Question

Water is filled in $\text{U}$ shaped tube up to height $H$ on both sides. If we displace a liquid column of any one side by $x\text{cm}$. What is the time period of oscillation of liquid column ?
Mass of water per unit length $=50\text{g}$.

• A. $3.2\text{s}$
• B. $2.8\text{s}$
• C. $2.0\text{s}$
• D. $1.4\text{s}$

Answer 1

The correct option is B $2.8\text{s}$

Given:

$\lambda =50\text{g/m}$

After small displacement $x$


The weight of the liquid column of length $2x$ in $\text{U}$ tube will provide a restoring force to the liquid

$\therefore$ Restoring force on liquid,

$F_{restoring}=-\lambda g\left(2x\right)$

Also, $a=\frac{F_{restoring}}{m}$

$\Rightarrow a=\frac{-\lambda g\left(2x\right)}{\lambda \left(2H\right)}$

$\Rightarrow a=\frac{-g}{H}x$

On comparing with simple harmonic motion, $a=-{\omega }^{2}x$

$\therefore \omega =\sqrt{\frac{g}{H}}$

$\therefore T=2\pi \sqrt{\frac{H}{g}}$

$\therefore T=2\pi \sqrt{\frac{2}{10}}$

So that time period

$T=2\pi \sqrt{\frac{1}{5}}$

$=2\pi \sqrt{0.2}$

$T=2.8\text{s}$

Hence, option $\left(B\right)$ is correct.