Pressure in a fluid

What you'll learn

What pressure means and how to calculate it from force and area.
Why liquids and gases push at right angles to surfaces.
How pressure in a liquid depends on depth, density and gravitational field strength.
How pressure differences create upthrust, floating and sinking.

Pressure: force spread over an area

Pressure tells you how concentrated a force is. The same force can make a large pressure if it acts over a small area, or a smaller pressure if it is spread over a larger area.

A force that acts normal to a surface acts at right angles to that surface.

Definition

Pressure

Pressure is the force acting normally on each square metre of a surface. It is calculated using:

\frac{F}{A}

where $p$ is pressure in pascals (Pa), $F$ is force in newtons (N), and $A$ is area in square metres (m²).

One pascal is one newton per square metre.

Tip

Area units matter

If the area is given in cm², convert it to m² before using the equation. A common conversion is: 10 000 cm² = 1 m².

Example

Calculating pressure from force and area

A student presses down on the floor with a force of 600 N. The area of their shoe in contact with the floor is 0.030 m². Calculate the pressure on the floor.

Choose the pressure equation because you know force and area:
$\frac{F}{A}$
Substitute the values with units:
$\frac{600 \ \text{N}}{0.030 \ \text{m}^2}$
Calculate the pressure:
$20\,000 \ \text{Pa}$

So the pressure on the floor is 20 000 Pa.

What is a fluid?

A fluid is any substance that can flow. That means both liquids and gases are fluids.

Definition

Fluid

A fluid is a liquid or a gas. Fluids can flow and take the shape of their container.

Fluids exert pressure on surfaces. In a stationary fluid, pressure at a point acts equally in all directions. However, the force caused by that pressure on a surface acts normal to the surface.

For example, water in a tank pushes sideways on the walls, downwards on the base, and upwards on anything submerged in it.

Key Idea

Fluid pressure causes normal forces

Pressure in a fluid causes a force at right angles to any surface it touches.

Common Mistake

Pressure vs force

Pressure is not usually drawn as one single arrow in one direction. The force caused by pressure is what acts at right angles to a surface.

Why liquid pressure increases with depth

In a liquid, pressure increases as you go deeper. This is because a deeper point has a taller column of liquid above it. That column has weight, so it presses down more.

The diagram shows that pressure is greater at point B than at point A because B is deeper below the liquid surface.

Diagram of liquid pressure increasing with depth

Definition

Density

Density, $ρ\rho$ , is mass per unit volume. A denser liquid has more mass packed into each cubic metre, so it produces a larger pressure at the same depth.

The other factor is gravitational field strength, $g$ . This tells you the force of gravity on each kilogram of mass. On Earth it is usually about 9.8 N/kg, but in GCSE questions you may be told to use 10 N/kg.

The pressure difference due to a column of liquid is:

\rho g

where:

$p$ is the pressure difference in pascals (Pa)
$h$ is the vertical depth or height of the liquid column in metres (m)
$ρ\rho$ is the density of the liquid in kg/m³
$g$ is gravitational field strength in N/kg

Key Idea

What affects liquid pressure

Liquid pressure increases if the depth increases, the density of the liquid increases, or the gravitational field strength increases.

Building the equation

You can understand $h\rho g$ from earlier ideas:

A taller column of liquid has more volume.
More volume means more mass.
More mass means more weight.
More weight acting on the same area means more pressure.

So the equation is really a combination of density, weight and pressure.

Common Mistake

Using the wrong height

In $h\rho g$ , $h$ is the vertical depth below the surface, or the vertical height difference between two points. It is not the length of a sloping pipe or the width of the container.

Example

Calculating pressure difference between two depths

A diver moves from 0.40 m below the surface to 2.20 m below the surface. The density of water is 1000 kg/m³ and $\ \text{N/kg}$ . Calculate the increase in water pressure.

Find the vertical height difference:
$\ \text{m} - 0.40 \ \text{m} = 1.80 \ \text{m}$
Substitute into the liquid pressure equation:
$h\rho g = 1.80 \times 1000 \times 10$
Calculate the pressure difference:
$18\,000 \ \text{Pa}$

The water pressure has increased by 18 000 Pa.

Common Mistake

Pressure difference, not always total pressure

The equation $h\rho g$ gives the extra pressure due to the liquid column. If a question asks for total pressure below an open surface, add atmospheric pressure if it is supplied.

Same depth, same pressure

In the same liquid, points at the same vertical depth have the same pressure. This is true even if the container is wide, narrow or oddly shaped.

This is why the bottom of a dam has to be especially strong: the water pressure is greatest at the greatest depth.

Key Idea

Only vertical depth matters

For a stationary liquid, pressure depends on vertical depth, not on the total amount of liquid or the shape of the container.

Upthrust from pressure differences

An object in a fluid has pressure acting on all its surfaces.

The sides of the object are usually at similar depths, so the sideways pressure forces balance. But the bottom of the object is deeper than the top, so the pressure on the bottom is larger than the pressure on the top.

