Question

Show that if the room temperature changes by a small amount from T to $T+\Delta T$, the fundamental frequency of an organ pipe changes from v to $v+\Delta v$, where $\frac{\Delta v}{v}=\frac{1}{2}\frac{\Delta T}{T}.$

Answer 1

We know that, $f\alpha \sqrt{T}$

According to the question

$f+\Delta f\alpha \sqrt{\Delta T+T}$

$\Rightarrow \frac{f+\Delta f}{f}=\sqrt{\frac{\Delta T+T}{T}}$

$\Rightarrow 1+\frac{\Delta f}{f}={(1+\frac{\Delta T}{T})}^{\frac{1}{2}}$

$=1+\frac{1}{2}\frac{\Delta T}{T}$.....(neglecting other terms)

$\Rightarrow \frac{\Delta f}{f}=\left(\frac{1}{2}\right)\frac{\Delta T}{T}$