Pressure in fluids and upthrust

What you'll learn

What pressure means, and how it depends on force and area.
Why pressure in liquids and gases acts in all directions and is normal to surfaces.
Why liquid pressure increases with depth and density.
How upthrust explains floating, sinking, and objects feeling lighter in fluids.

This Edexcel section is marked with a P, so it is Separate Physics content. Some parts, especially the liquid pressure equation and upthrust explanations, are Higher Tier only in the specification.

The starting point: fluids and pressure

A fluid is a substance that can flow. Liquids and gases are both fluids.

A fluid contains particles that move around and collide with surfaces. Each collision gives a tiny push. The total effect of many tiny pushes over an area is called pressure.

Definition

Pressure

Pressure is the force acting normally on each square metre of a surface. It is measured in pascals, Pa.

One pascal means one newton per square metre.

Pressure can be caused by:

a solid pressing on a surface, such as a shoe on the ground
a liquid pressing on the sides of a container
a gas pressing on surfaces, such as air pressing on your body

Force, area and pressure

Pressure depends on two things:

the size of the force
the area over which the force is spread

If the same force acts over a smaller area, the pressure is larger. This is why a sharp knife cuts better than a blunt one: the force is concentrated over a tiny area.

Key Idea

Force spread over area

A larger force increases pressure, but a larger area decreases pressure.

The equation is:

P=FAP=\frac{F}{A}

where:

$P$ is pressure in pascals, Pa
$F$ is force normal to the surface in newtons, N
$A$ is area in square metres, m²

For Edexcel, this is a recall and use equation, so you should learn it.

Example

Calculating pressure from force and area

A student pushes a drawing pin into a board with a force of 12 N. The sharp tip has an area of 0.000004 m². Calculate the pressure on the board.

Choose the pressure equation because the question gives force and area:
$P=FAP=\frac{F}{A}$
Substitute the values, keeping the area in square metres:
$P=120.000004P=\frac{12}{0.000004}$
Calculate the pressure:
$PaP=3000000\text{ Pa}$
So the pressure is 3,000,000 Pa, or $3.0 \times 10^6\text{ Pa}$ .

Common Mistake

Forgetting to convert area

Area must be in square metres, m². If an area is given in cm², convert it before using $P=FAP=\frac{F}{A}$ . Remember: 1 cm² is 0.0001 m².

Pressure acts normal to a surface

In physics, normal means at right angles, or perpendicular, to a surface.

Definition

Normal force

A force normal to a surface acts at 90 degrees to that surface.

Pressure in a fluid causes a force normal to any surface it touches. For example:

water pushes sideways on the walls of a container
air pushes normally on your skin
water pushes upwards on the bottom of a boat

Pressure in a fluid is not just “downwards”. It acts in all directions, but the force on any particular surface is perpendicular to that surface.

The diagram shows fluid pressure increasing with depth and acting normally to the container wall.

Diagram of fluid pressure increasing with depth and acting normal to surfaces

Atmospheric pressure

The atmosphere is the layer of air around Earth. Air is a gas, so it is a fluid.

Definition

Atmospheric pressure

Atmospheric pressure is the pressure caused by the weight of the air above a surface.

A simple model is to imagine the atmosphere as many layers of air stacked above Earth’s surface.

At sea level, there is a tall column of air above you. The air below has to support the weight of all the air above it, so atmospheric pressure is relatively high.

Higher up, such as on a mountain, there is less air above you. The weight of air above is smaller, so atmospheric pressure is lower. The air is also less dense higher up, which adds to the decrease in pressure.

Example

Explaining lower pressure on a mountain

Why is atmospheric pressure lower at the top of a mountain than at sea level?

Compare the amount of air above each place: at sea level there is more air above you; on the mountain there is less air above you.
Link pressure to weight: atmospheric pressure is caused by the weight of the air above the surface.
Conclude from the model: less air above means less weight pressing down, so the atmospheric pressure is lower on the mountain.

Pressure in a fluid

The total pressure at a point in a fluid can include:

pressure due to the fluid itself
atmospheric pressure acting on the surface of the fluid

For example, in an open swimming pool, atmospheric pressure acts on the water surface. The water pressure then increases as you go deeper.

Key Idea

Pressure in an open liquid

At depth in an open liquid, the pressure is due to both the water above that point and the atmosphere pushing on the surface.

Why liquid pressure increases with depth

As you go deeper in a liquid, there is more liquid above you. That liquid has weight. The deeper point has to support a taller column of liquid, so the pressure is greater.

