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The Forces of Fluids on Sides of Tanks

Instructor: Michael Gundlach
Feeling under pressure? Worried about getting forced into doing calculus? This lesson will allow you to learn more about how to find the force of fluids on the sides of a tank, an important application of integrals to physics.

Finding Forces

Imagine you work for an aquarium. You're building a tank to hold a whale, and you're trying to determine what type of glass to use on the viewing window into the whale's tank. You want to make sure the glass is strong enough to last, so you need to know how much force the water will be putting on the glass. This lesson will help you figure that out.

Suppose the viewing window you're buying glass for starts 4 meters below the surface of the water and is 8 meters wide and 4 meters tall, as in the picture below.


Water tank view


You look in a physics textbook and find out that the force of a fluid on a surface at a certain depth is given by the equation:

F = ρgdA

where:

ρ is the density of water

g is acceleration due to gravity

A is the area of the surface

d is the depth of the surface

Feeling triumphant, you go to calculate the force on the glass when you realize something very important--your window isn't all at the same depth! The top of the window is only 4 meters deep, while the bottom is 8 meters deep! This is where calculus comes to the rescue.

Integrating to Find Force

Since the window isn't all at the same depth, we're going to imagine cutting the window into an infinite number of horizontal slices. These slices are each going to be so thin that the top and bottom of the slice are at the same depth, and they'll all be rectangles. Let's see if we can find the force on each of these small slices, and then try to add them all up in order to find the total force.

The density of water is 1000 kg/m³. Acceleration due to gravity is 9.8 m/s². This gives us ρ = 1000 and g = 9.8. Now, let's find A and d. Since we're trying to find the force on a slice at an arbitrary depth, I'm just going to represent it with a variable, x. Since our slice is rectangular, we just need to find its width and height to find the area. Since we're taking horizontal slices of a rectangle, each slice will be 8 m wide. Since their infinitely thin, we're going to use dx to represent the height of a slice. Putting that all together, the force on one slice is:

(1000)(9.8)x(8)dx = 78400xdx.

Since we have an infinite number of slices, we use an integral to add up all of the slices. Since the depth of the window goes from 4 m to 8 m, we can find the force with the following integral.


Integral


Thus, the total force on the window is 1,881,600 N.

Triangular Window

Now, suppose your boss at the aquarium comes to you and tells you she wants to do a more interesting window design, as seen in the picture below.


Triangular Window


What is the force going to be on this window? You may think it's going to be half the force, but it's actually not! Let's check how much force is on each slice again.

The density of water, the acceleration due to gravity, the depth of a slice, and the height of a slice are all the same as with the rectangular window. The width of a slice changes as we get deeper, though. Let's see if we can come up with a function that will tell us the width of a slice at a depth x. Let w represent this width.

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