1 0:00:00 --> 00:00:06 We're going to discuss today 2 00:00:03 --> 00:00:09 resistive forces and drag forces. 3 00:00:06 --> 00:00:12 When you move an object through a medium, 4 00:00:09 --> 00:00:15 whether it's a gas or whether it's a liquid, 5 00:00:12 --> 00:00:18 it experiences a drag force. 6 00:00:14 --> 00:00:20 This drag force depends on the shape of the object, 7 00:00:17 --> 00:00:23 the size of the object, 8 00:00:19 --> 00:00:25 the medium through which you move it 9 00:00:21 --> 00:00:27 and the speed of the object. 10 00:00:23 --> 00:00:29 The medium is immediately obvious. 11 00:00:25 --> 00:00:31 If it's air and you move through air, 12 00:00:27 --> 00:00:33 you feel the wind through your hair-- that's a drag force. 13 00:00:31 --> 00:00:37 If you swim in water, you feel this drag force. 14 00:00:35 --> 00:00:41 In oil, the drag force would be even larger. 15 00:00:38 --> 00:00:44 This drag force, this resistive force is very, very different 16 00:00:42 --> 00:00:48 from the friction that we have discussed earlier 17 00:00:46 --> 00:00:52 when two surfaces move relative to each other. 18 00:00:49 --> 00:00:55 There, the kinetic friction coefficient remains constant 19 00:00:54 --> 00:01:00 independent of the speed. 20 00:00:56 --> 00:01:02 With the drag forces and the resistive forces, 21 00:01:00 --> 00:01:06 they are not at all independent of the speed. 22 00:01:03 --> 00:01:09 In very general terms, 23 00:01:05 --> 00:01:11 the resistive force can be written 24 00:01:10 --> 00:01:16 as k1 times the velocity plus k2 times the velocity squared 25 00:01:22 --> 00:01:28 and always in the opposite direction 26 00:01:25 --> 00:01:31 of the velocity vector. 27 00:01:26 --> 00:01:32 This v here is the speed, 28 00:01:28 --> 00:01:34 so all these signs-- k1, v and k2, and obviously v squared-- 29 00:01:34 --> 00:01:40 they all are positive values. 30 00:01:36 --> 00:01:42 And the k values depend 31 00:01:38 --> 00:01:44 on the shape and the size of the object 32 00:01:41 --> 00:01:47 and on the kind of medium that I have. 33 00:01:44 --> 00:01:50 Today I will restrict myself exclusively to spheres. 34 00:01:49 --> 00:01:55 And when we deal with spheres, we're going to get 35 00:01:55 --> 00:02:01 that the force, the magnitude of the force-- 36 00:01:59 --> 00:02:05 so that's this part-- 37 00:02:01 --> 00:02:07 equals C1 times r times the speed 38 00:02:07 --> 00:02:13 plus C2 times r squared times v squared. 39 00:02:12 --> 00:02:18 And again, it's always opposing the velocity vector. 40 00:02:18 --> 00:02:24 C1 in our unit is kilograms per meters per second 41 00:02:24 --> 00:02:30 and C2 has the dimension 42 00:02:27 --> 00:02:33 of density kilogram per cubic meters. 43 00:02:32 --> 00:02:38 We call this the viscous term, 44 00:02:35 --> 00:02:41 and we call this the pressure term. 45 00:02:40 --> 00:02:46 The viscous term has to do with the stickiness of the medium. 46 00:02:46 --> 00:02:52 If you take, for instance, liquids-- 47 00:02:48 --> 00:02:54 water and oil and tar-- 48 00:02:50 --> 00:02:56 there is a huge difference in stickiness. 49 00:02:53 --> 00:02:59 Physicists also refer to that as viscosity. 50 00:02:57 --> 00:03:03 If you have a high viscosity, it's very sticky, 51 00:03:01 --> 00:03:07 then this number, C1, will be very high. 52 00:03:04 --> 00:03:10 So this we call the viscous term, 53 00:03:08 --> 00:03:14 and this we call the pressure term. 54 00:03:14 --> 00:03:20 The C1 is a strong function of temperature. 55 00:03:17 --> 00:03:23 We all know that if you take tar and you heat it 56 00:03:21 --> 00:03:27 that the viscosity goes down. 57 00:03:23 --> 00:03:29 It is way more sticky when it is cold. 58 00:03:26 --> 00:03:32 C2 is not very dependent on the temperature. 59 00:03:31 --> 00:03:37 60 00:03:32 --> 00:03:38 It's not so easy to see 61 00:03:33 --> 00:03:39 why this pressure term here has a v square. 62 00:03:36 --> 00:03:42 Later in the course when we deal with transfer of momentum, 63 00:03:40 --> 00:03:46 we will understand why there is a v-square term there. 64 00:03:44 --> 00:03:50 But the r square is very easy to see, 65 00:03:46 --> 00:03:52 because if you have a sphere and there is some fluid-- 66 00:03:51 --> 00:03:57 gas or liquid-- streaming onto it, 67 00:03:54 --> 00:04:00 then this has a cross-sectional area 68 00:03:57 --> 00:04:03 which is proportional to r squared, 69 00:03:59 --> 00:04:05 and so it's easy to see 70 00:04:01 --> 00:04:07 that the force that this object experiences-- 71 00:04:03 --> 00:04:09 we call it the pressure term-- 72 00:04:05 --> 00:04:11 is proportional to r square, so that's easy to see. 73 00:04:09 --> 00:04:15 Two liquids with the very same density would have... 74 00:04:15 --> 00:04:21 they could have very different values for C1. 75 00:04:18 --> 00:04:24 They could differ by ten... 76 00:04:20 --> 00:04:26 not ten, by four or five orders of magnitude. 77 00:04:23 --> 00:04:29 But if they have the same density, the liquids, 78 00:04:26 --> 00:04:32 then the C2 is very much the same. 79 00:04:29 --> 00:04:35 C2 is almost the density rho of the liquid-- 80 00:04:33 --> 00:04:39 not quite but almost-- but there is 81 00:04:36 --> 00:04:42 a very strong correlation between the C2 and the density. 82 00:04:41 --> 00:04:47 If I drop an object and I just let it go, 83 00:04:44 --> 00:04:50 I take an object and I let it fall-- 84 00:04:47 --> 00:04:53 we're only dealing with spheres today-- 85 00:04:50 --> 00:04:56 then what you will see, I have a mass m, 86 00:04:53 --> 00:04:59 and so there is a force mg-- that is gravity. 87 00:04:59 --> 00:05:05 And as it picks up speed, the resistive force will grow, 88 00:05:06 --> 00:05:12 and it will grow and it will grow, 89 00:05:07 --> 00:05:13 and there comes a time... because the speed increases, 90 00:05:11 --> 00:05:17 so the resistive force will grow, 91 00:05:13 --> 00:05:19 and there comes a time that the two are equal. 92 00:05:16 --> 00:05:22 And when the two are equal, then there is no longer acceleration, 93 00:05:20 --> 00:05:26 so the object has a constant speed, 94 00:05:22 --> 00:05:28 and we call that the terminal velocity, 95 00:05:25 --> 00:05:31 and that will be the case 96 00:05:27 --> 00:05:33 when mg equals C1 r v plus C2 r squared v squared. 97 00:05:40 --> 00:05:46 And then we have here terminal velocity. 98 00:05:46 --> 00:05:52 99 00:05:49 --> 00:05:55 If you know what m is, the mass of an object, the radius, 100 00:05:55 --> 00:06:01 and you know the values for C1 or C2 101 00:05:57 --> 00:06:03 of that medium in which you move it, 102 00:05:59 --> 00:06:05 then you can calculate what the terminal velocity is. 103 00:06:02 --> 00:06:08 It is a quadratic equation, so you get two solutions 104 00:06:06 --> 00:06:12 of which one of them is nonphysical, 105 00:06:08 --> 00:06:14 so you can reject that one. 106 00:06:10 --> 00:06:16 107 00:06:12 --> 00:06:18 Very often will we work in a domain, in a regime 108 00:06:18 --> 00:06:24 whereby this viscous term is dominating. 109 00:06:22 --> 00:06:28 I call that regime one, but it also happens-- 110 00:06:25 --> 00:06:31 and I will show you examples today-- that we're working 111 00:06:29 --> 00:06:35 in a regime where really this force is dominating. 