1 00:00:00,636 --> 00:00:02,001 - [Instructor] So in the previous couple videos, 2 00:00:02,001 --> 00:00:05,203 we defined all these new rotational motion variables 3 00:00:05,203 --> 00:00:07,825 and we defined them exactly the same way we defined 4 00:00:07,825 --> 00:00:09,886 all these linear motion variables. 5 00:00:09,886 --> 00:00:12,307 So for instance, this angular displacement 6 00:00:12,307 --> 00:00:14,967 was defined the exact same way we defined 7 00:00:14,967 --> 00:00:18,374 regular displacement, it's just this is the angular position 8 00:00:18,374 --> 00:00:21,943 as opposed to the position, the regular position. 9 00:00:21,943 --> 00:00:24,255 Similarly this angular velocity was the 10 00:00:24,255 --> 00:00:27,088 angular displacement per time just like velocity 11 00:00:27,088 --> 00:00:29,317 was the regular displacement over time. 12 00:00:29,317 --> 00:00:31,604 And the angular acceleration was the change in the 13 00:00:31,604 --> 00:00:35,036 angular velocity per time, just like regular acceleration 14 00:00:35,036 --> 00:00:37,906 was the change in regular velocity per time. 15 00:00:37,906 --> 00:00:40,678 And so because these definitions are exactly the same 16 00:00:40,678 --> 00:00:43,230 except for the fact that the linear motion variable 17 00:00:43,230 --> 00:00:45,534 is replaced with its angular counterpart, 18 00:00:45,534 --> 00:00:49,362 all the equations results in principles we found and derived 19 00:00:49,362 --> 00:00:52,816 for the linear motion variables will also hold true 20 00:00:52,816 --> 00:00:56,050 for the rotational motions variables as long as you replace 21 00:00:56,050 --> 00:00:59,151 the linear motion variable in that equation with its 22 00:00:59,151 --> 00:01:01,519 rotational motion variable counterpart. 23 00:01:01,519 --> 00:01:03,043 And it even works with graphs. 24 00:01:03,043 --> 00:01:05,308 So let's say you had a velocity versus time graph 25 00:01:05,308 --> 00:01:06,805 and it looked like this. 26 00:01:06,805 --> 00:01:10,121 Since we already know from 1D motion that the slope of this 27 00:01:10,121 --> 00:01:13,641 velocity versus time graph is equal to the acceleration, 28 00:01:13,641 --> 00:01:16,853 that means on an angular velocity versus time graph, 29 00:01:16,853 --> 00:01:20,245 the slope is going to represent the angular acceleration, 30 00:01:20,245 --> 00:01:22,632 because the relationship between omega and alpha 31 00:01:22,632 --> 00:01:25,954 is the same as the relationship between v and a. 32 00:01:25,954 --> 00:01:28,237 Similarly the area underneath the curve 33 00:01:28,237 --> 00:01:30,443 on a velocity versus time graph 34 00:01:30,443 --> 00:01:32,242 represented the displacement. 35 00:01:32,242 --> 00:01:34,455 So that means that the area under the curve on 36 00:01:34,455 --> 00:01:37,080 a omega versus time graph, an angular velocity versus 37 00:01:37,080 --> 00:01:40,723 time graph is gonna represent the angular displacement. 38 00:01:40,723 --> 00:01:43,523 And so if you remember from 1D motion, the way we 39 00:01:43,523 --> 00:01:46,791 derived a lot of the 1D kinematic formulas that related 40 00:01:46,791 --> 00:01:49,761 these linear motion variables, was by looking for areas 41 00:01:49,761 --> 00:01:51,297 under a velocity graph. 42 00:01:51,297 --> 00:01:52,834 We could do the same thing 43 00:01:52,834 --> 00:01:54,260 for the rotational motion variables. 