1 0:00:01 --> 00:00:07 Okay, you did have some problems with physical pendulums, 2 00:00:04 --> 00:00:10 and I want to talk a little bit more about physical pendulums. 3 00:00:07 --> 00:00:13 Let's first look at the picture in very general terms. 4 00:00:13 --> 00:00:19 I have here a solid object, which is rotating about point P 5 00:00:18 --> 00:00:24 about an axis vertical to the blackboard, 6 00:00:20 --> 00:00:26 and here at C is the center of mass. 7 00:00:24 --> 00:00:30 The object has a mass M, and so there is here a force Mg, 8 00:00:30 --> 00:00:36 and let the separation be here b. 9 00:00:34 --> 00:00:40 I'm going to offset it over an angle theta, 10 00:00:37 --> 00:00:43 and I'm going to oscillate it. 11 00:00:39 --> 00:00:45 Clearly, there has to be a force at the pin. 12 00:00:42 --> 00:00:48 If there were no force at the pin, 13 00:00:44 --> 00:00:50 this object would be accelerated down with acceleration g, 14 00:00:47 --> 00:00:53 and that's not what's going to happen. 15 00:00:50 --> 00:00:56 But I don't care about that force 16 00:00:51 --> 00:00:57 because I'm going to take the torque about point P. 17 00:00:54 --> 00:01:00 Remember when we had a spring, just a one-dimensional case, 18 00:00:59 --> 00:01:05 we had F equals ma, and that, for the spring, became minus kx, 19 00:01:04 --> 00:01:10 and the minus sign indicates that it's a restoring force. 20 00:01:08 --> 00:01:14 So we now get something very similar. 21 00:01:12 --> 00:01:18 In rotation, force becomes torque, 22 00:01:14 --> 00:01:20 mass becomes moment of inertia, 23 00:01:16 --> 00:01:22 and acceleration becomes angular acceleration. 24 00:01:20 --> 00:01:26 So now we have minus r cross F, 25 00:01:25 --> 00:01:31 and the minus sign indicates that it is restoring. 26 00:01:28 --> 00:01:34 So if I take the torque relative to point P, then I have... 27 00:01:32 --> 00:01:38 This is the position vector, which has magnitude b. 28 00:01:36 --> 00:01:42 The force is Mg, and I have to multiply by the sine of theta. 29 00:01:40 --> 00:01:46 So I have b times Mg times the sine of theta 30 00:01:46 --> 00:01:52 and that now equals minus... I can bring the minus here-- 31 00:01:52 --> 00:01:58 minus the moment of inertia about point P times alpha, 32 00:01:56 --> 00:02:02 and alpha is the angular acceleration, 33 00:01:58 --> 00:02:04 which is theta double dot. 34 00:02:01 --> 00:02:07 I bring them together, 35 00:02:03 --> 00:02:09 and I use the small-angle approximation, small angles. 36 00:02:09 --> 00:02:15 Then the sine of theta is approximately theta 37 00:02:13 --> 00:02:19 if theta is in radians. 38 00:02:15 --> 00:02:21 And so I bring this all on one side, 39 00:02:18 --> 00:02:24 so I get theta double dot plus bMg 40 00:02:24 --> 00:02:30 divided by the moment of inertia about that point P 41 00:02:28 --> 00:02:34 times theta-- now this is my small-angle approximation-- 42 00:02:31 --> 00:02:37 equals zero. 43 00:02:32 --> 00:02:38 And this is a well-known equation. 44 00:02:35 --> 00:02:41 It is clearly a simple harmonic oscillation in theta 45 00:02:39 --> 00:02:45 because this is a constant. 46 00:02:41 --> 00:02:47 And so we're going to get as a solution 47 00:02:44 --> 00:02:50 that theta equals theta maximum-- 48 00:02:48 --> 00:02:54 you can call that the angular amplitude-- 49 00:02:50 --> 00:02:56 times cosine omega t plus phi. 50 00:02:55 --> 00:03:01 This omega is the angular frequency 51 00:02:58 --> 00:03:04 It is a constant. 52 00:03:01 --> 00:03:07 Omega here, theta dot, is the angular velocity, 53 00:03:05 --> 00:03:11 which isnot a constant. 54 00:03:07 --> 00:03:13 The two are completely different. 55 00:03:10 --> 00:03:16 That is the angular frequency. 56 00:03:12 --> 00:03:18 So we know that the solution to this differential equation 57 00:03:15 --> 00:03:21 gives me omega is the square root of this constant. 58 00:03:21 --> 00:03:27 So it is bMg divided by the moment of inertia 59 00:03:27 --> 00:03:33 about that point P. 60 00:03:28 --> 00:03:34 And so the period of oscillation is two pi divided by omega, 61 00:03:35 --> 00:03:41 and so that is two pi times the square root of I relative to point P divided by bMg. 62 00:03:44 --> 00:03:50 And let's hang on to this for a large part of this lecture 63 00:03:48 --> 00:03:54 because I'm going to apply this to various geometries. 64 00:03:52 --> 00:03:58 Make sure that I have it correct-- yes, I do. 65 00:03:55 --> 00:04:01 This is independent of the mass of the object. 66 00:03:58 --> 00:04:04 Even though you will say there is an M here, 67 00:04:00 --> 00:04:06 you will see that in all cases 68 00:04:02 --> 00:04:08 when we calculate the moment of inertia about point P 69 00:04:04 --> 00:04:10 that there is always a mass up here. 70 00:04:07 --> 00:04:13 So the mass will disappear, as you will see very shortly. 71 00:04:11 --> 00:04:17 I have four objects here, 72 00:04:13 --> 00:04:19 and they all have different moments of inertia. 73 00:04:17 --> 00:04:23 They're all going to rotate 74 00:04:19 --> 00:04:25 about an axis perpendicular to the blackboard, so to speak, 75 00:04:24 --> 00:04:30 and we're going to massage each one of them 76 00:04:28 --> 00:04:34 to predict their periods. 77 00:04:31 --> 00:04:37 Let's first go to the rod. 78 00:04:35 --> 00:04:41 So we first do the rod. 79 00:04:39 --> 00:04:45 We have the rod here. 80 00:04:43 --> 00:04:49 This is point P, and here is the center of the rod. 81 00:04:47 --> 00:04:53 The rod has mass M and it has length L. 82 00:04:53 --> 00:04:59 So we have here Mg. 83 00:04:58 --> 00:05:04 I don't have to worry about this anymore. 84 00:05:00 --> 00:05:06 I simply go to this equation, and I want to know 85 00:05:03 --> 00:05:09 what the period is of this rod, of this oscillating rod. 86 00:05:06 --> 00:05:12 All I have to know now is 87 00:05:08 --> 00:05:14 what is the moment of inertia about P. 88 00:05:10 --> 00:05:16 And I know already that b in that equation equals one-half L. 89 00:05:17 --> 00:05:23 So, what is the moment of inertia 90 00:05:19 --> 00:05:25 of oscillation about point P? 91 00:05:21 --> 00:05:27 I have to apply now the parallel axis theorem-- 92 00:05:24 --> 00:05:30 which you also had to do during the exam-- 93 00:05:27 --> 00:05:33 which says it is the moment of inertia 94 00:05:30 --> 00:05:36 of rotation about the center of mass, 95 00:05:32 --> 00:05:38 which, in our case, is C-- the axes have to be parallel, 96 00:05:36 --> 00:05:42 so there is this axis perpendicular to the blackboard, 97 00:05:40 --> 00:05:46 and this axis perpendicular to the blackboard-- 98 00:05:43 --> 00:05:49 plus the mass of the rod 99 00:05:45 --> 00:05:51 times the distance between P and c squared-- 100 00:05:49 --> 00:05:55 plus M times this distance squared-- 101 00:05:53 --> 00:05:59 so that is b squared, 102 00:05:55 --> 00:06:01 and b squared is one-quarter L squared. 