This is why:

your ears feel more pressure when you dive deeper underwater
dams are built thicker at the bottom
submarines must withstand very large pressures deep in the sea

Why liquid pressure depends on density

Density tells you how much mass there is in a certain volume.

Definition

Density

Density is mass per unit volume. It is measured in kilograms per cubic metre, kg/m³.

A denser liquid has more mass in the same volume. That means a column of denser liquid has a greater weight than the same-sized column of a less dense liquid. So, at the same depth, a denser liquid causes a greater pressure.

For Higher Tier, you need to be able to calculate liquid pressure using:

P=hρgP=h\rho g

where:

$P$ is pressure due to the liquid column in pascals, Pa
$h$ is height or depth of the liquid column in metres, m
$ρ\rho$ is density in kilograms per cubic metre, kg/m³
$g$ is gravitational field strength in newtons per kilogram, N/kg

The specification says use this equation, so you should be confident applying it when it is given or available.

Example

Calculating pressure at a depth

Calculate the pressure due to water at a depth of 4.0 m. Take the density of water as 1000 kg/m³ and $N/kgg=10\text{ N/kg}$ .

Choose the liquid pressure equation because the pressure depends on depth, density and gravitational field strength:
$P=hρgP=h\rho g$
Substitute the values:
$\times 1000 \times 10$
Calculate the pressure due to the water:
$PaP=40000\text{ Pa}$
The pressure due to the water is 40,000 Pa.

Tip

Depth means vertical depth

In $P=hρgP=h\rho g$ , the height $h$ is the vertical depth below the surface, not the sloping distance along the side of a container.

Pressure differences in a liquid

Often, you need the difference in pressure between two depths. You can calculate the pressure at each depth and subtract, or just use the difference in height.

If two points in the same liquid are separated by a vertical height difference $Δh\Delta h$ , then:

ΔP=Δhρg\Delta P=\Delta h\rho g

Example

Calculating a pressure difference

Two points in oil are at depths of 1.5 m and 5.5 m. The density of the oil is 800 kg/m³ and $N/kgg=10\text{ N/kg}$ . Calculate the difference in pressure between the two points.

Find the vertical difference in depth:
$m\Delta h=5.5-1.5=4.0\text{ m}$
Use the pressure difference equation for the same liquid:
$ΔP=Δhρg\Delta P=\Delta h\rho g$
Substitute and calculate:
$Pa\Delta P=4.0 \times 800 \times 10=32000\text{ Pa}$
The deeper point has 32,000 Pa more pressure.

Upthrust

When an object is in a fluid, the fluid pushes on it from all directions. The pressure is greater lower down because pressure increases with depth.

So, for a submerged object:

the pressure on the bottom is greater than the pressure on the top
the upward force on the bottom is greater than the downward force on the top
the sideways forces usually balance
the resultant force from the fluid is upwards

This upward resultant force is called upthrust.

Definition

Upthrust

Upthrust is the upward force exerted by a fluid on an object in the fluid.

The diagram shows why upthrust happens for a fully immersed object.

Diagram showing upthrust on an immersed block caused by greater pressure on the bottom face

Displaced fluid

When an object is placed in a fluid, it pushes some of the fluid out of the way. This is called displacing fluid.

Definition

Displaced fluid

Displaced fluid is the fluid that has been moved out of the space now occupied by the object.

For Higher Tier, you need to recall this important result:

Key Idea

Archimedes' principle

The upthrust on an object is equal to the weight of fluid displaced by the object.

This applies to:

objects fully immersed in a liquid or gas
objects partly immersed in a liquid, such as a floating boat

If an object displaces more fluid, the upthrust is larger. If the fluid is denser, the same displaced volume has more weight, so the upthrust is larger.

Example

Calculating upthrust from displaced water

A fully submerged object displaces 0.003 m³ of water. The density of water is 1000 kg/m³ and $N/kgg=10\text{ N/kg}$ . Calculate the upthrust.

Calculate the mass of water displaced using density:
$kgm=\rho V=1000 \times 0.003=3.0\text{ kg}$
Calculate the weight of the displaced water:
$\times 10=30\text{ N}$
Use the rule that upthrust equals the weight of fluid displaced:
$NU=30\text{ N}$
The upthrust is 30 N.

Floating and sinking

Whether an object floats or sinks depends on the comparison between its weight and the upthrust.

If weight is greater than upthrust, the object sinks.
If upthrust is greater than weight, the object rises.
If upthrust equals weight, the object stays at the same level or floats steadily.

