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INSTRUCTOR: Welcome to this lesson.

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In this video, I'm just going to do a brief proof on

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how to find the height that a pendulum swings,

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given length of the pendulum and theta.

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I'm just going to be doing a simple pendulum here.

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We're not doing a pendulum with a mass on string. There's some pendulum.

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There's a bob here hanging by a rope with a length, l, and this rope swings through an arc,

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and it goes to a particular height over here.

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We might want to know what exactly that height is.

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Let me just give you the variables that we would start with here.

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Let's say I have theta here,

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and I have some length here like this,

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I'll call that l and it's the same length here.

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This is l also because it just swings and it swings across some arc here, back and forth.

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It'll just swing back and forth, the pendulum.

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Let's just say this is the height that it swings.

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I drew a ground down here just to emphasize something,

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because our lowest working point is going to be right here and that's

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the lowest point that the center of

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mass passes, and that's what we're going to call our datum.

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That's our lowest working point.

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We're going to make that our zero point.

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Now, I drew some ground below it just to

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emphasize that it doesn't matter if the ground is lower than that point.

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We're going to make that our lowest working point.

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Let's say that I want to find out the height that this thing swings up here.

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In other words, I want to know how

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high does this pendulum swing when it comes up to here?

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Because maybe I'm doing a potential energy problem.

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Maybe I want to find out the potential energy at this point,

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the gravitational potential energy.

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Maybe I want to know what is the MGH at this point.

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Let's just say when it swings to its max height.

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Well, I need to know the height to do that, right?

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Let's just do a little proof here to calculate this,

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so you can use this anytime you're using pendulums.

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There's your height and we want to know what is that height.

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How do we find it given the length and theta? That's all we know.

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What I did here is I drew these two parallel lines.

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You can see them right here,

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and what I'm going to do is I'm going to make this a 90-degree triangle.

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I'm going to start drawing some components of this.

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I'm going to choose a different color, I'm going to choose the color red.

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I'm going to break this triangle into components.

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This is the hypotenuse,

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this is the angle that I'm working with,

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and this is the 90 degree.

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If I want to know the opposite side,

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this is just going to be l times sine of theta.

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That gives me this total length all the way across from here to here,

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this length right here, all the way across to here. I know that side.

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I'm just doing the geometry of this triangle here.

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I'm going to move my little l out here and I'm going to draw this component here.

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What if I wanted to find out this particular part of the triangle right here?

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That's the adjacent side

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because this is the length and this is theta.

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We know that cosine theta is adjacent over hypotenuse.

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This side right here is going to be

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l times cosine theta and that's from here all the way down to here.

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That's important because we need to

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define some geometry when we're working with this problem.

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Remember, I'm trying to find this height here.

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I'm going to go ahead and clone this variable and I'm just going to move it over here.

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Just remember this is the height here that we're looking for.

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But we also know something that the total length of this is still

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l. I'll go ahead and I'm

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going to draw that in just a different color just to emphasize something here.

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This is still l. This total length all the way from here to here, all the way down.

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What does that mean? It simply means this—that

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the height plus l times cosine theta equals the length.

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We can put that together in just a very basic proof here.

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We can make a formula for the height.

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Using dimensions, much like an engineer would do.

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If we knew that two sides added up to the total length,

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we can actually set this as an equation.

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Right here, I can begin and I can say

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this length equals l cosine theta plus the height.

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What am I interested in looking for?

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Remember, whatever we're looking for in an equation,

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I always want you to circle that.

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I'm looking for the height here.

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That's what I'm looking for.

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I'm going to solve this equation in terms of the height.

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The height is going to equal l minus l cosine theta.

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Then what we can do is we can factor out an l and we can say that

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the height equals l times 1 minus cosine theta.

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That will work on any simple pendulum that has

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a massless rod and the center of mass is concentrated here at the center.

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That will always give you the height that

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the pendulum swings and it's very reliable formula.

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You can use that for potential energy or any other application that

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you want to use just to find out the relative height that that pendulum swings.

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Again, just to summarize,

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we found the height because maybe I want to find the potential energy.

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Let's say I want to find MGH, a simple application.

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I know that there's a length of a pendulum here, a length of a pendulum here, they're equal.

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It swings across an arc here.

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If I do a parallel line here and draw a 90-degree,

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the opposite side here is l sine theta.

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The adjacent side is l cosine theta of this.

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If I move the height over, I can see that

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the height plus l cosine theta equals the length.

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I simply set those equal to each other in an equation.

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This total length here equals

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the l cosine theta plus the height right here and we want to solve for the height.

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The height is going to be l minus l cosine theta, or very simply,

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we can factor out the l,

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and we can say the height equals l times 1 minus cosine theta.

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That's all I got for you tonight. Thanks for watching and I'll talk to you soon.