1 00:00:00,000 --> 00:00:01,303 - [Instructor] Alright, so we know how to find 2 00:00:01,303 --> 00:00:03,028 the torque now, but who cares? 3 00:00:03,028 --> 00:00:04,175 What good is torque? 4 00:00:04,175 --> 00:00:05,952 What good is it gonna do for us? 5 00:00:05,952 --> 00:00:07,294 Well here's what it can do. 6 00:00:07,294 --> 00:00:09,712 We know from Newton's second law 7 00:00:09,712 --> 00:00:12,743 that the acceleration is proportional to the force. 8 00:00:12,743 --> 00:00:14,905 What we would like to have is some sort 9 00:00:14,905 --> 00:00:17,474 of rotational analog of this formula. 10 00:00:17,474 --> 00:00:18,937 Something that would tell us alright, 11 00:00:18,937 --> 00:00:21,764 we'll get a certain amount of angular acceleration 12 00:00:21,764 --> 00:00:23,548 for a certain amount of torque. 13 00:00:23,548 --> 00:00:24,970 And you could probably guess 14 00:00:24,970 --> 00:00:27,493 that this angular acceleration's gonna have 15 00:00:27,493 --> 00:00:29,980 probably something with torque on top 'cause torque 16 00:00:29,980 --> 00:00:32,406 is gonna cause something to angularly accelerate. 17 00:00:32,406 --> 00:00:35,373 And then on the bottom, maybe it's mass, maybe it isn't. 18 00:00:35,373 --> 00:00:36,481 That's what we need here. 19 00:00:36,481 --> 00:00:39,791 If we had this formula, this rotational analog 20 00:00:39,791 --> 00:00:42,621 of Newton's second law, then by knowing the torque 21 00:00:42,621 --> 00:00:45,532 we could figure out what the angular acceleration is 22 00:00:45,532 --> 00:00:47,738 just like up here by knowing force, 23 00:00:47,738 --> 00:00:50,135 we could tell what the regular acceleration is. 24 00:00:50,135 --> 00:00:51,659 So that's what I want to do in this video. 25 00:00:51,659 --> 00:00:54,591 I want to derive this rotational analog 26 00:00:54,591 --> 00:00:57,340 of Newton's second law for an object 27 00:00:57,340 --> 00:01:00,245 that's rotating in a circle like this cue ball. 28 00:01:00,245 --> 00:01:02,373 And not just rotating in a circle. 29 00:01:02,373 --> 00:01:04,546 Something that's angularly accelerating. 30 00:01:04,546 --> 00:01:06,548 So it would be speeding up in its rotation 31 00:01:06,548 --> 00:01:08,953 or it'd be slowing down in its rotation. 32 00:01:08,953 --> 00:01:11,469 So let's do this, let's derive this formula 33 00:01:11,469 --> 00:01:14,143 so that if we know the torque we could determine 34 00:01:14,143 --> 00:01:16,567 the angular acceleration just like we determine 35 00:01:16,567 --> 00:01:18,808 regular acceleration by knowing the force 36 00:01:18,808 --> 00:01:20,094 and Newton's second law. 37 00:01:20,094 --> 00:01:20,944 So how do we do this? 38 00:01:20,944 --> 00:01:22,718 In order to have an angular acceleration 39 00:01:22,718 --> 00:01:26,962 we're gonna need a force that's tangential to the circle. 40 00:01:26,962 --> 00:01:29,656 So in order to go angularly accelerate something 41 00:01:29,656 --> 00:01:31,572 you need a force that's tangential 42 00:01:31,572 --> 00:01:34,540 because this force is gonna cause a torque. 43 00:01:34,540 --> 00:01:36,809 So let's say this is the force causing the torque, 44 00:01:36,809 --> 00:01:37,941 we know how to find it now. 45 00:01:37,941 --> 00:01:40,815 Remember torque is R times F times sine theta, 46 00:01:40,815 --> 00:01:42,063 but let's make it simple. 47 00:01:42,063 --> 00:01:44,844 Let's say the angle's 90 so that sine theta 48 00:01:44,844 --> 00:01:47,784 will end up being one 'cause sine of 90 is one. 49 00:01:47,784 --> 00:01:49,522 And let's make it simple too in this way, 50 00:01:49,522 --> 00:01:51,454 let's say this force is the net force. 