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  4. AU -tube of uniform bore is placed with arms vertical. The length of the liquid in the two arms of the U-tube is L. The time period T of the oscillation of the liquid column when it is displaced by y is

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Q. AU -tube of uniform bore is placed with arms vertical. The length of the liquid in the two arms of the U-tube is L. The time period T of the oscillation of the liquid column when it is displaced by y is

Oscillations

Solution:

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$\frac{d^{2} y}{d t^{2}}=-\frac{F}{m}=-\left(\frac{2 y A \rho g}{L A \rho}\right)$
$ \Rightarrow \frac{d^{2} y}{d t^{2}}=-\left(\frac{2 g}{L}\right) y$
or $\omega^{2}=\frac{2 g}{L} \Rightarrow T=2 \pi \sqrt{\frac{L}{2 g}}$