This creates a resultant upward force called upthrust.

Diagram showing upthrust caused by higher pressure on the bottom of a submerged object

Definition

Upthrust

Upthrust is the resultant upward force on an object in a fluid, caused by the fluid pressure being greater on the bottom surface than on the top surface.

Key Idea

Where upthrust comes from

Upthrust is not a separate “mystery force”. It is the result of pressure forces, with the upward force on the bottom being larger than the downward force on the top.

Floating and sinking

A submerged or floating object has two important vertical forces:

Weight acts downwards because of gravity.
Upthrust acts upwards because of the fluid pressure difference.

An object floats when the upthrust equals its weight. The resultant force is then zero, so the object stays at rest or moves at constant speed.

An object sinks if its weight is greater than the maximum upthrust the fluid can provide.

Upthrust depends on the fluid displaced. Displaced fluid means fluid pushed out of the way by the object. A larger displaced volume or a denser fluid gives a larger upthrust.

Common Mistake

Floating is not force-free

A floating object still has forces acting on it. Its weight and upthrust are equal in size and opposite in direction, so the resultant force is zero.

Example

Deciding whether an object sinks

A small object has a weight of 5.0 N. When it is fully submerged in water, the upthrust on it is 3.5 N. Decide whether it floats or sinks.

Compare the downward force and upward force: weight is 5.0 N downwards, upthrust is 3.5 N upwards.
Calculate the resultant force:
$\ \text{N} - 3.5 \ \text{N} = 1.5 \ \text{N}$
The resultant force is 1.5 N downwards, so the object sinks.

Exam technique

In the exam

Decide which pressure equation fits the information: use $\frac{F}{A}$ for force and area, and $h\rho g$ for liquid depth.
Check units carefully: area in m², depth in m, density in kg/m³, and pressure in Pa.
For liquid pressure, use the vertical depth or vertical height difference, not the length along a slope.
For floating and sinking, compare weight downwards with upthrust upwards.

Self review

Check yourself

Why does pressure in a liquid increase with depth?
What does the symbol $ρ\rho$ mean in $h\rho g$ ?
For an object floating at rest, how are weight and upthrust related?

Pressure in a fluid

What you'll learn

What pressure means and how to calculate it from force and area.
Why liquids and gases push at right angles to surfaces.
How pressure in a liquid depends on depth, density and gravitational field strength.
How pressure differences create upthrust, floating and sinking.

Pressure: force spread over an area

Pressure tells you how concentrated a force is. The same force can make a large pressure if it acts over a small area, or a smaller pressure if it is spread over a larger area.

A force that acts normal to a surface acts at right angles to that surface.

Definition

Pressure

Pressure is the force acting normally on each square metre of a surface. It is calculated using:

\frac{F}{A}

where $p$ is pressure in pascals (Pa), $F$ is force in newtons (N), and $A$ is area in square metres (m²).

One pascal is one newton per square metre.

Tip

Area units matter

If the area is given in cm², convert it to m² before using the equation. A common conversion is: 10 000 cm² = 1 m².

Example

Calculating pressure from force and area

A student presses down on the floor with a force of 600 N. The area of their shoe in contact with the floor is 0.030 m². Calculate the pressure on the floor.

Choose the pressure equation because you know force and area:
$\frac{F}{A}$
Substitute the values with units:
$\frac{600 \ \text{N}}{0.030 \ \text{m}^2}$
Calculate the pressure:
$20\,000 \ \text{Pa}$

So the pressure on the floor is 20 000 Pa.

What is a fluid?

A fluid is any substance that can flow. That means both liquids and gases are fluids.

Definition

Fluid

A fluid is a liquid or a gas. Fluids can flow and take the shape of their container.

Fluids exert pressure on surfaces. In a stationary fluid, pressure at a point acts equally in all directions. However, the force caused by that pressure on a surface acts normal to the surface.

For example, water in a tank pushes sideways on the walls, downwards on the base, and upwards on anything submerged in it.

Key Idea

Fluid pressure causes normal forces

Pressure in a fluid causes a force at right angles to any surface it touches.

Common Mistake

Pressure vs force

Pressure is not usually drawn as one single arrow in one direction. The force caused by pressure is what acts at right angles to a surface.

Why liquid pressure increases with depth

In a liquid, pressure increases as you go deeper. This is because a deeper point has a taller column of liquid above it. That column has weight, so it presses down more.

The diagram shows that pressure is greater at point B than at point A because B is deeper below the liquid surface.

Diagram of liquid pressure increasing with depth

Definition

Density

Density, $ρ\rho$ , is mass per unit volume. A denser liquid has more mass packed into each cubic metre, so it produces a larger pressure at the same depth.

The pressure difference due to a column of liquid is:

\rho g

where:

$p$ is the pressure difference in pascals (Pa)
$h$ is the vertical depth or height of the liquid column in metres (m)
$ρ\rho$ is the density of the liquid in kg/m³
$g$ is gravitational field strength in N/kg

Key Idea

What affects liquid pressure

Liquid pressure increases if the depth increases, the density of the liquid increases, or the gravitational field strength increases.

