Question

A hollow metal pipe of length $L$ is closed at one end and produces resonance when sounded with a tuning fork of frequency $n$. The entire tube is then heated carefully so that at equilibrium temperature its length changes by $l$. If the changes in velocity $V$ of the sound is $v$, then the resonance will occur with a tuning fork of frequency

• A. $\frac{V+v}{4(L+l)}$
• B. $\frac{V+v}{4(L-l)}$
• C. $\frac{V-v}{4(L+l)}$
• D. $\frac{V-v}{4(L-l)}$

Answer 1

The correct option is A $\frac{V+v}{4(L+l)}$

Since the temperature of the metal pipe is increasing so its length also increases. and its new length $L_{new}$ becomes $L+l$.
now we will consider the effect of temperature on the speed of sound. when the temperature of the pipe increases air molecules have more kinetic energy which results in the increase in the speed of sound. so its new speed
$V_{new}$ becomes $V+v$
so now the resonance will occur with a tuning fork of frequency $=\frac{V_{new}}{4L_{new}}$
$\Rightarrow f_{new}=\frac{V_{new}}{4L_{new}}=\frac{V+v}{4(L+l)}$