1 00:00:00,544 --> 00:00:02,149 - [Instructor] The rotational kinematic formulas 2 00:00:02,149 --> 00:00:04,434 allow us to relate the five different 3 00:00:04,434 --> 00:00:06,497 rotational motion variables 4 00:00:06,497 --> 00:00:08,989 and they look just like the regular kinematic formulas 5 00:00:08,989 --> 00:00:10,756 except instead of displacement, 6 00:00:10,756 --> 00:00:12,299 there's angular displacement. 7 00:00:12,299 --> 00:00:13,750 Instead of initial velocity 8 00:00:13,750 --> 00:00:15,792 there's initial angular velocity. 9 00:00:15,792 --> 00:00:17,107 Instead of final velocity 10 00:00:17,107 --> 00:00:19,108 there's final angular velocity. 11 00:00:19,108 --> 00:00:22,187 Instead of acceleration there's angular acceleration 12 00:00:22,187 --> 00:00:24,177 and the time is still just the time. 13 00:00:24,177 --> 00:00:25,759 You only get the first two of these 14 00:00:25,759 --> 00:00:27,776 on the AP exam formula sheet. 15 00:00:27,776 --> 00:00:29,523 You do not get three and four. 16 00:00:29,523 --> 00:00:31,837 And just like the regular kinematic formulas, 17 00:00:31,837 --> 00:00:33,592 these rotational kinematic formulas 18 00:00:33,592 --> 00:00:37,517 are only true if the angular acceleration is constant. 19 00:00:37,517 --> 00:00:40,379 What do each of these rotational variables mean? 20 00:00:40,379 --> 00:00:42,792 Well, the angular displacement is the amount of 21 00:00:42,792 --> 00:00:45,241 angle the object has rotated through 22 00:00:45,241 --> 00:00:47,209 in the certain amount of time t. 23 00:00:47,209 --> 00:00:49,914 Angular velocity is defined to be the amount of 24 00:00:49,914 --> 00:00:52,644 angle you rotated through per time 25 00:00:52,644 --> 00:00:56,299 just like regular velocity is the displacement per time, 26 00:00:56,299 --> 00:00:58,549 and the angular acceleration is defined 27 00:00:58,549 --> 00:01:02,038 to be the amount of change in angular velocity per time. 28 00:01:02,038 --> 00:01:04,132 Just like regular acceleration is the change 29 00:01:04,132 --> 00:01:06,188 in regular velocity per time. 30 00:01:06,188 --> 00:01:07,821 For something rotating in a circle, 31 00:01:07,821 --> 00:01:10,017 technically the angular velocity points 32 00:01:10,017 --> 00:01:12,773 perpendicular to that plane of rotation 33 00:01:12,773 --> 00:01:14,434 but it's easiest to just think of omega 34 00:01:14,434 --> 00:01:17,060 as being counterclockwise or clockwise. 35 00:01:17,060 --> 00:01:18,769 And there's relationships between these 36 00:01:18,769 --> 00:01:21,395 angular variables and their linear counterparts. 37 00:01:21,395 --> 00:01:24,744 To get the arc length s the object has traveled through 38 00:01:24,744 --> 00:01:27,194 you just multiply the radius of the path 39 00:01:27,194 --> 00:01:29,257 by the amount of angular displacement. 40 00:01:29,257 --> 00:01:30,942 To get the speed of the object 41 00:01:30,942 --> 00:01:32,744 just multiply the radius of the path 42 00:01:32,744 --> 00:01:34,758 by the angular speed of the object. 43 00:01:34,758 --> 00:01:36,868 To get the tangential acceleration 44 00:01:36,868 --> 00:01:38,544 multiply the radius of the path 45 00:01:38,544 --> 00:01:40,552 by the angular acceleration. 46 00:01:40,552 --> 00:01:43,318 Note that this is the tangential acceleration. 47 00:01:43,318 --> 00:01:45,171 This acceleration causes the object 48 00:01:45,171 --> 00:01:47,560 to speed up or slow down. 49 00:01:47,560 --> 00:01:50,096 It's the centripetal component of the acceleration 50 00:01:50,096 --> 00:01:52,680 that causes the object to change directions 51 00:01:52,680 --> 00:01:54,241 and the formula for that is still 52 00:01:54,241 --> 00:01:56,054 just v squared over r. 53 00:01:56,054 --> 00:01:57,326 If an object's moving in a circle 54 00:01:57,326 --> 00:01:59,375 it must have centripetal acceleration 55 00:01:59,375 --> 00:02:01,144 because it's changing directions 56 00:02:01,144 --> 00:02:03,389 but only if it's speeding up or slowing down 57 00:02:03,389 --> 00:02:05,500 will it have tangential acceleration 58 00:02:05,500 --> 00:02:07,508 and angular acceleration. 59 00:02:07,508 --> 00:02:08,961 What's an example problem involving 60 00:02:08,961 --> 00:02:11,327 the angular motion variables look like? 61 00:02:11,327 --> 00:02:13,560 Let's say an object is rotating in a circle 62 00:02:13,560 --> 00:02:14,880 at a constant rate. 63 00:02:14,880 --> 00:02:16,133 Which would best describe 64 00:02:16,133 --> 00:02:19,395 the three different types of accelerations of the object? 