For a floating object, the object settles at a depth where the upthrust equals its weight. A heavier boat sits lower in the water because it must displace more water to get enough upthrust.

Example

Deciding whether an object floats or sinks

An object has a weight of 18 N. When fully submerged in water, the maximum upthrust on it is 12 N. Decide what happens to the object.

Compare the two forces: weight is 18 N downward, while upthrust is 12 N upward.
Find the resultant force direction: the downward force is larger by 6 N.
Conclude the motion: because weight is greater than the maximum upthrust, the object sinks.

Common Mistake

Thinking floating means no weight

A floating object still has weight. It floats because the upthrust balances its weight, not because gravity has disappeared.

Density and floating

Density also helps predict floating.

An object tends to float in a fluid if its average density is less than the density of the fluid. It tends to sink if its average density is greater than the density of the fluid.

This is why:

wood often floats in water
steel can sink as a solid block
a steel ship can float because its hollow shape gives it a low average density and lets it displace a large volume of water

Exam technique

In the exam

For pressure calculations, check whether you need $P=FAP=\frac{F}{A}$ or $P=hρgP=h\rho g$ , then convert area to m² and depth to metres before substituting.
For explanations of fluid pressure, link your answer to weight of fluid above, depth, density, and forces acting normal to surfaces.
For floating and sinking, compare upthrust with weight, and remember that upthrust equals the weight of fluid displaced.

Self review

Check yourself

Why does atmospheric pressure decrease as height above Earth’s surface increases?
A force is spread over a larger area. What happens to the pressure?
Why is the upward force on the bottom of a submerged block larger than the downward force on the top?

Pressure in fluids and upthrust

What you'll learn

What pressure means, and how it depends on force and area.
Why pressure in liquids and gases acts in all directions and is normal to surfaces.
Why liquid pressure increases with depth and density.
How upthrust explains floating, sinking, and objects feeling lighter in fluids.

This Edexcel section is marked with a P, so it is Separate Physics content. Some parts, especially the liquid pressure equation and upthrust explanations, are Higher Tier only in the specification.

The starting point: fluids and pressure

A fluid is a substance that can flow. Liquids and gases are both fluids.

A fluid contains particles that move around and collide with surfaces. Each collision gives a tiny push. The total effect of many tiny pushes over an area is called pressure.

Definition

Pressure

Pressure is the force acting normally on each square metre of a surface. It is measured in pascals, Pa.

One pascal means one newton per square metre.

Pressure can be caused by:

a solid pressing on a surface, such as a shoe on the ground
a liquid pressing on the sides of a container
a gas pressing on surfaces, such as air pressing on your body

Force, area and pressure

Pressure depends on two things:

the size of the force
the area over which the force is spread

If the same force acts over a smaller area, the pressure is larger. This is why a sharp knife cuts better than a blunt one: the force is concentrated over a tiny area.

Key Idea

Force spread over area

A larger force increases pressure, but a larger area decreases pressure.

The equation is:

P=FAP=\frac{F}{A}

where:

$P$ is pressure in pascals, Pa
$F$ is force normal to the surface in newtons, N
$A$ is area in square metres, m²

For Edexcel, this is a recall and use equation, so you should learn it.

Example

Calculating pressure from force and area

A student pushes a drawing pin into a board with a force of 12 N. The sharp tip has an area of 0.000004 m². Calculate the pressure on the board.

Choose the pressure equation because the question gives force and area:
$P=FAP=\frac{F}{A}$
Substitute the values, keeping the area in square metres:
$P=120.000004P=\frac{12}{0.000004}$
Calculate the pressure:
$PaP=3000000\text{ Pa}$
So the pressure is 3,000,000 Pa, or $3.0 \times 10^6\text{ Pa}$ .

Common Mistake

Forgetting to convert area

Area must be in square metres, m². If an area is given in cm², convert it before using $P=FAP=\frac{F}{A}$ . Remember: 1 cm² is 0.0001 m².

Pressure acts normal to a surface

In physics, normal means at right angles, or perpendicular, to a surface.

Definition

Normal force

A force normal to a surface acts at 90 degrees to that surface.

Pressure in a fluid causes a force normal to any surface it touches. For example:

water pushes sideways on the walls of a container
air pushes normally on your skin
water pushes upwards on the bottom of a boat

Pressure in a fluid is not just “downwards”. It acts in all directions, but the force on any particular surface is perpendicular to that surface.

The diagram shows fluid pressure increasing with depth and acting normally to the container wall.

