Question

Estimate the difference in air pressure between the top and the bottom of the Empire State Building in New York City. It is 380 m tall and is located at sea level. Express as a fraction of atmospheric pressure at sea level.

Short answer

The fraction of atmospheric pressure at sea level is\(4.42\% \).

Step-by-step solution

Understanding the tension force

The term hydrostatic pressure exists when a specific stationary liquid is stored in a container. The liquid applies pressure at the bottom surface of the container, and the value of this pressure varies with the liquid height linearly.

Identification of given data

The height of the Empire state building is \(h = 380\;{\rm{m}}\).

Determination of the fraction of atmospheric pressure at sea level

The expression of the pressure difference between top and bottom of the building is given by,

\(\Delta P = \rho gh\).

Here, \(\Delta P\) is the pressure difference between top and bottom of the building, \(\rho \) is the density of air is \(\left( {\rho = 1.2\;{\rm{kg/}}{{\rm{m}}^{\rm{3}}}} \right)\) and \(g\) is the gravitational acceleration\(\left( {g = 9.81\;{\rm{m/}}{{\rm{s}}^{\rm{2}}}} \right)\).

Substitute all the known values in the above expression,

\(\begin{array}{c}\Delta P = \left( {1.2\;{\rm{kg/}}{{\rm{m}}^{\rm{3}}}} \right)\left( {9.81\;{\rm{m/}}{{\rm{s}}^{\rm{2}}}} \right)\left( {380\;{\rm{m}}} \right)\\ = 4473.36\;{\rm{Pa}}\end{array}\)

The value of atmospheric pressure is \(1\;{\rm{atm}} = 1.01325 \times {10^5}\;{\rm{Pa}}\).

So, the fraction of atmospheric pressure at sea level can be calculated as,

\(\begin{array}{c}\frac{{\Delta P}}{{1\;{\rm{atm}}}} = \left( {\frac{{4473.36\;{\rm{Pa}}}}{{1.01325 \times {{10}^5}\;{\rm{Pa}}}} \times 100} \right)\\ = 4.42\% \end{array}\)

Thus, the fraction of atmospheric pressure at sea level is\(4.42\% \).