65 00:02:19,395 --> 00:02:21,147 Well, if an object's moving in a circle at all 66 00:02:21,147 --> 00:02:23,583 there has to be centripetal acceleration 67 00:02:23,583 --> 00:02:25,297 so that's got to be non-zero. 68 00:02:25,297 --> 00:02:27,143 And if it's rotating at a constant rate 69 00:02:27,143 --> 00:02:29,053 there's no change in omega 70 00:02:29,053 --> 00:02:32,271 and that's means the angular acceleration is gonna be zero. 71 00:02:32,271 --> 00:02:34,104 If the angular acceleration is zero 72 00:02:34,104 --> 00:02:37,768 the tangential acceleration would also be zero. 73 00:02:37,768 --> 00:02:40,459 Only when the object is speeding up or slowing down 74 00:02:40,459 --> 00:02:42,206 do you have angular acceleration 75 00:02:42,206 --> 00:02:44,156 and tangential acceleration. 76 00:02:44,156 --> 00:02:45,771 This change the speed 77 00:02:45,771 --> 00:02:49,007 and centripetal acceleration changes the direction. 78 00:02:49,007 --> 00:02:50,589 What does torque mean? 79 00:02:50,589 --> 00:02:53,532 Just like force is what causes acceleration, 80 00:02:53,532 --> 00:02:56,718 torque is what causes angular acceleration. 81 00:02:56,718 --> 00:02:59,147 In order for an object to speed up or slow down 82 00:02:59,147 --> 00:03:00,595 in its angular motion, 83 00:03:00,595 --> 00:03:03,094 there's got to be a net torque on the object. 84 00:03:03,094 --> 00:03:04,618 What causes a torque? 85 00:03:04,618 --> 00:03:06,228 Forces cause torque. 86 00:03:06,228 --> 00:03:08,881 In order to have a torque you have to have a force 87 00:03:08,881 --> 00:03:11,291 but the same force could exert a different torque 88 00:03:11,291 --> 00:03:13,660 depending on where that force is exerted. 89 00:03:13,660 --> 00:03:16,720 If the force is exerted far from the axis of rotation, 90 00:03:16,720 --> 00:03:19,711 you'll get more torque for that given amount of force 91 00:03:19,711 --> 00:03:21,416 compared to forces that are exerted 92 00:03:21,416 --> 00:03:23,053 near the axis of rotation. 93 00:03:23,053 --> 00:03:26,279 This r represents how far that force is applied 94 00:03:26,279 --> 00:03:27,625 from the axis. 95 00:03:27,625 --> 00:03:28,917 And to maximize that force 96 00:03:28,917 --> 00:03:32,272 you would actually wanna point it perpendicular to this r 97 00:03:32,272 --> 00:03:35,416 since sine of 90 degrees is equal to one. 98 00:03:35,416 --> 00:03:38,023 In other words, to maximize the amount of torque you get 99 00:03:38,023 --> 00:03:41,502 exert the force as far away as possible from the axis 100 00:03:41,502 --> 00:03:43,287 and exert that force perpendicular 101 00:03:43,287 --> 00:03:46,341 to the line from the axis to that force. 102 00:03:46,341 --> 00:03:48,656 You might get many angles in a problem 103 00:03:48,656 --> 00:03:50,721 but this angle here is always the angle 104 00:03:50,721 --> 00:03:53,064 between the r and the F. 105 00:03:53,064 --> 00:03:55,895 And just like an object is in translational equilibrium 106 00:03:55,895 --> 00:03:57,625 if the net force is zero, 107 00:03:57,625 --> 00:04:00,226 we say that an object is in rotational equilibrium 108 00:04:00,226 --> 00:04:02,006 if the net torque is zero. 109 00:04:02,006 --> 00:04:04,829 This would cause the angular acceleration to be zero 110 00:04:04,829 --> 00:04:06,527 just like translational equilibrium 111 00:04:06,527 --> 00:04:08,786 causes the acceleration to be zero. 112 00:04:08,786 --> 00:04:11,280 Torque is a vector so it has a direction. 113 00:04:11,280 --> 00:04:13,276 Typically it's easiest to think of the direction 114 00:04:13,276 --> 00:04:16,088 as just being counterclockwise or clockwise 115 00:04:16,088 --> 00:04:17,642 based on which way that force 116 00:04:17,642 --> 00:04:19,642 would cause the object to rotate. 117 00:04:19,642 --> 00:04:21,516 And since torque is r times F, 118 00:04:21,516 --> 00:04:25,093 the units are gonna be meters times newton or newton meters. 119 00:04:25,093 --> 00:04:27,539 What's an example problem involving torque look like? 120 00:04:27,539 --> 00:04:30,333 Let's say you had this rod with this axis here 121 00:04:30,333 --> 00:04:32,622 and there were forces applied as shown. 122 00:04:32,622 --> 00:04:35,768 We wanna know how large would the force F have to be 123 00:04:35,768 --> 00:04:38,721 in order for this rod to be in rotational equilibrium. 124 00:04:38,721 --> 00:04:40,536 Remember, rotational equilibrium means 125 00:04:40,536 --> 00:04:42,884 that the net torque is equal to zero. 126 00:04:42,884 --> 00:04:45,925 In other words, all the torque that's pointing clockwise 127 00:04:45,925 --> 00:04:47,619 would have to equal all the torque 128 00:04:47,619 --> 00:04:49,442 that points counterclockwise 129 00:04:49,442 --> 00:04:51,406 in order for the system to be balanced. 130 00:04:51,406 --> 00:04:53,222 The three newton force and the one newton force, 131 00:04:53,222 --> 00:04:55,577 you're trying to rotate this system clockwise 132 00:04:55,577 --> 00:04:57,826 and the unknown force F is trying to rotate 133 00:04:57,826 --> 00:04:59,640 the system counterclockwise. 