1 00:00:00,406 --> 00:00:02,255 - [Voiceover] So we saw in previous videos 2 00:00:02,255 --> 00:00:05,699 that a ball of mass m rotating in a circle 3 00:00:05,699 --> 00:00:08,224 of radius r at a speed v has 4 00:00:08,224 --> 00:00:10,449 what we call angular momentum, 5 00:00:10,449 --> 00:00:11,282 and the symbol we use 6 00:00:11,282 --> 00:00:14,221 for angular momentum is a capital L, 7 00:00:14,221 --> 00:00:15,889 and the amount of angular momentum 8 00:00:15,889 --> 00:00:17,666 that it would have would be the mass 9 00:00:17,666 --> 00:00:19,580 of the ball times the speed of the ball, 10 00:00:19,580 --> 00:00:22,274 so that means this is basically just the magnitude 11 00:00:22,274 --> 00:00:24,401 of the momentum, but then we multiply 12 00:00:24,401 --> 00:00:27,063 by the radius of the circle it's traveling in, 13 00:00:27,063 --> 00:00:29,186 and that gives us the angular momentum 14 00:00:29,186 --> 00:00:30,944 of this ball going in a circle, 15 00:00:30,944 --> 00:00:32,837 which is great and good to know, 16 00:00:32,837 --> 00:00:34,809 but sometimes you don't have a ball going 17 00:00:34,809 --> 00:00:37,644 in a circle and you wanna know the angular momentum. 18 00:00:37,644 --> 00:00:40,000 So, for instance, instead of this case, 19 00:00:40,000 --> 00:00:41,408 let's say you have this case, 20 00:00:41,408 --> 00:00:42,776 where instead of a ball going around 21 00:00:42,776 --> 00:00:45,814 in a circle, you've got a rod of mass m 22 00:00:45,814 --> 00:00:48,757 and radius R and the whole rod rotates 23 00:00:48,757 --> 00:00:50,196 around in a circle. 24 00:00:50,196 --> 00:00:52,913 Let's say the outside edge travels at a speed v 25 00:00:52,913 --> 00:00:54,445 just like the ball did. 26 00:00:54,445 --> 00:00:55,278 So the question is, 27 00:00:55,278 --> 00:00:58,091 will this rod also have an angular momentum 28 00:00:58,091 --> 00:01:00,647 that's equal to mvR and it won't. 29 00:01:00,647 --> 00:01:01,949 You can probably convince yourself of that 30 00:01:01,949 --> 00:01:04,367 because for the ball all the mass was traveling 31 00:01:04,367 --> 00:01:06,856 at a speed v and all the mass was 32 00:01:06,856 --> 00:01:09,259 at the outside edge of the circle that it traces out. 33 00:01:09,259 --> 00:01:11,918 In other words, all this mass is traveling 34 00:01:11,918 --> 00:01:13,652 at a radius of R. 35 00:01:13,652 --> 00:01:16,241 But for this rod, some of the mass, in fact, 36 00:01:16,241 --> 00:01:18,879 only part of the mass, only this outside edge 37 00:01:18,879 --> 00:01:22,303 of the mass, is actually traveling at a radius R. 38 00:01:22,303 --> 00:01:25,539 That's the part that travels at the full radius R. 39 00:01:25,539 --> 00:01:28,345 The rest of these pieces of mass, like this one in here, 40 00:01:28,345 --> 00:01:29,890 traces out a circle. 41 00:01:29,890 --> 00:01:31,695 It definitely traces out a circle, 42 00:01:31,695 --> 00:01:33,814 but the circle it traces out is not equal 43 00:01:33,814 --> 00:01:35,195 to the radius R. 44 00:01:35,195 --> 00:01:37,002 It's got a diminished R value. 45 00:01:37,002 --> 00:01:39,418 So how do we determine the angular momentum 46 00:01:39,418 --> 00:01:42,651 of an object whose mass is distributed in a way 47 00:01:42,651 --> 00:01:44,795 that some of the mass is close to the axis 48 00:01:44,795 --> 00:01:47,277 and some of the mass is far away from the axis. 49 00:01:47,277 --> 00:01:48,912 That's what we're gonna do in this video. 