Past exam of the medicine course of the University of Oxford / 1865 / First examination / Physics / 4

Original question.

What is the principle of a barometer ? What corrections have to be applied to the reading of a cistern barometer ?

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- Solution 4

Solution

A mercury barometer balances atmospheric pressure against the hydrostatic head between the open cistern surface and the mercury surface in a closed tube. Writing

p_{t}

for the pressure above the column, the ideal relation is

p_{atm} = p_{t} + ρ_{Hg} (T) g_{local} h .

(1)

The relevant

h

is the vertical difference between the two mercury levels. It is not simply the length of filled glass, and the top space need not be an absolutely perfect vacuum.

A cistern barometer needs the following corrections or controlled adjustments:

Cistern level and scale zero. When mercury rises in a fixed tube, the reservoir level falls. A fixed-zero scale must allow for both motions. For uniform tube area $A_{t}$ and effective cistern free area $A_{c}$ , volume conservation gives $A_{c} Δ z_{c} = - A_{t} Δ z_{t}$ , so $Δ h = (1 + A_{t} / A_{c}) Δ z_{t}$ . In a Fortin barometer, an adjustable cistern brings its surface to a fixed index before the column is read.
Temperature. Both mercury density and scale length change. To express a corrected physical height $h$ as an equivalent height at reference density $ρ_{0}$ and standard gravity $g_{s}$ , use $h_{s} = ρ (T) g_{local} h / (ρ_{0} g_{s})$ . If a scale marked at the reference temperature expands with coefficient $α_{s}$ , while mercury has volume-expansion coefficient $β$ , a reading $R$ at temperature difference $Δ T$ has the first-order thermal factor $1 + (α_{s} - β) Δ T$ . For an ordinary metal scale warmer than its reference, the mercury-density effect dominates and the thermal correction reduces the reading.
Gravity. The same physical mercury height represents a different pressure where $g$ differs. Reduction to a conventional mercury pressure unit includes $g_{local} / g_{s}$ ; this does not assert that local gravity has physically changed.
Capillarity, scale and sighting. Correct the capillary depression and calibrated scale/index errors, read the proper part of the meniscus, avoid parallax, and set the instrument vertical. Mercury contamination and changes in the meniscus can compromise a previously determined capillary correction.
Top-space pressure. Trapped gas and mercury vapour add $p_{t}$ . A significant trapped-air error cannot be cured by pretending the top space is empty.

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Past exam of the medicine course of the University of Oxford / 1865 / First examination / Physics / 4

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