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Question
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.
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Solution
Let p0 be the density of air on the surface of the earth.
As per the question, pressure ∝ density
⇒ `p/p_0 = ρ/ρ_0`
⇒ `ρ = ρ_0/p_0 p`
∴ `dp = - (ρ_0g)/p_0 pdh` ......[∵ dp = – ρgdh]
⇒ `(dp)/p = - (ρ_0g)/p_0 dh`
⇒ `int_(p_0)^p (dp)/p = - (ρ_0 g)/p_0 int_0^h dh` .....`[(∵ at h = p, r = p_0),(and at h = h, p = p)]`
⇒ In `p/ρ_0 = - (ρ_0 g)/p_0 h`
By removing log, `p = p_0e (- (ρ_0gh)/p_0)`
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