51 00:01:51,454 --> 00:01:53,557 Let's say there's only one force on this object, 52 00:01:53,557 --> 00:01:54,809 and it's this force here. 53 00:01:54,809 --> 00:01:56,738 Well we know that the net force 54 00:01:56,738 --> 00:01:59,603 has to be equal to the mass of the object 55 00:01:59,603 --> 00:02:01,922 times the acceleration of the object. 56 00:02:01,922 --> 00:02:04,038 And you're probably like, big whoop. 57 00:02:04,038 --> 00:02:05,097 We already knew this. 58 00:02:05,097 --> 00:02:06,084 What's new here? 59 00:02:06,084 --> 00:02:08,241 Well remember, we want to relate torque 60 00:02:08,241 --> 00:02:10,164 to the angular acceleration, 61 00:02:10,164 --> 00:02:11,999 so let's write down the torque formula. 62 00:02:11,999 --> 00:02:14,223 How do you find the torque from a force? 63 00:02:14,223 --> 00:02:15,928 Remember that the torque from a force 64 00:02:15,928 --> 00:02:19,603 is gonna be equal to the force exerting that torque 65 00:02:19,603 --> 00:02:23,162 times R, the distance from the axis 66 00:02:23,162 --> 00:02:24,968 to the point where the force is applied. 67 00:02:24,968 --> 00:02:27,423 Now in this case, that's the entire radius 68 00:02:27,423 --> 00:02:30,140 'cause we applied this force all the way at the edge. 69 00:02:30,140 --> 00:02:32,543 If this force was applied inward somewhere, 70 00:02:32,543 --> 00:02:35,014 it would be only that distance from the axis 71 00:02:35,014 --> 00:02:36,328 to the point where the force is. 72 00:02:36,328 --> 00:02:37,563 But we applied it at the very edge 73 00:02:37,563 --> 00:02:39,647 so this would F times the entire radius. 74 00:02:39,647 --> 00:02:44,103 And then there's also a sine of the angle between F and R, 75 00:02:44,103 --> 00:02:47,662 but the angle between F and R is 90 degrees here, 76 00:02:47,662 --> 00:02:50,755 and the sine of 90 degrees is just one, 77 00:02:50,755 --> 00:02:52,020 so we can get rid of that. 78 00:02:52,020 --> 00:02:54,674 So this is simple, the torque exerted by this force F 79 00:02:54,674 --> 00:02:56,745 is gonna be F times R. 80 00:02:56,745 --> 00:02:58,121 What do we do with this? 81 00:02:58,121 --> 00:03:00,689 Well look at down here, we've already got an F down here. 82 00:03:00,689 --> 00:03:02,562 If you're creative you might be like, 83 00:03:02,562 --> 00:03:05,676 well let's just multiply both sides by R down here. 84 00:03:05,676 --> 00:03:07,886 That way we'll get torque into this formula. 85 00:03:07,886 --> 00:03:10,382 In other words, if I multiply the left side by R 86 00:03:10,382 --> 00:03:13,390 I'll get R times F, and now that's gonna equal 87 00:03:13,390 --> 00:03:15,114 R times the right-hand side. 88 00:03:15,114 --> 00:03:18,301 So it's gonna be R times M times the acceleration. 89 00:03:18,301 --> 00:03:20,716 And this was good, look at now we have R times F. 90 00:03:20,716 --> 00:03:22,127 That's just the torque. 91 00:03:22,127 --> 00:03:24,325 Torque is R times F, or F times R. 92 00:03:24,325 --> 00:03:28,036 So I've got torque equals R times M, 93 00:03:28,036 --> 00:03:30,914 times the acceleration, but that's no good. 94 00:03:30,914 --> 00:03:32,582 Remember over here we want a formula 95 00:03:32,582 --> 00:03:35,676 that relates torque to angular acceleration, 96 00:03:35,676 --> 00:03:38,829 not a formula that relates torque to regular acceleration. 97 00:03:38,829 --> 00:03:41,588 So what could I replace regular acceleration with 98 00:03:41,588 --> 00:03:43,829 in order to get angular acceleration? 99 00:03:43,829 --> 00:03:45,141 Maybe you remember when we talked 100 00:03:45,141 --> 00:03:47,056 about angular motion variables. 101 00:03:47,056 --> 00:03:50,839 The tangential acceleration is always gonna equal 102 00:03:50,839 --> 00:03:53,500 the distance from the axis to that object 103 00:03:53,500 --> 00:03:55,688 that's got the tangential acceleration, 104 00:03:55,688 --> 00:03:59,642 multiplied by the angular acceleration alpha. 