44 00:01:54,260 --> 00:01:56,952 We could find this area, relate it to omega and alpha 45 00:01:56,952 --> 00:01:59,331 and we'd get the rotational kinematic formulas, 46 00:01:59,331 --> 00:02:01,430 but we already know since these are all defined 47 00:02:01,430 --> 00:02:04,122 the same way the linear motion variables are defined, 48 00:02:04,122 --> 00:02:07,243 we're gonna get the exact same equations, just with 49 00:02:07,243 --> 00:02:09,892 the linear motion variable replaced with its rotational 50 00:02:09,892 --> 00:02:10,920 motion variable. 51 00:02:10,920 --> 00:02:12,028 So let's right those down. 52 00:02:12,028 --> 00:02:15,026 First we'll right down the linear motion kinematic formulas. 53 00:02:15,026 --> 00:02:17,136 If you remember they looked like this. 54 00:02:17,136 --> 00:02:18,316 So there they are. 55 00:02:18,316 --> 00:02:20,786 These are the four kinematic formulas that relate 56 00:02:20,786 --> 00:02:22,273 the linear motion variables. 57 00:02:22,273 --> 00:02:25,663 But remember this only works, these equations only work 58 00:02:25,663 --> 00:02:28,609 if the acceleration is constant. 59 00:02:28,609 --> 00:02:30,183 But if the acceleration is constant, 60 00:02:30,183 --> 00:02:33,545 these four kinematic formulas are a convenient way to relate 61 00:02:33,545 --> 00:02:36,183 all these kinematic linear motion variables. 62 00:02:36,183 --> 00:02:38,737 Now if you wanted rotational kinematic formulas, 63 00:02:38,737 --> 00:02:40,948 you could go though the trouble that we went through 64 00:02:40,948 --> 00:02:43,558 with these to derive them using areas under curves, 65 00:02:43,558 --> 00:02:46,437 but since we know the relationship between all these 66 00:02:46,437 --> 00:02:48,922 rotational motion variables is the same as the 67 00:02:48,922 --> 00:02:51,229 relationship between the linear motion variables, 68 00:02:51,229 --> 00:02:53,789 I can make rotational motion kinematic formulas 69 00:02:53,789 --> 00:02:57,118 simply by replacing all of these linear variables 70 00:02:57,118 --> 00:03:00,369 with their rotational motion variable counterparts. 71 00:03:00,369 --> 00:03:01,202 So let's do that. 72 00:03:01,202 --> 00:03:03,634 So in other words, instead of V, the velocity, 73 00:03:03,634 --> 00:03:05,856 the final velocity, I would have omega, 74 00:03:05,856 --> 00:03:07,649 the final angular velocity. 75 00:03:07,649 --> 00:03:10,771 Instead of V initial, the initial velocity, 76 00:03:10,771 --> 00:03:13,279 I'd have the initial angular velocity. 77 00:03:13,279 --> 00:03:17,347 Instead of acceleration, I'd have the angular acceleration. 78 00:03:17,347 --> 00:03:19,223 And time is just time. 79 00:03:19,223 --> 00:03:22,164 So there's no such thing as angular time or linear time. 80 00:03:22,164 --> 00:03:25,530 As far as we know, there's only one time and that's t 81 00:03:25,530 --> 00:03:27,236 and that works in either equation. 82 00:03:27,236 --> 00:03:29,449 So you could probably guess, when are these rotational 83 00:03:29,449 --> 00:03:31,610 motion kinematic formulas gonna be true? 84 00:03:31,610 --> 00:03:33,285 It's gonna be when the alpha, 85 00:03:33,285 --> 00:03:35,437 the angular acceleration is constant. 86 00:03:35,437 --> 00:03:36,871 And so you can keep goin' through, 87 00:03:36,871 --> 00:03:39,518 wherever you had an x, that was the regular position, 88 00:03:39,518 --> 00:03:42,295 you'd replace it with theta, the angular position. 89 00:03:42,295 --> 00:03:44,742 So I'll replace all these x's with thetas. 90 00:03:44,742 --> 00:03:46,907 We replace all of our accelerations 91 00:03:46,907 --> 00:03:48,546 with angular accelerations. 