1 00:00:00,262 --> 00:00:01,351 - [Instructor] I found that for many people 2 00:00:01,351 --> 00:00:05,293 the hardest part about solving a rotational motion problem 3 00:00:05,293 --> 00:00:07,452 is just keeping track of all the new names 4 00:00:07,452 --> 00:00:10,128 for all the rotational quantities that there are. 5 00:00:10,128 --> 00:00:11,870 So in this video, I want to go over all the different 6 00:00:11,870 --> 00:00:14,882 rotational variables like angular displacement, 7 00:00:14,882 --> 00:00:17,563 angular velocity, and angular acceleration. 8 00:00:17,563 --> 00:00:19,869 We'll explain what they mean, how they're defined, 9 00:00:19,869 --> 00:00:22,171 and how you can solve for them, so let's do this. 10 00:00:22,171 --> 00:00:23,417 Let's consider this example. 11 00:00:23,417 --> 00:00:25,836 Say take a tennis ball, you tie a string to it, 12 00:00:25,836 --> 00:00:28,710 and you whirl the tennis ball around in a circle. 13 00:00:28,710 --> 00:00:31,456 If you did this and you wanted to start defining 14 00:00:31,456 --> 00:00:33,855 motion variables that would describe the rotational motion 15 00:00:33,855 --> 00:00:36,538 of this tennis ball, maybe the most basic quantity 16 00:00:36,538 --> 00:00:39,404 you'd come up with would just be how much angle 17 00:00:39,404 --> 00:00:42,672 has this tennis ball swung through during its motion. 18 00:00:42,672 --> 00:00:44,997 So if we imagine the tennis ball starting there 19 00:00:44,997 --> 00:00:47,758 and it rotates over to here, we could define a quantity 20 00:00:47,758 --> 00:00:51,247 that just says how much angle has this thing gone through. 21 00:00:51,247 --> 00:00:54,514 And that would be what's called the angular displacement. 22 00:00:54,514 --> 00:00:57,547 And it's given the symbol delta theta, 23 00:00:57,547 --> 00:01:01,393 because theta is the angle and delta theta is the change 24 00:01:01,393 --> 00:01:02,864 in the angle, so this is really 25 00:01:02,864 --> 00:01:06,182 theta final minus theta initial. 26 00:01:06,182 --> 00:01:09,959 For instance if we started the tennis ball over here at zero 27 00:01:09,959 --> 00:01:13,162 and we ended it at 180, theta final would be 180, 28 00:01:13,162 --> 00:01:15,719 theta initial would be zero, so our angular displacement 29 00:01:15,719 --> 00:01:19,421 would be 180 degrees or pie radians. 30 00:01:19,421 --> 00:01:21,388 And if we started at zero and went through 31 00:01:21,388 --> 00:01:24,727 an entire circle all the way, and then another circle 32 00:01:24,727 --> 00:01:28,153 all the way, our angular displacement wouldn't be zero. 33 00:01:28,153 --> 00:01:30,590 It would technically be two whole revolutions, 34 00:01:30,590 --> 00:01:34,689 which would be either 720 degrees or four pie radians. 35 00:01:34,689 --> 00:01:36,471 And we don't even have to start at the zero. 36 00:01:36,471 --> 00:01:38,620 Our theta initial could be over here at 180, 37 00:01:38,620 --> 00:01:41,055 and we'd go down to 270, in which case the angular 38 00:01:41,055 --> 00:01:45,042 displacement would be 90 degrees or pie over two radians. 39 00:01:45,042 --> 00:01:47,391 So this is how we define the angular displacement 40 00:01:47,391 --> 00:01:49,569 and we typically measure it in radians, 41 00:01:49,569 --> 00:01:52,911 as opposed to degrees for reasons that I'll show you 42 00:01:52,911 --> 00:01:54,002 in just a second. 43 00:01:54,002 --> 00:01:56,869 And the name for this symbol here is theta. 44 00:01:56,869 --> 00:01:58,873 And we should mention that this is analogous 45 00:01:58,873 --> 00:02:01,489 to how we defined the regular displacement, 46 00:02:01,489 --> 00:02:03,009 so if you imagine a tennis ball just going 47 00:02:03,009 --> 00:02:05,320 in a straight line, the regular displacement 48 00:02:05,320 --> 00:02:07,614 was a defined b, the final position 49 00:02:07,614 --> 00:02:11,424 minus the initial positions, which we called delta x. 50 00:02:11,424 --> 00:02:13,714 And that's just usually called the displacement, 51 00:02:13,714 --> 00:02:15,668 which is measured in meters. 