Application of the integral: Pressure and force of a fluid

carlos84 (72)in Popular STEM • yesterday

Is it important for us to know what is the pressure of a fluid?

Suppose we have no idea what it is, however when we dive into a pool we realize that as we dive deeper and deeper into the pool the pressure we feel is greater, this is because the pressure of a fluid is defined as the force exerted by a fluid per unit area, so a column of fluid the longer it is, the more pressure it will generate on the object that is immersed.

That is why the pressure of an object immersed in a column of a fluid where the depth is denoted by h, the pressure p is:

p = wh, where:

h: is the depth
w: is the density of the fluid

Pascal's principle for fluid pressure calculation

The pressure exerted by a fluid at a depth h is exactly equal in all directions and since:

P = f / A, implies that if we clear f:

f = P A

Definition of force exerted by a fluid applying the integral

The force f exerted by a fluid of constant weight-density w (per unit volume) on a vertical plane region submerged from y = c to y = d is:

Exercise: Force of a fluid on a vertical surface

A gate of a vertical dam on a levee has the shape of a trapezoid, with 8 feet at the top and 6 feet at the bottom with a height of 5 feet, as shown in the following figure:

What is the fluid force on the gate when the top is 4 feet below the water surface?

Solution to the exercise

The first thing to do is to find the functions h(y) y L(y).

For this purpose, we must formulate a mathematical model that allows us to obtain these equations:

For this purpose it is convenient to place the y-axis bisecting the floodgate and to place the x-axis at the water surface, as shown in the following figure:

As shown in the previous figure, the depth is:

h(y) = -y

The mathematical model helps us to find L(y), which in turn is the length of one side of the trapezoid, for this we find the equation of the line that forms the right side of the trapezoid, of that side we know the coordinates of the points P2(4 ; -4) and P1(3 ; -9), so to find the equation of the line we apply the following formula:

We clear the variable x:

In the figure of the submerged trapezoid that is the gate of a vertical dam it can be seen that the length of the region at y is 2x:

Length = L(y) = 2x; we substitute the value of x = (y + 24) / 5:

As we already have:

h(y) = -y
L(y) = 2/5 (y + 24)

The definition of force exerted by a fluid is applied by applying the integral:

If we assume that the fluid in which the gate is immersed is water, then its density in units of pounds / Cubic Inches is 62.4, which implies that:

We take out the common factor -1:

We solve the definite integral:

The force exerted by the water on the gate, considering a water density of 62.4 Pounds / cubic inches and considering the length dimensions of the gate is 13936 pounds force.

The calculations given at the end of the exercise have to consider that the units of cubic inches of the density cancel with the cubic inches represented by Volume = h(y)L(y), where: The force exerted by the water on the gate, considering a water density of 62.4 Pounds / cubic inches and considering the length dimensions of the gate is 13936 pounds force.

The calculations given at the end of the exercise have to consider that the units of cubic inches of the density cancel with the cubic inches represented by Volume = h(y)L(y), where:

h(y): Depth measured in inches.
L(y): Area measured in square inches.