1 00:00:00,232 --> 00:00:02,515 - [Instructor] Alright, we should talk about oscillators. 2 00:00:02,515 --> 00:00:05,167 And what an oscillator is is an object 3 00:00:05,167 --> 00:00:07,849 or variable that can move back and forth 4 00:00:07,849 --> 00:00:10,930 or increase and decrease, go up and down, 5 00:00:10,930 --> 00:00:13,002 left and right, over and over and over. 6 00:00:13,002 --> 00:00:15,376 So for instance, a mass on a spring here 7 00:00:15,376 --> 00:00:18,112 is an oscillator if we pull this mass back, 8 00:00:18,112 --> 00:00:20,346 it's gonna oscillate back and forth, 9 00:00:20,346 --> 00:00:21,953 and that's what we mean by an oscillator. 10 00:00:21,953 --> 00:00:24,217 Or another common example is a pendulum, 11 00:00:24,217 --> 00:00:26,758 and a pendulum is just a mass connected to a string, 12 00:00:26,758 --> 00:00:28,158 and you pull the mass back 13 00:00:28,158 --> 00:00:29,959 and then it swings back and forth. 14 00:00:29,959 --> 00:00:31,732 So you've got something going back and forth, 15 00:00:31,732 --> 00:00:32,807 that's an oscillator. 16 00:00:32,807 --> 00:00:34,401 These are the two most common types. 17 00:00:34,401 --> 00:00:35,875 Masses on springs, pendulum, 18 00:00:35,875 --> 00:00:37,462 but there's many other examples 19 00:00:37,462 --> 00:00:40,379 and all those examples share one common 20 00:00:40,379 --> 00:00:43,226 feature of why they're an oscillator. 21 00:00:43,226 --> 00:00:44,563 So you could ask why do these things 22 00:00:44,563 --> 00:00:46,223 oscillate in the first place, 23 00:00:46,223 --> 00:00:48,245 and it's because they all share this common fact, 24 00:00:48,245 --> 00:00:50,815 that they all have a restoring force. 25 00:00:50,815 --> 00:00:53,808 And a restoring force, like the name suggests, 26 00:00:53,808 --> 00:00:56,246 tries to restore this system, 27 00:00:56,246 --> 00:00:57,925 but restore it to what? 28 00:00:57,925 --> 00:01:00,722 Restore the system to the equilibrium position. 29 00:01:00,722 --> 00:01:03,402 So every oscillator has an equilibrium position, 30 00:01:03,402 --> 00:01:05,193 and that would be the point at which 31 00:01:05,193 --> 00:01:09,440 there's no net force on the object that's oscillating. 32 00:01:09,440 --> 00:01:11,503 So for instance, for this mass, 33 00:01:11,503 --> 00:01:13,175 if this mass on the spring was sitting 34 00:01:13,175 --> 00:01:14,705 at the equilibrium position, 35 00:01:14,705 --> 00:01:17,365 the net force on that mass would be 0 because 36 00:01:17,365 --> 00:01:19,956 that's what we mean by the equilibrium position. 37 00:01:19,956 --> 00:01:21,344 In other words, if you just sat 38 00:01:21,344 --> 00:01:22,916 the mass there it would just stay there 39 00:01:22,916 --> 00:01:24,242 because there's no net force on it. 40 00:01:24,242 --> 00:01:26,941 However, if I pull this mass to the right, 41 00:01:26,941 --> 00:01:28,273 the spring's like uh uh, 42 00:01:28,273 --> 00:01:30,665 now I'm gonna try and restore this mass 43 00:01:30,665 --> 00:01:32,753 back to the equilibrium position, 44 00:01:32,753 --> 00:01:34,460 the spring would pull to the left. 45 00:01:34,460 --> 00:01:36,242 If I push this mass to the left, 46 00:01:36,242 --> 00:01:37,682 the spring's like uh uh, 47 00:01:37,682 --> 00:01:40,791 we're movin' this thing back to the equilibrium position, 48 00:01:40,791 --> 00:01:42,013 we're trying to push it back there. 49 00:01:42,013 --> 00:01:44,380 So if I push left, the spring pushes right. 50 00:01:44,380 --> 00:01:46,194 And if I pull the mass right, 51 00:01:46,194 --> 00:01:47,313 the spring pulls left. 52 00:01:47,313 --> 00:01:49,668 It tries to restore always, it tries to restore 53 00:01:49,668 --> 00:01:51,483 mass back to the equilibrium position. 54 00:01:51,483 --> 00:01:52,655 Sam for the pendulum. 