52 00:02:15,668 --> 00:02:17,557 Okay, so now we know how to quantify the amount 53 00:02:17,557 --> 00:02:20,365 of angle that this ball has rotated through, 54 00:02:20,365 --> 00:02:22,100 but another quantity that might be useful 55 00:02:22,100 --> 00:02:25,663 is the rate at which it's traveling through that angle. 56 00:02:25,663 --> 00:02:28,668 Just like up here, knowing about the displacement is good, 57 00:02:28,668 --> 00:02:29,828 but you might want to know about 58 00:02:29,828 --> 00:02:31,605 the rate that it's being displaced. 59 00:02:31,605 --> 00:02:34,335 In terms of regular linear quantities that was called 60 00:02:34,335 --> 00:02:36,522 the velocity of the ball, and it was defined 61 00:02:36,522 --> 00:02:39,225 to be the displacement per time. 62 00:02:39,225 --> 00:02:40,936 So down here we'll define a similar quantity, 63 00:02:40,936 --> 00:02:43,390 but it's going to be the angular velocity, 64 00:02:43,390 --> 00:02:46,315 which is defined analogously to the regular velocity. 65 00:02:46,315 --> 00:02:48,784 If regular velocity is displacement per time, 66 00:02:48,784 --> 00:02:51,310 the angular velocity is going to be the angular 67 00:02:51,310 --> 00:02:53,023 displacement per time. 68 00:02:53,023 --> 00:02:55,304 And the symbol we used to represent angular velocity 69 00:02:55,304 --> 00:02:58,424 is the Greek letter omega, which looks like a w, 70 00:02:58,424 --> 00:03:01,214 but it's really the Greek letter omega. 71 00:03:01,214 --> 00:03:03,296 And the units of omega, angular velocity, 72 00:03:03,296 --> 00:03:05,147 are going to be radians per seconds. 73 00:03:05,147 --> 00:03:08,424 Since delta theta, the angular displacement 74 00:03:08,424 --> 00:03:10,727 is in radians, and the time is in seconds. 75 00:03:10,727 --> 00:03:12,908 Just like how regular velocity had units 76 00:03:12,908 --> 00:03:15,945 of meters per second, angular velocity has units 77 00:03:15,945 --> 00:03:17,922 of radians per second. 78 00:03:17,922 --> 00:03:19,660 What is angular velocity mean? 79 00:03:19,660 --> 00:03:21,111 What is this omega? 80 00:03:21,111 --> 00:03:22,979 It represents the rate at which an object 81 00:03:22,979 --> 00:03:25,127 is changing its angle in time. 82 00:03:25,127 --> 00:03:27,086 So let's say the tennis ball starts here, 83 00:03:27,086 --> 00:03:30,660 and it's going through a circle at this leisurely rate, 84 00:03:30,660 --> 00:03:33,036 that means the rate at which it's changing its angle 85 00:03:33,036 --> 00:03:36,072 is very small and it has a very small omega. 86 00:03:36,072 --> 00:03:38,678 Whereas if you had this tennis ball going through a circle 87 00:03:38,678 --> 00:03:41,358 very fast, the rate at which it's going in a circle 88 00:03:41,358 --> 00:03:45,170 would be large and that means the angular velocity 89 00:03:45,170 --> 00:03:46,862 and omega would also be large. 90 00:03:46,862 --> 00:03:49,674 So the velocity and the angular velocity are related, 91 00:03:49,674 --> 00:03:51,911 they're not equal because the velocity gives you 92 00:03:51,911 --> 00:03:54,223 how many meters per second something is going through, 93 00:03:54,223 --> 00:03:55,790 and the angular velocity gives you how many 94 00:03:55,790 --> 00:03:57,812 radians per second it's going through, 95 00:03:57,812 --> 00:04:00,041 but if it's got a larger angular velocity, 96 00:04:00,041 --> 00:04:01,971 it's going to have a larger velocity as well. 97 00:04:01,971 --> 00:04:03,941 And just like velocity is a vector, 98 00:04:03,941 --> 00:04:06,452 angular velocity is also a vector, 99 00:04:06,452 --> 00:04:08,468 so I'll put an arrow over this omega. 100 00:04:08,468 --> 00:04:09,593 Which way does it point? 101 00:04:09,593 --> 00:04:11,974 Technically speaking, you'd use the same right hand rule 102 00:04:11,974 --> 00:04:13,628 you use to determine the direction 103 00:04:13,628 --> 00:04:15,229 of the angular displacement. 