1 00:00:00,184 --> 00:00:01,472 - [Instructor] Let's say you've got a mass 2 00:00:01,472 --> 00:00:03,181 connected to a spring and the mass is sitting 3 00:00:03,181 --> 00:00:04,889 on a friction, the surface. 4 00:00:04,889 --> 00:00:06,543 If the mass is sitting at a point 5 00:00:06,543 --> 00:00:09,217 where the spring is just at the spring's natural length, 6 00:00:09,217 --> 00:00:10,481 the mass isn't going to go anywhere 7 00:00:10,481 --> 00:00:12,762 because when the spring is at its natural length, 8 00:00:12,762 --> 00:00:15,090 it is content with its place in the universe. 9 00:00:15,090 --> 00:00:17,029 It neither pushes nor pulls. 10 00:00:17,029 --> 00:00:18,623 It has no spring energy. 11 00:00:18,623 --> 00:00:20,968 This is like me most days in the summer. 12 00:00:20,968 --> 00:00:23,489 So we call this point where the spring neither pushes 13 00:00:23,489 --> 00:00:26,423 nor pulls, the spring's natural length. 14 00:00:26,423 --> 00:00:28,851 For a mass on a horizontal spring, 15 00:00:28,851 --> 00:00:32,003 this is gonna be the equilibrium position. 16 00:00:32,003 --> 00:00:34,549 What we mean by equilibrium position is the point 17 00:00:34,549 --> 00:00:37,778 where the net force on the mass is zero. 18 00:00:37,778 --> 00:00:39,988 So for a mass on a horizontal spring, 19 00:00:39,988 --> 00:00:42,102 the equilibrium position is at the point 20 00:00:42,102 --> 00:00:44,639 where the spring is at its natural length 21 00:00:44,639 --> 00:00:45,947 because the spring wouldn't be pushing 22 00:00:45,947 --> 00:00:47,121 to the right or the left. 23 00:00:47,121 --> 00:00:49,189 If you just put the mass there at that point, 24 00:00:49,189 --> 00:00:52,269 it would just stay there forever at rest. 25 00:00:52,269 --> 00:00:53,341 That would be boring. 26 00:00:53,341 --> 00:00:55,133 So let's say we pull the mass to the right 27 00:00:55,133 --> 00:00:56,268 at distance d. 28 00:00:56,268 --> 00:01:00,002 If we do this, we give this spring spring potential energy. 29 00:01:00,002 --> 00:01:03,096 If we release the mass from rest while the spring has 30 00:01:03,096 --> 00:01:05,787 spring potential energy, the spring's gonna pull the mass 31 00:01:05,787 --> 00:01:08,639 back to the left and that mass is gonna move 32 00:01:08,639 --> 00:01:11,671 through the equilibrium position with the sum speed. 33 00:01:11,671 --> 00:01:13,346 We can figure out what that speed is 34 00:01:13,346 --> 00:01:16,322 just by using conservation of energy and it's not that hard. 35 00:01:16,322 --> 00:01:18,240 The potential energy, the spring with start with 36 00:01:18,240 --> 00:01:21,678 would be 1/2 k, the spring constant, times d, 37 00:01:21,678 --> 00:01:24,880 the amount of spring has been stretched, squared. 38 00:01:24,880 --> 00:01:26,862 There would be no kinetic energy to start 39 00:01:26,862 --> 00:01:29,299 because we release the mass from rest. 40 00:01:29,299 --> 00:01:31,235 As this mass flies to the left, 41 00:01:31,235 --> 00:01:33,996 it would start gaining kinetic energy, the spring energy 42 00:01:33,996 --> 00:01:35,922 would start turning into kinetic energy. 43 00:01:35,922 --> 00:01:38,774 When the mass gets to the equilibrium position, 44 00:01:38,774 --> 00:01:39,891 the d would be zero. 45 00:01:39,891 --> 00:01:41,817 So at that point there would be no spring energy 46 00:01:41,817 --> 00:01:44,699 and all of the spring energy would have turned 47 00:01:44,699 --> 00:01:47,757 into kinetic energy and you get this simple relationship 48 00:01:47,757 --> 00:01:49,832 that says all the spring energy equals 49 00:01:49,832 --> 00:01:52,318 all the kinetic energy at the equilibrium position. 