 So let me ask you a question. If you were to compare two hot air balloons, a large one and a small one, do you think the large one would be easier to fly or the small one would be easier to fly? Now to answer that question, you're going to have to ask yourself, well, what is it that we think about when we're talking about what makes the hot air balloon fly? Well, first of all, what keeps it from flying? In the case of a hot air balloon or anything, what keeps us from rising into the sky is our weight. And if we think about that weight, what is it that contributes to the weight? Well, the main thing that contributes to the weight of a balloon is the balloon itself. What's the balloon made out of? What material is it made out of? If it's just a small balloon, maybe it's made out of some sort of rubber if it was like a regular balloon. But if it's a hot air balloon, it's made out of some sort of non-flammable material. But we can assume that the material is the same for either of our balloons if we want to compare the two balloons. If we assume the material is the same, what is it that really contributes to the material? Well, the weight of the material basically depends on the surface area. In other words, how much the balloon encompasses it. The area of the balloon determines the weight. If you want a bigger balloon, you have more area to cover. And therefore, it weighs more. So that's what brings the balloon down. What gets the balloon to go up? Well, in this case, that's the buoyant force, the buoyancy. And the buoyancy, depending on what we know about the buoyant force, buoyant force is a relationship between volume, how much air there is, and the difference between the things you have in there. In this case, the difference between hot air. And actually, it's the other way around. The difference between the cool air outside and the hot air inside. As you change the density of the two things, that difference between the density of the cold air outside and the density of the air inside, or in this case, the specific weight of those two things, as you change those, you get more buoyancy. Well, once again, we can assume that the changes inside are the same, that we have the same temperatures inside, and those same temperatures lead to same density differences. So the main difference between the two balloons and the main thing that our buoyant force depends on is the volume of the balloon. The bigger balloon has more volume, the smaller balloon has less volume. Both of those values depend on the scale of whatever we're building. We want a bigger or smaller. That term is generally called scale. And one of the really important concepts in engineering is how various quantities change with assumptions of scale. Now, if we wanted to compare the two balloons, one of the things we could do is compare the tendency to go up from buoyant force to the tendency to go down based on weight. And one way to do that is just to find a number that relates these two things or relates the volume to the surface area. Notice if that number is a big number, then we're going to tend to float. That means volume is dominant. There's more volume contribution in surface area. And it's going to tend to float. And if that number is a small number, then it's going to tend to sink. So let's calculate this ratio for an example of a big balloon and a small balloon. However, I'm going to make the math easy by assuming the balloons are cube shaped. Well, balloons aren't really generally cube shaped, are they? Well, no, they could be. That's kind of hard to build in some cases. You have to keep the structure in place, but you could build it as a cube. But we're going to do it just for easy math. So let's check it out. I'm going to assume that I have a cube, a small cube, small balloon, and we're going to assume that the small balloon has dimensions of one meter by one meter by one meter. Now that's much bigger than I'm drawing it, one meter by one meter by one meter. But we'll use it to sort of figure out our values here. If we do that, we're going to find that we have a volume of length times width times height of one meter times meter times meter or one meter cubed. And then if I take my surface area, the surface area, well, one of these sides is a meter times a meter, and there are six of them. So six times one square meter is going to be six meters squared. And so that value, cancelling out two meters on both, depends on being one sixth of a meter. Ooh, it's a length measurement, because it's measured in meters. But it's a relationship, and we don't really care what it means except in comparison to something else. The whole purpose of finding that, even though it depends on a length, was to compare it to a larger balloon. So let's go ahead and design a slightly larger balloon that's two meters by two meters by two meters. Now, some of you are already saying, well, that balloon is getting larger in many ways. Why don't we just make it two meters taller? It's easy to see that in the case of the cube, but that's one of the main concepts of scale, is that as we're talking about scale, often our assumption, when I talk about a big balloon and a small balloon, is we assume that the balloons keep a similar shape. That no matter what dimension I look at it, they still look like each other. I'm not looking at a big balloon. I'm not making one tall and skinny. I'm not stacking two balloons on top of each other. When I double in one dimension, I double in the other dimensions. In other words, we maintain something called the aspect ratio that the relationships in each dimension stay the same. That is one of the assumptions of working in scale. So we're gonna make that assumption here, and let's see how that affects our tendency to float. Well, if I take the volume, let's find the volume here. Two