104 00:04:15,229 --> 00:04:17,683 But again if it's rotating counter clockwise, 105 00:04:17,683 --> 00:04:19,903 we can just consider that to be positive, 106 00:04:19,903 --> 00:04:21,969 and if it's rotating clockwise, we can consider 107 00:04:21,969 --> 00:04:25,109 that to be a negative omega, or a negative angular velocity. 108 00:04:25,109 --> 00:04:27,122 So let me get rid of these, and let's define 109 00:04:27,122 --> 00:04:29,914 our last angular motion variable. 110 00:04:29,914 --> 00:04:31,399 You can probably guess what it is. 111 00:04:31,399 --> 00:04:33,593 There's regular displacement 112 00:04:33,593 --> 00:04:34,785 and there's angular displacement. 113 00:04:34,785 --> 00:04:37,170 There's regular velocity and there's angular velocity. 114 00:04:37,170 --> 00:04:39,708 And then the next logical step in this motion 115 00:04:39,708 --> 00:04:42,476 variable sequence would be the acceleration, 116 00:04:42,476 --> 00:04:45,157 which was defined for regular variables to be 117 00:04:45,157 --> 00:04:47,990 the change in velocity over the change in time. 118 00:04:47,990 --> 00:04:50,447 So we'll define an analogous angular quantity 119 00:04:50,447 --> 00:04:53,002 that would be the angular acceleration. 120 00:04:53,002 --> 00:04:54,588 And it's going to be defined to be, instead of 121 00:04:54,588 --> 00:04:57,106 change in velocity over change in time, it's going to 122 00:04:57,106 --> 00:04:59,490 be the change in the angular velocity 123 00:04:59,490 --> 00:05:00,951 over the change in time. 124 00:05:00,951 --> 00:05:03,606 And the letter we use to denote angular acceleration 125 00:05:03,606 --> 00:05:05,603 is this Greek letter alpha, 126 00:05:05,603 --> 00:05:07,239 so this is the Greek letter alpha. 127 00:05:07,239 --> 00:05:09,567 It looks like a little fishy, and that represents 128 00:05:09,567 --> 00:05:12,262 the angular acceleration of an object. 129 00:05:12,262 --> 00:05:14,964 So what does this angular acceleration mean? 130 00:05:14,964 --> 00:05:17,401 Well, looking at the units, helps us to figure this out, 131 00:05:17,401 --> 00:05:20,811 so the units of regular acceleration were meters per second 132 00:05:20,811 --> 00:05:23,561 per second, so regular acceleration represented 133 00:05:23,561 --> 00:05:26,415 the rate at which the velocity is changing, 134 00:05:26,415 --> 00:05:28,619 and that's the same analogous definition down here. 135 00:05:28,619 --> 00:05:31,699 The units down here are going to be radians per second 136 00:05:31,699 --> 00:05:33,527 per second, so this is going to represent 137 00:05:33,527 --> 00:05:35,829 this angular acceleration is going to represent 138 00:05:35,829 --> 00:05:39,732 the rate at which the angular velocity is changing. 139 00:05:39,732 --> 00:05:40,986 What would that look like? 140 00:05:40,986 --> 00:05:43,828 Well if we've got this ball rotating in a circle, 141 00:05:43,828 --> 00:05:46,149 if it's rotating at a constant rate, 142 00:05:46,149 --> 00:05:50,026 there's no angular acceleration since the omega, 143 00:05:50,026 --> 00:05:52,878 the angular velocity wouldn't be changing. 144 00:05:52,878 --> 00:05:55,672 So in other words, if it's rotating at a constant rate, 145 00:05:55,672 --> 00:05:58,520 there's no change in the angular velocity, 146 00:05:58,520 --> 00:06:01,786 and that means there's no angular acceleration. 147 00:06:01,786 --> 00:06:05,223 But conversely, if it starts off moving slowly, 148 00:06:05,223 --> 00:06:09,183 and it speeds up its angular velocity is increasing, 149 00:06:09,183 --> 00:06:11,828 then there is an angular acceleration because 150 00:06:11,828 --> 00:06:15,146 there's a change in the angular velocity of this ball. 151 00:06:15,146 --> 00:06:18,013 And just like any acceleration, this angular acceleration 152 00:06:18,013 --> 00:06:22,154 can increase the angular velocity and speed something up. 153 00:06:22,154 --> 00:06:24,811 Or it can slow the object down and decrease 154 00:06:24,811 --> 00:06:26,159 the angular velocity. 