1 00:00:00,283 --> 00:00:01,637 - [Instructor] We should talk some more about 2 00:00:01,637 --> 00:00:04,136 the moment of inertia, 'cause this is something 3 00:00:04,136 --> 00:00:06,261 that people get confused about a lot. 4 00:00:06,261 --> 00:00:08,647 So remember, first of all this moment of inertia 5 00:00:08,647 --> 00:00:11,950 is really just the rotational inertia. 6 00:00:11,950 --> 00:00:13,746 In other words, how much something's going 7 00:00:13,746 --> 00:00:17,070 to resist being angularly accelerated, 8 00:00:17,070 --> 00:00:20,329 so being sped up in its rotation, or slowed down. 9 00:00:20,329 --> 00:00:23,307 So if it has a, if this system has a large moment 10 00:00:23,307 --> 00:00:25,621 of inertia, it's going to be very difficult 11 00:00:25,621 --> 00:00:27,774 to try to get this thing accelerating, 12 00:00:27,774 --> 00:00:30,148 but if the moment of inertia is small, 13 00:00:30,148 --> 00:00:33,002 it should be very easy, relatively easy 14 00:00:33,002 --> 00:00:35,046 to get this thing angularly accelerating. 15 00:00:35,046 --> 00:00:36,630 So that's what this number is good for, 16 00:00:36,630 --> 00:00:38,894 the reason why you wanna know the moment of inertia 17 00:00:38,894 --> 00:00:40,349 is 'cause it'll let you determine 18 00:00:40,349 --> 00:00:43,681 how difficult it'll be to angularly accelerate something, 19 00:00:43,681 --> 00:00:46,351 and remember it shows up in the angular version 20 00:00:46,351 --> 00:00:48,200 of Newton's second law, that says that 21 00:00:48,200 --> 00:00:51,207 the angular acceleration is gonna be equal to 22 00:00:51,207 --> 00:00:54,774 the net torque divided by the moment of inertia, 23 00:00:54,774 --> 00:00:57,944 or the rotational inertia, since they're the same thing. 24 00:00:57,944 --> 00:01:00,207 So that should make sense, we're dividing by the moment 25 00:01:00,207 --> 00:01:03,212 of inertia, we're dividing by the rotational inertia 26 00:01:03,212 --> 00:01:06,018 because that means if this rotational inertia is big, 27 00:01:06,018 --> 00:01:07,795 look it, this is in the denominator. 28 00:01:07,795 --> 00:01:09,472 You've got a big denominator, 29 00:01:09,472 --> 00:01:11,341 you're gonna have a small value, 30 00:01:11,341 --> 00:01:13,486 that means this alpha is gonna be small, 31 00:01:13,486 --> 00:01:16,147 it's gonna be a small angular acceleration, 32 00:01:16,147 --> 00:01:18,177 but if this moment of inertia were small, 33 00:01:18,177 --> 00:01:19,805 then it's gonna be easier to rotate, 34 00:01:19,805 --> 00:01:22,934 and you'll get a relatively larger angular acceleration 35 00:01:22,934 --> 00:01:25,494 'cause you're now dividing by a smaller number. 36 00:01:25,494 --> 00:01:27,566 So it does serve the same role that mass did, 37 00:01:27,566 --> 00:01:31,334 it serves as this inertia term for angular acceleration, 38 00:01:31,334 --> 00:01:33,061 and we figured out how to determine 39 00:01:33,061 --> 00:01:35,123 the moment of inertia for a point mass, 40 00:01:35,123 --> 00:01:36,511 and you'll hear people say this a lot, 41 00:01:36,511 --> 00:01:38,545 "point mass," I'm gonna say it a lot. 42 00:01:38,545 --> 00:01:41,633 By point mass I just mean a mass you could treat 43 00:01:41,633 --> 00:01:43,999 as if all the mass were rotating at 44 00:01:43,999 --> 00:01:46,132 the same distance from the axis, 45 00:01:46,132 --> 00:01:47,112 and that's what's happening here. 46 00:01:47,112 --> 00:01:49,656 If you've got a heavy ball connected to a string, 47 00:01:49,656 --> 00:01:51,938 a very light string that has very little mass, 48 00:01:51,938 --> 00:01:53,686 you can neglect the mass here. 49 00:01:53,686 --> 00:01:56,357 If all the mass is rotating at the same radius 50 00:01:56,357 --> 00:01:58,745 like this is, we determined last time 51 00:01:58,745 --> 00:02:00,939 that the moment of inertia of a point mass going in 52 00:02:00,939 --> 00:02:04,077 a circle is just the mass times how far 53 00:02:04,077 --> 00:02:07,653 that mass is from the axis, squared. 