Pressure in a liquid increases with depth because the weight of liquid above a point grows as depth increases. The pressure at any point depends only on the depth, the liquid's density, and gravitational field strength — not on the shape of the container. This has practical consequences for dam design, deep-sea exploration, and hydraulic systems.
Why does pressure increase with depth in a liquid?
Consider a horizontal surface at depth h below the surface of a liquid. The liquid directly above this surface has a column of height h pressing down on it. The pressure at that depth equals the weight of that liquid column per unit area.
Derivation of the formula:
- Volume of liquid column above unit area = h × 1 = h m³
- Mass of column = density × volume = ρ × h
- Weight of column = mass × g = ρ × h × g
- Pressure = force per unit area = ρhg / 1 = ρhg
P = ρgh
Where:
- P = pressure (Pa, pascals)
- ρ (rho) = density of the liquid (kg/m³)
- g = gravitational field strength (10 N/kg at Earth's surface for GCSE)
- h = depth below the surface (m)
This formula gives the gauge pressure — the pressure due to the liquid above, not including atmospheric pressure. The total pressure at depth h is P = P_atm + ρgh, where P_atm ≈ 101,000 Pa.
Worked examples
Example 1: Calculate the pressure due to water at a depth of 4.0 m. (Density of water = 1000 kg/m³, g = 10 N/kg.)
P = ρgh = 1000 × 10 × 4.0 = 40,000 Pa = 40 kPa
Example 2: A submarine is at a depth of 200 m in seawater (density = 1025 kg/m³). Calculate the total pressure on the submarine's hull. (g = 10 N/kg, P_atm = 101,000 Pa.)
Pressure due to seawater: P = ρgh = 1025 × 10 × 200 = 2,050,000 Pa Total pressure = 2,050,000 + 101,000 = 2,151,000 Pa ≈ 2.15 MPa
This is about 21 times atmospheric pressure — which explains why submarine hulls must be extraordinarily strong.
How does liquid pressure act in all directions?
An important property of liquid (and gas) pressure: it acts equally in all directions at a given depth. If you are at depth h in water, the pressure is ρgh regardless of whether you measure it acting upward, downward, or sideways on any surface.
This is why:
- A sealed bag full of water at depth h has the same pressure pressing on all faces.
- Holes in the side of a container at the same depth let water out with the same speed, regardless of where on the perimeter the hole is.
- Diver suits experience the same crushing pressure on all surfaces.
The fact that pressure acts in all directions is what makes hydraulic systems possible.
What are hydraulic systems and how do they work?
A hydraulic system transmits pressure through an incompressible fluid (usually oil) to produce a mechanical advantage — a small force can produce a much larger force at another point in the system.
Principle: When pressure is applied to an enclosed liquid, it is transmitted equally to all parts of the liquid (Pascal's law). If you push with a small force on a small piston (small area A₁), the pressure P = F₁/A₁ is transmitted through the fluid. A larger piston (area A₂) experiences the same pressure P, but over a larger area, so it exerts a much larger force: F₂ = P × A₂.
Key equation:
P₁ = P₂ → F₁/A₁ = F₂/A₂
Worked example: A hydraulic jack has an input piston of area 0.002 m² and an output piston of area 0.10 m². A force of 50 N is applied to the input. What force is produced at the output?
P = F₁/A₁ = 50 / 0.002 = 25,000 Pa F₂ = P × A₂ = 25,000 × 0.10 = 2500 N
The hydraulic jack multiplies the force by a factor of 50 (= A₂/A₁ = 0.10/0.002).
How does pressure vary with the density of the liquid?
The formula P = ρgh shows that denser liquids exert greater pressure at the same depth. This has practical implications:
| Liquid | Density (kg/m³) | Pressure at 10 m depth (Pa) |
|---|---|---|
| Fresh water | 1000 | 100,000 |
| Seawater | 1025 | 102,500 |
| Mercury | 13,600 | 1,360,000 |
Mercury's very high density is why it was used in traditional barometers — atmospheric pressure (101,000 Pa) supports a column of mercury only 76 cm tall, whereas a water barometer would require a column over 10 m high.
Frequently asked questions
What is the formula for pressure in a liquid at GCSE?
The formula is P = ρgh, where P is the pressure in pascals (Pa), ρ is the density of the liquid in kg/m³, g is the gravitational field strength (10 N/kg for GCSE), and h is the depth below the surface in metres. This gives the pressure due to the liquid column above the point, not including atmospheric pressure. For total pressure at depth h, add atmospheric pressure: P_total = P_atm + ρgh.
Why does pressure in a liquid not depend on the shape of the container?
Pressure at a given depth depends only on the weight of liquid directly above — which is determined by depth and density, not by the container's shape. In a wide container and a narrow tube connected at the base, the liquid settles at the same level and the pressure at the base is the same in both. This is the principle behind connected vessels: liquid always finds its own level, meaning the pressure at equal depths must be equal.
How does a hydraulic system create a mechanical advantage?
A hydraulic system uses an incompressible fluid to transmit pressure. A small force applied to a small-area piston creates a pressure that is transmitted equally throughout the fluid. This same pressure acts on a larger-area piston, producing a proportionally larger force (F = P × A). The mechanical advantage (force multiplication) equals the ratio of the output piston area to the input piston area. Examples include car brake systems, hydraulic jacks, and the lifts in vehicle service centres.
Why must submarine hulls withstand enormous pressure?
Pressure in seawater increases by approximately 10,000 Pa (0.1 atm) for every metre of depth, using P = ρgh ≈ 1025 × 10 × h. At 200 m depth this is about 2 MPa on top of atmospheric pressure — more than 20 times the pressure at sea level. This force acts inward on all surfaces of the hull. To prevent crushing, submarine hulls are made from high-strength steel or titanium alloys, shaped as cylinders or spheres (which distribute pressure evenly), and tested to withstand pressures well beyond their maximum operating depth.
For Socratic GCSE physics with Professor Newton — deriving ρgh from a column of liquid before applying it to hydraulics, so the formula is never just a memory — visit aitutors.me.