1 00:00:03,440 --> 00:00:05,340 We now would like to analyze the force 2 00:00:05,340 --> 00:00:10,850 law on an idealized spring, a force law called Hooke's law. 3 00:00:10,850 --> 00:00:15,630 Let's begin by drawing a ideal spring connected 4 00:00:15,630 --> 00:00:18,150 to an object on a frictionless surface. 5 00:00:18,150 --> 00:00:21,450 So we'll draw our object. 6 00:00:21,450 --> 00:00:25,300 We'll draw our idealized spring. 7 00:00:25,300 --> 00:00:28,900 This surface here is going to be frictionless. 8 00:00:32,400 --> 00:00:36,090 And this is a wall, and we'll draw it like that. 9 00:00:36,090 --> 00:00:40,670 And now what we'd like to do is to introduce the force law 10 00:00:40,670 --> 00:00:45,150 for a spring, a force law, which will be called Hooke's law. 11 00:00:45,150 --> 00:00:47,710 So the way we'd like to analyze this 12 00:00:47,710 --> 00:00:50,100 is by drawing two pictures. 13 00:00:50,100 --> 00:00:52,930 Our first picture will have the spring 14 00:00:52,930 --> 00:00:54,970 at an equilibrium picture. 15 00:00:54,970 --> 00:00:59,470 So let's draw the wall again and let's 16 00:00:59,470 --> 00:01:03,360 draw the spring and our object. 17 00:01:03,360 --> 00:01:07,300 And this is called the equilibrium picture, 18 00:01:07,300 --> 00:01:12,039 so I'll write equilibrium picture. 19 00:01:12,039 --> 00:01:14,640 And now what we like to do is draw 20 00:01:14,640 --> 00:01:18,670 a picture, a dynamic picture at some arbitrary time t 21 00:01:18,670 --> 00:01:24,760 so here we have a dynamic picture at time t. 22 00:01:24,760 --> 00:01:27,740 And I'll draw it in a second. 23 00:01:27,740 --> 00:01:32,090 So we draw the same object, and now we're 24 00:01:32,090 --> 00:01:36,630 going to move our object to some arbitrary position. 25 00:01:36,630 --> 00:01:39,479 In this case, the spring has been stretched, 26 00:01:39,479 --> 00:01:41,950 wall, frictionless surface. 27 00:01:41,950 --> 00:01:45,020 And now, I'm in position to choose a coordinate system. 28 00:01:45,020 --> 00:01:50,690 So what I'll do is I'll choose my origin at the edge right 29 00:01:50,690 --> 00:01:54,070 here, this is called x equal 0. 30 00:01:54,070 --> 00:01:56,430 I'll show it in the dynamic picture. 31 00:01:56,430 --> 00:01:59,160 And my coordinate function for the object 32 00:01:59,160 --> 00:02:04,820 here, I'm going to refer to that as x sub t. 33 00:02:04,820 --> 00:02:07,380 Now, notice that this coordinate function is also 34 00:02:07,380 --> 00:02:11,420 an indication of how much the spring has been displaced. 35 00:02:11,420 --> 00:02:15,160 As far as directions go, we'll have an axis. 36 00:02:15,160 --> 00:02:17,340 And this is our plus x direction, 37 00:02:17,340 --> 00:02:22,450 so usually we indicate that with the unit vector i hat. 38 00:02:22,450 --> 00:02:26,060 And now Hooke's law is the statement 39 00:02:26,060 --> 00:02:33,240 that the force on the spring, that the force on the object F 40 00:02:33,240 --> 00:02:37,920 is proportional to how much the object has displaced, 41 00:02:37,920 --> 00:02:40,040 which represents either the stretching 42 00:02:40,040 --> 00:02:42,770 or the compressing of the spring. 43 00:02:42,770 --> 00:02:48,950 So it's equal to minus kx I hat. 44 00:02:48,950 --> 00:02:51,410 Now, what does this minus sign mean? 45 00:02:51,410 --> 00:02:56,829 Well, this is an example of what we call a restoring force. 