Physics · Lesson 10

Pressure, liquids at rest, Archimedes & flotation

JAMBWAECNECO

What the syllabus asks

JAMB: pressure, liquids at rest, equilibrium of forces in fluids, Archimedes' principle, flotation, density and relative density

WAEC: pressure in solids, liquids and gases; P = hρg; transmission of pressure (Pascal's principle) and the hydraulic press, car brakes; atmospheric pressure, barometers and manometers; Archimedes' principle, upthrust, relative density; law of flotation, hydrometer, submarines, balloons

NECO: pressure in solids, liquids and gases; P = hρg; transmission of pressure (Pascal's principle) and the hydraulic press, car brakes; atmospheric pressure, barometers and manometers; Archimedes' principle, upthrust, relative density; law of flotation, hydrometer, submarines, balloons

What you will learn

Define pressure and calculate pressures exerted by solids.
Derive and use P = hρg for liquids and explain its consequences.
State Pascal's principle and solve hydraulic press and brake problems.
Explain atmospheric pressure and describe the mercury barometer, aneroid barometer and manometer.
State Archimedes' principle and use it to find upthrust, volume, density and relative density.
State the law of flotation and apply it to ships, hydrometers, submarines and balloons.
Use Hare's apparatus and the U-tube to compare densities of liquids.

Key points

P = F/A in pascals (N/m²); small area, big pressure.
Liquid pressure P = hρg; increases with depth and density; same in all directions; independent of vessel shape.
Total pressure at depth = atmospheric + hρg.
Pascal's principle: F₁/A₁ = F₂/A₂ (hydraulic press, jack, brakes); work in = work out.
Atmospheric pressure ≈ 1.0 × 10⁵ Pa ≈ 760 mmHg ≈ 10 m of water; falls with altitude.
Manometer: gas pressure = atmospheric + hρg; balancing liquids: ρ₁h₁ = ρ₂h₂; Hare's RD = h(water)/h(liquid).
Archimedes: upthrust = weight of fluid displaced = loss in weight.
RD of solid = weight in air ÷ loss in water; RD of liquid = loss in liquid ÷ loss in water.
Flotation: floating body displaces its own weight; fraction immersed = ρ(body)/ρ(fluid).
Applications: ships, Plimsoll line, hydrometer, submarine, balloons, siphon, water supply.

Start here: What pressure is

Why does a sharp knife cut tomatoes easily, but a blunt one squashes them? Why do lorries carrying heavy loads have many wide tyres? Why does a woman in stiletto heels sink into a sandy compound while a man much heavier in flat sandals does not? The answer to all three is pressure.

Pressure = force acting normally (at right angles) ÷ area, P = F/A. Unit: pascal (Pa), where 1 Pa = 1 N/m². Pressure is a scalar.

For a given force, a smaller area gives a bigger pressure. The knife edge is narrow; the stiletto heel is tiny. For a given force, a larger area gives a smaller pressure, which is why camels have broad feet for sand, tractors have wide tyres, and a nail has a flat head where you hit it.

Worked example 10.1

A 600 N woman stands on one stiletto heel of area 1 cm². Then she changes into flat shoes with a total area of 150 cm². Find the pressure in each case.

Convert areas to m², because the pascal is N/m²: 1 cm² = 1 × 10⁻⁴ m²; 150 cm² = 0.015 m².
Heel: P = 600/(1 × 10⁻⁴) = 6 × 10⁶ Pa.
Flat shoes: P = 600/0.015 = 4 × 10⁴ Pa.
The heel gives 150 times more pressure, enough to dent a wooden floor.

6 × 10⁶ Pa and 4 × 10⁴ Pa

Worked example 10.2 (maximum and minimum pressure)

A block weighing 400 N measures 0.5 m × 0.4 m × 0.1 m. Find the greatest and least pressures it can exert on the floor.

Least pressure on the largest face: 0.5 × 0.4 = 0.2 m² → 400/0.2 = 2000 Pa.
Greatest on the smallest face: 0.4 × 0.1 = 0.04 m² → 400/0.04 = 10,000 Pa.

10,000 Pa and 2000 Pa

⚠️ Common mistake: using areas in cm² with forces in N and calling the answer Pa. Always convert to m².

Try it yourself. A 50 kg box rests on a base of 0.25 m². Find the pressure (g = 10).

Show solution

F = 500 N; P = 500/0.25 = 2000 Pa.

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Practice questions

1. A force of 200 N acts on an area of 0.05 m². Find the pressure.

Show worked answer

4000 Pa.

2. Find the pressure due to water at a depth of 8 m.

Show worked answer

8 × 1000 × 10 = 8.0 × 10⁴ Pa.

3. Hydraulic press pistons have areas 0.002 m² and 0.4 m². What force on the small piston lifts 8000 N?

Show worked answer

8000 × 0.002/0.4 = 40 N.

4. A stone weighs 6 N in air and 4 N in water. Find its RD and volume.

Show worked answer

RD = 3; V = 2/(1000 × 10) = 2 × 10⁻⁴ m³.

5. A block of density 900 kg/m³ floats in water. What percentage is above the surface?

Show worked answer

10%.

Key words

Pressure: Force acting normally per unit area; unit pascal (Pa) = N/m².
Pascal (Pa): A pressure of one newton per square metre.
Density ρ: Mass per unit volume; unit kg/m³.
Relative density: Density of a substance ÷ density of water; no unit.
Atmospheric pressure: The pressure exerted by the weight of the air above us.
Barometer: An instrument for measuring atmospheric pressure.
Manometer: A U-tube instrument for measuring gas pressure relative to the atmosphere.
Pascal's principle: Pressure applied to an enclosed fluid is transmitted equally to every part of the fluid.
Hydraulic press: A machine that uses Pascal's principle to multiply force.
Upthrust: The upward force a fluid exerts on a body immersed in it.
Archimedes' principle: A body wholly or partly immersed in a fluid experiences an upthrust equal to the weight of fluid displaced.
Law of flotation: A floating body displaces a weight of fluid equal to its own weight.

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