105 00:03:59,642 --> 00:04:02,142 So this is the relationship between alpha 106 00:04:02,142 --> 00:04:03,978 and the tangential acceleration. 107 00:04:03,978 --> 00:04:06,375 Is this tangential acceleration? 108 00:04:06,375 --> 00:04:08,826 It is 'cause this was the tangential force. 109 00:04:08,826 --> 00:04:10,767 So since we took the tangential force, 110 00:04:10,767 --> 00:04:13,957 that's gonna be proportional to the tangential acceleration. 111 00:04:13,957 --> 00:04:15,586 These are both tangential here, 112 00:04:15,586 --> 00:04:17,461 and these forces are all tangential. 113 00:04:17,461 --> 00:04:20,567 That means I can rewrite the tangential acceleration 114 00:04:20,567 --> 00:04:23,260 as R times alpha, and that's what I'm gonna do. 115 00:04:23,260 --> 00:04:26,483 I'm gonna rewrite this side as R times alpha 116 00:04:26,483 --> 00:04:30,151 'cause R alpha is the tangential acceleration. 117 00:04:30,151 --> 00:04:31,433 So this whole term right here 118 00:04:31,433 --> 00:04:33,241 was just tangential acceleration, 119 00:04:33,241 --> 00:04:34,234 and now look what we've got. 120 00:04:34,234 --> 00:04:36,142 We've got torque is gonna be equal 121 00:04:36,142 --> 00:04:38,933 to R times M, times R times alpha. 122 00:04:38,933 --> 00:04:40,965 I can combine the two Rs and just write this 123 00:04:40,965 --> 00:04:45,132 as M times R squared times alpha, the angular acceleration. 124 00:04:46,594 --> 00:04:47,554 And now we're close. 125 00:04:47,554 --> 00:04:49,486 If I wanted a form of Newton's second law 126 00:04:49,486 --> 00:04:51,094 I could leave it like this or I could put it 127 00:04:51,094 --> 00:04:53,160 in this form over here and just solve for alpha, 128 00:04:53,160 --> 00:04:54,491 and get the alpha. 129 00:04:54,491 --> 00:04:57,526 The angular acceleration of this mass 130 00:04:57,526 --> 00:05:01,056 is gonna equal the torque exerted on that mass 131 00:05:01,056 --> 00:05:02,802 divided by this weird term, 132 00:05:02,802 --> 00:05:06,037 this M the mass, times R squared. 133 00:05:06,037 --> 00:05:07,349 And this is what we were looking for. 134 00:05:07,349 --> 00:05:08,549 This is what we were looking for over here. 135 00:05:08,549 --> 00:05:09,868 I'm gonna write it in this box. 136 00:05:09,868 --> 00:05:13,733 The rotational analog of Newton's second law for rotation 137 00:05:13,733 --> 00:05:17,558 is this torque divided by this term here. 138 00:05:17,558 --> 00:05:19,328 This M R squared, what is that? 139 00:05:19,328 --> 00:05:20,862 Well it's serving the same role 140 00:05:20,862 --> 00:05:23,748 that mass did for regular acceleration 141 00:05:23,748 --> 00:05:25,721 and the regular Newton's second law. 142 00:05:25,721 --> 00:05:28,198 And remember, this mass was proportional 143 00:05:28,198 --> 00:05:30,613 to the inertia of an object. 144 00:05:30,613 --> 00:05:34,516 It told you how hard it was to get that object accelerating. 145 00:05:34,516 --> 00:05:36,780 How sluggish an object is. 146 00:05:36,780 --> 00:05:40,009 How resistive it is to being accelerated. 147 00:05:40,009 --> 00:05:42,100 That's what this term down here's gonna be. 148 00:05:42,100 --> 00:05:45,345 People usually call this the moment of inertia, 149 00:05:45,345 --> 00:05:48,114 but that's gotta be the most complicated name 150 00:05:48,114 --> 00:05:50,369 for any physics idea I've ever heard of. 151 00:05:50,369 --> 00:05:51,374 I don't even know what this means. 152 00:05:51,374 --> 00:05:52,575 Moment of inertia. 153 00:05:52,575 --> 00:05:53,980 That just sounds strange. 