Physics: Finding the Height of a Pendulum Given Length and Theta

INSTRUCTOR: Welcome to this lesson.

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

how to find the height that a pendulum swings,

given length of the pendulum and theta.

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

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

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

and it goes to a particular height over here.

We might want to know what exactly that height is.

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

Let's say I have theta here,

and I have some length here like this,

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

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

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

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

I drew a ground down here just to emphasize something,

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

the lowest point that the center of

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

That's our lowest working point.

We're going to make that our zero point.

Now, I drew some ground below it just to

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

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

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

In other words, I want to know how

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

Because maybe I'm doing a potential energy problem.

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

the gravitational potential energy.

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

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

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

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

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

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

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

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

You can see them right here,

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

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

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

I'm going to break this triangle into components.

This is the hypotenuse,

this is the angle that I'm working with,

and this is the 90 degree.

If I want to know the opposite side,

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

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

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

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

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

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

That's the adjacent side

because this is the length and this is theta.

We know that cosine theta is adjacent over hypotenuse.

This side right here is going to be

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

That's important because we need to

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

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

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

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

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

l. I'll go ahead and I'm

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

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

What does that mean? It simply means this—that

the height plus l times cosine theta equals the length.

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

We can make a formula for the height.

Using dimensions, much like an engineer would do.

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

we can actually set this as an equation.

Right here, I can begin and I can say

this length equals l cosine theta plus the height.

What am I interested in looking for?

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

I always want you to circle that.

I'm looking for the height here.

That's what I'm looking for.

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

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

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

the height equals l times 1 minus cosine theta.

That will work on any simple pendulum that has

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

That will always give you the height that

the pendulum swings and it's very reliable formula.

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

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

Again, just to summarize,

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

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

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

It swings across an arc here.

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

the opposite side here is l sine theta.

The adjacent side is l cosine theta of this.

If I move the height over, I can see that

the height plus l cosine theta equals the length.

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

This total length here equals

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

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

we can factor out the l,

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

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