54 00:02:07,653 --> 00:02:11,078 This is the term for a point mass going in a circle 55 00:02:11,078 --> 00:02:12,660 for what the moment of inertia is, 56 00:02:12,660 --> 00:02:15,975 how difficult it's going to be to angularly accelerate. 57 00:02:15,975 --> 00:02:19,437 This is the rotational inertia, mr squared, 58 00:02:19,437 --> 00:02:21,704 but you get more complicated problems too, 59 00:02:21,704 --> 00:02:23,133 so you could be like, "All right, what happens 60 00:02:23,133 --> 00:02:26,641 "if we don't have a single point mass, we've got the three?" 61 00:02:26,641 --> 00:02:28,092 Well we did this last time as well, 62 00:02:28,092 --> 00:02:30,481 if you have multiple point masses, 63 00:02:30,481 --> 00:02:31,862 all you need to do is say that all right, 64 00:02:31,862 --> 00:02:35,813 for multiple point masses, just add up all 65 00:02:35,813 --> 00:02:39,267 the contributions from each individual point mass. 66 00:02:39,267 --> 00:02:40,994 So if we're careful here, mathematically, 67 00:02:40,994 --> 00:02:43,676 we should put an i subscript, 68 00:02:43,676 --> 00:02:44,805 but don't let that freak you out, 69 00:02:44,805 --> 00:02:46,469 this just really means all them all up. 70 00:02:46,469 --> 00:02:50,115 So this would be m one times r one squared, 71 00:02:50,115 --> 00:02:52,505 so you take the mass one times its distance from 72 00:02:52,505 --> 00:02:56,588 the axis squared, plus m two times r two squared, 73 00:02:57,614 --> 00:02:59,704 you take mass two times its distance from 74 00:02:59,704 --> 00:03:01,199 the axis squared, and then you do 75 00:03:01,199 --> 00:03:03,912 the same for m three, and if you had more masses, 76 00:03:03,912 --> 00:03:05,375 you would just keep adding 'em up. 77 00:03:05,375 --> 00:03:07,884 If you have a whole bunch of point masses 78 00:03:07,884 --> 00:03:11,410 that you can treat as if all the mass were rotating 79 00:03:11,410 --> 00:03:13,732 at the same distance from the axis, 80 00:03:13,732 --> 00:03:15,488 and you might object, you might say, "Wait, 81 00:03:15,488 --> 00:03:16,890 "different masses here are rotating 82 00:03:16,890 --> 00:03:19,239 "at different distances from the axis," 83 00:03:19,239 --> 00:03:21,797 but all of that particular mass, 84 00:03:21,797 --> 00:03:26,592 all of m one is rotating at the same radius from the axis, 85 00:03:26,592 --> 00:03:29,059 so we can use this formula for point masses 86 00:03:29,059 --> 00:03:30,178 and we can add them up. 87 00:03:30,178 --> 00:03:33,115 The total amount is gonna be the total rotational inertia, 88 00:03:33,115 --> 00:03:34,863 so in other words, for this case here, 89 00:03:34,863 --> 00:03:36,802 if we really wanted to do it, we would say 90 00:03:36,802 --> 00:03:39,657 that the moment of inertia for these objects, 91 00:03:39,657 --> 00:03:41,588 and this system in total would be, 92 00:03:41,588 --> 00:03:42,775 all right, let's take 'em in order. 