50 00:01:52,318 --> 00:01:55,691 So if we solve for v, we can get that the v of this mass 51 00:01:55,691 --> 00:01:58,021 at the equilibrium position would be the square root 52 00:01:58,021 --> 00:02:00,416 of k over m times d squared. 53 00:02:00,416 --> 00:02:01,857 You could call this d out 54 00:02:01,857 --> 00:02:03,366 because d squared and square root. 55 00:02:03,366 --> 00:02:04,387 but this is the idea. 56 00:02:04,387 --> 00:02:06,589 This is the speed you would get of the mass 57 00:02:06,589 --> 00:02:08,443 passing through the equilibrium position. 58 00:02:08,443 --> 00:02:09,276 Let me ask you this. 59 00:02:09,276 --> 00:02:10,925 What if it was a vertical spring? 60 00:02:10,925 --> 00:02:12,510 And this mass is sitting here. 61 00:02:12,510 --> 00:02:15,117 We come find a vertical spring with a mass hanging on it. 62 00:02:15,117 --> 00:02:16,622 We're like hey, I wanna pull this down, 63 00:02:16,622 --> 00:02:17,880 the distance d. 64 00:02:17,880 --> 00:02:19,726 If I pull it down to distance d, 65 00:02:19,726 --> 00:02:22,767 when this mass reaches the equilibrium position again, 66 00:02:22,767 --> 00:02:27,427 will it also be going root k over m times d squared? 67 00:02:27,427 --> 00:02:29,167 Or, would it be going at some different speed 68 00:02:29,167 --> 00:02:30,918 because now it's hanging vertically? 69 00:02:30,918 --> 00:02:33,496 Well it turns the spring constant's the same 70 00:02:33,496 --> 00:02:35,000 and you pull it down from the point 71 00:02:35,000 --> 00:02:36,212 where the mass is hanging. 72 00:02:36,212 --> 00:02:39,344 This exact same procedure is gonna hold over here 73 00:02:39,344 --> 00:02:41,924 and you can find the speed in the exact same way 74 00:02:41,924 --> 00:02:43,438 and that should be surprising. 75 00:02:43,438 --> 00:02:44,844 That should not be obvious. 76 00:02:44,844 --> 00:02:46,596 Because when the mass is hanging over here, 77 00:02:46,596 --> 00:02:47,883 you don't just have a spring force. 78 00:02:47,883 --> 00:02:49,961 You got a gravitational force. 79 00:02:49,961 --> 00:02:52,769 You don't just have a spring energy and kinetic energy. 80 00:02:52,769 --> 00:02:53,602 Think about it. 81 00:02:53,602 --> 00:02:55,065 This mass is moving up and down. 82 00:02:55,065 --> 00:02:57,908 You got changes and gravitational potential energy. 83 00:02:57,908 --> 00:02:59,967 So why don't we have to taken into account 84 00:02:59,967 --> 00:03:02,073 the gravitational potential energy 85 00:03:02,073 --> 00:03:04,815 when we're doing conservation of energy in this equation? 86 00:03:04,815 --> 00:03:05,781 Well that's what I wanna prove to you 87 00:03:05,781 --> 00:03:06,973 in the rest of this video. 88 00:03:06,973 --> 00:03:09,090 If all you wanted was the result, if all you, 89 00:03:09,090 --> 00:03:10,283 if you're good, if you're like, 90 00:03:10,283 --> 00:03:12,542 "Man, all right, I can do the same thing above cases, 91 00:03:12,542 --> 00:03:14,252 "don't even tell me anything else." 92 00:03:14,252 --> 00:03:15,124 You're good. 93 00:03:15,124 --> 00:03:16,618 But I suggest you watch the rest of it 94 00:03:16,618 --> 00:03:19,830 because knowing why you can ignore the m g h 95 00:03:19,830 --> 00:03:22,257 in this calculation gives you better insight 96 00:03:22,257 --> 00:03:26,424 into what we really mean by this d and this h and this v 97 00:03:27,531 --> 00:03:30,653 as well as what we really mean by the equilibrium position 98 00:03:30,653 --> 00:03:33,083 and that will conceptually aid you 99 00:03:33,083 --> 00:03:35,529 if you get a problem that's more challenging. 