1 00:00:00,494 --> 00:00:01,625 - [Instructor] So we saw last time 2 00:00:01,625 --> 00:00:03,988 that there's two types of kinetic energy, 3 00:00:03,988 --> 00:00:06,072 translational and rotational, 4 00:00:06,072 --> 00:00:07,349 but these kinetic energies 5 00:00:07,349 --> 00:00:10,263 aren't necessarily proportional to each other. 6 00:00:10,263 --> 00:00:13,096 In other words, the amount of translational kinetic energy 7 00:00:13,096 --> 00:00:15,920 isn't necessarily related to the amount 8 00:00:15,920 --> 00:00:17,998 of rotational kinetic energy. 9 00:00:17,998 --> 00:00:21,010 However, there's a whole class of problems. 10 00:00:21,010 --> 00:00:22,955 A really common type of problem 11 00:00:22,955 --> 00:00:25,120 where these are proportional. 12 00:00:25,120 --> 00:00:26,827 So that's what we're gonna talk about today 13 00:00:26,827 --> 00:00:28,580 and that comes up in this case. 14 00:00:28,580 --> 00:00:29,613 So, imagine this. 15 00:00:29,613 --> 00:00:32,104 Imagine we, instead of pitching this baseball, 16 00:00:32,104 --> 00:00:35,523 we roll the baseball across the concrete. 17 00:00:35,523 --> 00:00:37,020 So, say we take this baseball 18 00:00:37,020 --> 00:00:39,389 and we just roll it across the concrete. 19 00:00:39,389 --> 00:00:40,306 What's it gonna do? 20 00:00:40,306 --> 00:00:43,865 It's gonna rotate as it moves forward, 21 00:00:43,865 --> 00:00:46,372 and so, it's gonna do something that we call, 22 00:00:46,372 --> 00:00:48,706 rolling without slipping. 23 00:00:48,706 --> 00:00:51,312 At least that's what this baseball's most likely gonna do. 24 00:00:51,312 --> 00:00:54,418 I mean, unless you really chucked this baseball hard 25 00:00:54,418 --> 00:00:56,432 or the ground was really icy, 26 00:00:56,432 --> 00:00:58,580 it's probably not gonna skid across the ground 27 00:00:58,580 --> 00:01:01,280 or even if it did, that would stop really quick 28 00:01:01,280 --> 00:01:02,634 because it would start rolling 29 00:01:02,634 --> 00:01:04,694 and that rolling motion would just keep up 30 00:01:04,694 --> 00:01:06,094 with the motion forward. 31 00:01:06,094 --> 00:01:08,979 So when you have a surface like leather against concrete, 32 00:01:08,979 --> 00:01:10,691 it's gonna be grippy enough, 33 00:01:10,691 --> 00:01:13,036 grippy enough that as this ball moves forward, 34 00:01:13,036 --> 00:01:16,647 it rolls, and that rolling motion just keeps up so that 35 00:01:16,647 --> 00:01:19,857 the surfaces never skid across each other. 36 00:01:19,857 --> 00:01:22,679 In other words, this ball's gonna be moving forward, 37 00:01:22,679 --> 00:01:24,745 but it's not gonna be slipping across the ground. 38 00:01:24,745 --> 00:01:26,585 There's gonna be no sliding motion 39 00:01:26,585 --> 00:01:28,327 at this bottom surface here, 40 00:01:28,327 --> 00:01:29,970 which means, at any given moment, 41 00:01:29,970 --> 00:01:31,218 this is a little weird to think about, 42 00:01:31,218 --> 00:01:32,135 at any given moment, 43 00:01:32,135 --> 00:01:34,097 this baseball rolling across the ground, 44 00:01:34,097 --> 00:01:37,017 has zero velocity at the very bottom. 45 00:01:37,017 --> 00:01:40,169 This bottom surface right here isn't actually moving 46 00:01:40,169 --> 00:01:41,829 with respect to the ground 47 00:01:41,829 --> 00:01:43,501 because otherwise, it'd be slipping 48 00:01:43,501 --> 00:01:45,330 or sliding across the ground, 49 00:01:45,330 --> 00:01:47,030 but this point right here, 50 00:01:47,030 --> 00:01:48,679 that's in contact with the ground, 51 00:01:48,679 --> 00:01:51,315 isn't actually skidding across the ground 52 00:01:51,315 --> 00:01:53,340 and that means this point right here on the baseball 53 00:01:53,340 --> 00:01:55,111 has zero velocity. 