93 00:03:42,775 --> 00:03:46,279 M one is gonna contribute m one times its distance from 94 00:03:46,279 --> 00:03:49,602 the axis squared would be a, so we do a squared, 95 00:03:49,602 --> 00:03:51,511 and let's say b is just the length of this string, 96 00:03:51,511 --> 00:03:53,299 so b just represents that length, 97 00:03:53,299 --> 00:03:55,900 and similarly c represents that length, 98 00:03:55,900 --> 00:03:59,546 and we're gonna assume the radii of these masses are small. 99 00:03:59,546 --> 00:04:01,946 I had to draw 'em big so we could see 'em, but it's easiest 100 00:04:01,946 --> 00:04:03,679 if you consider them to be small, 'cause then 101 00:04:03,679 --> 00:04:07,086 we don't have to take into account their actual radius. 102 00:04:07,086 --> 00:04:10,113 So we'd add to this, that's this m one a squared 103 00:04:10,113 --> 00:04:12,824 is just the contribution to the moment of inertia 104 00:04:12,824 --> 00:04:14,891 that's being contributed by just m one, 105 00:04:14,891 --> 00:04:16,180 so we have to figure out the contributions 106 00:04:16,180 --> 00:04:17,683 from each of these other masses, 107 00:04:17,684 --> 00:04:21,620 so we'll have m two times its distance from the axis. 108 00:04:21,620 --> 00:04:23,753 It isn't gonna be b, it's gonna be all the way, 109 00:04:23,753 --> 00:04:26,383 so that's gonna be a plus b squared, 110 00:04:26,383 --> 00:04:27,319 and then if you wanted to find 111 00:04:27,319 --> 00:04:30,053 the contribution from m three so that you'd get the total, 112 00:04:30,053 --> 00:04:32,134 you'd have m three times, well, 113 00:04:32,134 --> 00:04:36,555 it'd be a plus b plus c squared, this would be 114 00:04:36,555 --> 00:04:39,684 the total moment of inertia for the entire system, 115 00:04:39,684 --> 00:04:41,826 which says it's gonna be more difficult, right? 116 00:04:41,826 --> 00:04:44,060 The more mass you add into the system, 117 00:04:44,060 --> 00:04:46,417 the more sluggish it is to acceleration, 118 00:04:46,417 --> 00:04:48,481 the more difficult it is to rotate. 119 00:04:48,481 --> 00:04:50,896 So how could we make this three mass system 120 00:04:50,896 --> 00:04:52,037 easier to rotate? 121 00:04:52,037 --> 00:04:55,012 Let's say you were tired of requiring so much torque 122 00:04:55,012 --> 00:04:57,328 to move this thing, you wanna make it easier to rotate. 123 00:04:57,328 --> 00:04:59,258 One thing you can always do is just take your masses 124 00:04:59,258 --> 00:05:01,655 and move them toward the axis, 125 00:05:01,655 --> 00:05:03,901 i.e. just move these toward the center. 126 00:05:03,901 --> 00:05:07,498 If you do that notice all of these rs are gonna get smaller, 127 00:05:07,498 --> 00:05:11,087 if you reduce the r you're gonna get less moment of inertia, 128 00:05:11,087 --> 00:05:13,434 and that object's gonna be easier to rotate, 129 00:05:13,434 --> 00:05:15,283 easier to angularly accelerate, 130 00:05:15,283 --> 00:05:18,311 you can whip this thing around easier if 131 00:05:18,311 --> 00:05:21,114 the mass is more toward the axis. 132 00:05:21,114 --> 00:05:22,871 So this makes sense, think about a baseball bat. 133 00:05:22,871 --> 00:05:24,659 If you had a baseball bat, so if you got 134 00:05:24,659 --> 00:05:26,804 this baseball bat, this is not the best drawing of 135 00:05:26,804 --> 00:05:28,418 a baseball bat, but you've got a baseball bat. 136 00:05:28,418 --> 00:05:31,883 If you swing it from this end where this is the axis, 137 00:05:31,883 --> 00:05:33,379 it's hard to rotate, right? 