100 00:03:35,529 --> 00:03:38,517 So let's prove this and figure out why we can get away 101 00:03:38,517 --> 00:03:40,573 with ignoring this m g h right here. 102 00:03:40,573 --> 00:03:41,947 So let me get rid of all these. 103 00:03:41,947 --> 00:03:42,844 Let's start fresh. 104 00:03:42,844 --> 00:03:46,310 Let's just say we have a spring hanging from the ceiling 105 00:03:46,310 --> 00:03:47,931 of spring constant k. 106 00:03:47,931 --> 00:03:49,544 Let's say this spring is light. 107 00:03:49,544 --> 00:03:52,745 If it was heavy, it might pull itself down by its own weight 108 00:03:52,745 --> 00:03:54,548 so I'm gonna assume this is very light spring 109 00:03:54,548 --> 00:03:55,615 and it's hanging right here. 110 00:03:55,615 --> 00:03:57,647 There's no mass connected to it initially 111 00:03:57,647 --> 00:03:59,870 so it's just hanging at its natural length. 112 00:03:59,870 --> 00:04:02,704 It's neither pulling up nor pushing down 113 00:04:02,704 --> 00:04:04,279 as you see it right here 114 00:04:04,279 --> 00:04:06,033 because it's at its natural length 115 00:04:06,033 --> 00:04:07,737 but we connect the mass m to it. 116 00:04:07,737 --> 00:04:10,349 When we do that, we lower the mass with our hand. 117 00:04:10,349 --> 00:04:12,016 We don't just let it fall and start oscillating. 118 00:04:12,016 --> 00:04:13,662 We first lower the mass. 119 00:04:13,662 --> 00:04:14,977 We connect it and lower it. 120 00:04:14,977 --> 00:04:18,829 We find the point where the mass is just gonna stay at rest. 121 00:04:18,829 --> 00:04:21,609 That would be the new equilibrium position. 122 00:04:21,610 --> 00:04:23,792 So this right here is essentially 123 00:04:23,792 --> 00:04:26,190 our new equilibrium position. 124 00:04:26,190 --> 00:04:28,585 In other words, that's the point where the net force 125 00:04:28,585 --> 00:04:30,595 on the mass would be zero. 126 00:04:30,595 --> 00:04:33,739 But this time that's not at the spring's natural length. 127 00:04:33,739 --> 00:04:35,649 The way it was on the horizontal spring. 128 00:04:35,649 --> 00:04:38,607 This time the equilibrium position is this place, 129 00:04:38,607 --> 00:04:42,416 the distance a away from the spring's natural length 130 00:04:42,416 --> 00:04:45,683 because right now it's battling the force of gravity. 131 00:04:45,683 --> 00:04:49,734 In other words the spring force exerted upward k x 132 00:04:49,734 --> 00:04:53,635 minus the gravitational force which is m times g 133 00:04:53,635 --> 00:04:56,172 has got to equal zero in order for this mass 134 00:04:56,172 --> 00:04:57,423 to be an equilibrium. 135 00:04:57,423 --> 00:04:59,727 So we can actually figure out what this distance a 136 00:04:59,727 --> 00:05:01,824 would have to be in terms of given variables. 137 00:05:01,824 --> 00:05:04,864 Since at the equilibrium position, x, the distance 138 00:05:04,864 --> 00:05:07,673 the spring has been stretched has just gonna have to equal, 139 00:05:07,673 --> 00:05:10,756 m g divided by the spring constant k. 140 00:05:11,594 --> 00:05:13,233 This is what a is gonna equal. 141 00:05:13,233 --> 00:05:15,562 So the distance, the mass hangs down 142 00:05:15,562 --> 00:05:18,492 at the equilibrium position from the natural length 143 00:05:18,492 --> 00:05:21,097 of the spring is just gonna be m g over k. 144 00:05:21,097 --> 00:05:23,877 This is a in this diagram. 