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Pressure and Buoyancy Explained

Why does a steel ship float while a steel coin sinks? Why does your ears pop in a plane or when you dive deep in a swimming pool? Why can a hydraulic car jack lift a tonne of metal with a small push? All of these questions are answered by the physics of pressure and buoyancy — two ideas that emerge from the behaviour of fluids under forces.

What Is Pressure?

Pressure is defined as the force acting perpendicularly on a unit area of a surface:

P = F / A

where P is pressure in pascals (Pa), F is the force in newtons (N), and A is the area in square metres (m²). One pascal equals one newton per square metre. This definition immediately explains why sharp knives cut better than blunt ones: for the same applied force, a smaller contact area means a higher pressure at the cutting edge, which more easily separates the material.

Other common units include: kilopascals (kPa = 1000 Pa), atmospheres (1 atm = 101,325 Pa), and bar (1 bar = 100,000 Pa). Standard atmospheric pressure at sea level is approximately 101 kPa or 1 atmosphere — the weight of the column of air above each square metre of Earth’s surface.

Pressure in Fluids

Fluids (liquids and gases) transmit pressure in all directions. The pressure at any point in a static fluid depends on the weight of the fluid above that point. The formula for pressure due to a column of fluid is:

P = ρ g h

where ρ (rho) is the fluid density (kg/m³), g is gravitational field strength (9.8 m/s²), and h is the depth below the surface (m). Three consequences follow:

• Pressure increases with depth. The pressure 10 m below the surface of water is about 98,000 Pa more than at the surface — nearly one additional atmosphere for every 10 m of depth.
• Pressure acts equally in all directions at a given depth — a fluid does not push down more than sideways.
• Pressure depends on the density of the fluid, not on the shape of the container. Water in a narrow tube and water in a wide lake have the same pressure at the same depth (if the tube is tall enough).

Pascal’s Law and Hydraulics

Pascal’s Law states that pressure applied to an enclosed fluid is transmitted unchanged to every part of that fluid and to the walls of its container. This principle underlies the operation of hydraulic systems. In a hydraulic press, a small force F₁ is applied to a small piston of area A₁, creating a pressure P = F₁/A₁. That pressure is transmitted through the hydraulic fluid to a larger piston of area A₂, producing a larger output force F₂ = P × A₂ = F₁(A₂/A₁).

A hydraulic car jack with a piston area ratio of 50 multiplies the input force by 50 — so a person pushing with 200 N can lift a car exerting 10,000 N. This force multiplication does not violate conservation of energy: the smaller piston moves a much greater distance than the larger one, so work done (force × distance) is the same on both sides (ignoring friction).

Archimedes’ Principle and Buoyancy

Archimedes’ Principle states that any object fully or partially submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces. In other words:

F_buoyancy = ρ_fluid × V_displaced × g

This buoyant force arises because pressure increases with depth: the upward pressure on the bottom of a submerged object is always greater than the downward pressure on its top, producing a net upward force. Archimedes reputedly discovered this principle while stepping into an overflowing bath and noticing the water displaced — a moment of insight reportedly leading him to run through the streets shouting “Eureka!” (I have found it).

Why Objects Float or Sink

Whether an object floats or sinks depends on the comparison between its weight and the maximum buoyant force the fluid can exert:

• If the object’s average density is less than the fluid’s density, the buoyant force when fully submerged exceeds the object’s weight. The object rises until only the fraction of its volume needed to displace its own weight is submerged — it floats. A wooden log floats because wood is less dense than water.
• If the object’s average density is greater than the fluid’s density, the maximum buoyant force (when fully submerged) is less than the weight. The object sinks. A solid steel coin sinks because steel is about 8 times denser than water.
• A steel ship floats because its hull encloses a large volume of air, making its average density far less than water. If water floods the hull, the average density increases above that of water and the ship sinks.

Frequently Asked Questions

Why is it easier to swim in the sea than in a freshwater pool?

Seawater is denser than freshwater (about 1025 kg/m³ versus 1000 kg/m³) because dissolved salts increase its density. According to Archimedes’ Principle, a submerged body displaces a greater weight of the denser seawater, so the buoyant force is larger, making it easier to stay afloat.

Why do your ears hurt when you dive deep in a pool?

Water pressure increases with depth (ρgh). At a depth of 3 m, the water exerts about 30,000 Pa extra pressure on your eardrums. The trapped air in the middle ear is at a lower pressure, causing the eardrum to be pushed inward painfully. Equalising (swallowing or pinching the nose and blowing gently) opens the Eustachian tube and equalises the pressure.

Does the shape of a container affect fluid pressure?

No. Pressure at a given depth depends only on ρ, g, and h — not on the shape of the container. This is demonstrated by connected U-tubes: liquid finds the same level in all connected arms regardless of their shape or width.

Summary

Pressure equals force per unit area (P = F/A) and is measured in pascals. In fluids, pressure increases with depth according to P = ρgh. Pascal’s Law states that pressure is transmitted equally through an enclosed fluid, allowing small forces to be amplified in hydraulic systems. Archimedes’ Principle states that the buoyant force equals the weight of displaced fluid: objects float if their average density is less than the fluid’s, and sink if it is greater. These ideas link directly to forces and momentum and underpin engineering from submarines to hydraulic brakes.