[Script Info]
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[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.05,0:00:02.16,Default,,0000,0000,0000,,INSTRUCTOR: Welcome to this lesson.
Dialogue: 0,0:00:02.16,0:00:05.08,Default,,0000,0000,0000,,In this video, I'm just going to do a brief proof on
Dialogue: 0,0:00:05.08,0:00:08.54,Default,,0000,0000,0000,,how to find the height that a pendulum swings,
Dialogue: 0,0:00:08.54,0:00:11.32,Default,,0000,0000,0000,,given length of the pendulum and theta.
Dialogue: 0,0:00:11.32,0:00:13.76,Default,,0000,0000,0000,,I'm just going to be doing a simple pendulum here.
Dialogue: 0,0:00:13.76,0:00:17.64,Default,,0000,0000,0000,,We're not doing a pendulum with a mass on string. There's some pendulum.
Dialogue: 0,0:00:17.64,0:00:23.08,Default,,0000,0000,0000,,There's a bob here hanging by a rope with a length, l, and this rope swings through an arc,
Dialogue: 0,0:00:23.08,0:00:26.58,Default,,0000,0000,0000,,and it goes to a particular height over here.
Dialogue: 0,0:00:26.58,0:00:31.33,Default,,0000,0000,0000,,We might want to know what exactly that height is.
Dialogue: 0,0:00:31.33,0:00:34.18,Default,,0000,0000,0000,,Let me just give you the variables that we would start with here.
Dialogue: 0,0:00:34.18,0:00:36.24,Default,,0000,0000,0000,,Let's say I have theta here,
Dialogue: 0,0:00:36.24,0:00:39.01,Default,,0000,0000,0000,,and I have some length here like this,
Dialogue: 0,0:00:39.01,0:00:42.27,Default,,0000,0000,0000,,I'll call that l and it's the same length here.
Dialogue: 0,0:00:42.27,0:00:47.01,Default,,0000,0000,0000,,This is l also because it just swings and it swings across some arc here, back and forth.
Dialogue: 0,0:00:47.01,0:00:48.83,Default,,0000,0000,0000,,It'll just swing back and forth, the pendulum.
Dialogue: 0,0:00:48.83,0:00:50.49,Default,,0000,0000,0000,,Let's just say this is the height that it swings.
Dialogue: 0,0:00:50.49,0:00:53.41,Default,,0000,0000,0000,,I drew a ground down here just to emphasize something,
Dialogue: 0,0:00:53.41,0:00:57.35,Default,,0000,0000,0000,,because our lowest working point is going to be right here and that's
Dialogue: 0,0:00:57.35,0:00:58.89,Default,,0000,0000,0000,,the lowest point that the center of
Dialogue: 0,0:00:58.89,0:01:02.80,Default,,0000,0000,0000,,mass passes, and that's what we're going to call our datum.
Dialogue: 0,0:01:02.80,0:01:04.68,Default,,0000,0000,0000,,That's our lowest working point.
Dialogue: 0,0:01:04.68,0:01:06.26,Default,,0000,0000,0000,,We're going to make that our zero point.
Dialogue: 0,0:01:06.26,0:01:09.50,Default,,0000,0000,0000,,Now, I drew some ground below it just to
Dialogue: 0,0:01:09.50,0:01:13.60,Default,,0000,0000,0000,,emphasize that it doesn't matter if the ground is lower than that point.
Dialogue: 0,0:01:13.60,0:01:15.58,Default,,0000,0000,0000,,We're going to make that our lowest working point.
Dialogue: 0,0:01:15.58,0:01:19.90,Default,,0000,0000,0000,,Let's say that I want to find out the height that this thing swings up here.
Dialogue: 0,0:01:19.90,0:01:22.73,Default,,0000,0000,0000,,In other words, I want to know how
Dialogue: 0,0:01:22.73,0:01:26.91,Default,,0000,0000,0000,,high does this pendulum swing when it comes up to here?
Dialogue: 0,0:01:26.91,0:01:29.87,Default,,0000,0000,0000,,Because maybe I'm doing a potential energy problem.
Dialogue: 0,0:01:29.87,0:01:33.65,Default,,0000,0000,0000,,Maybe I want to find out the potential energy at this point,
Dialogue: 0,0:01:33.65,0:01:35.33,Default,,0000,0000,0000,,the gravitational potential energy.
Dialogue: 0,0:01:35.33,0:01:39.71,Default,,0000,0000,0000,,Maybe I want to know what is the MGH at this point.
Dialogue: 0,0:01:39.71,0:01:42.45,Default,,0000,0000,0000,,Let's just say when it swings to its max height.
Dialogue: 0,0:01:42.45,0:01:44.66,Default,,0000,0000,0000,,Well, I need to know the height to do that, right?
Dialogue: 0,0:01:44.66,0:01:47.47,Default,,0000,0000,0000,,Let's just do a little proof here to calculate this,
Dialogue: 0,0:01:47.47,0:01:51.51,Default,,0000,0000,0000,,so you can use this anytime you're using pendulums.
