# Hydrostatic Pressure: Definition, Equation, and Calculations

## Pressure

When you step on a scale, you get a reading of your **weight**, which is simply the force due to gravity. Your weight on the scale will read the same no matter how you stand on it - with both feet on the scale, with one foot in the air or even if you do a handstand!

What's different is the **pressure** you exert on the scale in each of these situations because this is the force exerted over a given area, or in equation form, *P* = *F*/*A*. Your weight is the force, but the pressure depends on how much area that weight is applied over, be it both feet, one foot or your two hands.

Pressure in a liquid is also the force exerted over a given area, but the difference is that a fluid's pressure pushes on the walls of the surrounding container, as well as on all parts of the fluid itself. This is true for both liquids and gases because they are both fluids, but pressure in a liquid is a little different from that of a gas.

## Pressure in a Liquid

Gas particles are not very friendly. They spread out to fill the entire space of their container, enjoying their personal space and freedom. But as gas particles fly around, they sometimes collide with each other, as well as the walls of the container. These interactions create pressure in the container, and in a gas, this pressure is the same throughout the entire fluid.

But you can clearly see that this is not the case for liquids because they do not fill their entire container like gases do. This is because of the bonds between the liquid's molecules, which are what hold them together. When you pour a liquid into a container, it fills the bottom because gravity pulls it down. This force due to gravity is the same as your scale reading - it's the liquid's weight and is what creates pressure in that liquid.

The pressure in the liquid also increases with depth because of gravity. The liquid at the bottom has to bear the weight of all the liquid above it, as well as all of the air above that! You don't notice the weight of the air around you because your body is 'pressurized' the same as the atmosphere, but any liquid under that atmosphere definitely feels it.

You can experience this change in pressure when you swim to the bottom of a pool. As you go deeper underwater, you feel the pressure increasing because there is more and more weight on top of you. But the pressure doesn't just build up on top of you. Because you're in a fluid, you'll feel that pressure increase all around you.

## Calculating Liquid Pressure

When a liquid is at rest, meaning that it is not flowing, we can determine its pressure at a given depth known as **hydrostatic pressure**. The way we determine this is through an equation: *P* = *rho* * *g* * *d*, where *P* is the pressure, *rho* is the density of the liquid, *g* is gravity and *d* is the depth.

You may also see the hydrostatic equation written as P = *rho* * *g* * *h*, where the *h* stands for height. This may be used because sometimes we want to calculate the pressure of a liquid as it fills a column (like when measuring barometric pressure), so we need to know the height of the fluid. It's like taking the depth and flipping it upside down. As long as you use the appropriate measurement, either letter is okay to use, but it might help to stick with the letter that best represents what you're measuring - either the depth or the height.

It's important to remember that the density of the liquid doesn't change with depth any more than the density of a candy bar changes when you break it into separate pieces. Liquids are not compressible, meaning their molecules are already about as close together as they can be. It's also a good time to take note of that *g* in the equation. It acts as a constant reminder of how gravity plays a crucial role in the pressure of a liquid at any given depth.

## Examples

Now that we know how to calculate hydrostatic pressure, let's put it into action. Let's say we want to calculate the pressure of water at the bottom of a pool that's four meters deep. Luckily, you don't need to memorize the densities of various fluids since those can be looked up, and the density of water is 1,000 kg/m^3. We know that *g* is always 9.8 m/s^2, so it looks like we have everything we need to find the pressure.

Plugging in our values, we get: *P* = 1,000kg/m^3 * 9.8 m/s^2 * 4 m. Our pressure then is 39,200 kg/m-s^2. These units of pressure are perfectly acceptable, but we can also write them as **Pascal**. This is represented by the letters 'Pa,' which is the standard unit of pressure and is named after the French mathematician Blaise Pascal. A Pascal is the same as 1 kg/m-s^2, but writing Pa sure takes a lot less time!

We can also rearrange this equation to determine other information about the liquid. Say, for example, that we already know the pressure and the density of a liquid, but we want to find the depth at which this pressure occurs. All we have to do is move the variables around in the equation and then calculate the depth. Let's say our pressure is 10,000 Pa (same as 10,000 kg/m-s^2) and our liquid this time is milk, which has a density of about 1,035 kg/m^3.

Our equation needs to be rearranged so that depth is alone, so we simply divide the pressure by the density of the liquid and *g*. Plugging in our variables, we get: 10,000 Pa/(1,035 kg/m^3 * 9.8 m/s^2) = d. Once we do the math, we find that this pressure occurs in our milk at a depth of 0.986 m.

This same principle can be used to find the density of the liquid if the pressure and depth are known. In fact, since *g* is always 9.8 m/s^2, as long as you know two of the other variables, you can easily calculate the third. All it takes is a little rearranging followed by some quick math.

## Lesson Summary

In a liquid, pressure pushes not only on the container that holds the liquid but also on all parts of the fluid itself. Pressure in a liquid is caused by the **weight** of the liquid, which is the force due to gravity. As the depth increases, so does the pressure because there is more weight (or force) coming from above.

The pressure in a liquid at a given depth is called the **hydrostatic pressure**. This can be calculated using the hydrostatic equation: *P* = *rho* * *g* * *d*, where *P* is the pressure, *rho* is the density of the liquid, *g* is gravity (9.8 m/s^2) and *d* is the depth (or height) of the liquid.

Using this equation, we can determine the pressure at any given depth within a liquid as long as we know the liquid's density. We can also find the density or depth of the liquid, as long as we know the other variables and rearrange the equation appropriately.

## Learning Outcomes

When you get to the end of this lesson, you might have the capacity to:

- Define hydrostatic pressure
- Understand the characteristics of pressure in a liquid
- Calculate the pressure of any liquid using the hydrostatic equation

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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.

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