103 00:05:59 --> 00:06:05 What is the moment of inertia 104 00:06:01 --> 00:06:07 for rotation of a rod about this axis? 105 00:06:03 --> 00:06:09 I looked it up in a table. 106 00:06:05 --> 00:06:11 I happen to remember it now, because I am lecturing 801. 107 00:06:09 --> 00:06:15 Two months from now, I will have forgotten. 108 00:06:11 --> 00:06:17 So I remember now that it is 1/12 ML squared 109 00:06:17 --> 00:06:23 plus one-quarter ML squared. 110 00:06:21 --> 00:06:27 That becomes one-third ML squared. 111 00:06:24 --> 00:06:30 And so the period T becomes two pi 112 00:06:31 --> 00:06:37 times the square root of this moment of inertia, 113 00:06:34 --> 00:06:40 which is the one-third ML squared, divided by bMg, 114 00:06:42 --> 00:06:48 and b is one-half L for this geometry. 115 00:06:45 --> 00:06:51 So one-half LMg, and notice, indeed, as I anticipated, 116 00:06:52 --> 00:06:58 you always lose your M, you also lose one L here, 117 00:06:57 --> 00:07:03 and so you get two pi times 118 00:07:00 --> 00:07:06 the square root of two-thirds L divided by g. 119 00:07:05 --> 00:07:11 So that is the period that we predict for the rod. 120 00:07:08 --> 00:07:14 So let's write that under here, 121 00:07:11 --> 00:07:17 because we are going to compare them shortly. 122 00:07:14 --> 00:07:20 So this is two pi times the square root of two-thirds L 123 00:07:23 --> 00:07:29 divided by g. 124 00:07:25 --> 00:07:31 125 00:07:28 --> 00:07:34 The rod that we have here is designed in such a way 126 00:07:32 --> 00:07:38 that the period is very close to one second. 127 00:07:36 --> 00:07:42 That was our goal. 128 00:07:38 --> 00:07:44 So T is as close as we can get it to 1.00 seconds. 129 00:07:45 --> 00:07:51 And so if you substitute in this equation T equals one, 130 00:07:50 --> 00:07:56 you will find that the length of this rod, 131 00:07:53 --> 00:07:59 if it is really pivoting at the very end, 132 00:07:56 --> 00:08:02 should be about 37.2 centimeters. 133 00:07:58 --> 00:08:04 134 00:08:02 --> 00:08:08 So we did the best we can when we made this rod. 135 00:08:05 --> 00:08:11 There is always an uncertainty, of course-- 136 00:08:08 --> 00:08:14 how you drill the holes and where you drill the holes-- 137 00:08:11 --> 00:08:17 so I would say the value that we actually achieved is 37.2, 138 00:08:15 --> 00:08:21 probably with an uncertainty of about three millimeters, 139 00:08:19 --> 00:08:25 so 0.3 centimeters. 140 00:08:21 --> 00:08:27 That's what we have. 141 00:08:23 --> 00:08:29 That is an error of one part in 370. 142 00:08:26 --> 00:08:32 Let's make that... round that off. 143 00:08:29 --> 00:08:35 That's a one-percent error in the length. 144 00:08:32 --> 00:08:38 Since the length is under the square root, 145 00:08:35 --> 00:08:41 the one-percent error becomes half a percent error, 146 00:08:39 --> 00:08:45 so the period that I then would predict 147 00:08:41 --> 00:08:47 is about 1 plus half a percent, 148 00:08:47 --> 00:08:53 so that is 0.005 seconds. 149 00:08:51 --> 00:08:57 So that, then, has a one-half-percent error. 150 00:08:55 --> 00:09:01 So this is my predicted period. 151 00:08:57 --> 00:09:03 152 00:09:00 --> 00:09:06 And so we're going to make ten oscillations 153 00:09:03 --> 00:09:09 of the observed oscillations. 154 00:09:06 --> 00:09:12 We're going to get a number. 155 00:09:08 --> 00:09:14 My reaction time is not much better than a tenth of a second. 156 00:09:13 --> 00:09:19 So we're going to get a number there. 157 00:09:16 --> 00:09:22 We divide this number by ten. 158 00:09:18 --> 00:09:24 And so we can always calculate the period, then, 159 00:09:21 --> 00:09:27 and then we get a much improved error 160 00:09:23 --> 00:09:29 of a hundredth of a second, 161 00:09:25 --> 00:09:31 because we will divide this number by ten, 162 00:09:28 --> 00:09:34 so this also is going to be divided by ten. 163 00:09:31 --> 00:09:37 And let's see how close we were able to get this 164 00:09:34 --> 00:09:40 to the 1.0 seconds. 165 00:09:37 --> 00:09:43 166 00:09:39 --> 00:09:45 So, here is the rod. 167 00:09:42 --> 00:09:48 Turn this on, the timer. 168 00:09:44 --> 00:09:50 169 00:09:47 --> 00:09:53 We'll offset the rod, 170 00:09:49 --> 00:09:55 and I will start it when it stops somewhere. 171 00:09:52 --> 00:09:58 Now-- one, two, three, four, five, six, seven, eight, nine, 172 00:10:02 --> 00:10:08 ten... not bad. 173 00:10:06 --> 00:10:12 9.92-- well within the prediction. 174 00:10:13 --> 00:10:19 9.92, and so this becomes 0.992 plus or minus .01, 175 00:10:21 --> 00:10:27 and you see that is well within the prediction that I made. 176 00:10:27 --> 00:10:33 Now, all these four objects were designed in such a way 177 00:10:30 --> 00:10:36 that they have exactly the same period of one second. 178 00:10:34 --> 00:10:40 And now comes the question, 179 00:10:36 --> 00:10:42 how do the dimensions relate to each other 180 00:10:40 --> 00:10:46 in order to get them a period of one second? 181 00:10:44 --> 00:10:50 So let's now calculate the period 182 00:10:49 --> 00:10:55 for the other three objects: 183 00:10:51 --> 00:10:57 for the ring, for the disk and for the pendulum. 184 00:10:53 --> 00:10:59 Let's start with the pendulum. 185 00:10:55 --> 00:11:01 186 00:10:58 --> 00:11:04 For the pendulum, center of mass is here. 187 00:11:02 --> 00:11:08 Here is point P. 188 00:11:05 --> 00:11:11 I give the pendulum length little l-- 189 00:11:07 --> 00:11:13 you see that on the right blackboard there. 190 00:11:10 --> 00:11:16 So b equals l-- that's the separation between P and C. 191 00:11:15 --> 00:11:21 The moment of inertia about point P is very easy now. 192 00:11:20 --> 00:11:26 This has no mass-- the mass is all here, mass capital M, 193 00:11:26 --> 00:11:32 so that must be M times l squared. 194 00:11:29 --> 00:11:35 So I go to this equation, 195 00:11:31 --> 00:11:37 and so I ask, what is the period of a pendulum? 196 00:11:35 --> 00:11:41 And you're not surprised that you find 197 00:11:37 --> 00:11:43 two pi times the square root of l over g. 198 00:11:40 --> 00:11:46 We've seen that before, but, of course, it also comes out 199 00:11:44 --> 00:11:50 if you do it in this more complicated way. 200 00:11:47 --> 00:11:53 So here we get two pi times the square root of l over g. 201 00:11:55 --> 00:12:01 Let's now do the ring. 202 00:12:00 --> 00:12:06 203 00:12:05 --> 00:12:11 This is the ring. 204 00:12:07 --> 00:12:13 205 00:12:09 --> 00:12:15 Pivot about point P here. 206 00:12:11 --> 00:12:17 Here is the center of mass 207 00:12:13 --> 00:12:19 right in the middle, in the middle of nowhere. 208 00:12:16 --> 00:12:22 Right here is the center of mass. 209 00:12:20 --> 00:12:26 And so the distance b is the radius of the ring, 210 00:12:25 --> 00:12:31 so we have to calculate the moment of inertia about point P. 211 00:12:30 --> 00:12:36 Again, we have to use now the parallel axis theorem. 