154 00:05:53,980 --> 00:05:56,033 It's represented with a letter I, 155 00:05:56,033 --> 00:05:57,892 and it's serving the same role. 156 00:05:57,892 --> 00:06:00,249 It's in this denominator just like mass is, 157 00:06:00,249 --> 00:06:01,820 and it's serving the same role. 158 00:06:01,820 --> 00:06:05,561 It's serving as the rotational inertia 159 00:06:05,561 --> 00:06:07,539 of the system in question. 160 00:06:07,539 --> 00:06:11,528 So in other words, something with a big rotational inertia 161 00:06:11,528 --> 00:06:14,778 is gonna be sluggish to angular acceleration, 162 00:06:14,778 --> 00:06:17,560 just like something with a big regular inertia 163 00:06:17,560 --> 00:06:19,981 is sluggish to regular acceleration. 164 00:06:19,981 --> 00:06:22,030 So if this ball, and we can see what it depends on. 165 00:06:22,030 --> 00:06:24,188 Look at, for a ball on the end of a string, 166 00:06:24,188 --> 00:06:27,670 the moment of inertia for a ball on the end of the string 167 00:06:27,670 --> 00:06:29,220 was just M R squared. 168 00:06:29,220 --> 00:06:30,958 This was the denominator. 169 00:06:30,958 --> 00:06:34,469 This was the term serving as the rotational inertia 170 00:06:34,469 --> 00:06:36,612 for this mass on a string. 171 00:06:36,612 --> 00:06:39,085 And what that means is if you had a bigger mass, 172 00:06:39,085 --> 00:06:41,249 or if the radius were bigger, 173 00:06:41,249 --> 00:06:44,367 this object would be harder to angularly accelerate. 174 00:06:44,367 --> 00:06:47,206 So it would be difficult to get this thing going 175 00:06:47,206 --> 00:06:48,912 and start speeding it up. 176 00:06:48,912 --> 00:06:51,292 But on the other hand, if the mass were small, 177 00:06:51,292 --> 00:06:52,958 or the radius were small, 178 00:06:52,958 --> 00:06:56,168 it'd be much easier to angularly accelerate. 179 00:06:56,168 --> 00:06:57,585 You could whip it around like crazy. 180 00:06:57,585 --> 00:07:00,300 But if the mass were very big or the radius were big, 181 00:07:00,300 --> 00:07:03,035 this moment of inertia term would get much bigger. 182 00:07:03,035 --> 00:07:05,503 This is the moment of inertia for a mass 183 00:07:05,503 --> 00:07:08,719 on the end of a string, and that's what the I is here. 184 00:07:08,719 --> 00:07:11,152 So you could think about it as the rotational inertia. 185 00:07:11,152 --> 00:07:12,556 That's a much better name for it. 186 00:07:12,556 --> 00:07:14,203 People are coming around and realizing 187 00:07:14,203 --> 00:07:15,644 that you should just call it this 188 00:07:15,644 --> 00:07:16,856 'cause that's what it really is. 189 00:07:16,856 --> 00:07:19,511 This moment of inertia is kind of a historical term. 190 00:07:19,511 --> 00:07:21,489 It stuck around, it's not a very good one. 191 00:07:21,489 --> 00:07:24,137 Rotational inertia is much more descriptive 192 00:07:24,137 --> 00:07:26,487 of what this I really is. 193 00:07:26,487 --> 00:07:28,775 And we should note the units of this moment of inertia, 194 00:07:28,775 --> 00:07:31,155 since it's mass times radius squared, 195 00:07:31,155 --> 00:07:34,441 the units are gonna be kilgram meters squared. 196 00:07:34,441 --> 00:07:36,854 These are the units of moment of inertia, 197 00:07:36,854 --> 00:07:39,456 and this is the formula if you just have a point mass. 198 00:07:39,456 --> 00:07:42,644 And by that I just mean a mass where all of the mass 199 00:07:42,644 --> 00:07:45,651 is traveling at the same radius in a circle. 200 00:07:45,651 --> 00:07:48,033 It doesn't have to be tied to a string. 201 00:07:48,033 --> 00:07:50,109 This could be the moon going around the Earth. 202 00:07:50,109 --> 00:07:53,319 But as long as all of the mass is at the same radius 203 00:07:53,319 --> 00:07:54,973 and traveling around in a circle, 204 00:07:54,973 --> 00:07:57,116 or at least mostly at the same radius. 205 00:07:57,116 --> 00:07:59,542 Let's assume this little radius of the sphere 206 00:07:59,542 --> 00:08:02,745 is really small compared to this radius of the string. 