Pressure and Forces in Fluids

Learn how pressure works in liquids and gases — from why dams are thicker at the bottom to how a hydraulic press multiplies force.

What is Pressure?

Pressure is the force applied perpendicular to a surface, spread over an area.

Pressure

p = \frac{F}{A}

F = force (N), A = area (m²), p = pressure (Pa = N/m²).

A sharp knife cuts because the area is tiny — the same force creates enormous pressure.

Hydrostatic Pressure

Pressure in a liquid increases with depth.

Hydrostatic Pressure

p = \rho g h

ρ = density (kg/m³), g ≈ 9.8 m/s², h = depth (m).

Note At the same depth, pressure is equal in all directions (Pascal's principle).

Atmospheric Pressure

The weight of Earth's air creates atmospheric pressure at the surface:

$p_0 \approx 101\,325 \text{ Pa} \approx 101.3 \text{ kPa}$

Pressure decreases as you go higher — that is why airplane cabins are pressurised.

Pascal's Law

Any pressure applied to a confined fluid is transmitted equally throughout:

Hydraulic press

\frac{F_1}{A_1} = \frac{F_2}{A_2}

A small force on a small piston creates a large force on a large piston.

1
Identify force and area
Write down F (in N) and A (in m²).
2
Apply p = F / A
Calculate pressure in pascals (Pa).
3
Check units
1 Pa = 1 N/m². 1 kPa = 1000 Pa.

A box weighs 500 N and has a base of 0.25 m². Find the pressure it exerts.

$p = \frac{500}{0.25} = 2000 \text{ Pa}$

Real world: hydraulic car brakes

In a car braking system, the driver applies a small force on a narrow master cylinder piston (area $A_1 = 0.001\text{ m}^2$ ). The pressure transmits through brake fluid to wider caliper pistons ( $A_2 = 0.01\text{ m}^2$ ). Output force: $F_2 = F_1 \times \frac{A_2}{A_1} = 150 \times 10 = 1500\text{ N}$ . Pascal's law multiplies the foot force ten-fold to squeeze the brake disc.

Real world: dam wall thickness

A dam holding back a reservoir has thicker walls at the base because hydrostatic pressure $p = \rho g h$ increases with depth $h$ . At 20 m depth: $p = 1000 \times 9.8 \times 20 = 196\,000\text{ Pa}$ . Engineers calculate this pressure distribution to design walls thick enough to resist the force at every depth.

Real world: sharp knife versus blunt knife

A sharp knife has a blade edge area of approximately $0.00001\text{ m}^2$ . Applying 5 N of force creates pressure $p = \frac{5}{0.00001} = 500\,000\text{ Pa}$ , easily cutting through food. A blunt edge has a much larger area, so the same force creates far less pressure — and cuts poorly. This is why regular sharpening matters.

Tip Pressure is force per unit area:

p = \frac{F}{A}

. Increasing the area (e.g. snowshoes on snow) reduces pressure; decreasing the area (e.g. a needle) increases pressure for the same applied force.

Practice Problems

A force of 300 N acts on an area of 0.05 m². What is the pressure?

Show answer

p = 300 / 0.05 = 6 000 Pa.

Find the hydrostatic pressure at 4 m depth in water (ρ = 1000 kg/m²).

Show answer

p = 1000 × 9.8 × 4 = 39 200 Pa ≈ 39.2 kPa.

A hydraulic press has pistons of area 0.01 m² and 0.5 m². A 100 N force is applied to the small piston. What force acts on the large piston?

Show answer

F₂ = 100 × (0.5 / 0.01) = 5 000 N.