54 00:01:55,111 --> 00:01:56,684 So this is weird, zero velocity, 55 00:01:56,684 --> 00:01:58,002 and what's weirder, 56 00:01:58,002 --> 00:01:59,889 that's means when you're driving down the freeway, 57 00:01:59,889 --> 00:02:00,893 at a high speed, 58 00:02:00,893 --> 00:02:02,594 no matter how fast you're driving, 59 00:02:02,594 --> 00:02:05,891 the bottom of your tire has a velocity of zero. 60 00:02:05,891 --> 00:02:08,573 It's not actually moving with respect to the ground. 61 00:02:08,573 --> 00:02:11,019 It's just, the rest of the tire 62 00:02:11,019 --> 00:02:13,283 that rotates around that point. 63 00:02:13,283 --> 00:02:14,682 So that point kinda sticks there 64 00:02:14,682 --> 00:02:16,540 for just a brief, split second. 65 00:02:16,540 --> 00:02:18,548 That makes it so that the tire can push itself 66 00:02:18,548 --> 00:02:19,750 around that point, 67 00:02:19,750 --> 00:02:22,699 and then a new point becomes the point that doesn't move, 68 00:02:22,699 --> 00:02:25,131 and then, it gets rotated around that point, 69 00:02:25,131 --> 00:02:27,140 and then, a new point is the point that doesn't move. 70 00:02:27,140 --> 00:02:29,148 So, they all take turns, it's very nice of them. 71 00:02:29,148 --> 00:02:30,082 Other points are moving. 72 00:02:30,082 --> 00:02:32,822 This point up here is going crazy fast on your tire, 73 00:02:32,822 --> 00:02:34,444 relative to the ground, 74 00:02:34,444 --> 00:02:36,673 but the point that's touching the ground, 75 00:02:36,673 --> 00:02:38,664 unless you're driving a little unsafely, 76 00:02:38,664 --> 00:02:40,261 you shouldn't be skidding here, 77 00:02:40,261 --> 00:02:42,304 if all is working as it should, 78 00:02:42,304 --> 00:02:44,307 under normal operating conditions, 79 00:02:44,307 --> 00:02:45,822 the bottom part of your tire 80 00:02:45,822 --> 00:02:47,801 should not be skidding across the ground 81 00:02:47,801 --> 00:02:50,112 and that means that bottom point on your tire 82 00:02:50,112 --> 00:02:53,182 isn't actually moving with respect to the ground, 83 00:02:53,182 --> 00:02:55,394 which means it's stuck for just a split second. 84 00:02:55,394 --> 00:02:56,979 It has no velocity. 85 00:02:56,979 --> 00:02:59,759 So that's what we mean by rolling without slipping. 86 00:02:59,759 --> 00:03:01,425 Why is this a big deal? 87 00:03:01,425 --> 00:03:02,708 I'll show you why it's a big deal. 88 00:03:02,708 --> 00:03:05,489 This implies that these two kinetic energies 89 00:03:05,489 --> 00:03:07,549 right here, are proportional, 90 00:03:07,549 --> 00:03:10,487 and moreover, it implies that these two velocities, 91 00:03:10,487 --> 00:03:13,343 this center mass velocity and this angular velocity 92 00:03:13,343 --> 00:03:14,928 are also proportional. 93 00:03:14,928 --> 00:03:16,472 So that's what I wanna show you here. 94 00:03:16,472 --> 00:03:17,574 So, how do we prove that? 95 00:03:17,574 --> 00:03:19,670 How do we prove that the center mass velocity 96 00:03:19,670 --> 00:03:22,532 is proportional to the angular velocity? 97 00:03:22,532 --> 00:03:25,045 Well imagine this, imagine we coat the outside 98 00:03:25,045 --> 00:03:26,845 of our baseball with paint. 99 00:03:26,845 --> 00:03:28,749 So I'm about to roll it on the ground, right? 100 00:03:28,749 --> 00:03:30,252 Roll it without slipping. 101 00:03:30,252 --> 00:03:32,574 Let's say I just coat this outside with paint, 102 00:03:32,574 --> 00:03:34,182 so there's a bunch of paint here. 103 00:03:34,182 --> 00:03:36,750 Now let's say, I give that baseball a roll forward, 104 00:03:36,750 --> 00:03:38,102 well what are we gonna see on the ground? 105 00:03:38,102 --> 00:03:41,104 We're gonna see that it just traces out a distance 106 00:03:41,104 --> 00:03:43,275 that's equal to however far it rolled. 