momentum. 812 01:00:20,280 --> 01:00:22,620 PROFESSOR: Angular momentum. 813 01:00:22,620 --> 01:00:24,710 She's asking, shouldn't there be a theta component 814 01:00:24,710 --> 01:00:27,870 of angular momentum? 815 01:00:27,870 --> 01:00:31,080 So we compute our angular momentum with this formula 816 01:00:31,080 --> 01:00:32,360 at the top. 817 01:00:32,360 --> 01:00:43,350 It's an r cross P. So the r consists of the z part, 818 01:00:43,350 --> 01:00:46,680 and the r part is exactly this right here. 819 01:00:46,680 --> 01:00:49,080 And the P is only into the board. 820 01:00:49,080 --> 01:00:51,350 It's only in the theta hat direction. 821 01:00:51,350 --> 01:00:58,000 So you have a term that's r hat cross theta hat gives you a k, 822 01:00:58,000 --> 01:01:01,915 and you have a term k cross theta 823 01:01:01,915 --> 01:01:03,350 hat, which gives you an r. 824 01:01:03,350 --> 01:01:05,300 There just are no cross products that 825 01:01:05,300 --> 01:01:09,020 come out of this that are in the theta hat direction. 826 01:01:09,020 --> 01:01:09,994 Yeah. 827 01:01:09,994 --> 01:01:12,388 AUDIENCE: So if it's just at this one position, 828 01:01:12,388 --> 01:01:13,792 then you don't have it. 829 01:01:13,792 --> 01:01:15,196 But as it spins-- 830 01:01:15,196 --> 01:01:16,520 PROFESSOR: Ah. 831 01:01:16,520 --> 01:01:20,210 In this case, that's why polar coordinates are nice 832 01:01:20,210 --> 01:01:23,830 because as it spins, the theta hat's 833 01:01:23,830 --> 01:01:26,380 just constantly going with it. 834 01:01:26,380 --> 01:01:29,920 The r hat's constantly going with it. 835 01:01:29,920 --> 01:01:32,720 And so the beauty of this thing is 836 01:01:32,720 --> 01:01:36,730 this is an axially symmetric problem. 837 01:01:36,730 --> 01:01:43,890 It goes round and round, and the torques are given out 838 01:01:43,890 --> 01:01:47,550 in this rotating frame. 839 01:01:47,550 --> 01:01:49,250 I think maybe what's confusing you, 840 01:01:49,250 --> 01:01:52,400 is if you wanted to know the torques in a fixed inertial 841 01:01:52,400 --> 01:01:56,879 frame, you'd have to break them down into ijk components, 842 01:01:56,879 --> 01:01:57,670 which you could do. 843 01:01:57,670 --> 01:01:59,200 A little tedious. 844 01:01:59,200 --> 01:02:03,730 But the answers in this one came out in r hat, theta hat, k hat 845 01:02:03,730 --> 01:02:04,230 terms. 846 01:02:12,100 --> 01:02:14,540 Happy to answer. 847 01:02:14,540 --> 01:02:17,760 This is good stuff, but thick. 848 01:02:17,760 --> 01:02:20,155 So keep-- other questions? 849 01:02:31,790 --> 01:02:35,270 So what do you think will happen in that final homework problem 850 01:02:35,270 --> 01:02:39,010 with the monkey running-- now he has some velocity. 851 01:02:39,010 --> 01:02:43,390 How will that problem differ from what we've done? 852 01:02:43,390 --> 01:02:44,406 Do you have a question? 853 01:02:44,406 --> 01:02:45,530 Do you want to answer that? 854 01:02:45,530 --> 01:02:46,372 Yeah. 855 01:02:46,372 --> 01:02:48,330 AUDIENCE: Well, [INAUDIBLE] monkey [INAUDIBLE]. 856 01:02:51,790 --> 01:02:53,040 PROFESSOR: It's going to what? 857 01:02:53,040 --> 01:02:53,910 AUDIENCE: Look like a circle. 858 01:02:53,910 --> 01:02:55,730 PROFESSOR: It's going to look like a circle, OK. 859 01:02:55,730 --> 01:02:57,360 He'll be going in a circle, at any-- 860 01:02:57,360 --> 01:02:59,857 AUDIENCE: [INAUDIBLE] as in a spiral. 861 01:02:59,857 --> 01:03:00,690 PROFESSOR: A spiral. 862 01:03:00,690 --> 01:03:02,980 He'll be going like a helix, huh? 863 01:03:02,980 --> 01:03:04,000 All right. 864 01:03:04,000 --> 01:03:07,290 Yeah, the monkey will be going in a helix, yeah. 865 01:03:07,290 --> 01:03:08,230 AUDIENCE: [INAUDIBLE]. 866 01:03:08,230 --> 01:03:11,862 PROFESSOR: What forces will act on that monkey? 867 01:03:11,862 --> 01:03:13,502 AUDIENCE: [INAUDIBLE]. 868 01:03:13,502 --> 01:03:14,960 PROFESSOR: In what direction do you 869 01:03:14,960 --> 01:03:17,890 think there will be forces acting on that-- he's hanging 870 01:03:17,890 --> 01:03:19,060 on for dear life, you know. 871 01:03:19,060 --> 01:03:20,143 This thing's going around. 