My daughter is busy with Calculus AB and she asked me about a problem that she was having. In one of her multiple choice quizzes, the examiner asked about the force on the side of a half full cylinder that is lying on it's side. The cylinder is 20 feet long and has a diameter of 12 feet. The fluid has a density of 50.
The cylinder would look something like this:
My daughter got the right answer but only when she ignored the length of the cylinder. That seems wrong to me. If a dam is 1 foot wide then the force of the water on the dam wall would be a lot less than it would be for a dam that was 10000 feet wide.
It's been a while but looking up the formula in her notes says that it should be this:
$$ \int_a^b \rho \cdot h(y) \cdot L(y) dy $$
The formula for the circle is:
$$ x^2+(y-6)^2=6^2 $$
In terms of x that would be:
$$ x=\sqrt{6^{2}-\left(y-6\right)^{2}} $$
h(y) is the depth of the slice. The L(y) dy is the area of each slice. In this case, the slice is 20 long and has a width of 2 times x (the formula above).
which is:
$$ \int_0^6 \biggl( 50 \cdot (6-y) \cdot 20 \cdot 2 \cdot \sqrt{6^{2}-\left(y-6\right)^{2}}\biggl) dy $$
Is that correct or is the length of the cylinder supposed to be ignored? Is the area a 2d slice that only includes the side wall itself or is the area a slice that includes a full horizontal slice of the container?
The area under pressure in question does not depend on the length of the cylinder. Think about a diver, the pressure only depends on the depth of submersion. It does not matter whether the diver is in a tank of water or in the ocean. The length would be important if we needed to find force acting on the side of the tank.