112 00:06:32 --> 00:06:38 I call that regime two. 113 00:06:36 --> 00:06:42 114 00:06:39 --> 00:06:45 Where one and two are the same-- 115 00:06:41 --> 00:06:47 where the force due to the viscous force 116 00:06:44 --> 00:06:50 and the pressure force are the same-- 117 00:06:47 --> 00:06:53 we can make these terms the same, 118 00:06:49 --> 00:06:55 so you get C1 r v equals C2 r square v squared, 119 00:06:54 --> 00:07:00 and that velocity we call the critical velocity, 120 00:06:59 --> 00:07:05 even though there is nothing critical about it. 121 00:07:02 --> 00:07:08 It's not critical at all; 122 00:07:03 --> 00:07:09 it's simply the speed at which the two terms are equal. 123 00:07:07 --> 00:07:13 That's all it means. 124 00:07:08 --> 00:07:14 And that, of course, then equals C1 divided by C2 divided by r. 125 00:07:13 --> 00:07:19 126 00:07:15 --> 00:07:21 Now we're going to make a clear distinction 127 00:07:18 --> 00:07:24 between the domains one and two, the regimes one and two. 128 00:07:22 --> 00:07:28 Regime one is when the speed is much, much less 129 00:07:29 --> 00:07:35 than the critical velocity. 130 00:07:31 --> 00:07:37 So we then have that mg equals C1 r v terminal, 131 00:07:40 --> 00:07:46 and therefore the terminal velocity equals 132 00:07:45 --> 00:07:51 mg divided by C1 r. 133 00:07:50 --> 00:07:56 If you take objects of the same material-- 134 00:07:55 --> 00:08:01 that means they have the same density, 135 00:07:57 --> 00:08:03 the density of the objects that you drop in the liquid 136 00:08:00 --> 00:08:06 or that you drop in the gas-- 137 00:08:02 --> 00:08:08 so that m equals 4/3 pi rho r cubed-- 138 00:08:09 --> 00:08:15 this is now the rho, the density of the object; 139 00:08:11 --> 00:08:17 it's not the density of the medium-- 140 00:08:14 --> 00:08:20 then you can immediately see, since you get an r cubed here, 141 00:08:18 --> 00:08:24 that this is proportional to the square of the radius 142 00:08:21 --> 00:08:27 if you drop objects in there with the same density. 143 00:08:28 --> 00:08:34 Regime two is the case when v is much larger than v critical, 144 00:08:33 --> 00:08:39 so then mg equals C2 r squared v squared 145 00:08:41 --> 00:08:47 if this is the terminal velocity. 146 00:08:44 --> 00:08:50 So the terminal velocity is then 147 00:08:47 --> 00:08:53 the square root of mg divided by C2 r squared... 148 00:08:52 --> 00:08:58 mg divided by C2 r squared. 149 00:08:55 --> 00:09:01 And if you take objects with the same density 150 00:08:58 --> 00:09:04 and you compare their radii, m is proportional to r cubed 151 00:09:02 --> 00:09:08 so this is now proportional to the square root of r. 152 00:09:08 --> 00:09:14 So this separates these two regimes, 153 00:09:11 --> 00:09:17 and we will see examples of that 154 00:09:13 --> 00:09:19 that sometimes you really work exclusively in one 155 00:09:17 --> 00:09:23 and sometimes you work in the other. 156 00:09:20 --> 00:09:26 157 00:09:23 --> 00:09:29 I have for you a view graph that is on the Web 158 00:09:27 --> 00:09:33 so you do not have to copy it. 159 00:09:30 --> 00:09:36 It summarizes what I have just told you. 160 00:09:33 --> 00:09:39 It has all the key equations. 161 00:09:35 --> 00:09:41 You see there on top the resistive force, 162 00:09:38 --> 00:09:44 the magnitude of the resistive force. 163 00:09:40 --> 00:09:46 You see then the critical velocity. 164 00:09:43 --> 00:09:49 There's nothing critical about it; 165 00:09:46 --> 00:09:52 it's just the speed at which this term has 166 00:09:49 --> 00:09:55 the same magnitude as this term. 167 00:09:51 --> 00:09:57 Then you see the condition here, 168 00:09:54 --> 00:10:00 which I call equation one for terminal speed, 169 00:09:58 --> 00:10:04 and then we have regime one, 170 00:10:00 --> 00:10:06 whereby the speed is way less than the critical speed 171 00:10:04 --> 00:10:10 and then you get the terminal velocity, 172 00:10:07 --> 00:10:13 as you see on the blackboard, 173 00:10:08 --> 00:10:14 which then is proportional to r squared if you only look 174 00:10:12 --> 00:10:18 at objects which have a particular given density. 175 00:10:16 --> 00:10:22 And if the velocity, 176 00:10:17 --> 00:10:23 if the speed is way larger than the critical speed, 177 00:10:20 --> 00:10:26 you are in regime two 178 00:10:21 --> 00:10:27 and then you have a dependence with the square root of r. 179 00:10:26 --> 00:10:32 180 00:10:28 --> 00:10:34 I'm going to do a demonstration and some measurements 181 00:10:35 --> 00:10:41 with ball bearings, 182 00:10:36 --> 00:10:42 which have very precise radii-- very well known-- 183 00:10:41 --> 00:10:47 which I'm going to drop into syrup. 184 00:10:44 --> 00:10:50 And we have chosen for that Karo light corn syrup. 185 00:10:50 --> 00:10:56 It may interest you that two tablespoons is 180 calories. 186 00:10:57 --> 00:11:03 I needed to know more for this demonstration 187 00:11:01 --> 00:11:07 and so I had to do my own homework on it, 188 00:11:04 --> 00:11:10 but at least you see here what this Karo syrup can do for you. 189 00:11:09 --> 00:11:15 You see your 180 calories per two tablespoons... 190 00:11:13 --> 00:11:19 or per tablespoon. 191 00:11:14 --> 00:11:20 It's very low fat, 192 00:11:15 --> 00:11:21 and the rest may interest you before you use it. 193 00:11:19 --> 00:11:25 I had to know C1, which I calculated; I measured it. 194 00:11:23 --> 00:11:29 In fact, 195 00:11:24 --> 00:11:30 the kind of demonstration you and I will be doing today, 196 00:11:27 --> 00:11:33 you can derive it, 197 00:11:28 --> 00:11:34 but it's very strongly temperature-dependent. 198 00:11:31 --> 00:11:37 It could be different yesterday from today. 199 00:11:34 --> 00:11:40 I measured C2 to a reasonable accuracy. 200 00:11:37 --> 00:11:43 Notice that the density of the syrup 201 00:11:41 --> 00:11:47 in terms of kilograms per cubic meter 202 00:11:43 --> 00:11:49 is very close to C2-- I mentioned that earlier. 203 00:11:46 --> 00:11:52 They're very close, not exactly but very close. 204 00:11:49 --> 00:11:55 These steel ball bearings have a density 205 00:11:51 --> 00:11:57 of about 7,800 kilograms per cubic meter. 206 00:11:57 --> 00:12:03 And I'm going to drop in that Karo syrup four ball bearings, 207 00:12:00 --> 00:12:06 and they have diameters of an eighth of an inch, 208 00:12:04 --> 00:12:10 5/32, 3/16 and a quarter of an inch. 209 00:12:11 --> 00:12:17 And what I calculated was the terminal velocity 210 00:12:15 --> 00:12:21 as a function of radius of these ball bearings. 211 00:12:20 --> 00:12:26 All of this is on the Web. 212 00:12:24 --> 00:12:30 And so what you see here, it is a logarithmic plot-- 213 00:12:27 --> 00:12:33 this is a log scale and this is a log scale. 214 00:12:29 --> 00:12:35 Here you see the speed, 215 00:12:30 --> 00:12:36 and here you see the radius in meters of the ball bearings. 216 00:12:33 --> 00:12:39 And this is my solution to equation number one 217 00:12:38 --> 00:12:44 when I substitute various values of r in there. 218 00:12:42 --> 00:12:48 I get the terminal speed like this. 219 00:12:45 --> 00:12:51 And this is the critical speed, 220 00:12:48 --> 00:12:54 which has a 1-over-r relationship. 221 00:12:50 --> 00:12:56 If you look at this black dot here, 222 00:12:53 --> 00:12:59 then the terminal speed is ten times larger here 223 00:12:57 --> 00:13:03 than the critical speed. 224 00:12:59 --> 00:13:05 And so notice that when you are at speeds above that 225 00:13:03 --> 00:13:09 that you are exclusively in domain two 226 00:13:06 --> 00:13:12 and your terminal speed is proportional 227 00:13:08 --> 00:13:14 to the square root of the radius of the ball bearings. 