134 00:04:59,640 --> 00:05:01,244 This green one newton force 135 00:05:01,244 --> 00:05:03,074 isn't actually exerting any torque 136 00:05:03,074 --> 00:05:05,772 even though the r value is not zero. 137 00:05:05,772 --> 00:05:08,589 The angle between the force and the r value 138 00:05:08,589 --> 00:05:10,409 is gonna be 180 degrees 139 00:05:10,409 --> 00:05:12,889 and the sine of 180 degrees is zero. 140 00:05:12,889 --> 00:05:14,248 Which makes sense since this force 141 00:05:14,248 --> 00:05:16,609 isn't actually causing this rod to rotate 142 00:05:16,609 --> 00:05:18,542 clockwise or counterclockwise. 143 00:05:18,542 --> 00:05:20,630 The torque and the clockwise direction would be 144 00:05:20,630 --> 00:05:22,795 one meter times three newtons 145 00:05:22,795 --> 00:05:24,967 to find the torque from the three newton force. 146 00:05:24,967 --> 00:05:27,141 Plus you wouldn't use two meters 147 00:05:27,141 --> 00:05:29,130 for the r of the one newton force. 148 00:05:29,130 --> 00:05:31,008 You have to find the r from the axis 149 00:05:31,008 --> 00:05:34,188 which is gonna be three meters times one newton force 150 00:05:34,188 --> 00:05:37,500 which gives a total clockwise torque of six newton meters, 151 00:05:37,500 --> 00:05:38,508 and we can write the torque 152 00:05:38,508 --> 00:05:42,636 applied by the unknown force F as one meter times F. 153 00:05:42,636 --> 00:05:45,938 In order for six newton meters to equal one times F, 154 00:05:45,938 --> 00:05:49,117 the force F just has to equal six newtons. 155 00:05:49,117 --> 00:05:51,160 What's rotational inertia mean? 156 00:05:51,160 --> 00:05:53,564 Well, an object with a large rotational inertia 157 00:05:53,564 --> 00:05:57,347 will be hard to get rotating and harder to stop rotating. 158 00:05:57,347 --> 00:05:59,458 Basically the rotational inertia tells you 159 00:05:59,458 --> 00:06:03,008 how much an object will resist angular acceleration. 160 00:06:03,008 --> 00:06:04,722 Just like regular inertia tells you 161 00:06:04,722 --> 00:06:07,677 how much an object will resist regular acceleration. 162 00:06:07,677 --> 00:06:09,802 And this rotational inertia is often referred to 163 00:06:09,802 --> 00:06:11,625 as the Moment of inertia. 164 00:06:11,625 --> 00:06:13,847 How do you make the rotational inertia large? 165 00:06:13,847 --> 00:06:15,978 Well you can increase the rotational inertia 166 00:06:15,978 --> 00:06:19,452 if you place the mass far from the axis of rotation 167 00:06:19,452 --> 00:06:21,738 and you can make the rotational inertia smaller 168 00:06:21,738 --> 00:06:24,826 if you place the mass close to the axis of rotation. 169 00:06:24,826 --> 00:06:26,317 In other words if you could push the mass 170 00:06:26,317 --> 00:06:28,663 closer to the axis of rotation 171 00:06:28,663 --> 00:06:31,420 which is the point about which the object rotates, 172 00:06:31,420 --> 00:06:34,222 you can make the moment of inertia smaller and smaller. 173 00:06:34,222 --> 00:06:36,868 To find the moment of inertia or rotational inertia 174 00:06:36,868 --> 00:06:39,505 of an object whose entire mass rotates 175 00:06:39,505 --> 00:06:41,433 at the same radius r, 176 00:06:41,433 --> 00:06:43,453 you can just use the formula I 177 00:06:43,453 --> 00:06:45,514 equals the mass that's rotating, 178 00:06:45,514 --> 00:06:48,729 times how far it's rotating from the axis squared. 179 00:06:48,729 --> 00:06:51,075 This formula's not given, you have to memorize it. 180 00:06:51,075 --> 00:06:53,131 I equals mr squared. 181 00:06:53,131 --> 00:06:55,567 And if you had many masses rotating at different r's 182 00:06:55,567 --> 00:06:57,525 you could just add up all the contributions 183 00:06:57,525 --> 00:06:59,458 from each single mass. 184 00:06:59,458 --> 00:07:01,319 Now if you have a continuous object 185 00:07:01,319 --> 00:07:05,008 whose mass is not all at the same radius from the axis, 186 00:07:05,008 --> 00:07:07,262 the formulas are a little more complicated. 187 00:07:07,262 --> 00:07:09,600 For a rod rotating about its center, 188 00:07:09,600 --> 00:07:11,979 the moment of inertia would be 1/12 189 00:07:11,979 --> 00:07:13,253 the mass of the rod times 190 00:07:13,253 --> 00:07:15,500 the entire length of the rod squared. 191 00:07:15,500 --> 00:07:17,729 For rod rotating about one end, 192 00:07:17,729 --> 00:07:19,473 the moment of inertia is gonna be larger 193 00:07:19,473 --> 00:07:22,829 since more mass is distributed farther from the axis 194 00:07:22,829 --> 00:07:25,779 and this formula is 1/3 the mass of the rod 195 00:07:25,779 --> 00:07:27,903 times the entire length of the rod squared. 