Question

A tuning fork of frequency $340Hz$ is vibrated just above the tube of $120cm$ height. Water is poured slowly in the tube. What is the minimum height of water necessary for the resonance (speed of sound in the air $=340m$/sec)?

• A. $25cm$
• B. $15cm$
  
• C. $30cm$
• D. $45cm$

Answer 1

The correct option is D $45cm$

The tuning fork is in resonance with air column in the pipe closed at one end.
The frequency $\Rightarrow n=\frac{(2N-1)v}{4l}\dots \left(i\right)$

Here, $N=1,2,3,\dots$(corresponds to different modes of vibration)
A tuning fork of frequency, $n=340Hz$
speed of sound in the air ,$v=340m/sec$

The length of air column in the pipe can be expressed as from equation $\left(i\right)$,
$\Rightarrow l=\frac{(2N-1)v}{4n}$

$\Rightarrow l=\frac{(2N-1)340}{4\times 340}$

$\Rightarrow l=\frac{(2N-1)}{4}m$

$\Rightarrow l=\frac{(2N-1)\times 100}{4}cm$

For $N=1,$
$\Rightarrow l=\frac{\left(2\right(1)-1)\times 100}{4}cm=\frac{100}{4}cm$
$=25cm$

For $N=2,$
$\Rightarrow l=\frac{\left(2\right(2)-1)\times 100}{4}cm=\frac{300}{4}cm$
$=75cm$

For $N=3,$
$\Rightarrow l=\frac{\left(2\right(3)-1)\times 100}{4}cm=\frac{500}{4}cm$
$=125cm$

As the length of the tube is $120cm,$
length of the air column after water is poured in it should be $25cm,75cm,125cm.$

The corresponding length of water column in the tube will be,
$\Rightarrow$ length of the tube $-$ length of the air column after water is poured
$\Rightarrow (120cm-25cm)=95cm$
$\Rightarrow (120cm-75cm)=45cm$
$\Rightarrow (120cm-125cm)=-5cm$(not possible)

$\therefore$ The minimum length of water column is $45cm$

Final Answer: (d)