WEBVTT 00:00:00.050 --> 00:00:02.160 INSTRUCTOR: Welcome to this lesson. 00:00:02.160 --> 00:00:05.080 In this video, I'm just going to do a brief proof on 00:00:05.080 --> 00:00:08.540 how to find the height that a pendulum swings, 00:00:08.540 --> 00:00:11.320 given length of the pendulum and theta. 00:00:11.320 --> 00:00:13.760 I'm just going to be doing a simple pendulum here. 00:00:13.760 --> 00:00:17.640 We're not doing a pendulum with a mass on string. There's some pendulum. 00:00:17.640 --> 00:00:23.080 There's a bob here hanging by a rope with a length, l, and this rope swings through an arc, 00:00:23.080 --> 00:00:26.575 and it goes to a particular height over here. 00:00:26.575 --> 00:00:31.330 We might want to know what exactly that height is. 00:00:31.330 --> 00:00:34.175 Let me just give you the variables that we would start with here. 00:00:34.175 --> 00:00:36.245 Let's say I have theta here, 00:00:36.245 --> 00:00:39.010 and I have some length here like this, 00:00:39.010 --> 00:00:42.270 I'll call that l and it's the same length here. 00:00:42.270 --> 00:00:47.010 This is l also because it just swings and it swings across some arc here, back and forth. 00:00:47.010 --> 00:00:48.830 It'll just swing back and forth, the pendulum. 00:00:48.830 --> 00:00:50.490 Let's just say this is the height that it swings. 00:00:50.490 --> 00:00:53.410 I drew a ground down here just to emphasize something, 00:00:53.410 --> 00:00:57.350 because our lowest working point is going to be right here and that's 00:00:57.350 --> 00:00:58.890 the lowest point that the center of 00:00:58.890 --> 00:01:02.795 mass passes, and that's what we're going to call our datum. 00:01:02.795 --> 00:01:04.680 That's our lowest working point. 00:01:04.680 --> 00:01:06.260 We're going to make that our zero point. 00:01:06.260 --> 00:01:09.500 Now, I drew some ground below it just to 00:01:09.500 --> 00:01:13.600 emphasize that it doesn't matter if the ground is lower than that point. 00:01:13.600 --> 00:01:15.575 We're going to make that our lowest working point. 00:01:15.575 --> 00:01:19.900 Let's say that I want to find out the height that this thing swings up here. 00:01:19.900 --> 00:01:22.730 In other words, I want to know how 00:01:22.730 --> 00:01:26.910 high does this pendulum swing when it comes up to here? 00:01:26.910 --> 00:01:29.870 Because maybe I'm doing a potential energy problem. 00:01:29.870 --> 00:01:33.650 Maybe I want to find out the potential energy at this point, 00:01:33.650 --> 00:01:35.330 the gravitational potential energy. 00:01:35.330 --> 00:01:39.710 Maybe I want to know what is the MGH at this point. 00:01:39.710 --> 00:01:42.450 Let's just say when it swings to its max height. 00:01:42.450 --> 00:01:44.655 Well, I need to know the height to do that, right? 00:01:44.655 --> 00:01:47.470 Let's just do a little proof here to calculate this, 00:01:47.470 --> 00:01:51.510 so you can use this anytime you're using pendulums. 00:01:52.070 --> 00:01:57.435 There's your height and we want to know what is that height. 00:01:57.435 --> 00:02:02.675 How do we find it given the length and theta? That's all we know. 00:02:02.675 --> 00:02:05.470 What I did here is I drew these two parallel lines. 00:02:05.470 --> 00:02:07.055 You can see them right here, 00:02:07.055 --> 00:02:12.815 and what I'm going to do is I'm going to make this a 90-degree triangle. 00:02:12.815 --> 00:02:16.115 I'm going to start drawing some components of this. 00:02:16.115 --> 00:02:19.205 I'm going to choose a different color, I'm going to choose the color red. 00:02:19.205 --> 00:02:21.630 I'm going to break this triangle into components. 00:02:21.630 --> 00:02:23.160 This is the hypotenuse, 00:02:23.160 --> 00:02:24.950 this is the angle that I'm working with, 00:02:24.950 --> 00:02:26.520 and this is the 90 degree. 00:02:26.520 --> 00:02:28.390 If I want to know the opposite side, 00:02:28.390 --> 00:02:33.820 this is just going to be l times sine of theta. 00:02:33.860 --> 00:02:38.790 That gives me this total length all the way across from here to here, 00:02:38.790 --> 00:02:44.685 this length right here, all the way across to here. I know that side. 00:02:44.685 --> 00:02:47.050 I'm just doing the geometry of this triangle here. 00:02:47.050 --> 00:02:52.150 I'm going to move my little l out here and I'm going to draw this component here. 00:02:52.150 --> 00:02:57.340 What if I wanted to find out this particular part of the triangle right here? 00:02:57.340 --> 00:02:59.010 That's the adjacent side 00:02:59.010 --> 00:03:01.295 because this is the length and this is theta. 