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Sandbox Physics
How physics instruments measure

Inside the instrument

Which pressure does a liquid column measure?

Bend a clear tube into a U, add water and leave both ends open. The surfaces settle at the same height. Blow gently into one end and its surface falls as the other rises. Their separation tells you the difference between the two pressures. To recover the absolute pressure at the end you blew into, you must identify what is connected to the other end.

Sandbox Physics ·

The stronger push lowers its own surface

Imagine a small piece of water at the bottom of the bend. If the pressure on its left were larger than the pressure on its right, it would move. Once the liquid rests, pressures at the same height in the connected liquid agree. This is how the tube compares the two ends.

Extra gas pressure on the left pushes that surface down and lifts the right surface. The longer column on the right supplies more weight above the bottom. The added weight balances the extra push on the left. A denser liquid needs a smaller height difference to balance the same pressure difference.

Different tube widths change how far each surface moves. A wide limb can supply the volume needed to lift a narrow limb with only a small downward movement. The final pressure calculation still uses the entire vertical separation between the surfaces.

Sources: [1]

Turning eight centimetres of water into pressure

Take a water-like liquid with density 998.2 kg/m3998.2\,\mathrm{kg/m^3} and gravity 9.80665 m/s29.80665\,\mathrm{m/s^2}. A separation of about 81.7 mm81.7\,\mathrm{mm} balances roughly 800 Pa800\,\mathrm{Pa}. A pascal is one newton of force per square metre.

Read the full separation. In equal-width limbs, one surface falls about four centimetres and the other rises about four centimetres. Either movement alone gives only half the required height. Two horizontal sight lines make the relevant distance clear.

Moving the entire ruler upward decreases both readings together. Their difference stays the same. This provides a useful check on the calculation: choosing another common height origin should leave the pressure difference unchanged.

pL=pR+ρghp_L=p_R+\rho g h
The left pressure is the right reference pressure plus the pressure balanced by the liquid column. The symbol rho denotes liquid density, g denotes gravity, and h is the right surface height minus the left surface height, in metres. A lower right surface gives a negative h.

Sources: [1]

An open end still feels the atmosphere

The open end is pressed by the surrounding air. It provides an atmospheric reference, so the height difference measures how far the unknown pressure lies above or below atmospheric pressure. This difference is called gauge pressure. Equal surfaces mean equal pressures at the two ends.

If an independent atmospheric reading is 101325 Pa101325\,\mathrm{Pa} and the liquid gives a positive gauge pressure of 800 Pa800\,\mathrm{Pa}, the unknown absolute pressure is approximately 102125 Pa102125\,\mathrm{Pa}. Absolute pressure starts from a vacuum, so the atmospheric reference must be added back.

Suction reverses the surfaces. The left side rises as the right side falls, and the gauge pressure becomes negative. This says that the gas pushes less strongly than the atmosphere. Its absolute pressure can still be positive.

Sources: [2]

A trapped gas and an evacuated end give different references

Capping the right end traps a pocket of gas. When the right surface rises, it squeezes that pocket into a smaller volume. The gas pushes back more strongly. Its initial pressure, volume and temperature are needed to calculate the reference pressure after compression.

A barometer instead leaves a nearly evacuated space above its liquid column. Very little gas pushes back there. The atmosphere at the other surface supports the column, whose height then measures almost the full atmospheric pressure.

With a selected dense-liquid value of 13546 kg/m313546\,\mathrm{kg/m^3}, one standard atmosphere requires about 763 mm763\,\mathrm{mm} of head. The water-like liquid would require about 10.35 m10.35\,\mathrm{m}. A short tabletop tube cannot contain that water column. For precise work, any vapour above the liquid contributes to the reference pressure too.

Sources: [2][1][3]

A zero check compares the two height readings

Temporarily join the two gas connections so that both surfaces experience the same pressure. They should settle at equal heights. Any remaining difference between the ruler readings is a differential zero offset. Save it and subtract it from later height differences. This check determines the difference between the reading offsets.

A differential offset of 1 mm1\,\mathrm{mm} corresponds to about 9.8 Pa9.8\,\mathrm{Pa} for the water-like liquid. Repeating the same biased reading preserves that shift. Temperature belongs with the measurement too: a changed density changes how much pressure a given head balances.

Keep both original height readings, their common datum, the liquid density and temperature, and the zero check used at the time. The reference connection belongs with those readings. Together, they explain where the final pressure starts and how the liquid adds to it.

Sources: [1]

Sources and further reading

  1. AMETEK, Precision Pressure Measurement, P-CP-2010-0200 (October 2016)

    Chapter 5, printed pages 34–39: U-tube balance, unequal limbs, density corrections and reading the surface.

  2. OpenStax, University Physics Volume 1, section 14.2 (2016)

    Equation 14.11 and figures 14.12–14.13 distinguish atmospheric reference from an evacuated barometer reference.

  3. OpenStax, University Physics Volume 2, section 2.1 (2016)

    Equations 2.1–2.2: fixed gas amount, absolute pressure and Kelvin temperature.

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