107 00:03:43,275 --> 00:03:44,917 So if it rolled to this point, 108 00:03:44,917 --> 00:03:48,842 in other words, if this baseball rotates that far, 109 00:03:48,842 --> 00:03:50,490 it's gonna have moved forward 110 00:03:50,490 --> 00:03:53,399 exactly that much arc length forward, right? 111 00:03:53,399 --> 00:03:56,080 'Cause if this baseball's rolling without slipping, 112 00:03:56,080 --> 00:03:58,629 then, as this baseball rotates forward, 113 00:03:58,629 --> 00:04:00,138 it will have moved forward 114 00:04:00,138 --> 00:04:03,580 exactly this much arc length forward. 115 00:04:03,580 --> 00:04:06,257 So in other words, if you unwind this purple shape, 116 00:04:06,257 --> 00:04:08,428 or if you look at the path that traces out on the ground, 117 00:04:08,428 --> 00:04:11,563 it would trace out exactly that arc length forward, 118 00:04:11,563 --> 00:04:12,602 and why do we care? 119 00:04:12,602 --> 00:04:15,609 Why do we care that it travels an arc length forward? 120 00:04:15,609 --> 00:04:18,848 'Cause that means the center of mass of this baseball 121 00:04:18,848 --> 00:04:21,582 has traveled the arc length forward. 122 00:04:21,582 --> 00:04:23,939 So the center of mass of this baseball 123 00:04:23,939 --> 00:04:25,814 has moved that far forward. 124 00:04:25,814 --> 00:04:28,903 That's the distance the center of mass has moved 125 00:04:28,903 --> 00:04:30,894 and we know that's equal to the arc length. 126 00:04:30,894 --> 00:04:32,461 What's the arc length? 127 00:04:32,461 --> 00:04:33,738 Remember we got a formula for that. 128 00:04:33,738 --> 00:04:36,159 If something rotates through a certain angle. 129 00:04:36,159 --> 00:04:38,574 So if we consider the angle from there to there 130 00:04:38,574 --> 00:04:40,501 and we imagine the radius of the baseball, 131 00:04:40,501 --> 00:04:42,033 the arc length is gonna equal 132 00:04:42,033 --> 00:04:44,030 r times the change in theta, 133 00:04:44,030 --> 00:04:46,491 how much theta this thing has rotated through, 134 00:04:46,491 --> 00:04:48,064 but note that this is not true 135 00:04:48,064 --> 00:04:49,771 for every point on the baseball. 136 00:04:49,771 --> 00:04:51,199 Consider this point at the top, 137 00:04:51,199 --> 00:04:53,898 it was both rotating around the center of mass, 138 00:04:53,898 --> 00:04:56,592 while the center of mass was moving forward, 139 00:04:56,592 --> 00:04:59,439 so this took some complicated curved path through space. 140 00:04:59,439 --> 00:05:00,913 It might've looked like that. 141 00:05:00,913 --> 00:05:02,864 This distance here is not necessarily 142 00:05:02,864 --> 00:05:04,199 equal to the arc length, 143 00:05:04,199 --> 00:05:06,643 but the center of mass was not rotating around 144 00:05:06,643 --> 00:05:07,769 the center of mass, 145 00:05:07,769 --> 00:05:09,255 'cause it's the center of mass. 146 00:05:09,255 --> 00:05:10,869 The center of mass here at this baseball 147 00:05:10,869 --> 00:05:12,471 was just going in a straight line 148 00:05:12,471 --> 00:05:13,969 and that's why we can say 149 00:05:13,969 --> 00:05:16,749 the center mass of the baseball's distance traveled 150 00:05:16,749 --> 00:05:19,774 was just equal to the amount of arc length 151 00:05:19,774 --> 00:05:21,265 this baseball rotated through. 152 00:05:21,265 --> 00:05:23,175 In other words it's equal to the length 153 00:05:23,175 --> 00:05:25,288 painted on the ground, so to speak, 154 00:05:25,288 --> 00:05:26,339 and so, why do we care? 155 00:05:26,339 --> 00:05:27,697 Why do we care that the distance 156 00:05:27,697 --> 00:05:29,218 the center of mass moves 157 00:05:29,218 --> 00:05:30,884 is equal to the arc length? 