 The cylindrical can shown in the figure below floats such that the top of the can floats three centimeters above the surface of the water around it. What is the weight of this can in Newtons? We will be approaching this problem by considering Archimedes principle. Archimedes principle states that a an object partially or completely submerged in a fluid, either a gas or a liquid, will create a buoyant force. That is equal to the weight of the fluid that is displaced. So for our purposes here, I can label this FB and FB, the buoyant force, is the weight of the water that has been displaced by the can. If this is an equilibrium, then I can say that the sum of forces in the y direction has to equal zero. Y here is defined in the upward direction, let's say. Therefore, the forces up must equal the forces down. The force up would be the buoyant force. The force down would be the weight of the can, which is the thing that we're actually solving for. Then weight is equal to buoyant force, which is equal to the weight of the water displaced. Therefore, the weight of this can is going to be equal to the volume of this cylinder of water multiplied by the density of water to get the mass of the water displaced multiplied by gravity. We weren't told gravity, so we can reasonably assume its standard gravitational acceleration, 9.81 meters per second squared. We weren't told anything about the water, so the best course of action is to assume that it's regular water at 20 degrees Celsius in one atmosphere. For that, we can get the density of water from table A1 in our textbook. On table A1, we can read the density of water at a variety of temperatures at one atmosphere. 20 degrees Celsius is going to be 998 kilograms per cubic meter, and then the volume displaced is going to be the volume of a cylinder that has the cross-sectional area of a circle multiplied by height. The area of that circle is going to be 5 or 4 times the diameter of this circle times height. The diameter of the circle is 9 centimeters, the height of the water displaced is 8 centimeters, and with that we have enough information to compute an answer. I'm going to bring that calculation over here, so we have room for units. So we have pi over 4 times 9 squared centimeters squared. That's the diameter of the can, multiplied by 8 centimeters. That's the height of the can, so so far we have volume displaced, then we're multiplying by density, 998 kilograms per cubic meter, and then we're multiplying by gravity, 9.81 meters per second squared. And we want an answer in Newtons. A Newton is a kilogram meter per second squared. So far kilograms cancel kilograms, second squared cancels second squared, meters cancels meters. So in order to get the rest of the units to cancel, I have to say there are 100 centimeters in one meter, and then cube everything. Square centimeters times centimeters, cancels cubic centimeters, cubic meters, cancels cubic meters, and I'm left with Newtons. So calculator if you would help us out. That is pi over 4, and that was the wrong direction of parentheses there, calculator. It's okay, I understand. 9 squared, I'm gonna write that as 9 times 9 because I don't want to grab the mouse and drag it over to the carat, and then I'm multiplying by 998 times 9.81 times 1 cubed, which is 1. Then we are dividing by 4 times 100 cubed, and I get 4.98 Newtons. So the weight of the can is equal to the weight of the displaced water, and that is 4.98 Newtons. So now that we've answered that question, let me pose you another question. What if this were sea water? Instead of regular water, what if this water had some salt in it? Would the can float higher, lower, or at the same level? Well, if we go back and do our appendix, and we navigate over to table A3 this time, we can see the density of a variety of different liquids at one atmosphere and 20 degrees Celsius. I have sea water here, which is 30% salinity, and that would be a density of 1025 kilograms per cubic meter. So the logic here is it would take less volume to make the same weight as the can. Again, it would take less volume of sea water to equal 5 Newtons, therefore the can would float higher in the water. We could figure out how much higher if we wanted to, but I'll leave that as an exercise for the viewer. And if it were something crazy dense, like say mercury, then that can would float so high that it might look like it's just resting on the surface of the water. That's why it's easier to swim in places like the Dead Sea the high salt content makes the density of that water higher, which means that it takes less volume to support your weight, therefore you are more buoyant in the Dead Sea.