1 00:00:00,384 --> 00:00:01,366 - [Instructor] What's up everybody? 2 00:00:01,366 --> 00:00:03,192 I wanna show you something kind of crazy. 3 00:00:03,192 --> 00:00:05,545 When I first heard about this, it really bothered me. 4 00:00:05,545 --> 00:00:07,859 So if you have that reaction, it's natural, 5 00:00:07,859 --> 00:00:09,180 but I hope you get over it, 6 00:00:09,180 --> 00:00:11,415 and it'll hopefully makes sense by the end of this. 7 00:00:11,415 --> 00:00:13,149 The crazy thing is this. 8 00:00:13,149 --> 00:00:14,312 If you got an object, 9 00:00:14,312 --> 00:00:16,421 say this ball, say it's a bouncy ball, 10 00:00:16,421 --> 00:00:18,214 and it's going in a straight line, 11 00:00:18,214 --> 00:00:21,065 it can have angular momentum. 12 00:00:21,065 --> 00:00:22,282 And I'll say that again. 13 00:00:22,282 --> 00:00:25,104 A ball traveling in a straight line 14 00:00:25,104 --> 00:00:26,843 can have angular momentum. 15 00:00:26,843 --> 00:00:28,042 When I first heard this, 16 00:00:28,042 --> 00:00:29,005 I was like "What? 17 00:00:29,005 --> 00:00:31,334 "There's no way this ball can have angular momentum. 18 00:00:31,334 --> 00:00:32,509 "It's moving in a straight line. 19 00:00:32,509 --> 00:00:34,390 "It's not even rotating. 20 00:00:34,390 --> 00:00:37,075 "Don't things have to have some sort of rotational motion 21 00:00:37,075 --> 00:00:39,053 "in order to have angular momentum?" 22 00:00:39,053 --> 00:00:40,408 And it turns out, they don't. 23 00:00:40,408 --> 00:00:41,914 But, the proper response, 24 00:00:41,914 --> 00:00:43,154 it gets a little weirder. 25 00:00:43,154 --> 00:00:44,584 Before I show you how it makes sense, 26 00:00:44,584 --> 00:00:45,534 let's me show you this. 27 00:00:45,534 --> 00:00:47,854 It doesn't even have to have angular momentum. 28 00:00:47,854 --> 00:00:50,597 It could be that this has no angular momentum. 29 00:00:50,597 --> 00:00:53,036 And before you get really confused, let me explain. 30 00:00:53,036 --> 00:00:55,561 The proper response to someone saying, 31 00:00:55,561 --> 00:00:57,698 "Does this ball have angular momentum?" 32 00:00:57,698 --> 00:01:01,726 is to respond with, "Angular momentum about which axis?" 33 00:01:01,726 --> 00:01:03,979 So you have to specify the axis. 34 00:01:03,979 --> 00:01:06,292 So the axis is the point about which 35 00:01:06,292 --> 00:01:08,322 you're gonna consider the rotation. 36 00:01:08,322 --> 00:01:11,309 So if I said, "Does this ball have angular momentum? 37 00:01:11,309 --> 00:01:13,771 "If the ball is moving in a straight line, 38 00:01:13,771 --> 00:01:17,479 "does it have angular momentum relative to this axis?" 39 00:01:17,479 --> 00:01:18,764 that's a proper question. 40 00:01:18,764 --> 00:01:19,983 But if it just ask you, 41 00:01:19,983 --> 00:01:22,235 "Does this ball have angular momentum?" 42 00:01:22,235 --> 00:01:24,094 and don't specify the axis, 43 00:01:24,094 --> 00:01:25,948 then it's not even a meaningful question. 44 00:01:25,948 --> 00:01:28,226 So let's try to figure this out. 45 00:01:28,226 --> 00:01:31,754 Why does this ball have angular momentum at all, 46 00:01:31,754 --> 00:01:33,454 regardless of any axis, right? 47 00:01:33,454 --> 00:01:34,971 That's the confusing part. 48 00:01:34,971 --> 00:01:36,079 It's not even rotating. 49 00:01:36,079 --> 00:01:38,240 How does a ball moving in a straight line 50 00:01:38,240 --> 00:01:40,058 have angular momentum? 51 00:01:40,058 --> 00:01:41,527 We know it has regular momentum, 52 00:01:41,527 --> 00:01:44,889 because objects with mass and velocity have momentum. 53 00:01:44,889 --> 00:01:45,988 But it's not even rotating. 54 00:01:45,988 --> 00:01:48,187 How can it have angular momentum? 