92 00:03:48,546 --> 00:03:50,273 And then I'll finish cleaning up 93 00:03:50,273 --> 00:03:52,836 these v initial and v finals. 94 00:03:52,836 --> 00:03:54,127 And then we've got 'em. 95 00:03:54,127 --> 00:03:56,580 These are the rotational kinematic formulas. 96 00:03:56,580 --> 00:03:59,604 The are only true if the angular acceleration is constant, 97 00:03:59,604 --> 00:04:02,847 but if it is constant, these are a convenient way 98 00:04:02,847 --> 00:04:05,236 to relate all these rotational motion variables 99 00:04:05,236 --> 00:04:07,103 and you can solve a ton a problems 100 00:04:07,103 --> 00:04:10,632 using these rotational kinematic formulas. 101 00:04:10,632 --> 00:04:13,761 And in fact, you use these, the exact same way 102 00:04:13,761 --> 00:04:16,231 you used these regular kinematic formulas. 103 00:04:16,231 --> 00:04:18,611 You identify the variables that you know. 104 00:04:18,611 --> 00:04:20,951 You identify the variable that you wanna find 105 00:04:20,951 --> 00:04:23,736 and you use one of the formulas that lets you solve 106 00:04:23,736 --> 00:04:25,472 for that unknown variable. 107 00:04:25,472 --> 00:04:26,691 So let me show you some examples. 108 00:04:26,691 --> 00:04:29,180 Let's do a couple examples using these formulas, 109 00:04:29,180 --> 00:04:31,846 cause it takes a while before you get the swing of 'em. 110 00:04:31,846 --> 00:04:32,895 So let me copy these. 111 00:04:32,895 --> 00:04:34,266 We're about to use these. 112 00:04:34,266 --> 00:04:36,365 And let's tackle a couple examples 113 00:04:36,365 --> 00:04:39,146 of rotational kinematic formula problems. 114 00:04:39,146 --> 00:04:42,172 So let me get rid of all this and let's tackle this problem. 115 00:04:42,172 --> 00:04:44,099 Let's say you had a four meter long bar, 116 00:04:44,099 --> 00:04:46,110 that's why I've had this bar here the whole time, 117 00:04:46,110 --> 00:04:47,210 to show that it can rotate. 118 00:04:47,210 --> 00:04:50,461 It starts from rest and it rotates through five 119 00:04:50,461 --> 00:04:53,637 revolutions with a constant angular acceleration 120 00:04:53,637 --> 00:04:56,767 of 30 radians per second squared. 121 00:04:56,767 --> 00:05:00,331 And the question is, how long did it take for this bar 122 00:05:00,331 --> 00:05:02,263 to make the five revolutions? 123 00:05:02,263 --> 00:05:03,406 So what do we do? 124 00:05:03,406 --> 00:05:04,941 How do we tackle these problems? 125 00:05:04,941 --> 00:05:07,551 You first identify all the variables that you know. 126 00:05:07,551 --> 00:05:10,379 So it said that it revolved five revolutions, 127 00:05:10,379 --> 00:05:13,398 that's the amount of angle that it's gone through, 128 00:05:13,398 --> 00:05:14,489 but it's in weird units. 129 00:05:14,489 --> 00:05:16,048 This is in units for revolutions. 130 00:05:16,048 --> 00:05:18,008 So we know what the delta theta is, 131 00:05:18,008 --> 00:05:19,244 five revolutions. 132 00:05:19,244 --> 00:05:22,231 But we want our delta theta always to be in radians, 133 00:05:22,231 --> 00:05:23,618 cause look it, our acceleration 134 00:05:23,618 --> 00:05:25,569 was given in radians per seconds squared. 135 00:05:25,569 --> 00:05:28,252 You've gotta make sure you compare apples to apples. 136 00:05:28,252 --> 00:05:30,247 I can't have revolutions for delta theta 137 00:05:30,247 --> 00:05:32,154 and radians for acceleration. 138 00:05:32,154 --> 00:05:34,289 You've gotta pick one unit to go with 139 00:05:34,289 --> 00:05:36,873 and the unit we typically go with is radians. 