Pressure and Total Force
14 Calculating Total Force
Click play on the following audio player to listen along as you read this section.
So far we’ve determined that pressure is a force per unit area and that pressure can be stated in different ways with pounds per square inch being the most common. We calculated the pressure at the base of an object or piping system filled with water.
Now it’s time to calculate the total force at the base of an object or piping system given the pressure exerted at its base. As before we will use water as the fluid in the example questions.
Take a look at the figure below and then read the explanation below that.
Start with an object filled with water. In this case a cylinder or possibly something like a hot water tank. The object will be three dimensional and have a height, width and depth.
This is the base of the object. As we are calculating total force at the base of this object we have to decide what the base consists of and define its parameters.
The base has been expanded for visual purposes to allow for the introduction of a few square inches inserted into the base. As stated previously we are calculating pressure in pounds per square inch therefore it’s going to be important to find the area at the base in square inches.
After all that we can put everything together to get our total force formula.
[latex]\text{total force} = \text{ area} \times \text{ psi}[/latex]
There are a couple things to note here.
1) The area must be in square inches
2) The height used to determine psi must be in feet.
This concept occasionally mixes people up as one part of the equation is calculated using inches and the other part is calculated using feet. This is generally opposite from what we’ve been saying in most of this math journey where we’ve made sure all the variables must be in the same unit. Let’s head off to an example to put all this together.
Example
Calculate the force at the base of the tank pictured above.
Step 1: Write down the formula you’ll be working with.
Note that in this case we have a rectangular tank with the area of the base being the length multiplied by the width.
[latex]\text{total force} = (l \times w) \times (height \times 0.433)[/latex]
Step 2: Write down the variables and put them in proper units to work with.
Step 3: Plug the reworked variables into the equation.
[latex]\begin{array}{c} \text{total force} &= (l \times w) \times (height \times 0.433) \\ \text{total force} &= (32in \times 8 in) \times (1.08 ft \times 0.433 \text{ psi/ft}) \\ \text{total force} &= (256 in^2) \times (0.464 \text{ psi}) \\ \text{total force} &= 118.78 \text{ pounds} \end{array}[/latex]
[latex]118.78 lbs[/latex]
Note the units that we end up with. We end up with pounds as our total force. The inches in the area at the base and the inches in the pounds per square inch cancel each other out. This leaves us with just pounds as an answer.
The total force at the base of the tank works out to be 188.78 pounds. There are 256