145 00:05:23,877 --> 00:05:24,833 This is gonna be key. 146 00:05:24,833 --> 00:05:26,851 So we're gonna hold on to this result right here. 147 00:05:26,851 --> 00:05:27,909 Well let's do this. 148 00:05:27,909 --> 00:05:30,894 Let's ask if we take this mass and pull it down 149 00:05:30,894 --> 00:05:34,929 an extra amount b from the new equilibrium position, 150 00:05:34,929 --> 00:05:37,194 well at that point the forces won't be equal. 151 00:05:37,194 --> 00:05:39,053 The spring's gonna be stretched extra. 152 00:05:39,053 --> 00:05:40,705 It's gonna be pulling up with more force 153 00:05:40,705 --> 00:05:42,046 in the force of gravity. 154 00:05:42,046 --> 00:05:43,763 So this mass is gonna accelerate upward. 155 00:05:43,763 --> 00:05:47,058 It's gonna reach this equilibrium position with some speed. 156 00:05:47,058 --> 00:05:48,204 It's gonna shoo pass that. 157 00:05:48,204 --> 00:05:50,865 Now the spring force is less than the force of gravity 158 00:05:50,865 --> 00:05:52,790 and so gravity wins in that case 159 00:05:52,790 --> 00:05:54,355 and it keeps going back and forth. 160 00:05:54,355 --> 00:05:56,020 We're gonna ask ourselves the same question we did before. 161 00:05:56,020 --> 00:05:58,124 If we pull this down, the distance b, 162 00:05:58,124 --> 00:06:01,094 what is the speed of the mass 163 00:06:01,094 --> 00:06:05,153 when it passes through the equilibrium position? 164 00:06:05,153 --> 00:06:07,811 Again we're gonna use conservation of energy to answer this. 165 00:06:07,811 --> 00:06:10,812 So we're gonna say that the initial energy in our system 166 00:06:10,812 --> 00:06:13,098 is gonna equal to final energy in our system. 167 00:06:13,098 --> 00:06:14,826 We're gonna choose two points. 168 00:06:14,826 --> 00:06:16,764 Let's choose initially this point down here. 169 00:06:16,764 --> 00:06:18,949 We release the mass from rest 170 00:06:18,949 --> 00:06:21,211 when it's pulled down at distance b 171 00:06:21,211 --> 00:06:23,520 below the new equilibrium position. 172 00:06:23,520 --> 00:06:25,302 Then our final point is gonna be right here 173 00:06:25,302 --> 00:06:26,749 at the equilibrium position 174 00:06:26,749 --> 00:06:29,466 because that's where we wanna know the speed of the mass. 175 00:06:29,466 --> 00:06:31,555 So let's try to figure out how much energy there is 176 00:06:31,555 --> 00:06:32,993 in the system initially 177 00:06:32,993 --> 00:06:35,916 if I pull this mass down and let it go. 178 00:06:35,916 --> 00:06:38,294 Well initially if I'm just letting this mass go, 179 00:06:38,294 --> 00:06:40,175 the mass is starting from rest. 180 00:06:40,175 --> 00:06:42,162 If the mass starts from rest, it's got no speed 181 00:06:42,162 --> 00:06:44,891 and if it's got no speed, it's got no kinetic energy. 182 00:06:44,891 --> 00:06:47,120 So there's no kinetic energy to start with 183 00:06:47,120 --> 00:06:48,801 if this mass is starting from rest 184 00:06:48,801 --> 00:06:50,783 but there is gonna be spring potential energy 185 00:06:50,783 --> 00:06:53,115 and there's gonna be a lot of spring potential energy. 186 00:06:53,115 --> 00:06:54,086 Because think about it. 187 00:06:54,086 --> 00:06:56,416 Not only is this mass stretching the spring passed 188 00:06:56,416 --> 00:06:59,371 the new equilibrium position by an amount b 189 00:06:59,371 --> 00:07:01,960 but the new equilibrium position itself is stretched 190 00:07:01,960 --> 00:07:04,820 from the spring's natural length a. 191 00:07:04,820 --> 00:07:07,072 This formula, when you have 1/2, 192 00:07:07,072 --> 00:07:09,618 the spring constant times the length 193 00:07:09,618 --> 00:07:11,213 that the spring has been stretched, 194 00:07:11,213 --> 00:07:13,835 that's the total amount the spring has been stretched. 