158 00:05:30,884 --> 00:05:32,149 Here's why we care, check this out. 159 00:05:32,149 --> 00:05:35,282 We can just divide both sides by the time that that took, 160 00:05:35,282 --> 00:05:36,886 and look at what we get, we get the distance, 161 00:05:36,886 --> 00:05:39,150 the center of mass moved, over the time that that took. 162 00:05:39,150 --> 00:05:41,345 That's just the speed of the center of mass, 163 00:05:41,345 --> 00:05:42,964 and we get that that equals 164 00:05:42,964 --> 00:05:45,846 the radius times delta theta over deltaT, 165 00:05:45,846 --> 00:05:47,930 but that's just the angular speed. 166 00:05:47,930 --> 00:05:50,600 So this shows that the speed of the center of mass, 167 00:05:50,600 --> 00:05:52,940 for something that's rotating without slipping, 168 00:05:52,940 --> 00:05:55,250 is equal to the radius of that object 169 00:05:55,250 --> 00:05:59,314 times the angular speed about the center of mass. 170 00:05:59,314 --> 00:06:00,811 So the speed of the center of mass 171 00:06:00,811 --> 00:06:03,865 is equal to r times the angular speed 172 00:06:03,865 --> 00:06:06,320 about that center of mass, 173 00:06:06,320 --> 00:06:07,557 and this is important. 174 00:06:07,557 --> 00:06:09,252 This you wanna commit to memory 175 00:06:09,252 --> 00:06:12,445 because when a problem says something's rotating 176 00:06:12,445 --> 00:06:14,122 or rolling without slipping, 177 00:06:14,122 --> 00:06:18,006 that's basically code for V equals r omega, 178 00:06:18,006 --> 00:06:20,037 where V is the center of mass speed 179 00:06:20,037 --> 00:06:24,002 and omega is the angular speed about that center of mass. 180 00:06:24,002 --> 00:06:25,070 Now, you might not be impressed. 181 00:06:25,070 --> 00:06:26,407 You might be like, "Wait a minute. 182 00:06:26,407 --> 00:06:27,585 "Didn't we already know this? 183 00:06:27,585 --> 00:06:30,842 "Didn't we already know that V equals r omega?" 184 00:06:30,842 --> 00:06:32,722 We did, but this is different. 185 00:06:32,722 --> 00:06:35,927 This V we showed down here is the V of the center of mass, 186 00:06:35,927 --> 00:06:37,808 the speed of the center of mass. 187 00:06:37,808 --> 00:06:40,083 This V up here was talking about the speed 188 00:06:40,083 --> 00:06:41,482 at some point on the object, 189 00:06:41,482 --> 00:06:43,462 a distance r away from the center, 190 00:06:43,462 --> 00:06:45,726 and it was relative to the center of mass. 191 00:06:45,726 --> 00:06:46,678 So, in other words, 192 00:06:46,678 --> 00:06:48,738 say we've got some baseball that's rotating, 193 00:06:48,738 --> 00:06:49,571 if we wanted to know, 194 00:06:49,571 --> 00:06:51,948 okay at some distance r away from the center, 195 00:06:51,948 --> 00:06:55,245 how fast is this point moving, V, 196 00:06:55,245 --> 00:06:57,648 compared to the angular speed? 197 00:06:57,648 --> 00:07:01,056 Well if this thing's rotating like this, 198 00:07:01,056 --> 00:07:03,254 that's gonna have some speed, V, 199 00:07:03,254 --> 00:07:06,458 but that's the speed, V, relative to the center of mass. 200 00:07:06,458 --> 00:07:08,670 What we found in this equation's different. 201 00:07:08,670 --> 00:07:11,401 This is the speed of the center of mass. 202 00:07:11,401 --> 00:07:14,327 This tells us how fast is that center of mass going, 203 00:07:14,327 --> 00:07:17,101 not just how fast is a point on the baseball moving, 204 00:07:17,101 --> 00:07:19,412 relative to the center of mass. 205 00:07:19,412 --> 00:07:21,176 This gives us a way to determine, 206 00:07:21,176 --> 00:07:23,150 what was the speed of the center of mass? 207 00:07:23,150 --> 00:07:25,333 And it turns out that is really useful 208 00:07:25,333 --> 00:07:26,958 and a whole bunch of problems 209 00:07:26,958 --> 00:07:28,193 that I'm gonna show you right now. 210 00:07:28,193 --> 00:07:29,546 Let's do some examples. 