212 00:12:33 --> 00:12:39 It is the moment of inertia for rotation about this axis 213 00:12:37 --> 00:12:43 through point C, through a center of mass. 214 00:12:40 --> 00:12:46 That is MR squared, if R is the radius of the ring. 215 00:12:45 --> 00:12:51 All the mass is at the circumference, 216 00:12:48 --> 00:12:54 all at a distance R from the center of mass. 217 00:12:52 --> 00:12:58 And then we have to add to that-- 218 00:12:55 --> 00:13:01 according to the parallel axis theorem-- 219 00:12:57 --> 00:13:03 the mass times this distance squared, and this distance is R, 220 00:13:01 --> 00:13:07 so you get MR squared. 221 00:13:04 --> 00:13:10 So it is 2MR squared. 222 00:13:07 --> 00:13:13 So what is the period of an oscillation? 223 00:13:13 --> 00:13:19 I of P, that is 2MR squared, divided by bMg-- b is R-- RMg. 224 00:13:24 --> 00:13:30 225 00:13:26 --> 00:13:32 I lose one R, I lose an M, and we have seen this before. 226 00:13:31 --> 00:13:37 I derived this in lecture number 21. 227 00:13:35 --> 00:13:41 I remember everything by lecture number, believe it or not. 228 00:13:40 --> 00:13:46 So the period, now, 229 00:13:41 --> 00:13:47 is the same as a pendulum with length two R. 230 00:13:46 --> 00:13:52 This is the ring. 231 00:13:48 --> 00:13:54 And so here we have 232 00:13:50 --> 00:13:56 two pi times the square root of 2R divided by g. 233 00:13:55 --> 00:14:01 I'll make a comparison very shortly. 234 00:13:57 --> 00:14:03 I just want to finish them all. 235 00:13:59 --> 00:14:05 And I now would like to do the disk, last but not least. 236 00:14:07 --> 00:14:13 237 00:14:09 --> 00:14:15 For the disk, all we have to do now 238 00:14:11 --> 00:14:17 is calculate the moment of inertia. 239 00:14:13 --> 00:14:19 This was very close to your... the problem you had on the exam. 240 00:14:18 --> 00:14:24 So, here you have the disk, a solid disk. 241 00:14:21 --> 00:14:27 This is point P, this is the center of mass, 242 00:14:24 --> 00:14:30 but now it's solid, so again, b is R, 243 00:14:28 --> 00:14:34 the separation between c and b. 244 00:14:31 --> 00:14:37 And the moment of inertia for rotation about point P 245 00:14:36 --> 00:14:42 is now the moment of inertia 246 00:14:38 --> 00:14:44 for rotation about the center of mass, 247 00:14:40 --> 00:14:46 which you look up in a table-- 248 00:14:41 --> 00:14:47 in the case of your exam, it was given on the front cover-- 249 00:14:44 --> 00:14:50 one-half MR squared. 250 00:14:48 --> 00:14:54 That's the moment of inertia 251 00:14:50 --> 00:14:56 for rotation about the center of mass. 252 00:14:52 --> 00:14:58 And now we have to add Mb squared, and b equals R, 253 00:14:58 --> 00:15:04 so we have to add MR squared. 254 00:15:02 --> 00:15:08 So we find three-halves MR squared. 255 00:15:08 --> 00:15:14 So what is the period of oscillation? 256 00:15:11 --> 00:15:17 Two pi times three-halves MR squared 257 00:15:18 --> 00:15:24 divided by bMg-- b is R-- 258 00:15:22 --> 00:15:28 RMg, and that equals two pi times the square root... 259 00:15:32 --> 00:15:38 I lose my M as always, I lose an R. 260 00:15:36 --> 00:15:42 I get here three-halves R divided by g. 261 00:15:40 --> 00:15:46 And so let's write that down here, so we have 262 00:15:45 --> 00:15:51 two pi times the square root of three-halves R 263 00:15:54 --> 00:16:00 divided by g. 264 00:15:56 --> 00:16:02 So we have them all four there, 265 00:15:58 --> 00:16:04 and so we can now make a meaningful comparison. 266 00:16:04 --> 00:16:10 We want the periods to be the same, 267 00:16:07 --> 00:16:13 so we can hang on to those numbers, 268 00:16:09 --> 00:16:15 so we don't need this anymore. 269 00:16:11 --> 00:16:17 We want the periods to be the same. 270 00:16:14 --> 00:16:20 We already have established 271 00:16:16 --> 00:16:22 that the period of the rod is close to the one second, 272 00:16:19 --> 00:16:25 so we're not going to measure them anymore. 273 00:16:22 --> 00:16:28 We just want to compare them-- 274 00:16:24 --> 00:16:30 whether, indeed, they have the same period of oscillation-- 275 00:16:28 --> 00:16:34 by making them oscillate in unison. 276 00:16:31 --> 00:16:37 But we want to know 277 00:16:32 --> 00:16:38 what the relative ratios are of these dimensions 278 00:16:34 --> 00:16:40 the way we designed it. 279 00:16:35 --> 00:16:41 So let's first go to the rod, then, and make a comparison 280 00:16:39 --> 00:16:45 between the rod and the pendulum. 281 00:16:41 --> 00:16:47 So if you look at the rod-- 282 00:16:43 --> 00:16:49 and we use the pendulum as our standard, with length little l-- 283 00:16:48 --> 00:16:54 then you see the period will be the same if 1½ R is little l... 284 00:16:53 --> 00:16:59 Oh, sorry, the rod. 285 00:16:55 --> 00:17:01 If two-thirds capital L is little l. 286 00:16:58 --> 00:17:04 So, for the rod, two-thirds L equals little l, 287 00:17:06 --> 00:17:12 so capital L is 1½ l. 288 00:17:10 --> 00:17:16 It has to be exactly 1½ times the length of the pendulum, 289 00:17:15 --> 00:17:21 so this length has to be exactly 1½ times this length 290 00:17:18 --> 00:17:24 to the center of mass, 291 00:17:20 --> 00:17:26 which is the center of that billiard ball. 292 00:17:22 --> 00:17:28 And that's what we tried, to the best that we could. 293 00:17:24 --> 00:17:30 So let's now go to the ring. 294 00:17:27 --> 00:17:33 For the ring to have the same period, one second, 295 00:17:32 --> 00:17:38 2R has to be the same as the length of the pendulum. 296 00:17:37 --> 00:17:43 So 2R equals l, so we can put in here for L, 297 00:17:47 --> 00:17:53 this has to be 1½ times l, 298 00:17:51 --> 00:17:57 and here, this now, which is 2R, has to be l. 299 00:18:00 --> 00:18:06 Very nonintuitive, that this length here 300 00:18:03 --> 00:18:09 is the same as the diameter of the ring. 301 00:18:06 --> 00:18:12 Not at all obvious. 302 00:18:09 --> 00:18:15 And so now we go for the disk. 303 00:18:13 --> 00:18:19 So now we require that three-halves R, 1½ R... 304 00:18:20 --> 00:18:26 We want that to be l, so we want R to be 2l divided by three. 305 00:18:26 --> 00:18:32 So we want the diameter 2R to be 4l divided by three. 306 00:18:31 --> 00:18:37 And when you look at the disk, 307 00:18:33 --> 00:18:39 it's hard to see that it is exactly four-thirds, 308 00:18:36 --> 00:18:42 but you can see that it is longer than the pendulum-- 309 00:18:39 --> 00:18:45 it should be one-third longer-- but it is not as long as the rod 310 00:18:44 --> 00:18:50 because the rod is 1½ times the length of the pendulum, 311 00:18:48 --> 00:18:54 and so we can now complete that picture. 312 00:18:51 --> 00:18:57 And we have now that the diameter here-- 313 00:18:58 --> 00:19:04 2R-- is now four-thirds times l. 314 00:19:04 --> 00:19:10 And so what we can do now, we can play with them. 315 00:19:07 --> 00:19:13 We can oscillate them simultaneously 316 00:19:09 --> 00:19:15 and just see how well they track each other. 317 00:19:12 --> 00:19:18 There's no sense in giving you the dimensions. 318 00:19:16 --> 00:19:22 The rod was 37.2 centimeters. 