Manometry & Pressure Measurement

Turn the hydrostatic equation into a real measuring instrument — no electronics required.

Fluid StaticsMechanical Engineering Year 1Free preview
⏱️ About 18 min

Before pressure transducers existed, engineers measured pressure with nothing more than a bent glass tube and a denser fluid — and the method is still one of the most reliable, calibration-free ways to measure pressure today.

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The big idea: A manometer works because you can apply the hydrostatic equation p = p0 + ρgh across a chain of connected fluid columns: start at a known pressure, add or subtract ρgh for every fluid layer you pass through, and arrive at the unknown pressure.
🎯 By the end, you'll be able to
  • Explain how a simple U-tube manometer measures pressure
  • Apply the hydrostatic equation step by step across a manometer's fluid columns
  • Solve for an unknown gauge pressure given a manometer fluid and column-height difference
📎 Helpful to know first

The idea: walk the hydrostatic equation across the tube

A simple U-tube manometer connects the pressure you want to measure (say, a pipe carrying water) to the open atmosphere, using a denser "manometer fluid" (classically mercury) inside a U-shaped tube. Because the fluid is connected and at rest, you can apply p = p0 + ρgh one step at a time as you move down one leg of the tube and up the other, tracking every fluid layer you cross.

The trick that makes manometers so reliable: at any two points at the same height within the same connected fluid, the pressure must be equal (that's just the hydrostatic equation applied twice, since h is the same for both points). This lets you "jump across" the bottom of the U-tube for free.

\[ p_{pipe} \;=\; \rho_{manometer}\, g\, h \]
This compact form holds ONLY for the configuration shown: the pipe centerline lines up exactly with the manometer fluid's surface on the connected side, so there is no extra elevation term. If the pipe sits higher or lower than that meniscus, you must add the ρgh contribution of the pipe fluid over that elevation difference — always work point-to-point through the tube rather than memorizing this shortcut.
🔑 Why mercury?

Manometers classically use mercury (ρ ≈ 13,600 kg/m³) because its high density means a large pressure produces only a small, easy-to-read column height — measuring the same pressure with a water-filled manometer would require an impractically tall tube (roughly 13.6 times taller).

🎮 Interactive: manometer reading LIVE
Predict first: How much taller would the mercury column need to be to read double the pressure?

An interactive slider tool showing the gauge pressure measured by a manometer, as a function of the manometer fluid's density and the column height difference.

p = ρ·g·h. Slide the manometer fluid density and column-height difference to see the measured gauge pressure update live.
📝 Worked example: A U-tube mercury manometer (ρ_Hg = 13{,}600 kg/m³) is connected to a water pipe. The pipe centerline lines up with the mercury surface in the connected leg. The mercury in the open leg stands 0.35 m higher than in the connected leg. Find the gauge pressure in the pipe. Use g = 9.81 m/s².
  1. Since the pipe connects directly at the mercury surface (no water column offset to account for), the pressure balance is simply between the pipe pressure and the mercury column height difference.
  2. p_pipe = ρ_Hg × g × h = (13{,}600)(9.81)(0.35).
  3. Compute: 13{,}600 × 9.81 = 133{,}416. Then 133{,}416 × 0.35 = 46{,}695.6 Pa.
✓ p_pipe ≈ 46.7 kPa (gauge)
✏️ Practice: Using the same setup (mercury manometer, pipe centerline at the mercury surface), the height difference is 0.2 m instead. Find the gauge pressure in the pipe, in kPa.
kPa
Solution
  1. p = ρ_Hg × g × h = (13{,}600)(9.81)(0.2).
  2. p = 26{,}683.2 Pa ≈ 26.7 kPa.

Check your understanding

1. Why do manometers traditionally use mercury instead of water?
Mercury's much higher density means a given pressure produces a much shorter, easier-to-read column than water would.
2. At two points at the same height within the same connected static fluid, the pressures are:
This follows directly from the hydrostatic equation: p = p0 + ρgh depends only on depth h, so equal heights in the same fluid give equal pressures.
🔎 Related lessons
Pressure Variation with Depth — the Hydrostatic Equation Hydrostatic Forces on Plane Surfaces · soon
✅ Key takeaways
  • A manometer measures pressure by applying the hydrostatic equation across connected fluid columns.
  • Equal heights in the same connected static fluid always have equal pressure — the key trick for solving manometer problems.
  • Dense fluids like mercury are used so the resulting column height stays small and practical to read.
➡️ With pressure at a point and pressure measurement covered, the next lessons scale up to the total force a fluid exerts on an entire submerged surface.
Want to test yourself on this? Try the Mechanical Aptitude test →
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