55 00:01:48,187 --> 00:01:51,509 First, let me explain conceptually why that makes sense. 56 00:01:51,509 --> 00:01:53,780 So imagine you have this bar here, right? 57 00:01:53,780 --> 00:01:55,828 And this bar, say this is a bird's eye view. 58 00:01:55,828 --> 00:01:57,366 We're looking down. 59 00:01:57,366 --> 00:02:01,014 This bar is attached to an axis that can rotate, 60 00:02:01,014 --> 00:02:03,669 and we're looking down. 61 00:02:03,669 --> 00:02:04,940 This is on a table top. 62 00:02:04,940 --> 00:02:06,564 Let's say this is all happening on a table top, 63 00:02:06,564 --> 00:02:08,199 and we're looking down at it. 64 00:02:08,199 --> 00:02:10,235 So imagine we throw this ball, right? 65 00:02:10,235 --> 00:02:12,544 This ball moving in a straight line 66 00:02:12,544 --> 00:02:14,923 hits the edge of this bar. 67 00:02:14,923 --> 00:02:16,790 And this bar, what is it gonna do? 68 00:02:16,790 --> 00:02:17,881 We know what it's gonna do, 69 00:02:17,881 --> 00:02:18,714 it's gonna rotate. 70 00:02:18,714 --> 00:02:20,979 It's gonna rotate about its axis. 71 00:02:20,979 --> 00:02:22,306 And now the question is, 72 00:02:22,306 --> 00:02:25,081 this bar initially had no angular momentum, 73 00:02:25,081 --> 00:02:27,495 because the bar was just sitting here at rest. 74 00:02:27,495 --> 00:02:31,324 Then it did have angular momentum after the ball hit it, 75 00:02:31,324 --> 00:02:33,316 because objects rotating around in a circle 76 00:02:33,316 --> 00:02:35,057 have angular momentum. 77 00:02:35,057 --> 00:02:38,521 Where did this bar get its angular momentum? 78 00:02:38,521 --> 00:02:41,149 Well, the only thing that that bar interacted with 79 00:02:41,149 --> 00:02:42,211 was this ball. 80 00:02:42,211 --> 00:02:43,951 So this ball must've come in 81 00:02:43,951 --> 00:02:46,675 and transferred some angular momentum 82 00:02:46,675 --> 00:02:47,796 to the bar. 83 00:02:47,796 --> 00:02:49,304 Because where else would the bar 84 00:02:49,304 --> 00:02:51,131 get the angular momentum from? 85 00:02:51,131 --> 00:02:54,309 I mean, if we believe in conservation of angular momentum, 86 00:02:54,309 --> 00:02:56,516 that angular momentum has to come from somewhere. 87 00:02:56,516 --> 00:02:58,332 It can't just pop out at nowhere. 88 00:02:58,332 --> 00:03:01,673 So the only place that angular momentum that the bar got 89 00:03:01,673 --> 00:03:02,855 had to be from the ball 90 00:03:02,855 --> 00:03:05,203 because that was the only other thing in the problem. 91 00:03:05,203 --> 00:03:09,230 So this ball had to come in with its own angular momentum 92 00:03:09,230 --> 00:03:11,559 even though it was traveling in a straight line, 93 00:03:11,559 --> 00:03:14,057 which is kind of weird, but that's the case. 94 00:03:14,057 --> 00:03:15,100 That's physics. 95 00:03:15,100 --> 00:03:17,204 And that's also why it makes sense 96 00:03:17,204 --> 00:03:19,452 why it depends on where the axis is. 97 00:03:19,452 --> 00:03:21,307 Because if I take this bar, 98 00:03:21,307 --> 00:03:23,972 and I move this bar over to here instead, 99 00:03:23,972 --> 00:03:27,089 So now we can set our axis over this point. 100 00:03:27,089 --> 00:03:31,377 Well, imagine the ball hitting the bar at the axis point. 101 00:03:31,377 --> 00:03:32,599 It just hits right there. Boink! 102 00:03:32,599 --> 00:03:34,796 It's not even gonna cause that bar to rotate 103 00:03:34,796 --> 00:03:36,869 because it's hitting it at the axis. 104 00:03:36,869 --> 00:03:39,651 So the location of the axis is gonna determine 105 00:03:39,651 --> 00:03:42,403 how much angular momentum an object has 106 00:03:42,403 --> 00:03:43,529 that's moving in a straight line 107 00:03:43,529 --> 00:03:46,545 because if it hits this bar at the axis, 108 00:03:46,545 --> 00:03:48,624 it's gonna transfer no angular momentum. 