319 00:19:19 --> 00:19:25 Let me write that down, because we calculated that. 320 00:19:23 --> 00:19:29 So this was 37.2 centimeters, and I think that translates 321 00:19:30 --> 00:19:36 into-- for the pendulum-- 24.8 centimeters. 322 00:19:36 --> 00:19:42 But for the others 323 00:19:38 --> 00:19:44 I leave it up to you to calculate the dimensions. 324 00:19:41 --> 00:19:47 Yes, 25... 24.8 is correct. 325 00:19:44 --> 00:19:50 So timing is not useful anymore. 326 00:19:47 --> 00:19:53 Let's just see how these two go together. 327 00:19:51 --> 00:19:57 So we offset them, and then we let them go. 328 00:19:54 --> 00:20:00 And they go pretty much in unison. 329 00:19:56 --> 00:20:02 If you wait long enough, of course, 330 00:19:57 --> 00:20:03 you will see there is a difference. 331 00:19:59 --> 00:20:05 You can never make them exactly the same, 332 00:20:02 --> 00:20:08 but they track each other nicely. 333 00:20:05 --> 00:20:11 We can now also use the rod and the disk. 334 00:20:07 --> 00:20:13 335 00:20:12 --> 00:20:18 They track each other beautifully. 336 00:20:13 --> 00:20:19 They're both very close to 1.00 seconds. 337 00:20:20 --> 00:20:26 And we can have the disk versus the pendulum. 338 00:20:25 --> 00:20:31 339 00:20:27 --> 00:20:33 And you see they track each other very nicely. 340 00:20:31 --> 00:20:37 But wait long enough and you will see 341 00:20:33 --> 00:20:39 that, of course, the periods will be different. 342 00:20:36 --> 00:20:42 343 00:20:39 --> 00:20:45 So, this is my last word on physical pendulums, 344 00:20:41 --> 00:20:47 but you may see it again on the final. 345 00:20:43 --> 00:20:49 Not maybe-- you can almost count on that, I'm telling you. 346 00:20:48 --> 00:20:54 Okay, I want to discuss now some other interesting oscillation-- 347 00:20:52 --> 00:20:58 again, simple harmonic-- 348 00:20:54 --> 00:21:00 and that is liquid in a u-tube, which you see there. 349 00:21:01 --> 00:21:07 If I have here a tube, which has everywhere... 350 00:21:07 --> 00:21:13 it's open on both sides 351 00:21:10 --> 00:21:16 and everywhere the same cross-section, 352 00:21:16 --> 00:21:22 and I put a liquid in here, in equilibrium, just like that, 353 00:21:23 --> 00:21:29 and the liquid has mass M. 354 00:21:27 --> 00:21:33 It has density rho. 355 00:21:29 --> 00:21:35 The area of the tube is A, and the length of the liquid is l. 356 00:21:39 --> 00:21:45 So this is l. 357 00:21:44 --> 00:21:50 358 00:21:46 --> 00:21:52 I'm going to offset it, the liquid, 359 00:21:49 --> 00:21:55 and I wanted to see it oscillate, and I wanted to see 360 00:21:53 --> 00:21:59 whether I could calculate the period of the oscillation. 361 00:21:58 --> 00:22:04 The total mass of the liquid that I have... 362 00:22:03 --> 00:22:09 the total mass is the volume, 363 00:22:06 --> 00:22:12 which is the area times the length times rho. 364 00:22:11 --> 00:22:17 I'm going to offset it 365 00:22:13 --> 00:22:19 so that this is higher over a distance y. 366 00:22:19 --> 00:22:25 So this, then, is lower over a distance y. 367 00:22:24 --> 00:22:30 368 00:22:27 --> 00:22:33 So this distance is also y, same as that. 369 00:22:31 --> 00:22:37 So the liquid now is here, and then I release it, 370 00:22:34 --> 00:22:40 and it will start to oscillate. 371 00:22:36 --> 00:22:42 372 00:22:39 --> 00:22:45 Well, when it starts to oscillate, 373 00:22:45 --> 00:22:51 there comes a time that the liquid, the whole liquid 374 00:22:48 --> 00:22:54 is going to slosh back and forth and so everywhere in the tube, 375 00:22:53 --> 00:22:59 the velocity at any moment in time will be the same 376 00:22:57 --> 00:23:03 because the cross-section is not changing-- 377 00:22:59 --> 00:23:05 it's the same everywhere. 378 00:23:01 --> 00:23:07 See, if there is a certain velocity here v, 379 00:23:03 --> 00:23:09 then it's the same as the velocity here, 380 00:23:06 --> 00:23:12 as the velocity there, as the velocity there. 381 00:23:10 --> 00:23:16 And that, of course, is y dot. 382 00:23:13 --> 00:23:19 That's the first derivative of that position here. 383 00:23:19 --> 00:23:25 I'm going to write down 384 00:23:21 --> 00:23:27 the conservation of total energy, mechanical energy. 385 00:23:24 --> 00:23:30 I assume that there is no energy loss, 386 00:23:27 --> 00:23:33 although there probably is some. 387 00:23:29 --> 00:23:35 Friction inside the liquid will probably generate some heat 388 00:23:32 --> 00:23:38 and that will cause some damping. 389 00:23:34 --> 00:23:40 You will see that when we do the experiment. 390 00:23:37 --> 00:23:43 For now, I will assume that that's not the case. 391 00:23:40 --> 00:23:46 So what, now, is then the total energy of the system-- that is, 392 00:23:43 --> 00:23:49 the sum of the kinetic energy plus the potential energy? 393 00:23:47 --> 00:23:53 And if we assume that that's constant, 394 00:23:49 --> 00:23:55 we will be able to find the period of the oscillation 395 00:23:53 --> 00:23:59 very shortly, as you will see. 396 00:23:55 --> 00:24:01 The kinetic energy of the liquid is easy. 397 00:23:59 --> 00:24:05 That is one-half M times the velocity squared, 398 00:24:03 --> 00:24:09 and the velocity, we agreed, is y dot squared. 399 00:24:09 --> 00:24:15 Now the potential energy. 400 00:24:11 --> 00:24:17 I call the potential energy here, I call that u equals zero. 401 00:24:16 --> 00:24:22 When the liquid is standing here and the liquid is standing here, 402 00:24:22 --> 00:24:28 I call that potential energy zero. 403 00:24:23 --> 00:24:29 404 00:24:27 --> 00:24:33 The mass that is now above this level here, I call that delta M, 405 00:24:32 --> 00:24:38 and delta M is the area times y times rho. 406 00:24:40 --> 00:24:46 This is how much mass there is here. 407 00:24:42 --> 00:24:48 It was taken away from here and was put here. 408 00:24:46 --> 00:24:52 How much work do you have to do 409 00:24:48 --> 00:24:54 to take this liquid and put it there? 410 00:24:50 --> 00:24:56 Well, that's the same when you take this liquid 411 00:24:54 --> 00:25:00 and put it here. 412 00:24:55 --> 00:25:01 And when you bring this liquid which was therehere, 413 00:24:59 --> 00:25:05 then you have moved it up over a distance y, 414 00:25:03 --> 00:25:09 and so the gravitational potential energy increases 415 00:25:08 --> 00:25:14 by delta M-- this is the amount of mass here-- 416 00:25:12 --> 00:25:18 times g times h, and h is y. 417 00:25:14 --> 00:25:20 Mgh, remember? 418 00:25:16 --> 00:25:22 That's the increase in potential energy. 419 00:25:18 --> 00:25:24 And so I move an amount of mass which is delta M... 420 00:25:23 --> 00:25:29 I move it over a distance y. 421 00:25:26 --> 00:25:32 I bring it here, but that makes no difference, of course. 422 00:25:29 --> 00:25:35 And so this is the total energy, and this is now a constant. 