228 00:13:12 --> 00:13:18 This black dot is a factor of ten below the critical speed, 229 00:13:15 --> 00:13:21 and so you see when you are at lower speeds, 230 00:13:18 --> 00:13:24 when you work here, again, you see that you fall into... 231 00:13:22 --> 00:13:28 exclusively in domain one, and you see 232 00:13:25 --> 00:13:31 that the terminal speed is proportional to r square. 233 00:13:27 --> 00:13:33 This slope here is plus two in this diagram 234 00:13:31 --> 00:13:37 and this slope here is plus one-half. 235 00:13:35 --> 00:13:41 Our ball bearings are all here, 236 00:13:37 --> 00:13:43 and so we are exclusively operating in regime one 237 00:13:41 --> 00:13:47 where the viscous term dominates. 238 00:13:44 --> 00:13:50 Now you could say, 239 00:13:45 --> 00:13:51 "Well, what is the meaning of this critical speed here 240 00:13:48 --> 00:13:54 if they never reach that speed anyhow?" 241 00:13:51 --> 00:13:57 Well, this critical speed for a small ball bearing 242 00:13:54 --> 00:14:00 would be some hundred meters per second. 243 00:13:56 --> 00:14:02 That's about 200 miles per hour. 244 00:13:58 --> 00:14:04 There is nothing wrong with injecting a ball bearing 245 00:14:02 --> 00:14:08 with 400 miles per hour into this syrup, 246 00:14:04 --> 00:14:10 in which case, if you injected it with 400 miles per hour, 247 00:14:08 --> 00:14:14 you would be above the critical speed 248 00:14:10 --> 00:14:16 and so for a short while 249 00:14:12 --> 00:14:18 would the motion be controlled by the pressure term. 250 00:14:16 --> 00:14:22 But of course when gravity takes over, 251 00:14:19 --> 00:14:25 then you ultimately end up in regime one. 252 00:14:21 --> 00:14:27 So that's the meaning of the critical velocity. 253 00:14:24 --> 00:14:30 If you could give the ball bearing such a high speed, 254 00:14:29 --> 00:14:35 then the two terms are equal. 255 00:14:31 --> 00:14:37 That's all it means. 256 00:14:33 --> 00:14:39 Very well. 257 00:14:35 --> 00:14:41 Now we are going to look at the various ball bearings, 258 00:14:40 --> 00:14:46 the various sizes, and I'll show you how we do the experiment. 259 00:14:45 --> 00:14:51 You will shortly see on the screen there seven marks 260 00:14:51 --> 00:14:57 which are one centimeter apart. 261 00:14:53 --> 00:14:59 They are in the liquid-- 262 00:14:55 --> 00:15:01 one, two, three, four, five, six, seven. 263 00:14:58 --> 00:15:04 So here is the liquid, 264 00:14:59 --> 00:15:05 and the ball bearings are dropped from above. 265 00:15:02 --> 00:15:08 When it reaches this line, I will start my timer. 266 00:15:06 --> 00:15:12 And when it crosses one, two, three, four... 267 00:15:10 --> 00:15:16 when it crosses this line, I will stop my timer. 268 00:15:13 --> 00:15:19 And each mark is about one centimeter apart, 269 00:15:16 --> 00:15:22 so this is a journey of about four centimeters. 270 00:15:21 --> 00:15:27 And we will measure the time that it takes 271 00:15:23 --> 00:15:29 to go from here to here. 272 00:15:25 --> 00:15:31 And the terminal velocity is given. 273 00:15:29 --> 00:15:35 It's clearly regime one, 274 00:15:31 --> 00:15:37 so you see the terminal velocity right there. 275 00:15:35 --> 00:15:41 Now, the time that it will take is of course this distance-- 276 00:15:41 --> 00:15:47 let me call it h-- that the ball bearings travel 277 00:15:45 --> 00:15:51 divided by the terminal velocity, 278 00:15:47 --> 00:15:53 and that is proportional, since you're in regime one, 279 00:15:52 --> 00:15:58 by 1 over r squared. 280 00:15:54 --> 00:16:00 And now I will give you the... 281 00:15:56 --> 00:16:02 not the radii but I will give you 282 00:15:58 --> 00:16:04 the diameters of these ball bearings. 283 00:16:01 --> 00:16:07 That's the way they come. 284 00:16:03 --> 00:16:09 So we're going to get a list here 285 00:16:05 --> 00:16:11 of the diameters of the ball bearings, 286 00:16:08 --> 00:16:14 and the diameters is in inches. 287 00:16:11 --> 00:16:17 My smallest one has a diameter of an eighth of an inch. 288 00:16:14 --> 00:16:20 Then I have 5/32, I have 3/16-- 289 00:16:20 --> 00:16:26 all these things come in inches; that's one of those things-- 290 00:16:22 --> 00:16:28 and I have one-quarter-inch diameter. 291 00:16:25 --> 00:16:31 292 00:16:28 --> 00:16:34 If I plot here a plot, 293 00:16:30 --> 00:16:36 if I give you here the diameter in terms of 1/32s of an inch, 294 00:16:35 --> 00:16:41 then this is four, five, six, and eight-- easy numbers. 295 00:16:41 --> 00:16:47 Clearly if the time that it takes 296 00:16:43 --> 00:16:49 is proportional to 1 over r squared, 297 00:16:45 --> 00:16:51 it will also be proportional to 1 over d squared-- 298 00:16:48 --> 00:16:54 of course, that's the same. 299 00:16:52 --> 00:16:58 What I'm going to plot is not 1 over d squared, 300 00:16:56 --> 00:17:02 but to get some nice numbers 301 00:16:58 --> 00:17:04 I'm going to plot for you 100 over d squared, 302 00:17:03 --> 00:17:09 whereby d is then in these units, 1/32 of an inch, 303 00:17:06 --> 00:17:12 and that gives me some nice numbers. 304 00:17:09 --> 00:17:15 Then I get a 6.25 here, I get a 4.00 here. 305 00:17:14 --> 00:17:20 You can see 100 divided by 25 is exactly four. 306 00:17:19 --> 00:17:25 I get 2.78 here, and my last number is 1.56. 307 00:17:25 --> 00:17:31 And now I'm going to time it, and my timing uncertainty 308 00:17:29 --> 00:17:35 is of course dictated by my reaction time. 309 00:17:33 --> 00:17:39 That should be at least 0.1 seconds. 310 00:17:37 --> 00:17:43 However, you will see 311 00:17:38 --> 00:17:44 when I reach the quarter-inch ball bearing 312 00:17:41 --> 00:17:47 that it goes so fast 313 00:17:43 --> 00:17:49 that my error could well be 2/10 of a second. 314 00:17:46 --> 00:17:52 It goes in a flash. 315 00:17:47 --> 00:17:53 So I would allow here 2/10 of a second, 316 00:17:51 --> 00:17:57 and here I really don't know-- 317 00:17:53 --> 00:17:59 maybe 1/10, maybe 2/10 of a second. 318 00:17:55 --> 00:18:01 You may ask me, 319 00:17:57 --> 00:18:03 "Why didn't you give us the error in the diameter?" 320 00:17:59 --> 00:18:05 which ultimately, of course, translates 321 00:18:02 --> 00:18:08 into the error in the mass. 322 00:18:03 --> 00:18:09 The reason is that these are so precise the way you buy them 323 00:18:07 --> 00:18:13 that the uncertainty is completely negligible 324 00:18:09 --> 00:18:15 compared to the timing error that I make, 325 00:18:12 --> 00:18:18 so I won't even take that into account. 326 00:18:16 --> 00:18:22 All right, so now we can start the demonstration, 327 00:18:22 --> 00:18:28 and I'll have to switch to this unit here. 328 00:18:27 --> 00:18:33 Here is this container with the Karo syrup. 329 00:18:36 --> 00:18:42 It is very sticky indeed. 330 00:18:41 --> 00:18:47 You see there are seven marks, and just for my own convenience, 331 00:18:45 --> 00:18:51 I have put there two black marks so that I can easily see 332 00:18:51 --> 00:18:57 the moment that I have to start my timer 333 00:18:54 --> 00:19:00 and the moment that I stop it. 334 00:18:56 --> 00:19:02 There are so many lines, I may get confused if I don't do that. 335 00:18:59 --> 00:19:05 And we're going to time this together, 336 00:19:03 --> 00:19:09 and we'll see how these objects... 337 00:19:07 --> 00:19:13 how long it takes for them to go through. 338 00:19:09 --> 00:19:15 I will start with my one-eighth-of-an-inch diameter. 339 00:19:14 --> 00:19:20 I have one here-- tweezer. 340 00:19:16 --> 00:19:22 341 00:19:19 --> 00:19:25 I release it at zero, three... oh, oh, you can't see that. 342 00:19:24 --> 00:19:30 You should see that: three, two, one, zero. 343 00:19:28 --> 00:19:34 Look how beautiful it's working it's way through! 