**Uncertainty Analysis of Determining a Building
Height using a Barometer**

In the analysis of the **previous
page** we determined that the height of the column of air
was given by the the following formula:

Our first task is to determine to what accuracy
can we infer the pressure difference ΔP from the mercury
barometer difference in height. Note that the error can be either
negative or positive, thus we develop the analysis in terms of
a **Root Mean Square** error in order to remove the sign from possibly cancelling
the total error.

A very important lesson learned: Whenever a
result is dependent upon the difference between two large values,
evaluate it very carefully, since the absolute value of the error
does not change. We now continue to evaluate the error in the
height, which is a function of ΔP, the air density ρ_{air},
and the acceleration due to gravity g. We noticed above that the
error in g has a minimal effect and can be neglected.

The problem here is that the nominal value
of ρ_{air} = 1.18 [kg/m^{}3] refers to a very
pleasant day of 25°C. Unfortunately the possible variation
in temperature (0°C - 40°C) has a significant effect on
the air density.

At this stage we combine both errors (ΔP
and ρ_{air}) to determine the final error in height:

Niels Bohr, we salute you! (recall the Legend
of Niels Bohr from the **Virtual
Teacher**).

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Engineering Thermodynamics by Israel Urieli is licensed under a
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