423 00:25:36 --> 00:25:42 So I'm going to substitute in there the A, l and rho, 424 00:25:40 --> 00:25:46 so I get one-half Al rho velocity squared plus A rho g. 425 00:25:52 --> 00:25:58 And I get a y squared equals a constant, 426 00:25:56 --> 00:26:02 because I have a y here and I have a y there. 427 00:26:00 --> 00:26:06 We've done this before-- this is the conservation of energy, 428 00:26:03 --> 00:26:09 and in order to find 429 00:26:05 --> 00:26:11 the period of the simple harmonic oscillation, 430 00:26:09 --> 00:26:15 we take the time derivative. 431 00:26:11 --> 00:26:17 By the way, before we do that, 432 00:26:14 --> 00:26:20 this is... this was delta M and this is an A, right? Yeah. 433 00:26:19 --> 00:26:25 A, rho-- yeah, that's it. 434 00:26:22 --> 00:26:28 So we lose A, we lose rho, 435 00:26:25 --> 00:26:31 and we continue with what we have. 436 00:26:29 --> 00:26:35 And so we're going to take 437 00:26:31 --> 00:26:37 the derivative versus time of this equation. 438 00:26:34 --> 00:26:40 That gives me one-half l. 439 00:26:36 --> 00:26:42 The two pops out, and the two becomes a one, 440 00:26:40 --> 00:26:46 so I get 2y dot, then I apply the chain rule, 441 00:26:45 --> 00:26:51 so I get y double dot. 442 00:26:47 --> 00:26:53 Here I get plus g. 443 00:26:49 --> 00:26:55 The two pops out, becomes 2y, 444 00:26:52 --> 00:26:58 and then I get the chain rule, y dot, and that equals zero. 445 00:26:57 --> 00:27:03 I lose my y dot, because I have y dot in both terms. 446 00:27:02 --> 00:27:08 This two eats up this two. 447 00:27:05 --> 00:27:11 And so I find that y double dot plus 2g 448 00:27:11 --> 00:27:17 divided by l times y equals zero, and that was my goal, 449 00:27:18 --> 00:27:24 because this is clearly a simple harmonic oscillation, 450 00:27:22 --> 00:27:28 because this is a constant. 451 00:27:23 --> 00:27:29 And it'll oscillate in the following way: 452 00:27:26 --> 00:27:32 y equals y max times the cosine of omega t plus phi, 453 00:27:30 --> 00:27:36 then this is the angular frequency, 454 00:27:33 --> 00:27:39 which is directly related to the period. 455 00:27:37 --> 00:27:43 Omega, angular frequency equals the square root of 2g over l, 456 00:27:43 --> 00:27:49 and so the period will be 457 00:27:45 --> 00:27:51 two pi times the square root of l over 2g. 458 00:27:49 --> 00:27:55 So this is the period for an oscillating liquid. 459 00:27:55 --> 00:28:01 Notice that it is the period that you would have had... 460 00:27:58 --> 00:28:04 would have obtained from a pendulum 461 00:28:00 --> 00:28:06 if the length of the pendulum were l... l over two-- 462 00:28:03 --> 00:28:09 not at all obvious, not at all intuitive. 463 00:28:08 --> 00:28:14 You see our setup here. 464 00:28:11 --> 00:28:17 I have to know what l is, and that is not so easy. 465 00:28:16 --> 00:28:22 Because of this radius here, if I measure l on the outside, 466 00:28:23 --> 00:28:29 it's substantially larger than on the inside. 467 00:28:25 --> 00:28:31 You may not think it's a big difference, 468 00:28:27 --> 00:28:33 but it's huge-- it's a nine-centimeter difference 469 00:28:30 --> 00:28:36 between the outside and the inside. 470 00:28:32 --> 00:28:38 If I take the average value between the two... 471 00:28:36 --> 00:28:42 if I take the average, I find 72 centimeters, 472 00:28:41 --> 00:28:47 and I could be off by one. 473 00:28:43 --> 00:28:49 474 00:28:47 --> 00:28:53 If I use this number for l and I substitute it in this equation, 475 00:28:53 --> 00:28:59 then I find my predicted period, which is 1.204, 476 00:29:02 --> 00:29:08 and because of this error that I have of one, that would give me 477 00:29:05 --> 00:29:11 an uncertainty of about 0.01 seconds. 478 00:29:11 --> 00:29:17 However, before we start measuring it-- 479 00:29:13 --> 00:29:19 and I will do ten oscillations 480 00:29:15 --> 00:29:21 to get a reasonable, accurate result-- 481 00:29:18 --> 00:29:24 I want to warn you. 482 00:29:21 --> 00:29:27 I make a prediction-- 483 00:29:22 --> 00:29:28 that the period that we will measure 484 00:29:25 --> 00:29:31 will probably be larger than this, 485 00:29:27 --> 00:29:33 and I can think of two reasons. 486 00:29:29 --> 00:29:35 The first is that the damping of this liquid will be huge. 487 00:29:33 --> 00:29:39 You will see how quickly it damps. 488 00:29:35 --> 00:29:41 In the past, we have never taken damping into account, 489 00:29:38 --> 00:29:44 and we won't do that in 801. 490 00:29:40 --> 00:29:46 But the damping has the effect on making the period longer. 491 00:29:43 --> 00:29:49 We've always ignored that, 492 00:29:45 --> 00:29:51 and in most of the demonstrations that we did-- 493 00:29:47 --> 00:29:53 like just now-- that was acceptable. 494 00:29:49 --> 00:29:55 It may not be acceptable for the liquid. 495 00:29:52 --> 00:29:58 But now there is a second point that I want you to think about. 496 00:29:56 --> 00:30:02 Is it correct that I take the average length, 497 00:30:00 --> 00:30:06 namely the average value 498 00:30:02 --> 00:30:08 between the outer length and the inner length? 499 00:30:06 --> 00:30:12 I don't think it is. 500 00:30:08 --> 00:30:14 I want you to think about why that is not correct. 501 00:30:11 --> 00:30:17 Look carefully 502 00:30:12 --> 00:30:18 where that l comes into my differential equation 503 00:30:15 --> 00:30:21 and you will probably come up with the right answer. 504 00:30:18 --> 00:30:24 And I claim that the actual l that we should have taken 505 00:30:21 --> 00:30:27 is a little bit larger-- 506 00:30:22 --> 00:30:28 I don't know how much larger, but it's a little bit larger-- 507 00:30:26 --> 00:30:32 and so that will also make the observed period 508 00:30:28 --> 00:30:34 become larger than the predicted one. 509 00:30:30 --> 00:30:36 So I'm not too optimistic 510 00:30:32 --> 00:30:38 that we will go and hit this the way we want to hit it, 511 00:30:35 --> 00:30:41 but that's good, because that's where the physics lies-- 512 00:30:38 --> 00:30:44 that you see that there are other factors 513 00:30:40 --> 00:30:46 that have to be taken into account. 514 00:30:42 --> 00:30:48 I'll turn this one on. 515 00:30:43 --> 00:30:49 Is it on now? Is it zeroed? 516 00:30:44 --> 00:30:50 I'll make it completely dark in the classroom, 517 00:30:47 --> 00:30:53 because you're going to see... 518 00:30:49 --> 00:30:55 otherwise, you can't see the liquids. 519 00:30:51 --> 00:30:57 520 00:30:52 --> 00:30:58 So you see the liquid now. 521 00:30:56 --> 00:31:02 Oh, you see these equations, too. 522 00:30:59 --> 00:31:05 Okay, it's zeroed. 523 00:31:03 --> 00:31:09 So, let me try to give this a swing, a large swing. 524 00:31:07 --> 00:31:13 It damps so enormously 525 00:31:09 --> 00:31:15 that I really want to get a very large swing. 526 00:31:12 --> 00:31:18 527 00:31:14 --> 00:31:20 That's nice. 528 00:31:16 --> 00:31:22 Now-- one... two... three... four... five... six... 529 00:31:24 --> 00:31:30 seven... eight... nine... 530 00:31:27 --> 00:31:33 Ten! 531 00:31:30 --> 00:31:36 Ah, not bad. 532 00:31:33 --> 00:31:39 12.18-- not bad. 533 00:31:37 --> 00:31:43 We'll get a little bit of light. 