195 00:07:13,835 --> 00:07:15,644 So the total amount the spring has been stretched 196 00:07:15,644 --> 00:07:19,739 from its natural length is gonna be a plus b. 197 00:07:19,739 --> 00:07:21,532 I've got to square that whole term. 198 00:07:21,532 --> 00:07:24,328 This is how much spring potential energy there's gonna be 199 00:07:24,328 --> 00:07:26,010 in the system initially. 200 00:07:26,010 --> 00:07:27,962 So how much gravitational potential energy 201 00:07:27,962 --> 00:07:28,962 we're gonna start with? 202 00:07:28,962 --> 00:07:30,305 Well, that's kind of hard to us 203 00:07:30,305 --> 00:07:32,651 because you can always choose where you want your h 204 00:07:32,651 --> 00:07:34,788 equals zero reference line to be. 205 00:07:34,788 --> 00:07:37,456 In other words, I'm gonna choose this lowest point here 206 00:07:37,456 --> 00:07:40,454 because that's often convenient when choose this to be 207 00:07:40,454 --> 00:07:42,792 the h equals zero reference line. 208 00:07:42,792 --> 00:07:45,096 We'll measure all heights from that point. 209 00:07:45,096 --> 00:07:47,651 This is allowed because it's only really differences 210 00:07:47,651 --> 00:07:50,014 in gravitational potential energy that matter 211 00:07:50,014 --> 00:07:50,901 so you can do this. 212 00:07:50,901 --> 00:07:52,654 You just have to be consistent with your choice. 213 00:07:52,654 --> 00:07:55,273 But with that choice where this is h equals zero, 214 00:07:55,273 --> 00:07:57,522 the height my mass has at the initial position 215 00:07:57,522 --> 00:07:58,476 is gonna be zero. 216 00:07:58,476 --> 00:08:01,167 So that means the gravitational potential energy 217 00:08:01,167 --> 00:08:03,866 which is given by m g h is also gonna be zero 218 00:08:03,866 --> 00:08:05,153 at that initial point. 219 00:08:05,153 --> 00:08:07,627 So in terms of initial energies, this is all I've got. 220 00:08:07,627 --> 00:08:09,402 This is my total initial energy, 221 00:08:09,402 --> 00:08:11,283 just the energy from the spring. 222 00:08:11,283 --> 00:08:13,944 Now we can set that equal to our final energies 223 00:08:13,944 --> 00:08:16,329 so we're gonna have any spring potential energy here 224 00:08:16,329 --> 00:08:18,048 in our final position. 225 00:08:18,048 --> 00:08:19,737 You might think no because the final position 226 00:08:19,737 --> 00:08:21,880 is at the new equilibrium position 227 00:08:21,880 --> 00:08:25,136 but remember this new equilibrium position still displays 228 00:08:25,136 --> 00:08:27,260 from the spring's natural length. 229 00:08:27,260 --> 00:08:30,373 So what I have, I'm gonna have 1/2 k times 230 00:08:30,373 --> 00:08:32,470 the amount the spring has been stretched 231 00:08:32,470 --> 00:08:34,227 from its natural position. 232 00:08:34,227 --> 00:08:35,818 At this new equilibrium position, 233 00:08:35,818 --> 00:08:37,921 the amount that has been stretched is just a. 234 00:08:37,922 --> 00:08:39,586 So I'm gonna write a here 235 00:08:39,586 --> 00:08:42,105 because that's how far the spring has stretched 236 00:08:42,105 --> 00:08:43,520 at this new equilibrium position. 237 00:08:43,520 --> 00:08:44,611 I have to square that 238 00:08:44,611 --> 00:08:46,450 because that's 1/2 k x squared. 