344 00:19:31 --> 00:19:37 You see, it's building up. 345 00:19:32 --> 00:19:38 You see that nice air bubble at the top? 346 00:19:35 --> 00:19:41 It's going very slowly, 347 00:19:36 --> 00:19:42 but just wait when it has broken through the surface. 348 00:19:39 --> 00:19:45 There it goes. 349 00:19:41 --> 00:19:47 Now! 350 00:19:43 --> 00:19:49 One centimeter, two centimeter, three centimeter. 351 00:19:47 --> 00:19:53 Now! 352 00:19:48 --> 00:19:54 Okay, what is that? 353 00:19:50 --> 00:19:56 (students responding ) 354 00:19:51 --> 00:19:57 LEWIN: Nothing. 355 00:19:52 --> 00:19:58 What happened? 356 00:19:54 --> 00:20:00 STUDENT: 5.93. 357 00:19:55 --> 00:20:01 LEWIN: I didn't see it. 358 00:19:57 --> 00:20:03 I want to do another one. 359 00:19:58 --> 00:20:04 360 00:19:59 --> 00:20:05 Did I... did I clean... did I erase it? 361 00:20:02 --> 00:20:08 (student answers ) 362 00:20:04 --> 00:20:10 LEWIN: How much was it? 363 00:20:05 --> 00:20:11 STUDENTS: 5.93. 364 00:20:06 --> 00:20:12 LEWIN: 5.93. 365 00:20:08 --> 00:20:14 366 00:20:11 --> 00:20:17 Keep that in mind. 367 00:20:12 --> 00:20:18 It's nice to see whether they reproduce, actually. 368 00:20:14 --> 00:20:20 369 00:20:17 --> 00:20:23 Okay, there it goes. 370 00:20:19 --> 00:20:25 Now! 371 00:20:20 --> 00:20:26 One centimeter, two centimeter, three centimeter. 372 00:20:25 --> 00:20:31 Now! 373 00:20:26 --> 00:20:32 5.66-- that shows you the uncertainty in my timing. 374 00:20:30 --> 00:20:36 So we had a 5.93, and we have now a 5.66. 375 00:20:36 --> 00:20:42 It's not so bad, 5.9, 5.7-- 376 00:20:39 --> 00:20:45 timing error a tenth of a second. 377 00:20:41 --> 00:20:47 My timing error could be 378 00:20:42 --> 00:20:48 a little bit larger than a tenth of a second. 379 00:20:44 --> 00:20:50 You don't have very much time. 380 00:20:46 --> 00:20:52 So now we go to the 5/32. 381 00:20:49 --> 00:20:55 382 00:20:53 --> 00:20:59 Okay? 383 00:20:54 --> 00:21:00 5/32. 384 00:20:56 --> 00:21:02 385 00:21:00 --> 00:21:06 It takes some time to break through the surface. 386 00:21:02 --> 00:21:08 Isn't that funny? 387 00:21:04 --> 00:21:10 Because a thin film has formed on the surface of the syrup 388 00:21:06 --> 00:21:12 due to its exposure to air. 389 00:21:09 --> 00:21:15 It's wonderful-- that lets us wait patiently. 390 00:21:12 --> 00:21:18 But now it goes-- there it goes. 391 00:21:13 --> 00:21:19 Now! 392 00:21:14 --> 00:21:20 One, two, three. 393 00:21:17 --> 00:21:23 Now! 394 00:21:19 --> 00:21:25 3.80. 395 00:21:21 --> 00:21:27 Is that what you have? 396 00:21:24 --> 00:21:30 397 00:21:27 --> 00:21:33 It's going to be tougher and tougher for me. 398 00:21:29 --> 00:21:35 3/16 of an inch. 399 00:21:33 --> 00:21:39 It's actually a good thing 400 00:21:34 --> 00:21:40 that it stays for a while at the surface 401 00:21:36 --> 00:21:42 so that I can get ready. 402 00:21:39 --> 00:21:45 403 00:21:42 --> 00:21:48 Really, that really helps, doesn't it? 404 00:21:44 --> 00:21:50 If you do this in water, it goes (whoosh ). 405 00:21:47 --> 00:21:53 You don't even see it. 406 00:21:49 --> 00:21:55 407 00:21:51 --> 00:21:57 Come on... There we go. 408 00:21:53 --> 00:21:59 Now! 409 00:21:54 --> 00:22:00 410 00:21:56 --> 00:22:02 Now! 411 00:21:57 --> 00:22:03 So you see, that's very hard for me, 412 00:21:58 --> 00:22:04 and so I could easily have a substantial error. 413 00:22:00 --> 00:22:06 2.69. 414 00:22:02 --> 00:22:08 415 00:22:04 --> 00:22:10 And now we have the quarter-inch, the real big one. 416 00:22:10 --> 00:22:16 417 00:22:14 --> 00:22:20 I have to do that again. 418 00:22:16 --> 00:22:22 I don't trust this at all. 419 00:22:18 --> 00:22:24 It went through the surface too fast. 420 00:22:21 --> 00:22:27 421 00:22:27 --> 00:22:33 (timer button clicking ) 422 00:22:29 --> 00:22:35 I... can't do it very accurately. 423 00:22:30 --> 00:22:36 What was the first number, by the way? 424 00:22:32 --> 00:22:38 (students respond ) 425 00:22:33 --> 00:22:39 LEWIN: One point...? 426 00:22:34 --> 00:22:40 (student responds ) 427 00:22:35 --> 00:22:41 LEWIN: Six eight. 428 00:22:36 --> 00:22:42 And this is 1.40. 429 00:22:38 --> 00:22:44 1.68 and 1.40. 430 00:22:42 --> 00:22:48 So you see I wasn't kidding 431 00:22:44 --> 00:22:50 when I said that my uncertainty could easily be .2. 432 00:22:47 --> 00:22:53 Now comes the acid test. 433 00:22:50 --> 00:22:56 And the acid test is that if I'm... 434 00:22:54 --> 00:23:00 if the measurements were done correctly 435 00:22:55 --> 00:23:01 and if we really work in that regime, 436 00:22:57 --> 00:23:03 then if I plot 100 divided by d squared versus t 437 00:23:02 --> 00:23:08 on linear paper, then it should be a straight line. 438 00:23:06 --> 00:23:12 439 00:23:10 --> 00:23:16 All right. 440 00:23:12 --> 00:23:18 Here I have a plot which I prepared, 441 00:23:16 --> 00:23:22 and I'm going to put these numbers in there. 442 00:23:19 --> 00:23:25 So first we're going to get six point... five point... 443 00:23:24 --> 00:23:30 let's put in 5.8 seconds for the smallest ball bearing. 444 00:23:31 --> 00:23:37 This is the smallest one. 445 00:23:33 --> 00:23:39 Don't be misled, because this is 100 divided by d squared, 446 00:23:37 --> 00:23:43 so this is the smallest one-- 5.8. 447 00:23:42 --> 00:23:48 So we are here on this line, and we are at 5.8-- somewhere here. 448 00:23:51 --> 00:23:57 That's it. 449 00:23:53 --> 00:23:59 Notice the point is lower than where I expected it, 450 00:23:56 --> 00:24:02 and the reason is the temperature went up. 451 00:23:59 --> 00:24:05 And if the temperature went up, 452 00:24:00 --> 00:24:06 then the viscosity goes down and they go faster. 453 00:24:02 --> 00:24:08 But that's okay, that doesn't worry me. 454 00:24:05 --> 00:24:11 The next one, 3.80. 455 00:24:10 --> 00:24:16 Four, 3.80. 456 00:24:13 --> 00:24:19 Ah! 457 00:24:14 --> 00:24:20 You see that? 458 00:24:16 --> 00:24:22 I predicted that-- straight line. 459 00:24:18 --> 00:24:24 460 00:24:20 --> 00:24:26 Isn't that a straight line? 461 00:24:22 --> 00:24:28 (students respond ) 462 00:24:24 --> 00:24:30 LEWIN: It's not a straight line? 463 00:24:26 --> 00:24:32 (students respond ) 464 00:24:27 --> 00:24:33 LEWIN: What's wrong? 465 00:24:30 --> 00:24:36 466 00:24:31 --> 00:24:37 Okay, we'll put in a third point. 467 00:24:34 --> 00:24:40 2.69. 468 00:24:35 --> 00:24:41 469 00:24:37 --> 00:24:43 Two point... 470 00:24:41 --> 00:24:47 This is 2.7-- 2.69. 471 00:24:46 --> 00:24:52 I can hardly put in the error of the timing, 472 00:24:48 --> 00:24:54 because it is not much larger than the size of my dots. 473 00:24:52 --> 00:24:58 And now we have the last one. 474 00:24:54 --> 00:25:00 One point... let's take the average-- 1.55. 475 00:25:00 --> 00:25:06 1.55... 476 00:25:03 --> 00:25:09 with an error of about 2/10 of a second, 477 00:25:07 --> 00:25:13 and this has an error of about 2/10 of a second. 478 00:25:10 --> 00:25:16 All right, there we go. 479 00:25:12 --> 00:25:18 Now... is this a straight line or is it not? 480 00:25:19 --> 00:25:25 481 00:25:21 --> 00:25:27 A gorgeous straight line. 482 00:25:23 --> 00:25:29 And so you see 483 00:25:24 --> 00:25:30 you are really working here in the regime of 1 over... 484 00:25:30 --> 00:25:36 in the regime where the terminal velocity is proportional 485 00:25:35 --> 00:25:41 to the radius squared. 486 00:25:37 --> 00:25:43 Okay, we'll give you your lights back. 487 00:25:41 --> 00:25:47 Now comes a question which is relevant to this experiment, 488 00:25:45 --> 00:25:51 and that is, 489 00:25:46 --> 00:25:52 how long does it take for the terminal speed to be reached? 