534 00:31:41 --> 00:31:47 12.18. 535 00:31:43 --> 00:31:49 So observed... 536 00:31:46 --> 00:31:52 Ten T observed... is it 12.18? 537 00:31:52 --> 00:31:58 Okay, my reaction time is 0.1. 538 00:31:55 --> 00:32:01 So T observed is 1.2... 539 00:32:01 --> 00:32:07 let's make it two plus or minus 0.01 seconds. 540 00:32:06 --> 00:32:12 Oh, that's not bad. 541 00:32:07 --> 00:32:13 It actually... there is an overlap. 542 00:32:09 --> 00:32:15 If you add this one here, you get one-to-one, 543 00:32:11 --> 00:32:17 and if you subtract it here, you get also one-to-one. 544 00:32:14 --> 00:32:20 So it's not bad. 545 00:32:15 --> 00:32:21 I expected it to be a little higher, 546 00:32:16 --> 00:32:22 but it's close enough to be happy. 547 00:32:19 --> 00:32:25 Think about why l should have been taken a little larger. 548 00:32:24 --> 00:32:30 549 00:32:27 --> 00:32:33 Now one more very interesting oscillation-- 550 00:32:31 --> 00:32:37 a torsional pendulum. 551 00:32:33 --> 00:32:39 There's a wire there, 2½-meter steel wire, 552 00:32:35 --> 00:32:41 and there's hanging something on the bottom, 553 00:32:37 --> 00:32:43 which we're going to offset, 554 00:32:38 --> 00:32:44 and then it's going to oscillate back and forth. 555 00:32:41 --> 00:32:47 That's called a torsional pendulum. 556 00:32:45 --> 00:32:51 And we're going to calculate 557 00:32:46 --> 00:32:52 the period of oscillation of the torsional pendulum, 558 00:32:49 --> 00:32:55 and they have wonderful properties. 559 00:32:51 --> 00:32:57 They are in a way like a spring, like a one-dimensional spring. 560 00:32:56 --> 00:33:02 Remember the one-dimensional spring 561 00:32:58 --> 00:33:04 that we had a period which was independent of the amplitude? 562 00:33:03 --> 00:33:09 Well, within reason, of course. 563 00:33:05 --> 00:33:11 If you make the amplitude too large, 564 00:33:07 --> 00:33:13 then you get permanent deformation of the spring. 565 00:33:10 --> 00:33:16 But you never had to make 566 00:33:11 --> 00:33:17 any small-angle approximation with the spring 567 00:33:15 --> 00:33:21 as we had to do with the pendulum. 568 00:33:17 --> 00:33:23 Here is the pendulum, the torsional pendulum. 569 00:33:23 --> 00:33:29 Here's a bar, and there is a weight here 570 00:33:26 --> 00:33:32 and there's a weight here-- 571 00:33:27 --> 00:33:33 I'll tell you more about that later. 572 00:33:29 --> 00:33:35 It's hanging here from the ceiling. 573 00:33:31 --> 00:33:37 It has a certain length l. 574 00:33:33 --> 00:33:39 This is point P. 575 00:33:34 --> 00:33:40 And we're going to twist it 576 00:33:37 --> 00:33:43 and then we are going to let it oscillate-- this bar-- 577 00:33:41 --> 00:33:47 in a horizontal plane. 578 00:33:43 --> 00:33:49 So when you look from above, you will see the bar here-- 579 00:33:49 --> 00:33:55 see point P here-- 580 00:33:50 --> 00:33:56 and then we can offset it over an angle theta, 581 00:33:55 --> 00:34:01 and then it will oscillate back and forth. 582 00:34:00 --> 00:34:06 The torque relative to point P is now very similar 583 00:34:10 --> 00:34:16 to what we had with a spring. 584 00:34:15 --> 00:34:21 We have a minus sign; 585 00:34:19 --> 00:34:25 again, that illustrates that it is restoring. 586 00:34:22 --> 00:34:28 Instead of a k now, we have kappa, 587 00:34:25 --> 00:34:31 which is what we call the torsional spring constant, 588 00:34:29 --> 00:34:35 and now we have an angle which we call theta. 589 00:34:33 --> 00:34:39 So we generate a torque which is proportional to the angle, 590 00:34:38 --> 00:34:44 very similar to the linear spring 591 00:34:39 --> 00:34:45 whereby we generate a force 592 00:34:41 --> 00:34:47 which is proportional to the linear displacement. 593 00:34:44 --> 00:34:50 Now you generate a torque 594 00:34:45 --> 00:34:51 which is linearly proportional to the angle. 595 00:34:48 --> 00:34:54 And this is 596 00:34:50 --> 00:34:56 the moment of inertia about point P times alpha, 597 00:34:53 --> 00:34:59 and alpha is theta double dot. 598 00:34:56 --> 00:35:02 So we're going to get 599 00:34:57 --> 00:35:03 that theta double dot plus kappa times theta 600 00:35:03 --> 00:35:09 divided by I of P equals zero. 601 00:35:06 --> 00:35:12 Kappa, by the way, is the torsional constant. 602 00:35:14 --> 00:35:20 603 00:35:16 --> 00:35:22 So we have a differential equation. 604 00:35:18 --> 00:35:24 It's clear that you're going to see 605 00:35:21 --> 00:35:27 a simple harmonic oscillation. 606 00:35:24 --> 00:35:30 This is a constant, 607 00:35:25 --> 00:35:31 and so you're going to get theta equals theta maximum 608 00:35:30 --> 00:35:36 times the cosine omega t plus phi. 609 00:35:32 --> 00:35:38 It's getting boring. 610 00:35:34 --> 00:35:40 This is the angular frequency, angular frequency... 611 00:35:38 --> 00:35:44 And angular frequency is the square root of kappa 612 00:35:42 --> 00:35:48 divided by the moment of inertia about point P. 613 00:35:46 --> 00:35:52 And therefore the period-- which is two pi divided by omega-- 614 00:35:51 --> 00:35:57 equals two pi times the square root 615 00:35:54 --> 00:36:00 moment of inertia about point P divided by kappa. 616 00:35:58 --> 00:36:04 617 00:36:01 --> 00:36:07 Well... how about kappa? 618 00:36:06 --> 00:36:12 Kappa is a function 619 00:36:08 --> 00:36:14 of the cross-sectional area here A and the length l. 620 00:36:14 --> 00:36:20 And it's also a function of what kind of material you have. 621 00:36:17 --> 00:36:23 Whether you have steel or nylon makes a big difference. 622 00:36:19 --> 00:36:25 That's very intuitive, of course. 623 00:36:22 --> 00:36:28 Remember that in an earlier lecture 624 00:36:25 --> 00:36:31 when we stressed a wire to the point that it was breaking. 625 00:36:30 --> 00:36:36 We dealt with Young's modulus. 626 00:36:33 --> 00:36:39 We had a wire 627 00:36:34 --> 00:36:40 and we had a mass hanging at the end of the wire. 628 00:36:38 --> 00:36:44 And we discussed the vertical oscillation. 629 00:36:41 --> 00:36:47 We could stress it and let it go, 630 00:36:43 --> 00:36:49 and then we would get an oscillation, like a spring, 631 00:36:47 --> 00:36:53 and that spring constant that we found, then, was Young's modulus 632 00:36:51 --> 00:36:57 times the cross-sectional area here divided by the length. 633 00:36:55 --> 00:37:01 And that was kind of pleasing. 634 00:36:57 --> 00:37:03 The thicker the bar, the stiffer it is; 635 00:37:00 --> 00:37:06 the longer the wire, the less stiff it is. 636 00:37:03 --> 00:37:09 Well, there is something very similar here, 637 00:37:05 --> 00:37:11 but I don't want to go into the details 638 00:37:07 --> 00:37:13 of exactly how you derive here kappa. 639 00:37:10 --> 00:37:16 It's a little bit more complicated. 640 00:37:12 --> 00:37:18 But indeed, the same is true. 641 00:37:15 --> 00:37:21 If you make the wire thicker, then kappa will go up, 642 00:37:20 --> 00:37:26 and if you make the wire longer, kappa will go down. 643 00:37:23 --> 00:37:29 That's immediately obvious. 