Consider a long narrow cylinder of cross-section A filled with a compressible liquid upto a height h whose density ρ is a function of pressure P(z) as ρ(z)=ρ02(1+P(z)P0), where P0 and ρ0 are constants. The depth z is measured from the free surface of the liquid where pressure is equal to atmospheric pressure (Patm). Choose the correct options.

ρ(z)=ρ02(1+P(z)P0) P(z)=∫ρ(z)gdz+Patm=∫0xρ02(1+p(z)P0)gdz+Patm dP(z)dz=ρ02(1+P(z)P0)g log(P0+P(z)P0+Patm)=ρ0gz2P0 ⇒ P(z)=[(P0+Patm)eρ0gz/2P0−P0] ρ(z)=ρ0(P0+Patm)eρ0gz/2P0 ρ(z)=ρ0(P0+Patm)2P0eρ0gz/2P0 Mass of the liquid M=∫0hρ(z)Adz=Ag(P0+Patm)(eρ0gh/2P0−1)

Consider a long narrow cylinder of cross-section A filled with a compressible liquid upto a height h whose density ρ is a function of pressure P(z) as ρ(z)=ρ02(1+P(z)P0), where P0 and ρ0 are constants. The depth z is measured from the free surface of the liquid where pressure is equal to atmospheric pressure (Patm). Choose the correct options.

ρ(z)=ρ02(1+P(z)P0) P(z)=∫ρ(z)gdz+Patm=∫0xρ02(1+p(z)P0)gdz+Patm dP(z)dz=ρ02(1+P(z)P0)g log(P0+P(z)P0+Patm)=ρ0gz2P0 ⇒ P(z)=[(P0+Patm)eρ0gz/2P0−P0] ρ(z)=ρ0(P0+Patm)eρ0gz/2P0 ρ(z)=ρ0(P0+Patm)2P0eρ0gz/2P0 Mass of the liquid M=∫0hρ(z)Adz=Ag(P0+Patm)(eρ0gh/2P0−1)

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Consider a long narrow cylinder of cross-section A filled with a compressible liquid upto a height h whose density $ρ$ is a function of pressure $P (z)$ as $ρ (z) = \frac{ρ_{0}}{2} (1 + \frac{P (z)}{P_{0}})$ , where $P_{0}$ and $ρ_{0}$ are constants. The depth z is measured from the free surface of the liquid where pressure is equal to atmospheric pressure ( $P_{a t m}$ ). Choose the correct options.

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Mass of the liquid in container is $\frac{A}{g} (P_{0} + P_{a t m}) (e^{ρ_{0} g h / 2 P_{0}} - 1)$

$\frac{d P (z)}{d z} = \frac{ρ_{0}}{2} (1 + \frac{P (z)}{P_{0}}) g$

Pressure varies with depth as $P (z) = P_{a t m} e^{ρ_{0} g z / 2 P_{0}} - P_{0}$

Density varies with depth as $ρ (z) = \frac{ρ_{0} (P_{0} + P_{a t m})}{2 P_{0}} e^{ρ_{0} g z / 2 P_{0}}$

answer is A, B, D.

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Detailed Solution

$ρ (z) = \frac{ρ_{0}}{2} (1 + \frac{P (z)}{P_{0}}) P (z) = \int ρ (z) g d z + P_{a t m} = \int_{0}^{x} \frac{ρ_{0}}{2} (1 + \frac{p (z)}{P_{0}}) g d z + P_{a t m} \frac{d P (z)}{d z} = \frac{ρ_{0}}{2} (1 + \frac{P (z)}{P_{0}}) g \log (\frac{P_{0} + P (z)}{P_{0} + P_{a t m}}) = \frac{ρ_{0} g z}{2 P_{0}} \Rightarrow P (z) = [(P_{0} + P_{a t m}) e^{ρ_{0} g z / 2 P_{0} - P_{0}}] ρ (z) = ρ_{0} (P_{0} + P_{a t m}) e^{ρ_{0} g z / 2 P_{0}} ρ (z) = \frac{ρ_{0} (P_{0} + P_{a t m})}{2 P_{0}} e^{ρ_{0} g z / 2 P_{0}}$