 Okay, so there was a request to do review of Chapter 3. I'm trying a new camera here so if things just go terribly wrong, audio or video, I don't know. It's all my fault. Okay, the first thing in Chapter 3 is this gravitational and electric force. So if I have a mass here in 1 and a mass here in 2, and I know the locations of those, I like this marker better, let's call this R1, that marker is actually not better, and R2, then I can find the location, the vector from mass 1 to mass 2, we'll call that R, such that R equals R2 minus R1. That's your basic vector stuff. Then I can find the gravitational force, let's say on mass 1, I would say F, this marker is really inferior to that one. Let's try this one. I'm just going to throw that one away. That's nice. F, and let's see, how does the book call this? Let me see, I just want to use this notation. 1, 1 by 2, let's put it like that notation, which we'll usually drop, but it's going to be equal to G mass 1, mass 2 over the magnitude of R squared, R hat. So in this case G is a gravitational constant, 6.67 times 10 to the negative 11th Newton's meter squared per kilogram squared. I don't expect you to memorize that, I'll give that to you. It's just an experimentally determined constant that relates to the force between these masses based on the distance between them. So that's the mass, 1 mass 2, that's the distance between them squared, and R hat is this vector, that unit vector. So mass 2 pulls on mass 1 that way. What if I wanted to redo the experiment, or the calculation for F, 2 by 1, the force on 2 by 1? Well, I could change it around, and I could say R prime is equal to R1 minus R2. So this would be R prime and R prime. But the magnitude of R prime is the same as the magnitude of R. The only difference is R prime hat would be in the negative direction. So I could write F2 by 1 equals negative F1 by 2. That notation is a little awkward. But this, because if I switch the mass around, it doesn't matter, they're just scalar values, right? So the magnitude of these are the same, the only thing difference is the direction. So 1 pulls on 2, the same as 2 pulls on 1, just in opposite directions. Let me check the camera. Okay, so that's one of these really important ideas about force. Force is always an interaction between two objects. So if one interacts with two, two interacts with one the same way. Forces are like two ends of the same string. Or force is like, if I go from Los Angeles to New York, that's the same distance as New York to Los Angeles. It's the same thing, it's the distance between New York and Los Angeles. Force is the same way, and this is very important. Okay, the other force that's very similar to this is called the Coulomb force. I'll just write down the formula and then mention something about it. If I have, the gravitational force is an interaction between objects that have mass. The Coulomb force, F, and again I could say 1 on 2, the force on 1 by 2, but this is the Coulomb force. 1 over 4 pi epsilon knot, that's epsilon knot, q1, q2, r squared, r hat. Actually this would be negative. That's one difference. So here we have this constant, 1 over 4 pi epsilon knot equals 9 times 10 to the 9th Newton's meter squared per Coulomb squared. Coulomb is a unit of charge. q1 and q2 are the charges of the two objects. It doesn't matter about their mass, it matters about their charges. But other than that, it looks very similar. This negative sign means that if these two have the same value of charge, then the force will be pushing that way. It'll be pushing it away. If one of these is negative and one's positive, then this whole thing will be positively pulling them together. The opposite charges attract, light charges repel, which you don't have with gravity. Gravity always attracts. Typically 9 times 10 to the 9th, it depends on the value of the charge, but compare that to 6.67 times 10 to the negative 11th. The gravitational force is much weaker than the electric force. Okay, let me say something about forces. The way we look at forces, we actually have two kinds of forces. And I'll say this again later. Let's look at some of the forces we've seen so far. We've seen the gravitational force. I'll just call it fg. And that's both this gravitational force and mass times g, which is just an approximation. We have the Coulomb force. I'll call it fc. We also saw the force by a spring. We also saw the air resistance force. But all of these have something in common. They all can be calculated based on the situation. If I know the position of objects for any of these three, I can calculate the force. I just need to know where they're at and I calculate the force. That one, if I just know how fast and direction it's going, I can calculate the force. Now there's other forces. These are forces that we can determine. But there's other forces that I'll call constraint forces. Suppose I have a table and have a block sitting on that table. In this case I have two forces, fn, I'll call it the normal force of the table pushing up. And the gravitational force I'll call that mg pulling down. So if I set that block on the table, then the table is going to push up whatever it needs to do to keep that block from going through it. It's constraining the block to the surface. I can't say what the force at that table exerts. I have to know something about its motion. I have to know something about how it's constrained. If I take my finger and I push down on it with the greater force, if the table doesn't push up more, then this is not going to be an equilibrium. And you'll see that if you push down on a book, it stays there if it's on a table. So the table will adjust its force to make it stay stationary in equilibrium. Up to a point, I mean, if the Hulk pushes down on it, the table is going to break. But that's a different kind of force. We can't just say what it is. We have to know something about the constraint in order to determine that. So that's a little different situation. Okay, I've said it. Okay, let's look at one other quick thing. Let's suppose I have a ball one, ball two, and they're going towards each other and they're going to collide. Maybe they're going to hit. Maybe they're going to interact with an electric force. It doesn't really matter. But they do hit. That's just like that. Okay, and then afterwards, no, I want to draw it. They're actually hitting. Okay, one, two. And then afterwards, there's ball one and there's ball two. And I'm going to use some notation here that's maybe not the best, but just to get my point across. I'll call this P1 initial. That's the momentum of that one beforehand. P2 initial. And this is going to be P2 final. And P1 final. And then during the collision, what's going to happen? Well, ball two is going to push on ball one. So it's going to be a force this way, F1 on two. Is that what I said one by? The force on one by two. Yeah, so put one two. And then one's going to push back on two. F2, one. So let me write down the... Determine the change in momentum for ball one. So if I write down the momentum principle, and this is the only force, then I can say F1 on two is a change in momentum of one over the change in time. That's the momentum principle. That's what it says. If that's the only force, then that's the change in momentum. Now I can do the same thing for ball two. F2 on one is the change in P2 over the change in T. That's the momentum principle. Now, if these forces are the same interaction, which they are, ball one pushing on ball two, it's the same as two pushing on one, and they happen for the same time, I can say F1 on two equals negative F2 on one, just like with gravity. So that means that I can write this is equal to the negative of that. So I have delta P1 over delta T equals negative delta P2 over delta T. And the times are the same, so those cancel. So I get, let me just write it all on one side, delta P1 plus delta P2. That's a delta equal to zero vector. Zero vector meaning it's still a vector. Zero, zero, zero. Or I could write this as delta P total equals zero. So the change momentum for this system is zero. And this is where you get that conservation momentum idea. This says that there's no external forces on the system. Then the momentum of the stuff before is the momentum of the stuff afterwards. I could also write this as P total initial equals P total final. Same thing. That's the same as saying that. Now, what if there is an external force? What if these are two cars and there's friction? Well, if delta T is very, very small, then the change momentum from right before the collision to right after the collision is still very, very, very small. And I can still approximate this as saying the change momentum is zero. So this is the very first introduction you get to collisions. You'll look at this a lot more detail later. Okay, that's really the most important things for Chapter 3.