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  • The motion of an oscillating liquid column in a U-tube is (a) periodic but not simple harmonic

The motion of an oscillating liquid column in a U-tube is
(a) periodic but not simple harmonic
(b) non-periodic
(c) simple harmonic and time period is independent of the density of the  liquid
(d) simple harmonic and time period is directly proportional to the density of the liquid

Answers (1)

The answer is the option (c) Simple harmonic and time period is independent of the density of the liquid.

Explanation: Let us take a test tube viz., filled with a liquid of density \rho up to height ‘h’.

When the liquid is lifted in arm Q to a height ‘y’ from A to B, the liquid in arm P drops by the height ‘y’ from A’ to C’. The height difference between the two arms is 2y.

Here, the hydrostatic pressure provides the restoring force, thus,

F = -V. \rho.g

 = -A.2y\rho g , where,

A = Area of a cross-section of tube, & F \propto -y,

Thus, it is simple harmonic motion.

T = \sqrt{\frac{2\pi m (inertia)}{k (spring)}}= \sqrt{\frac{2\pi A (2h)}{\rho 2 A \rho g}}

Thus, T = 2\pi \sqrt{\frac{h}{g}}

Thus, the motion is harmonic as the time period is independent of density.

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