Diagram of fluid pressure increasing with depth and acting normal to surfaces

Atmospheric pressure

The atmosphere is the layer of air around Earth. Air is a gas, so it is a fluid.

Definition

Atmospheric pressure

Atmospheric pressure is the pressure caused by the weight of the air above a surface.

A simple model is to imagine the atmosphere as many layers of air stacked above Earth’s surface.

At sea level, there is a tall column of air above you. The air below has to support the weight of all the air above it, so atmospheric pressure is relatively high.

Example

Explaining lower pressure on a mountain

Why is atmospheric pressure lower at the top of a mountain than at sea level?

Compare the amount of air above each place: at sea level there is more air above you; on the mountain there is less air above you.
Link pressure to weight: atmospheric pressure is caused by the weight of the air above the surface.
Conclude from the model: less air above means less weight pressing down, so the atmospheric pressure is lower on the mountain.

Pressure in a fluid

The total pressure at a point in a fluid can include:

pressure due to the fluid itself
atmospheric pressure acting on the surface of the fluid

For example, in an open swimming pool, atmospheric pressure acts on the water surface. The water pressure then increases as you go deeper.

Key Idea

Pressure in an open liquid

At depth in an open liquid, the pressure is due to both the water above that point and the atmosphere pushing on the surface.

Why liquid pressure increases with depth

As you go deeper in a liquid, there is more liquid above you. That liquid has weight. The deeper point has to support a taller column of liquid, so the pressure is greater.

This is why:

your ears feel more pressure when you dive deeper underwater
dams are built thicker at the bottom
submarines must withstand very large pressures deep in the sea

Why liquid pressure depends on density

Density tells you how much mass there is in a certain volume.

Definition

Density

Density is mass per unit volume. It is measured in kilograms per cubic metre, kg/m³.

For Higher Tier, you need to be able to calculate liquid pressure using:

P=hρgP=h\rho g

where:

$P$ is pressure due to the liquid column in pascals, Pa
$h$ is height or depth of the liquid column in metres, m
$ρ\rho$ is density in kilograms per cubic metre, kg/m³
$g$ is gravitational field strength in newtons per kilogram, N/kg

The specification says use this equation, so you should be confident applying it when it is given or available.

Example

Calculating pressure at a depth

Calculate the pressure due to water at a depth of 4.0 m. Take the density of water as 1000 kg/m³ and $N/kgg=10\text{ N/kg}$ .

Choose the liquid pressure equation because the pressure depends on depth, density and gravitational field strength:
$P=hρgP=h\rho g$
Substitute the values:
$\times 1000 \times 10$
Calculate the pressure due to the water:
$PaP=40000\text{ Pa}$
The pressure due to the water is 40,000 Pa.

Tip

Depth means vertical depth

In $P=hρgP=h\rho g$ , the height $h$ is the vertical depth below the surface, not the sloping distance along the side of a container.

Pressure differences in a liquid

Often, you need the difference in pressure between two depths. You can calculate the pressure at each depth and subtract, or just use the difference in height.

If two points in the same liquid are separated by a vertical height difference $Δh\Delta h$ , then:

ΔP=Δhρg\Delta P=\Delta h\rho g

Example

Calculating a pressure difference

Two points in oil are at depths of 1.5 m and 5.5 m. The density of the oil is 800 kg/m³ and $N/kgg=10\text{ N/kg}$ . Calculate the difference in pressure between the two points.

Find the vertical difference in depth:
$m\Delta h=5.5-1.5=4.0\text{ m}$
Use the pressure difference equation for the same liquid:
$ΔP=Δhρg\Delta P=\Delta h\rho g$
Substitute and calculate:
$Pa\Delta P=4.0 \times 800 \times 10=32000\text{ Pa}$
The deeper point has 32,000 Pa more pressure.

Upthrust

When an object is in a fluid, the fluid pushes on it from all directions. The pressure is greater lower down because pressure increases with depth.

So, for a submerged object:

the pressure on the bottom is greater than the pressure on the top
the upward force on the bottom is greater than the downward force on the top
the sideways forces usually balance
the resultant force from the fluid is upwards

This upward resultant force is called upthrust.

Definition

Upthrust

Upthrust is the upward force exerted by a fluid on an object in the fluid.

The diagram shows why upthrust happens for a fully immersed object.

Diagram showing upthrust on an immersed block caused by greater pressure on the bottom face

Displaced fluid

When an object is placed in a fluid, it pushes some of the fluid out of the way. This is called displacing fluid.