211 00:07:29,546 --> 00:07:30,771 Let's get rid of all this. 212 00:07:30,771 --> 00:07:31,827 So let's do this one right here. 213 00:07:31,827 --> 00:07:34,405 Let's say you took a cylinder, a solid cylinder 214 00:07:34,405 --> 00:07:37,301 of five kilograms that had a radius of two meters 215 00:07:37,301 --> 00:07:39,368 and you wind a bunch of string around it 216 00:07:39,368 --> 00:07:41,864 and then you tie the loose end to the ceiling 217 00:07:41,864 --> 00:07:45,568 and you let go and you let this cylinder unwind downward. 218 00:07:45,568 --> 00:07:48,110 As it rolls, it's gonna be moving downward. 219 00:07:48,110 --> 00:07:50,403 Let's say you drop it from a height of four meters, 220 00:07:50,403 --> 00:07:51,257 and you wanna know, 221 00:07:51,257 --> 00:07:53,915 how fast is this cylinder gonna be moving? 222 00:07:53,915 --> 00:07:56,620 How fast is this center of mass gonna be moving 223 00:07:56,620 --> 00:07:58,606 right before it hits the ground? 224 00:07:58,606 --> 00:07:59,645 That's what we wanna know. 225 00:07:59,645 --> 00:08:01,195 We're calling this a yo-yo, 226 00:08:01,195 --> 00:08:02,989 but it's not really a yo-yo. 227 00:08:02,989 --> 00:08:04,509 A yo-yo has a cavity inside 228 00:08:04,509 --> 00:08:07,592 and maybe the string is wound around a tiny axle 229 00:08:07,592 --> 00:08:09,147 that's only about that big. 230 00:08:09,147 --> 00:08:11,481 We're winding our string around the outside edge 231 00:08:11,481 --> 00:08:12,569 and that's gonna be important 232 00:08:12,569 --> 00:08:14,362 because this is basically a case 233 00:08:14,362 --> 00:08:16,289 of rolling without slipping. 234 00:08:16,289 --> 00:08:18,640 You might be like, "this thing's not even rolling at all", 235 00:08:18,640 --> 00:08:20,237 but it's still the same idea, 236 00:08:20,237 --> 00:08:22,472 just imagine this string is the ground. 237 00:08:22,472 --> 00:08:25,026 It's as if you have a wheel or a ball 238 00:08:25,026 --> 00:08:26,767 that's rolling on the ground 239 00:08:26,767 --> 00:08:29,426 and not slipping with respect to the ground, 240 00:08:29,426 --> 00:08:32,741 except this time the ground is the string. 241 00:08:32,741 --> 00:08:36,135 This cylinder is not slipping with respect to the string, 242 00:08:36,135 --> 00:08:37,807 so that's something we have to assume. 243 00:08:37,808 --> 00:08:40,414 We're gonna assume this yo-yo's unwinding, 244 00:08:40,414 --> 00:08:43,380 but the string is not sliding across 245 00:08:43,381 --> 00:08:45,012 the surface of the cylinder 246 00:08:45,012 --> 00:08:47,578 and that means we can use our previous derivation, 247 00:08:47,578 --> 00:08:50,753 that the speed of the center of mass of this cylinder, 248 00:08:50,753 --> 00:08:53,702 is gonna have to equal the radius of the cylinder 249 00:08:53,702 --> 00:08:56,152 times the angular speed of the cylinder, 250 00:08:56,152 --> 00:08:58,149 since the center of mass of this cylinder 251 00:08:58,149 --> 00:09:02,294 is gonna be moving down a distance equal to the arc length 252 00:09:02,294 --> 00:09:05,608 traced out by the outside edge of the cylinder, 253 00:09:05,608 --> 00:09:07,251 but this doesn't let us solve, 'cause look, 254 00:09:07,251 --> 00:09:08,911 I don't know the speed of the center of mass 255 00:09:08,911 --> 00:09:10,444 and I don't know the angular velocity, 256 00:09:10,444 --> 00:09:12,882 so we need another equation, another idea in here, 257 00:09:12,882 --> 00:09:15,111 and that idea is gonna be conservation of energy. 258 00:09:15,111 --> 00:09:17,073 This problem's crying out to be solved 259 00:09:17,073 --> 00:09:19,331 with conservation of energy, so let's do it. 260 00:09:19,331 --> 00:09:20,887 So we're gonna put everything in our system. 261 00:09:20,887 --> 00:09:22,518 We're gonna say energy's conserved. 