00:03:01.295 --> 00:03:04.110 We know that cosine theta is adjacent over hypotenuse. 00:03:04.110 --> 00:03:05.565 This side right here is going to be 00:03:05.565 --> 00:03:15.730 l times cosine theta and that's from here all the way down to here. 00:03:17.030 --> 00:03:19.750 That's important because we need to 00:03:19.750 --> 00:03:23.220 define some geometry when we're working with this problem. 00:03:23.350 --> 00:03:25.670 Remember, I'm trying to find this height here. 00:03:25.670 --> 00:03:28.350 I'm going to go ahead and clone this variable and I'm just going to move it over here. 00:03:28.350 --> 00:03:31.730 Just remember this is the height here that we're looking for. 00:03:31.730 --> 00:03:37.235 But we also know something that the total length of this is still 00:03:37.235 --> 00:03:41.490 l. I'll go ahead and I'm 00:03:41.490 --> 00:03:46.030 going to draw that in just a different color just to emphasize something here. 00:03:46.030 --> 00:03:54.550 This is still l. This total length all the way from here to here, all the way down. 00:03:56.210 --> 00:03:59.250 What does that mean? It simply means this—that 00:03:59.250 --> 00:04:03.300 the height plus l times cosine theta equals the length. 00:04:03.300 --> 00:04:09.260 We can put that together in just a very basic proof here. 00:04:09.260 --> 00:04:11.965 We can make a formula for the height. 00:04:11.965 --> 00:04:14.450 Using dimensions, much like an engineer would do. 00:04:14.450 --> 00:04:18.020 If we knew that two sides added up to the total length, 00:04:18.020 --> 00:04:21.605 we can actually set this as an equation. 00:04:21.605 --> 00:04:26.610 Right here, I can begin and I can say 00:04:34.630 --> 00:04:44.790 this length equals l cosine theta plus the height. 00:04:45.860 --> 00:04:48.290 What am I interested in looking for? 00:04:48.290 --> 00:04:51.310 Remember, whatever we're looking for in an equation, 00:04:51.310 --> 00:04:52.765 I always want you to circle that. 00:04:52.765 --> 00:04:54.450 I'm looking for the height here. 00:04:54.450 --> 00:04:56.335 That's what I'm looking for. 00:04:56.335 --> 00:04:59.450 I'm going to solve this equation in terms of the height. 00:04:59.450 --> 00:05:08.140 The height is going to equal l minus l cosine theta. 00:05:09.110 --> 00:05:13.910 Then what we can do is we can factor out an l and we can say that 00:05:13.910 --> 00:05:21.085 the height equals l times 1 minus cosine theta. 00:05:21.085 --> 00:05:24.900 That will work on any simple pendulum that has 00:05:24.900 --> 00:05:30.080 a massless rod and the center of mass is concentrated here at the center. 00:05:30.080 --> 00:05:32.520 That will always give you the height that 00:05:32.520 --> 00:05:36.155 the pendulum swings and it's very reliable formula. 00:05:36.155 --> 00:05:40.980 You can use that for potential energy or any other application that 00:05:40.980 --> 00:05:45.885 you want to use just to find out the relative height that that pendulum swings. 00:05:45.885 --> 00:05:48.045 Again, just to summarize, 00:05:48.045 --> 00:05:51.500 we found the height because maybe I want to find the potential energy. 00:05:51.500 --> 00:05:54.100 Let's say I want to find MGH, a simple application. 00:05:54.100 --> 00:05:57.840 I know that there's a length of a pendulum here, a length of a pendulum here, they're equal. 00:05:57.840 --> 00:06:00.035 It swings across an arc here. 00:06:00.035 --> 00:06:03.960 If I do a parallel line here and draw a 90-degree, 00:06:03.960 --> 00:06:06.400 the opposite side here is l sine theta. 00:06:06.400 --> 00:06:09.880 The adjacent side is l cosine theta of this. 00:06:09.880 --> 00:06:11.780 If I move the height over, I can see that 00:06:11.780 --> 00:06:15.080 the height plus l cosine theta equals the length. 00:06:15.080 --> 00:06:18.260 I simply set those equal to each other in an equation. 00:06:18.260 --> 00:06:21.560 This total length here equals 00:06:21.560 --> 00:06:28.335 the l cosine theta plus the height right here and we want to solve for the height. 00:06:28.335 --> 00:06:32.835 The height is going to be l minus l cosine theta, or very simply, 00:06:32.835 --> 00:06:35.580 we can factor out the l, 00:06:35.580 --> 00:06:39.345 and we can say the height equals l times 1 minus cosine theta. 00:06:39.345 --> 00:06:43.380 That's all I got for you tonight. Thanks for watching and I'll talk to you soon.