109 00:03:48,624 --> 00:03:51,982 But if it hits this bar far away from the axis, 110 00:03:51,982 --> 00:03:54,078 it can transfer a lot of angular momentum 111 00:03:54,078 --> 00:03:56,082 because the bar is gonna rotate a lot, 112 00:03:56,082 --> 00:03:58,879 because there was a large amount of torque applied 113 00:03:58,879 --> 00:04:01,606 since this force was applied at a distance 114 00:04:01,606 --> 00:04:03,475 that was far away from the axis. 115 00:04:03,475 --> 00:04:05,296 Now, you still might not be all that impressed. 116 00:04:05,296 --> 00:04:06,129 You might be like, 117 00:04:06,129 --> 00:04:07,028 "That just sounded 118 00:04:07,028 --> 00:04:09,420 "like a bunch of physics witchcraft to me. 119 00:04:09,420 --> 00:04:11,837 "How do you calculate this exactly? 120 00:04:11,837 --> 00:04:15,045 "How do you exactly define what we mean 121 00:04:15,045 --> 00:04:18,860 "by the fact that this ball has angular momentum?" 122 00:04:18,860 --> 00:04:20,321 So let's define this exactly. 123 00:04:20,321 --> 00:04:23,352 Let's say this ball has a speed or a velocity v 124 00:04:23,352 --> 00:04:25,055 and the mass of the ball, 125 00:04:25,055 --> 00:04:27,012 we'll say the mass of this ball is m. 126 00:04:27,012 --> 00:04:30,402 And the distance from the axis to the ball, 127 00:04:30,402 --> 00:04:32,106 let's just draw that on here, 128 00:04:32,106 --> 00:04:34,688 so that's gonna be from the axis to the ball, 129 00:04:34,688 --> 00:04:36,599 we'll call that little r. 130 00:04:36,599 --> 00:04:39,400 And now we can define precisely what we mean 131 00:04:39,400 --> 00:04:42,306 by the angular momentum of a point mass. 132 00:04:42,306 --> 00:04:44,340 The angular momentum of a point mass. 133 00:04:44,340 --> 00:04:46,991 L is the symbol for angular momentum. 134 00:04:46,991 --> 00:04:49,582 It's gonna be m, the mass of the ball. 135 00:04:49,582 --> 00:04:52,495 So the mass of the object that has the angular momentum 136 00:04:52,495 --> 00:04:54,572 times v, the speed of the ball, 137 00:04:54,572 --> 00:04:56,214 and this is looking pretty familiar 138 00:04:56,214 --> 00:04:58,698 because, m times v is just momentum. 139 00:04:58,698 --> 00:05:00,705 So this is regular momentum. 140 00:05:00,705 --> 00:05:02,061 But if it just left you like this, 141 00:05:02,061 --> 00:05:03,297 that'd just be regular momentum. 142 00:05:03,297 --> 00:05:04,869 We have to turn this into angular momentum 143 00:05:04,869 --> 00:05:07,821 and we do that by multiplying by r. 144 00:05:07,821 --> 00:05:09,972 And r is defining to be 145 00:05:09,972 --> 00:05:13,617 the distance from the axis or your origin 146 00:05:13,617 --> 00:05:15,505 to the mass that you've considering. 147 00:05:15,505 --> 00:05:17,570 And so, this is the total distance r. 148 00:05:17,570 --> 00:05:18,615 But you're not done yet. 149 00:05:18,615 --> 00:05:20,677 You need one more term in here. 150 00:05:20,677 --> 00:05:23,246 That's can be sine of the angle, 151 00:05:23,246 --> 00:05:26,694 sine of the angle between the velocity and the r vector. 152 00:05:26,694 --> 00:05:30,668 So you're gonna have sine of this angle right here, 153 00:05:30,668 --> 00:05:34,081 the angle between the velocity and the r vector. 154 00:05:34,081 --> 00:05:35,404 Now I'm being a little sloppy. 155 00:05:35,404 --> 00:05:38,028 So, people paying attention out there that already know this 156 00:05:38,028 --> 00:05:39,149 might be a little concerned. 