644 00:37:24 --> 00:37:30 If you have a very short rod and you try to twist that rod... 645 00:37:28 --> 00:37:34 It's clamped at the top, and you twist it and it's very short, 646 00:37:32 --> 00:37:38 you would need a tremendous torque for ten degrees. 647 00:37:35 --> 00:37:41 If you make the wire a hundred meters long 648 00:37:37 --> 00:37:43 and you want to twist it ten degrees, it takes nothing. 649 00:37:40 --> 00:37:46 So you can immediately see that, of course, the value for kappa-- 650 00:37:44 --> 00:37:50 the torsional constant-- is a function of the length. 651 00:37:48 --> 00:37:54 It will go down when the length goes up. 652 00:37:51 --> 00:37:57 We have a wire here which is 2½ meters long, 653 00:37:54 --> 00:38:00 and the thickness of the wire, the diameter-- 654 00:37:57 --> 00:38:03 according to the manufacturer, it's a piano string-- 655 00:38:01 --> 00:38:07 the thickness is 25/1,000 of an inch. 656 00:38:04 --> 00:38:10 And if I calculate kappa to the best of my ability, 657 00:38:08 --> 00:38:14 well, I find that kappa should be very close 658 00:38:12 --> 00:38:18 to four times ten to the minus four newton-meters per radian. 659 00:38:21 --> 00:38:27 And so all we have to do now 660 00:38:22 --> 00:38:28 is to calculate the moment of inertia of the system, 661 00:38:25 --> 00:38:31 and then we can predict 662 00:38:27 --> 00:38:33 what the period of this pendulum is going to be-- 663 00:38:31 --> 00:38:37 which is not my goal. 664 00:38:32 --> 00:38:38 You will see, my goal is going to be a different one. 665 00:38:36 --> 00:38:42 Look at the bar and look at the wire. 666 00:38:40 --> 00:38:46 The wire is so thin 667 00:38:41 --> 00:38:47 that the moment of inertia relative to point P of the wire 668 00:38:46 --> 00:38:52 is very close to zero. 669 00:38:48 --> 00:38:54 Remember, it is proportional to our square. 670 00:38:51 --> 00:38:57 But you can forget about that. 671 00:38:53 --> 00:38:59 Almost all moment of inertia is in this system. 672 00:38:59 --> 00:39:05 I'll blow up that system for you-- here it is. 673 00:39:02 --> 00:39:08 You'll see it there, and it has on both sides... 674 00:39:06 --> 00:39:12 it has 200 grams. 675 00:39:09 --> 00:39:15 It has 0.2 kilograms and it has here 0.2 kilograms. 676 00:39:16 --> 00:39:22 And this mass is almost negligible. 677 00:39:19 --> 00:39:25 And this distance here is 30 centimeters 678 00:39:24 --> 00:39:30 and this distance is 30 centimeters. 679 00:39:29 --> 00:39:35 So to a very good approximation 680 00:39:32 --> 00:39:38 the moment of inertia for rotation about that point P-- 681 00:39:37 --> 00:39:43 this rotation-- will be this mass times radius squared 682 00:39:41 --> 00:39:47 plus this mass times radius squared. 683 00:39:45 --> 00:39:51 So that will be twice, because I have two masses, 684 00:39:48 --> 00:39:54 times the mass times the radius squared, 685 00:39:53 --> 00:39:59 and that is about 0.036 kilogram meters squared. 686 00:40:02 --> 00:40:08 And so when I use that into our equation-- 687 00:40:05 --> 00:40:11 so I know now what kappa is, 688 00:40:06 --> 00:40:12 at least I have a reasonable idea what kappa is-- 689 00:40:09 --> 00:40:15 and I know what I of P is-- 690 00:40:11 --> 00:40:17 that's really almost exclusively determined by that cross-bar-- 691 00:40:14 --> 00:40:20 I will find, then, using that equation 692 00:40:17 --> 00:40:23 that the period is very close to 60 seconds. 693 00:40:21 --> 00:40:27 694 00:40:23 --> 00:40:29 My goal is not to prove to you that it is close to 60 seconds. 695 00:40:27 --> 00:40:33 My goal is to show you 696 00:40:29 --> 00:40:35 that for this dimension, which is very thin and very long, 697 00:40:33 --> 00:40:39 that we can make that angle theta maximum-- 698 00:40:37 --> 00:40:43 this angle-- amazingly large. 699 00:40:40 --> 00:40:46 You're not talking about ten degrees or 30 degrees. 700 00:40:43 --> 00:40:49 We can go much further. 701 00:40:45 --> 00:40:51 And what I want to test with you is how far we can go. 702 00:40:49 --> 00:40:55 This is one of the great things in life 703 00:40:52 --> 00:40:58 for you and for me-- a challenge: 704 00:40:54 --> 00:41:00 How far can you go and get away with it? 705 00:40:57 --> 00:41:03 There comes a time that if we make the angle too large 706 00:41:01 --> 00:41:07 that we permanently deform the wire-- 707 00:41:04 --> 00:41:10 it will not come back to its original position. 708 00:41:07 --> 00:41:13 The same happens with a spring. 709 00:41:10 --> 00:41:16 If you take a spring, it is true 710 00:41:12 --> 00:41:18 that the period of oscillation is independent of the amplitude 711 00:41:16 --> 00:41:22 but only up to a point. 712 00:41:18 --> 00:41:24 If you go too far, that Hooke's Law no longer holds-- 713 00:41:21 --> 00:41:27 that you deform it permanently-- then, of course, 714 00:41:24 --> 00:41:30 the period will become a function of the extension, 715 00:41:27 --> 00:41:33 and the same is true here. 716 00:41:29 --> 00:41:35 So if we twist it too much up, 717 00:41:31 --> 00:41:37 then, of course, we will permanently deform it, 718 00:41:35 --> 00:41:41 and then the period will not be independent of theta maximum. 719 00:41:40 --> 00:41:46 Having said that, I would like to start asking you for advice. 720 00:41:46 --> 00:41:52 What kind of angle in this direction shall we start with? 721 00:41:51 --> 00:41:57 What do you think is reasonable 722 00:41:54 --> 00:42:00 without total torture for the wire? 723 00:41:57 --> 00:42:03 And then we'll write down the times. 724 00:42:00 --> 00:42:06 So we're not really interested in testing the 60 seconds, 725 00:42:04 --> 00:42:10 but we would rather like to compare 726 00:42:07 --> 00:42:13 the various angles that we give it. 727 00:42:09 --> 00:42:15 So what is the first one we will try? Any idea? 728 00:42:12 --> 00:42:18 (echoing class ): 30 degrees? What? 729 00:42:14 --> 00:42:20 STUDENT: Six pi. 730 00:42:16 --> 00:42:22 LEWIN: The first try?! You are cruel! 731 00:42:21 --> 00:42:27 Man! The first one, you want six pi?! 732 00:42:25 --> 00:42:31 You're out of your mind. 733 00:42:27 --> 00:42:33 (class laughs ) 734 00:42:29 --> 00:42:35 LEWIN: I'm willing to go one rotation, okay? 735 00:42:32 --> 00:42:38 You think that's nothing-- it's peanuts for you, right? 736 00:42:35 --> 00:42:41 Okay, I would like to go theta maximum of 360 degrees-- 737 00:42:42 --> 00:42:48 so two pi-- and measure the period. 738 00:42:47 --> 00:42:53 In fact, to measure the period takes a minute, 739 00:42:52 --> 00:42:58 and it's not necessary. 740 00:42:53 --> 00:42:59 We can measure half a period. 741 00:42:55 --> 00:43:01 Namely, we wait until the pendulum stops, 742 00:42:57 --> 00:43:03 and we measure the time until it stops again; 743 00:42:59 --> 00:43:05 that's half a period. 