Show that if the room temperature changes by a small amount from … - Vidyadip

Question

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

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Answer

We know that $\text{f}\propto\sqrt{\text{T}}$ According to the question $\text{f}+\triangle\text{f}\propto\sqrt{\triangle\text{T}}+\text{T}$$\Rightarrow\frac{\text{f}+\triangle\text{f}}{\text{f}}=\sqrt{\frac{\triangle\text{t}+\text{T}}{\text{T}}}$
$\Rightarrow1+\frac{\triangle\text{f}}{\text{f}}=\Big(1+\frac{\triangle\text{T}}{\text{T}}\Big)^{\frac{1}{2}}=1+\frac{1}{2}\frac{\triangle\text{T}}{\text{T}}+\ \dots$ (neglecting other terms)
$\Rightarrow\frac{\triangle\text{f}}{\text{f}}=\Big(\frac{1}{2}\Big)\frac{\triangle\text{T}}{\text{T}}$

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