55 00:01:52,655 --> 00:01:54,023 If I pull the pendulum to the right, 56 00:01:54,023 --> 00:01:55,430 gravity is the restoring force 57 00:01:55,430 --> 00:01:56,826 trying to bring it back to the left. 58 00:01:56,826 --> 00:01:58,496 But if I pull the mass to the left, 59 00:01:58,496 --> 00:02:00,173 gravity tries to pull it back to the right, 60 00:02:00,173 --> 00:02:01,501 always trying to restore this mass 61 00:02:01,501 --> 00:02:03,729 back to the equilibrium position. 62 00:02:03,729 --> 00:02:06,345 That's what we mean by a restoring force. 63 00:02:06,345 --> 00:02:08,154 Now there's lots of oscillators, 64 00:02:08,154 --> 00:02:10,269 but only some of those oscillators 65 00:02:10,269 --> 00:02:13,234 are really special, and we give those a special name. 66 00:02:13,234 --> 00:02:15,954 We call them Simple Harmonic Oscillators. 67 00:02:15,954 --> 00:02:17,017 And you might be thinking, 68 00:02:17,017 --> 00:02:18,817 that's a pretty dumb name because that 69 00:02:18,817 --> 00:02:20,809 doesn't sound very simple. 70 00:02:20,809 --> 00:02:21,992 But they're something called the 71 00:02:21,992 --> 00:02:24,602 Simple Harmonic Oscillator. 72 00:02:24,602 --> 00:02:26,333 So what makes Simple Harmonic Oscillator's 73 00:02:26,333 --> 00:02:28,157 so special is that even though 74 00:02:28,157 --> 00:02:30,893 all oscillators have a restoring force, 75 00:02:30,893 --> 00:02:33,709 Simple Harmonic Oscillators have a restoring force 76 00:02:33,709 --> 00:02:37,024 that's proportional to the amount of displacement. 77 00:02:37,024 --> 00:02:38,345 So what that means is if I pull 78 00:02:38,345 --> 00:02:41,277 this mass to the right there will be a restoring force, 79 00:02:41,277 --> 00:02:43,289 but if it's proportional to the displacement, 80 00:02:43,289 --> 00:02:46,113 if I pulled this mass back twice as much, 81 00:02:46,113 --> 00:02:48,603 I'd get twice the restoring force. 82 00:02:48,603 --> 00:02:50,619 And if I pulled it back three times as much, 83 00:02:50,619 --> 00:02:52,828 I'd get three times the restoring force. 84 00:02:52,828 --> 00:02:53,916 Same down here. 85 00:02:53,916 --> 00:02:55,053 If I pulled this pendulum back 86 00:02:55,053 --> 00:02:56,708 with two times the angle, 87 00:02:56,708 --> 00:02:58,810 I'd get two times the restoring force. 88 00:02:58,810 --> 00:03:01,522 If that's the case, then you've got what 89 00:03:01,522 --> 00:03:04,322 we call a Simple Harmonic Oscillator. 90 00:03:04,322 --> 00:03:05,927 And you still might not be impressed, 91 00:03:05,927 --> 00:03:07,664 you might be like who cares if the 92 00:03:07,664 --> 00:03:10,340 restoring force is proportional to the displacement. 93 00:03:10,340 --> 00:03:11,975 Why should I care about that? 94 00:03:11,975 --> 00:03:13,598 You should care about that because these 95 00:03:13,598 --> 00:03:15,978 satisfy some very special rules that 96 00:03:15,978 --> 00:03:17,609 I'll show you throughout this video 97 00:03:17,609 --> 00:03:19,424 and it turns out that even though 98 00:03:19,424 --> 00:03:21,172 this doesn't sound very simple, 99 00:03:21,172 --> 00:03:23,729 they are much simpler than the alternative 100 00:03:23,729 --> 00:03:26,113 of Non-Simple Harmonic Oscillators. 101 00:03:26,113 --> 00:03:27,638 So these are what we typically study in 102 00:03:27,638 --> 00:03:29,664 introductory physics classes, 103 00:03:29,664 --> 00:03:31,570 and it turns out a mass on a spring 104 00:03:31,570 --> 00:03:33,734 is a Simple Harmonic Oscillator, 105 00:03:33,734 --> 00:03:36,928 and a pendulum also for small oscillations, 106 00:03:36,928 --> 00:03:38,239 here you have to make a caveat, 107 00:03:38,239 --> 00:03:41,185 you have to say only for small angles, 108 00:03:41,185 --> 00:03:42,879 but for those small angles, 109 00:03:42,879 --> 00:03:45,727 the pendulum is a Simple Harmonic Oscillator as well. 110 00:03:45,727 --> 00:03:47,638 Now in this video, we're just going to look 111 00:03:47,638 --> 00:03:49,810 at the mass on the spring to make it simple. 