140 00:05:36,873 --> 00:05:39,910 So how many radians would five revolutions be? 141 00:05:39,910 --> 00:05:43,659 One revolution is two pi radians, cause one time around 142 00:05:43,659 --> 00:05:45,907 the entire circle is two pi radians. 143 00:05:45,907 --> 00:05:48,192 That means that five revolutions would be 144 00:05:48,192 --> 00:05:51,833 five times two pi radians, 145 00:05:51,833 --> 00:05:54,203 which gives us 10 pi radians. 146 00:05:54,203 --> 00:05:57,445 So we've got our angular displacement, what else do we know? 147 00:05:57,445 --> 00:06:00,415 It tells us this 30 radians per second squared. 148 00:06:00,415 --> 00:06:02,989 That is the angular acceleration. 149 00:06:02,989 --> 00:06:06,897 So we know that alpha is 30 radians per second squared. 150 00:06:06,897 --> 00:06:09,602 You can write the radian, you can leave it off. 151 00:06:09,602 --> 00:06:11,284 Sometimes people write the radian, 152 00:06:11,284 --> 00:06:12,509 sometimes they leave it blank. 153 00:06:12,509 --> 00:06:14,876 So you can write one over second squared if you wanted to. 154 00:06:14,876 --> 00:06:16,428 That's why I left this blank over here, 155 00:06:16,428 --> 00:06:18,473 but we could write radians if we wanted to. 156 00:06:18,473 --> 00:06:20,383 And should this be alpha be positive or negative? 157 00:06:20,383 --> 00:06:22,481 Well, since this object is speeding up, 158 00:06:22,481 --> 00:06:24,851 it started from rest, that means it sped up. 159 00:06:24,851 --> 00:06:27,175 So our direction of the angular displacement, 160 00:06:27,175 --> 00:06:30,404 has to be the same direction as this angular acceleration. 161 00:06:30,404 --> 00:06:32,579 In other words, if something's speeding up, 162 00:06:32,579 --> 00:06:34,862 you have to make sure that your angular acceleration 163 00:06:34,862 --> 00:06:37,281 has the same sign as your angular velocity, 164 00:06:37,281 --> 00:06:39,965 and your angular velocity'll have the same sign 165 00:06:39,965 --> 00:06:41,234 as your angular displacement. 166 00:06:41,234 --> 00:06:43,652 So since we called this positive 10 pi radians 167 00:06:43,652 --> 00:06:46,354 and the object sped up, we're gonna call this positive 168 00:06:46,354 --> 00:06:48,296 30 radians per second squared. 169 00:06:48,296 --> 00:06:51,751 If this bar would have slowed down, we'd of had to make sure 170 00:06:51,751 --> 00:06:54,631 that this alpha has the opposite sign 171 00:06:54,631 --> 00:06:55,735 as our angular velocity. 172 00:06:55,735 --> 00:06:58,337 But that's only two rotational kinematic variables. 173 00:06:58,337 --> 00:07:01,885 You always need three in order to solve for a fourth. 174 00:07:01,885 --> 00:07:04,930 So what's our third rotational kinematic variable? 175 00:07:04,930 --> 00:07:06,092 It's this. 176 00:07:06,092 --> 00:07:08,135 That it says the object started from rest. 177 00:07:08,135 --> 00:07:10,019 So this is code. 178 00:07:10,019 --> 00:07:13,550 This is code word for omega initial is zero. 179 00:07:13,550 --> 00:07:17,484 Initial angular velocity is zero, cause it starts from rest. 180 00:07:17,484 --> 00:07:19,067 So that's what we can say down here. 181 00:07:19,067 --> 00:07:21,269 That's our third known variable. 182 00:07:21,269 --> 00:07:22,410 And now we can solve. 183 00:07:22,410 --> 00:07:24,510 We've got three, we can solve for a fourth. 184 00:07:24,510 --> 00:07:25,929 Which one do we want to solve for? 185 00:07:25,929 --> 00:07:28,586 It says how long, so that's the time. 186 00:07:28,586 --> 00:07:30,425 We wanna know the time that it took. 187 00:07:30,425 --> 00:07:32,442 All right so these are the variables that are involved. 