Building the equation

You can understand $h\rho g$ from earlier ideas:

A taller column of liquid has more volume.
More volume means more mass.
More mass means more weight.
More weight acting on the same area means more pressure.

So the equation is really a combination of density, weight and pressure.

Common Mistake

Using the wrong height

In $h\rho g$ , $h$ is the vertical depth below the surface, or the vertical height difference between two points. It is not the length of a sloping pipe or the width of the container.

Example

Calculating pressure difference between two depths

A diver moves from 0.40 m below the surface to 2.20 m below the surface. The density of water is 1000 kg/m³ and $\ \text{N/kg}$ . Calculate the increase in water pressure.

Find the vertical height difference:
$\ \text{m} - 0.40 \ \text{m} = 1.80 \ \text{m}$
Substitute into the liquid pressure equation:
$h\rho g = 1.80 \times 1000 \times 10$
Calculate the pressure difference:
$18\,000 \ \text{Pa}$

The water pressure has increased by 18 000 Pa.

Common Mistake

Pressure difference, not always total pressure

The equation $h\rho g$ gives the extra pressure due to the liquid column. If a question asks for total pressure below an open surface, add atmospheric pressure if it is supplied.

Same depth, same pressure

In the same liquid, points at the same vertical depth have the same pressure. This is true even if the container is wide, narrow or oddly shaped.

This is why the bottom of a dam has to be especially strong: the water pressure is greatest at the greatest depth.

Key Idea

Only vertical depth matters

For a stationary liquid, pressure depends on vertical depth, not on the total amount of liquid or the shape of the container.

Upthrust from pressure differences

An object in a fluid has pressure acting on all its surfaces.

This creates a resultant upward force called upthrust.

Diagram showing upthrust caused by higher pressure on the bottom of a submerged object

Definition

Upthrust

Upthrust is the resultant upward force on an object in a fluid, caused by the fluid pressure being greater on the bottom surface than on the top surface.

Key Idea

Where upthrust comes from

Upthrust is not a separate “mystery force”. It is the result of pressure forces, with the upward force on the bottom being larger than the downward force on the top.

Floating and sinking

A submerged or floating object has two important vertical forces:

Weight acts downwards because of gravity.
Upthrust acts upwards because of the fluid pressure difference.

An object floats when the upthrust equals its weight. The resultant force is then zero, so the object stays at rest or moves at constant speed.

An object sinks if its weight is greater than the maximum upthrust the fluid can provide.

Upthrust depends on the fluid displaced. Displaced fluid means fluid pushed out of the way by the object. A larger displaced volume or a denser fluid gives a larger upthrust.

Common Mistake

Floating is not force-free

A floating object still has forces acting on it. Its weight and upthrust are equal in size and opposite in direction, so the resultant force is zero.

Example

Deciding whether an object sinks

A small object has a weight of 5.0 N. When it is fully submerged in water, the upthrust on it is 3.5 N. Decide whether it floats or sinks.

Compare the downward force and upward force: weight is 5.0 N downwards, upthrust is 3.5 N upwards.
Calculate the resultant force:
$\ \text{N} - 3.5 \ \text{N} = 1.5 \ \text{N}$
The resultant force is 1.5 N downwards, so the object sinks.

Exam technique

In the exam

Decide which pressure equation fits the information: use $\frac{F}{A}$ for force and area, and $h\rho g$ for liquid depth.
Check units carefully: area in m², depth in m, density in kg/m³, and pressure in Pa.
For liquid pressure, use the vertical depth or vertical height difference, not the length along a slope.
For floating and sinking, compare weight downwards with upthrust upwards.

Self review

Check yourself

Why does pressure in a liquid increase with depth?
What does the symbol $ρ\rho$ mean in $h\rho g$ ?
For an object floating at rest, how are weight and upthrust related?

Pressure in a fluid

What you'll learn

Pressure: force spread over an area

What is a fluid?

Why liquid pressure increases with depth

Building the equation

Same depth, same pressure

Upthrust from pressure differences

Floating and sinking

You've reached the end

Pressure in a fluid

What you'll learn

Pressure: force spread over an area

What is a fluid?

Why liquid pressure increases with depth

Building the equation

Same depth, same pressure

Upthrust from pressure differences

Floating and sinking

You've reached the end

Pressure in a fluid

What you'll learn

Pressure: force spread over an area

What is a fluid?

Why liquid pressure increases with depth

Building the equation

Same depth, same pressure

Upthrust from pressure differences

Floating and sinking

You've reached the end

Pressure in a fluid Revision Guide

Pressure in a fluid

What you'll learn

Pressure: force spread over an area

What is a fluid?

Why liquid pressure increases with depth

Building the equation

Same depth, same pressure

Upthrust from pressure differences

Floating and sinking

You've reached the end

Pressure in a fluid Revision Guide