 Okay, springs, springs and energy, that's what we're going to talk about right now. So let's just say that we have a mass connected to a spring, and this is an ideal spring, so it's perfect, and we can let this all sit back and forth with no friction. So it's not too difficult, first of all we know that the force of spring exerts is the magnitude is K times S, where K is the spring constant and S is the amount that's either stretched or displaced. It's not too difficult to show that if you pull it, the work done by the spring and therefore the potential energy stored in the spring is going to involve the opposite of that. It would be one-half K S squared. Okay, so now let's choose our work energy principle. So let's say that we take this as our system, the mass plus the spring. In that case I can write work is change in kinetic plus change in potential energy. Now, what's the work done on our system? The spring doesn't do any work because it's part of the system. What about gravity? When gravity doesn't do any work, if the mass just moves back and forth this way, then delta S is always in this plane and G, or gravity is down, so they're perpendicular and no work done. And the same thing with the normal force pushing up, no work done. So I get 0 equals delta K plus delta U. So I could rewrite that as 0 equals K final minus K initial plus U final minus U initial, or I could write it as 0 equals K final plus U final minus K initial plus U initial, which I could then write as 0 equals, or I'm sorry, K final plus U final equals K initial plus U initial equals some constant, E total. So this is what we call a closed system because there's no work done on it. And so in that case the energy is constant. So the energy at the end of some time is equal, or position is beginning and anywhere between. At any instant of this mass, oscillating back and forth, the total energy is the same. When it's at the farthest point over here, then it's not moving, it just has potential energy. And it gets back to the equilibrium point. It has no potential energy, so it's just kinetic energy. One way to represent this is with an energy diagram. So let's say that I have, can you see that? Yes? So this is energy and this is s, not time. And this is the representation of the spring potential energy of an ideal spring. Real springs aren't quite like that, but the cool thing is that any spring, if you get near the middle, usually can be represented as an ideal spring. So that's the spring potential energy. And let's say that I have, I pull this over here, a distance a, and I let go. Well, at that instant it's not moving and it would be at a location a, over here, and we could represent as that dot right there. And that dot represents this total energy because there's no kinetic energy, so e total equals ua plus zero kinetic. Well, anyway, what's going to happen now is that mass is going to start speeding up and moving towards the center, and when it gets to the center, it'd be right there, and it would have no potential energy. But it'd still have the same total kinetic energy. So we could represent this distance as our kinetic energy. And then the mass will continue on over here and get to the other farthest point, a, away in the negative direction, and it'll stop again. So we could just have that line right there represent the total energy of the system. It's going to move back and forth between those points, but that's the total energy. So it's a very useful way of looking at a system because we can get an idea of what's happening in terms of energy without having to worry about time. And so in this case, at any particular instant, this distance on the graph represents a kinetic energy. And so you can see as it gets closer, this over to this end, that kinetic energy gets smaller and smaller until it stops, and it comes back the other way. Now, I couldn't have a mass over here. I couldn't have the particle over here because then u is above it, so it'd have to have a negative kinetic energy. And kinetic energy is 1 half mv squared. And so if that's negative, either the mass is negative or v squared is negative, and both those aren't very good situations. So let me just draw this one more way. Let me represent energy as a function of time. So when I let it go, it had no kinetic energy and just potential energy, and then it would start to speed up. So it would start losing potential energy. If I plotted that, it would actually look like this. And then the kinetic energy would look like, here's a red marker. Let's see if this works. It's old. It'd start off with no kinetic energy and it'd go up, back down. And then if you add those two together, you get the total energy. Okay, so that's a similar representation to this. That's time representation. Okay, so that's a simple example, or not I didn't give you numbers, but I'm showing a mass on a spring using an energy diagram and showing that the total energy is constant.