155 00:06:26,159 --> 00:06:28,891 But if the angular velocity is remaining constant, 156 00:06:28,891 --> 00:06:31,789 in other words it's rotating in a circle at a constant rate, 157 00:06:31,789 --> 00:06:34,291 then the angular acceleration is zero and that means 158 00:06:34,291 --> 00:06:36,333 alpha equals zero. 159 00:06:36,333 --> 00:06:38,477 And just like the rest of these motion variables, 160 00:06:38,477 --> 00:06:41,351 angular acceleration is a vector, 161 00:06:41,351 --> 00:06:43,661 just like regular acceleration is a vector. 162 00:06:43,661 --> 00:06:46,522 And the direction that the angular acceleration points 163 00:06:46,522 --> 00:06:48,787 will be in the direction of the change 164 00:06:48,787 --> 00:06:50,290 in the angular velocity. 165 00:06:50,290 --> 00:06:53,211 So in other words, if this tennis ball is speeding up, 166 00:06:53,211 --> 00:06:55,022 then the angular acceleration is pointed 167 00:06:55,022 --> 00:06:57,944 in the same direction as the angular velocity. 168 00:06:57,944 --> 00:07:00,915 And if the angular velocity is slowing down, 169 00:07:00,915 --> 00:07:03,437 the angular acceleration points in the opposite direction 170 00:07:03,437 --> 00:07:05,442 to the angular velocity. 171 00:07:05,442 --> 00:07:07,742 At this point, I wouldn't blame you if you weren't like 172 00:07:07,742 --> 00:07:10,112 why, why do we need to define all these new 173 00:07:10,112 --> 00:07:12,817 angular variables when we already had all these 174 00:07:12,817 --> 00:07:14,352 regular variables up here. 175 00:07:14,352 --> 00:07:16,801 And the answer is that it's the same reason we define 176 00:07:16,801 --> 00:07:18,715 most variables in physics, because it turns out 177 00:07:18,715 --> 00:07:21,229 to be really convenient to do so, 178 00:07:21,229 --> 00:07:22,717 and these angular variables are going to be 179 00:07:22,717 --> 00:07:25,129 way more convenient to describe an object 180 00:07:25,129 --> 00:07:27,884 that's rotating than these regular variables. 181 00:07:27,884 --> 00:07:30,711 For this reason, imagine you wanted to describe 182 00:07:30,711 --> 00:07:32,760 not just the ball on the end of the string, 183 00:07:32,760 --> 00:07:34,962 but all points on the string as well. 184 00:07:34,962 --> 00:07:37,128 If you limited yourself to only these regular 185 00:07:37,128 --> 00:07:40,097 motion variables, you'd run into a problem. 186 00:07:40,097 --> 00:07:42,629 You'd realize that his ball goes through a circle 187 00:07:42,629 --> 00:07:44,880 in a certain amount of time, but every point 188 00:07:44,880 --> 00:07:48,115 on this string also goes through a circumference 189 00:07:48,115 --> 00:07:51,708 in that same amount of time, so the velocity of the ball 190 00:07:51,708 --> 00:07:54,333 is going to be greater than the velocity of points 191 00:07:54,333 --> 00:07:56,775 on the string that are closer to the center. 192 00:07:56,775 --> 00:07:58,795 Because everything's taking the same amount of time 193 00:07:58,795 --> 00:08:01,154 to go through one circle, but the circle the ball 194 00:08:01,154 --> 00:08:03,042 goes through has a larger circumference 195 00:08:03,042 --> 00:08:06,061 than the circle that points nearer to the center do. 196 00:08:06,061 --> 00:08:08,092 And so all points on this string are going to have 197 00:08:08,092 --> 00:08:09,747 a different velocity the closer you get 198 00:08:09,747 --> 00:08:11,275 to the center of the string. 