112 00:03:49,810 --> 00:03:51,196 We could look at the pendulum later. 113 00:03:51,196 --> 00:03:52,508 So I'm going to get rid of the pendulum 114 00:03:52,508 --> 00:03:55,096 so we can focus on this mass on a spring. 115 00:03:55,096 --> 00:03:56,508 Now you might not be convinced, 116 00:03:56,508 --> 00:03:57,646 you might be like how do we know 117 00:03:57,646 --> 00:03:59,393 this mass on a spring is really a 118 00:03:59,393 --> 00:04:01,401 Simple Harmonic Oscillator? 119 00:04:01,401 --> 00:04:03,624 Well we can prove it because the force that's 120 00:04:03,624 --> 00:04:07,387 providing the restoring force in this case is the spring. 121 00:04:07,387 --> 00:04:10,113 So the spring is the restoring force in this case, 122 00:04:10,113 --> 00:04:12,689 and we know the formula for the force from a spring, 123 00:04:12,689 --> 00:04:14,310 that's given by Hooke's Law. 124 00:04:14,310 --> 00:04:16,344 And Hooke's Law says that the spring force, 125 00:04:16,344 --> 00:04:17,872 the force provided by the spring, 126 00:04:17,872 --> 00:04:18,930 is going to be negative. 127 00:04:18,930 --> 00:04:22,643 The spring constant times x, the spring displacement, 128 00:04:22,643 --> 00:04:24,520 so x is going to be positive if 129 00:04:24,520 --> 00:04:27,282 the spring has been displaced to the right 130 00:04:27,282 --> 00:04:28,964 because the spring's going to get longer. 131 00:04:28,964 --> 00:04:30,684 So this would be a positive x amount. 132 00:04:30,684 --> 00:04:32,136 And if you compress the spring, 133 00:04:32,136 --> 00:04:34,504 the length of the spring gets smaller, 134 00:04:34,504 --> 00:04:36,926 that's going to count as a negative x value. 135 00:04:36,926 --> 00:04:37,759 But think about it, 136 00:04:37,759 --> 00:04:39,839 if I compress the spring to the left, 137 00:04:39,839 --> 00:04:41,937 my x is going to be negative, 138 00:04:41,937 --> 00:04:43,602 and that negative combines with this negative 139 00:04:43,602 --> 00:04:46,462 to be a positive so I'd get a positive force. 140 00:04:46,462 --> 00:04:49,514 That means the spring is there's a force to the right. 141 00:04:49,514 --> 00:04:50,393 And that makes sense. 142 00:04:50,393 --> 00:04:53,214 Restoring, it means it opposes what you do. 143 00:04:53,214 --> 00:04:54,622 If you push the mass to the left, 144 00:04:54,622 --> 00:04:56,335 the spring is going to push to the right. 145 00:04:56,335 --> 00:04:57,271 And if we did it the other way, 146 00:04:57,271 --> 00:04:58,721 if we pulled the mass to the right, 147 00:04:58,721 --> 00:05:00,694 now that would be a positive x value. 148 00:05:00,694 --> 00:05:02,379 If I have a positive x value in here 149 00:05:02,379 --> 00:05:03,967 and combine that with a negative, 150 00:05:03,967 --> 00:05:06,093 I'd get a negative spring force. 151 00:05:06,093 --> 00:05:07,634 And that means the spring would 152 00:05:07,634 --> 00:05:08,611 be pulling to the left, 153 00:05:08,611 --> 00:05:11,901 it's restoring this mass back to the equilibrium position. 154 00:05:11,901 --> 00:05:14,337 And that's exactly what an oscillator does. 155 00:05:14,337 --> 00:05:15,368 And look at it up here, 156 00:05:15,368 --> 00:05:17,259 this spring force, this restoring force, 157 00:05:17,259 --> 00:05:20,101 is proportional to the displacement. 158 00:05:20,101 --> 00:05:22,387 So x is the displacement, this is a force 159 00:05:22,387 --> 00:05:24,410 that's proportional to the displacement. 160 00:05:24,410 --> 00:05:25,732 And that's the definition. 161 00:05:25,732 --> 00:05:28,565 That was what we meant by Simple Harmonic Oscillator. 162 00:05:28,565 --> 00:05:30,195 So that's why masses on springs 163 00:05:30,195 --> 00:05:32,812 are going to be Simple Harmonic Oscillators, 164 00:05:32,812 --> 00:05:34,448 because the restoring force is 165 00:05:34,448 --> 00:05:36,770 proportional to the displacement. 