For these next few problems, we're going to introduce a new tool for us to work with. Springs! Springs are a whole lot of fun, and we can use them to make all sorts of fun things like trampoline and spring scale. We can even use them to protect people if elevator cable snapped, and springs add all sort of fun problems to physics, so we're going to do a few those now. But first, I just want to introduce a few of the concepts we are going to need to understand how springs behave. To make sure we're all in the same page for our discussion of springs, I want to go over some of the assumptions we make when we're talking about springs. We are going to assume that we can either compress springs like this or stretch them like this, but we're not going to deal with springs being pulled and angled like this. Now, you can actually handle problems like this in physics but that's a bit beyond a scope of this class. We're going to assume that the spring only moves in one dimension. It can either be squished or it can be pulled on to extend it. We're also going to assume that the mass our spring is 0 so the mass was springs and that doesn't seem very realistic, but it's really a helpful assumption for solving our problems. And it turns out that in many cases, the mass of the spring is pretty small compared to the masses of the other things we're dealing within the problem. This assumption isn't the terrible one but just know that if you're doing super realistic problems, we couldn't say the springs are massless. One important concept we'll need in our discussion of spring is the idea of equilibrium length, and to see what this is, let's look at a few examples. First imagine that we have a spring here that no forces are acting on. Nobody's pulling it or pushing it or anything like that. Even though it's not being pushed or pulled, it has some natural length that it likes to be at. It may be 10th of a meter or something like that. We called this natural length of the spring, the equilibrium length. Now, if we pull on the spring or push on the spring to compress it or extend it, then we're going to change the length of the spring away from the equilibrium length. And the amount that the length of the string changes, we called Δx. Δx is the change in the length of the spring away from the equilibrium length. It is important to note that both extending the spring and compressing the spring can change the length of the spring away from the equilibrium length. All right. Now, we're ready to talk about energy. If you push a block against the spring, you'll notice that you have to fight keep it in that same spot. It wants to bounce back in the other direction. Similarly, if I try and pull on the block extending the spring, it tries to pull back to the other direction. Now, if I let the block go, in either the case when I'm compressing the spring or in the case where I've extended the spring, the block is going to accelerate and go bouncing back and forth, which means that it's gaining kinetic energy. And if the blocks gained kinetic energy, that energy must to come from somewhere, and it turns out that energy was potential energy stored in the spring, and you'll notice that the more I stretch the spring, the more dramatically, the block will rebound. In other words, it will have more kinetic energy if I stretch the spring further or compress it further. This means that the energy stored in the spring, the potential energy, should probably scale with the distance away from the equilibrium length, and in fact, it does. It turns out that we can write the potential energy stored in the spring as 1/2 times some constant k times Δx² and remember that Δx is the difference between the spring's current length and it's equilibrium length. Now, this constant k is something that's particular to the spring itself. It's a measure of how stretchy or stiff the spring is. Then you may be wondering whether this equation came from. I'm afraid I'm not actually going to derive it for you because that requires calculus, but I will say that it is not a mathematical coincidence that this equation looks a lot like the equation for kinetic energy. That's the topic we can discuss more on the forms. All right. Now, I think we are ready to do a few problems using springs.