744 00:43:01 --> 00:43:07 Like with the spring-- 745 00:43:02 --> 00:43:08 if it stops here and it stops there, that's half a period. 746 00:43:06 --> 00:43:12 So we'll measure half a period. 747 00:43:09 --> 00:43:15 748 00:43:12 --> 00:43:18 Now, I don't know what my reaction time is going to be, 749 00:43:14 --> 00:43:20 It may be another tenth of a second, 750 00:43:16 --> 00:43:22 because the moment that it stops 751 00:43:18 --> 00:43:24 is not so well defined, you will see. 752 00:43:20 --> 00:43:26 I'm just guessing-- 753 00:43:22 --> 00:43:28 probably a little larger than one-tenth of a second. 754 00:43:24 --> 00:43:30 Let's give it a shot. 755 00:43:27 --> 00:43:33 Let's try first 360 degrees. 756 00:43:31 --> 00:43:37 757 00:43:35 --> 00:43:41 You see, this is black and this is a little red, 758 00:43:38 --> 00:43:44 so we will rotate it one rotation. 759 00:43:43 --> 00:43:49 This is back here where it was. 760 00:43:45 --> 00:43:51 Yeah, have you seen that? 360 degrees. 761 00:43:47 --> 00:43:53 Okay, now, I will first let it go and wait until it stops. 762 00:43:50 --> 00:43:56 I always do that, and then I start the time. 763 00:43:53 --> 00:43:59 And then when it stops again, I'll stop the time, 764 00:43:56 --> 00:44:02 and then we have half a period. 765 00:43:58 --> 00:44:04 So let it first do its own thing... 766 00:44:01 --> 00:44:07 very slowly, very gently. 767 00:44:05 --> 00:44:11 It should take roughly 30 seconds for half a rotation. 768 00:44:09 --> 00:44:15 so you'll see now that it... it's now at equilibrium again, 769 00:44:14 --> 00:44:20 because we wound it up one rotation 770 00:44:17 --> 00:44:23 and so back to equilibrium. 771 00:44:19 --> 00:44:25 And now it's going to stop very shortly, 772 00:44:22 --> 00:44:28 and when it stops, I want to start the timer. 773 00:44:25 --> 00:44:31 Now! 774 00:44:27 --> 00:44:33 Okay, so now it goes back, and we'll wait until it stops again. 775 00:44:33 --> 00:44:39 That gives us half a period. 776 00:44:35 --> 00:44:41 777 00:44:48 --> 00:44:54 Okay. 778 00:44:49 --> 00:44:55 779 00:44:54 --> 00:45:00 Now! 780 00:44:56 --> 00:45:02 28.75. 781 00:44:59 --> 00:45:05 782 00:45:05 --> 00:45:11 28.8. 783 00:45:08 --> 00:45:14 What are we going to do now? 784 00:45:10 --> 00:45:16 785 00:45:12 --> 00:45:18 Three rotations? Five rotations? Three. 786 00:45:16 --> 00:45:22 Are we in favor of three? 787 00:45:18 --> 00:45:24 Who is responsible for permanent damage to the wire? 788 00:45:22 --> 00:45:28 Do you accept responsibility in life? 789 00:45:24 --> 00:45:30 Three is a lot. 790 00:45:26 --> 00:45:32 Three is six pi, man, six. 791 00:45:30 --> 00:45:36 9I can start it, because it will take a while before it stops. 792 00:45:34 --> 00:45:40 Three rotations-- first we have to be sure 793 00:45:37 --> 00:45:43 that it is more or less back at equilibrium, 794 00:45:40 --> 00:45:46 which is always a difficult thing, because it's so slow. 795 00:45:43 --> 00:45:49 796 00:45:49 --> 00:45:55 Yeah, close enough. 797 00:45:51 --> 00:45:57 Okay, three-- shall we go clockwise or counterclockwise? 798 00:45:54 --> 00:46:00 It should make no difference. 799 00:45:55 --> 00:46:01 I went this way first; shall we go back? 800 00:45:57 --> 00:46:03 Yeah? You can sleep with that? 801 00:45:59 --> 00:46:05 Okay? Okay. 802 00:46:03 --> 00:46:09 One... two... three. 803 00:46:10 --> 00:46:16 804 00:46:12 --> 00:46:18 Six pi. 805 00:46:13 --> 00:46:19 806 00:46:16 --> 00:46:22 The piano wire. 807 00:46:18 --> 00:46:24 Okay, let's go. 808 00:46:20 --> 00:46:26 I will start the timer when it stops. 809 00:46:24 --> 00:46:30 We have some time. 810 00:46:26 --> 00:46:32 Six pi. 811 00:46:27 --> 00:46:33 812 00:46:39 --> 00:46:45 If we rotate it six... three times, 813 00:46:41 --> 00:46:47 then it will rotate six times back before it stops. 814 00:46:44 --> 00:46:50 I hope you realize that. 815 00:46:45 --> 00:46:51 (student speaks up in background ) 816 00:46:48 --> 00:46:54 817 00:46:50 --> 00:46:56 LEWIN: Was I too late? 818 00:46:52 --> 00:46:58 Thank you for pointing that out. 819 00:46:54 --> 00:47:00 So if you rotate it three times and then let it go, 820 00:46:57 --> 00:47:03 it goes first three times back, then it's at equilibrium, 821 00:47:01 --> 00:47:07 and then it winds three times up again before it stops. 822 00:47:04 --> 00:47:10 Notice it's going much faster now, but the time-- 823 00:47:07 --> 00:47:13 that's the whole thing-- 824 00:47:10 --> 00:47:16 should be very close to that number again. 825 00:47:13 --> 00:47:19 826 00:47:18 --> 00:47:24 28.5. 827 00:47:19 --> 00:47:25 Oh! Not bad. 828 00:47:21 --> 00:47:27 829 00:47:24 --> 00:47:30 Shall we now go all the way? 830 00:47:27 --> 00:47:33 What do you want to do now? 831 00:47:29 --> 00:47:35 Break the wire or... 832 00:47:32 --> 00:47:38 833 00:47:34 --> 00:47:40 Ten rotations? 834 00:47:35 --> 00:47:41 835 00:47:37 --> 00:47:43 You'd love to see that, right? 836 00:47:40 --> 00:47:46 It will go like mad, ten rotations. 837 00:47:44 --> 00:47:50 Isn't it amazing how much faster it goes? 838 00:47:46 --> 00:47:52 (imitates whirring ) 839 00:47:48 --> 00:47:54 It's still 30 seconds. 840 00:47:50 --> 00:47:56 I must make sure that I have my equilibrium. 841 00:47:53 --> 00:47:59 This was not equilibrium. 842 00:47:55 --> 00:48:01 843 00:47:57 --> 00:48:03 I know it's somewhere here. 844 00:48:00 --> 00:48:06 845 00:48:02 --> 00:48:08 No, it wasn't equilibrium. 846 00:48:05 --> 00:48:11 847 00:48:08 --> 00:48:14 I think this is it. 848 00:48:10 --> 00:48:16 Okay-- ten, right? 849 00:48:13 --> 00:48:19 Ready? 850 00:48:15 --> 00:48:21 One... two... three... four... 851 00:48:23 --> 00:48:29 five... six... seven... eight... nine... 852 00:48:33 --> 00:48:39 Ten-- poor wire. 853 00:48:35 --> 00:48:41 We'll let it go and we'll see what happens. 854 00:48:39 --> 00:48:45 When it stops, I'll start the time. 855 00:48:42 --> 00:48:48 856 00:48:44 --> 00:48:50 (student speaks up in background ) 857 00:48:46 --> 00:48:52 LEWIN: Excuse me? 858 00:48:47 --> 00:48:53 (student repeats ) 859 00:48:48 --> 00:48:54 LEWIN: Thank you. 860 00:48:49 --> 00:48:55 Thank you for pointing that out. 861 00:48:51 --> 00:48:57 Look how fast it's going. 862 00:48:53 --> 00:48:59 I mean, it's really going wacko. 863 00:48:55 --> 00:49:01 It has to do all that in 30 seconds. 864 00:48:57 --> 00:49:03 865 00:49:04 --> 00:49:10 Now! 866 00:49:05 --> 00:49:11 867 00:49:07 --> 00:49:13 So now it has to go back to its stopping. 868 00:49:10 --> 00:49:16 It has to make 20 rotations now, 20 rotations in 30 seconds-- 869 00:49:13 --> 00:49:19 ten back to equilibrium and then ten to come to a halt. 870 00:49:19 --> 00:49:25 871 00:49:23 --> 00:49:29 This is going to be 872 00:49:24 --> 00:49:30 your Thanksgiving farewell demonstration. 873 00:49:27 --> 00:49:33 874 00:49:32 --> 00:49:38 Now! 875 00:49:35 --> 00:49:41 29.2-- fantastic. 876 00:49:38 --> 00:49:44 Okay, have a good Thanksgiving. 877 00:49:40 --> 00:49:46