199 00:08:11,275 --> 00:08:13,974 So trying to describe its motion with just velocity 200 00:08:13,974 --> 00:08:16,963 might be a nightmare, whereas if you were just going to use 201 00:08:16,963 --> 00:08:20,657 angular velocity, note that every point on the string, 202 00:08:20,657 --> 00:08:23,265 including the ball moved through the same amount 203 00:08:23,265 --> 00:08:25,842 of angle in the same amount of time. 204 00:08:25,842 --> 00:08:26,997 They don't move through the same amount 205 00:08:26,997 --> 00:08:29,858 of meters per second, but they do move through 206 00:08:29,858 --> 00:08:32,077 the same amount of radians per second 207 00:08:32,077 --> 00:08:35,260 because when the ball has rotated through two pie radians, 208 00:08:35,260 --> 00:08:37,760 once full circle, every point on this string 209 00:08:37,760 --> 00:08:40,229 has rotated through two pie radians. 210 00:08:40,229 --> 00:08:42,011 If this ball and string are going to maintain 211 00:08:42,011 --> 00:08:43,340 the same shape. 212 00:08:43,341 --> 00:08:45,613 So that's the great thing about these angular motion 213 00:08:45,613 --> 00:08:49,022 variables, every point on a rigid object is going 214 00:08:49,022 --> 00:08:51,473 to have the same angular displacement, 215 00:08:51,473 --> 00:08:53,118 the same angular velocity, 216 00:08:53,118 --> 00:08:55,054 and the same angular acceleration. 217 00:08:55,054 --> 00:08:57,305 It won't matter what point you're talking about. 218 00:08:57,305 --> 00:08:59,210 The angular displacement, angular velocity, 219 00:08:59,210 --> 00:09:01,238 and angular acceleration will be the same 220 00:09:01,238 --> 00:09:04,215 for every point on that rotating object. 221 00:09:04,215 --> 00:09:05,744 Alright, so before this gets too abstract, 222 00:09:05,744 --> 00:09:07,243 let's try a sample problem. 223 00:09:07,243 --> 00:09:09,879 Let's say that the ball starts over here at rest, 224 00:09:09,879 --> 00:09:13,592 and it rotates all the way to this point in four seconds. 225 00:09:13,592 --> 00:09:16,141 So it started over here at rest, and it took 226 00:09:16,141 --> 00:09:19,170 four seconds for it to rotate over to this point. 227 00:09:19,170 --> 00:09:21,053 And let's say when the ball makes it over to this side, 228 00:09:21,053 --> 00:09:23,970 it's going 1.57 radians per second. 229 00:09:25,565 --> 00:09:28,341 Let's say that's the final angular velocity. 230 00:09:28,341 --> 00:09:30,144 So let's just go through and try to figure these out. 231 00:09:30,144 --> 00:09:32,892 What would the angular displacement be for this example? 232 00:09:32,892 --> 00:09:36,060 Well if the ball started here and it made it over to here, 233 00:09:36,060 --> 00:09:38,573 the angular displacement would be pie radians, 234 00:09:38,573 --> 00:09:40,057 or 180 degrees. 235 00:09:40,057 --> 00:09:42,158 What would the angular velocity be? 236 00:09:42,158 --> 00:09:45,752 Well it started at rest, so initially omega was zero 237 00:09:45,752 --> 00:09:48,360 at this point here, and then finally it tells us 238 00:09:48,360 --> 00:09:51,346 what the final omega would be, 1.57. 239 00:09:51,346 --> 00:09:53,734 So you might wonder, what would we do with this formula? 240 00:09:53,734 --> 00:09:55,845 What if we just used this formula, what would we get? 241 00:09:55,845 --> 00:09:57,262 Well if we used that formula there, 242 00:09:57,262 --> 00:09:59,582 we're going to get that it went pie radians, 243 00:09:59,582 --> 00:10:01,499 and it did it in four seconds, 244 00:10:01,499 --> 00:10:05,662 which gives us 0.785 radians per second. 245 00:10:05,662 --> 00:10:07,038 So you might be like wait a minute, 246 00:10:07,038 --> 00:10:09,564 this omega doesn't correspond to the initial omega 247 00:10:09,564 --> 00:10:11,327 or the final omega. 248 00:10:11,327 --> 00:10:13,217 What is this corresponding to? 249 00:10:13,217 --> 00:10:15,367 Well this would be the average omega. 250 00:10:15,367 --> 00:10:17,884 This is the average angular velocity 251 00:10:17,884 --> 00:10:19,696 between this initial and final point. 252 00:10:19,696 --> 00:10:22,265 This at rest initially shows that the omega 253 00:10:22,265 --> 00:10:23,779 started off as zero. 254 00:10:23,779 --> 00:10:26,413 The instantaneous omega was zero, 255 00:10:26,413 --> 00:10:29,787 and the instantaneous omega, or the final angular velocity 256 00:10:29,787 --> 00:10:32,745 would be 1.57, so you got to be careful. 