157 00:05:39,149 --> 00:05:43,106 Technically, r goes from the axis to the mass, 158 00:05:43,106 --> 00:05:46,335 so this isn't the angle between v and r. 159 00:05:46,335 --> 00:05:48,007 Technically, you have to imagine 160 00:05:48,007 --> 00:05:50,180 r being extended out this way, 161 00:05:50,180 --> 00:05:51,861 and then that's the angle. 162 00:05:51,861 --> 00:05:53,971 But because we're taking sine, 163 00:05:53,971 --> 00:05:55,733 sine of either these angles, 164 00:05:55,733 --> 00:05:57,585 these angles are supplementary. 165 00:05:57,585 --> 00:05:58,608 The sine of either of them 166 00:05:58,608 --> 00:06:00,247 are gonna give you the same number, 167 00:06:00,247 --> 00:06:02,844 so you're safe by just taking any angle here 168 00:06:02,844 --> 00:06:04,143 between v and r, 169 00:06:04,143 --> 00:06:05,809 and you'll get the right answer. 170 00:06:05,809 --> 00:06:06,985 But this is complicated. 171 00:06:06,985 --> 00:06:08,756 If I were you, I'd be like, "Oh man, 172 00:06:08,756 --> 00:06:09,906 "mvr sine theta. 173 00:06:09,906 --> 00:06:11,072 "I don't wanna have to figure out 174 00:06:11,072 --> 00:06:13,040 "what the angles are between things." 175 00:06:13,040 --> 00:06:14,031 And you don't. 176 00:06:14,031 --> 00:06:15,713 There's a trick, so check this out, 177 00:06:15,713 --> 00:06:19,530 if you just consider what is r sine theta even mean. 178 00:06:19,530 --> 00:06:20,363 What is this? 179 00:06:20,363 --> 00:06:21,196 Right? 180 00:06:21,196 --> 00:06:22,944 Visually, what does this represent on here? 181 00:06:22,944 --> 00:06:24,720 Well, here's the total amount r. 182 00:06:24,720 --> 00:06:25,645 Here's theta. 183 00:06:25,645 --> 00:06:26,478 Think about it. 184 00:06:26,478 --> 00:06:30,129 r times sine theta, imagine making the triangle out of this. 185 00:06:30,129 --> 00:06:32,309 I'll make a triangle that goes from to there, 186 00:06:32,309 --> 00:06:33,513 then here to here. 187 00:06:33,513 --> 00:06:35,504 So you've got this triangle here. 188 00:06:35,504 --> 00:06:38,372 r times sine theta, it's just this right here, 189 00:06:38,372 --> 00:06:39,460 so it's just this. 190 00:06:39,460 --> 00:06:42,924 This length right here, this opposite side, because... 191 00:06:42,924 --> 00:06:44,835 All right, if you didn't catch that, 192 00:06:44,835 --> 00:06:46,376 that might be a little bit weird. 193 00:06:46,376 --> 00:06:50,075 So sine of theta is always opposite over hypotenuse, 194 00:06:50,075 --> 00:06:53,909 and our opposite side opposite to that angle is just R, 195 00:06:53,909 --> 00:06:55,432 so that's just R. 196 00:06:55,432 --> 00:06:57,865 And then the hypotenuse, excuse me, 197 00:06:57,865 --> 00:07:00,350 is just this little r, this pink r. 198 00:07:00,350 --> 00:07:02,138 Let me call that little r. 199 00:07:02,138 --> 00:07:04,423 So, if I multiply both sides by little r, 200 00:07:04,423 --> 00:07:06,755 I'll get that r times sine theta, 201 00:07:06,755 --> 00:07:09,445 and this theta here is this theta here, 202 00:07:09,445 --> 00:07:12,377 so this is that theta right there. 203 00:07:12,377 --> 00:07:16,069 r sine theta is just equal to this right here. 204 00:07:16,069 --> 00:07:16,902 And what is this? 205 00:07:16,902 --> 00:07:19,753 This is just this point of closest approach. 206 00:07:19,753 --> 00:07:22,409 So when the ball makes it to this point right here, 207 00:07:22,409 --> 00:07:24,004 when the ball gets to this point, 208 00:07:24,004 --> 00:07:26,111 going this way at some speed v, 209 00:07:26,111 --> 00:07:27,970 it's gonna be R away. 210 00:07:27,970 --> 00:07:29,238 So all you really need to do 211 00:07:29,238 --> 00:07:32,903 to find the angular momentum of an object of a point mass, 212 00:07:32,903 --> 00:07:34,830 even if that point mass is going in a straight line 213 00:07:34,830 --> 00:07:36,973 is take the mass times