50 00:01:48,912 --> 00:01:51,344 That's the goal and the approach physicists take 51 00:01:51,344 --> 00:01:53,778 to this is almost always the same. 52 00:01:53,778 --> 00:01:55,648 We say, well, I've got the formula 53 00:01:55,648 --> 00:01:58,269 for the angular momentum of a single particle 54 00:01:58,269 --> 00:01:59,662 traveling at a single radius, 55 00:01:59,662 --> 00:02:02,758 so let's just imagine our continuous object 56 00:02:02,758 --> 00:02:06,292 being composed of a bunch of single masses 57 00:02:06,292 --> 00:02:08,973 all traveling at a single radius. 58 00:02:08,973 --> 00:02:10,993 So if I break this continuous mass up 59 00:02:10,994 --> 00:02:13,258 into individual pieces, right? 60 00:02:13,258 --> 00:02:14,816 So if I imagine it being broken up 61 00:02:14,816 --> 00:02:16,682 into all these little pieces, 62 00:02:16,682 --> 00:02:19,729 then if I found the angular momentum of each piece 63 00:02:19,729 --> 00:02:22,976 and added it up, I'd get the total angular momentum 64 00:02:22,976 --> 00:02:24,401 for the whole object. 65 00:02:24,401 --> 00:02:25,704 So let's try this. 66 00:02:25,704 --> 00:02:28,875 So the angular momentum of some piece of the object, 67 00:02:28,875 --> 00:02:30,337 let's say that little piece of mass, 68 00:02:30,337 --> 00:02:33,151 is gonna be, well the mass of that little piece, 69 00:02:33,151 --> 00:02:34,539 I'm gonna write m. 70 00:02:34,539 --> 00:02:37,659 It's not the entire mass of this entire rod, 71 00:02:37,659 --> 00:02:40,630 it'd just be the mass of that small piece times the speed 72 00:02:40,630 --> 00:02:44,147 of that piece times the radius that it is at. 73 00:02:44,147 --> 00:02:45,517 So to make this clear, let me just write this as 74 00:02:45,517 --> 00:02:49,350 like piece one, so this would be m1, v1 and R1 75 00:02:51,090 --> 00:02:53,653 and this would be the angular momentum 76 00:02:53,653 --> 00:02:55,557 of that small piece, and you could do this 77 00:02:55,557 --> 00:02:58,473 for mass two over here, and you would get 78 00:02:58,473 --> 00:02:59,380 that the angular momentum 79 00:02:59,380 --> 00:03:02,297 of mass two would be m2, v2 and R2. 80 00:03:05,049 --> 00:03:08,190 Now keep in mind that these v's are all gonna be different 81 00:03:08,190 --> 00:03:09,519 so the speed out here 82 00:03:09,519 --> 00:03:11,518 at the outside edge is gonna be fastest. 83 00:03:11,518 --> 00:03:13,644 This speed's not gonna be as great, 84 00:03:13,644 --> 00:03:16,526 and this speed closer to the middle is even smaller 85 00:03:16,526 --> 00:03:18,709 because they're tracing out smaller circles 86 00:03:18,709 --> 00:03:20,024 in the same amount of time 87 00:03:20,024 --> 00:03:23,557 as these outside pieces trace out larger circles 88 00:03:23,557 --> 00:03:24,859 in the same amount of time. 89 00:03:24,859 --> 00:03:26,246 So at this point, you might be worried. 90 00:03:26,246 --> 00:03:28,054 You might be like, "This is gonna be really hard. 91 00:03:28,054 --> 00:03:29,408 We're gonna have to add all these up. 92 00:03:29,408 --> 00:03:31,008 They've all got different speeds. 93 00:03:31,008 --> 00:03:32,636 They're all at different radii. 94 00:03:32,636 --> 00:03:33,897 How are we gonna do this?" 95 00:03:33,897 --> 00:03:35,248 Well, you gotta have faith 96 00:03:35,248 --> 00:03:37,769 and something magical is about to happen. 