138 00:05:33,379 --> 00:05:36,751 You've got all this heavy mass over here at the end, 139 00:05:36,751 --> 00:05:40,347 but if you swing it instead where this is the axis, 140 00:05:40,347 --> 00:05:42,862 if you just turn it around and swing it from this end, 141 00:05:42,862 --> 00:05:46,101 where this is the axis, now you've made it so most of 142 00:05:46,101 --> 00:05:49,352 the mass is near the axis, and if you do that, 143 00:05:49,352 --> 00:05:52,208 the radius of that mass is gonna be smaller, 144 00:05:52,208 --> 00:05:54,925 and if the radius is smaller it's gonna contribute less 145 00:05:54,925 --> 00:05:58,184 to the moment of inertia, less to the rotational inertia, 146 00:05:58,184 --> 00:05:59,064 it's gonna be easier to swing. 147 00:05:59,064 --> 00:06:00,649 So you can swing a baseball bat really easy 148 00:06:00,649 --> 00:06:03,827 if you hold it by the fat end, compared to the actual end 149 00:06:03,827 --> 00:06:06,054 you're supposed to hold, you can swing this faster. 150 00:06:06,054 --> 00:06:06,895 It's probably not a good idea, 151 00:06:06,895 --> 00:06:08,968 you've probably not gonna hit the ball very far, 152 00:06:08,968 --> 00:06:10,727 but you'll be able to swing it much faster 153 00:06:10,727 --> 00:06:13,002 'cause that moment of inertia's gonna be smaller. 154 00:06:13,002 --> 00:06:14,221 And then the other thing we could do, 155 00:06:14,221 --> 00:06:15,773 we could always just reduce the masses. 156 00:06:15,773 --> 00:06:17,603 If you can make the mass less you reduce 157 00:06:17,603 --> 00:06:20,215 the moment of inertia, and if you can move 158 00:06:20,215 --> 00:06:23,547 those masses toward the axis, you reduce the r, 159 00:06:23,547 --> 00:06:26,786 you reduce the moment of inertia or the rotational inertia. 160 00:06:26,786 --> 00:06:28,960 But what if you don't have point masses at all? 161 00:06:28,960 --> 00:06:30,931 I mean, we don't always have situations where 162 00:06:30,931 --> 00:06:34,405 the thing that's rotating are a bunch of point masses, 163 00:06:34,405 --> 00:06:36,904 what if you had something more like this, 164 00:06:36,904 --> 00:06:40,112 where it was like a rod that had its mass evenly distributed 165 00:06:40,112 --> 00:06:44,605 throughout the entire rod, and it rotated in a circle. 166 00:06:44,605 --> 00:06:46,567 I mean, we couldn't use this formula now 167 00:06:46,567 --> 00:06:49,594 because this assumes that all the mass is rotating 168 00:06:49,594 --> 00:06:52,710 at some radius, r, but for this rod, 169 00:06:52,710 --> 00:06:55,474 only the mass at the end of the rod is rotating at 170 00:06:55,474 --> 00:06:57,521 the full length of the rod. 171 00:06:57,521 --> 00:06:59,736 The mass that's closer to the axis 172 00:06:59,736 --> 00:07:00,837 is gonna have a smaller radius, 173 00:07:00,837 --> 00:07:03,284 it'll only be rotating at part of the length. 174 00:07:03,284 --> 00:07:05,737 This would only have a radius of L over two, 175 00:07:05,737 --> 00:07:07,252 and this part right here would only have 176 00:07:07,252 --> 00:07:09,506 a radius of maybe, L over eight. 177 00:07:09,506 --> 00:07:10,339 So how do we figure this out? 178 00:07:10,339 --> 00:07:13,305 We can't just say the total mass of this rod, 179 00:07:13,305 --> 00:07:15,621 if this rod has a total mass m, 180 00:07:15,621 --> 00:07:18,335 and a total length L, we cannot say 181 00:07:18,335 --> 00:07:21,716 that the moment of inertia of this rod about its end 182 00:07:21,716 --> 00:07:24,958 is gonna be mL squared, that's just a lie. 183 00:07:24,958 --> 00:07:29,009 This total mass is not rotating all at a radius of length L, 184 00:07:29,009 --> 00:07:31,236 only the little piece at the end 185 00:07:31,236 --> 00:07:33,398 is rotating with a radius of length L. 186 00:07:33,398 --> 00:07:35,889 The rest of this mass is having its contribution 187 00:07:35,889 --> 00:07:38,347 to the rotational inertia diminished by 188 00:07:38,347 --> 00:07:41,210 the fact that these masses are getting closer and closer 189 00:07:41,210 --> 00:07:43,351 to the axis, so what do we do? 190 00:07:43,351 --> 00:07:45,841 Well we can't use this, let's get rid of this. 191 00:07:45,841 --> 00:07:47,458 That's not possible. 