[MUSIC] So we can store energy in, for example, the gravitational interaction. I do positive work, MGH against gravity. Now I do negative work minus MGH. Gravity does negative work, gravity does positive work. When we get back to where we started, gravity has done zero work. That makes gravity a conservative force. It does zero work around a closed path. A closed path is one that returns to its starting point. I input energy, but I get it back. Same with a spring. I do positive work against the spring. Now the spring does positive work for me. No work done around a closed path. Gravity and spring forces are both conservative forces. We can store potential energy in these interactions. But what about friction? I do work against friction. I do more work against friction. Friction does negative work. Friction does more negative work. Friction is a non-conservative force. It does negative work around a closed path. When I do work against friction, or air resistance, it doesn't go into potential energy. It's turned into heat. And we don't get it back. Here's another. I run up the stairs. Where did the energy come from? Well obviously from my muscles. It comes from my internal energy. And again that's not reversible. Muscular forces are non-conservative. By the way, non-conservative forces often involve transfer of energy between the molecular and the macroscopic levels. Friction produces heat, which we later discover is the macroscopic kinetic energy of molecules. Chemical energy is electrical potential energy and mechanical energy of the molecules. It can be converted to macroscopic mechanical energy by muscles or by motors. The Work Energy Theorem says that the work done by all the forces equals the increase in kinetic energy. And forces are either conservative or non-conservative. So we can write, work by conservative forces plus work by non-conservative forces equals delta K. And the work done by conservative forces is minus 1 times the increase in their potential energy. If gravity does work, gravitational potential energy decreases. So the increase in potential energy plus the increase in kinetic energy, equals the work done by the non-conservative forces. Suppose that muscles don't do any work, and that friction and air resistance don't do any work either. Then, delta U plus delta K equals 0. Next we define the mechanical energy. Mechanical energy E, is the potential energy plus kinetic energy. If non-conservative forces do no work, then, mechanical energy is conserved. A convenient way to write that conservation is, U initial plus K initial equals U final plus K final. [LAUGH] An important principle. So remember, if non-conservative forces do no work, then mechanical energy is conserved. But, a big warning. Never, ever write potential energy equals kinetic energy. It just isn't true, and you wouldn't want to tell a lie, would you? Let's set the 0 of height and of potential energy here. So the car initially has potential energy mgh. We release it, and let's use two successive frames to estimate the final velocity. Let's assume that the wheels turn freely. So little work is lost in non-conservative forces. Therefore, mechanical energy is approximately conserved. We write that, U initial plus K initial equals U final plus K final. Initially, U is mgh, and K is 0. Finally, U is 0 and K is a half mv final squared. Cancelling m and rearranging, v final equals the square root of 2 gh. Note that this is independent of the path. That's the cool thing about conservation of mechanical energy. Provided that it applies, we need only look at the initial and final states, which often makes analysis easier. Let's do a few problems before we go any further. [LAUGH] See you back here soon.