v. 214 00:07:36,973 --> 00:07:38,452 And then if you don't wanna have to worry about 215 00:07:38,452 --> 00:07:40,110 sine theta and all of that mess, 216 00:07:40,110 --> 00:07:42,681 just multiply it by R, 217 00:07:42,681 --> 00:07:46,604 which is the distance of closest approach to the axis. 218 00:07:46,604 --> 00:07:47,624 So that's what this R is. 219 00:07:47,624 --> 00:07:50,858 This R is the distance of closest approach 220 00:07:50,858 --> 00:07:52,039 to the axis. 221 00:07:52,039 --> 00:07:53,104 And that's just this right here. 222 00:07:53,104 --> 00:07:54,742 That's just this distance right here, 223 00:07:54,742 --> 00:07:56,997 between the axis and 224 00:07:56,997 --> 00:08:00,270 the point where the ball will be closest to the axis, 225 00:08:00,270 --> 00:08:01,956 so that's this distance right here. 226 00:08:01,956 --> 00:08:03,188 It turns out this r sine theta 227 00:08:03,188 --> 00:08:05,181 is always just equal to that, 228 00:08:05,181 --> 00:08:06,316 so you could make your life easy. 229 00:08:06,316 --> 00:08:08,602 Just imagine, when this ball comes in, 230 00:08:08,602 --> 00:08:11,370 at what point is it closest to the axis? 231 00:08:11,370 --> 00:08:12,246 That would be this point. 232 00:08:12,246 --> 00:08:15,091 And then how far is it when it is closest? 233 00:08:15,091 --> 00:08:16,782 That gives you this R value. 234 00:08:16,782 --> 00:08:18,426 You can take mvR. 235 00:08:18,426 --> 00:08:21,329 That gives you the angular momentum of this point mass. 236 00:08:21,329 --> 00:08:23,504 It tells you the total amount of angular momentum 237 00:08:23,504 --> 00:08:25,800 that thing could transfer to something else 238 00:08:25,800 --> 00:08:28,035 if it lost all of its angular momentum. 239 00:08:28,035 --> 00:08:29,730 That's how much angular momentum something, 240 00:08:29,730 --> 00:08:31,837 like that rod, could get. 241 00:08:31,837 --> 00:08:32,889 So let's try an example. 242 00:08:32,889 --> 00:08:34,240 Let's do this example. 243 00:08:34,240 --> 00:08:36,900 It's actually a classic, a ball hitting a rod. 244 00:08:36,900 --> 00:08:39,037 Man, I'm telling you, physics teachers and professors, 245 00:08:39,037 --> 00:08:39,933 they love this thing. 246 00:08:39,933 --> 00:08:40,816 You should know how to do this. 247 00:08:40,816 --> 00:08:41,878 Let's get you prepared here. 248 00:08:41,878 --> 00:08:43,283 So let's say this ball comes in. 249 00:08:43,283 --> 00:08:44,835 It hits a rod, right? 250 00:08:44,835 --> 00:08:46,475 And so the ball is gonna come in. 251 00:08:46,475 --> 00:08:47,872 Ball is gonna hit a rod, 252 00:08:47,872 --> 00:08:49,341 and let's put some numbers on this thing, 253 00:08:49,341 --> 00:08:51,442 so we can actually solve this example. 254 00:08:51,442 --> 00:08:53,613 let's say the ball had a mass of five kilograms. 255 00:08:53,613 --> 00:08:55,355 It was going eight meters per second, 256 00:08:55,355 --> 00:08:56,992 hits the end of the rod, 257 00:08:56,992 --> 00:08:58,998 and the rod is 10 kilograms, 258 00:08:58,998 --> 00:09:00,596 four meters long. 259 00:09:00,596 --> 00:09:04,062 Let's assume this rod has uniform density, 260 00:09:04,062 --> 00:09:07,048 so this rod has a nice mass distributed 261 00:09:07,048 --> 00:09:08,359 evenly throughout it, 262 00:09:08,359 --> 00:09:10,166 and it can rotate around the end. 263 00:09:10,166 --> 00:09:11,591 So when the ball gets in here, 264 00:09:11,591 --> 00:09:13,098 strikes the end of the rod, 265 00:09:13,098 --> 00:09:16,255 the rod is gonna rotate around its axis. 266 00:09:16,255 --> 00:09:17,588 And let's make another assumption. 267 00:09:17,588 --> 00:09:20,801 Let's assume when this ball does hit the rod, 268 00:09:20,801 --> 00:09:22,275 the ball stops. 