Fluid Pressure

Last updated on June 28th, 2023 at 10:07 am

If we dive to the bottom of a swimming pool, we will feel the water pressure pushing into our ears. This pressure is created by the weight of the water above us, and the deeper we go, the greater this pressure. Similarly, the weight of the atmosphere produces an air pressure of about 1.0 × 10⁵ Pa at the Earth’s surface. All these are examples of Fluid Pressure.

In a fluid of constant density variation of pressure happens with depth. In this post, we will find out how pressure varies with depth in a fluid of constant density, and then derive the fluid pressure formula.

Variation of fluid pressure with depth (in a fluid of constant density)

Fluid pressure is directly proportional to the depth (or the height of the fluid column). If 2 points in a fluid are in depths h1 and h2 respectively, and h2>h1, then fluid pressure at depth h2 > fluid pressure at depth h1.

Derivation of fluid pressure formula | Fluid pressure formula derivation

Let’s consider a fluid of constant density.

Let’s take:
The density of the fluid = ρ
The volume of the fluid = V
Area of the bottom surface =A
and height of the fluid column = h
weight of the fluid = w
and mass of the fluid =m

The pressure at the bottom of the container is due to the pressure of the atmosphere (P₀) plus the pressure due to the weight of the fluid.

The pressure due to the fluid is equal to the weight of the fluid divided by the area.

Now, the weight of the fluid is equal to its mass times the acceleration due to gravity

Since the density is constant, the weight of the fluid can be calculated using the density:
w = mg = ρVg = ρAhg. ……………… (equation 1)

Note: We used these formulas to get equation 1 given above,
mass = density x volume
=> m = ρV
&
The volume of the fluid (V)= (area of the bottom surface) (height of the fluid)
=> V = A h

Now we will use equation 1 above to get the equation of the pressure of a fluid.

Pressure due to the weight of the fluid = weight of the fluid /area = w/A = ρAhg/A = ρhg

Pressure due to the weight of the fluid = ρhg

This equation thus shows how the pressure varies with depth in a fluid of constant density.

Fluid pressure formula | equation of the fluid pressure

Pressure due to the weight of the fluid = ρhg

Total pressure at the bottom of the container containing fluid = atmospheric pressure P₀ + the Pressure due to the weight of the fluid
=> P = P₀ + ρhg