239 00:08:46,450 --> 00:08:47,805 We're gonna have kinetic energy. 240 00:08:47,805 --> 00:08:51,373 This mass is gonna gain speed as it flies upward. 241 00:08:51,373 --> 00:08:52,790 It's gonna be moving with some speed 242 00:08:52,790 --> 00:08:54,079 when it gets up to that point. 243 00:08:54,079 --> 00:08:57,562 So the kinetic energy is gonna be 1/2 times the mass, 244 00:08:57,562 --> 00:08:59,151 times the speed the mass has 245 00:08:59,151 --> 00:09:01,380 at the equilibrium position squared. 246 00:09:01,380 --> 00:09:02,672 That's what we want to determine. 247 00:09:02,672 --> 00:09:04,811 But there's also gonna be gravitational potential energy. 248 00:09:04,811 --> 00:09:07,096 We said h is zero down here. 249 00:09:07,096 --> 00:09:08,793 So if the mass is not there, 250 00:09:08,793 --> 00:09:10,868 it's gonna have potential energy due to gravity. 251 00:09:10,868 --> 00:09:13,143 If it's b above this point, 252 00:09:13,143 --> 00:09:14,433 look at it, we pulled it down b 253 00:09:14,433 --> 00:09:16,718 so when it gets back to the equilibrium position, 254 00:09:16,718 --> 00:09:18,621 if this is h equals zero, 255 00:09:18,621 --> 00:09:22,476 that's gonna be h equals b above where it started. 256 00:09:22,476 --> 00:09:26,967 So I have to put in m times g times the b value. 257 00:09:26,967 --> 00:09:28,407 This length right here. 258 00:09:28,407 --> 00:09:32,251 Since in moving up this mass gained m g times b 259 00:09:32,251 --> 00:09:34,046 a gravitational potential energy. 260 00:09:34,046 --> 00:09:35,272 So how do we make progress here? 261 00:09:35,272 --> 00:09:37,167 I wanna solve for the speed v 262 00:09:37,167 --> 00:09:38,401 but I've got this mess over here. 263 00:09:38,401 --> 00:09:40,328 Look, I've got a plus b squared 264 00:09:40,328 --> 00:09:41,658 so I better handle that first. 265 00:09:41,658 --> 00:09:44,853 So let's say we do the 1/2 k and then we square this out. 266 00:09:44,853 --> 00:09:45,979 Remember we do FOIL. 267 00:09:45,979 --> 00:09:47,940 So it's First, Outer, Inner, Last. 268 00:09:47,940 --> 00:09:50,684 I'm gonna get a quantity of a squared 269 00:09:50,684 --> 00:09:51,836 and then this cross term, 270 00:09:51,836 --> 00:09:54,503 I'm get plus two times a times b 271 00:09:55,400 --> 00:09:57,491 and then plus b squared. 272 00:09:57,491 --> 00:10:00,231 That's what happens if I square this whole term right here. 273 00:10:00,231 --> 00:10:02,681 These start looking really bad but don't despair. 274 00:10:02,681 --> 00:10:05,531 Something great, something wonderful is about to happen 275 00:10:05,531 --> 00:10:08,465 because I'm gonna set this equal to the right hand side. 276 00:10:08,465 --> 00:10:10,337 So if we multiply out on the left hand side, 277 00:10:10,337 --> 00:10:14,405 we're gonna get a 1/2 times k times a squared 278 00:10:14,405 --> 00:10:18,572 plus a 1/2 times k times this 2 a b term, 279 00:10:19,966 --> 00:10:21,512 plus I'm gonna get another one, 280 00:10:21,512 --> 00:10:25,093 1/2 k times this b squared term. 281 00:10:25,093 --> 00:10:26,819 We can say that that's supposed to equal 282 00:10:26,819 --> 00:10:28,730 everything on the right hand side. 283 00:10:28,730 --> 00:10:31,023 So we can already see something that we can cancel out. 284 00:10:31,023 --> 00:10:33,906 I've got a 1/2 k a squared on each side. 285 00:10:33,906 --> 00:10:35,912 So if we subtract that from both sides, 286 00:10:35,912 --> 00:10:37,080 I can get rid of that. 287 00:10:37,080 --> 00:10:39,453 This 1/2 here is gonna cancel this two 288 00:10:39,453 --> 00:10:42,731 and I'm left with k a b on the left hand side 289 00:10:42,731 --> 00:10:44,732 plus 1/2 k b squared. 