Variation of Pressure With Depth

It’s reasonable to assume that the deeper a person travels into a liquid or gas, the higher the pressure exerted on him by the surrounding fluid. The reason for the increasing pressure is that the deeper a person goes into a fluid, the more fluid he has over top of him, and therefore the more weight he has.

Pressure: The ratio of the force applied to the surface area over which the force is applied is known as the pressure. The pressure imposed by liquids is known as hydrostatic pressure. The SI unit of pressure is Pascal.

Fluid Pressure

Solid things do not change form when pressure is applied, which is obviously not the case with fluids. In a closed container, fluid pressure can be generated by gravity, acceleration, or forces. The fluid acts equally in all directions since it has no set form. When you fill a bottle with water, the weight of the water acts evenly on both sides of the bottle.

The force is always exerted perpendicular to the container’s surface. This may be seen in a balloon. As you fill the balloon with air, you’ll note that it grows evenly, with no one side inflating more than the other. Liquids in a container also show this behaviour.

Hydrostatic Pressure

The hydrostatic pressure is the pressure exerted by a fluid in equilibrium owing to gravity at any given period. When a downward force is applied, hydrostatic pressure is proportional to the depth measured from the surface as the weight of the fluid increases.

Fluids exert equal pressure in all directions. Another intriguing event occurs as a result of this rule. When we examine the layer of water on top of a bottle, the pressure it exerts acts on the edges of the container, the air surface on top, and the layer of water at the bottom. The pressure imposed by the top layer on the bottom increases as we go down the bottle from top to bottom.

The fluid at bottom of the container receives higher pressure than the fluid above it as a result of this process.

Variation of Pressure with Depth

Take a look at the figure below for an example of a container. The weight of fluid inside it is supported by its bottom. Let’s see how much pressure the weight of liquids exerts on the bottom. The weight of fluid ‘mg’ divided by the area ‘A’ equals the pressure.

Weight of the fluid, W = m g

Mass of the fluid is equal to product of volume and density of substance, i.e., m = ρ V

Volume of the fluid is equal to the dimension of the container, i.e., V = A h

Combine the last two equations for mass.

m = ρ A h

Therefore, weight of the fluid, W = ρ A h g

The pressure exerted on the bottom of the container is given as:

P = W ⁄ A

P = (ρ A h g) ⁄ A

P = ρ h g

where,

• ρ is the density of fluid
• A is the surface area of container
• h is the height upto the fluid is filled in container
• V is the volume of fluid
• m is the mass of fluid
• g is the acceleration due to gravity
• W is the weight of fluid
• P is the pressure exerted on the bottom of the container.

This is the pressure created by a fluid’s weight. Beyond the specific conditions under which it is derived here, the equation has generic validity. The surrounding fluid would exert this pressure even if the container was not there, keeping the fluid static. This equation remains true to deep depths for liquids that are practically incompressible. This equation may be used for gases that are very compressible as long as the density variations are minimal across the depth covered.

Sample Problems

Problem 1: Define the relationship between the pressure and height of the liquid column.

Answer:

The pressure exerted by a liquid depends on the height of the liquid column.

Pressure can be written as P = ρ g h where h is height and ρ is density. The formula shows the direct relation between the pressure and height of the column. 

Therefore, as the height increases, pressure will also increase .

Problem 2: Calculate the force exerted by water on the base of a tank of area 3 m² when filled with water up to a height of 2 m. (Density of water is 1000 kg m⁻³ and g = 10 m s⁻²).

Solution:

Given:

Area of base of tank, A = 3 m²

Height of the tank filled with water, h = 2 m

Density of the fluid, ρ = 1000 kg m⁻³

The formula of the pressure is given as:

P = ρ g h

= (1000 × 10 × 2) N m⁻²

= 20000 N m⁻²

The force exerted by the water,

F = P A

= (20000 × 3) N

= 60000 N

Hence, the force exerted by the water is 60000 N.

Problem 3: Why the dam of the water reservoir is thick at the bottom?

Answer:

The dam of a water reservoir is thick at the bottom because the pressure of water is highest at the maximum depth, and the dam must be strong at the bottom to withstand this maximum pressure.

Problem 4: What is hydrostatic pressure?

Answer:

The hydrostatic pressure is the pressure exerted by a fluid in equilibrium owing to gravity at any given period. When a downward force is applied, hydrostatic pressure is proportional to the depth measured from the surface as the weight of the fluid increases.

Problem 5: What is the pressure acting on the water at a depth of 2 m at 4°C?

Solution:

Given:

The depth of water column, h = 2 m

The density of water at 4°C, ρ = 1000 kg ⁄ m³

The formula of the pressure is given as:

P = ρgh

= (1000 × 9.81 × 1) Pa

= 9810 Pa.

Hence, the pressure acting on the water at a depth of 2 m is 9810 Pa.