Pressure in Solids, Liquids and Gases, Archimedes' Principle and Flotation

Why this topic matters

WAEC and NECO set pressure questions every year: liquid pressure at a depth, the hydraulic press, Archimedes' principle and relative density. These questions use a few formulas over and over, so they are easy marks with practice.

Pressure

Pressure = force ÷ area, so P = F ÷ A. The unit is the pascal (Pa) = N m⁻².

• The same force over a smaller area gives a larger pressure.
• This is why knives are sharpened, nails are pointed, and lorries and tractors have wide tyres. A woman in stiletto heels presses harder on the floor than an elephant does with its wide feet.

Pressure in liquids

P = hρg, where h is the depth, ρ the density of the liquid and g the acceleration due to gravity.

Pressure in a liquid:

• increases with depth;
• increases with the density of the liquid;
• acts equally in all directions at a given depth;
• does not depend on the shape or cross-section of the container.

This is why dam walls are thicker at the bottom, and water spurts further from a hole lower down a tank.

Atmospheric pressure

The air above us exerts pressure: about 1.0 × 10⁵ Pa at sea level.

• A mercury barometer holds up a column of about 76 cm (760 mm) of mercury: P = 0.76 × 13 600 × 9.8 ≈ 1.01 × 10⁵ Pa.
• An aneroid barometer has no liquid. It is used in altimeters, because atmospheric pressure falls with height.
• Atmospheric pressure is used by drinking straws, syringes, rubber suckers and lift pumps.

Pascal's principle and hydraulic machines

Pascal's principle: pressure applied to an enclosed fluid is transmitted equally throughout the fluid.

In a hydraulic press: F₁ ÷ A₁ = F₂ ÷ A₂. A small force on a small piston gives a large force on a large piston. This is used in car brakes, hydraulic jacks and barber's chairs.

U-tube manometer: measures gas pressure. Gas pressure = atmospheric pressure + hρg, where h is the difference in liquid levels.

Density and relative density

• Density = mass ÷ volume, in kg m⁻³. For water, ρ = 1000 kg m⁻³ (1 g cm⁻³).
• Relative density = density of the substance ÷ density of water. It has no unit, and is measured with a hydrometer or a density bottle.

Archimedes' principle and upthrust

Archimedes' principle: when a body is wholly or partly immersed in a fluid, it experiences an upthrust equal to the weight of fluid displaced.

• Upthrust = weight in air − weight in the liquid (the apparent loss in weight).
• Relative density = weight in air ÷ upthrust in water (the loss in weight in water).

Flotation

Law of flotation: a floating body displaces its own weight of the fluid it floats in.

• A body floats if its average density is less than that of the fluid.
• A steel ship floats because its hollow shape displaces a large weight of water.
• A submarine sinks or rises by letting water into its ballast tanks, or pumping it out.
• A hydrometer floats higher in denser liquids, so its stem can be marked with relative densities.
• Hot-air balloons rise because hot air is less dense than the cold air around them.

Top exam traps

• In P = hρg, h is the depth below the surface, not the height from the bottom.
• Use SI units: convert cm to m and g cm⁻³ to kg m⁻³ (1 g cm⁻³ = 1000 kg m⁻³).
• Upthrust is the loss in weight, not the weight in water.
• Relative density has no unit.