Hey, guys. Remember when we talked about the energy of mass spring systems? We're going to do the same thing for pendulums in this one. So remember that just a little refresher about mass spring systems, you have this object that's moving back and forth and it's oscillating between due to different kinds of energies. We've got the elastic potential energy that's the maximum at the endpoints, and then you've got the kinetic energy that's maximum at the equilibrium position. But for pendulums, it's the same thing. So as you pull this thing back and it's swinging back and forth, it's got 2 types of energy. It's got kinetic energy depending on how fast it's moving, and because it's going up and then coming back down again, it's changing this vertical height, it's also got gravitational potential energy. And those are the 2 things we're going to be looking at in this video.

So as this thing is swinging back and forth, right, you're going to take this pendulum that's length l, and you're going to pull on it so that it's on some θ angle, and you're going to release it. But when you release it, that θ is equal to its maximum value because it's just going to swing back and forth between those angles. So what does the gravitational potential look like? Well, the gravitational potential we always have to establish like a zero point. And for the zero point in pendulums, it's going to be at the bottom of the swing. So that's h0 = 0. So now what happens is that when you pull this thing all the way out to θ maximum, now you have the maximum height right here, where the height is measured relative to that zero point. So that means at the very, very top, the gravitational potential energy is maximum and the kinetic energy is equal to 0. It's not moving.

So let's take a look at the equilibrium position. The equilibrium is the exact opposite. So now you're at the bottom of the swing right here. You're at the bottom of the swing. So your gravitational potential energy when θ=0 is just 0, and then your kinetic energy is at its maximum value.

So that means the total mechanical energy for these things, in the first case, is just mghmax. And then for this guy, it's just 12 mvmax^2. And then at any other point in between, I've got this, like, point right here, θp. So this is θ p, and the energy is just going to be a combination of both of them. So I'm just going to have whatever the height is at that specific point, plus whatever the kinetic energy is at that specific point, and that's it. But a lot of the times, what happens in problems is we don't have what the height is specifically. We're actually going to see that in the example that we're going to get to. So how do we figure out what that height is if we're only just given l and the θ angle itself? Here between the zero point and wherever high we are. So let's figure that out.

If you've got the total entire length of the pendulum, and that's equal to l, and I'm trying to figure out what this highlighted distance is, then all I need to do is just figure out what this little distance is in the triangle. So using SOHCAHTOA, I'm going to have l × cosθ = that little vertical piece right there. So that means that this little highlighted distance is actually going to be the difference between the length and the lcosθ. So that means that for any θ, the height is just equal to l−lcosθ. This is sometimes called the pendulum equation. And the other way you might see it is l1−cosθ, and that's it.

So let's rewrite our mechanical energy formula. So if all of these things are conserved between all these cases, then that means that mghmax = 12 mvmax^2, and that's just equal to mgh@acertainpoint, and then plus 12 mvv@acertainpoint^2. And that's basically the energy conservation formula for pendulums. So that's it. Let's take a look at an example. So like I said before, a lot of times in problems, you won't see what the actual height of this thing is because then you could just use normal energy conservation. So notice how in our energy conservation for pendulums, it's exactly the same thing as what we did for energy conservation for, like, gravity and kinetic energy and stuff.

Okay. So we've got this mass and it's attached to this pendulum. It's some length l. So let me go ahead and start filling stuff in. So I've got this as l and it's pulled up an angle θ and then it's released. So that means that that angle is already at its maximum. It's just going to swing back and forth. That's the amplitude. And we're supposed to figure out what the maximum speed is. We're supposed to derive an expression for the max speed. So here's what I'm going to do. I'm going to go up here and I'm just going to copy down my energy conservation formula. I'm going to put that guy, like, right over here. So that's my energy conservation formula. And what am I looking for? So this is equals, this is equal, and this is a plus sign. So I'm actually looking for what the V_{max} is. And what am I given? Am I given the energy at a specific point, or am I given the amplitude's energy or the energy at the amplitude? Well, let's see. When θ = its maximum, now we just have to establish what our 0, potential energies are. So let me just move this over a second. And so I've got the zero energy is going to be at the bottom of the swing, and the maximum energy is going to be at hmax. So if I've got hmax because I've got θmax, I'm going to use these two equations right here. The problem is that I'm not asked for hmax, and I'm not given what hmax is. I'm supposed to get this in terms of l and θ.

So here's what I'm going to do. I'm going to take my pendulum equation that I just derived when h=l one - cosθ. And when θ = maximum, so hmax = when 1 - cosθ is at its maximum value. So now I'm just going to take this expression here and plug it in for H_{Max}. So let's go ahead and do that. So I've got mg, and this is going to be l one - cosθmax = 12 m vmax^2. And that's what I'm looking for, v_{max}. So if we're looking for this vmaximum right here, take a look. I've got these m's that actually cancel, and then I can move the one half to the other side. So that means I'm going to get 2 g l one -cosθ max is equal to vmax^2. Now all I have to do is just take the square root. So if I move that over, I get that v_{max} = √2 g l one -cosθ maximum. So this is how problems will usually go. They'll add they'll tell you the length of the pendulum, and they'll tell you the angle that they pull it back, and you can find out what the energy is based on you just using the energy conservation formula. So let me know if you guys have any questions. Let's keep moving on for now.