Question

The pressure $P(z)$ as the function of $z$ is

• A. $(p_0 + P_{atm})e^{\rho_0 gz/2P_0} - P_0$
• B. $(P_0 + P_{atm})e^{-\rho_0 gz/2P_0} - P_0$
• C. $(P_0 + P_{atm})e^{\rho_0 gz/P_0} - P_0$
• D. $(P_0 + P_{atm})e^{-\rho_0 gz/P_0} - P_0$

Answer 1

Answer: (D)

(P_0 + P_{atm})e^{-\rho_0 gz/P_0} - P_0

Answer 2

The pressure variation with height $z$ is given by $dP = -\rho g dz$. Assuming a linear relationship between density $\rho$ and pressure $P$ of the form $\rho = aP + b$, the integration of the differential equation leads to a pressure function of the form $P(z) = A e^{-agz} - b/a$. Comparing this with the given options, specifically option 4, $P(z) = (P_0 + P_{atm})e^{-\rho_0 gz/P_0} - P_0$, we can identify $b/a = P_0$, $ag = \rho_0 g/P_0$, and $A = P_0 + P_{atm}$. From $b/a = P_0$, we get $b = aP_0$. The density relation becomes $\rho = aP + aP_0 = a(P + P_0)$. From $ag = \rho_0 g/P_0$, we get $a = \rho_0/P_0$. Substituting $a = \rho_0/P_0$ into the density relation gives $\rho = (\rho_0/P_0)(P + P_0) = (\rho_0/P_0)P + \rho_0$. The term $A = P_0 + P_{atm}$ is consistent with the boundary condition $P(0) = P_{atm}$, as $A = \frac{1}{a}(aP(0)+b) = P(0) + b/a = P(0) + P_0$, which implies $P(0) = P_{atm}$. This shows that option 4 is derived from the differential equation $dP = -\rho g dz$ with the density-pressure relationship $\rho = (\rho_0/P_0)P + \rho_0$ and the boundary condition $P(0) = P_{atm}$.