Question

A source of frequency $340Hz$ is kept above a certical cylindrical tube closed at lower end. The length of the tube is $120cm$. Water is slowly poured in just enough to produce resonance. Then the minimum height (velocity of sound $=340m/s$) of the water level in the tube for the resonance is

• A. $0.75m$
• B. $0.25m$
• C. $0.95m$
• D. $0.45m$

Answer 1

The correct option is D $0.45m$

Given : $\nu =340Hz$ $v=340m/s$

Length of tube, $L=120$ cm $=1.20$ m

Let the length of air column in the tube after water is being poured be $l$.

Frequency of sound in a closed pipe, $\nu =\frac{(2n-1)v}{4l}$

where $\nu =340Hz$

$\therefore$ $340=\frac{(2n-1)\left(340\right)}{4l}$

$⟹2n-1=4l$

For minimum height of water level, the length of air column must be longest i.e $l$ must be largest

Thus for $n=1$, $l=\frac{2\left(1\right)-1}{4}=0.25$ m

For $n=2$, $l=\frac{2\left(2\right)-1}{4}=0.75$ m

For $n=3$, $l=\frac{2\left(3\right)-1}{4}=1.25$ m (Not possible)

Hence, possible longest length of air column, $l=0.75$ m

$⟹$ Minimum length of water level for resonance, $L_w=1.20-0.75=0.45$ m