Question

A liquid of mass $m$ is set into oscillations in a U-tube of cross section $A$. Its time period recorded is $T$, where $T=2\pi \sqrt{l/2g}$ and $l$ is the length of the liquid column. If the liquid of same mass is set into oscillations in U-tube of cross-section $A/16$, then determine time period of oscillation.

• A. $T^{\prime }=2T$
• B. $T^{\prime }=4T$
• C. $T^{\prime }=T/2$
• D. $T^{\prime }=T/4$

Answer 1

The correct option is B $T^{\prime }=4T$

As mass is constant
$\therefore \text{Volume}\times \text{density}=\text{constant}(\because \text{Density}=\frac{\text{Mass}}{\text{Volume}})$
$\Rightarrow V_{1}d=V_{2}d$
where, $V_1$ and $V_2$ are the volume of liquid in U-tube in two cases respectively.
$\left(Al\right)d=\left|\frac{A}{16}{l}^{\prime }\right|d\Rightarrow l^{\prime }=16l$

$\therefore \frac{T^{\prime }}{T}=\sqrt{\frac{l^{\prime }}{l}}=\sqrt{\frac{16l}{l}}=4\Rightarrow T^{\prime }=4T$