Question

A hollow metallic tube of length $L$ closed at one end produces resonance (in its fundamental mode) with a tuning fork of frequency $n$. The entire tube is then heated carefully, so that at equilibrium temperature, its length changes by $\ell$. If the change in velocity of sound is $v$, resonance (in fundamental mode) will now be produced by the tuning fork of frequency:
[Neglect end corrections]

• A. $\frac{(V+v)}{\left[4\right(L+\ell \left)\right]}$
• B. $\frac{(V+v)}{\left[4\right(L-\ell \left)\right]}$
• C. $\frac{(V-v)}{\left[4\right(L+\ell \left)\right]}$
• D. $\frac{(V-v)}{\left[4\right(L-\ell \left)\right]}$

Answer 1

The correct option is A $\frac{(V+v)}{\left[4\right(L+\ell \left)\right]}$


For tube closed at one end,
Fundamental frequency $f_0=\frac{v}{4L}$

Before increasing temperature,

Frequency of the tuning fork, $n=\frac{v}{4L}$

As the temperature increases, length increases

$L_{new}=L+\ell$

As the temperature increases, air molecules have more kinetic energy which results in increase in speed of sound.

$V_{new}=V+v$

Therefore, required frequency of the tuning fork for resonance,
$n^{\prime }=\frac{V+v}{\left[4\right(L+\ell \left)\right]}$

Thus, option (a) is the correct answer.