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# Hydrostatic Pressure: Equation and Example Calculations

Betsy Chesnutt, Elizabeth Friedl, Kathryn Boddie
• Author
Betsy Chesnutt

Betsy has a Ph.D. in biomedical engineering from the University of Memphis, M.S. from the University of Virginia, and B.S. from Mississippi State University. She has over 10 years of experience developing STEM curriculum and teaching physics, engineering, and biology.

• Instructor
Elizabeth Friedl

Elizabeth, a Licensed Massage Therapist, has a Master's in Zoology from North Carolina State, one in GIS from Florida State University, and a Bachelor's in Biology from Eastern Michigan University. She has taught college level Physical Science and Biology.

• Expert Contributor
Kathryn Boddie

Kathryn has taught high school or university mathematics for over 10 years. She has a Ph.D. in Applied Mathematics from the University of Wisconsin-Milwaukee, an M.S. in Mathematics from Florida State University, and a B.S. in Mathematics from the University of Wisconsin-Madison.

What is Hydrostatic Pressure? Learn about Pascal's Law equation and the calculations to understand the pressure variation in fluids at rest with depth. Updated: 11/05/2021

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## What Is Pressure?

Pressure is defined as the force exerted divided by the area over which the force is applied. Pressure is typically measured in units of Pascals (Pa), and 1 Pa is equal to 1 N/m2.

$$P=\frac{force}{area}=\frac{F}{A}$$

For example, when a person is standing on the ground, there is always pressure exerted by the person's feet on the ground. The amount of pressure depends not only on the person's weight, but also on the type of shoes they are wearing and whether they are standing on one or two feet. For example, walking in high heels creates a much higher pressure than walking in flat shoes. This is because, when walking in high heels, the weight of the person is distributed over a very small area.

Pressure is also exerted by fluids like water and even gases like air. In a fluid, pressure is exerted on all sides of the container that holds the fluid, as well as on the particles within the fluid.

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## Hydrostatic Pressure

On Earth, gravity pulls down on everything including all the liquid water contained in oceans and lakes throughout the world. Gravity even exerts forces on the fluids contained in a cup of coffee or glass of water. Within fluids, these gravitational forces create a type of pressure known as hydrostatic pressure. To define hydrostatic pressure, the word hydrostatic is a combination of two Greek words, Hydro, meaning water, and static, meaning to stand or be firm. So, the hydrostatic pressure definition is the pressure that exists within standing or stationary fluids like water.

At the surface of a body of water, air in the atmosphere exerts a downward force on the particles in the fluid. This is known as atmospheric pressure (Patm).

Although the only pressure exerted on a fluid at its surface is atmospheric pressure, hydrostatic pressure increases as the depth of the fluid increases. This happens because the fluid particles at the bottom feel pressure not only from the atmosphere, but also because of the weight of all the rest of the fluid above. This change in pressure with depth means that, in the ocean, the pressure at the bottom of the deep ocean is very high. It is also apparent in smaller bodies of water like swimming pools. As swimmers swim down toward the bottom of a pool, they can feel the increased hydrostatic pressure pressing on their bodies.

### Hydrostatic Pressure Equation

The hydrostatic pressure in a fluid depends on the depth of the water (h), the gravitational constant (g=9.8 m/s2), and the density of the fluid {eq}\rho {/eq}, which for water is about 1000 kg/m3. The hydrostatic pressure in a fluid is calculated using the hydrostatic pressure equation, which is also known as the hydrostatic pressure formula or sometimes the water pressure formula:

$$P=\rho gh+P_{atm}$$

Sometimes, the depth of water is written using a d instead of an h, so the hydrostatic pressure equation can also be written as:

$$P=\rho gd+P_{atm}$$

It's often important to determine how much the hydrostatic pressure changes from one point to another within a fluid. In that case, it is the change in depth ({eq}\Delta h {/eq}) that is important. In an equation or formula, the symbol delta ({eq}\Delta {/eq}) means change in, so the change in pressure ({eq}\Delta P {/eq}) can be calculated using a version of the hydrostatic pressure equation:

$$\Delta P=\rho g\Delta h=\rho g\Delta d$$

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## Comparing Hydrostatic Pressure in Different Liquids

In the following examples, students will calculate the hydrostatic pressure of a liquid at a certain depth, given the liquid's density. Students will also get practice finding the depth of the liquid, given the density and the hydrostatic pressure as well as practice finding the density of the liquid, given the depth and the hydrostatic pressure. After completing these beginning examples, students will be more comfortable manipulating the equation for hydrostatic pressure to solve for different variables. Then students can move on to the challenge example, where they will solve to find what depth for different liquids are needed so that the liquids have the same hydrostatic pressure.

## Beginning Examples

• Find the hydrostatic pressure of water at the bottom of a 7 meter pool. The density of water is 1000 kg/m^3.
• At what depth is the hydrostatic pressure of gasoline equal to 50,000 Pa? The density of gasoline is 725 kg/m^3.
• What is the density of diesel, given that the hydrostatic pressure at a depth of 4 meters is 33,712 Pa?

## Solutions

• We will use the formula P = rho * g * d where rho is the density of the liquid, g is gravity, and d is the depth of the liquid. We have P = 1000 * 9.8 * 7 = 68,600 Pa for the hydrostatic pressure of water at the bottom of a 7 meter pool.
• Using the same formula P = rho*g*d, we have 50,000 = 725 * 9.8 * d, and so we have 50,000 / 7105 = d. The depth is about 7.04 meters.
• Using the same formula P = rho * g * d, we have 33,712 = rho * 9.8 * 4 and so we have 33,712 / 39.2 = rho. The density of diesel is 860 kg/m^3.

## Challenge Problem

• At what depth is the hydrostatic pressure for gasoline equal to the hydrostatic pressure for water at the bottom of a 10 meter pool? Remember that the density of water is 1000 kg/m^3 and the density of gasoline is 725 kg/m^3.

## Solution

• First, find the hydrostatic pressure of the water at a depth of 10 meters. We have P = 1000 * 9.8 * 10 = 98,000 Pa. We now need to find what depth is needed for the hydrostatic pressure of gasoline to also equal 98,000 Pa. Using the formula again, we have 98,000 = 725 * 9.8 * d and so 98,000 / 7105 = d. The depth is about 13.79 meters. This means that if you are submerged 13.79 meters in gasoline, the pressure will feel the same as the pressure of being submerged 10 meters in water.

#### How is hydrostatic pressure calculated?

The hydrostatic pressure in a fluid is calculated using the hydrostatic pressure equation, and it depends on the density of the fluid, the depth, and the gravitational constant, g.

P=density×depth×g = ρ×g×h

#### What is the unit for hydrostatic pressure?

The units of hydrostatic pressure are the same as the units for any kind of pressure. Pressure is defined as force/area, so the units are N/m² or Pa.

#### What causes hydrostatic pressure?

Hydrostatic pressure is caused by the force of gravity acting on all the particles within a fluid. As the depth of the fluid increases, the weight of the fluid above that depth also increases, which increases the hydrostatic pressure.

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