196 00:07:27,903 --> 00:07:29,593 The rotational inertia of a sphere 197 00:07:29,593 --> 00:07:31,949 rotating about an axis through its center 198 00:07:31,949 --> 00:07:34,617 would be 2/5 the mass of the sphere 199 00:07:34,617 --> 00:07:36,598 times the radius of the sphere squared. 200 00:07:36,598 --> 00:07:39,670 And the rotational inertia of a cylinder or a disk 201 00:07:39,670 --> 00:07:41,716 rotating about an axis through its center 202 00:07:41,716 --> 00:07:44,343 would be 1/2 the mass of the disk 203 00:07:44,343 --> 00:07:46,753 times the radius of that disk squared. 204 00:07:46,753 --> 00:07:48,233 Another example that comes up a lot 205 00:07:48,233 --> 00:07:50,904 that you wouldn't be given the formula for is a hoop. 206 00:07:50,904 --> 00:07:53,400 That is to say where all the mass is distributed 207 00:07:53,400 --> 00:07:56,335 around the center point with a hollow center. 208 00:07:56,335 --> 00:07:59,761 Since all the mass is at the same radius r, 209 00:07:59,761 --> 00:08:02,534 the formula for the rotational inertia of a hoop 210 00:08:02,534 --> 00:08:05,162 is just the same as the formula for the rotational inertia 211 00:08:05,162 --> 00:08:08,313 of a single mass rotating at radius r. 212 00:08:08,313 --> 00:08:10,791 The fact that the mass is distributed in a circle 213 00:08:10,791 --> 00:08:12,605 doesn't actually matter since the mass 214 00:08:12,605 --> 00:08:15,325 still stayed at the same radius r away. 215 00:08:15,325 --> 00:08:17,225 Rotational inertia is not a vector 216 00:08:17,225 --> 00:08:19,328 so it's always positive or zero 217 00:08:19,328 --> 00:08:22,136 and the units since it's mr squared would be 218 00:08:22,136 --> 00:08:24,250 kilograms times meter squared. 219 00:08:24,250 --> 00:08:25,735 What's an example problem involving 220 00:08:25,735 --> 00:08:27,718 rotational inertia look like? 221 00:08:27,718 --> 00:08:29,673 Let's say two cylinders are allowed to roll 222 00:08:29,673 --> 00:08:32,328 without slipping from rest down a hill. 223 00:08:32,328 --> 00:08:34,438 The mass of cylinder A is distributed 224 00:08:34,438 --> 00:08:36,979 evenly throughout the entire cylinder. 225 00:08:36,979 --> 00:08:39,751 Cylinder B is made from a more dense material 226 00:08:39,751 --> 00:08:41,347 and it has a hollow center 227 00:08:41,347 --> 00:08:44,410 with the mass distributed around that hollow center. 228 00:08:44,410 --> 00:08:47,921 If the masses and radii of the cylinders are the same, 229 00:08:47,921 --> 00:08:50,972 which cylinder would reach the bottom of the hill first? 230 00:08:50,972 --> 00:08:52,215 To figure out which cylinder gets 231 00:08:52,215 --> 00:08:53,640 to the bottom of the hill first 232 00:08:53,640 --> 00:08:56,404 we have to ask which one would roll more readily. 233 00:08:56,404 --> 00:08:58,982 The cylinder with the least moment of inertia 234 00:08:58,982 --> 00:09:00,545 is gonna be easier to rotate. 235 00:09:00,545 --> 00:09:02,365 That means it would roll more readily 236 00:09:02,365 --> 00:09:04,471 and get to the bottom of the hill faster. 237 00:09:04,471 --> 00:09:08,165 Whenever the mass is distributed farther away from the axis, 238 00:09:08,165 --> 00:09:10,568 the object's gonna have a larger moment of inertia 239 00:09:10,568 --> 00:09:13,073 so since the mass of cylinder B overall 240 00:09:13,073 --> 00:09:16,529 is farther away from the axis compared to cylinder A, 241 00:09:16,529 --> 00:09:19,134 cylinder B has a larger moment of inertia, 242 00:09:19,134 --> 00:09:20,836 that means it's harder to rotate. 243 00:09:20,836 --> 00:09:22,795 It will take longer to get down the hill 244 00:09:22,795 --> 00:09:24,916 and cylinder A is gonna win. 245 00:09:24,916 --> 00:09:27,670 What's the angular version of the Newton's Second Law? 246 00:09:27,670 --> 00:09:29,017 Well, Newton's Second Law says 247 00:09:29,017 --> 00:09:31,568 that the acceleration is equal to the net force 248 00:09:31,568 --> 00:09:33,124 divided by the mass 249 00:09:33,124 --> 00:09:35,285 and the angular version of Newton's Second Law says 250 00:09:35,285 --> 00:09:38,722 that the angular acceleration is equal to the net torque 251 00:09:38,722 --> 00:09:40,840 divided by the rotational inertia. 252 00:09:40,840 --> 00:09:44,191 M tells you how much an object resist acceleration 253 00:09:44,191 --> 00:09:46,336 and the moment of inertia or rotational inertia 254 00:09:46,336 --> 00:09:49,665 tells you how much an object resists angular acceleration. 255 00:09:49,665 --> 00:09:51,359 Just like when you add up force vectors 256 00:09:51,359 --> 00:09:53,729 you have to be careful with positive and negative signs. 257 00:09:53,729 --> 00:09:55,767 The same holds true with the torque vectors. 