 The hemispherical dome in the figure below weighs 30 kilonewtons, is filled with water, and is attached to the floor with six equally spaced bolts. What is the force in each bolt required to hold down the dome? So even though this is a hydrostatic forces on curved surfaces problem and not a buoyancy problem, we're going to approach it like a buoyancy problem. We can look at this and consider the weight of the water that is quote displaced, and we can figure out what resistive force is going to be present instead. So what we're going to do is we're going to calculate how much water there would be around the dome and the tube, figure out the weight of that water, and that will give us our upward force supplied by the hydrostatic pressure of the water, and then to represent the volume of the displaced water, I'm going to start with the volume of a cylinder that has a radius of two meters and a height of six meters, then I'm going to subtract the volume of a very poorly drawn hemisphere that has a radius of two meters, and then I'm going to subtract the volume of that connecting tube on the top, which is a smaller cylinder that has a diameter of three centimeters and a height of four meters. So again, the volume displaced here is going to be the volume of the entire cylinder minus the volume that actually has water in it in reality. So the volume of the total cylinder would be the cross-sectional area of that cylinder, which is going to be pi over four times diameter squared, or pi times radius squared times height. The height is six meters, and then I'm going to subtract the volume of a hemisphere. I know the volume of a full sphere is four-thirds pi r cubed. Therefore, half a sphere would be half of this. So four-thirds would become four divided by three times two, which is four-sixths, and that would be two-thirds. The two-thirds times pi times the radius, which is two meters, cubed, and then I'm subtracting out that little straw tube at the top, and that is pi over four times three centimeters, because I have a diameter, not a radius, squared times the height of that cylinder, which is four meters. So just to recap, this is the volume of the cylinder that would enclose this apparatus minus the volume that actually has water in it. You can think of it like the volume of the air around the hemisphere and the tube, or you can think of it like the volume of water that is displaced. So if I take pi times two meters squared times six meters, I will be left with a quantity in cubic meters, and then if I subtract a constant times a constant times two meters cubed, I'll be left with a quantity in meters cubed, and then if I multiply pi over four times three centimeters squared times four meters, then I'm going to be left with a quantity in square centimeters times meters. So I will have to multiply this last term with 100 centimeters, excuse me, one meter divided by 100 centimeters squared. So I'm going to be dividing by 100 squared in that last quantity. So here we go, calculator. I want to take pi multiplied by two times two times six, and then I want to subtract two divided by three multiplied by pi again. Multiply by two times two times two minus pi over four multiplied by three times three times four times one over 100 squared. And why did I finally break out the carrot for that one and not any of the other squared or cubes? I don't know. And I get 559,973 pi divided by 30,000. Thank you, calculator. Which is 56.64 cubic meters. And then the weight here is going to be the volume displaced multiplied by the density of water multiplied by gravity. I don't know gravity, so I'm going to assume standard gravitational acceleration. I don't know the properties of the water here, so I'm going to assume it is H2O NSTP. So from table A1 or A3, I can grab the density of water at 20 degrees Celsius in one atmosphere. And I see that it is 998, which is a comforting number to keep coming back to. And then I have my volume in cubic meters already. So 56.64 cubic meters multiplied by 998 kilograms per cubic meter multiplied by 9.81 meters per second squared and a Newton is defined as a kilogram meter per second squared. cubic meters, cancels cubic meters, meters, cancels meters, second squared, cancels second squared, kilograms, cancels kilograms, leaving me with Newtons. And that quantity in Newtons is this quantity multiplied by 998 multiplied by 9.81 and we get 574.110 kilo Newtons. So that force is the weight of the water that is displaced in the downward direction. Therefore, the hydrostatic force is going to be in the upward direction and it is going to have a magnitude of 574.110 Newtons. And again, I know that the sum of forces in the y-direction has to equal zero. Therefore, the forces up must equal the forces down. The force up is 574.110 Newtons. There are two forces down. One of them is the weight of the apparatus itself, which is 30 kilonewtons. And I'm going to make sure that future john knows this is a four and not a nine. And there are more forces down, too, because the bolts are holding the apparatus down. There are six bolts and I'm assuming that the hydrostatic force is perfectly balanced as all things should be. And I'm going to say that each bolt is supporting the same weight. Therefore, this is six times the force of the bolt. Therefore, F-bolt is going to equal 574.110 minus 30,000 divided by six Newtons. So, calculator, are you still there? 574. No, let's just grab the number because it might have hidden decimal places. Calculator strikes again. Fickle with its number of displayed digits, divided by six. So each bolt, F-bolt, is supporting 90,685 newtons. Or 90.685 kilonewtons.