# Hydrostatic Pressure: Equation and Example Calculations

## 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.

## 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:

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

### Hydrostatic Paradox

As can be seen from the hydrostatic pressure equation, the hydrostatic pressure in a fluid only depends on the depth of the fluid and not on the size or shape of the container that holds the fluid. This is known as the **hydrostatic paradox**.

This means that the pressure 1 meter under the surface of the ocean is exactly the same as the pressure at the bottom of a 1 meter deep swimming pool even though the ocean is MUCH bigger!

## Hydrostatic Pressure: Examples

The hydrostatic pressure equation can be used to calculate the hydrostatic pressure at varying depths, as well as to find the density of a fluid if the pressure and depth are known. Let's look at a couple of examples of how to use it.

First, the hydrostatic pressure equation can be used to calculate how much the pressure at the bottom of a 5 meter deep diving well in a swimming pool increases compared to the pressure at the surface of the water. The pool is filled with water with a density of 1000 kg/m3, and g is always 9.8 m/s2 on Earth, so the pressure at the bottom of the pool is given by:

$$\Delta P=\rho g\Delta h=(1000\; \frac{kg}{m^{3}})(9.8\; \frac{m}{s^{2}})(5\; m)=49,000 \; Pa $$

What if the pool were drained and then filled with a mystery fluid that is NOT water? If the pressure difference between the surface and the bottom of the diving well is 60,000 Pa, what is the density of this mystery fluid?

To find the density of the mystery fluid, rearrange the hydrostatic force equation to solve for the density.

$$\rho =\frac{\Delta P}{g\Delta h} $$

Then, use the measured values to find the density of the fluid.

$$\rho =\frac{60,000\; Pa}{(9.8 \; \frac{m}{s^{2}})(5\; m)}=1,220 \; \frac{kg}{m^{3}} $$

## Lesson Summary

**Pressure** is defined as force divided by the area over which the force is applied {eq}P=\frac{force}{area}=\frac{F}{A} {/eq}. **Hydrostatic pressure** is the pressure that exists within standing or stationary fluids like water. It is determined by:

- the
**atmospheric pressure**that is exerted by the surrounding atmosphere on the surface of the fluid - the depth within the fluid (
*d*or*h*)

The hydrostatic pressure in a fluid is calculated using the **hydrostatic pressure equation**:

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

The change in pressure ({eq}\Delta P {/eq}) within a fluid can also be calculated using a version of the hydrostatic pressure equation, {eq}\Delta P=\rho g\Delta h=\rho g\Delta d {/eq}. As the hydrostatic pressure equation demonstrates, the hydrostatic pressure in a fluid only depends on the depth of the fluid and not on the size or shape of the container that holds the fluid, which is known as the **hydrostatic paradox**.

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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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