Definition

Displaced fluid

Displaced fluid is the fluid that has been moved out of the space now occupied by the object.

For Higher Tier, you need to recall this important result:

Key Idea

Archimedes' principle

The upthrust on an object is equal to the weight of fluid displaced by the object.

This applies to:

objects fully immersed in a liquid or gas
objects partly immersed in a liquid, such as a floating boat

If an object displaces more fluid, the upthrust is larger. If the fluid is denser, the same displaced volume has more weight, so the upthrust is larger.

Example

Calculating upthrust from displaced water

A fully submerged object displaces 0.003 m³ of water. The density of water is 1000 kg/m³ and $N/kgg=10\text{ N/kg}$ . Calculate the upthrust.

Calculate the mass of water displaced using density:
$kgm=\rho V=1000 \times 0.003=3.0\text{ kg}$
Calculate the weight of the displaced water:
$\times 10=30\text{ N}$
Use the rule that upthrust equals the weight of fluid displaced:
$NU=30\text{ N}$
The upthrust is 30 N.

Floating and sinking

Whether an object floats or sinks depends on the comparison between its weight and the upthrust.

If weight is greater than upthrust, the object sinks.
If upthrust is greater than weight, the object rises.
If upthrust equals weight, the object stays at the same level or floats steadily.

For a floating object, the object settles at a depth where the upthrust equals its weight. A heavier boat sits lower in the water because it must displace more water to get enough upthrust.

Example

Deciding whether an object floats or sinks

An object has a weight of 18 N. When fully submerged in water, the maximum upthrust on it is 12 N. Decide what happens to the object.

Compare the two forces: weight is 18 N downward, while upthrust is 12 N upward.
Find the resultant force direction: the downward force is larger by 6 N.
Conclude the motion: because weight is greater than the maximum upthrust, the object sinks.

Common Mistake

Thinking floating means no weight

A floating object still has weight. It floats because the upthrust balances its weight, not because gravity has disappeared.

Density and floating

Density also helps predict floating.

An object tends to float in a fluid if its average density is less than the density of the fluid. It tends to sink if its average density is greater than the density of the fluid.

This is why:

wood often floats in water
steel can sink as a solid block
a steel ship can float because its hollow shape gives it a low average density and lets it displace a large volume of water

Exam technique

In the exam

For pressure calculations, check whether you need $P=FAP=\frac{F}{A}$ or $P=hρgP=h\rho g$ , then convert area to m² and depth to metres before substituting.
For explanations of fluid pressure, link your answer to weight of fluid above, depth, density, and forces acting normal to surfaces.
For floating and sinking, compare upthrust with weight, and remember that upthrust equals the weight of fluid displaced.

Self review

Check yourself

Why does atmospheric pressure decrease as height above Earth’s surface increases?
A force is spread over a larger area. What happens to the pressure?
Why is the upward force on the bottom of a submerged block larger than the downward force on the top?

Pressure in fluids and upthrust

What you'll learn

The starting point: fluids and pressure

Force, area and pressure

Pressure acts normal to a surface

Atmospheric pressure

Pressure in a fluid

Why liquid pressure increases with depth

Why liquid pressure depends on density

Pressure differences in a liquid

Upthrust

Displaced fluid

Floating and sinking

Density and floating

Recap questions

Pressure in fluids and upthrust

What you'll learn

The starting point: fluids and pressure

Force, area and pressure

Pressure acts normal to a surface

Atmospheric pressure

Pressure in a fluid

Why liquid pressure increases with depth

Why liquid pressure depends on density

Pressure differences in a liquid

Upthrust

Displaced fluid

Floating and sinking

Density and floating

Recap questions

Pressure in fluids and upthrust

What you'll learn

The starting point: fluids and pressure

Force, area and pressure

Pressure acts normal to a surface

Atmospheric pressure

Pressure in a fluid

Why liquid pressure increases with depth

Why liquid pressure depends on density

Pressure differences in a liquid

Upthrust

Displaced fluid

Floating and sinking

Density and floating

Recap questions

Pressure in fluids and upthrust Revision Guide

Pressure in fluids and upthrust

What you'll learn

The starting point: fluids and pressure

Force, area and pressure

Pressure acts normal to a surface

Atmospheric pressure

Pressure in a fluid

Why liquid pressure increases with depth

Why liquid pressure depends on density

Pressure differences in a liquid

Upthrust

Displaced fluid

Floating and sinking

Density and floating

Recap questions

Pressure in fluids and upthrust Revision Guide