 So we're going to talk a little bit about balance today. Some of the basic fundamental theories and the ideas of balance. And that's what this little seesaw model is here that we're going to play with. We start here with five newtons of weight on either side of the seesaw. And notice, I can balance the seesaw in more than one way. I can have them both at the same distance, but I could take one of these weights and move it into six. Well, oh, wait a minute. When I move the weight, even though we have the same amount of weight, if I move one in, I now change whether it's balanced or not. So notice that apparently, when you're trying to balance something, it doesn't just take into account the weight of what you're trying to balance. It also takes into account where it is, what the position is. If I wanted to rebalance this on the other end, notice to get it rebalanced, I simply need to put them at the same distance away. But let's say we actually have something different. What if, instead of our weights being five and five, what if our weights are unequal? Let's do three and one. So I have three on the left and one on the right. So there's a child who's trying to balance with another child in the seesaw, but one of them is significantly heavier than the other one, or maybe an adult who's trying to balance. If I'm a dad, I might want to balance with my son on the seesaw. How am I going to make this happen? So somebody, go ahead and balance your seesaw. If you have three and one, how are you going to balance your seesaw? All right, anybody getting any numbers? I'll put the heavier weight at 18. At 18, oh, so we did six and 18. Let's see here, six and 18. Hey, that balanced out. Anybody else have a different way of doing that? You did five and 15. So five and 15 balances. Whoops, five, five, that's six again, and 15. Anybody else do anything different? You would have ended one and three, all right? So you did one, I assume, for the bigger one, and three for the smaller one. So could somebody describe for me if I'm trying to describe this mathematically? How do I figure this out? If I have two weights, how do I figure it out? How would I figure out this relationship or describe this relationship? OK, so one's three times heavier, so the other side has to be three times further. OK, seems to make sense that there's a relationship there. So if that works out, let's try seven and three. How does the re-apply here? Can you explain it to me? OK, so I would do one and 2.3. OK, so how am I going to make this balance? So the heavier one goes to three, that's four, three, and the lighter one goes to, let's see if I can actually get to three here. There we go, three and seven. That's six, seven. Ta-da. Did anybody have any other numbers that balance? How'd you get that? You played around with the scale? OK, so six and 14. Whoops, let's see here, six and 14. Do you want to make a comment, or did you have another set of numbers? It's because the distance will always be proportional. OK, proportional to what? So it'll be in distance to get your number. If you enter the distance over your weight, it'll become balanced, but then you find other distances, if it's proportional, it will always be balanced. OK, so in other words, we had seven over three, and you said the inverse, so that's three over seven, right? So we can think about as a fraction of three over seven. Well, if it's proportional, three over seven, six over 14 reduces to three over seven. Nine over 21, if we could do that, would reduce to three over seven. So any of the relationships would actually work out. So OK, so we seem to have a sense for how this is going to go about balancing. Notice we're assuming that there's sort of a balance in the middle. However, it might be a little bit more complex. What if I actually want to balance something, but I don't exactly know where the middle is? Then what do I have to do? Is there a different way of doing this? For example, this time I'm going to go ahead and draw my two boxes, OK? But if instead of, let's say I have a weight of, we'll do the weight of six and, oh, let's not do six. Let's do five. We'll call this a weight of five newtons, and this will be a weight of two newtons, OK? So if that's the case, if this is five newtons and two newtons, instead of me setting them on something that's equal, I want to know instead where to put the fulcrum, where do I put the point to make it balance? Where does it go? Does it go directly in between, or where does it need to go to make it balance? One of the things I might want to know is how long my seesaw is here, OK? Let's say my seesaw, we'll assume we're in the middle of the weight, so let's assume my seesaw is 14 feet long. Where do I need to put this to make it balance? OK, so how did you get that? OK, so if we look, we do know that we want our relationship to be, instead of five to two, we want it to be two to five. And of course, four to 10 is two to five, and those add up together to give us 14. But that might be a little bit harder or a little bit more complicated, particularly if I said, what about if I do this? What happens now if I actually want to put my fulcrum in a particular place? Now it gets to be a little bit more complex, because now we have to find some relationships. And that relationship there, we're going to say, for example, let's say that this is actually four, we'll say that this is four feet here. Where would we actually put it then? Well, that's going to make it a little bit more complicated, with where we actually would locate our fulcrum point. But here's the idea. So this leads to a concept of something called the center of gravity. C, G, the center of gravity. The center of gravity is basically that point that if you take the weight of something, that if you want to hold up the weight, that you also hold it up. I mean, you could hold up the weight at any place. But it's the place that you would want to hold it, that if I was trying to balance something on the tip of my finger, everything has a center of gravity, which basically is where's its balance point. And we want to be able to figure out that center of gravity, try to figure out where it exists. In particular, the reason why we want to figure it out for a plane is because we want, what's holding the plane up? What holds up an airplane? The wings. So we need to make sure that the center of gravity of the plane ends up where the wings are. Otherwise, the wings are going to push it up in the wrong place, and the airplane is going to tend to want to spin. And that's generally not good for our flight characteristics. So let's talk about how to calculate this center of gravity. And the idea of the center of gravity leads us to something called torque or moment. In physics, they call it torque in engineering class. We often call it moment. So I'm going to write the word torque. But I'm also going to call this moment. And the idea here is this ratio that we talked about before when we were playing with the seesaw, it's actually a little bit easier way of sort of thinking about this ratio. We know that how much you want to balance depends both on your weight and on your distance. So we're going to define something that is a relationship between the weight and distance. And we also learned that it was an inverse relationship, but that when one went up, the other went down. One of the ways to write an inverse relationship would be say that x times y is equal to some constant. Well, that's basically what we're going to do is we're going to say that our moment, this thing m, is going to be equal to the weight times the distance. I'm going to call this x, but this fancy word is called the moment arm. Moment arm. In other words, if we keep the same moment, if the weight goes up, the moment arm must go down and vice versa. So that gives us our inverse relationship that I was sort of talking about here. So let's sort of talk about that from the perspective of our seesaw again. Here's my seesaw. And before, when I had two things and I had my seesaw in the middle, what we were doing is we were measuring the moment arm from the seesaw. And if I had something that was seven Newtons and I had something that was three Newtons, what I could do is there was a lot of different distances I could have put them. One distance I could have put them was something like three, would say maybe it's three meters, and that would balance with seven meters. I didn't draw this very well, okay? Let me draw it a little bit more proportionally here. Okay, we knew those worked out. Well, let's find the moment for each of these. The moment on the left, in this case, is seven Newtons times three meters. Okay, so that's gonna be 21 Newton meters. And if I go to the other side, I have seven meters times three Newtons, which is again 21 Newton meters. Notice, to balance them, the moments are equal. If I instead decide to put it out further, let's say I say it was six meters, balancing with 14 meters. Well, now I have seven times six, that's 42, and three times 14, that's 42. So we can start thinking about this as being a balance of these things called moments, how much spin they're actually going to have. Let's try something where I do, I have a weight of nine on one side, wait up if I use the right keyboard, and a weight of two on the other side, okay? And let's say with my nine, I'm actually gonna put it at a distance of four. Where do I have to put my other one? Now I can do some ratio calculations, but if I use my little moment idea, and I wanna balance the moment, I should be able to figure this out pretty quickly. What's the moment on my left side, if it's at nine and four? What's the moment on my left side? Nine times four, 36. So if I have 36 on my left, what do I need to do to get 36 on my right? Two times 18. So the left side is 36, the right side is 36. The product of both sides, which is the weight times the moment arm is equal.