188 00:07:32,442 --> 00:07:33,789 We wanna know the time. 189 00:07:33,789 --> 00:07:35,340 We know the top three. 190 00:07:35,340 --> 00:07:37,820 The way I figure out what kinematic formula to use 191 00:07:37,820 --> 00:07:41,354 is that I just ask which variable's left out of all these? 192 00:07:41,354 --> 00:07:43,012 I've got my three knowns and my one unknown 193 00:07:43,012 --> 00:07:44,813 that I want to find. 194 00:07:44,813 --> 00:07:46,188 Which variable isn't involved? 195 00:07:46,188 --> 00:07:47,991 And it's omega final. 196 00:07:47,991 --> 00:07:50,664 So omega final is not involved here at all. 197 00:07:50,664 --> 00:07:53,362 So I'm gonna use the rotational kinematic formula 198 00:07:53,362 --> 00:07:56,045 that does not involve omega final. 199 00:07:56,045 --> 00:07:57,152 I'll put these over here. 200 00:07:57,152 --> 00:07:58,255 So I'll look through 'em. 201 00:07:58,255 --> 00:07:59,595 First one's got omega final. 202 00:07:59,595 --> 00:08:01,006 I don't wanna use that one 203 00:08:01,006 --> 00:08:02,276 cause I wouldn't know what to plug in here 204 00:08:02,276 --> 00:08:03,468 and I don't wanna solve for it anyway. 205 00:08:03,468 --> 00:08:04,601 I don't want the second one. 206 00:08:04,601 --> 00:08:07,238 This third one has no omega final, 207 00:08:07,238 --> 00:08:08,799 so I'm gonna use that one. 208 00:08:08,799 --> 00:08:11,119 So let's just take this, we'll put it over here. 209 00:08:11,119 --> 00:08:12,207 So we know delta theta. 210 00:08:12,207 --> 00:08:14,677 Delta theta was 10 pi radians. 211 00:08:14,677 --> 00:08:17,572 And we know omega initial was zero. 212 00:08:17,572 --> 00:08:19,183 So this whole term is zero. 213 00:08:19,183 --> 00:08:22,480 Zero times t is still zero, so that's all zero. 214 00:08:22,480 --> 00:08:24,124 And we have 1/2. 215 00:08:24,124 --> 00:08:25,722 The angular acceleration was 30 216 00:08:25,722 --> 00:08:28,366 and the time is what we wanna know. 217 00:08:28,366 --> 00:08:29,846 And you can't for get that that's squared. 218 00:08:29,846 --> 00:08:32,251 So now we just solve this algebraically for time. 219 00:08:32,251 --> 00:08:34,274 We multiply both side by two. 220 00:08:34,274 --> 00:08:35,852 That would give us 20 pi. 221 00:08:35,852 --> 00:08:37,299 Then we divide by 30. 222 00:08:37,299 --> 00:08:39,662 And that'll end up giving us 20 pi, 223 00:08:39,663 --> 00:08:42,520 and technically that is 20 pi radians 224 00:08:42,520 --> 00:08:45,853 divided by 30 radians per second squared 225 00:08:46,997 --> 00:08:48,645 and then you have to take the square root, 226 00:08:48,645 --> 00:08:50,168 because it's t squared. 227 00:08:50,168 --> 00:08:51,849 And if you solve all this for t, 228 00:08:51,849 --> 00:08:56,047 I get that the time ended up taking about 1.45 seconds. 229 00:08:56,047 --> 00:08:58,312 And our units all canceled out the way should here. 230 00:08:58,312 --> 00:08:59,759 Radians canceled radians. 231 00:08:59,759 --> 00:09:01,718 You ended up with seconds squared on the top. 232 00:09:01,718 --> 00:09:03,254 You too the square root. 233 00:09:03,254 --> 00:09:04,750 That gives you seconds to end with. 234 00:09:04,750 --> 00:09:06,325 Now this second part, part b, says 235 00:09:06,325 --> 00:09:09,390 what was the angular velocity after rotating 236 00:09:09,390 --> 00:09:10,645 for five revolutions? 237 00:09:10,645 --> 00:09:12,333 Now there's a couple ways we could solve this. 238 00:09:12,333 --> 00:09:15,930 Because we solved for the time, we know every variable 239 00:09:15,930 --> 00:09:19,161 except for the final angular velocity. 