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Motion of an oscillating liquid in a $U$ tube is
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simple harmonic and time period is independent of the density of the liquid.
Motion of an oscillating liquid column in a $\mathrm{U}$ tube is SHM with period, $T=2 \pi \sqrt{\frac{h}{g}}$, where $h$ is the height of liquid column in one arm of $U$ tube in equilibrium position of liquid. Therefore, $T$ is independent of density of liquid.
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