Describe, qualitatively, how the pressure beneath the surface of a liquid changes with depth and density of the liquid

Pressure in Liquids – Topic 1.8

1. Definition of pressure (AO 1)

Pressure is the force exerted perpendicular to a surface per unit area.

\(p = \dfrac{F}{A}\)

  • F = normal force (N)

  • A = area over which the force acts (m²)

SI unit: pascal (Pa), where 1 Pa = 1 N m⁻².

For quick conversion in IGCSE work: 1 Pa ≈ 1 × 10⁻⁵ atm.

What‑if box:

If a force of 10 N is applied to an area of 0.01 m², the pressure is

\(p = \dfrac{10}{0.01}=1000\;{\rm Pa}\).

Halving the area to 0.005 m² while keeping the same force doubles the pressure:

\(p = \dfrac{10}{0.005}=2000\;{\rm Pa}\).

2. Pascal’s principle (AO 3)

In a fluid at rest the pressure at any point is the same in all directions. This is known as Pascal’s principle. It can be demonstrated with a simple U‑tube manometer: when equal pressures are applied to both arms the liquid level remains the same.

3. How pressure varies with depth (core content)

The weight of the liquid column above a point adds to the atmospheric pressure acting on the free surface.

\(\displaystyle \Delta p = \rho\,g\,h\)

  • \(\rho\) = density of the liquid (kg m⁻³)

  • \(g\) ≈ 9.81 m s⁻² (acceleration due to gravity)

  • \(h\) = vertical depth below the free surface (m)

The total pressure at depth \(h\) is therefore

\(p{\text{total}} = p{\text{atm}} + \rho\,g\,h\)

4. Qualitative description (exact syllabus wording)

“Pressure beneath the surface of a liquid increases linearly with depth and with the liquid’s density**.”

  • Depth (\(h\)): A greater depth means a taller column of liquid above the point, so the weight per unit area is larger → pressure rises in a straight‑line relationship.

  • Density (\(\rho\)): For a given depth, a denser liquid contains more mass per unit volume, giving a larger weight per unit area → the *slope* of the pressure‑vs‑depth line is steeper.

  • Atmospheric pressure (\(p_{\text{atm}}\)): Provides a constant base pressure at the surface; it adds the same amount to the pressure at every depth but does not affect the rate of increase with depth.

5. Practical tip – measuring pressure with a manometer (AO 3)

  1. Set up a U‑tube filled with a coloured liquid (e.g., water).

  2. Connect one arm to the point whose pressure you wish to measure (e.g., a submerged opening).

  3. Record the difference in liquid heights, \(\Delta h\), between the two arms.

  4. Calculate the pressure difference using \(\Delta p = \rho g \Delta h\).

  5. To obtain the absolute pressure, add the known atmospheric pressure (measured with a barometer).

When analysing data, keep the same number of significant figures as the measured height (usually 2 sf for a ruler reading).

6. Worked examples (include significant figures)

  1. Water vs. oil at the same depth (4 m)

    • Water (\(\rho = 1000\;{\rm kg\,m^{-3}}\)): \(\Delta p = 1000 \times 9.81 \times 4 = 3.92 \times 10^{4}\;{\rm Pa}\) (3 sf).

    • Oil (\(\rho = 800\;{\rm kg\,m^{-3}}\)): \(\Delta p = 800 \times 9.81 \times 4 = 3.14 \times 10^{4}\;{\rm Pa}\) (3 sf).

    Because water is denser, the pressure increase is larger.

  2. Total pressure 5 m below the surface of fresh water

    \(\Delta p = 1000 \times 9.81 \times 5 = 4.91 \times 10^{4}\;{\rm Pa}\) (3 sf).

    Adding atmospheric pressure \(p_{\text{atm}} \approx 1.01 \times 10^{5}\;{\rm Pa}\):

    \(p_{\text{total}} \approx 1.01 \times 10^{5} + 4.91 \times 10^{4} = 1.50 \times 10^{5}\;{\rm Pa}\) (3 sf), i.e. ≈ 1.48 atm.

7. Pressure‑depth graph (interpretation skill)

Sketch a graph with depth (\(h\)) on the horizontal axis and pressure (\(p\)) on the vertical axis. For a single liquid the plot is a straight line passing through p = patm at \(h = 0\) and having a slope of \(\rho g\). Different liquids give parallel lines with steeper slopes for higher densities.

Typical exam question: “Draw and label the pressure‑depth graph for oil (ρ = 800 kg m⁻³). Indicate the pressure at 3 m depth if atmospheric pressure is 101 kPa.”

8. Summary table

ParameterEffect on pressure beneath the surface
Depth (\(h\))Pressure increases linearly with depth; deeper → higher pressure (slope = ρg).
Density (\(\rho\))For a given depth, higher density gives a larger pressure increase (steeper slope).
Atmospheric pressure (\(p_{\text{atm}}\))Adds a constant base pressure at the surface; same offset at all depths.

9. Suggested diagram (to be drawn by students)

Vertical column of liquid

  • Free surface labelled \(p_{\text{atm}}\)

  • Depth \(h\) measured from the surface

  • Arrow showing weight of the liquid column (\(\rho g h\))

  • Resulting pressure at the bottom: \(p_{\text{atm}} + \rho g h\)

  • Labels for \(\rho\), \(g\) and \(h\)

10. Link to other topics

While this section deals with liquids, the idea that pressure is caused by particles colliding with surfaces also underpins the kinetic‑particle model of gases (Core 1.4). Recognising this connection helps when you later study gas laws and sound propagation.