 So now let's consider a pop culture example of buoyancy. Some of you may have seen the movie Up, in which case an old man takes his house and he inflates a whole bunch of helium balloons and it lifts his entire house off the ground and they float away to a special land somewhere. Or perhaps you thought when you were a kid at some point you saw the clown holding the entire bag or the whole bunch of balloons and you wondered how many would have to happen before that clown would float away or if he handed you too many when you were little, whether you would float away or not. So that's a good question. How much helium would be necessary to lift you up? So we all have different weights and masses, etc. So I'm going to use myself as an example here. But what we're interested here is finding the relationship between our buoyant force, which is going to act in an upward direction, and our weight. There's our weight, which is going to act in a downward direction. So in this particular case, if we take those two things, we're going to do an equilibrium where we take our buoyant force, which acts positive direction, and our weight in the negative direction, and we want those two things to add up to be some value greater than zero. If that value is positive, we'll float. If that value is negative, we'll stay rooted to the ground. So we want to understand those conditions. Well, we've already studied and discussed the buoyant force, our relationship between the buoyant force. So let's go ahead and put those pieces in. For the buoyant force, we know the buoyant force is going to be the volume that the helium is taking up times our specific weight, or the specific weight of the air that we remove and replace with helium. Remember that gives us the positive value. But then we have to subtract the helium in the balloons itself. So our specific weight of helium will subtract off there. Then we're going to take that value and we're going to compare it to our weight. Now notice if I move that weight value to the other side of the equation by adding it to both sides, it becomes positive. So basically this is the relationship we want to satisfy that our volume, our specific weights need to be greater than the weight we're actually trying to lift. So the question how much is the question of how much volume. So I can look up values for the specific weights and I can select a value for the weight that I want to lift. I'm going to go ahead and use my own personal weight here, which depending on the day is roughly about 180 pounds. Then I can also go ahead and plug in some values for my specific weights of air. Now the volume is what I'm trying to solve for. The specific weight of air at standard temperature and pressures, I'm going to use a value of 0.0765, 0.0765 what? What units would I need to be using? Well I would want to use some English units. In this case we'll do pounds per cubic feet, feet cubed. Also I could abbreviate that as PCF, typically abbreviated as PCF. Alright, I'm going to subtract a specific weight value for helium. The value I'll use there is 0.011 PCF. Now typically when you're doing a problem you won't be writing two different ways. We might write PCF and PCF, but just to demonstrate that these two things are the same things. Pound per cubic feet, PCF are the same thing. Now it's just simply a matter of doing a little bit of math. Take my volume, subtract these things. We get 0.0665 PCF. We realize that that has to be greater than our 180 pounds. Divide both sides by the 0.0665 PCF. And we get a value for our volume. The volume of the balloons, or the volume of the helium in the balloons I should say, needs to be greater than in order to make me float 2,750. Now what are my units here? Well PCF is pounds per cubic foot. We'll realize that the pounds will cancel. The cubic feet, which are in the denominator, will move to the numerator. And so our final answer will be in 2,750 cubic feet. Now notice, this did take an account. This would be how much helium necessary to lift me up. But in order to encase all that helium, we would actually have to include the mass of the balloons themselves. So that's the next step that we'll look at in just a bit. So earlier we tried to answer the question how much helium would be necessary to lift you up if you were to grab a bunch of balloons and float away. And we answered that question by calculating a volume of helium for a 180 pound person. For a 180 pound person, that volume would be 2,750 cubic feet of helium. However, that didn't account for the balloons that would be necessary to contain that helium. So if we were really going to talk about how many balloons would be necessary to float somebody away, we'd have to do a little bit extra work. In fact, we're going to have to do a little bit of algebra in this case because we have an unknown volume that we might need to solve for. So we started with our original equation, this sort of buoyancy equation. And now what we also want to do in this particular case is take into account that we have an additional weight that we need to account for, the weight of the balloons themselves, which acts in a downward fashion. So maybe I'll abbreviate that WB. So now we actually have to include the weight of our balloons in our equation. But notice the weight of the balloons will depend on the volume of the balloons necessary to fill them up. So we'll need to know how much, some measurements for the balloon. So I took a standard 12-inch balloon that I bought at a party store. And with that 12-inch balloon, I did some measurements. So first thing I did, okay, so I have a 12-inch balloon and that 12-inch balloon had a mass, when I measured it on my gram balance, of 2.8 grams. Now so far we've done all of our measurements here in English units. So we should probably go ahead and continue to do that. Not only that, when we're dealing with buoyancy, typically we're dealing with specific weights as opposed to densities and our standard measurements of mass don't necessarily apply. Now they're interchangeable here, but let's just stick with specific weights. So I'm going to go ahead and convert those 2.8 grams. Actually I'll be a little bit more careful with my