258 00:09:55,767 --> 00:09:57,969 You've got to treat either counterclockwise 259 00:09:57,969 --> 00:09:59,511 or clockwise as positive 260 00:09:59,511 --> 00:10:01,128 and then be consistent with it. 261 00:10:01,128 --> 00:10:02,471 What's an example problem involving 262 00:10:02,471 --> 00:10:04,960 the angular version of Newton's Second Law look like? 263 00:10:04,960 --> 00:10:06,352 Let's say the rod shown below 264 00:10:06,352 --> 00:10:09,754 has a rotational inertia of two kilogram meter squared 265 00:10:09,754 --> 00:10:12,226 and has the forces acting on it as shown. 266 00:10:12,226 --> 00:10:13,455 We wanna know what the magnitude 267 00:10:13,455 --> 00:10:16,115 of the angular acceleration is of the rod. 268 00:10:16,115 --> 00:10:18,255 We use Newton's Second Law in angular form 269 00:10:18,255 --> 00:10:20,998 which says that the angular acceleration is the net torque 270 00:10:20,998 --> 00:10:22,750 divided by the rotational inertia. 271 00:10:22,750 --> 00:10:24,387 We got the rotational inertia, 272 00:10:24,387 --> 00:10:25,767 we just need the net torque. 273 00:10:25,767 --> 00:10:27,076 We have to figure out the total torque 274 00:10:27,076 --> 00:10:28,444 from all these forces. 275 00:10:28,444 --> 00:10:30,364 The torque from the one newton force 276 00:10:30,364 --> 00:10:32,896 would be r which is gonna be three meters 277 00:10:32,896 --> 00:10:36,077 from the axis to that force one newton. 278 00:10:36,077 --> 00:10:37,820 And since it's applied perpendicular, 279 00:10:37,820 --> 00:10:39,745 the sine of 90 is gonna be one. 280 00:10:39,745 --> 00:10:41,432 The torque from the one newton force 281 00:10:41,432 --> 00:10:43,161 would be three newton meters 282 00:10:43,161 --> 00:10:45,265 in the counterclockwise direction. 283 00:10:45,265 --> 00:10:46,865 And the torque from the four newton force 284 00:10:46,865 --> 00:10:48,116 would be one meter 285 00:10:48,116 --> 00:10:50,565 since it's applied one meter from the axis 286 00:10:50,565 --> 00:10:53,750 times four newtons and we get four newton meters 287 00:10:53,750 --> 00:10:55,541 in the clockwise direction. 288 00:10:55,541 --> 00:10:57,621 That means the total net torque 289 00:10:57,621 --> 00:11:00,813 when you have four newton meters in the clockwise direction 290 00:11:00,813 --> 00:11:03,550 and three newton meters in the counterclockwise direction 291 00:11:03,550 --> 00:11:05,697 would just be one newton meter 292 00:11:05,697 --> 00:11:07,156 in the clockwise direction 293 00:11:07,156 --> 00:11:09,990 since four is one unit bigger than three. 294 00:11:09,990 --> 00:11:11,715 And now we divide by the rotational inertia 295 00:11:11,715 --> 00:11:12,612 which was two 296 00:11:12,612 --> 00:11:16,676 which gives us an angular acceleration of 1/2 or 0.5. 297 00:11:16,676 --> 00:11:18,783 What's rotational kinetic energy mean? 298 00:11:18,783 --> 00:11:20,745 Well if an object is rotating or spinning, 299 00:11:20,745 --> 00:11:22,925 we say it has rotational kinetic energy. 300 00:11:22,925 --> 00:11:25,189 If the center of mass of an object is moving 301 00:11:25,189 --> 00:11:26,928 and the object is rotating, 302 00:11:26,928 --> 00:11:28,417 we typically say that object has 303 00:11:28,417 --> 00:11:31,975 translational kinetic energy and rotational kinetic energy. 304 00:11:31,975 --> 00:11:33,192 They're both kinetic energies, 305 00:11:33,192 --> 00:11:35,186 this is just a convenient way to delineate 306 00:11:35,186 --> 00:11:37,298 between two types of kinetic energy 307 00:11:37,298 --> 00:11:38,991 and a particularly convenient way 308 00:11:38,991 --> 00:11:40,569 to find the total kinetic energy 309 00:11:40,569 --> 00:11:42,940 for something that's moving and rotating. 310 00:11:42,940 --> 00:11:44,905 The formula for rotational kinetic energy 311 00:11:44,905 --> 00:11:47,203 is 1/2 times the moment of inertia 312 00:11:47,203 --> 00:11:50,473 or the rotational inertia times the angular speed squared. 313 00:11:50,473 --> 00:11:51,875 Which makes sense because the formula 314 00:11:51,875 --> 00:11:55,504 for regular kinetic energy is 1/2 times the regular inertia, 315 00:11:55,504 --> 00:11:58,046 the mass times the regular speed squared. 316 00:11:58,046 --> 00:11:59,693 Again, if an object is rotating, 317 00:11:59,693 --> 00:12:01,590 it's got rotational kinetic energy. 318 00:12:01,590 --> 00:12:03,729 If the center of mass of an object is moving 319 00:12:03,729 --> 00:12:06,170 it's got regular translational kinetic energy. 320 00:12:06,170 --> 00:12:07,961 And if the center of mass is moving 321 00:12:07,961 --> 00:12:09,711 and the object is rotating, 322 00:12:09,711 --> 00:12:12,236 then we say that object has both rotational energy 323 00:12:12,236 --> 00:12:13,763 and translational energy. 324 00:12:13,763 --> 00:12:16,033 Rotational kinetic energy is not a vector. 