240 00:09:19,161 --> 00:09:21,317 So I could use any of these now. 241 00:09:21,317 --> 00:09:23,470 To me, this first one's the simplest. 242 00:09:23,470 --> 00:09:25,161 There's no squares involved. 243 00:09:25,161 --> 00:09:28,002 There's not even a ratio or anything, so let's use this. 244 00:09:28,002 --> 00:09:30,122 We could say that omega final is gonna equal 245 00:09:30,122 --> 00:09:32,945 omega initial, that was just zero, plus 246 00:09:32,945 --> 00:09:35,397 the angular acceleration was 30, 247 00:09:35,397 --> 00:09:37,486 and now that we know the time we could say that this 248 00:09:37,486 --> 00:09:39,902 time was 1.45 seconds. 249 00:09:39,902 --> 00:09:42,183 And that gives me a final angular velocity 250 00:09:42,183 --> 00:09:44,350 of 43.5 radian per second. 251 00:09:46,085 --> 00:09:48,556 That's how fast this thing was revolving in a circle 252 00:09:48,556 --> 00:09:51,199 the moment it hit five revolutions. 253 00:09:51,199 --> 00:09:52,615 So that was one example. 254 00:09:52,615 --> 00:09:53,794 Let's do another one. 255 00:09:53,794 --> 00:09:56,284 Let's carry our kinematic formulas with us. 256 00:09:56,284 --> 00:09:57,515 We could use those. 257 00:09:57,515 --> 00:09:58,587 So we get rid of all that. 258 00:09:58,587 --> 00:09:59,767 Let's check this one out. 259 00:09:59,767 --> 00:10:02,748 Says this four meter long bar is gonna start, 260 00:10:02,748 --> 00:10:04,298 this time it doesn't start from rest. 261 00:10:04,298 --> 00:10:06,705 This time it starts with an angular velocity, 262 00:10:06,705 --> 00:10:08,792 oh, we're not gonna rotate that. 263 00:10:08,792 --> 00:10:11,562 Whoa, that'd be a more difficult problem, 264 00:10:11,562 --> 00:10:12,618 we're gonna rotate this. 265 00:10:12,618 --> 00:10:15,538 This four meter long bar starts with an angular velocity 266 00:10:15,538 --> 00:10:19,801 of 40 radians per second, but it decelerates to a stop 267 00:10:19,801 --> 00:10:21,983 after it rotates 20 revolutions. 268 00:10:21,983 --> 00:10:24,881 And the first question is, how fast is the edge of the bar 269 00:10:24,881 --> 00:10:27,793 moving initially in meters per second? 270 00:10:27,793 --> 00:10:30,476 So in other words, this point on the bar right here, 271 00:10:30,476 --> 00:10:32,565 is gonna have some velocity this way. 272 00:10:32,565 --> 00:10:35,110 We wanna know, what is that velocity initially 273 00:10:35,110 --> 00:10:36,648 in meters per second? 274 00:10:36,648 --> 00:10:38,107 Well this isn't too hard. 275 00:10:38,107 --> 00:10:40,497 We've got a formula that relates the speed 276 00:10:40,497 --> 00:10:41,770 to the angular speed. 277 00:10:41,770 --> 00:10:44,229 You just take the distance from the axis, 278 00:10:44,229 --> 00:10:46,646 to the point that you want to determine the speed 279 00:10:46,646 --> 00:10:48,773 and then you multiply it by the angular velocity 280 00:10:48,773 --> 00:10:51,704 and that gives you what the speed of that point is. 281 00:10:51,704 --> 00:10:53,404 So this r, let's be careful, 282 00:10:53,404 --> 00:10:55,182 this is always from the axis. 283 00:10:55,182 --> 00:10:58,439 And in this case this is the axis right there. 284 00:10:58,439 --> 00:11:01,029 The distance from the axis to the point we wanna find 285 00:11:01,029 --> 00:11:03,871 is in fact the entire length of this bar, 286 00:11:03,871 --> 00:11:05,637 so this will be four meters. 