192 00:07:47,458 --> 00:07:50,333 The truth is you have to use calculus to derive 193 00:07:50,333 --> 00:07:53,582 the formula for these continuous objects, and it's fun. 194 00:07:53,582 --> 00:07:55,321 You can do integrals and you can solve 195 00:07:55,321 --> 00:07:56,978 for these moments of inertia, that's one of 196 00:07:56,978 --> 00:07:59,904 my favorite calculations to do, it's kinda like a puzzle. 197 00:07:59,904 --> 00:08:01,163 You can solve for the moments of inertia, 198 00:08:01,163 --> 00:08:03,509 but if you don't know calculus, that would just look 199 00:08:03,509 --> 00:08:06,516 like witchcraft to you, so I suggest you learn calculus 200 00:08:06,516 --> 00:08:08,255 and try it, 'cause it's really fun, 201 00:08:08,255 --> 00:08:09,511 but I'm just gonna give you the result. 202 00:08:09,511 --> 00:08:13,375 It turns out the moment of inertia for this rod is gonna be, 203 00:08:13,375 --> 00:08:15,609 and without knowing the exact answer, we should be able 204 00:08:15,609 --> 00:08:17,711 to say, is it gonna be bigger than, 205 00:08:17,711 --> 00:08:20,465 less than or equal to mL squared. 206 00:08:20,465 --> 00:08:22,152 We should be able to say, it's gotta be less 207 00:08:22,152 --> 00:08:25,851 than mL squared, it's not going to be mL squared, 208 00:08:25,851 --> 00:08:28,216 it's gonna be less than this because mL squared 209 00:08:28,216 --> 00:08:30,207 would be if all of the mass were at 210 00:08:30,207 --> 00:08:33,580 the full length of the rod for their radius. 211 00:08:33,580 --> 00:08:35,472 Then you would put mL squared. 212 00:08:35,472 --> 00:08:38,967 If you could melt this rod down into just a ball, 213 00:08:38,967 --> 00:08:41,416 and put that ball at the very far end, 214 00:08:41,417 --> 00:08:44,058 you'd be maximizing its rotational inertia, 215 00:08:44,058 --> 00:08:45,713 'cause you'd put all of the mass with 216 00:08:45,713 --> 00:08:49,533 the same largest radius r, but some of this mass is in here. 217 00:08:49,533 --> 00:08:51,707 Some of this mass is only at L over two, 218 00:08:51,707 --> 00:08:53,698 or L over four, or at L over eight. 219 00:08:53,698 --> 00:08:55,275 So those little pieces of mass are having 220 00:08:55,275 --> 00:08:58,099 their rotational inertia contribution diminished, 221 00:08:58,099 --> 00:08:59,898 so we're gonna have less than Ml squared. 222 00:08:59,898 --> 00:09:00,935 How much less? 223 00:09:00,935 --> 00:09:05,462 Turns out for a rod about its end, it's 1/3 mL squared, 224 00:09:05,462 --> 00:09:06,920 and if you do the integral, 225 00:09:06,920 --> 00:09:08,586 that's where this 1/3 comes from. 226 00:09:08,586 --> 00:09:13,001 So this is for a rod with the axis at the end of the rod. 227 00:09:13,001 --> 00:09:15,726 So that's the moment of inertia for a rod rotating about 228 00:09:15,726 --> 00:09:18,611 an axis that's at one of the ends of the rod, 229 00:09:18,611 --> 00:09:21,618 but what if we move this axis to the center? 230 00:09:21,618 --> 00:09:23,621 What if we move the axis here so that 231 00:09:23,621 --> 00:09:27,023 this whole rod rotates around a point in its center. 