207 00:08:02,745 --> 00:08:05,526 If that's the case, where basically all the mass 208 00:08:05,526 --> 00:08:08,213 is traveling around in a circle at the same radius, 209 00:08:08,213 --> 00:08:11,449 this would be the formula to find the moment of inertia. 210 00:08:11,449 --> 00:08:12,923 So how does this ever get harder? 211 00:08:12,923 --> 00:08:14,397 What do you have to look out for? 212 00:08:14,397 --> 00:08:16,726 Well we only considered one force. 213 00:08:16,726 --> 00:08:19,593 You could imagine maybe there's many forces on this object. 214 00:08:19,593 --> 00:08:21,526 Maybe there's some other force this way. 215 00:08:21,526 --> 00:08:24,632 Well in that case, you just have the net force here 216 00:08:24,632 --> 00:08:26,367 to make sure it's M times A, 217 00:08:26,367 --> 00:08:27,627 and you just have to make sure 218 00:08:27,627 --> 00:08:29,995 you use the net torque up here. 219 00:08:29,995 --> 00:08:33,525 So this formula will still work if you have multiple torques 220 00:08:33,525 --> 00:08:35,273 on this object or this system. 221 00:08:35,273 --> 00:08:37,626 You just have to use the net torque up here. 222 00:08:37,626 --> 00:08:40,014 You add up all of the torques where torque's 223 00:08:40,015 --> 00:08:42,331 trying to rotate it one way would be positive, 224 00:08:42,331 --> 00:08:44,871 and torque trying to rotate it the other direction 225 00:08:44,871 --> 00:08:46,735 would be negative, so you'd have to make sure 226 00:08:46,735 --> 00:08:48,448 signs are correct up here. 227 00:08:48,448 --> 00:08:49,981 And what about rotational inertia? 228 00:08:49,981 --> 00:08:53,703 What if your object isn't as simple as a single mass? 229 00:08:53,703 --> 00:08:54,818 What do you do then? 230 00:08:54,818 --> 00:08:55,651 Let's look at that. 231 00:08:55,651 --> 00:08:57,877 Let's take this formula here, I'm gonna copy that. 232 00:08:57,877 --> 00:08:59,655 Let's get rid of all of this, 233 00:08:59,655 --> 00:09:01,767 and let's say you had this crazy problem. 234 00:09:01,767 --> 00:09:04,535 You had three masses now, and one force 235 00:09:04,535 --> 00:09:07,572 on this mass two was 20 newtons downward, 236 00:09:07,572 --> 00:09:11,286 and one force was upward 50 newtons on this mass one. 237 00:09:11,286 --> 00:09:14,809 And they're all separated by three meters, and can rotate. 238 00:09:14,809 --> 00:09:16,695 We're stepping it up, this is complicated. 239 00:09:16,695 --> 00:09:19,327 It can rotate in a circle, but we can do it. 240 00:09:19,327 --> 00:09:21,908 We can do it with the formula we just derived. 241 00:09:21,908 --> 00:09:23,057 Let's use that. 242 00:09:23,057 --> 00:09:23,965 This is gonna be useful. 243 00:09:23,965 --> 00:09:27,304 Let's say the question is what's the angular acceleration 244 00:09:27,304 --> 00:09:31,205 for these masses in this particular set up of forces? 245 00:09:31,205 --> 00:09:33,642 We're gonna use this formula for Newton's second law. 246 00:09:33,642 --> 00:09:36,789 In angular form we'll say that the angular acceleration 247 00:09:36,789 --> 00:09:40,226 if that's what we want, is gonna equal the net torque. 248 00:09:40,226 --> 00:09:41,525 How do we find the net torque? 249 00:09:41,525 --> 00:09:42,960 Now there's two forces. 250 00:09:42,960 --> 00:09:43,793 Well it's not that bad. 251 00:09:43,793 --> 00:09:46,661 You just find the torque from each one individually 252 00:09:46,661 --> 00:09:47,639 and you add 'em up. 253 00:09:47,639 --> 00:09:49,728 Just like you would do with any net vector, 254 00:09:49,728 --> 00:09:51,893 find each individually and add 'em up. 255 00:09:51,893 --> 00:09:54,022 But it's not gonna be 50 minus 20. 256 00:09:54,022 --> 00:09:55,417 These are torques. 