Oscillation of a Liquid Column in a U-Tube with Adiabatic Change

Problem Statement

A U-tube with uniform cross-sectional area $S$S is fixed horizontally. The left tube is closed at the top and the right tube is open at the top. A liquid of density $\rho$ρ is poured into the U-tube, trapping an ideal gas (monatomic) in the closed left tube. Let the atmospheric pressure be $P_0$P0​ and the magnitude of gravitational acceleration be $g$g.

[State 0] In the initial state, the trapped gas has temperature $T_0$T0​ and the liquid levels in both tubes are at the same height. At this time, the length of the trapped gas column is $l_0$l0​.

[State 1] The gas on the left is slowly heated, causing it to expand and push the liquid to the right. When the gas column length reaches $l_1$l1​ ($l_1 > l_0$l1​>l0​), heating stops and thermal equilibrium is reached. Let the gas pressure at this point be $P_1$P1​.

[State 2 (Small Oscillations)] Starting from state 1, the liquid column is displaced slightly along the tube and then released, causing the liquid column to oscillate. The period of this oscillation is sufficiently short that the state change of the trapped gas can be treated as an adiabatic process.

Let the total length of the liquid column be $L$L. All energy dissipation other than the kinetic energy of the liquid column motion (viscosity, friction with the tube, etc.) is negligible. The mass of the gas is negligible compared to the liquid.

Find the square of the angular frequency $\omega^2$ω2 of this small oscillation of the liquid column.

Constraints

• Atmospheric pressure: $P_0 = 1.013 \times 10^5 \text{ Pa}$P0​=1.013×105 Pa
• Gravitational acceleration: $g = 9.80 \text{ m/s}^2$g=9.80 m/s2
• Liquid density: $\rho = 1.00 \times 10^3 \text{ kg/m}^3$ρ=1.00×103 kg/m3
• Total length of liquid column: $L = 2.00 \text{ m}$L=2.00 m
• Length of gas column in state 0: $l_0 = 0.400 \text{ m}$l0​=0.400 m
• Length of gas column in state 1: $l_1 = 0.500 \text{ m}$l1​=0.500 m
• Heat capacity ratio of monatomic ideal gas: $\gamma = \frac{5}{3}$γ=35​

Input Format

Compute $\omega^2$ω2 in units of $\text{rad}^2/\text{s}^2$rad2/s2 and give the answer as a positive integer equal to the value multiplied by $10$10.