Dialogue: 0,0:01:52.07,0:01:57.44,Default,,0000,0000,0000,,There's your height and we want to know what is that height.
Dialogue: 0,0:01:57.44,0:02:02.68,Default,,0000,0000,0000,,How do we find it given the length and theta? That's all we know.
Dialogue: 0,0:02:02.68,0:02:05.47,Default,,0000,0000,0000,,What I did here is I drew these two parallel lines.
Dialogue: 0,0:02:05.47,0:02:07.06,Default,,0000,0000,0000,,You can see them right here,
Dialogue: 0,0:02:07.06,0:02:12.82,Default,,0000,0000,0000,,and what I'm going to do is I'm going to make this a 90-degree triangle.
Dialogue: 0,0:02:12.82,0:02:16.12,Default,,0000,0000,0000,,I'm going to start drawing some components of this.
Dialogue: 0,0:02:16.12,0:02:19.20,Default,,0000,0000,0000,,I'm going to choose a different color, I'm going to choose the color red.
Dialogue: 0,0:02:19.20,0:02:21.63,Default,,0000,0000,0000,,I'm going to break this triangle into components.
Dialogue: 0,0:02:21.63,0:02:23.16,Default,,0000,0000,0000,,This is the hypotenuse,
Dialogue: 0,0:02:23.16,0:02:24.95,Default,,0000,0000,0000,,this is the angle that I'm working with,
Dialogue: 0,0:02:24.95,0:02:26.52,Default,,0000,0000,0000,,and this is the 90 degree.
Dialogue: 0,0:02:26.52,0:02:28.39,Default,,0000,0000,0000,,If I want to know the opposite side,
Dialogue: 0,0:02:28.39,0:02:33.82,Default,,0000,0000,0000,,this is just going to be l times sine of theta.
Dialogue: 0,0:02:33.86,0:02:38.79,Default,,0000,0000,0000,,That gives me this total length all the way across from here to here,
Dialogue: 0,0:02:38.79,0:02:44.68,Default,,0000,0000,0000,,this length right here, all the way across to here. I know that side.
Dialogue: 0,0:02:44.68,0:02:47.05,Default,,0000,0000,0000,,I'm just doing the geometry of this triangle here.
Dialogue: 0,0:02:47.05,0:02:52.15,Default,,0000,0000,0000,,I'm going to move my little l out here and I'm going to draw this component here.
Dialogue: 0,0:02:52.15,0:02:57.34,Default,,0000,0000,0000,,What if I wanted to find out this particular part of the triangle right here?
Dialogue: 0,0:02:57.34,0:02:59.01,Default,,0000,0000,0000,,That's the adjacent side
Dialogue: 0,0:02:59.01,0:03:01.30,Default,,0000,0000,0000,,because this is the length and this is theta.
Dialogue: 0,0:03:01.30,0:03:04.11,Default,,0000,0000,0000,,We know that cosine theta is adjacent over hypotenuse.
Dialogue: 0,0:03:04.11,0:03:05.56,Default,,0000,0000,0000,,This side right here is going to be
Dialogue: 0,0:03:05.56,0:03:15.73,Default,,0000,0000,0000,,l times cosine theta and that's from here all the way down to here.
Dialogue: 0,0:03:17.03,0:03:19.75,Default,,0000,0000,0000,,That's important because we need to
Dialogue: 0,0:03:19.75,0:03:23.22,Default,,0000,0000,0000,,define some geometry when we're working with this problem.
Dialogue: 0,0:03:23.35,0:03:25.67,Default,,0000,0000,0000,,Remember, I'm trying to find this height here.
Dialogue: 0,0:03:25.67,0:03:28.35,Default,,0000,0000,0000,,I'm going to go ahead and clone this variable and I'm just going to move it over here.
Dialogue: 0,0:03:28.35,0:03:31.73,Default,,0000,0000,0000,,Just remember this is the height here that we're looking for.
Dialogue: 0,0:03:31.73,0:03:37.24,Default,,0000,0000,0000,,But we also know something that the total length of this is still
Dialogue: 0,0:03:37.24,0:03:41.49,Default,,0000,0000,0000,,l. I'll go ahead and I'm
Dialogue: 0,0:03:41.49,0:03:46.03,Default,,0000,0000,0000,,going to draw that in just a different color just to emphasize something here.
Dialogue: 0,0:03:46.03,0:03:54.55,Default,,0000,0000,0000,,This is still l. This total length all the way from here to here, all the way down.
Dialogue: 0,0:03:56.21,0:03:59.25,Default,,0000,0000,0000,,What does that mean? It simply means this—that
Dialogue: 0,0:03:59.25,0:04:03.30,Default,,0000,0000,0000,,the height plus l times cosine theta equals the length.
Dialogue: 0,0:04:03.30,0:04:09.26,Default,,0000,0000,0000,,We can put that together in just a very basic proof here.
Dialogue: 0,0:04:09.26,0:04:11.96,Default,,0000,0000,0000,,We can make a formula for the height.
Dialogue: 0,0:04:11.96,0:04:14.45,Default,,0000,0000,0000,,Using dimensions, much like an engineer would do.