262 00:09:22,518 --> 00:09:24,788 Starts off at a height of four meters. 263 00:09:24,788 --> 00:09:26,895 That means it starts off with potential energy. 264 00:09:26,895 --> 00:09:29,258 So I'm gonna say that this starts off with mgh, 265 00:09:29,258 --> 00:09:30,907 and what does that turn into? 266 00:09:30,907 --> 00:09:33,025 Well this cylinder, when it gets down to the ground, 267 00:09:33,025 --> 00:09:34,842 no longer has potential energy, 268 00:09:34,842 --> 00:09:37,379 as long as we're considering the lowest most point, 269 00:09:37,379 --> 00:09:38,691 as h equals zero, 270 00:09:38,691 --> 00:09:40,073 but it will be moving, 271 00:09:40,073 --> 00:09:41,477 so it's gonna have kinetic energy 272 00:09:41,477 --> 00:09:44,107 and it won't just have translational kinetic energy. 273 00:09:44,107 --> 00:09:46,772 So, it will have translational kinetic energy, 274 00:09:46,772 --> 00:09:49,831 'cause the center of mass of this cylinder 275 00:09:49,831 --> 00:09:51,166 is going to be moving. 276 00:09:51,166 --> 00:09:53,372 The center of mass of the cylinder is gonna have a speed, 277 00:09:53,372 --> 00:09:55,897 but it's also gonna have rotational kinetic energy 278 00:09:55,897 --> 00:09:57,726 because the cylinder's gonna be rotating 279 00:09:57,726 --> 00:09:59,490 about the center of mass, 280 00:09:59,490 --> 00:10:02,462 at the same time that the center of mass is moving downward, 281 00:10:02,462 --> 00:10:06,525 so we have to add 1/2, I omega, squared 282 00:10:06,525 --> 00:10:08,766 and it still seems like we can't solve, 283 00:10:08,766 --> 00:10:11,502 'cause look, we don't know V and we don't know omega, 284 00:10:11,502 --> 00:10:12,785 but this is the key. 285 00:10:12,785 --> 00:10:14,736 This is why you needed to know this formula 286 00:10:14,736 --> 00:10:17,197 and we spent like five or six minutes deriving it. 287 00:10:17,197 --> 00:10:20,204 This is the link between V and omega. 288 00:10:20,204 --> 00:10:21,975 So, we can put this whole formula here, 289 00:10:21,975 --> 00:10:23,594 in terms of one variable, 290 00:10:23,594 --> 00:10:27,809 by substituting in for either V or for omega. 291 00:10:27,809 --> 00:10:29,904 Now, I'm gonna substitute in for omega, 292 00:10:29,904 --> 00:10:31,744 because we wanna solve for V. 293 00:10:31,744 --> 00:10:33,300 So, I'm just gonna say that omega, 294 00:10:33,300 --> 00:10:34,786 you could flip this equation around 295 00:10:34,786 --> 00:10:37,880 and just say that, "Omega equals the speed 296 00:10:37,880 --> 00:10:40,609 "of the center of mass divided by the radius." 297 00:10:40,609 --> 00:10:41,787 So I'm gonna use it that way, 298 00:10:41,787 --> 00:10:44,144 I'm gonna plug in, I just solve this for omega, 299 00:10:44,144 --> 00:10:46,431 I'm gonna plug that in for omega over here. 300 00:10:46,431 --> 00:10:47,795 Let's just see what happens 301 00:10:47,795 --> 00:10:49,560 when you get V of the center of mass, 302 00:10:49,560 --> 00:10:51,098 divided by the radius, 303 00:10:51,098 --> 00:10:52,509 and you can't forget to square it, 304 00:10:52,509 --> 00:10:53,478 so we square that. 305 00:10:53,478 --> 00:10:54,784 So after we square this out, 306 00:10:54,784 --> 00:10:56,572 we're gonna get the same thing over again, 307 00:10:56,572 --> 00:10:58,679 so I'm just gonna copy that, paste it again, 308 00:10:58,679 --> 00:11:00,299 but this whole term's gonna be squared. 309 00:11:00,299 --> 00:11:02,632 So I'm gonna have a V of the center of mass, squared, 310 00:11:02,632 --> 00:11:04,258 over radius, squared, 311 00:11:04,258 --> 00:11:05,721 and so, now it's looking much better. 