257 00:10:32,745 --> 00:10:35,738 The instantaneous values are not necessarily equal 258 00:10:35,738 --> 00:10:37,586 to the average value. 259 00:10:37,586 --> 00:10:39,626 You can get the average value by taking the change 260 00:10:39,626 --> 00:10:42,293 in theta over the change in time, but it doesn't necessarily 261 00:10:42,293 --> 00:10:44,919 give you the instantaneous angular velocity 262 00:10:44,919 --> 00:10:47,082 at a specific point on the trip. 263 00:10:47,082 --> 00:10:49,417 And we can find the angular acceleration as well 264 00:10:49,417 --> 00:10:51,548 if we use this formula. 265 00:10:51,548 --> 00:10:53,800 The change in omega over the change in time. 266 00:10:53,800 --> 00:10:55,618 That would be the angular acceleration, 267 00:10:55,618 --> 00:10:59,690 our omega final minus omega initial over time 268 00:10:59,690 --> 00:11:04,141 would come out to be 1.57 as our final angular velocity 269 00:11:04,141 --> 00:11:07,027 minus zero was our initial angular velocity, 270 00:11:07,027 --> 00:11:09,291 and that took four seconds to accomplish, 271 00:11:09,291 --> 00:11:11,365 so our angular acceleration would come out to 272 00:11:11,365 --> 00:11:14,615 be 0.393 radians per second per second, 273 00:11:17,014 --> 00:11:20,116 or you can write that as radians per second squared. 274 00:11:20,116 --> 00:11:22,740 Now technically that is also the average 275 00:11:22,740 --> 00:11:24,364 angular acceleration during this trip, 276 00:11:24,364 --> 00:11:26,842 but if the angular acceleration was constant 277 00:11:26,842 --> 00:11:29,389 during this trip, which in almost all cases 278 00:11:29,389 --> 00:11:32,638 we're going to look at, the angular acceleration 279 00:11:32,638 --> 00:11:33,471 is going to be constant. 280 00:11:33,471 --> 00:11:34,739 If that's the case, this would be both 281 00:11:34,739 --> 00:11:37,886 the average value and the instantaneous value 282 00:11:37,886 --> 00:11:41,072 of the angular acceleration at every point on the trip 283 00:11:41,072 --> 00:11:44,406 since the angular acceleration would be remaining constant. 284 00:11:44,406 --> 00:11:46,266 So in this example, we can say that the angular 285 00:11:46,266 --> 00:11:48,756 displacement was pie radians. 286 00:11:48,756 --> 00:11:53,146 The average angular velocity was 0.785 radians per second. 287 00:11:53,146 --> 00:11:56,194 The initial angular velocity was zero. 288 00:11:56,194 --> 00:11:59,227 The final angular velocity was 1.57, 289 00:11:59,227 --> 00:12:01,780 and the angular acceleration was 290 00:12:01,780 --> 00:12:05,032 0.393 radians per second squared. 291 00:12:05,032 --> 00:12:07,297 So recapping, the angular displacement represents 292 00:12:07,297 --> 00:12:10,263 the angle through which an object is rotated. 293 00:12:10,263 --> 00:12:11,985 It's typically measured in radians, 294 00:12:11,985 --> 00:12:14,503 and it's represented with a delta theta. 295 00:12:14,503 --> 00:12:17,148 The angular velocity represents the rate 296 00:12:17,148 --> 00:12:19,000 at which an object is rotating. 297 00:12:19,000 --> 00:12:20,667 It's measured in radians per second, 298 00:12:20,667 --> 00:12:23,080 and it's represented with a Greek letter omega. 299 00:12:23,080 --> 00:12:25,404 And the angular acceleration represents the rate 300 00:12:25,404 --> 00:12:28,891 at which an object is changing its angular velocity, 301 00:12:28,891 --> 00:12:31,131 so if an object rotates at a constant rate, 302 00:12:31,131 --> 00:12:33,644 there is zero angular acceleration, 303 00:12:33,644 --> 00:12:36,528 but conversely, if an object's rotation is speeding up 304 00:12:36,528 --> 00:12:40,729 or slowing down, there must be angular acceleration. 305 00:12:40,729 --> 00:12:43,679 It's measured in units of radians per second per second 306 00:12:43,679 --> 00:12:45,511 or radians per second squared, 307 00:12:45,511 --> 00:00:00,000 and it's represented with the Greek letter alpha.