[MUSIC] So that's where the work done by the total force goes, it changes the kinetic energy. Positive work, [SOUND] increases it, and negative work, [SOUND] decreases it. Now, we'll look at the work done by individual forces. Here, suppose I'm moving mass so slowly that the change kinetic is negligible. No change in kinetic energy, so the total work done by all forces is approximately zero. Two forces. My hand pushing up, gravity pulls down. My hand pushes up, it moves up, I do positive work. Gravity pulls down, it moves up, gravity does negative work. And the two work expressions add to zero. Where did the work go when I did work against gravity? Note that I can get it back [SOUND] well, minus a bit for friction that we'll deal with later. This gives us the idea of stored energy or potential energy. This raised mass has the potential to do work for me. I have stored work in the gravitational indirection between this mass and the Earth. How much work do I store in this gravitational indirection? Again, moving slowly so acceleration's negligible. So the force that I apply equals the weight of the object. Over a distance s, I do work, Wme equals Fs cos theta. But from this triangle, S cos theta is just the change in height till the H. So, I do work if delta H. Going up, I do work in G delta h against, the gravitational interaction between the mass N and the earth. I'll call this work that I've done against gravity this stored work, the change in potential energy with symbol U. Because it's equal to work, the units of energy are joules. We've defined potential energy by changing delta U. Let's choose a convenient reference height, say the level of this bench top. And call that the zero of height, and the zero gravitational potential energy. Then we can write an important equation. Gravitational potential energy equals MGH. A warning, I've assumed that g is constant, which is approximately true near the earth's surface, if I lifted an object 5,000 kilometers, that wouldn't be true. The gravitational force becomes weaker if we go a long way from Earth, more about that in week eight. Remember that the gravitational potential energy is the work done against gravity in raising the object to a height h from the reference height. We can get that energy back. For example, by accelerating an object. [LAUGH] Let's check we've got that. Where else can we store potential energy? A spring is one place. Once again, I do work storing energy and we can recover as say, [SOUND] kinetic energy. How much energy is stored in a spring? Here's Hooke's law from last week. X is the extension, K is the spring constant. As I extend it, the spring force is in the negative X direction, but I pull with a positive force. Calculating the work I store is trickier here this time, because unlike gravity, the force keeps changing. Here's force times distance when the force is small, but I do more work for the same distance when the spring is more stretched. So, for the total work done in extending it from zero to X, I want lots of thin rectangles like that. And that gives me the area under this line. That area is easy enough to find, it's just half FX, which is half, KX squared. The spring does work minus half squared. So that gives us the potential energy stored in a spring, half KX squared, and as usual, we check the units, substitute for K and for X. Yes, a Newton times a meter is a joule. There's much more we'll do with energy, but we already have enough to answer the questions coming up in the quiz.