490 00:25:52 --> 00:25:58 Well, the object has a certain mass, 491 00:25:56 --> 00:26:02 so there's a gravitational force on it, 492 00:26:01 --> 00:26:07 and the gravitational force equals mg. 493 00:26:05 --> 00:26:11 And then there is a resistive force, which in the case-- 494 00:26:10 --> 00:26:16 because we are operating in regime one exclusively-- 495 00:26:14 --> 00:26:20 that resistive force equals C1 r v in terms of magnitude, 496 00:26:20 --> 00:26:26 C1 r v, because we deal with regime one. 497 00:26:25 --> 00:26:31 And so if I call this the increasing value of y, 498 00:26:29 --> 00:26:35 the second... Newton's Second Law would give me 499 00:26:32 --> 00:26:38 ma equals mg minus C1 r times v, and this equals m dv dt, 500 00:26:43 --> 00:26:49 so I have here a differential equation in v, 501 00:26:47 --> 00:26:53 and that can be solved. 502 00:26:48 --> 00:26:54 And you're going to solve it on your assignment number four. 503 00:26:52 --> 00:26:58 What you're going to see is that the speed as a function of time 504 00:26:58 --> 00:27:04 is going to build up to a maximum value... 505 00:27:02 --> 00:27:08 this is the... 506 00:27:03 --> 00:27:09 to a maximum value, which is the terminal velocity-- 507 00:27:08 --> 00:27:14 or you may want to call it terminal speed-- 508 00:27:11 --> 00:27:17 and it's going to build up in some fashion 509 00:27:14 --> 00:27:20 and then it's going to asymptotically approach 510 00:27:18 --> 00:27:24 the terminal speed. 511 00:27:19 --> 00:27:25 And this is what I'm asking you on your third... 512 00:27:23 --> 00:27:29 on your fourth assignment, to calculate that. 513 00:27:25 --> 00:27:31 If there were no drag force at all, I hope you realize 514 00:27:29 --> 00:27:35 that the velocity would increase linearly, 515 00:27:34 --> 00:27:40 so you would get something like this. 516 00:27:36 --> 00:27:42 So there's no drag. 517 00:27:40 --> 00:27:46 So the behavior is extremely different due to the drag. 518 00:27:45 --> 00:27:51 And I calculated already 519 00:27:46 --> 00:27:52 something that is part of your assignment-- 520 00:27:50 --> 00:27:56 how long does it take for the quarter-inch ball bearing... 521 00:27:54 --> 00:28:00 how long does it take in time 522 00:27:56 --> 00:28:02 to reach a speed which is about 99% of the terminal speed? 523 00:28:02 --> 00:28:08 And I calculated that, 524 00:28:03 --> 00:28:09 and you will go through that calculation for yourself. 525 00:28:05 --> 00:28:11 That is only nine milliseconds. 526 00:28:07 --> 00:28:13 In other words, once it has broken through the surface-- 527 00:28:10 --> 00:28:16 that takes a while because of the thin film-- 528 00:28:12 --> 00:28:18 then in nine milliseconds 529 00:28:14 --> 00:28:20 will I already be at 99% of the terminal speed, 530 00:28:16 --> 00:28:22 and so there was no problem at all; 531 00:28:19 --> 00:28:25 when I waited for the object to cross the first mark, 532 00:28:23 --> 00:28:29 it was already clearly going at the terminal speed. 533 00:28:27 --> 00:28:33 So that was fine. 534 00:28:30 --> 00:28:36 Now I want to turn to air. 535 00:28:33 --> 00:28:39 Air, of course, behaves in an extremely different way. 536 00:28:37 --> 00:28:43 The principle is the same, 537 00:28:39 --> 00:28:45 but the values for C1 and C2 are vastly different. 538 00:28:43 --> 00:28:49 If we take air at one atmospheres, 539 00:28:47 --> 00:28:53 and we take it at room temperature, 540 00:28:50 --> 00:28:56 then C1 is about 3.1 times ten to the minus four in our units 541 00:28:57 --> 00:29:03 and C2 is about 0.85. 542 00:29:01 --> 00:29:07 This is very close to the density of air, 543 00:29:03 --> 00:29:09 which is about one kilogram per cubic meter, 544 00:29:05 --> 00:29:11 which I told you earlier, C2 and rho are very strongly related. 545 00:29:09 --> 00:29:15 And so the critical speed, which is C1 divided by C2 by r, 546 00:29:18 --> 00:29:24 is about 3.7 times ten to the minus four divided by r meters per second. 547 00:29:25 --> 00:29:31 And that is about 400 times lower 548 00:29:29 --> 00:29:35 than the critical speed in syrup, in the Karo syrup 549 00:29:34 --> 00:29:40 for the same value of r. 550 00:29:36 --> 00:29:42 So if I compare the quarter-inch ball bearing 551 00:29:41 --> 00:29:47 and I drop it in the Karo syrup, 552 00:29:45 --> 00:29:51 then the terminal velocity in the Karo syrup 553 00:29:48 --> 00:29:54 isway below the critical velocity of the Karo syrup. 554 00:29:51 --> 00:29:57 The critical velocity of the Karo syrup 555 00:29:53 --> 00:29:59 would be 100 miles per hour for a quarter-inch ball bearing. 556 00:29:56 --> 00:30:02 So it's way below. 557 00:29:57 --> 00:30:03 Here, in air, the critical velocity is 558 00:30:00 --> 00:30:06 something like 11 centimeters per second. 559 00:30:03 --> 00:30:09 This is 11 centimeters in one second, and we know 560 00:30:06 --> 00:30:12 when you drop a one-quarter-inch ball bearing in air 561 00:30:09 --> 00:30:15 that the speed is way larger, and therefore 562 00:30:12 --> 00:30:18 in the case of air, a quarter- inch ball bearing would have 563 00:30:16 --> 00:30:22 a speed way above the critical speed, 564 00:30:18 --> 00:30:24 and so you are now exclusively in regime two. 565 00:30:23 --> 00:30:29 That's the regime two. 566 00:30:26 --> 00:30:32 Almost all spheres that you drop in air operate exclusively 567 00:30:33 --> 00:30:39 in regime two. 568 00:30:35 --> 00:30:41 Whether it is a raindrop 569 00:30:37 --> 00:30:43 or whether it is a baseball that you hit, 570 00:30:39 --> 00:30:45 or a golf ball, or even a beach ball, 571 00:30:42 --> 00:30:48 or you throw a pebble off a high building, 572 00:30:45 --> 00:30:51 or whether you jump out of an airplane, 573 00:30:48 --> 00:30:54 with or without a parachute, makes no difference, 574 00:30:51 --> 00:30:57 you're always dominated by the pressure term, 575 00:30:54 --> 00:31:00 by the v-square term, and you always are in a range 576 00:30:59 --> 00:31:05 whereby the terminal speed is proportional 577 00:31:02 --> 00:31:08 to the square root of the radius 578 00:31:04 --> 00:31:10 for a given density of the object. 579 00:31:08 --> 00:31:14 If you take a pebble with a radius of about one centimeter 580 00:31:12 --> 00:31:18 and you throw it off a high building, 581 00:31:14 --> 00:31:20 it will reach a speed which will not exceed 75 miles per hour 582 00:31:18 --> 00:31:24 because of the air drag. 583 00:31:20 --> 00:31:26 If you jump out of a plane and you have no parachute 584 00:31:25 --> 00:31:31 and I make the assumption 585 00:31:28 --> 00:31:34 that your mass is about 70 kilograms-- 586 00:31:32 --> 00:31:38 I want rough numbers-- if I can approximate you 587 00:31:35 --> 00:31:41 by a sphere with a radius of about 40 centimeters-- 588 00:31:38 --> 00:31:44 that's also an approximation, you're not really like a sphere, 589 00:31:41 --> 00:31:47 but I want to get some rough numbers-- 590 00:31:43 --> 00:31:49 then the terminal velocity is 150 miles per hour. 591 00:31:51 --> 00:31:57 So if you jump out of a plane and you have no parachute, 592 00:31:56 --> 00:32:02 you will not go much faster than 150 miles per hour. 593 00:32:00 --> 00:32:06 I just read an article yesterday 594 00:32:02 --> 00:32:08 about sky divers who jump out of planes, 595 00:32:04 --> 00:32:10 and they want to open the parachute 596 00:32:05 --> 00:32:11 at the very last possible, 597 00:32:07 --> 00:32:13 and they reach terminal velocities 598 00:32:09 --> 00:32:15 of 120 miles per hour, which doesn't surprise me. 599 00:32:11 --> 00:32:17 It's very close to this rough number that I calculated. 