Question

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 is independent of the density of the liquid
• D. simple harmonic and time period is directly proportional to the density of the liquid

Answer 1

The correct option is C simple harmonic and time is independent of the density of the liquid

Consider the diagram in which a liquid column oscillates.
In this case, restoring force acts on the liquid due to gravity. Acceleration of the liquid column, can be calculated in terms of restoring force.

Let $h$ be height of liquid in each limb of U tube,
$y$ be drop in height of of liquid column in one of the limbs
Restoring force $f=$ Weight of the column of height $2y$
$\Rightarrow f=-(A\times 2y\times \rho )\times g=-2A\rho gy[\because m=\rho v]$
$\Rightarrow ma=-2A\rho gy[\because m=\rho v]$
$\Rightarrow a\propto -y$
Motion is SHM with force consant $k=2A\rho g$.
$\Rightarrow \text{Time period}T=2\pi \sqrt{\frac{m}{k}}=2\pi \sqrt{\frac{A\times 2h\times \rho }{2A\rho g}}=2\pi \sqrt{\frac{h}{g}}$
$T=2\pi \sqrt{\frac{l}{g}}$, where $l=h$
Which is independent of the density of the liquid.