232 00:09:27,023 --> 00:09:29,205 Do you think the moment of inertia of this rod 233 00:09:29,205 --> 00:09:31,736 that's the same mass and length that it was, 234 00:09:31,736 --> 00:09:33,511 we're just rotating it about the center, 235 00:09:33,511 --> 00:09:36,243 do you think this moment of inertia is gonna be bigger than, 236 00:09:36,243 --> 00:09:39,394 smaller than or equal to what the moment of inertia was 237 00:09:39,394 --> 00:09:41,560 for a rod rotated about the end. 238 00:09:41,560 --> 00:09:42,461 And the way I would think about it, 239 00:09:42,461 --> 00:09:45,011 I'd just ask myself this question, "Is more of 240 00:09:45,011 --> 00:09:47,771 "the mass farther away now, or closer to the axis?", 241 00:09:47,771 --> 00:09:50,259 'cause we know if we can decrease these rs, 242 00:09:50,259 --> 00:09:52,035 we decrease the moment of inertia, 243 00:09:52,035 --> 00:09:54,736 and in this case we did decrease the rs. 244 00:09:54,736 --> 00:09:57,419 Think about it, the farthest some piece of mass will be 245 00:09:57,419 --> 00:09:59,553 from the axis now is L over two. 246 00:09:59,553 --> 00:10:01,928 It's L over two this way, and L over two that way, 247 00:10:01,928 --> 00:10:04,782 whereas before, where the axis was at one end, 248 00:10:04,782 --> 00:10:07,943 some of the mass was at L away, so that'd be L squared, 249 00:10:07,943 --> 00:10:10,489 but now you're only gonna have L over two squared for 250 00:10:10,489 --> 00:10:13,557 the farthest some piece of this mass is gonna be, 251 00:10:13,557 --> 00:10:16,201 and that's gonna decrease the moment of inertia even more, 252 00:10:16,201 --> 00:10:18,832 because more of this mass is closer to 253 00:10:18,832 --> 00:10:20,874 the axis when you move it to the center, 254 00:10:20,874 --> 00:10:23,120 so it's gonna be less than 1/3 mL squared. 255 00:10:23,120 --> 00:10:27,396 Turns out if you do the integral you get 1/12 mL squared, 256 00:10:27,396 --> 00:10:30,739 so this is for a rod with the axis at its center. 257 00:10:30,739 --> 00:10:32,913 So what's another common geometry? 258 00:10:32,913 --> 00:10:35,411 Well if we get rid of that, another case that comes up a lot 259 00:10:35,411 --> 00:10:38,081 is a cylinder, or sometimes it's called a disc. 260 00:10:38,081 --> 00:10:41,311 So let's say you have a cylinder, a solid cylinder of mass m 261 00:10:41,311 --> 00:10:45,042 and it has a radius r, what would this moment of inertia be? 262 00:10:45,042 --> 00:10:46,545 Well you can probably tell by now, all right, 263 00:10:46,545 --> 00:10:49,808 so it's not gonna be the total mr squared, 264 00:10:49,808 --> 00:10:52,051 and it's not gonna be the total mr squared 265 00:10:52,051 --> 00:10:55,334 because all of the mass is not rotating at 266 00:10:55,334 --> 00:10:57,618 the full radius of the cylinder, right, 267 00:10:57,618 --> 00:10:59,602 so it's gonna be less than this. 268 00:10:59,602 --> 00:11:00,647 How much less? 269 00:11:00,647 --> 00:11:02,174 If you do that integral it turns out 270 00:11:02,174 --> 00:11:05,628 that you get 1/2 mr squared, so it turns out 271 00:11:05,628 --> 00:11:08,554 the fact that some of these masses are closer to the axis 272 00:11:08,554 --> 00:11:10,545 than the full radius of the cylinder, 273 00:11:10,545 --> 00:11:12,097 makes it so that the total moment of inertia 274 00:11:12,097 --> 00:11:15,197 is 1/2 the total mass of the cylinder 275 00:11:15,197 --> 00:11:18,283 times the total radius of the cylinder squared. 