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Question
Motion of an oscillating liquid column in a U-tube is ______.
Options
periodic but not simple harmonic.
non-periodic.
simple harmonic and time period is independent of the density of the liquid.
simple harmonic and time-period is directly proportional to the density of the liquid.
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Solution
Motion of an oscillating liquid column in a U-tube is simple harmonic and time period is independent of the density of the liquid.
Explanation:
If the liquid in U-tube is filled to a height of h and the cross-section of the tube is uniform and the liquid is incompressible and non-viscous. Initially, the level of liquid in the two limbs will be at the same height equal to h. If the liquid is pressed by y in one limb, it will rise by y along the length of the tube in the other limb, so the restoring force will be developed by the hydrostatic pressure difference
Here, the hydrostatic pressure provides the restoring force, thus,
F = – V.p.g
= – A.2ypg, where,
A = Area of a cross-section of tube and F ∝ – y,
Thus, it is a simple harmonic motion.
T = `sqrt((2πm("inertia"))/(k("spring"))`
= `sqrt((2πA(2h))/(p2Apg))`
T = `2π sqrt(h/g)`
Thus, the motion is harmonic as the time period is independent of density.
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