In force calculations it is usually assumed that the pressure in still air is constant.

The objective is to examine the limitations of that assumption.

(a) Derive an expression for P(z) in static air, with the z axis pointing upward and P(0) = P₀. Use the ideal-gas law to find ρ(z), assuming that the temperature is constant at T₀.

(b) As an application of the result from part (a), consider the pressure force on a vertical plane of height H and width W. If T₀ = 15 °C, how large must H be for the pressure variation to affect the force by 1%?

(c) Repeat part (a), assuming now that T(z) = T₀ − Gz, where G is a positive constant.

(d) On average, the atmospheric temperature on Earth declines linearly from about 15 °C at z = 0 to –25 °C at z = 10 km. (The height of the troposphere, the lower part of the atmosphere, is 17 km in the middle latitudes.) Plot P(z)/P₀ for z up to 10 km. Where does P first deviate from P₀ by 1%?