166 00:05:36,770 --> 00:05:38,524 Now to be completely honest, 167 00:05:38,524 --> 00:05:42,027 it has to be negatively proportional to the displacement. 168 00:05:42,027 --> 00:05:44,856 If you just had f equals kx with no negative, 169 00:05:44,856 --> 00:05:46,433 then if you displaced it to the right, 170 00:05:46,433 --> 00:05:48,006 the force would be to the right 171 00:05:48,006 --> 00:05:49,774 which would displace it more to the right, 172 00:05:49,774 --> 00:05:52,227 which would create a larger force to the right, 173 00:05:52,227 --> 00:05:53,696 this would be a runaway solution, 174 00:05:53,696 --> 00:05:55,365 this thing would blow up, that wouldn't be good. 175 00:05:55,365 --> 00:05:58,570 So it's really forces that have a negative 176 00:05:58,570 --> 00:06:00,838 proportionality to the displacement. 177 00:06:00,838 --> 00:06:03,007 That way it's going to restore 178 00:06:03,007 --> 00:06:04,789 back to the equilibrium position 179 00:06:04,789 --> 00:06:06,688 and if this is proportional, 180 00:06:06,688 --> 00:06:08,666 you get a Simple Harmonic Oscillator. 181 00:06:08,666 --> 00:06:09,762 And so we should talk about this, 182 00:06:09,762 --> 00:06:12,070 what the heck do we mean by simple? 183 00:06:12,070 --> 00:06:13,824 Like what is simple about this? 184 00:06:13,824 --> 00:06:15,521 It turns out that what's simple is that 185 00:06:15,521 --> 00:06:17,863 these types of oscillators are going to be 186 00:06:17,863 --> 00:06:20,652 described by sin and cosin functions. 187 00:06:20,652 --> 00:06:22,825 So Simple Harmonic Oscillators will be described 188 00:06:22,825 --> 00:06:26,248 by sin and cosin and that should make sense 189 00:06:26,248 --> 00:06:28,212 because think about sin and cosin, 190 00:06:28,212 --> 00:06:29,330 what do those look like? 191 00:06:29,330 --> 00:06:30,789 Sin and cosin look like this. 192 00:06:30,789 --> 00:06:32,281 So here's what sin looks like, 193 00:06:32,281 --> 00:06:34,403 it's a function that oscillates back and forth. 194 00:06:34,403 --> 00:06:37,358 And cosin looks like this, it starts up here, 195 00:06:37,358 --> 00:06:40,000 so it's also a function that oscillates back and forth. 196 00:06:40,000 --> 00:06:42,674 And so these are simple, turns out those are very simple 197 00:06:42,674 --> 00:06:44,769 functions that oscillate back and forth. 198 00:06:44,769 --> 00:06:46,547 And because of that, we like those. 199 00:06:46,547 --> 00:06:48,179 In physics, we love things that are 200 00:06:48,179 --> 00:06:49,596 described by sin and cosin, 201 00:06:49,596 --> 00:06:51,065 it turns out they're pretty easy 202 00:06:51,065 --> 00:06:52,703 to deal with mathematically. 203 00:06:52,703 --> 00:06:54,239 Maybe you don't feel that way, 204 00:06:54,239 --> 00:06:56,512 but they're much easier than the alternatives 205 00:06:56,512 --> 00:06:58,378 of other things that could oscillate. 206 00:06:58,378 --> 00:07:00,555 So that's what Simple Harmonic Oscillators mean. 207 00:07:00,555 --> 00:07:02,251 But let's try to get some intuition, 208 00:07:02,251 --> 00:07:05,160 what is really going on for this mass on a spring? 209 00:07:05,160 --> 00:07:07,835 So let's imagine we pull the mass back, right? 210 00:07:07,835 --> 00:07:09,543 So the mass, if the mass just continues 211 00:07:09,543 --> 00:07:11,380 to sit at the equilibrium position, 212 00:07:11,380 --> 00:07:13,461 it's a pretty boring problem because 213 00:07:13,461 --> 00:07:15,425 the net force right there would be 0 and 214 00:07:15,425 --> 00:07:16,873 it would just continue to sit there. 215 00:07:16,873 --> 00:07:18,688 So let's say we pull the mass back, 216 00:07:18,688 --> 00:07:20,882 we pull it back by a certain amount. 