WEBVTT 00:00:00.047 --> 00:00:02.750 This is College Physics Answers with Shaun Dychko. 00:00:03.390 --> 00:00:05.914 In figure 7.10 part C, 00:00:06.234 --> 00:00:08.765 you can see a graph showing the force applied 00:00:09.821 --> 00:00:12.571 versus the amount of compression of the spring 00:00:13.688 --> 00:00:17.641 and the work that this force does is the area underneath this curve. 00:00:18.407 --> 00:00:23.537 So the area is this triangle and so given a compression of distance x, 00:00:24.305 --> 00:00:29.568 the area is going to be one half times kx being the height 00:00:30.023 --> 00:00:33.311 and that comes from this force formula, Hooks Law. 00:00:34.195 --> 00:00:37.052 I mean this negative sign there I suppose that just has to do with direction 00:00:37.077 --> 00:00:38.732 and we care only about magnitude here. 00:00:39.076 --> 00:00:41.170 So the magnitude of this height is kx 00:00:41.945 --> 00:00:45.890 and the base of this triangle is x. 00:00:46.578 --> 00:00:51.357 Then if you were to compress it by two times that distance -- let's draw that. 00:00:51.766 --> 00:00:56.420 We'll take something that is this length 00:00:57.414 --> 00:00:59.693 and we'll move it over here 00:01:01.054 --> 00:01:05.177 and so now the base of the triangle will be at this point. 00:01:07.740 --> 00:01:12.500 Then we'll go up to the point where it intersects this line here -- 00:01:16.253 --> 00:01:17.526 there, 00:01:18.190 --> 00:01:24.394 and so now we have a distance of 2x on the base 00:01:25.000 --> 00:01:28.727 and the height as well k, times 2x as well. 00:01:29.515 --> 00:01:34.042 But what we care about is the graphical interpretation of this 00:01:35.740 --> 00:01:36.575 and that is -- 00:01:36.600 --> 00:01:40.023 let's consider how much area is underneath this curve now. 00:01:41.671 --> 00:01:44.507 Well, if you had this triangle -- 00:01:44.898 --> 00:01:47.083 here is the area that we had before. 00:01:49.953 --> 00:01:51.849 If we move that triangle over here 00:01:53.227 --> 00:01:54.235 we can see that 00:01:56.475 --> 00:02:00.697 this new area is more than two times the old triangle 00:02:00.722 --> 00:02:03.918 because this new area which is in purple 00:02:06.773 --> 00:02:08.089 is the work done. 00:02:09.758 --> 00:02:11.785 If you consider the old area, 00:02:12.538 --> 00:02:16.379 the work done previously with a compression of one x, 00:02:16.937 --> 00:02:18.691 it equals this -- 00:02:19.148 --> 00:02:20.855 I guess I should try a different color -- 00:02:21.531 --> 00:02:23.144 it equals this green triangle 00:02:23.445 --> 00:02:26.293 and then two times the green triangle would be that green triangle 00:02:26.318 --> 00:02:29.433 plus that green triangle again here 00:02:30.234 --> 00:02:35.449 and you can see that two times the area before 00:02:35.660 --> 00:02:38.675 does not fill up the entire area under the curve 00:02:39.063 --> 00:02:42.941 when the spring is compressed twice what it was before. 00:02:43.687 --> 00:02:45.886 So this is just a way of illustrating that 00:02:46.464 --> 00:02:50.300 the work done is non-linear. 00:02:50.655 --> 00:02:54.136 The work done when you compress by twox 00:02:54.161 --> 00:02:57.910 is not two times the work done when you compress by a single x 00:02:58.250 --> 00:02:59.445 because in green, 00:02:59.886 --> 00:03:01.986 I've shaded the total area one would expect 00:03:02.011 --> 00:03:04.681 with two times the work done in the first case 00:03:05.064 --> 00:03:06.923 and it's not accounting for all the area. 00:03:06.948 --> 00:03:12.719 It's missing this portion here and that portion there -- 00:03:15.204 --> 00:03:20.079 well, if we were to take this triangle and it's a bit complicated to do, 00:03:20.104 --> 00:03:21.383 but if we were to reflect it, 00:03:22.156 --> 00:03:25.841 it would fill up this area and then this bit here 00:03:26.945 --> 00:03:29.982 would be of equal size to that triangle yet again. 00:03:30.295 --> 00:03:36.886 So there are one, two, three, four triangles fitting into this total space 00:03:37.199 --> 00:03:39.964 under the curve when you compress by 2x. 00:03:40.608 --> 00:03:44.308 So, the amount of work done in this case 00:03:44.333 --> 00:03:47.835 where you compress by 2x is four times as much as before. 00:03:48.445 --> 00:03:49.578 The answer is D.