276 00:11:18,283 --> 00:11:21,876 This is for a cylinder with the axis through the center, 277 00:11:21,876 --> 00:11:24,315 so the center, so it's rotating around a point right here, 278 00:11:24,315 --> 00:11:26,700 so it's rotating like this around this point here, 279 00:11:26,700 --> 00:11:27,768 and that's important to note. 280 00:11:27,768 --> 00:11:30,333 It's not enough to just say, "Hey, I gave you a rod, 281 00:11:30,333 --> 00:11:31,970 "what's the moment of inertia?", 282 00:11:31,970 --> 00:11:33,812 because you've gotta know, "Well where's the axis?" 283 00:11:33,812 --> 00:11:35,276 If someone just hands you something 284 00:11:35,276 --> 00:11:37,206 and says, "What's the moment of inertia of this?" 285 00:11:37,206 --> 00:11:39,674 You can't give them an answer until they've specified 286 00:11:39,674 --> 00:11:42,236 where they want you to rotate the object around. 287 00:11:42,236 --> 00:11:44,056 If you rotate the rod about its end, 288 00:11:44,056 --> 00:11:46,667 it's 1/3 mL squared for the moment of inertia. 289 00:11:46,667 --> 00:11:49,371 If you rotate the rod about the center it's 1/12, 290 00:11:49,371 --> 00:11:50,700 and again, the reason for that is 291 00:11:50,700 --> 00:11:53,685 'cause by rotating it around different axes, 292 00:11:53,685 --> 00:11:56,084 you've made it so some of the mass is at different rs 293 00:11:56,084 --> 00:11:57,965 from other axes that you could choose. 294 00:11:57,965 --> 00:12:00,859 So this was for a cylinder, also called a disc. 295 00:12:00,859 --> 00:12:02,545 Sometimes a sphere comes up, so this 296 00:12:02,545 --> 00:12:04,750 is another common example, say you had a sphere, 297 00:12:04,750 --> 00:12:06,852 also rotating around an axis, 298 00:12:06,852 --> 00:12:09,180 like the earth rotating on its axis, 299 00:12:09,180 --> 00:12:12,968 and let's say it also has a mass m and a radius r. 300 00:12:12,968 --> 00:12:15,966 Again, because some of this mass is closer to the axis, 301 00:12:15,966 --> 00:12:17,854 look it, this mass right here is only rotating in 302 00:12:17,854 --> 00:12:22,100 a circle like that, as opposed to at the full radius of 303 00:12:22,100 --> 00:12:25,150 the sphere, it's gonna have less than mr squared. 304 00:12:25,150 --> 00:12:26,002 How much less? 305 00:12:26,002 --> 00:12:28,238 Well for a sphere rotating about an axis 306 00:12:28,238 --> 00:12:30,035 that goes through its center, you get 307 00:12:30,035 --> 00:12:34,029 that the moment of inertia is 2/5 mr squared, 308 00:12:34,029 --> 00:12:36,759 so that was for a sphere rotating about 309 00:12:36,759 --> 00:12:38,824 an axis that goes through its center. 310 00:12:38,824 --> 00:12:39,846 And at this point you might object, 311 00:12:39,846 --> 00:12:41,123 you might say, "Wait a minute, 312 00:12:41,123 --> 00:12:43,114 "we had spheres when we had spheres before, 313 00:12:43,114 --> 00:12:46,233 "and we did mr squared," but that was for spheres 314 00:12:46,233 --> 00:12:48,391 that were rotating where all of their mass 315 00:12:48,391 --> 00:12:50,433 was rotating at the same radius. 316 00:12:50,433 --> 00:12:51,784 So if you have a sphere, in other words, 317 00:12:51,784 --> 00:12:53,570 if you have a sphere and you're gonna rotate 318 00:12:53,570 --> 00:12:56,489 this whole sphere around in a circle like this, 319 00:12:56,489 --> 00:12:58,512 if that's the case you're talking about then yeah, 320 00:12:58,512 --> 00:13:02,283 that total mass is all rotating at the same radius, 321 00:13:02,283 --> 00:13:03,586 but here that's not the case. 322 00:13:03,586 --> 00:13:06,329 This is a sphere rotating around its center. 