600 00:32:14 --> 00:32:20 Of course, then they open the parachute 601 00:32:16 --> 00:32:22 and then the air drag increases enormously 602 00:32:19 --> 00:32:25 and then they slow down even further. 603 00:32:22 --> 00:32:28 604 00:32:25 --> 00:32:31 I told you a raindrop... 605 00:32:27 --> 00:32:33 almost all raindrops operate in regime two when they fall. 606 00:32:32 --> 00:32:38 So the terminal velocity is dictated by the v-square term. 607 00:32:37 --> 00:32:43 However, if you make the drops exceedingly small, 608 00:32:41 --> 00:32:47 there comes a time that you really enter regime one. 609 00:32:45 --> 00:32:51 And on assignment number four 610 00:32:47 --> 00:32:53 I've asked you to calculate where that happens, 611 00:32:50 --> 00:32:56 and I can't do it for water 612 00:32:52 --> 00:32:58 because the radius of that water drop will be so small 613 00:32:56 --> 00:33:02 that it would evaporate immediately, 614 00:32:58 --> 00:33:04 so I chose oil for that. 615 00:32:59 --> 00:33:05 So I'm asking you in assignment four, take an oil drop, 616 00:33:02 --> 00:33:08 make it smaller and smaller and smaller and smaller, 617 00:33:05 --> 00:33:11 and there comes a time that you begin to enter regime one, 618 00:33:09 --> 00:33:15 and I want you to calculate where that crossover is 619 00:33:13 --> 00:33:19 between these two domains. 620 00:33:15 --> 00:33:21 621 00:33:16 --> 00:33:22 I have here a ball-- 622 00:33:18 --> 00:33:24 you may call it a balloon, but I call it a ball 623 00:33:22 --> 00:33:28 because there's no helium in it. 624 00:33:25 --> 00:33:31 And this ball-- ooh!-- weighs approximately 34 grams. 625 00:33:31 --> 00:33:37 626 00:33:34 --> 00:33:40 So let me erase some here, because I want, yeah... 627 00:33:40 --> 00:33:46 628 00:33:42 --> 00:33:48 So we know the mass and we know the radius. 629 00:33:47 --> 00:33:53 The mass is about 34 grams 630 00:33:51 --> 00:33:57 and the radius is about 35 centimeters. 631 00:33:54 --> 00:34:00 It's about 70 centimeters across. 632 00:33:57 --> 00:34:03 I can calculate what the terminal velocity is-- 633 00:34:01 --> 00:34:07 a better name would be terminal speed. 634 00:34:04 --> 00:34:10 I know I'm definitely going to be in this regime, 635 00:34:08 --> 00:34:14 so I know the mass, I know C2, which in air is 0.85, 636 00:34:12 --> 00:34:18 I know the radius and I know g, 637 00:34:14 --> 00:34:20 and so I find that I find about 1.8 meters per second. 638 00:34:20 --> 00:34:26 So if I drop it from a height of three meters, 639 00:34:23 --> 00:34:29 which I'm going to do, then you would think 640 00:34:25 --> 00:34:31 that the time that it takes to hit the floor 641 00:34:28 --> 00:34:34 would be about my three meters 642 00:34:30 --> 00:34:36 divided by 1.8 meters per second, 643 00:34:34 --> 00:34:40 which is about 1.7 seconds. 644 00:34:36 --> 00:34:42 That's not bad. 645 00:34:38 --> 00:34:44 That's not a bad approximation. 646 00:34:40 --> 00:34:46 However, it will, of course, take longer, 647 00:34:42 --> 00:34:48 and the reason why it will take longer 648 00:34:44 --> 00:34:50 is that the terminal velocity, 649 00:34:46 --> 00:34:52 the terminal speed is not achieved instantaneously. 650 00:34:51 --> 00:34:57 With the ball bearings, it was within nine milliseconds. 651 00:34:54 --> 00:35:00 I can assure you that here it will take a lot longer. 652 00:34:58 --> 00:35:04 Now, if you want to calculate the time that it takes 653 00:35:02 --> 00:35:08 to get close to terminal speed, that is not an easy task, 654 00:35:07 --> 00:35:13 because you are going to end up 655 00:35:09 --> 00:35:15 with a nasty differential equation. 656 00:35:12 --> 00:35:18 You're going to get mg. 657 00:35:13 --> 00:35:19 You're going to get the acceleration 658 00:35:16 --> 00:35:22 which is the result of... 659 00:35:17 --> 00:35:23 Let's go with the equation we have. 660 00:35:19 --> 00:35:25 You see, we have ma equals mg 661 00:35:26 --> 00:35:32 and then we get minus the resistive force. 662 00:35:31 --> 00:35:37 And the resistive force has a term in v 663 00:35:34 --> 00:35:40 and has a term in v squared. 664 00:35:36 --> 00:35:42 You see, v and v squared, 665 00:35:39 --> 00:35:45 and so this cannot be solved analytically. 666 00:35:41 --> 00:35:47 But I've asked my graduate student Dave Pooley, 667 00:35:44 --> 00:35:50 who is one of your instructors, 668 00:35:46 --> 00:35:52 to solve this for me numerically. 669 00:35:49 --> 00:35:55 And I'm going to show you the results. 670 00:35:52 --> 00:35:58 In fact, he prepared... he has a nice view graph, 671 00:35:55 --> 00:36:01 and you can see the effect of time on the ball 672 00:36:01 --> 00:36:07 if you drop it from three meters. 673 00:36:05 --> 00:36:11 674 00:36:07 --> 00:36:13 Here it is. 675 00:36:09 --> 00:36:15 All the numbers are there. 676 00:36:10 --> 00:36:16 This is on the Web, so don't copy anything. 677 00:36:13 --> 00:36:19 You have the values for C1 and C2 are given at the top. 678 00:36:16 --> 00:36:22 You may not be able to see them from your seat, 679 00:36:19 --> 00:36:25 but they are there, and what you see here 680 00:36:21 --> 00:36:27 is the height above the ground as a function of time-- 681 00:36:27 --> 00:36:33 this is one second, this is 1½ seconds; 682 00:36:29 --> 00:36:35 this is the three-meter mark. 683 00:36:31 --> 00:36:37 If there were no air drag-- 684 00:36:33 --> 00:36:39 remember we dropped an apple early on from three meters-- 685 00:36:37 --> 00:36:43 it will hit the floor at about 780 milliseconds. 686 00:36:41 --> 00:36:47 However, with the air drag, it will be about one second later, 687 00:36:45 --> 00:36:51 more like 1.8 seconds. 688 00:36:47 --> 00:36:53 So the 1.7 wasn't bad, as you see, but if you look here 689 00:36:52 --> 00:36:58 at how the speed builds up as a function of time, 690 00:36:56 --> 00:37:02 you see it takes about three-, four-tenths of a second 691 00:37:00 --> 00:37:06 to build up to that terminal speed, 692 00:37:02 --> 00:37:08 which is the 1.8 meters per second that you have there. 693 00:37:04 --> 00:37:10 And needless to say, of course, 694 00:37:06 --> 00:37:12 that the acceleration due to gravity 695 00:37:08 --> 00:37:14 does not remain constant but goes down very quickly, 696 00:37:11 --> 00:37:17 and when the acceleration reaches, approaches zero, 697 00:37:15 --> 00:37:21 then you have terminal speed, 698 00:37:17 --> 00:37:23 and then there is no longer any change in the velocity. 699 00:37:21 --> 00:37:27 So let's try this. 700 00:37:24 --> 00:37:30 We'll give more light. 701 00:37:27 --> 00:37:33 And we're going to throw this object, 702 00:37:29 --> 00:37:35 and I don't think you're going to get 1.8 seconds. 703 00:37:32 --> 00:37:38 You may get something that is larger than 1.8 seconds, 704 00:37:36 --> 00:37:42 and the reasons are the following. 705 00:37:38 --> 00:37:44 Number one, this is not a perfect sphere, 706 00:37:41 --> 00:37:47 and only for spheres do these calculations hold-- 707 00:37:44 --> 00:37:50 that's number one. 708 00:37:45 --> 00:37:51 Number two, this thing is very springy, 709 00:37:48 --> 00:37:54 so the moment that I let it go, 710 00:37:50 --> 00:37:56 it probably goes in some kind of oscillation. 711 00:37:52 --> 00:37:58 That doesn't help either. 