325 00:12:16,033 --> 00:12:18,417 It is always positive or zero 326 00:12:18,417 --> 00:12:19,526 and the units can be written as 327 00:12:19,526 --> 00:12:22,044 kilogram meter squared per second squared 328 00:12:22,044 --> 00:12:25,100 but it's an energy so we know that just has to equal joules. 329 00:12:25,100 --> 00:12:26,154 What's an example problem 330 00:12:26,154 --> 00:12:28,408 involving rotational kinetic energy look like? 331 00:12:28,408 --> 00:12:30,811 Let's say a constant torque is exerted on a cylinder 332 00:12:30,811 --> 00:12:32,210 that's initially at rest 333 00:12:32,210 --> 00:12:34,644 and can rotate about an axis through its center. 334 00:12:34,644 --> 00:12:36,444 Which of these curves would best give 335 00:12:36,444 --> 00:12:39,045 the rotational kinetic energy of the cylinder 336 00:12:39,045 --> 00:12:40,366 as a function of time? 337 00:12:40,366 --> 00:12:41,734 Well if there's a constant amount 338 00:12:41,734 --> 00:12:43,095 of torque on an object, 339 00:12:43,095 --> 00:12:45,830 that will cause a constant angular acceleration. 340 00:12:45,830 --> 00:12:47,821 And if the angular acceleration is constant, 341 00:12:47,821 --> 00:12:49,462 we can use the kinematic formulas 342 00:12:49,462 --> 00:12:52,135 to figure out the final velocity of this object. 343 00:12:52,135 --> 00:12:55,380 The final angular velocity if it's started at rest 344 00:12:55,380 --> 00:12:57,409 would just be alpha times t. 345 00:12:57,409 --> 00:13:00,666 That means the rotational kinetic energy of this object 346 00:13:00,666 --> 00:13:03,100 could be written as 1/2 the moment of inertia 347 00:13:03,100 --> 00:13:05,653 which is a constant times omega squared. 348 00:13:05,653 --> 00:13:08,165 Which in this case would be 1/2 I 349 00:13:08,165 --> 00:13:10,052 times alpha t squared. 350 00:13:10,052 --> 00:13:12,216 Since the function for kinetic energy 351 00:13:12,216 --> 00:13:14,894 is proportional to the time squared, 352 00:13:14,894 --> 00:13:17,608 if you graph kinetic energy as a function of time, 353 00:13:17,608 --> 00:13:19,162 it would look like a parabola 354 00:13:19,162 --> 00:13:21,993 so the correct answer would be B. 355 00:13:21,993 --> 00:13:23,630 What's angular momentum? 356 00:13:23,630 --> 00:13:25,487 Well, the reason we care about angular momentum 357 00:13:25,487 --> 00:13:28,007 is that it will be conserved for a system 358 00:13:28,007 --> 00:13:30,867 if there's no external torque on that system. 359 00:13:30,867 --> 00:13:34,634 And just like regular momentum is mass times velocity, 360 00:13:34,634 --> 00:13:37,476 angular momentum will be the rotational inertia 361 00:13:37,476 --> 00:13:39,478 times the angular velocity. 362 00:13:39,478 --> 00:13:40,918 And this is a convenient formula 363 00:13:40,918 --> 00:13:43,724 to find the angular momentum of an extended object 364 00:13:43,724 --> 00:13:46,308 whose mass is distributed at different points 365 00:13:46,308 --> 00:13:48,729 away from the axis of rotation. 366 00:13:48,729 --> 00:13:50,517 The strange thing about angular momentum 367 00:13:50,517 --> 00:13:53,638 is that even a point mass moving in a straight line 368 00:13:53,638 --> 00:13:55,284 can have angular momentum. 369 00:13:55,284 --> 00:13:57,416 To find the angular momentum of a point mass 370 00:13:57,416 --> 00:13:58,972 moving in a straight line, 371 00:13:58,972 --> 00:14:00,507 take the mass of the object 372 00:14:00,507 --> 00:14:02,466 times the velocity of that object, 373 00:14:02,466 --> 00:14:06,933 and either multiply by how far that object is from the axis 374 00:14:06,933 --> 00:14:09,135 times sine of the angle between 375 00:14:09,135 --> 00:14:11,580 the velocity vector and that R. 376 00:14:11,580 --> 00:14:15,140 Or the easier way to do it is to just multiply by 377 00:14:15,140 --> 00:14:17,114 the distance of closest approach 378 00:14:17,114 --> 00:14:19,995 which is how close that mass will ever get to 379 00:14:19,995 --> 00:14:23,100 or ever has been from the axis. 380 00:14:23,100 --> 00:14:25,066 In other words to determine the angular momentum 381 00:14:25,066 --> 00:14:27,063 of this mass moving in a straight line, 382 00:14:27,063 --> 00:14:29,195 draw a straight line along its trajectory 383 00:14:29,195 --> 00:14:31,298 and ask how close has it gotten 384 00:14:31,298 --> 00:14:33,763 or ever will get to the axis? 385 00:14:33,763 --> 00:14:35,690 That's the capital R I'm talking about. 386 00:14:35,690 --> 00:14:38,732 And if you take that times the mass times velocity, 387 00:14:38,732 --> 00:14:41,935 you'll get the angular momentum of that point mass. 388 00:14:41,935 --> 00:14:43,844 Angular momentum is a vector 389 00:14:43,844 --> 00:14:45,882 and it's easiest to just think about the direction 390 00:14:45,882 --> 00:14:47,336 of angular momentum as being 391 00:14:47,336 --> 00:14:49,982 either counterclockwise or clockwise 392 00:14:49,982 --> 00:14:52,529 depending on which way the object is rotating. 