Hydrostatics

Hydrostatics deals with fluids (liquids and gases) at rest and the pressures they exert.

Pressure

Pressure is the force acting per unit area perpendicular to a surface:

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

where $P$ is pressure in pascals (Pa), $F$ is force in newtons (N), and $A$ is the area in m². One pascal equals one newton per square metre (1 Pa = 1 N m⁻²).

The same force produces more pressure when applied to a smaller area. A drawing pin exerts far greater pressure than a thumb pushing with the same force, because the point has a much smaller area.

Fluid Pressure

A fluid at rest exerts pressure in all directions. The pressure at a depth $h$ below the surface of a fluid depends on the density of the fluid and the depth:

$\Delta P = \rho g \Delta h$

where $\rho$ is the fluid density (kg m⁻³), $g = 10$ N kg⁻¹, and $\Delta h$ is the depth (m).

Key consequences:

Pressure increases with depth. Deep-sea submarines need thick hulls to withstand it.
Pressure does not depend on the shape of the container, only on depth and fluid density.
At the same depth, a denser fluid exerts more pressure than a less dense one.

ExampleFluid pressure, diver (2022 Paper 02, Q4)

Calculate the pressure that a diver experiences at 0.5 km below the surface of the sea. [Density of seawater = 1025 kg m⁻³, g = 10 N kg⁻¹]

$\Delta P = \rho g \Delta h = 1025 \times 10 \times 500 = 5\,125\,000 \text{Pa} \approx 5.1 \times 10^6 \text{Pa}$

This is about 50 times atmospheric pressure.

Archimedes' Principle

When an object is fully or partially submerged in a fluid, it experiences an upthrust (buoyant force) equal to the weight of fluid it displaces:

$U = \rho_{\text{fluid}} \times g \times V_{\text{displaced}}$

where $V_{\text{displaced}}$ is the volume of fluid pushed aside by the object.

The upthrust acts upward and opposes the object's weight.

Archimedes' principle: upthrust equals weight of fluid displaced

Floating and Sinking

If weight > upthrust: the object sinks.
If weight = upthrust: the object floats in equilibrium (it hovers at some depth).
If weight < upthrust: the object accelerates upward (floats to the surface).

A floating object displaces fluid whose weight exactly equals the object's own weight. This means a floating object's density is less than or equal to the fluid's density.

Density of object < density of fluid: floats. Density of object > density of fluid: sinks.

A steel ship floats because its hollow structure means the average density of the ship-plus-air system is less than water.

ExampleArchimedes, floating boat (2022 Paper 02, Q4)

A boat weighs 83,000 N and floats in seawater (density 1025 kg m⁻³, g = 10 N kg⁻¹). Calculate the volume of seawater displaced.

When floating, upthrust = weight:

$U = 83\,000 \text{N}$

$U = \rho g V \implies V = \frac{U}{\rho g} = \frac{83\,000}{1025 \times 10} = \frac{83\,000}{10\,250} \approx 8.1 \text{m}^3$

Exam Tip

In pressure problems, convert depth to metres before substituting. A depth of 0.5 km is 500 m, not 0.5.

In Archimedes problems, the key identity is: for a floating object, upthrust = weight. State this explicitly before substituting, it earns a method mark.