287 00:11:05,637 --> 00:11:06,874 So to find the speed we could just say that that's 288 00:11:06,874 --> 00:11:09,079 equal to four meters, 289 00:11:09,079 --> 00:11:11,739 since you wanna know the speed of a point out here 290 00:11:11,739 --> 00:11:13,456 that's four meters from the axis, 291 00:11:13,456 --> 00:11:15,843 and we multiply by the angular velocity, 292 00:11:15,843 --> 00:11:18,779 which initially was 40 radians per second. 293 00:11:18,779 --> 00:11:20,958 And we get the speed of this point on the rod, 294 00:11:20,958 --> 00:11:25,142 four meters away from the axis is 160 meters per second. 295 00:11:25,142 --> 00:11:26,265 That's really fast. 296 00:11:26,265 --> 00:11:28,699 And that's the fastest point on this rod. 297 00:11:28,699 --> 00:11:30,842 If you were gonna ask what the speed of the rod 298 00:11:30,842 --> 00:11:33,424 would be halfway, that would be half as much. 299 00:11:33,424 --> 00:11:35,664 Because this would only be, this r right here, 300 00:11:35,664 --> 00:11:38,656 would only be two meters from the axis to that point, 301 00:11:38,656 --> 00:11:39,807 it's only two meters. 302 00:11:39,807 --> 00:11:42,724 And the closer in you go, the smaller the r will be, 303 00:11:42,724 --> 00:11:44,801 the smaller the speed will be. 304 00:11:44,801 --> 00:11:47,646 So these are gonna travel, these points on the rod down here 305 00:11:47,646 --> 00:11:49,317 don't travel very fast at all, 306 00:11:49,317 --> 00:11:51,063 because their r is so small. 307 00:11:51,063 --> 00:11:53,940 All these points have the same angular velocity. 308 00:11:53,940 --> 00:11:56,187 They're all rotating with the same number 309 00:11:56,187 --> 00:11:59,281 of radians per second, but the actual distance of the circle 310 00:11:59,281 --> 00:12:00,907 they're traveling through is different, 311 00:12:00,907 --> 00:12:02,763 which makes all of their speeds different. 312 00:12:02,763 --> 00:12:03,942 So that answers part a, 313 00:12:03,942 --> 00:12:05,670 we got how fast in meters per second. 314 00:12:05,670 --> 00:12:08,253 It was going 160 meters per second. 315 00:12:08,253 --> 00:12:11,225 And the next part asks, what was the angular acceleration 316 00:12:11,225 --> 00:12:12,058 of the bar? 317 00:12:12,058 --> 00:12:14,121 All right, this one we're gonna have to actually use 318 00:12:14,121 --> 00:12:15,642 a kinematic formula for. 319 00:12:15,642 --> 00:12:17,704 We'll bring these back, put 'em over here. 320 00:12:17,704 --> 00:12:20,620 Again the way you use these, you identify what you know. 321 00:12:20,620 --> 00:12:23,602 We know the initial angular velocity was 40. 322 00:12:23,602 --> 00:12:27,120 So this time we know omega initial 40 radians per second. 323 00:12:27,120 --> 00:12:29,704 Set it revolved 20 revolutions. 324 00:12:29,704 --> 00:12:33,512 That's delta theta, but again, we can't just write 20. 325 00:12:33,512 --> 00:12:35,665 We've gotta right this in terms of radians 326 00:12:35,665 --> 00:12:37,441 if we're gonna use these radians per second. 327 00:12:37,441 --> 00:12:38,825 They have to all be in the same unit. 328 00:12:38,825 --> 00:12:41,006 So it's gonna be 20 revolution times 329 00:12:41,006 --> 00:12:44,110 two pi radians per revolution. 330 00:12:44,110 --> 00:12:46,229 So that's 40 pi radians. 331 00:12:46,229 --> 00:12:47,777 What's our third known? 332 00:12:47,777 --> 00:12:50,298 You always need a third known to use a kinematic formula. 333 00:12:50,298 --> 00:12:51,542 It's this. 334 00:12:51,542 --> 00:12:55,201 It says it decelerates to a stop, which means it stops. 