269 00:09:22,275 --> 00:09:24,026 So after hitting the rod, 270 00:09:24,026 --> 00:09:25,558 the ball has stopped, 271 00:09:25,558 --> 00:09:26,978 and the rod moves on 272 00:09:26,978 --> 00:09:29,710 with all the angular momentum that the ball had. 273 00:09:29,710 --> 00:09:31,472 That will just make it a little easier. 274 00:09:31,472 --> 00:09:33,242 We'll talk about what to do if that doesn't happen. 275 00:09:33,242 --> 00:09:34,734 It's not that much harder. 276 00:09:34,734 --> 00:09:37,655 Let's just say that's the case initially or so. 277 00:09:37,655 --> 00:09:39,596 I'll move the ball back over to here. 278 00:09:39,596 --> 00:09:41,685 How do we solve this problem? 279 00:09:41,685 --> 00:09:43,137 Well, we're gonna try to use 280 00:09:43,137 --> 00:09:44,814 conservation of angular momentum. 281 00:09:44,814 --> 00:09:46,757 We're gonna say that even though 282 00:09:46,757 --> 00:09:48,847 there's an axis his exerting a force, 283 00:09:48,847 --> 00:09:52,657 the force that that axis is gonna exert on our system 284 00:09:52,657 --> 00:09:54,422 is gonna exert zero torque, 285 00:09:54,422 --> 00:09:56,078 because the r value. 286 00:09:56,078 --> 00:09:57,643 Torque is equal 287 00:09:57,643 --> 00:09:59,143 to r F sine theta. 288 00:10:00,201 --> 00:10:01,908 And if the r is zero, 289 00:10:01,908 --> 00:10:04,790 r is the distance from the axis to the force, 290 00:10:04,790 --> 00:10:06,198 if r is zero, 291 00:10:06,198 --> 00:10:08,776 there's gonna be no torque exerted by that axis. 292 00:10:08,776 --> 00:10:11,307 And if there's no torque exerted externally, 293 00:10:11,307 --> 00:10:13,805 there's no change in angular momentum of the system. 294 00:10:13,805 --> 00:10:15,449 So this system of ball and rod 295 00:10:15,449 --> 00:10:17,809 is gonna have no external torque on it. 296 00:10:17,809 --> 00:10:21,424 That means the angular momentum has to stay the same. 297 00:10:21,424 --> 00:10:23,638 This is a classic conservation 298 00:10:23,638 --> 00:10:24,840 of angular momentum problems. 299 00:10:24,840 --> 00:10:26,640 So we're gonna say that L initial, 300 00:10:26,640 --> 00:10:28,196 the initial angular momentum, 301 00:10:28,196 --> 00:10:30,703 has to equal the final angular momentum. 302 00:10:30,703 --> 00:10:32,517 And we'll just say, for our entire system, 303 00:10:32,517 --> 00:10:34,723 what had angular momentum initially? 304 00:10:34,723 --> 00:10:35,949 Well, it was this mass. 305 00:10:35,949 --> 00:10:38,500 So this mass had the angular momentum. 306 00:10:38,500 --> 00:10:39,333 And how do we find that? 307 00:10:39,333 --> 00:10:42,751 Remember it's m times v times R, 308 00:10:42,751 --> 00:10:46,036 and the r is that distance of closest approach. 309 00:10:46,036 --> 00:10:47,619 So we're gonna use this here 310 00:10:47,619 --> 00:10:50,574 for the whole four meters as this R. 311 00:10:50,574 --> 00:10:53,441 Yes, you can consider this hypotenuse R 312 00:10:53,441 --> 00:10:55,307 and a sine of the angle, 313 00:10:55,307 --> 00:10:56,637 but that's harder than it needs to be. 314 00:10:56,637 --> 00:10:59,229 You can find angular momentum, mvR, 315 00:10:59,229 --> 00:11:01,810 that's gonna equal the final angular momentum. 316 00:11:01,810 --> 00:11:03,568 Remember this ball stops. 317 00:11:03,568 --> 00:11:05,434 So since this ball comes to rest, 318 00:11:05,434 --> 00:11:08,726 and it's only the bar that has angular momentum afterward, 319 00:11:08,726 --> 00:11:10,612 we only have to worry about the angular momentum 320 00:11:10,612 --> 00:11:12,772 of the bar on the final side. 321 00:11:12,772 --> 00:11:14,320 And to find the angular momentum 322 00:11:14,320 --> 00:11:16,838 of an extended object, a rigid object, 323 00:11:16,838 --> 00:11:18,576 you can use I omega. 