A thin-walled cylinder of mass m, height h and cross-sectional area A is filled with a gas and floats on the surface of water. As a result of leakage from the lower part of the cylinder, the depth of its submergence has increased by $Δ h$ . Find the initial pressure $p_{1}$ of the gas in the cylinder if the atmospheric pressure is $p_{0}$ and the temperature remains constant.

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$p_{1} = (p_{0} + \frac{m g}{A}) (3 - \frac{Δ h}{h})$

$p_{1} = (p_{0} + \frac{m g}{A}) (2 - \frac{Δ h}{h})$

$p_{1} = (p_{0} + \frac{m g}{A}) (1 - \frac{Δ h}{h})$

$p_{1} = (p_{0} + \frac{m g}{A}) (4 - \frac{Δ h}{h})$

answer is D.

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Detailed Solution

$p_{2} = pressure at depth x$

$p_{1} V_{1} = p_{2} V_{2} or p_{1} (A h) = p_{2} (A) (h - Δ h) ∴ p_{1} = \frac{p_{2} (h - Δ h)}{h} Initially, m g = ρ g x A Now, p_{2} = p_{0} + ρ g x = p_{0} + \frac{m g}{A} Substituting in Eq. (i), we have p_{1} = (p_{0} + \frac{m g}{A}) (1 - \frac{Δ h}{h})$