290 00:10:44,732 --> 00:10:46,177 But, what is k a? 291 00:10:46,177 --> 00:10:47,823 If you're clever and you look up here, 292 00:10:47,823 --> 00:10:50,371 you're like, wait a minute, I remember what k a was. 293 00:10:50,371 --> 00:10:54,744 K times a, if we just multiply both sides by k here, 294 00:10:54,744 --> 00:10:57,864 it's gotta equal m g because that's what just the statement 295 00:10:57,864 --> 00:11:00,546 of equilibrium that at the equilibrium position 296 00:11:00,546 --> 00:11:03,606 k times a has to equal m times g. 297 00:11:03,606 --> 00:11:06,889 So I can replace k times a over on this left hand side 298 00:11:06,889 --> 00:11:08,609 with m times g. 299 00:11:08,609 --> 00:11:11,032 M g might be like, why would I ever wanna do that? 300 00:11:11,032 --> 00:11:13,048 Why would I wanna replace k times a 301 00:11:13,048 --> 00:11:14,753 and I still multiply by this b? 302 00:11:14,753 --> 00:11:16,688 Why would I replace k a with m g? 303 00:11:16,688 --> 00:11:20,870 Because now that term is gonna cancel with the other m g b 304 00:11:20,870 --> 00:11:22,694 on the other side of this equation. 305 00:11:22,694 --> 00:11:25,885 One half m b squared plus m g b 306 00:11:25,885 --> 00:11:27,220 but I've got m g b on the left 307 00:11:27,220 --> 00:11:28,918 and m g b on the right now. 308 00:11:28,918 --> 00:11:31,067 That's why I replace this k a with m g. 309 00:11:31,067 --> 00:11:33,154 I can subtract that from both sides 310 00:11:33,154 --> 00:11:36,466 and magically I just get the exact same relationship 311 00:11:36,466 --> 00:11:38,951 we had for the horizontal spring 312 00:11:38,951 --> 00:11:41,905 measured from the new equilibrium position. 313 00:11:41,905 --> 00:11:44,158 This is important so let me restate this. 314 00:11:44,158 --> 00:11:47,681 You can either when solving these vertical spring problems, 315 00:11:47,681 --> 00:11:51,126 measure your spring displacement from the natural 316 00:11:51,126 --> 00:11:54,492 unstretched length of the spring like we did right here. 317 00:11:54,492 --> 00:11:56,504 We had to add a plus b 318 00:11:56,504 --> 00:11:57,705 because that was the distance 319 00:11:57,705 --> 00:12:00,627 from the natural unstretched length of the spring 320 00:12:00,627 --> 00:12:02,481 all the way to where the mass was. 321 00:12:02,481 --> 00:12:06,069 You can do that and include gravitational potential energy 322 00:12:06,069 --> 00:12:07,583 and get the right answer. 323 00:12:07,583 --> 00:12:10,370 But what we just saw is that these terms always cancel 324 00:12:10,370 --> 00:12:13,160 so the alternative is that you can just measure 325 00:12:13,160 --> 00:12:16,048 the spring extension, the spring displacement 326 00:12:16,048 --> 00:12:19,416 from the new equilibrium position. 327 00:12:19,416 --> 00:12:22,626 If you do that, you just leave off all mentioned 328 00:12:22,626 --> 00:12:24,580 of gravitational potential energy 329 00:12:24,580 --> 00:12:26,334 and you get the same answer. 330 00:12:26,334 --> 00:12:28,602 You can think of gravity simply as shifting 331 00:12:28,602 --> 00:12:30,985 the equilibrium position down a bit 332 00:12:30,985 --> 00:12:32,552 and then the mass and spring behaving 333 00:12:32,552 --> 00:12:34,838 just as I would on a horizontal surface 334 00:12:34,838 --> 00:12:38,079 as long as you only think about spring displacement 335 00:12:38,079 --> 00:12:40,796 from that new equilibrium position. 