217 00:07:20,882 --> 00:07:22,664 Say we pull it back this far, 218 00:07:22,664 --> 00:07:24,242 and then we let go. 219 00:07:24,242 --> 00:07:26,067 So since we let go of the mass, 220 00:07:26,067 --> 00:07:27,692 we've released it at rest. 221 00:07:27,692 --> 00:07:29,355 So it started at rest. 222 00:07:29,355 --> 00:07:32,021 And that means the speed initially over here is 0. 223 00:07:32,021 --> 00:07:33,583 So it starts off with 0 speed, 224 00:07:33,583 --> 00:07:35,538 but the spring has been stretched. 225 00:07:35,538 --> 00:07:37,729 And so the spring is going to restore, 226 00:07:37,729 --> 00:07:39,102 right, the spring is always trying to 227 00:07:39,102 --> 00:07:42,617 restore the mass back to the equilibrium position. 228 00:07:42,617 --> 00:07:44,626 So the spring pulling the mass to the left, 229 00:07:44,626 --> 00:07:47,299 speeding it up, speeds the mass up 230 00:07:47,299 --> 00:07:49,231 until it gets to the equilibrium position, 231 00:07:49,231 --> 00:07:50,822 and then the spring realizes, 232 00:07:50,822 --> 00:07:52,563 oh crud, I messed up. 233 00:07:52,563 --> 00:07:54,027 I wanted to get the mass here but 234 00:07:54,027 --> 00:07:55,847 I pulled it so much this mass has a 235 00:07:55,847 --> 00:07:58,091 huge speed to the left now. 236 00:07:58,091 --> 00:08:00,116 And masses don't just stop on their own, 237 00:08:00,116 --> 00:08:01,752 They need some force to do that. 238 00:08:01,752 --> 00:08:03,333 So this mass has inertia, 239 00:08:03,333 --> 00:08:04,883 and according to Newton's First Law, 240 00:08:04,883 --> 00:08:06,398 it's going to try and keep moving. 241 00:08:06,398 --> 00:08:08,050 So even though the spring got the mass 242 00:08:08,050 --> 00:08:10,648 back to the equilibrium position, that was its goal, 243 00:08:10,648 --> 00:08:12,791 it got it back there with this huge speed 244 00:08:12,791 --> 00:08:14,714 and the mass continues straight through 245 00:08:14,714 --> 00:08:16,512 the equilibrium position and the spring 246 00:08:16,512 --> 00:08:18,386 starts getting compressed and the spring's like 247 00:08:18,386 --> 00:08:20,588 oh no, I've gotta start pushing this thing to the right. 248 00:08:20,588 --> 00:08:22,780 I want to get the mass back to the equilibrium position. 249 00:08:22,780 --> 00:08:25,059 So now the spring's pushing to the right, 250 00:08:25,059 --> 00:08:27,738 slowing the mass down until it stops it, 251 00:08:27,738 --> 00:08:29,069 but the spring is compressed, 252 00:08:29,069 --> 00:08:30,761 so it's going to keep pushing to the right. 253 00:08:30,761 --> 00:08:32,755 Now it's pushing in the direction the mass is moving. 254 00:08:32,755 --> 00:08:34,174 Now it's got it going back to 255 00:08:34,174 --> 00:08:35,619 the equilibrium position again, 256 00:08:35,620 --> 00:08:38,000 which is good, but again, same mistake, 257 00:08:38,000 --> 00:08:39,919 the spring gets this mass back 258 00:08:39,919 --> 00:08:43,020 to the equilibrium position with a huge speed to the right, 259 00:08:43,020 --> 00:08:44,530 and now the spring's like oh great, 260 00:08:44,530 --> 00:08:46,514 I did it again, I got this mass back 261 00:08:46,514 --> 00:08:48,469 where I wanted it, but this mass 262 00:08:48,469 --> 00:08:50,898 had a huge speed and it's got inertia, 263 00:08:50,898 --> 00:08:53,404 and so this mass is going to keep moving to the right, 264 00:08:53,404 --> 00:08:55,102 past the equilibrium position. 265 00:08:55,102 --> 00:08:57,106 And this is why the oscillation happens. 