significant figures. I feel confident in saying that this was 2.80 grams, but I can't have any more significant digits than that based on my measurement. So I will multiply that by our value of 2.20 pounds per 1,000 grams per kilogram. Notice there are more precise values to convert pounds to kilograms, but because I only have three significant figures here, it's really only necessary to have three significant figures in my conversion. And when I multiply those two things together, I get a weight 0.00616 pounds for a single balloon. So now my question is I want a relationship between the weight and the volume of the balloon because the volume of the balloon is what's going to determine how much lift I end up getting. So I'm going to assume that the balloons are roughly spherical. Roughly spherical. They're not quite, and I could probably be a little more specific here, but I'm going to assume they're roughly spherical. Okay, and so therefore I have a radius of 6 inches of my balloons and I'm going to use the volume of a sphere of 4 thirds pi r cubed. Okay, and since it's a 12 inch balloon, I basically have a radius of 6 inches, so I will put that in 4 thirds pi times 6 inches quantity cubed. And when I plug that into my calculator, I get 904 cubic inches. Now, here we have measurements in cubic feet. There we have a measurement in cubic inches, so I need to go ahead and do a conversion there. I'm going to be very specific here when we make our conversion 904 inches cubed. If I do my conversion, I want to convert from feet to feet from inches, so I create my relationship there, recognizing that there are 12 inches in one foot. A very common mistake is to do that conversion and simply divide by 12. But notice that doesn't work because we're talking about cubic inches, so I need to make sure I do that conversion three times for each of the dimensions in my three-dimensional volume. So I need to divide by 12 three times effectively to find the volume in cubic feet. And the number I get when I do that, let's see here, is 0.523 cubic feet. Now we have a very useful relationship here, effectively the specific weight of the balloons that I can relate the weight of a balloon. So I'm going to sit there and put specific weight of a balloon where we have the weight of the balloon, 0.00616 pounds divided by the volume of the balloon, 0.523 cubic feet. And when I pull out that value, I get 0.0118 pounds per cubic feet, or PCF. So now that we've established a relationship between the weight and the volume of our balloon material, let's go back and reevaluate our buoyancy equation, including that piece. So we'll rework our buoyancy, recognizing again that our buoyancy force is a relationship between the volume of all the helium in our balloons and the specific weights of the air and the helium. And we knew that that value needed to be greater than our overall weight. And now we're going to replace the weight of our balloons with the relationship here with the volume of those balloons and the specific weight that we calculated of this relationship between the weight of the balloon and the volume of the filled balloon. Notice this does not include what's in the balloon. This is just a relationship between the volume of the filled balloon and the weight of the balloon itself, the balloon plastic itself. So now it's just a matter of plugging in our numbers. Notice our unknown here is still this volume. And if I do a little manipulation, I'm probably going to want to take this value here and move it to the other side of the equation so we can compare it to the weight. So now I have a series of specific weights that I can go ahead and put in here. We have our .0765 PCF for air, our .011 PCF for helium. We're going to subtract our .0118 PCF for our balloon. All of that's multiplied by our volume. And we want all of that to be greater than our 180-pound weight that we're trying to lift to float away. If we work through all of that, we get a value of .0537 PCF times our volume. And then when we solve for our volume there, we get a volume that must be greater than 3,350 cubic feet. Notice that's about 600 cubic feet more than was necessary just to float the payload, just to float the person who was the payload in this case. We need an additional 600 cubic feet to account for the actual weight of the balloon material, the rubber that the balloons are made of themselves. Note that now that we've solved this problem for one payload or one weight, it's simply a matter of scaling that to solve for any other weight. We can see that there's a relationship now between our volume necessary to lift the weight and the weight itself. So if I just simply divide this by my 180 pounds, I get a relationship of roughly 18.6 cubic feet per pound. In other words, if I want to lift one pound, I need using 12-inch balloons, I need 18.6 cubic feet of helium in those balloons to do so. And now all you have to do if you wanted to find out for any other weight, you'd simply multiply by the number of pounds that you would actually want to lift. Another question might be exactly how many balloons is it? Well, we know our relationship, we know that each balloon filled approximately the volume of one balloon was equal to 0.523 cubic feet. So if we try to figure out how many balloons, we simply take this 3350 and divide it by the 0.523. That's approximately multiplying it by 2, but if we actually do it in a calculation in this case to lift my weight of 180 pounds, I would need 6480 balloons filled with helium. There are a few other things you might want to consider. If you look at your stand-in helium tank, you'll find out that helium isn't 100% helium, that there's actually usually 90% helium or some other value. So that would have some effects. And there might be some effects to our specific weight of air depending on where we are. Are we up high in the mountains? Are we near the surface of the earth, near ocean level? Something along those lines could have some effects and it changed your specific weight. But in general, this gives you an idea, makes a pretty good estimate for that question. How much helium in helium balloons would be necessary for you to float away into the sky?