336 00:12:40,796 --> 00:12:42,506 So in other words, if you were given a problem, 337 00:12:42,506 --> 00:12:43,707 let me get rid of all these. 338 00:12:43,707 --> 00:12:45,088 If you're just given a problem 339 00:12:45,088 --> 00:12:47,188 and you were told a three kilogram mass 340 00:12:47,188 --> 00:12:49,227 is hanging from a vertical spring 341 00:12:49,227 --> 00:12:51,953 of spring constant 50 newtons per meter 342 00:12:51,953 --> 00:12:55,329 and this line here represents the equilibrium position, 343 00:12:55,329 --> 00:12:57,199 it's just hanging out right there at rest. 344 00:12:57,199 --> 00:12:59,145 You pull this mass down from the point 345 00:12:59,145 --> 00:13:03,056 where it was hanging at rest, 0.3 meter and let go. 346 00:13:03,056 --> 00:13:05,100 You wanna figure out what speed will it be going 347 00:13:05,100 --> 00:13:07,242 when it reaches equilibrium position. 348 00:13:07,242 --> 00:13:08,133 You can just do this. 349 00:13:08,133 --> 00:13:09,389 You can say, all right, down here 350 00:13:09,389 --> 00:13:10,832 it's got spring potential energy, 351 00:13:10,832 --> 00:13:13,792 1/2 k is 50 newtons per meter 352 00:13:13,792 --> 00:13:15,807 times the amount the spring has been stretched 353 00:13:15,807 --> 00:13:17,636 but I'm just gonna consider stretches 354 00:13:17,636 --> 00:13:19,909 from the new equilibrium position. 355 00:13:19,909 --> 00:13:21,559 So I'm just gonna do 0.3. 356 00:13:21,559 --> 00:13:23,112 I'm not gonna worry about the fact 357 00:13:23,112 --> 00:13:25,929 that the spring has actually already been stretched 358 00:13:25,929 --> 00:13:28,072 to get to this equilibrium position. 359 00:13:28,072 --> 00:13:31,328 I'm just gonna put this stretching from equilibrium. 360 00:13:31,328 --> 00:13:33,419 I'm gonna set that equal to the kinetic energy 361 00:13:33,419 --> 00:13:36,119 the mass has at the equilibrium position. 362 00:13:36,119 --> 00:13:40,561 We know that m is three kilograms times v squared. 363 00:13:40,561 --> 00:13:42,477 I'm not gonna include information 364 00:13:42,477 --> 00:13:45,006 about the gravitational potential energy at all 365 00:13:45,006 --> 00:13:46,915 because I only measured the displacements 366 00:13:46,915 --> 00:13:49,273 from the new equilibrium position. 367 00:13:49,273 --> 00:13:51,359 So at this point, I can just solve for my v. 368 00:13:51,359 --> 00:13:53,488 If you solve that algebraically for v, 369 00:13:53,488 --> 00:13:55,108 you cancel out the twos. 370 00:13:55,108 --> 00:13:56,559 You divide both sides by three 371 00:13:56,559 --> 00:13:57,664 and take a square root. 372 00:13:57,664 --> 00:14:00,336 You get to the speed of the mass at equilibrium 373 00:14:00,336 --> 00:14:03,461 is gonna be 1.2 meters per second. 374 00:14:03,461 --> 00:14:05,512 So to recap, even though it seems initially 375 00:14:05,512 --> 00:14:07,677 like vertical springs would be much harder 376 00:14:07,677 --> 00:14:09,693 than horizontal springs because you've got 377 00:14:09,693 --> 00:14:12,735 gravitational forces and gravitational potential energy 378 00:14:12,735 --> 00:14:14,957 to worry about, if you measure the spring displacement 379 00:14:14,957 --> 00:14:17,431 from the new equilibrium position 380 00:14:17,431 --> 00:14:19,655 as opposed to the natural spring length, 381 00:14:19,655 --> 00:14:22,217 you can simply use conservation of energy 382 00:14:22,217 --> 00:00:00,000 without mention of gravitational potential energy at all.