97 00:03:37,769 --> 00:03:38,702 So let me show you what happens 98 00:03:38,702 --> 00:03:40,963 if we imagine adding all these up. 99 00:03:40,963 --> 00:03:42,218 I only draw two. 100 00:03:42,218 --> 00:03:44,153 You gotta imagine there's an infinite amount 101 00:03:44,153 --> 00:03:45,914 of these so that makes it seem even harder, 102 00:03:45,914 --> 00:03:48,808 but imagine breaking this up into an infinite amount 103 00:03:48,808 --> 00:03:50,719 of these little discrete masses 104 00:03:50,719 --> 00:03:54,036 and considering each individual angular momentum, 105 00:03:54,036 --> 00:03:56,277 they'd be very small because this m1 would be 106 00:03:56,277 --> 00:03:59,654 an infinitesimal very small piece of mass, 107 00:03:59,654 --> 00:04:02,072 and let's add them all up and see what we get. 108 00:04:02,072 --> 00:04:05,989 So if we add up all of the mvR's of every piece 109 00:04:07,139 --> 00:04:09,527 of mass on this rod, that would be 110 00:04:09,527 --> 00:04:12,937 the total angular momentum of the rod. 111 00:04:12,937 --> 00:04:14,252 So in other words, this is really 112 00:04:14,252 --> 00:04:17,919 just m1, v1, R1 m2, v2, R2 and so on. 113 00:04:22,568 --> 00:04:24,122 You'd have an infinite amount of them, right? 114 00:04:24,122 --> 00:04:24,955 I can't write them all out 115 00:04:24,955 --> 00:04:25,983 'cause there's an infinite amount. 116 00:04:25,983 --> 00:04:27,190 But just imagine that. 117 00:04:27,190 --> 00:04:29,210 So what can we possibly do with this? 118 00:04:29,210 --> 00:04:31,076 How do we clean this up? 119 00:04:31,076 --> 00:04:32,575 When you're doing a physics problem, 120 00:04:32,575 --> 00:04:34,785 you don't want to solve an infinite series 121 00:04:34,785 --> 00:04:36,509 by writing each term out infinitely. 122 00:04:36,509 --> 00:04:38,054 We want a clever way to deal with this, 123 00:04:38,054 --> 00:04:40,479 and there's a really clever way to deal with this. 124 00:04:40,479 --> 00:04:41,312 Watch this. 125 00:04:41,312 --> 00:04:44,883 So if we write this as L equals the sum of mvR. 126 00:04:44,883 --> 00:04:48,917 One problem we have is that each mass has a different v. 127 00:04:48,917 --> 00:04:51,567 If I can pull things out of this summation, 128 00:04:51,567 --> 00:04:53,819 it would help me out 'cause it would simplify things. 129 00:04:53,819 --> 00:04:55,031 I could just factor them out, 130 00:04:55,031 --> 00:04:57,520 but right now I can't factor out the R, 131 00:04:57,520 --> 00:05:00,446 'cause these all are different radii from the axis. 132 00:05:00,446 --> 00:05:03,150 You always measure your radius from the axis here, 133 00:05:03,150 --> 00:05:05,308 and they're all at different radii from the axis, 134 00:05:05,308 --> 00:05:06,982 and they all have different speeds. 135 00:05:06,982 --> 00:05:09,708 But, remember, we like writing quantities in terms 136 00:05:09,708 --> 00:05:13,426 of angular variables because the angular variables 137 00:05:13,426 --> 00:05:16,063 are the same for every point on this mass. 138 00:05:16,063 --> 00:05:20,230 So every point on this rotating rod has a different speed v 139 00:05:21,331 --> 00:05:24,711 but they all have the same angular speed omega, 140 00:05:24,711 --> 00:05:25,836 so that's a key. 141 00:05:25,836 --> 00:05:27,890 That's often what we do and we're gonna do that here. 142 00:05:27,890 --> 00:05:30,609 I'm gonna write this as summation of m, 143 00:05:30,609 --> 00:05:32,872 but instead of writing v, I'm gonna write this 144 00:05:32,872 --> 00:05:34,739 as R times omega. 