Question

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

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

Answer 1

The correct option is D $0.45m$

Because the tuning fork is in resonance with air column in the pipe closed at one end, the frequency is $n=\frac{(2N-1)v}{4l}$ where $N=1,2,3....$ corresponds to different mode of vibration putting $n=340Hz,v=340m/s,$ the length of air column in the pipe can be $l=\frac{(2N-1)340}{4\times 340}=\frac{(2N-1)}{4}m=\frac{(2N-1)\times 100}{4}cm$ For $N=1,2,3,...$we get $l=25cm$, $75cm,125cm...$As the tube is only $120$ cm long, length of air column after water is poured in it may be $25$ cm or $75cm$ only, $125cm$ is not possible, the corresponding length of water column in the tube will be $(120?25)$ cm = $95cm$ or $(120?75)cm=45cm$. Thus minimum length of water column is $45cm.$