266 00:08:57,106 --> 00:08:59,496 It's a constant fight between inertia 267 00:08:59,496 --> 00:09:01,709 of the mass wanting to keep moving 268 00:09:01,709 --> 00:09:04,133 because it's got mass and it's got velocity, 269 00:09:04,133 --> 00:09:06,871 and the restoring force that is desperately 270 00:09:06,871 --> 00:09:08,882 trying to get this mass back to the equilibrium 271 00:09:08,882 --> 00:09:10,876 position and they can never quite figure it out 272 00:09:10,876 --> 00:09:12,697 because they keep overshooting each other 273 00:09:12,697 --> 00:09:15,504 and this oscillation happens over and over and over. 274 00:09:15,504 --> 00:09:16,962 So just knowing the story, 275 00:09:16,962 --> 00:09:18,410 let's you say some really important 276 00:09:18,410 --> 00:09:20,423 things about the oscillation. 277 00:09:20,423 --> 00:09:22,819 One of them is that at these end points, 278 00:09:22,819 --> 00:09:25,098 at these points of maximum compression 279 00:09:25,098 --> 00:09:27,588 or extension, the speed is 0. 280 00:09:27,588 --> 00:09:29,684 So this mass is moving the slowest, 281 00:09:29,684 --> 00:09:31,923 i.e. it's not moving at all at 282 00:09:31,923 --> 00:09:35,190 these maximum points of compression or extension 283 00:09:35,190 --> 00:09:36,521 because that's where the spring 284 00:09:36,521 --> 00:09:38,744 has stopped the mass and started bringing 285 00:09:38,744 --> 00:09:40,531 it back in the other direction. 286 00:09:40,531 --> 00:09:43,231 Whereas in the middle, at the equilibrium position, 287 00:09:43,231 --> 00:09:45,106 you get the most speed. 288 00:09:45,106 --> 00:09:47,284 So this is where the mass is moving fastest, 289 00:09:47,284 --> 00:09:49,939 when the spring has got it back to the equilibrium position 290 00:09:49,939 --> 00:09:52,750 and the spring at that point realizes oh crap, 291 00:09:52,750 --> 00:09:54,404 this mass is going really fast, 292 00:09:54,404 --> 00:09:57,252 and the mass is coming at it or going away from it 293 00:09:57,252 --> 00:09:59,670 too fast for the spring to stop it immediately. 294 00:09:59,670 --> 00:10:02,084 So if the equilibrium point this mass has the 295 00:10:02,084 --> 00:10:04,541 most speed during the oscillation. 296 00:10:04,541 --> 00:10:06,080 So we could also ask where will the 297 00:10:06,080 --> 00:10:08,752 magnitude of the restoring force be biggest 298 00:10:08,752 --> 00:10:11,411 and where will it be least during this oscillation? 299 00:10:11,411 --> 00:10:12,903 And we've got a formula for that. 300 00:10:12,903 --> 00:10:15,230 Look at, the spring force is the restoring force. 301 00:10:15,230 --> 00:10:16,767 So we could just ask where will 302 00:10:16,767 --> 00:10:18,697 the spring force be biggest? 303 00:10:18,697 --> 00:10:21,895 That's going to be where this x is biggest or smallest. 304 00:10:21,895 --> 00:10:23,337 So if we wanted to know where the 305 00:10:23,337 --> 00:10:25,753 magnitude of this f is largest, 306 00:10:25,753 --> 00:10:27,400 we could just ask where will 307 00:10:27,400 --> 00:10:29,968 the magnitude of the x be largest? 308 00:10:29,968 --> 00:10:32,055 If we don't care about which way the force is, 309 00:10:32,055 --> 00:10:33,223 we just want to know where we'll 310 00:10:33,223 --> 00:10:34,660 get a really big force, 311 00:10:34,660 --> 00:10:35,807 we just try to figure out where 312 00:10:35,807 --> 00:10:37,102 will I get the biggest x? 313 00:10:37,102 --> 00:10:39,036 X is displacement. 314 00:10:39,036 --> 00:10:42,544 So the x value at the equilibrium position is 0. 315 00:10:42,544 --> 00:10:44,675 So there's no displacement of the spring right here, 316 00:10:44,675 --> 00:10:47,023 that's what it means to be the equilibrium position, 317 00:10:47,023 --> 00:10:49,482 this is the natural length of the spring. 318 00:10:49,482 --> 00:10:51,709 That's the length that the spring wants to be. 319 00:10:51,709 --> 00:10:53,875 If the spring has that shape right there, 320 00:10:53,875 --> 00:10:55,314 it doesn't push or pull. 321 00:10:55,314 --> 00:10:57,731 But if you've displaced it this way, 322 00:10:57,731 --> 00:11:00,745 or the other way, this would be positive displacement, 323 00:11:00,745 --> 00:11:02,569 and this would be negative displacement, 324 00:11:02,569 --> 00:11:04,195 now the spring's going to exert a force. 