145 00:05:34,739 --> 00:05:38,229 So, remember, for something rotating in the circle, 146 00:05:38,229 --> 00:05:42,634 the speed v is gonna be equal to R times omega 147 00:05:42,634 --> 00:05:44,393 and that's what I'm gonna substitute down here, 148 00:05:44,393 --> 00:05:47,070 so the speed at any point here is the radii 149 00:05:47,070 --> 00:05:49,601 of that point times the angular speed 150 00:05:49,601 --> 00:05:51,819 of this rod rotating in a circle, 151 00:05:51,819 --> 00:05:54,112 and I still have to multiply by the last R here, 152 00:05:54,112 --> 00:05:55,163 so this was v. 153 00:05:55,163 --> 00:05:57,599 We substituted in what v was, but we have to multiply 154 00:05:57,599 --> 00:05:59,762 by R and what do we get? 155 00:05:59,762 --> 00:06:02,244 We get that L is gonna be the summation 156 00:06:02,244 --> 00:06:03,911 of mR squared omega. 157 00:06:05,402 --> 00:06:06,544 And this is great. 158 00:06:06,544 --> 00:06:09,921 The omega is the same for every single mass in here. 159 00:06:09,921 --> 00:06:12,939 Every single mass travels at the same angular speed, 160 00:06:12,939 --> 00:06:15,911 so we could factor this out of the summation. 161 00:06:15,911 --> 00:06:18,375 Imagine all these terms would have an omega. 162 00:06:18,375 --> 00:06:20,266 We can factor that out, and I could just bring that 163 00:06:20,266 --> 00:06:22,134 outside of the summation. 164 00:06:22,134 --> 00:06:25,803 So I'll write this as the summation of mR squared, 165 00:06:25,803 --> 00:06:28,285 and to make this clear, I'm gonna put parentheses here. 166 00:06:28,285 --> 00:06:31,749 It's that summation and then that whole thing times omega, 167 00:06:31,749 --> 00:06:33,805 'cause we're just factoring out omega. 168 00:06:33,805 --> 00:06:35,086 And you might not be impressed. 169 00:06:35,086 --> 00:06:36,385 You might be like, "All right. 170 00:06:36,385 --> 00:06:37,218 Big deal. 171 00:06:37,218 --> 00:06:39,481 We've still got an infinite sum in here. 172 00:06:39,481 --> 00:06:40,782 What the heck am I gonna do with that?" 173 00:06:40,782 --> 00:06:42,772 You don't have to do anything with that. 174 00:06:42,772 --> 00:06:44,017 This is where the magic happens. 175 00:06:44,017 --> 00:06:45,685 Look at what sum you've got. 176 00:06:45,685 --> 00:06:48,555 You've got the sum of all the mR squareds. 177 00:06:48,555 --> 00:06:50,310 Remember what mR squared was? 178 00:06:50,310 --> 00:06:53,899 Mr squared was the moment of inertia of a point mass, 179 00:06:53,899 --> 00:06:57,096 and if I add up all the mR squareds, I get the moment 180 00:06:57,096 --> 00:07:00,574 of inertia of the entire mass, this entire object. 181 00:07:00,574 --> 00:07:02,420 I get its total moment of inertia. 182 00:07:02,420 --> 00:07:05,496 So what we found was a really handy way 183 00:07:05,496 --> 00:07:07,691 to write the angular momentum of an object. 184 00:07:07,691 --> 00:07:11,457 It's just the moment of inertia of an object, I, 185 00:07:11,457 --> 00:07:15,262 times the angular speed of that object. 186 00:07:15,262 --> 00:07:17,259 So this is a great formula, 187 00:07:17,259 --> 00:07:19,640 and it totally makes sense for this reason. 188 00:07:19,640 --> 00:07:21,403 Think about regular momentum, right. 189 00:07:21,403 --> 00:07:25,494 Regular momentum, p, was just equal to mv. 190 00:07:25,494 --> 00:07:28,404 Well, if you then told me, "Determine the angular momentum," 191 00:07:28,404 --> 00:07:30,274 and I didn't wanna go through this derivation, 192 00:07:30,274 --> 00:07:31,176 I mighta just been like, 193 00:07:31,176 --> 00:07:33,935 "All right, angular momentum, shoot." 194 00:07:33,935 --> 00:07:37,678 Well, I'm just gonna replace mass with angular mass, 195 00:07:37,678 --> 00:07:39,888 and angular mass, the angular inertia, 196 00:07:39,888 --> 00:07:41,432 is just the moment of inertia, 197 00:07:41,432 --> 00:07:44,914 and I'll just replace the speed with the angular speed, 198 00:07:44,914 --> 00:07:46,505 and look, I just get this formula. 