A uniform U-tube of constant cross-sectional area contains a liquid column of total length L measured from one free surface to the other through the tube. The liquid is displaced slightly and released. Neglect viscosity. What is the time period of the resulting small oscillation?
Options

Question

A uniform U-tube of constant cross-sectional area contains a liquid column of total length L measured from one free surface to the other through the tube. The liquid is displaced slightly and released. Neglect viscosity. What is the time period of the resulting small oscillation?

Options

G1 Physics · Accepted Answer

If the liquid shifts by x, one free surface rises by x and the other falls by x. The level difference is 2x. Therefore the pressure difference is ΔP = 2ρgx. Restoring force: F = −2ρgAx. The total moving mass is m = ρAL. Hence, ρAL ẍ = −2ρgAx. Therefore, ẍ+[2g/L]x = 0. Thus, ω² = 2g/L, and T = 2π√[L/(2g)]. Correct option: B.