meters times two meters times two meters is two times two times two is volume is eight meters cubed. What's the surface area? Well, in this case, my surface area is two times two is four, so I have four square meters, and then there are six of them, so that ends up being 24 meters squared. Let's do the math, meters squared cancels out, and we end up with one third of a meter. One third of a meter, well, again, that's a length. But remember, the whole reason for this ratio was just to figure out a relative number. So what happens? As we get bigger, my number also got bigger. My balloon got bigger, my number got bigger. What does a bigger number mean? It means that volume is more important, and this thing is easier to fly as a tendency for it to raise. So the short answer to the question at the beginning is it's easier to fly a large hot air balloon. So I made some assumptions about the size and shape of our initial balloon. Let's see if those assumptions hold true more generically. Let's go ahead and recreate my cubic balloon and see if we can have a more general sense for this. But instead of my cubic balloon being one meter or two meters, let's just make it a length L, some general length L. Well, if we do that, we get our volume, and our volume is L cubed. We look at our surface area, and our surface area is L times L, and there's six of them, so six L squared. And if we cancel that out, we end up with some value of L over six, length divided by six. Well, that kind of makes sense based on what we just saw, that if the length goes up, my ratio goes up, which means I'm more likely to go up. Okay, so that's pretty straightforward. Our relationship basically says that our tendency to float goes up proportionally to the size of my system, whether I call my size L. But then you say, wait a minute, you did a cube. Is that true for other shapes too? Well, we can figure that out. Let's try a shape that's more closely related to my balloon. Okay, let's try a sphere. Sphere. And in the case of a sphere, we're gonna assume that that sphere has an overall outer dimension that's the same as our length. So we're gonna say the diameter of my sphere is equal to this length L, okay? That also means that the radius of my sphere is equal to L over two. That's gonna be kind of important because most of the formulas for a sphere are in terms of the radius of the sphere. There's the radius is half of the sphere, and that's half of the overall length here. So let's go ahead and do this. Our volume, what's the volume of a sphere? Well, the volume of the sphere is four thirds pi R cubed. Well, if R is L over two, we're gonna end up with L cubed over eight. Hopefully you can see why that eight is there because we're cubing the over two as well as the L. Okay, all right, now let's check out the surface area. What's the surface area of a cube? Well, the surface area of a cube is four pi R squared. Sorry, four pi R squared. So let's put in there four pi, and now we have R squared but R is L over two, so we have L squared over four. Well, now that ratio could use a little bit of massaging here. Let's get rid of these fours, and let's get rid of those pies they can cancel out, and we can cancel out the L cubed in both of these Ls there, and we end up with something that looks like, oh, and there's an eight here that we can actually cancel out and make a two and cancel out that four. And when we're all set and done, low and behold, what do we get? The same ratio, L over six. All right, I'm doing some kind of strange magic here. Let's try one more shape. What if I assume that I have a cylinder? And in this case, that cylinder, we've got our radius here, but that radius is gonna be equal to a height of L over two, so we keep the same dimensions in those two dimensions. And we're also gonna have a height here, but we'll say that height is also a height so that even though these are different shapes, they're all roughly the same size. Let's see how that works out. Let's calculate our volume and our surface area for a cylinder. What's the volume of the cylinder? The volume of the cylinder is going to be the height, which we have here is L, that's our height, times pi R squared. Pi or R is L over two squared. So there is our volume of the cylinder. How about our surface area? Well, our surface area is gonna be two parts. First of all, we have this area here, pi R squared, but we have two of them. So we have two pi R squared, but then that pi R squared ends up being L squared over four. So there's our first part. And then what about the area of this cylinder? Well, if you think about it sort of as a paper towel holder, we can cut it and then flatten it out and that gives us a rectangle. That rectangle has height of L, but also a width that is around this encircle, which is two pi R. Or if we press L in, pi L. So what is the area of this piece? The area of that piece is going to be pi L squared. Pi L times L. Okay, let's see here. Let's do this addition down here. If we do this addition down here, this becomes one half pi L squared. And that's one pi L squared. So this is three halves pi L squared. Now you're probably beginning to see some things. Pi is canceled out. My L squared's canceled out. And now I have L over, let's see here, L over four times three halves. So interestingly, this scaling thing is not dependent upon the shape. Yes, I did make some assumptions that were similar lengths in all these dimensions, but it is independent on the shape. So what's the key point here? Well, as you scale something, as your balloon gets bigger, it's going to have a tendency to float more. The volume is going to be a bigger effect, and therefore it will be easier to fly a very large hot air balloon than a very small hot air balloon, at least just simply based on the buoyancy and weight balance.