To keep up with energy, we have to talk about work. Work is how you exchange energy. Following the financial analogy, it is the ATM of physics. Now that one might be original, I'm not sure. That's where you put money in, take money out. Because some people do put money into the ATM, the people that show up with all the checks and hold up the line, hey, you know who you are. So it is the ATM of physics. It is also in Scalar, it is also in Joules. It's the same as kinetic energy, in J, in joules. Units are the same, everything's the same, except you don't really have work. You do work and you do it to exchange energy. Let's see. To do work on an object, you push with force. We'll call it F, through displacement, and we'll go back to our Delta r notation for displacement. It doesn't have to be a mechanical push, it could be a different kind of force push. But for now, we'll just think about a mechanical version. The formula for at work, W. W it's a scalar and it's F, that force you just apply dotted with that displacement you just went through. If you do work on something, you will give it energy. The only kind we talked about so far is kinetic. I can show you an example of me giving this little car kinetic energy. We'll say, I'm going to push, I'm going to start here, and I'm going to push and apply force until I get here. So this is my displacement right there. If I do that and I push constant force, what did I do? I gave it kinetic energy, and I applied a force, a constant force through this displacement Delta r. It went a lot further, but the only part that counts is the displacement over which I applied the force. It doesn't matter that it keeps going. It's definitely going to keep going, I gave it kinetic energy. But it's this Delta r here, and it's the force F I applied that matter, F dot Delta r. Now, one thing about this, for now, this is only applies for a constant force. Later, we'll get into what to do if the force changes with position. Let's think a little bit. In case you haven't done one in a while or ever about the Dot product. Dot product multiplies two vectors. So far we have multiplied a scalar times a vector, Newton's second law. Some of the forces vector equals mass scalar times acceleration vector. But now you'll need to multiply two things that are both vectors. Dot product is one way of doing it. So to figure out a dot product, we need the two vectors. We can write the force vector like this, fxi hat plus yj hat plus fzk hat. Of course, the one I just did was in one dimension, but it could be written in three-dimensions. Then Delta r vector, well that's just the three components. Delta xi hat plus Delta yj hat plus Delta zk hat. So you need both vectors fully described to take a dot product. Then you can decide, if you want do it in Cartesian coordinates or Polar coordinates. Some problems are set up one way, some are set up another way. Cartesian is for when you have those components. If you're given these x, y, z components, then this is what you'd want to do. The Dot product is F dot Delta r. It's basically you just multiply the x components plus the y components plus the z components, it's no longer a vector. So a Dot product of two vectors makes a scalar. So it would be fx times Delta-x, multiply those plus fy times Delta-y plus fz times Delta-z. Whatever you get, that'll be a scalar and that'll be your Dot product. But actually, I'd say with most problems, you're given them in polar coordinates. You're often told, you have a 10 Newton force in this direction, and then maybe your displacement was 30 degrees away in this direction this far. In that case, you write the dot product this way. You say, F dot Delta r, which is going to be a scalar is the magnitude of the force, vector times the magnitude of Delta r vector, the displacement times the cosine of the angle between them. That also gives you a scalar value. That is also going to give you the same thing that we get here, and here in here. Just depends on what you're given as far as which one you want to use.