712 00:37:53 --> 00:37:59 That will probably also slow it down, 713 00:37:55 --> 00:38:01 because what causes, of course, this slowdown in regime two 714 00:37:58 --> 00:38:04 is really turbulence. 715 00:38:00 --> 00:38:06 Turbulence is extremely hard to understand and predict. 716 00:38:03 --> 00:38:09 And so almost anything I do, I will only add turbulence, 717 00:38:07 --> 00:38:13 and therefore I predict 718 00:38:09 --> 00:38:15 that the time that it will take from three meters 719 00:38:12 --> 00:38:18 will be probably larger than 1.8 seconds, 720 00:38:14 --> 00:38:20 but it will be substantially larger 721 00:38:16 --> 00:38:22 than the 780 milliseconds, 722 00:38:18 --> 00:38:24 which is what you would have seen if you drop an apple. 723 00:38:23 --> 00:38:29 So let's see how close we are. 724 00:38:25 --> 00:38:31 725 00:38:30 --> 00:38:36 So, turn on this timer. 726 00:38:32 --> 00:38:38 Make sure I zero it-- I did. 727 00:38:36 --> 00:38:42 And... 728 00:38:37 --> 00:38:43 729 00:38:45 --> 00:38:51 It's not so easy to release it, by the way, 730 00:38:48 --> 00:38:54 and start the timer at the same moment. 731 00:38:50 --> 00:38:56 And it's not even so easy for me to see when it hits the ground, 732 00:38:53 --> 00:38:59 so there's a huge uncertainty in this experiment. 733 00:38:58 --> 00:39:04 Okay. 734 00:39:00 --> 00:39:06 735 00:39:02 --> 00:39:08 Three, two, one, zero. 736 00:39:03 --> 00:39:09 737 00:39:07 --> 00:39:13 What do we see? 738 00:39:08 --> 00:39:14 739 00:39:11 --> 00:39:17 Did I see something? 740 00:39:12 --> 00:39:18 2.0-- that's not bad. 741 00:39:15 --> 00:39:21 See? The prediction was 1.8; you get 2.0. 742 00:39:18 --> 00:39:24 That's not bad, so this takes the air drag into account, 743 00:39:21 --> 00:39:27 and it is not even an approximation. 744 00:39:23 --> 00:39:29 It uses the entire term linear in v as well as in v square. 745 00:39:30 --> 00:39:36 But it's almost exclusively dominated by the v-square term. 746 00:39:35 --> 00:39:41 747 00:39:37 --> 00:39:43 I also asked Dave to show me what happens 748 00:39:40 --> 00:39:46 when I throw a pebble off the Empire State Building. 749 00:39:44 --> 00:39:50 And the pebble that we chose had a radius of one centimeter-- 750 00:39:48 --> 00:39:54 it's the kind of pebble that all of us could find-- 751 00:39:51 --> 00:39:57 I know roughly the density of pebbles, 752 00:39:53 --> 00:39:59 and when we throw it off the Empire State Building, 753 00:39:55 --> 00:40:01 we reach a terminal speed of about 75 miles per hour. 754 00:40:00 --> 00:40:06 Without the air drag, we would have reached 225 miles per hour. 755 00:40:05 --> 00:40:11 So I want to show you that, too. 756 00:40:08 --> 00:40:14 757 00:40:13 --> 00:40:19 So now you see this Empire State Building, 758 00:40:16 --> 00:40:22 which has a height of 475 meters, 759 00:40:19 --> 00:40:25 so that's where you start, at t0; this is one second, 760 00:40:22 --> 00:40:28 five seconds... ten seconds, five seconds, 15 seconds, 761 00:40:25 --> 00:40:31 and if there had been no air drag, 762 00:40:28 --> 00:40:34 it would hit the ground a little less than ten seconds, 763 00:40:31 --> 00:40:37 but now it will hit the ground more like 16, 17 seconds. 764 00:40:35 --> 00:40:41 And you see that the terminal speed builds up 765 00:40:38 --> 00:40:44 in about five, six seconds. 766 00:40:39 --> 00:40:45 It's very close to the final value, 767 00:40:41 --> 00:40:47 and if there had been no air drag, 768 00:40:44 --> 00:40:50 then the speed at which it would hit the ground 769 00:40:46 --> 00:40:52 would, of course, grow linearly, 770 00:40:48 --> 00:40:54 and when it hits the ground, it would be somewhere here, 771 00:40:52 --> 00:40:58 which is 225 miles per hour. 772 00:40:53 --> 00:40:59 So you see 773 00:40:54 --> 00:41:00 that it's even a pebble you wouldn't expect to be... 774 00:40:57 --> 00:41:03 to have a very large effect on air drag, 775 00:40:59 --> 00:41:05 it is huge, provided that you throw it from a high building. 776 00:41:04 --> 00:41:10 Now, you may remember 777 00:41:06 --> 00:41:12 that we dropped an apple from three meters 778 00:41:10 --> 00:41:16 and that we calculated the gravitational acceleration 779 00:41:16 --> 00:41:22 given the time that it takes to fall. 780 00:41:17 --> 00:41:23 That was one of your... 781 00:41:19 --> 00:41:25 one of the things you did in your assignment. 782 00:41:21 --> 00:41:27 We had 781 milliseconds, I think. 783 00:41:23 --> 00:41:29 And out of that you can calculate g, right? 784 00:41:27 --> 00:41:33 Because you know that three meters is one-half g t squared, 785 00:41:30 --> 00:41:36 so I give you the three, with an uncertainty, 786 00:41:33 --> 00:41:39 I give you the time-- 787 00:41:34 --> 00:41:40 781 milliseconds with an uncertainty of two milliseconds, 788 00:41:37 --> 00:41:43 which we allowed. 789 00:41:39 --> 00:41:45 Out pops g. 790 00:41:40 --> 00:41:46 So I asked Dave, 791 00:41:41 --> 00:41:47 "What is the effect of air drag on this apple? 792 00:41:45 --> 00:41:51 Was it a responsible thing for us to ignore that?" 793 00:41:50 --> 00:41:56 The apple has a mass of 134 grams. 794 00:41:56 --> 00:42:02 It's easy to weigh, of course. 795 00:41:58 --> 00:42:04 So this was our apple during our first lecture-- m is 134 grams. 796 00:42:05 --> 00:42:11 It's almost a sphere, really-- not quite but almost a sphere-- 797 00:42:09 --> 00:42:15 and the radius is about three centimeters. 798 00:42:12 --> 00:42:18 And that leads to a terminal velocity 799 00:42:15 --> 00:42:21 which you can calculate if you want to 800 00:42:18 --> 00:42:24 using the v-square term, 801 00:42:21 --> 00:42:27 but I was not interested in that. 802 00:42:23 --> 00:42:29 I wanted to know how many milliseconds 803 00:42:27 --> 00:42:33 is the touchdown delayed because of the air drag. 804 00:42:31 --> 00:42:37 And Dave made the calculations, 805 00:42:33 --> 00:42:39 and he found that that is two milliseconds from three meters. 806 00:42:37 --> 00:42:43 From 1½ meters it's almost nothing, 807 00:42:39 --> 00:42:45 and the reason why it's almost nothing 808 00:42:41 --> 00:42:47 from 1½ meters-- you see, 809 00:42:43 --> 00:42:49 when you throw an apple in air, it's really in regime two, 810 00:42:48 --> 00:42:54 so you're really dominated by the speed squared, 811 00:42:52 --> 00:42:58 and the first 1½ meters 812 00:42:54 --> 00:43:00 it doesn't get at very high speed yet. 813 00:42:56 --> 00:43:02 The speed grows linearly, and so it is the last portion 814 00:42:59 --> 00:43:05 where you really get hit by the air drag, by the v-square term. 815 00:43:04 --> 00:43:10 Two milliseconds from three meters, so if h is three meters, 816 00:43:10 --> 00:43:16 there is a two-millisecond... let's call it delay. 817 00:43:15 --> 00:43:21 So we were on the hairy edge of being lucky and unlucky. 818 00:43:19 --> 00:43:25 If you really want 819 00:43:20 --> 00:43:26 to recalculate the gravitational acceleration using our data, 820 00:43:24 --> 00:43:30 you should really subtract the two milliseconds from the time. 821 00:43:28 --> 00:43:34 On the other hand, 822 00:43:29 --> 00:43:35 since we allowed a two- millisecond uncertainty, 823 00:43:32 --> 00:43:38 we really weren't too far off. 824 00:43:35 --> 00:43:41 825 00:43:37 --> 00:43:43 Now comes my last part, 826 00:43:39 --> 00:43:45 and that is, how does air drag influence trajectories? 