312 00:11:05,721 --> 00:11:08,019 We just have one variable in here that we don't know, 313 00:11:08,019 --> 00:11:09,442 V of the center of mass. 314 00:11:09,442 --> 00:11:11,334 This I might be freaking you out, 315 00:11:11,334 --> 00:11:13,726 this is the moment of inertia, what do we do with that? 316 00:11:13,726 --> 00:11:15,409 With a moment of inertia of a cylinder, 317 00:11:15,409 --> 00:11:17,133 you often just have to look these up. 318 00:11:17,133 --> 00:11:18,695 The moment of inertia of a cylinder 319 00:11:18,695 --> 00:11:22,177 turns out to be 1/2 m, the mass of the cylinder, 320 00:11:22,177 --> 00:11:24,598 times the radius of the cylinder squared. 321 00:11:24,598 --> 00:11:26,612 So we can take this, plug that in for I, 322 00:11:26,612 --> 00:11:27,541 and what are we gonna get? 323 00:11:27,541 --> 00:11:29,451 If I just copy this, paste that again. 324 00:11:29,451 --> 00:11:31,297 If we substitute in for our I, 325 00:11:31,297 --> 00:11:32,603 our moment of inertia, 326 00:11:32,603 --> 00:11:34,136 and I'm gonna scoot this over just a little bit, 327 00:11:34,136 --> 00:11:37,090 our moment of inertia was 1/2 mr squared. 328 00:11:37,090 --> 00:11:40,033 So I'm gonna have 1/2, and this is in addition to this 1/2, 329 00:11:40,033 --> 00:11:41,699 so this 1/2 was already here. 330 00:11:41,699 --> 00:11:44,660 There's another 1/2, from the moment of inertia term, 331 00:11:44,660 --> 00:11:46,828 1/2mr squared, 332 00:11:46,828 --> 00:11:48,836 but this r is the same as that r, 333 00:11:48,836 --> 00:11:49,962 so look it, I've got a, 334 00:11:49,962 --> 00:11:52,406 I've got a r squared and a one over r squared, 335 00:11:52,406 --> 00:11:55,215 these end up canceling, and this is really strange, 336 00:11:55,215 --> 00:11:58,077 it doesn't matter what the radius of the cylinder was, 337 00:11:58,077 --> 00:11:59,911 and here's something else that's weird, 338 00:11:59,911 --> 00:12:01,757 not only does the radius cancel, 339 00:12:01,757 --> 00:12:03,423 all these terms have mass in it. 340 00:12:03,423 --> 00:12:05,757 So no matter what the mass of the cylinder was, 341 00:12:05,757 --> 00:12:07,574 they will all get to the ground 342 00:12:07,574 --> 00:12:09,472 with the same center of mass speed. 343 00:12:09,472 --> 00:12:12,101 In other words, all yo-yo's of the same shape 344 00:12:12,101 --> 00:12:13,924 are gonna tie when they get to the ground 345 00:12:13,924 --> 00:12:15,346 as long as all else is equal 346 00:12:15,346 --> 00:12:16,902 when we're ignoring air resistance. 347 00:12:16,902 --> 00:12:18,470 No matter how big the yo-yo, 348 00:12:18,470 --> 00:12:20,281 or have massive or what the radius is, 349 00:12:20,281 --> 00:12:23,386 they should all tie at the ground with the same speed, 350 00:12:23,386 --> 00:12:24,803 which is kinda weird. 351 00:12:24,803 --> 00:12:27,078 So now, finally we can solve for the center of mass. 352 00:12:27,078 --> 00:12:28,657 We've got this right hand side. 353 00:12:28,657 --> 00:12:30,439 The left hand side is just gh, 354 00:12:30,439 --> 00:12:32,994 that's gonna equal, so we end up with 1/2, 355 00:12:32,994 --> 00:12:34,886 V of the center of mass squared, 356 00:12:34,886 --> 00:12:38,833 plus 1/4, V of the center of mass squared. 357 00:12:38,833 --> 00:12:40,964 That's just equal to 3/4 358 00:12:40,964 --> 00:12:43,210 speed of the center of mass squared. 359 00:12:43,210 --> 00:12:46,031 If you take a half plus a fourth, you get 3/4. 360 00:12:46,031 --> 00:12:49,096 So if I solve this for the speed of the center of mass, 361 00:12:49,096 --> 00:12:52,765 I'm gonna get, if I multiply gh by four over three, 362 00:12:52,765 --> 00:12:54,088 and we take a square root, 363 00:12:54,088 --> 00:12:58,076 we're gonna get the square root of 4gh over 3, 364 00:12:58,076 --> 00:12:59,615 and so now, I can just plug in numbers. 