325 00:11:04,195 --> 00:11:06,136 So where will the force be greatest? 326 00:11:06,136 --> 00:11:07,139 It's where the spring has been 327 00:11:07,139 --> 00:11:09,624 compressed or stretched the most. 328 00:11:09,624 --> 00:11:11,701 So at these points here, at the points of 329 00:11:11,701 --> 00:11:14,050 maximum extension or compression, 330 00:11:14,050 --> 00:11:16,818 you're going to have the greatest amount of force. 331 00:11:16,818 --> 00:11:19,317 So greatest magnitude of force, 332 00:11:19,317 --> 00:11:21,260 because the spring is really stretched, 333 00:11:21,260 --> 00:11:23,571 it's going to pull with a great amount of force 334 00:11:23,571 --> 00:11:25,246 back toward the equilibrium position. 335 00:11:25,246 --> 00:11:26,802 And we can say which way it points, right? 336 00:11:26,802 --> 00:11:29,046 This spring's going to be pulling to the left, 337 00:11:29,046 --> 00:11:32,050 so there's going to be a great spring force to the left. 338 00:11:32,050 --> 00:11:33,932 Technically that'd be a negative force, 339 00:11:33,932 --> 00:11:36,435 so I mean, if you're taking sins into account, 340 00:11:36,435 --> 00:11:38,679 you could say that that's the least force 341 00:11:38,679 --> 00:11:40,310 because it's really negative. 342 00:11:40,310 --> 00:11:42,271 But if you're just worried about magnitude, 343 00:11:42,271 --> 00:11:44,789 that would be a great magnitude of force. 344 00:11:44,789 --> 00:11:45,901 And then also over here, 345 00:11:45,901 --> 00:11:47,778 at the maximum compression, 346 00:11:47,778 --> 00:11:50,705 this spring is really pushing the mass to the right, 347 00:11:50,705 --> 00:11:52,880 you get a great amount of force this way 348 00:11:52,880 --> 00:11:54,500 because your x, even though it's 349 00:11:54,500 --> 00:11:55,997 very negative at that point, 350 00:11:55,997 --> 00:11:57,831 it's going to give you a large amount of force. 351 00:11:57,831 --> 00:11:59,777 And so here you would also have a great 352 00:11:59,777 --> 00:12:01,624 amount of magnitude of force 353 00:12:01,624 --> 00:12:03,349 which can be confusing because look it. 354 00:12:03,349 --> 00:12:06,363 At these end points, you have the least speed, 355 00:12:06,363 --> 00:12:08,534 but the greatest force. 356 00:12:08,534 --> 00:12:10,012 Sometimes that freaks people out. 357 00:12:10,012 --> 00:12:11,880 They're like, how can you have a great force 358 00:12:11,880 --> 00:12:13,446 and your speed be so small? 359 00:12:13,446 --> 00:12:14,965 Well that's the point where the 360 00:12:14,965 --> 00:12:17,073 spring has stopped the mass and started 361 00:12:17,073 --> 00:12:18,987 pulling it in the other direction. 362 00:12:18,987 --> 00:12:20,928 So even though the speed is 0, 363 00:12:20,928 --> 00:12:22,515 the force is greatest. 364 00:12:22,515 --> 00:12:24,630 So, be careful, force does not have to be 365 00:12:24,630 --> 00:12:26,456 proportional to the speed. 366 00:12:26,456 --> 00:12:27,836 The force has to be proportional 367 00:12:27,836 --> 00:12:29,950 to the acceleration, right? 368 00:12:29,950 --> 00:12:31,485 Because we know net force, 369 00:12:31,485 --> 00:12:34,625 we could say that the net force is equal to ma. 370 00:12:34,625 --> 00:12:37,028 So wherever you have the largest amount of force, 371 00:12:37,028 --> 00:12:39,641 you'll have the largest amount of acceleration. 372 00:12:39,641 --> 00:12:41,563 So we could also say at these endpoints, 373 00:12:41,563 --> 00:12:44,248 you'll have not only the greatest magnitude of the force, 374 00:12:44,248 --> 00:12:47,623 but the greatest magnitude of acceleration as well. 375 00:12:47,623 --> 00:12:49,880 Because where you're pulling or pushing on something 376 00:12:49,880 --> 00:12:51,215 with the greatest amount of force, 377 00:12:51,215 --> 00:12:52,723 you're going to get the greatest amount 378 00:12:52,723 --> 00:12:55,267 of acceleration according to Newton's Second Law. 379 00:12:55,267 --> 00:12:56,362 So at these endpoints, 380 00:12:56,362 --> 00:13:00,360 the force is greatest, the acceleration is also greatest. 