827 00:43:45 --> 00:43:51 And that is also part of your assignment, 828 00:43:48 --> 00:43:54 but I'm going to help you a little bit with that. 829 00:43:51 --> 00:43:57 In your assignment number four, I'm asking you 830 00:43:56 --> 00:44:02 to evaluate quantitatively the motion of an object in liquid, 831 00:44:05 --> 00:44:11 but I give that object 832 00:44:07 --> 00:44:13 an initial speed in the x direction, 833 00:44:11 --> 00:44:17 and then there's the liquid below. 834 00:44:13 --> 00:44:19 Then gravity is there, and there is that initial speed. 835 00:44:16 --> 00:44:22 If there were no drag, 836 00:44:17 --> 00:44:23 then this, of course, would be a parabola, 837 00:44:20 --> 00:44:26 and the horizontal velocity would always be the same. 838 00:44:24 --> 00:44:30 There would never be any change. 839 00:44:26 --> 00:44:32 But that's not the case now. 840 00:44:28 --> 00:44:34 Because of the resistive forces, 841 00:44:31 --> 00:44:37 because of the drag by the liquid, 842 00:44:34 --> 00:44:40 the object is going to get a velocity in this direction, 843 00:44:38 --> 00:44:44 so there's going to be 844 00:44:39 --> 00:44:45 a component of the resistive force opposing it. 845 00:44:43 --> 00:44:49 It has a speed in this direction, 846 00:44:45 --> 00:44:51 so there's also going to be a component of the resistive force 847 00:44:49 --> 00:44:55 in this direction. 848 00:44:51 --> 00:44:57 And that will decrease the speed-- 849 00:44:53 --> 00:44:59 this component in the x direction. 850 00:44:55 --> 00:45:01 And so you can already see 851 00:44:57 --> 00:45:03 that the curve that you're going to see is a very different one. 852 00:45:01 --> 00:45:07 It's going to look something like this. 853 00:45:06 --> 00:45:12 And then ultimately, there is nothing left of this component, 854 00:45:10 --> 00:45:16 and ultimately, when you go vertically down, 855 00:45:13 --> 00:45:19 you have the terminal speed that you can find 856 00:45:17 --> 00:45:23 from dropping an object just into the liquid vertically. 857 00:45:23 --> 00:45:29 So that's something you're going to deal with 858 00:45:25 --> 00:45:31 in your assignment number four, 859 00:45:27 --> 00:45:33 and this is exclusively done in regime one, 860 00:45:30 --> 00:45:36 because we have an object with liquid, 861 00:45:33 --> 00:45:39 and with liquid, you almost always work with regime one. 862 00:45:39 --> 00:45:45 Suppose I take a tennis ball 863 00:45:42 --> 00:45:48 and I throw up a tennis ball in 26.100. 864 00:45:45 --> 00:45:51 There is air drag on that tennis ball. 865 00:45:48 --> 00:45:54 In the absence of any air drag, I would get a nice parabola 866 00:45:53 --> 00:45:59 which will be completely symmetric. 867 00:45:56 --> 00:46:02 I throw it up with a certain initial speed, v-- 868 00:46:03 --> 00:46:09 call it v0, I don't care-- 869 00:46:05 --> 00:46:11 and the horizontal component would never change. 870 00:46:10 --> 00:46:16 This would be v0x; it would always be the same. 871 00:46:12 --> 00:46:18 But now with air drag, you're going to see 872 00:46:16 --> 00:46:22 that there's going to be a force, an air drag force 873 00:46:20 --> 00:46:26 in the y direction. 874 00:46:22 --> 00:46:28 If the object goes up in this direction, then there will be 875 00:46:26 --> 00:46:32 a resistive force component in the y direction, 876 00:46:29 --> 00:46:35 and since it has a speed in this direction, 877 00:46:32 --> 00:46:38 there will also be a resistive force in the x direction, 878 00:46:37 --> 00:46:43 so this speed is going to be eaten up in the same way 879 00:46:41 --> 00:46:47 that this component was going to be eaten up. 880 00:46:44 --> 00:46:50 This component is going to suffer. 881 00:46:47 --> 00:46:53 It will not stay constant throughout, 882 00:46:49 --> 00:46:55 and as a result of that, 883 00:46:51 --> 00:46:57 you're going to get a trajectory that looks more like this. 884 00:46:55 --> 00:47:01 885 00:46:57 --> 00:47:03 It's asymmetric. 886 00:46:59 --> 00:47:05 Clearly you don't reach 887 00:47:00 --> 00:47:06 the highest point that you would have reached without air drag, 888 00:47:05 --> 00:47:11 for the reason that this resistive force in the y direction 889 00:47:09 --> 00:47:15 will not allow it to go as high-- that's obvious. 890 00:47:12 --> 00:47:18 You don't go as far as you would without air drag, 891 00:47:15 --> 00:47:21 for obvious reasons that this resistive force 892 00:47:18 --> 00:47:24 is going to kill this speed, 893 00:47:20 --> 00:47:26 but you also will get asymmetry in the curvature, 894 00:47:24 --> 00:47:30 and I want you to see that. 895 00:47:26 --> 00:47:32 I call this point O, this point P, and let's call this point S. 896 00:47:31 --> 00:47:37 So what I will do is I will throw up a tennis ball 897 00:47:34 --> 00:47:40 and then I will throw up a Styrofoam ball, 898 00:47:37 --> 00:47:43 and the Styrofoam ball has very closely the same radius 899 00:47:41 --> 00:47:47 as the tennis ball. 900 00:47:43 --> 00:47:49 That means the resistive force is the same on both, 901 00:47:47 --> 00:47:53 because the resistive force is only dictated 902 00:47:50 --> 00:47:56 by r squared and by v-- r squared v squared, remember? 903 00:47:56 --> 00:48:02 However, this has a way larger mass than this one, 904 00:48:00 --> 00:48:06 and even though the resistive forces will be closely the same 905 00:48:03 --> 00:48:09 if I throw them up with the same initial speed, 906 00:48:06 --> 00:48:12 it has a way larger effect on a smaller mass 907 00:48:08 --> 00:48:14 than on a larger mass. 908 00:48:10 --> 00:48:16 F = ma, right? 909 00:48:11 --> 00:48:17 So on a very large mass, 910 00:48:13 --> 00:48:19 the resistive force will have a much lower effect 911 00:48:16 --> 00:48:22 than on the smaller mass, 912 00:48:18 --> 00:48:24 even though the resistive forces are about the same. 913 00:48:21 --> 00:48:27 So, try to see 914 00:48:23 --> 00:48:29 that the tennis ball is very close to an ideal parabola. 915 00:48:27 --> 00:48:33 You will not even see any effect of asymmetry. 916 00:48:30 --> 00:48:36 It will not be the case for the Styrofoam, though. 917 00:48:33 --> 00:48:39 So, look at the tennis ball first. 918 00:48:36 --> 00:48:42 919 00:48:38 --> 00:48:44 I should really do it here. 920 00:48:41 --> 00:48:47 Did it look more or less symmetric? 921 00:48:43 --> 00:48:49 Okay, now I'll try this one. 922 00:48:45 --> 00:48:51 923 00:48:46 --> 00:48:52 Did it look asymmetric? 924 00:48:48 --> 00:48:54 Could you see it? 925 00:48:50 --> 00:48:56 Are you just saying yes, or you really saw it? 926 00:48:53 --> 00:48:59 Let me do it once more; I can throw it back. 927 00:48:55 --> 00:49:01 So now it should curve like this 928 00:48:58 --> 00:49:04 and then sort of come down like that. 929 00:49:00 --> 00:49:06 You ready? 930 00:49:01 --> 00:49:07 931 00:49:04 --> 00:49:10 You see the asymmetry? 932 00:49:06 --> 00:49:12 Okay, now comes my last question. 933 00:49:08 --> 00:49:14 I'm going to ask you the following, 934 00:49:09 --> 00:49:15 and there's a unique answer to that. 935 00:49:12 --> 00:49:18 I want you to think about it and I want you to be able 936 00:49:15 --> 00:49:21 to give an answer with total, 100% confidence. 937 00:49:18 --> 00:49:24 When this object goes from O to P, 938 00:49:21 --> 00:49:27 that takes a certain amount of time. 939 00:49:23 --> 00:49:29 When it goes from P to S, back to the ground, 940 00:49:27 --> 00:49:33 that takes also a certain amount of time. 941 00:49:30 --> 00:49:36 Is this time the same as this time? 942 00:49:34 --> 00:49:40 Or is this time longer than this time, or is this time shorter? 943 00:49:39 --> 00:49:45 Think about it. 944 00:49:41 --> 00:49:47 See you Friday. 945 00:49:43 --> 00:49:49 946 00:49:47 --> 00:49:53.000