393 00:14:52,529 --> 00:14:54,749 And as for the units if you multiply mass 394 00:14:54,749 --> 00:14:57,872 of kilograms times meters per second times meters, 395 00:14:57,872 --> 00:15:00,793 you'd get kilogram meter squared per seconds 396 00:15:00,793 --> 00:15:02,791 as the units of angular momentum. 397 00:15:02,791 --> 00:15:03,843 What's an example problem 398 00:15:03,843 --> 00:15:05,767 involving angular momentum look like? 399 00:15:05,767 --> 00:15:07,589 Let's say a clay sphere of mass M 400 00:15:07,589 --> 00:15:10,655 was heading toward a rod of mass three M and length L 401 00:15:10,655 --> 00:15:12,388 with a speed v. 402 00:15:12,388 --> 00:15:15,650 The rod is free to rotate about an axis around its end. 403 00:15:15,650 --> 00:15:18,342 If the clay sticks to the end of the rod, 404 00:15:18,342 --> 00:15:20,780 what would be the angular velocity of the rod 405 00:15:20,780 --> 00:15:22,815 after the clay sticks to the rod? 406 00:15:22,815 --> 00:15:24,280 And we're given that the moment of inertia 407 00:15:24,280 --> 00:15:27,685 of a rod about its end is 1/3 ML squared. 408 00:15:27,685 --> 00:15:28,612 Since there's gonna be no 409 00:15:28,612 --> 00:15:30,955 net external torque on this system, 410 00:15:30,955 --> 00:15:34,186 the angular momentum of this system is gonna be conserved. 411 00:15:34,186 --> 00:15:35,670 The only object in this system 412 00:15:35,670 --> 00:15:38,948 that has angular momentum initially is this clay sphere. 413 00:15:38,948 --> 00:15:41,326 Since this is a point mass moving in a straight line 414 00:15:41,326 --> 00:15:44,359 we'll use the formula M times the velocity, 415 00:15:44,359 --> 00:15:47,657 times the closest it will ever get to the axis which is L, 416 00:15:47,657 --> 00:15:49,032 the length of the rod. 417 00:15:49,032 --> 00:15:51,629 That's gonna have to equal the final angular momentum 418 00:15:51,629 --> 00:15:53,915 which we could write as I times omega. 419 00:15:53,915 --> 00:15:55,854 And this I would be the moment of inertia 420 00:15:55,854 --> 00:15:58,009 of both the rod and the clay 421 00:15:58,009 --> 00:16:00,411 that is now stuck to the end of the rod. 422 00:16:00,411 --> 00:16:03,872 We'd have MvL equals the total moment of inertia, 423 00:16:03,872 --> 00:16:07,367 moment of inertia of the rod is 1/3 mass of the rod 424 00:16:07,367 --> 00:16:10,645 which is three M times the length of the rod squared. 425 00:16:10,645 --> 00:16:13,864 Plus the moment of inertia of this piece of clay 426 00:16:13,864 --> 00:16:15,493 stuck to the end of the rod 427 00:16:15,493 --> 00:16:17,578 rotating in a circle is gonna be 428 00:16:17,578 --> 00:16:20,656 the mass of the clay times the radius of the circle 429 00:16:20,656 --> 00:16:23,764 that clay traces out which is the length of the rod. 430 00:16:23,764 --> 00:16:25,234 In other words, we're using the formula 431 00:16:25,234 --> 00:16:27,472 for the moment of inertia of a point mass 432 00:16:27,472 --> 00:16:29,377 whose entire mass is rotating 433 00:16:29,377 --> 00:16:31,548 at the same radius from the center. 434 00:16:31,548 --> 00:16:33,212 And we add that to the moment of inertia 435 00:16:33,212 --> 00:16:34,532 of the rod itself. 436 00:16:34,532 --> 00:16:36,282 We multiply by omega. 437 00:16:36,282 --> 00:16:39,571 The term in brackets comes out to be two ML squared. 438 00:16:39,571 --> 00:16:41,005 We can cancel the M's, 439 00:16:41,005 --> 00:16:42,953 we can cancel one of the L's 440 00:16:42,953 --> 00:16:47,319 and we get that omega is gonna equal v over two L. 441 00:16:47,319 --> 00:16:48,703 The last topic I wanna talk about 442 00:16:48,703 --> 00:16:50,294 is the more general formula 443 00:16:50,294 --> 00:16:52,590 for the gravitational potential energy. 444 00:16:52,590 --> 00:16:54,457 Why do we need a more general formula? 445 00:16:54,457 --> 00:16:55,556 Well, if you're in a region 446 00:16:55,556 --> 00:16:58,548 where the gravitational field little g is constant 447 00:16:58,548 --> 00:17:00,408 then you could just use our familiar formula 448 00:17:00,408 --> 00:17:03,596 mgh to find the gravitational potential energy. 449 00:17:03,596 --> 00:17:04,605 But if you're in a region 450 00:17:04,605 --> 00:17:06,940 where the gravitational field is varying 451 00:17:06,940 --> 00:17:09,015 then you have to use this more general formula 452 00:17:09,015 --> 00:17:12,012 which states that the gravitational potential energy 453 00:17:12,012 --> 00:17:14,617 between two masses, m one and m two, 454 00:17:14,617 --> 00:17:18,067 is gonna equal negative of the gravitational constant, 455 00:17:18,067 --> 00:17:20,784 big G, times the product of the two masses, 456 00:17:20,784 --> 00:17:23,566 divided by the center to center distance 457 00:17:23,566 --> 00:17:24,991 between the two masses. 