324 00:11:18,576 --> 00:11:20,351 And this would let us solve for what is 325 00:11:20,351 --> 00:11:22,768 the final angular velocity of 326 00:11:23,849 --> 00:11:25,537 this rod after the collision. 327 00:11:25,537 --> 00:11:26,763 So that's what we wanna figure out. 328 00:11:26,763 --> 00:11:29,642 What is the final angular velocity of the rod 329 00:11:29,642 --> 00:11:30,855 after the collision? 330 00:11:30,855 --> 00:11:31,793 Now we can figure it out. 331 00:11:31,793 --> 00:11:33,617 We know the mass the of the ball, m. 332 00:11:33,617 --> 00:11:35,466 We know the speed of the ball initially. 333 00:11:35,466 --> 00:11:37,119 We know the R, line of closest approach. 334 00:11:37,119 --> 00:11:38,655 That's four meters. 335 00:11:38,655 --> 00:11:40,452 What's the moment of inertia here? 336 00:11:40,452 --> 00:11:43,370 Well, it's just gonna be 1/3 m L squared. 337 00:11:43,370 --> 00:11:45,530 Let me clean this up a little bit. 338 00:11:45,530 --> 00:11:46,363 Let me take this. 339 00:11:46,363 --> 00:11:47,501 I'll just copy that. 340 00:11:47,501 --> 00:11:49,239 Put that right down over here, 341 00:11:49,239 --> 00:11:50,822 and we could say that the moment of inertia 342 00:11:50,822 --> 00:11:52,405 of a mass of a rod, 343 00:11:53,488 --> 00:11:55,431 it's rotating around its end, 344 00:11:55,431 --> 00:11:58,316 is always gonna be 1/3 m L squared. 345 00:11:58,316 --> 00:12:00,916 So 1/3 times the mass of the rod, 346 00:12:00,916 --> 00:12:02,459 times the length of the rod squared, 347 00:12:02,459 --> 00:12:04,462 which is gonna be the same as this R here, 348 00:12:04,462 --> 00:12:08,102 because this ball's line of closest approach 349 00:12:08,102 --> 00:12:10,202 was jus equal to the entire length of the rod, 350 00:12:10,202 --> 00:12:12,317 since it struck it at the very end, 351 00:12:12,317 --> 00:12:14,136 and then times omega. 352 00:12:14,136 --> 00:12:15,410 So we can solve this for omega now. 353 00:12:15,410 --> 00:12:16,243 We can say that omega. 354 00:12:16,243 --> 00:12:18,194 I'm gonna bring this down around here, 355 00:12:18,194 --> 00:12:19,198 so we go some room. 356 00:12:19,198 --> 00:12:22,117 Omega final of the rod is just gonna be, what? 357 00:12:22,117 --> 00:12:23,623 It's gonna be mass of the ball 358 00:12:23,623 --> 00:12:25,663 times the initial speed of the ball, 359 00:12:25,663 --> 00:12:28,059 times the line of closet approach. 360 00:12:28,059 --> 00:12:31,878 And then I'm gonna divide by 1/3 the mass of the rod 361 00:12:31,878 --> 00:12:33,064 times the length of the rod. 362 00:12:33,064 --> 00:12:35,104 I can just call that R, it's the same variable, 363 00:12:35,104 --> 00:12:37,003 length of the rod squared, 364 00:12:37,003 --> 00:12:37,836 and that's what I get. 365 00:12:37,836 --> 00:12:40,068 So I can cancel off one of these Rs, 366 00:12:40,068 --> 00:12:41,022 and then I can plug in numbers 367 00:12:41,022 --> 00:12:42,169 if I wanted to actually get a number. 368 00:12:42,169 --> 00:12:44,147 I could say that the final angular velocity 369 00:12:44,147 --> 00:12:45,722 of this rod 370 00:12:45,722 --> 00:12:47,135 is gonna be five kilograms, 371 00:12:47,135 --> 00:12:49,019 that was the mass of the ball, 372 00:12:49,019 --> 00:12:50,466 times eight meters per second, 373 00:12:50,466 --> 00:12:53,023 that was the initial speed of the ball, 374 00:12:53,023 --> 00:12:55,767 and then I'm gonna divide by 1/3 of 375 00:12:55,767 --> 00:12:58,406 the mass of the rod was 10 kilograms, 376 00:12:58,406 --> 00:12:59,755 and then the length of the rod, 377 00:12:59,755 --> 00:13:01,587 which is this line of closest approach, 378 00:13:01,587 --> 00:13:03,205 was four meters. 