257 00:09:55,417 --> 00:09:58,074 We've gotta plug torque in up here, not force. 258 00:09:58,074 --> 00:09:59,084 This has gotta be a torque, 259 00:09:59,084 --> 00:10:03,051 and until you multiply that force by an R it's just a force. 260 00:10:03,051 --> 00:10:05,442 So don't try to just stick this 50 up in here. 261 00:10:05,442 --> 00:10:07,433 It needs to get multiplied by an R. 262 00:10:07,433 --> 00:10:08,266 What R? 263 00:10:08,266 --> 00:10:10,880 Be careful, you might think three meters, but no. 264 00:10:10,880 --> 00:10:13,564 The R is always from the axis of rotation 265 00:10:13,564 --> 00:10:15,344 which is the center all the way 266 00:10:15,344 --> 00:10:16,714 to where the force is applied. 267 00:10:16,714 --> 00:10:20,528 So the torque from this 50 is gonna be nine meters 268 00:10:20,528 --> 00:10:22,021 times the 50 newtons. 269 00:10:22,021 --> 00:10:23,374 Now we've got a torque. 270 00:10:23,374 --> 00:10:26,723 It's not a torque until you multiply that force by an R. 271 00:10:26,723 --> 00:10:28,540 That was the torque from the 50 newtons. 272 00:10:28,540 --> 00:10:30,496 How about the torque from the 20 newtons? 273 00:10:30,496 --> 00:10:32,018 You might be like, alright I got it now. 274 00:10:32,018 --> 00:10:35,191 It's gonna be 20 newtons, but I can't just put 20, right? 275 00:10:35,191 --> 00:10:37,527 We gotta multiply it by an R. 276 00:10:37,527 --> 00:10:40,897 It's gonna be 20 newtons times, and it's not three. 277 00:10:40,897 --> 00:10:43,831 It's always distance from the axis, so it's from the center 278 00:10:43,831 --> 00:10:46,082 all the way to where this 20 newtons was applied, 279 00:10:46,082 --> 00:10:47,813 and that's gonna be six meters. 280 00:10:47,813 --> 00:10:50,479 And sometimes when the people get here they're just so happy 281 00:10:50,479 --> 00:10:52,776 they remember the R, they just do plus, 282 00:10:52,776 --> 00:10:54,227 and without thinking about it, 283 00:10:54,227 --> 00:10:55,151 but they're gonna get it wrong. 284 00:10:55,151 --> 00:10:56,020 You can't do that. 285 00:10:56,020 --> 00:10:58,999 Look at, this 50 newtons was trying to rotate 286 00:10:58,999 --> 00:11:01,559 this system counterclockwise, right? 287 00:11:01,559 --> 00:11:03,881 The 50 newton's trying to rotate it this way. 288 00:11:03,881 --> 00:11:06,369 The 20 newton is trying to rotate it that way. 289 00:11:06,369 --> 00:11:07,492 They're opposing each other. 290 00:11:07,492 --> 00:11:10,302 These are opposite signs of torque, 291 00:11:10,302 --> 00:11:12,629 so I've gotta make sure I represent that up here. 292 00:11:12,629 --> 00:11:14,831 I'm gonna represent this 20 newton torque 293 00:11:14,831 --> 00:11:16,425 as a negative torque, 294 00:11:16,425 --> 00:11:17,975 and that's the convention we usually pick. 295 00:11:17,975 --> 00:11:20,175 Counterclockwise is usually positive, 296 00:11:20,175 --> 00:11:21,726 and clockwise is usually negative, 297 00:11:21,726 --> 00:11:23,578 but no matter what convention you pick, 298 00:11:23,578 --> 00:11:25,129 they've gotta have different signs in here 299 00:11:25,129 --> 00:11:26,128 so be careful there. 300 00:11:26,128 --> 00:11:27,791 So that's our net torque up here. 301 00:11:27,791 --> 00:11:29,896 How do we find the rotational and inertia, 302 00:11:29,896 --> 00:11:31,175 or the moment of inertia? 303 00:11:31,175 --> 00:11:33,324 Well we know from the previous example 304 00:11:33,324 --> 00:11:35,483 the moment of inertia of a point mass 305 00:11:35,483 --> 00:11:37,525 that is a mass going in a circle 306 00:11:37,525 --> 00:11:41,457 where all of the mass is going at that particular radius 307 00:11:41,457 --> 00:11:43,000 is just M R squared. 