Problem $4$

The sealed, quarter cylinder tank shown has radius $R=0\mathrm{.}5m$ and width $L=2\mathrm{.}0m$ into the page. It is comprised of a flat, rigid horizontal wall of length $R,$ a rigid quarter circle wall of radius $R,$ and a flat, massless gate of length $R$ that can rotate freely about a frictionless hinge. The tank is filled with air and remains sealed as the gate rotates. The tank is initially exposed to standard atmospheric pressure air with the gate at equilibrium at an angle of $90$ as shown in the left image. The tank is submerged in water $($density $1000k\frac{g}{m^3})$ to a depth of $H=1\mathrm{.}5m,$ as shown, causing the gate angle to change to a new equilibrium state. The mass of air in the tank remains constant and the air, water, and tank maintain a constant temperature of $300K$ throughout the process. Calculate the equilibrium angle of the gate, $,$ when submerged. Note that air is an ideal gas with equation of state,

$PV=n{R}_{u}T$

where $P$ is absolute pressure, $V$ is volume, $n$ is moles of gas, $T$ is temperature, and $R_u=8\mathrm{.}314\frac{J}{m}$olK.

Atmospheric pressure air