Describe how pressure varies with force and area in the context of everyday examples

1.8 Pressure

Learning objective

Describe how pressure varies with force, area and depth, and how it is related to the density of a gas, using everyday examples, simple calculations and experimental techniques required by the Cambridge IGCSE 0625 syllabus.

Definition & symbol

Pressure (\(\mathbf{p}\)) is the normal force applied per unit area of a surface.

where \(p\) = pressure (Pa), \(F\) = normal force (N) and \(A\) = contact area (m²).

Units and symbols

• SI unit: pascal (Pa) = N · m⁻²
• Common multiples: 1 kPa = 10³ Pa, 1 MPa = 10⁶ Pa
• Other useful units: bar (1 bar = 100 kPa), atmosphere (1 atm ≈ 101.3 kPa), mm Hg (used for blood‑pressure readings)

1. Relationship between force, area and pressure

• For a given area, pressure is directly proportional to force: double the force → double the pressure.
• For a given force, pressure is inversely proportional to area: double the area → halve the pressure.
• Consequences:
  • To obtain a required pressure a larger force can be applied over a larger area.
  • To obtain the same force with less effort the contact area must be reduced (e.g. a knife edge).

Worked numeric example (forward calculation)

A 200 N weight is placed on a flat board of area 0.020 m². The pressure is

Reverse‑calculation examples (re‑arranging the formula)

• Required force for a given pressure and area:
  \[ F = pA \] If a gardener wants a pressure of 5 kPa under a foot‑plate of area 0.015 m², the needed force is \(F = 5\times10^{3}\,\text{Pa}\times0.015\,\text{m}^2 = 75\;\text{N}\).
• Required area for a given force and pressure:
  \[ A = \frac{F}{p} \] A 300 N load must not exceed 2 kPa on a delicate glass surface. The minimum safe area is \(A = 300\;\text{N}/(2\times10^{3}\,\text{Pa}) = 0.15\;\text{m}^2\).

2. Pressure variation with depth in a fluid (hydrostatic pressure)

In a fluid at rest the pressure increases with depth because of the weight of the fluid above.

• \(\rho\) = density of the fluid (kg m⁻³)
• g = acceleration due to gravity (≈ 9.8 m s⁻²)
• h = vertical depth below the free surface (m)
• Total pressure at depth: \(\displaystyle p_{\text{total}} = p_{\text{atm}} + \rho g h\)

Example – pressure at 5 m depth in fresh water

\[ \Delta p = (1000\;\text{kg m}^{-3})(9.8\;\text{m s}^{-2})(5\;\text{m}) = 4.9\times10^{4}\;\text{Pa}=49\;\text{kPa} \] \[ p_{\text{total}} = 101\;\text{kPa} + 49\;\text{kPa} \approx 150\;\text{kPa} \]

3. Pressure in gases

For a fixed temperature, the pressure of a gas is directly proportional to its density (or mass per unit volume):

Quick calculation – Compare two containers of equal volume at the same temperature. Container A contains air of mass 1.2 kg (ρ ≈ 1.2 kg m⁻³); container B contains carbon‑dioxide of mass 1.8 kg (ρ ≈ 1.8 kg m⁻³). The pressure in B is \(\frac{1.8}{1.2}=1.5\) times the pressure in A.

4. Practical skills – measuring pressure with a U‑tube manometer

Step‑by‑step checklist

1. Secure the U‑tube vertically on the stand.
2. Zero the instrument: ensure both arms are at the same level when both are open to atmospheric pressure; record this as 0 m.
3. Connect the arm that will sense the unknown pressure to the system (e.g., a gas container). Keep the other arm open to the atmosphere.
4. Allow the liquid to settle; read the height difference \(h\) between the two arms to the nearest 0.01 m.
5. Calculate the pressure difference using \(\Delta p = \rho_{\text{liquid}} g h\). Add atmospheric pressure if an absolute pressure value is required.
6. Record the result, the liquid used, temperature (optional) and any observations.

5. Safety considerations

• Never point the open end of a pressurised container at yourself or others.
• Wear appropriate personal protective equipment (PPE): safety goggles, lab coat and gloves.
• When working with hydraulic or pneumatic systems, release pressure slowly via the supplied bleed valve.
• Check for leaks before applying force to a hydraulic piston or before connecting a manometer.
• Dispose of fluids (especially coloured oils) according to the school’s waste‑disposal policy.

6. Everyday examples (mapped to syllabus sub‑points)

Example	Force (N)	Contact area (m²)	Resulting pressure (Pa)	What the example illustrates
High‑heeled shoe on sand	≈ 500	≈ 5 × 10⁻⁴	≈ 1.0 × 10⁶ Pa (1 MPa)	Force + very small area → high pressure → sinking (force & area)
Snowshoe on snow	≈ 500	≈ 0.25	≈ 2.0 × 10³ Pa (2 kPa)	Large area reduces pressure → prevents sinking (force & area)
Sharp kitchen knife cutting bread	≈ 20	≈ 1 × 10⁻⁴	≈ 2.0 × 10⁵ Pa (0.2 MPa)	Small edge area gives enough pressure to break material (force & area)
Hydraulic car jack – small piston	100	0.001	1.0 × 10⁵ Pa (100 kPa)	Pascal’s principle – same pressure acts on larger piston (hydraulic press)
Dam resisting 5 m water depth	—	—	≈ 5.0 × 10⁴ Pa (50 kPa) additional to atmospheric	Hydrostatic pressure \(\Delta p = \rho g h\) (depth‑pressure)
Human blood pressure (clinical measurement)	—	—	Systolic ≈ 120 mm Hg ≈ 1.6 × 10⁴ Pa (16 kPa)	Use of mm Hg unit, conversion to pascals; real‑world health context

7. Application – hydraulic press (Pascal’s principle)

In a closed, incompressible‑fluid system the pressure is transmitted equally in all directions.

• Input piston: force \(F_{1}\), area \(A_{1}\) → pressure \(p = \dfrac{F_{1}}{A_{1}}\).
• Output piston: area \(A_{2}\) → output force \(F_{2} = pA_{2} = \dfrac{F_{1}}{A_{1}}A_{2}\).

Numerical illustration – \(F_{1}=100\;\text{N}\), \(A_{1}=0.001\;\text{m}^2\), \(A_{2}=0.05\;\text{m}^2\):

Assumptions: the fluid is incompressible and there are no losses due to friction.

Key points to remember

• Pressure = force ÷ area. Doubling \(F\) doubles \(p\); doubling \(A\) halves \(p\).
• In fluids at rest, total pressure at depth is \(p_{\text{total}} = p_{\text{atm}} + \rho g h\).
• At constant temperature, gas pressure is directly proportional to density (\(p \propto \rho\)).
• Pascal’s principle: pressure applied to a confined incompressible fluid is transmitted unchanged in all directions.
• Pressure can be measured with a U‑tube manometer using the relation \(\Delta p = \rho g h\); follow the procedural checklist and observe safety rules.