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utkarshakash
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Homework Statement
A liquid (density ρ) is poured into a bent tube such that the two halves of the bent form angles α and β with the horizontal. The length of liquid column is l. If one of the liquid levels is depressed and released, the levels begin to vibrate. Find the period of vibration. Neglect capillary forces and the viscosity of the liquid.
The Attempt at a Solution
Let the cross-sectional area of bottom-most part of the tube that separates the two sides be S, left part be S1 and right part be S2.
S1 = S sec α
S2= S sec β
Let the equilibrium height of liquid surface be h. when the liquid is depressed, let the left surface of liquid descend a distance Δx1 and the right part ascend a distance Δx2.
Since volume is constant,
S1Δx1=S2Δx2
sec α Δx1 = sec β Δx2
Let the force on the right part of the liquid by the left part be F1 and that on left part by the right part be F2 at the bottommost point
[itex]F_1=(P_0 + \rho g (h-Δx_1) ) S \\
F_2 = (P_0 + \rho g (h+Δx_2) ) S \\
F_1 - F_2 = - \rho g S (Δx_1+Δx_2)\\
\ddot{x} =\dfrac{-\rho gS}{m} \left(\dfrac{\sec \beta}{\sec \alpha}+1 \right) x[/itex]
But this equation gives me the incorrect angular frequency.