U-tube manometer

1. Definition

A. Manometer

A manometer is an instrument used to measure a pressure difference by using the difference in height between two columns of liquid in a U-tube.

2. Key Ideas

• In a simple U-tube manometer (same liquid in both arms):
  • the higher-pressure side pushes the liquid down on its side
  • the pressure difference is: Δ p = ρ g Δ h
• Δ h is the vertical height difference between the two liquid levels.
• If one side is open to the atmosphere:
  • gas-side level lower → p_gas = pₐₜₘ + ρ gΔ h
  • gas-side level higher → p_gas = pₐₜₘ - ρ gΔ h
• Gauge pressure is p_gauge = p_gas-pₐₜₘ; it is negative when the gas pressure is below atmospheric pressure.
• Useful densities:
  • water: ρ ≈ 1000 kg m⁻³
  • mercury: ρ ≈ 13.6 × 10³ kg m⁻³
• Convert heights to metres (e.g. 15 cm = 0.15 m).

3. Detailed Explanations

A. How a U-tube manometer works

If both sides are at the same pressure (e.g. both open to air), the liquid levels are equal.

If one side has a higher pressure, it pushes the liquid down on that side and the levels become different.

Using hydrostatic pressure ( p = ρ gh), the pressure difference between the two sides is:

Δ p = ρ g Δ h

where ρ is the density of the manometer liquid and Δ h is the vertical height difference.

B. Quick sign check (which side has higher pressure?)

For a U-tube manometer:

• the side with the lower liquid level has the higher pressure
• always write the relationship as: p_high = p_low + ρ g Δ h

C. Why mercury is sometimes used

For the same pressure difference Δ p:

Δ h = (Δ p)/(ρ g)

A denser liquid (larger ρ) gives a smaller height difference, so it is useful for large pressure differences.

4. Common Mistakes

• Using the wrong Δ h (it is the vertical height difference between the two liquid levels).
• Measuring from the bottom of the U-tube instead of measuring the difference between the two levels.
• Forgetting to convert cm to m.
• Using the wrong density (e.g. using water’s density when the liquid is mercury).
• Adding pₐₜₘ when the question only asks for pressure difference.

5. Exam Tips

• Decide what the question wants:
  • pressure difference: use Δ p = ρ gΔ h
  • gas pressure (one side open to air): use p_gas = pₐₜₘ ± ρ gΔ h
• State which side has higher pressure by looking at which side’s liquid level is lower.
• Keep units consistent: Pa, kg m⁻³, N kg⁻¹, m.

6. Worked Examples

7. Mind Stretchers

8. Practice and next step

Complete the Pressure structured questions, then return to the Pressure hub for the full topic checklist or continue to Thermal Physics.