381 00:13:00,360 --> 00:13:02,745 The magnitude, the acceleration is also greatest 382 00:13:02,745 --> 00:13:06,595 even though the speed is 0 at those points. 383 00:13:06,595 --> 00:13:07,592 So those are the points where you 384 00:13:07,592 --> 00:13:09,872 get the greatest force and greatest acceleration. 385 00:13:09,872 --> 00:13:11,753 Where will you get the least amount 386 00:13:11,753 --> 00:13:14,332 of magnitude of force and magnitude of acceleration? 387 00:13:14,332 --> 00:13:15,239 Well look at up here. 388 00:13:15,239 --> 00:13:17,436 The least force will happen where you 389 00:13:17,436 --> 00:13:19,958 get the least possible displacement. 390 00:13:19,958 --> 00:13:21,641 And the least possible displacement's 391 00:13:21,641 --> 00:13:23,471 right here in the middle, this equilibrium position 392 00:13:23,471 --> 00:13:25,263 is when x equals 0. 393 00:13:25,263 --> 00:13:27,563 That's when the spring is not pushing or pulling. 394 00:13:27,563 --> 00:13:28,668 When it's at this point here. 395 00:13:28,668 --> 00:13:30,763 So when the mass is passing through 396 00:13:30,763 --> 00:13:34,398 the equilibrium position, there is 0 force. 397 00:13:34,398 --> 00:13:36,253 Right, that's the point where the mass got 398 00:13:36,253 --> 00:13:37,445 back there and the spring was like 399 00:13:37,445 --> 00:13:40,397 I'm glad I got it back to the equilibrium position 400 00:13:40,397 --> 00:13:42,812 and then the spring quickly realized, 401 00:13:42,812 --> 00:13:44,828 oh no, this mass, I got it back there, 402 00:13:44,828 --> 00:13:46,547 but the mass was moving really fast, 403 00:13:46,547 --> 00:13:48,350 so it shot straight through that point. 404 00:13:48,350 --> 00:13:50,114 But right at that moment, 405 00:13:50,114 --> 00:13:52,185 the spring had this glorious moment 406 00:13:52,185 --> 00:13:53,363 where it thought it had done it 407 00:13:53,363 --> 00:13:55,368 and it stopped exerting any force 408 00:13:55,368 --> 00:13:58,104 because at that point, the x is 0. 409 00:13:58,104 --> 00:14:00,801 And if x is 0, we know from up here, the force is 0. 410 00:14:00,801 --> 00:14:03,762 So this would be the least possible force. 411 00:14:03,762 --> 00:14:06,599 And I guess I should say it's actually 0 force, 412 00:14:06,599 --> 00:14:07,722 it's not just the least, 413 00:14:07,722 --> 00:14:10,259 there is 0 force exerted at this point. 414 00:14:10,259 --> 00:14:12,551 And if there's 0 force, by the same argument, 415 00:14:12,551 --> 00:14:15,565 we could say that there's 0 acceleration at that point. 416 00:14:15,565 --> 00:14:17,483 Hopefully that gives you some intuition 417 00:14:17,483 --> 00:14:20,018 about why oscillators do what they do 418 00:14:20,018 --> 00:14:22,424 and where you might find the largest speed 419 00:14:22,424 --> 00:14:24,418 or force at any given point. 420 00:14:24,418 --> 00:14:26,617 So recapping, objects with a restoring force 421 00:14:26,617 --> 00:14:29,173 that's negatively proportional to the displacement 422 00:14:29,173 --> 00:14:31,012 will be a Simple Harmonic Oscillator 423 00:14:31,012 --> 00:14:33,118 and for all Simple Harmonic Oscillators, 424 00:14:33,118 --> 00:14:35,269 at the equilibrium position you'll get 425 00:14:35,269 --> 00:14:38,701 the greatest speed but 0 restoring force 426 00:14:38,701 --> 00:14:40,383 and 0 acceleration. 427 00:14:40,383 --> 00:14:43,032 Whereas at the points of maximum displacement, 428 00:14:43,032 --> 00:14:46,064 you'll get the maximum magnitude of restoring force 429 00:14:46,064 --> 00:00:00,000 and acceleration but the least possible speed.