Dialogue: 0,0:04:14.45,0:04:18.02,Default,,0000,0000,0000,,If we knew that two sides added up to the total length,
Dialogue: 0,0:04:18.02,0:04:21.60,Default,,0000,0000,0000,,we can actually set this as an equation.
Dialogue: 0,0:04:21.60,0:04:26.61,Default,,0000,0000,0000,,Right here, I can begin and I can say
Dialogue: 0,0:04:34.63,0:04:44.79,Default,,0000,0000,0000,,this length equals l cosine theta plus the height.
Dialogue: 0,0:04:45.86,0:04:48.29,Default,,0000,0000,0000,,What am I interested in looking for?
Dialogue: 0,0:04:48.29,0:04:51.31,Default,,0000,0000,0000,,Remember, whatever we're looking for in an equation,
Dialogue: 0,0:04:51.31,0:04:52.76,Default,,0000,0000,0000,,I always want you to circle that.
Dialogue: 0,0:04:52.76,0:04:54.45,Default,,0000,0000,0000,,I'm looking for the height here.
Dialogue: 0,0:04:54.45,0:04:56.34,Default,,0000,0000,0000,,That's what I'm looking for.
Dialogue: 0,0:04:56.34,0:04:59.45,Default,,0000,0000,0000,,I'm going to solve this equation in terms of the height.
Dialogue: 0,0:04:59.45,0:05:08.14,Default,,0000,0000,0000,,The height is going to equal l minus l cosine theta.
Dialogue: 0,0:05:09.11,0:05:13.91,Default,,0000,0000,0000,,Then what we can do is we can factor out an l and we can say that
Dialogue: 0,0:05:13.91,0:05:21.08,Default,,0000,0000,0000,,the height equals l times 1 minus cosine theta.
Dialogue: 0,0:05:21.08,0:05:24.90,Default,,0000,0000,0000,,That will work on any simple pendulum that has
Dialogue: 0,0:05:24.90,0:05:30.08,Default,,0000,0000,0000,,a massless rod and the center of mass is concentrated here at the center.
Dialogue: 0,0:05:30.08,0:05:32.52,Default,,0000,0000,0000,,That will always give you the height that
Dialogue: 0,0:05:32.52,0:05:36.16,Default,,0000,0000,0000,,the pendulum swings and it's very reliable formula.
Dialogue: 0,0:05:36.16,0:05:40.98,Default,,0000,0000,0000,,You can use that for potential energy or any other application that
Dialogue: 0,0:05:40.98,0:05:45.88,Default,,0000,0000,0000,,you want to use just to find out the relative height that that pendulum swings.
Dialogue: 0,0:05:45.88,0:05:48.04,Default,,0000,0000,0000,,Again, just to summarize,
Dialogue: 0,0:05:48.04,0:05:51.50,Default,,0000,0000,0000,,we found the height because maybe I want to find the potential energy.
Dialogue: 0,0:05:51.50,0:05:54.10,Default,,0000,0000,0000,,Let's say I want to find MGH, a simple application.
Dialogue: 0,0:05:54.10,0:05:57.84,Default,,0000,0000,0000,,I know that there's a length of a pendulum here, a length of a pendulum here, they're equal.
Dialogue: 0,0:05:57.84,0:06:00.04,Default,,0000,0000,0000,,It swings across an arc here.
Dialogue: 0,0:06:00.04,0:06:03.96,Default,,0000,0000,0000,,If I do a parallel line here and draw a 90-degree,
Dialogue: 0,0:06:03.96,0:06:06.40,Default,,0000,0000,0000,,the opposite side here is l sine theta.
Dialogue: 0,0:06:06.40,0:06:09.88,Default,,0000,0000,0000,,The adjacent side is l cosine theta of this.
Dialogue: 0,0:06:09.88,0:06:11.78,Default,,0000,0000,0000,,If I move the height over, I can see that
Dialogue: 0,0:06:11.78,0:06:15.08,Default,,0000,0000,0000,,the height plus l cosine theta equals the length.
Dialogue: 0,0:06:15.08,0:06:18.26,Default,,0000,0000,0000,,I simply set those equal to each other in an equation.
Dialogue: 0,0:06:18.26,0:06:21.56,Default,,0000,0000,0000,,This total length here equals
Dialogue: 0,0:06:21.56,0:06:28.34,Default,,0000,0000,0000,,the l cosine theta plus the height right here and we want to solve for the height.
Dialogue: 0,0:06:28.34,0:06:32.84,Default,,0000,0000,0000,,The height is going to be l minus l cosine theta, or very simply,
Dialogue: 0,0:06:32.84,0:06:35.58,Default,,0000,0000,0000,,we can factor out the l,
Dialogue: 0,0:06:35.58,0:06:39.34,Default,,0000,0000,0000,,and we can say the height equals l times 1 minus cosine theta.
Dialogue: 0,0:06:39.34,0:06:43.38,Default,,0000,0000,0000,,That's all I got for you tonight. Thanks for watching and I'll talk to you soon.