46 00:02:56,829 --> 00:02:59,410 And let's look at just a couple of examples. 47 00:02:59,410 --> 00:03:03,440 When x is positive, that means that the object 48 00:03:03,440 --> 00:03:06,010 has been pulled out from the equilibrium position. 49 00:03:06,010 --> 00:03:08,780 The spring is undergoing tension, 50 00:03:08,780 --> 00:03:09,960 it's being stretched apart. 51 00:03:09,960 --> 00:03:13,430 The molecules in the spring that constitute the spring 52 00:03:13,430 --> 00:03:16,810 are being pulled apart, and there's a restoring force 53 00:03:16,810 --> 00:03:18,270 inside the spring. 54 00:03:18,270 --> 00:03:20,140 This is an atomic force in nature 55 00:03:20,140 --> 00:03:22,650 that's pulling the spring backwards, 56 00:03:22,650 --> 00:03:26,450 hence exerting a restoring force on the object. 57 00:03:26,450 --> 00:03:30,510 So this forces in the minus i hat direction. 58 00:03:30,510 --> 00:03:32,720 It's restoring in that direction. 59 00:03:32,720 --> 00:03:36,610 I'll draw the force like that. 60 00:03:36,610 --> 00:03:39,790 Now, suppose we drew another picture 61 00:03:39,790 --> 00:03:45,270 where the object is pushed in, compressing the spring. 62 00:03:45,270 --> 00:03:49,800 So let's draw once again a diagram. 63 00:03:49,800 --> 00:03:54,360 And let's show the spring under compression 64 00:03:54,360 --> 00:03:57,560 from our equilibrium position. 65 00:03:57,560 --> 00:04:03,620 Now here, we have, again, that x of t. 66 00:04:03,620 --> 00:04:06,202 But in this case, x of t is negative. 67 00:04:08,830 --> 00:04:16,106 So this is extension, so we'll call that case A. 68 00:04:16,106 --> 00:04:22,328 And case B, when x is less than 0, it's under compression. 69 00:04:25,470 --> 00:04:29,040 And then we see that with x negative 70 00:04:29,040 --> 00:04:31,423 and the additional minus sign, the force 71 00:04:31,423 --> 00:04:34,530 is in the positive direction. 72 00:04:34,530 --> 00:04:37,570 And so in both instances, the force 73 00:04:37,570 --> 00:04:40,230 is a restoring force back to equilibrium. 74 00:04:40,230 --> 00:04:42,480 So the restoring force, whether you're 75 00:04:42,480 --> 00:04:48,210 under extension or compression, is pointing, 76 00:04:48,210 --> 00:04:51,540 let's just call it, is the direction 77 00:04:51,540 --> 00:05:02,470 of the force is towards the equilibrium position. 78 00:05:02,470 --> 00:05:08,050 And this example of a restoring force 79 00:05:08,050 --> 00:05:11,950 is also only proportional to x. 80 00:05:11,950 --> 00:05:15,470 And so in that case, we can add the word linear, 81 00:05:15,470 --> 00:05:18,280 because it's just to the single power x. 82 00:05:18,280 --> 00:05:22,350 And this is an example of a linear restoring force. 83 00:05:22,350 --> 00:05:27,943 And this is a model for an ideal spring. 84 00:05:31,620 --> 00:05:36,130 Now, this constant k is called the spring constant. 85 00:05:38,810 --> 00:05:41,159 And the units of the spring constant, 86 00:05:41,159 --> 00:05:44,850 if you divide the units of force by the units of distance-- 87 00:05:44,850 --> 00:05:53,180 so we have SI units are Newton over meters, 88 00:05:53,180 --> 00:05:55,600 and that's the measure of the spring constant. 89 00:05:55,600 --> 00:05:59,340 Now, what we'll show next is an experiment in which we 90 00:05:59,340 --> 00:06:04,492 can figure out how to actually measure that spring constant k.