335 00:12:55,201 --> 00:12:58,915 That means omega final, the final angular velocity is zero. 336 00:12:58,915 --> 00:13:02,343 And we want the angular acceleration, that's alpha. 337 00:13:02,343 --> 00:13:03,839 So this is what we wanna know. 338 00:13:03,839 --> 00:13:04,906 We wanna know alpha. 339 00:13:04,906 --> 00:13:06,401 We know the rest of these variables. 340 00:13:06,401 --> 00:13:08,326 Again to figure out which equation to use, 341 00:13:08,326 --> 00:13:09,975 I figure out which one got left out. 342 00:13:09,975 --> 00:13:11,019 And that's the time. 343 00:13:11,019 --> 00:13:12,840 I was neither give the time 344 00:13:12,840 --> 00:13:15,067 nor was I asked to find the time. 345 00:13:15,067 --> 00:13:17,688 Since this was left out, I'm gonna look for the formula 346 00:13:17,688 --> 00:13:19,896 that doesn't use time at all. 347 00:13:19,896 --> 00:13:20,907 And that's not the first one. 348 00:13:20,907 --> 00:13:22,523 That's not the second or the third, 349 00:13:22,523 --> 00:13:23,728 it's actually the fourth. 350 00:13:23,728 --> 00:13:24,994 So I'm gonna use this fourth equation. 351 00:13:24,994 --> 00:13:26,081 So what do we know? 352 00:13:26,081 --> 00:13:27,518 We know omega final was zero. 353 00:13:27,518 --> 00:13:28,797 So I'm gonna put a zero squared. 354 00:13:28,797 --> 00:13:30,214 But zero squared is still zero, 355 00:13:30,214 --> 00:13:32,378 equals omega initial squared. 356 00:13:32,378 --> 00:13:34,746 That's 40 radians per second squared. 357 00:13:34,746 --> 00:13:37,679 And then it's gonna be plus two times alpha. 358 00:13:37,679 --> 00:13:39,906 We don't know alpha, but that's what we wanna find, 359 00:13:39,906 --> 00:13:41,532 so I'm gonna leave that as a variable. 360 00:13:41,532 --> 00:13:42,864 And then delta theta we know. 361 00:13:42,864 --> 00:13:46,681 Delta theta was 40 pi radians since it was 20 revolutions. 362 00:13:46,681 --> 00:13:49,614 And if you solve this algebraically for alpha, 363 00:13:49,614 --> 00:13:51,887 you move the 40 over to the other side. 364 00:13:51,887 --> 00:13:52,809 So you'll subtract it. 365 00:13:52,809 --> 00:13:55,792 You get a negative 40 radians per second squared. 366 00:13:55,792 --> 00:13:57,993 And then you gotta divide by this two 367 00:13:57,993 --> 00:14:00,527 as well as the 40 pi radians, 368 00:14:00,527 --> 00:14:04,694 which gives me negative 6.37 radians per second squared. 369 00:14:06,019 --> 00:14:07,282 Why is it negative? 370 00:14:07,282 --> 00:14:09,028 Well this thing slowed down to a stop. 371 00:14:09,028 --> 00:14:12,446 So this angular acceleration has gotta have the opposite 372 00:14:12,446 --> 00:14:15,168 sign to the initial angular velocity. 373 00:14:15,168 --> 00:14:17,015 We called this positive 40, 374 00:14:17,015 --> 00:14:19,141 that means our alpha's gonna be negative. 375 00:14:19,141 --> 00:14:22,264 So recapping, these are the rotational kinematic formulas 376 00:14:22,264 --> 00:14:25,364 that relate the rotational kinematic variables. 377 00:14:25,364 --> 00:14:29,533 They're only true if the angular acceleration is constant. 378 00:14:29,533 --> 00:14:33,028 But when it's constant, you can identify the three known 379 00:14:33,028 --> 00:14:36,456 variables and the one unknown that you're trying to find 380 00:14:36,456 --> 00:14:39,379 and then use the variable that got left out of the mix 381 00:14:39,379 --> 00:14:42,399 to identify which kinematic formula to use, 382 00:14:42,399 --> 00:14:44,982 since you would use the formula that does not 383 00:14:44,982 --> 00:00:00,000 involve that variable that was neither given nor asked for.