A uniform V-shaped tube contains a liquid column of total length L. The left arm makes an angle α with the horizontal and the right arm makes an angle β with the horizontal. Both free surfaces are exposed to the same pressure. Neglect viscosity. What is the angular frequency of small oscillations of the liquid along the tube?
Options

Question

A uniform V-shaped tube contains a liquid column of total length L. The left arm makes an angle α with the horizontal and the right arm makes an angle β with the horizontal. Both free surfaces are exposed to the same pressure. Neglect viscosity. What is the angular frequency of small oscillations of the liquid along the tube?

Options

G1 Physics · Accepted Answer

Suppose the liquid shifts by distance x along the tube. The vertical displacement of one free surface is x sinα, while the other changes by x sinβ. Hence the total vertical head difference is Δh = x(sinα+sinβ). The restoring force is F = −ρAgx(sinα+sinβ). The moving mass is ρAL. Thus, ρAL ẍ = −ρAg(sinα+sinβ)x. Therefore, ω² = g(sinα+sinβ)/L. Hence, ω = √[g(sinα+sinβ)/L]. Correct option: A.