458 00:17:24,991 --> 00:17:28,169 Note, this is center to center, not surface to surface 459 00:17:28,169 --> 00:17:31,395 and it's not squared like it is in the force formula. 460 00:17:31,395 --> 00:17:33,077 This one's just the distance. 461 00:17:33,077 --> 00:17:34,526 Gravitational potential energy 462 00:17:34,526 --> 00:17:37,707 is not a vector but because of this negative sign, 463 00:17:37,707 --> 00:17:39,279 the gravitational potential energy 464 00:17:39,279 --> 00:17:41,564 is always gonna be negative or zero. 465 00:17:41,564 --> 00:17:43,736 It'll only be zero when these spheres become 466 00:17:43,736 --> 00:17:45,257 infinitely far apart 467 00:17:45,257 --> 00:17:47,266 because then you'd divide by infinity 468 00:17:47,266 --> 00:17:49,321 and one over infinity would be zero. 469 00:17:49,321 --> 00:17:51,156 Otherwise, it's always negative. 470 00:17:51,156 --> 00:17:52,390 But even though this gravitational 471 00:17:52,390 --> 00:17:53,927 potential energy is negative, 472 00:17:53,927 --> 00:17:56,873 this energy could still get converted into kinetic energy, 473 00:17:56,873 --> 00:17:58,810 it's just that in order for this gravitational 474 00:17:58,810 --> 00:18:00,530 potential energy to decrease, 475 00:18:00,530 --> 00:18:02,875 it would have to become even more negative 476 00:18:02,875 --> 00:18:05,316 to convert that energy into kinetic energy. 477 00:18:05,316 --> 00:18:07,060 I'm bringing up this topic in this section 478 00:18:07,060 --> 00:18:09,773 because oftentimes when planets are orbiting each other 479 00:18:09,773 --> 00:18:11,260 in circular orbits, 480 00:18:11,260 --> 00:18:13,452 you have to use this formula to determine 481 00:18:13,452 --> 00:18:15,787 the gravitational potential energy between them. 482 00:18:15,787 --> 00:18:18,246 And since it's an energy, the units are joules, 483 00:18:18,246 --> 00:18:19,350 so what's an example problem 484 00:18:19,350 --> 00:18:21,087 involving this more general formula 485 00:18:21,087 --> 00:18:23,308 for the gravitational potential energy look like? 486 00:18:23,308 --> 00:18:26,045 Let's say two spheres of radius R and mass M 487 00:18:26,045 --> 00:18:27,302 are falling toward each other 488 00:18:27,302 --> 00:18:29,396 due to their gravitational attraction. 489 00:18:29,396 --> 00:18:31,248 If the surface to surface distance 490 00:18:31,248 --> 00:18:35,280 between them starts off as four R and ends up as two R, 491 00:18:35,280 --> 00:18:38,462 how much kinetic energy would be gained by this system? 492 00:18:38,462 --> 00:18:40,425 We'll include both masses in our system 493 00:18:40,425 --> 00:18:41,258 and that would mean 494 00:18:41,258 --> 00:18:43,272 that there's gonna be no external work done 495 00:18:43,272 --> 00:18:46,042 so the energy of this system is gonna be conserved. 496 00:18:46,042 --> 00:18:47,012 The system's gonna start off 497 00:18:47,012 --> 00:18:48,863 with gravitational potential energy 498 00:18:48,863 --> 00:18:51,959 negative big G both masses multiplied together 499 00:18:51,959 --> 00:18:54,546 which is M squared divided by the distance 500 00:18:54,546 --> 00:18:55,846 they start off from each other 501 00:18:55,846 --> 00:18:58,769 which is not four R, it's the center to center distance 502 00:18:58,769 --> 00:19:00,304 which is gonna six R. 503 00:19:00,304 --> 00:19:01,608 We'll assume they start from rest 504 00:19:01,608 --> 00:19:03,533 so there'll be no kinetic energy to start with 505 00:19:03,533 --> 00:19:04,962 and this is gonna equal the final 506 00:19:04,962 --> 00:19:07,597 gravitational potential energy negative big G, 507 00:19:07,597 --> 00:19:09,971 both masses multiplied M squared 508 00:19:09,971 --> 00:19:12,893 divided by the distance they end up which is not two R, 509 00:19:12,893 --> 00:19:15,445 it's center to center distance so that's four R, 510 00:19:15,445 --> 00:19:17,257 plus however much potential energy 511 00:19:17,257 --> 00:19:19,217 was converted into kinetic energy. 512 00:19:19,217 --> 00:19:20,658 If we solve this for kinetic energy 513 00:19:20,658 --> 00:19:24,494 we're gonna get negative big G, M squared over six R 514 00:19:24,494 --> 00:19:28,086 plus big G, M squared over four R. 515 00:19:28,086 --> 00:19:31,628 1/4 minus 1/6 is gonna be 1/12. 516 00:19:31,628 --> 00:19:32,861 The amount of potential energy 517 00:19:32,861 --> 00:19:34,605 that was converted into kinetic energy 518 00:19:34,605 --> 00:00:00,000 would have been big G, M squared over 12 R.