323 00:13:06,329 --> 00:13:09,377 So if you just have a sphere that spins in place, 324 00:13:09,377 --> 00:13:11,327 that's not the same case as this mass 325 00:13:11,327 --> 00:13:14,770 that's being whirled around, around some common axis 326 00:13:14,770 --> 00:13:16,270 all at the same radius. 327 00:13:16,270 --> 00:13:18,113 It's the difference between, this is like 328 00:13:18,113 --> 00:13:20,662 the moon rotating around the earth. 329 00:13:20,662 --> 00:13:22,450 If you wanna talk about the moment of inertia of 330 00:13:22,450 --> 00:13:24,299 the moon rotating about the earth, 331 00:13:24,299 --> 00:13:26,034 you could treat the moon as a point mass, 332 00:13:26,034 --> 00:13:28,658 and you'd use mr squared, but if you're talking about 333 00:13:28,658 --> 00:13:31,503 the earth rotating on its axis, right? 334 00:13:31,503 --> 00:13:32,724 Not the earth going around the sun, 335 00:13:32,724 --> 00:13:35,003 but the earth rotating on its axis, 336 00:13:35,003 --> 00:13:36,830 then you'd have to say that the moment of inertia 337 00:13:36,830 --> 00:13:40,466 for that amount of rotation is 2/5 mr squared, 338 00:13:40,466 --> 00:13:42,855 because it's a sphere rotating through an axis 339 00:13:42,855 --> 00:13:44,225 that goes through its center. 340 00:13:44,225 --> 00:13:46,356 All right, so recapping, the moment of inertia 341 00:13:46,356 --> 00:13:48,614 or the rotational inertia gives you a number 342 00:13:48,614 --> 00:13:50,373 that tells you how difficult it'll be 343 00:13:50,373 --> 00:13:52,709 to angularly accelerate an object. 344 00:13:52,709 --> 00:13:54,192 If you've just got a point mass where all 345 00:13:54,192 --> 00:13:56,244 the mass rotates at the same radius, 346 00:13:56,244 --> 00:13:57,837 you could use mr squared. 347 00:13:57,837 --> 00:13:59,434 If you've got a collection of point masses, 348 00:13:59,434 --> 00:14:01,795 you can just add up all the mr squareds. 349 00:14:01,795 --> 00:14:03,979 If you've got a rod rotating about its end, 350 00:14:03,979 --> 00:14:06,103 you could use 1/3 mL squared. 351 00:14:06,103 --> 00:14:09,293 A rod rotating about its center is 1/12 mL squared. 352 00:14:09,293 --> 00:14:12,815 A cylinder rotating about its center is 1/2 mr squared, 353 00:14:12,815 --> 00:14:14,300 and a sphere rotating with an axis 354 00:14:14,300 --> 00:14:16,718 through its center is 2/5 mr squared. 355 00:14:16,718 --> 00:14:18,567 The reason why all these shapes 356 00:14:18,567 --> 00:14:21,220 that have mass distributed through them have factors 357 00:14:21,220 --> 00:14:24,328 that make their moment of inertia less than mr squared 358 00:14:24,328 --> 00:14:27,405 or mL squared is because some of that mass for 359 00:14:27,405 --> 00:14:31,518 a distributed object has mass closer to the axis 360 00:14:31,518 --> 00:14:34,398 than a case where all the mass is at the end. 361 00:14:34,398 --> 00:14:36,489 So the fact that you've got some of these masses 362 00:14:36,489 --> 00:14:39,721 that are closer to the axis for a uniform object 363 00:14:39,721 --> 00:14:43,277 reduces the total moment of inertia since it reduces the r, 364 00:14:43,277 --> 00:14:45,274 and if you ever forget any of these formulas, 365 00:14:45,274 --> 00:14:47,251 there's often a chart in your textbook, 366 00:14:47,251 --> 00:14:50,195 or look up the chart online, they're all over the place, 367 00:14:50,195 --> 00:14:52,206 lists of all the moments of inertia 368 00:14:52,206 --> 00:14:55,141 of commonly-shaped objects, and the axis. 369 00:14:55,141 --> 00:14:56,807 You gotta check that it's the axis 370 00:14:56,807 --> 00:00:00,000 that you're concerned with as well.