How does pressure vary with depth in a fluid of constant density? | with the equation of the fluid pressure

Last updated on April 15th, 2021 at 03:08 pm

Variation of pressure with depth in a fluid of constant density or How does pressure varies with depth in a fluid of constant density? – this is the topic of this post.

Variation of pressure with depth in a fluid of constant density – Lab setup

Let’s consider a fluid of constant density as shown in the Figure below. Now let’s go ahead step by step towards our objective.

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

How does pressure varies with depth in a fluid of constant density? – Observations | equation of the fluid pressure

1 ) 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.

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

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

Since the density is constant, the weight can be calculated using the density:
w = mg = ρVg = ρAhg.

[ we used these formulas above,
mass = density x volume
=> m = ρV
&
Volume of the fluid (V)= (area of the bottom-surface) (height of the fluid)
=> V = A h
]

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

5)
The pressure at the bottom of the container is therefore equal to atmospheric pressure P₀ added to the Pressure due to the weight of the fluid:

P = P₀ + ρhg . This equation thus shows how the pressure varies with depth in a fluid of constant density.