Q. 21

# (a) Pressure decreases as one ascends the atmosphere. If the density of air is ρ, what is the change in pressure dp over a differential height dh?(b) Considering the pressure p to be proportional to the density, find the pressure p at a height h if the pressure on the surface of the earth is p0.(c) If p0 = 1.03×105 N m-2, ρ0 = 1.29 kg m-3 and g = 9.8 m s-2, at what height will the pressure drop to (1/10) the value at the surface of the earth?(d) This model of the atmosphere works for relatively small distances. Identify the underlying assumption that limits the model.

(a) As we go up in the atmosphere, pressure decreases.

Consider a layer of atmosphere at height h and area A.

Let the pressure at h be p + dp.

Let the pressure at h + dh be p.

This layer is in equilibrium and hence, upward and downward forces are balanced.

Pressure × Area = Force = ma =

(b) Here,

Hence,

Integrating on both sides,

(c)

(d) We can assume only under isothermal conditions which is valid near the surface of the earth and not at great heights.

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