IN04 · Pressure needs a reference

U-Tube Manometer & Barometer

Do two liquid heights tell you an absolute pressure or a pressure difference? The liquid-height difference measures the pressure difference between the two surfaces. An open end supplies atmospheric pressure as the reference; an evacuated end supplies a nearly zero reference. Add the reference pressure to recover the unknown absolute pressure.

Connect the U-tube to air, a trapped gas or an ideal vacuum. Move the pressure dial and ruler datum, check the zero and infer pressure from saved height pairs.

IN04 · A pressure has a reference

Read the two surfaces

Drag the hose plug, source dial or datum. Zero checking disconnects the reference; reconnecting the gas cell prepares its initial fill.

Reference connectionSource pressure above atmosphere

800\,\mathrm{Pa}

Manometric liquidRuler datum height

0.5\,\mathrm{m}

A calibrated example is already saved.

Temperature, tube & reading errors

Liquid and gas temperatureAtmospheric standardLeft / right cross-sectionDifferential reading offsetReading scatterNoise seed

Temperature changes the assumed density and fill volume. Each saved pair keeps its own temperature, reference and zero check.

Model assumptions

Static incompressible liquid and ideal gas share one vertical height scale; tube widths are enlarged. The two ports are at the same height. Gas weight, capillarity, vapour pressure, glass expansion and liquid oscillations are omitted. A closed gas pocket changes pressure with its volume and temperature. Density and fill volume use declared teaching expansion coefficients between ten and thirty-five degrees Celsius. Zero checking precedes a fresh reference connection. Each record freezes heights, noise specification, reference, temperature, density and differential zero. The reported height uncertainty excludes density, reference-pressure and material-model uncertainty.

Saved simulated heights

A difference becomes a pressure

Pressure difference

800\,\mathrm{Pa}

Reference pressure

800\,\mathrm{Pa}

Unknown absolute pressure

800\,\mathrm{Pa}

Height contribution to uncertainty

800\,\mathrm{Pa}

Inspect a saved pair

Raw ruler readings share the saved datum
No.	Left	Right	Zero	Head after zero

Calibration & reference provenance

p_L=p_R+\rho g\bigl[(r_R-r_L)-b_0\bigr]

An equal-pressure check measures the difference of the two reading offsets. The sealed pocket uses its recorded initial volume, initial absolute pressure and temperature. Its current volume follows from the measured head and liquid-volume conservation.

Model limits & sources

Static, incompressible liquid and ideal gas; both ports share a height. Liquid density and volume use a constant teaching expansion coefficient between ten and thirty-five degrees Celsius. The gas cell is prepared at twenty degrees each time it is connected; zero checking takes place before that connection. The vacuum reference has zero pressure and the model liquid has no vapour pressure. Gas weight, capillarity, tube expansion and dynamic sloshing are omitted. The height uncertainty includes reading scatter, ruler resolution and the saved zero-mean uncertainty; density, reference-pressure and material-model uncertainties remain separate. Tube widths are enlarged; vertical heights retain one scale.

Start with the essentials

Focus question: Do two liquid heights tell you an absolute pressure or a pressure difference?
One-sentence intuition: The liquid balances a difference; the reference supplies the missing absolute level.

Core mathematical model

Follow the connected liquid

p_L-p_R=\rho g(z_R-z_L)

Height increases upward. Extra pressure on the left lowers that surface and raises the right. All heights are converted to metres.

A trapped gas supplies a changing reference

\frac{p_R V_R}{T_R}=\frac{p_{R0}V_{R0}}{T_0}

The gas pocket was closed at a known initial pressure, volume and temperature. Kelvin temperatures and absolute pressures are required.

Correct the two scale readings

h=(r_R-r_L)-b_0

Both readings share a ruler datum. Equalizing the pressures measures only the difference of the reading offsets.

Common difficulties

Zero gauge pressure still has an absolute pressure

Typical misconceptionEqual liquid levels mean that neither surface is under pressure.

Better mental modelBoth surfaces can be pressed equally by the atmosphere. Their pressures cancel in the height difference, and the absolute pressure remains.

Run the experiment

01
Shift the ruler
Drag the gold datum rail, then save another pair.
What to observe: Both raw ruler readings shift together. Their difference and the inferred pressure stay nearly the same, within rounding.
02
Change the reference
Drag the gold hose plug from Air to Gas pocket.
What to observe: The right side compresses a gas pocket. The reference pushes back, so the same source pressure produces a smaller head.
03
Measure an absolute pressure
Choose A vacuum reference. Then change the liquid to Water-like liquid.
What to observe: The dense-liquid column fits the tube; the water column would need more than ten metres and cannot be saved in this apparatus.
04
Check a shifted zero
Change Differential reading offset, then choose Equalize & check zero and return to Positive gauge pressure.
What to observe: The new calibration changes later measurements. Earlier height pairs keep their original zero check.

U-Tube Manometer & Barometer

A difference becomes a pressure

The liquid compares two pushes

Core mathematical model

Follow the connected liquid

A trapped gas supplies a changing reference

Correct the two scale readings

Common difficulties

Zero gauge pressure still has an absolute pressure

Run the experiment

Shift the ruler

Change the reference

Measure an absolute pressure

Check a shifted zero

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