199 00:07:46,505 --> 00:07:47,782 So it makes sense because 200 00:07:47,782 --> 00:07:49,675 if you replace all the linear quantities 201 00:07:49,675 --> 00:07:51,656 with their angular counterpart, 202 00:07:51,656 --> 00:07:54,088 you indeed just get the angular momentum 203 00:07:54,088 --> 00:07:55,836 of a rotating object. 204 00:07:55,836 --> 00:07:56,707 So this is how you do it. 205 00:07:56,707 --> 00:07:58,776 If you've got an object, an extended object 206 00:07:58,776 --> 00:08:02,318 where the mass is distributed about the entire object, 207 00:08:02,318 --> 00:08:03,659 if you just take the moment of inertia 208 00:08:03,659 --> 00:08:06,436 of that object and multiply by its angular speed, 209 00:08:06,436 --> 00:08:08,233 you get its angular momentum. 210 00:08:08,233 --> 00:08:10,667 So, for instance, if this rod has a mass 211 00:08:10,667 --> 00:08:13,156 of let's say three kilograms, 212 00:08:13,156 --> 00:08:14,985 and that mass is evenly distributed. 213 00:08:14,985 --> 00:08:17,807 Let's say the radius of this object is two meters. 214 00:08:17,807 --> 00:08:21,193 So that's the distance from the axis to the outside edge. 215 00:08:21,193 --> 00:08:24,189 And let's say the angular speed of this object was, 216 00:08:24,189 --> 00:08:27,207 let's say 10 radians per second, 217 00:08:27,207 --> 00:08:30,430 we can figure out the angular momentum of this rod 218 00:08:30,430 --> 00:08:32,889 by saying that the angular momentum is gonna equal 219 00:08:32,889 --> 00:08:34,676 the moment of inertia. 220 00:08:34,676 --> 00:08:36,057 Well the moment of inertia of a rod 221 00:08:36,057 --> 00:08:40,767 about one end is equal to 1/3 mL squared. 222 00:08:40,767 --> 00:08:43,128 That's the moment of inertia of a rod about the end, 223 00:08:43,128 --> 00:08:47,076 and I then multiply by the angular speed of the object. 224 00:08:47,076 --> 00:08:49,014 So if I plug in numbers, I get the angular momentum 225 00:08:49,014 --> 00:08:51,479 of this rod is gonna be, I'll use purple here, 226 00:08:51,479 --> 00:08:55,969 1/3 times three kilograms times the length 227 00:08:55,969 --> 00:08:57,750 of the object was two meters, 228 00:08:57,750 --> 00:08:59,683 and we square that, and then we multiply 229 00:08:59,683 --> 00:09:03,250 by the angular speed and that was 10 radians per second, 230 00:09:03,250 --> 00:09:05,047 which give us an angular momentum 231 00:09:05,047 --> 00:09:08,464 of 40 kilogram meters squared per second. 232 00:09:09,517 --> 00:09:11,575 So recapping, if you've got a point mass 233 00:09:11,575 --> 00:09:13,875 where all the mass rotates at the same radius 234 00:09:13,875 --> 00:09:16,038 and you wanna find the angular momentum, 235 00:09:16,038 --> 00:09:17,517 the easiest way to get it is probably 236 00:09:17,517 --> 00:09:19,408 with the formula mvR. 237 00:09:19,408 --> 00:09:22,722 However, if you have a mass whose mass is distributed 238 00:09:22,722 --> 00:09:25,319 throughout the object so that different points 239 00:09:25,319 --> 00:09:27,826 on the object are at different radii, 240 00:09:27,826 --> 00:09:30,144 the easiest way to get the angular momentum 241 00:09:30,144 --> 00:09:31,827 of that object is most likely 242 00:09:31,827 --> 00:09:34,309 with the formula I omega, 243 00:09:34,309 --> 00:09:35,324 where I is the moment 244 00:09:35,324 --> 00:09:36,673 of inertia of the object 245 00:09:36,673 --> 00:00:00,000 and omega is the angular velocity of the object.