872 01:03:20,143 --> 01:03:22,460 They could throw him off, right? 873 01:03:22,460 --> 01:03:23,930 So what forces act? 874 01:03:23,930 --> 01:03:25,700 And if you could figure out what forces 875 01:03:25,700 --> 01:03:29,030 act-- so you draw a free body diagram of the monkey. 876 01:03:29,030 --> 01:03:31,490 There's going to be possibly forces in the theta hat 877 01:03:31,490 --> 01:03:36,780 direction, in the z direction, in the r direction. 878 01:03:36,780 --> 01:03:40,110 But just think physically where they come from. 879 01:03:40,110 --> 01:03:41,930 So we now know that certainly he's 880 01:03:41,930 --> 01:03:45,640 hanging on because he is undergoing 881 01:03:45,640 --> 01:03:48,080 centripetal acceleration. 882 01:03:48,080 --> 01:03:50,640 And in order to force him to go in that circle, 883 01:03:50,640 --> 01:03:54,870 there has to be an inward force applied by this shaft to him. 884 01:03:54,870 --> 01:03:56,400 So there's a force like that because 885 01:03:56,400 --> 01:04:00,260 of the centripetal acceleration. 886 01:04:00,260 --> 01:04:02,970 If it's speeding up, there's a force pushing 887 01:04:02,970 --> 01:04:05,370 him to make it go faster. 888 01:04:05,370 --> 01:04:06,700 But he's running up the shaft. 889 01:04:06,700 --> 01:04:07,710 What else is there? 890 01:04:13,056 --> 01:04:14,526 Are there any other accelerations? 891 01:04:17,819 --> 01:04:20,110 AUDIENCE: [INAUDIBLE] angular acceleration [INAUDIBLE]? 892 01:04:20,110 --> 01:04:21,540 PROFESSOR: Yeah, there's angular acceleration. 893 01:04:21,540 --> 01:04:22,890 That's the thing trying to speed up. 894 01:04:22,890 --> 01:04:25,306 And so he's hanging on because this thing is accelerating, 895 01:04:25,306 --> 01:04:27,970 like pushing you back in the car seat, right? 896 01:04:27,970 --> 01:04:29,270 Linear acceleration. 897 01:04:29,270 --> 01:04:31,690 Well, this is trying to-- at any instant in time, 898 01:04:31,690 --> 01:04:33,020 it's trying to go faster. 899 01:04:33,020 --> 01:04:35,420 So he's having to hang on because of that. 900 01:04:35,420 --> 01:04:38,400 So that's one of those terms. 901 01:04:38,400 --> 01:04:40,070 But now he's moving. 902 01:04:40,070 --> 01:04:42,820 He's also running up the shaft. 903 01:04:42,820 --> 01:04:45,410 Will that lead to any other accelerations? 904 01:04:45,410 --> 01:04:47,450 And force equals mass times acceleration. 905 01:04:47,450 --> 01:04:49,230 So every time you can-- if you can 906 01:04:49,230 --> 01:04:51,750 account for all the accelerations in the system, 907 01:04:51,750 --> 01:04:52,790 multiply it by m. 908 01:04:52,790 --> 01:04:58,000 You've accounted for all of the forces, sum 909 01:04:58,000 --> 01:05:00,580 of the forces of the mass times acceleration. 910 01:05:00,580 --> 01:05:03,320 If you add a new acceleration, you better add a new force. 911 01:05:03,320 --> 01:05:05,990 If you add a new force, you'll probably add a new torque. 912 01:05:05,990 --> 01:05:07,910 AUDIENCE: So we have to account for gravity? 913 01:05:07,910 --> 01:05:10,220 PROFESSOR: Well, gravity, yeah. 914 01:05:10,220 --> 01:05:12,241 What else? 915 01:05:12,241 --> 01:05:14,730 AUDIENCE: [INAUDIBLE]. 916 01:05:14,730 --> 01:05:17,250 PROFESSOR: So he's suggesting there might be a Coriolis 917 01:05:17,250 --> 01:05:19,490 acceleration. 918 01:05:19,490 --> 01:05:21,720 And the Coriolis acceleration, in order 919 01:05:21,720 --> 01:05:25,920 to make the monkey accelerate, according to that term, 920 01:05:25,920 --> 01:05:28,750 there will have to be yet another force. 921 01:05:28,750 --> 01:05:31,650 And I think-- so we'll see if that turns up 922 01:05:31,650 --> 01:05:34,584 in the calculation. 923 01:05:34,584 --> 01:05:35,750 You've got a couple minutes. 924 01:05:35,750 --> 01:05:38,680 I want you to do the money cards. 925 01:05:38,680 --> 01:05:45,050 Think about-- and then on your way out, 926 01:05:45,050 --> 01:05:47,900 I think just pile them up down here on the table, 927 01:05:47,900 --> 01:05:50,605 or hand them to me or one of the TAs. 928 01:05:53,570 --> 01:05:57,610 And that'll help me understand what you understood 929 01:05:57,610 --> 01:05:59,872 or didn't understand today.