Density, pressure in liquids and atmospheric pressure

Why does a sharp knife cut better than a blunt one? Why are the walls of a dam thicker at the bottom? Why do your ears pop when a plane takes off? Pressure answers all three — and pressure in liquids and air is one of the most examined topics in WASSCE Physics.

Learning objectives

By the end of this lesson you should be able to:

• define density and relative density and calculate them
• define pressure and calculate pressure in solids and liquids
• use $p = h\rho g$ and explain the properties of liquid pressure
• explain atmospheric pressure, the mercury barometer and the manometer
• state Pascal's principle and apply it to hydraulic machines

Density and relative density

Water has density 1000 kg m⁻³ (RD = 1). Mercury has RD 13.6, so its density is 13 600 kg m⁻³.

Pressure

The same force produces a larger pressure on a smaller area:

• a sharp knife or a needle has a tiny area, so it cuts or pierces easily
• tractors and lorries have wide tyres; camels have broad feet: large areas give low pressure, so they do not sink into soft ground
• a woman in stiletto heels exerts more pressure on a floor than an elephant standing on its broad feet

Pressure in liquids

Figure 1. Pressure in a liquid increases with depth. WASSCE Tutor diagram.

The pressure at depth $h$ in a liquid of density $\rho$ is

$$p = h\rho g$$

(plus atmospheric pressure acting on the surface, if the total pressure is wanted).

Properties of pressure in a liquid

1. Pressure increases with depth.
2. At a given depth, pressure acts equally in all directions.
3. Pressure depends on depth and density only — not on the shape or cross-section of the container.
4. A liquid finds its own level: the surface is at the same height in connected vessels.

Atmospheric pressure

The air above us has weight, so it exerts a pressure: atmospheric pressure, about $1.01 \times 10^{5}$ Pa at sea level (often called 1 atmosphere, or 760 mmHg). It decreases with altitude.

Evidence: the collapsing-can experiment (a can with a little water is boiled, sealed and cooled — it is crushed by atmospheric pressure as the steam condenses); drinking through a straw; a rubber sucker sticking to a wall.

The mercury barometer

Figure 2. A simple mercury barometer. WASSCE Tutor diagram.

Atmospheric pressure on the mercury in the dish supports the column. At sea level the column is about 760 mm high:

Tilting the tube does not change the vertical height of the column. Water is not used because the column would need to be about 10.3 m tall.

The manometer

A U-tube manometer measures the pressure of a gas supply. One arm is connected to the gas; the other is open to the air. The difference in liquid levels $h$ gives the excess pressure of the gas above atmospheric: $\Delta p = h\rho g$.

Pascal's principle and hydraulic machines

In a hydraulic press, a small force on a small piston produces a large force on a large piston: $\dfrac{F_1}{A_1} = \dfrac{F_2}{A_2}$. Applications: hydraulic jacks, hydraulic car brakes, barber's chairs, hydraulic lifts in garages.

Key points

• Density $\rho = m/V$; relative density compares with water (no unit).
• Pressure $p = F/A$ (Pa); smaller area → greater pressure.
• Liquid pressure $p = h\rho g$: increases with depth, same in all directions, independent of the container's shape.
• Atmospheric pressure ≈ $1.01 \times 10^{5}$ Pa ≈ 760 mmHg; measured with a barometer.
• Pascal's principle underlies hydraulic machines.

WASSCE examination focus

Quick check

1. Why do tractors have wide tyres?
2. Find the pressure at a depth of 5 m in water ($\rho = 1000$ kg m⁻³, $g = 10$ m s⁻²).
3. State one reason why mercury is used in barometers.

Answers: 1. To spread the weight over a large area, reducing the pressure so they do not sink into soft soil. 2. $5 \times 1000 \times 10 = 5.0 \times 10^{4}$ Pa. 3. For example: its high density gives a short, convenient column (about 0.76 m instead of about 10 m for water).