365 00:12:59,615 --> 00:13:02,186 If I wanted to, I could just say that this is gonna equal 366 00:13:02,186 --> 00:13:04,589 the square root of four 367 00:13:04,589 --> 00:13:07,961 times 9.8 meters per second squared, 368 00:13:07,961 --> 00:13:10,794 times four meters, that's where we started from, 369 00:13:10,794 --> 00:13:12,750 that was our height, divided by three, 370 00:13:12,750 --> 00:13:17,655 is gonna give us a speed of the center of mass of 7.23 371 00:13:17,655 --> 00:13:19,053 meters per second. 372 00:13:19,053 --> 00:13:20,475 Now, here's something to keep in mind, 373 00:13:20,475 --> 00:13:22,879 other problems might look different from this, 374 00:13:22,879 --> 00:13:24,986 but the way you solve them might be identical. 375 00:13:24,986 --> 00:13:27,232 For instance, we could just take this whole 376 00:13:27,232 --> 00:13:29,229 solution here, I'm gonna copy that. 377 00:13:29,229 --> 00:13:31,551 Let's try a new problem, it's gonna be easy. 378 00:13:31,551 --> 00:13:32,903 It's not gonna take long. 379 00:13:32,903 --> 00:13:34,767 Let's say we take the same cylinder 380 00:13:34,767 --> 00:13:36,497 and we release it from rest 381 00:13:36,497 --> 00:13:38,087 at the top of an incline 382 00:13:38,087 --> 00:13:40,234 that's four meters tall and we let it roll 383 00:13:40,234 --> 00:13:42,330 without slipping to the bottom of the incline, 384 00:13:42,330 --> 00:13:43,822 and again, we ask the question, 385 00:13:43,822 --> 00:13:46,875 "How fast is the center of mass of this cylinder 386 00:13:46,875 --> 00:13:49,801 "gonna be going when it reaches the bottom of the incline?" 387 00:13:49,801 --> 00:13:51,490 Well, it's the same problem. 388 00:13:51,490 --> 00:13:53,528 It looks different from the other problem, 389 00:13:53,528 --> 00:13:55,460 but conceptually and mathematically, 390 00:13:55,460 --> 00:13:57,411 it's the same calculation. 391 00:13:57,411 --> 00:14:00,963 This thing started off with potential energy, mgh, 392 00:14:00,963 --> 00:14:03,785 and it turned into conservation of energy says 393 00:14:03,785 --> 00:14:06,646 that that had to turn into rotational kinetic energy 394 00:14:06,646 --> 00:14:08,614 and translational kinetic energy. 395 00:14:08,614 --> 00:14:10,182 Again, if it's a cylinder, 396 00:14:10,182 --> 00:14:12,359 the moment of inertia's 1/2mr squared, 397 00:14:12,359 --> 00:14:14,140 and if it's rolling without slipping, 398 00:14:14,140 --> 00:14:17,397 again, we can replace omega with V over r, 399 00:14:17,397 --> 00:14:19,365 since that relationship holds 400 00:14:19,365 --> 00:14:21,490 for something that's rotating without slipping, 401 00:14:21,490 --> 00:14:22,825 the m's cancel as well, 402 00:14:22,825 --> 00:14:24,444 and we get the same calculation. 403 00:14:24,444 --> 00:14:26,453 This cylinder again is gonna be going 404 00:14:26,453 --> 00:14:28,728 7.23 meters per second. 405 00:14:28,728 --> 00:14:31,265 The center of mass is gonna be traveling that fast 406 00:14:31,265 --> 00:14:34,504 when it rolls down a ramp that was four meters tall. 407 00:14:34,504 --> 00:14:37,645 So recapping, even though the speed of the center of mass 408 00:14:37,645 --> 00:14:40,849 of an object, is not necessarily proportional 409 00:14:40,849 --> 00:14:43,316 to the angular velocity of that object, 410 00:14:43,316 --> 00:14:46,538 if the object is rotating or rolling without slipping, 411 00:14:46,538 --> 00:14:48,529 this relationship is true 412 00:14:48,529 --> 00:14:50,497 and it allows you to turn equations 413 00:14:50,497 --> 00:14:52,773 that would've had two unknowns in them, 414 00:14:52,773 --> 00:14:55,106 into equations that have only one unknown, 415 00:14:55,106 --> 00:14:56,639 which then, let's you solve 416 00:14:56,639 --> 00:00:00,000 for the speed of the center of mass of the object.