308 00:11:43,000 --> 00:11:44,691 But now we've got three masses 309 00:11:44,691 --> 00:11:47,332 so you might think this is hard, but it's not that hard. 310 00:11:47,332 --> 00:11:49,979 All we have to do is say that the total moment of inertia 311 00:11:49,979 --> 00:11:53,038 is gonna be the sum of all the individual 312 00:11:53,038 --> 00:11:54,203 moments of inertia. 313 00:11:54,203 --> 00:11:57,055 So we just add up all the individual moments of inertia. 314 00:11:57,055 --> 00:11:59,046 In other words, this is just gonna be 315 00:11:59,046 --> 00:12:01,072 the moment of inertia of mass one. 316 00:12:01,072 --> 00:12:04,397 If that's one kilogram, that's gonna be one kilogram 317 00:12:04,397 --> 00:12:05,972 times R squared. 318 00:12:05,972 --> 00:12:06,805 That's what this means. 319 00:12:06,805 --> 00:12:08,084 You take all the masses. 320 00:12:08,084 --> 00:12:11,623 M one, times R one squared, plus M two, 321 00:12:11,623 --> 00:12:15,850 times R two squared, plus M three, times R three squared. 322 00:12:15,850 --> 00:12:17,268 You'd keep going if you had more. 323 00:12:17,268 --> 00:12:19,015 You just add them all up and that would give you 324 00:12:19,015 --> 00:12:22,806 the total moment of inertia for a system of masses. 325 00:12:22,806 --> 00:12:24,784 So if we do 'em one at a time, 326 00:12:24,784 --> 00:12:27,693 this one kilogram times the R for that one 327 00:12:27,693 --> 00:12:30,370 would be nine meters 'cause that's distance 328 00:12:30,370 --> 00:12:32,412 from the axis to the mass. 329 00:12:32,412 --> 00:12:35,960 That'd be nine meters squared plus alright, mass two. 330 00:12:35,960 --> 00:12:37,667 If that's two kilograms, 331 00:12:37,667 --> 00:12:40,217 and that's gonna be times six squared. 332 00:12:40,217 --> 00:12:41,361 And now we keep going. 333 00:12:41,361 --> 00:12:44,002 We take this three kilogram mass and we also add 334 00:12:44,002 --> 00:12:47,038 its contribution to the moment of inertia, 335 00:12:47,038 --> 00:12:48,436 or the rotational inertia, 336 00:12:48,436 --> 00:12:51,468 and that'd be three kilograms times 337 00:12:51,468 --> 00:12:53,962 it's only three meters from the axis squared, 338 00:12:53,962 --> 00:12:56,877 so times three meters squared. 339 00:12:56,877 --> 00:12:58,240 And if we add all this up 340 00:12:58,240 --> 00:13:00,192 and plug all this into the calculator, 341 00:13:00,192 --> 00:13:03,064 we'll get that the alpha, the angular acceleration 342 00:13:03,064 --> 00:13:06,731 is gonna be 1.83 radians per second squared. 343 00:13:09,074 --> 00:13:11,459 So that's the rate at which this object 344 00:13:11,459 --> 00:13:14,293 would start accelerating if it started from rest. 345 00:13:14,293 --> 00:13:17,003 It would start to speed up in this direction 346 00:13:17,003 --> 00:13:19,424 and start speeding up over and over and over 347 00:13:19,424 --> 00:13:21,924 if these forces maintained the torque 348 00:13:21,924 --> 00:13:23,163 that they were exerting. 349 00:13:23,163 --> 00:13:25,694 So recapping, just like Newton's second law 350 00:13:25,694 --> 00:13:27,806 relates forces to acceleration, 351 00:13:27,806 --> 00:13:30,569 this angular version of Newton's second law 352 00:13:30,569 --> 00:13:33,710 relates torques to angular acceleration. 353 00:13:33,710 --> 00:13:36,675 And on the bottom of this denominator isn't the mass, 354 00:13:36,675 --> 00:13:40,247 it's the rotational inertia that tells you how difficult 355 00:13:40,247 --> 00:13:43,521 it's going to be to angularly accelerate an object. 356 00:13:43,521 --> 00:13:46,023 And you can find the moment of inertia of a point mass 357 00:13:46,023 --> 00:13:48,796 as M R squared, and you could find the moment of inertia 358 00:13:48,796 --> 00:13:51,729 of a collection of point masses by adding up 359 00:13:51,729 --> 00:00:00,000 all the contributions from each individual mass.