379 00:13:03,205 --> 00:13:04,038 And if you solve all that, 380 00:13:04,038 --> 00:13:06,991 you get three radians per second. 381 00:13:06,991 --> 00:13:10,213 So that's how much angular speed or angular velocity 382 00:13:10,213 --> 00:13:12,832 this rod had after the ball hit it, 383 00:13:12,832 --> 00:13:16,213 and transfer the ball's angular momentum into the rod, 384 00:13:16,213 --> 00:13:17,588 giving the rod angular momentum, 385 00:13:17,588 --> 00:13:19,931 causing it to spin around and rotate 386 00:13:19,931 --> 00:13:21,718 at three radians per second. 387 00:13:21,718 --> 00:13:24,958 Now what would be different if instead of getting stopped, 388 00:13:24,958 --> 00:13:26,717 the ball bounced to backward 389 00:13:26,717 --> 00:13:29,158 let's say at two meters per second? 390 00:13:29,158 --> 00:13:31,708 Well, now the final angular momentum 391 00:13:31,708 --> 00:13:34,756 wouldn't just be the angular momentum of the rod, 392 00:13:34,756 --> 00:13:35,589 you'd have to include 393 00:13:35,589 --> 00:13:37,632 the angular momentum of the ball. 394 00:13:37,632 --> 00:13:39,010 But here's the tricky part. 395 00:13:39,010 --> 00:13:41,275 If the ball is coming in this way initially, 396 00:13:41,275 --> 00:13:42,783 which would mean it has angular momentum 397 00:13:42,783 --> 00:13:45,246 essentially around this way, 398 00:13:45,246 --> 00:13:47,286 and the ball went backward the other way, 399 00:13:47,286 --> 00:13:49,893 now it has angular momentum around this way, 400 00:13:49,893 --> 00:13:51,634 you'd have to have a negative sine in here. 401 00:13:51,634 --> 00:13:55,089 In other words, when you include this angular momentum 402 00:13:55,089 --> 00:13:55,922 on the right hand side, 403 00:13:55,922 --> 00:13:57,413 you'd have to treat it as... 404 00:13:57,413 --> 00:13:58,710 Well, there's a couple of ways you could do it. 405 00:13:58,710 --> 00:14:01,147 You could just do plus if you wanted to, 406 00:14:01,147 --> 00:14:02,866 and then you do the mass of the ball 407 00:14:02,866 --> 00:14:06,240 times this negative two meters per second, 408 00:14:06,240 --> 00:14:10,800 times the same four meters as the line of closest approach. 409 00:14:10,800 --> 00:14:13,008 You could put the negative here with the plus out here, 410 00:14:13,008 --> 00:14:14,204 or you could put the negative out here 411 00:14:14,204 --> 00:14:15,104 with the plus in here. 412 00:14:15,104 --> 00:14:16,363 You could do it either way. 413 00:14:16,363 --> 00:14:19,737 But this term, for the final angular momentum of the ball 414 00:14:19,737 --> 00:14:21,876 would have to have the opposite sine 415 00:14:21,876 --> 00:14:25,119 as this term for the initial angular momentum of the ball. 416 00:14:25,119 --> 00:14:27,196 So recapping, a ball can have angular momentum 417 00:14:27,196 --> 00:14:29,271 even if it's moving in a straight line, 418 00:14:29,271 --> 00:14:30,322 and you could determine 419 00:14:30,322 --> 00:14:32,110 the angular momentum of that ball, 420 00:14:32,110 --> 00:14:34,523 by using m v r sine theta, 421 00:14:34,523 --> 00:14:36,835 where r is the distance from the axis 422 00:14:36,835 --> 00:14:38,270 to the point where the ball is, 423 00:14:38,270 --> 00:14:41,029 and theta is the angle between r and the velocity. 424 00:14:41,029 --> 00:14:43,325 Or if you don't wanna use r sine theta, 425 00:14:43,325 --> 00:14:45,403 you can use m v capital R, 426 00:14:45,403 --> 00:14:47,002 where this capital R represents 427 00:14:47,002 --> 00:14:49,419 the closest that ball will ever be 428 00:14:49,419 --> 00:00:00,000 to the axis as it's traveling along its straight line.