Sol. $\mathrm{f} \propto \frac{1}{\ell_1} \Rightarrow \mathrm{f}=\frac{\mathrm{K}}{\ell_1}$ ( $\ell_1 \Rightarrow$ initial length of pipe) $$ \left(\frac{\mathrm{V}}{\mathrm{~V}-\mathrm{V}_{\mathrm{T}}}\right) \mathrm{f}=\frac{\mathrm{k}}{\ell_2}\left\{\mathrm{~V}_{\mathrm{T}} \text { Speed of tuning fork, } \ell_2 \rightarrow \text { new length of pipe }\right\} $$ (1) ÷ (2) $$ \begin{aligned} & \frac{\mathrm{V}-\mathrm{V}_{\mathrm{T}}}{\mathrm{~V}}=\frac{\ell_2}{\ell_1} \\ & \frac{\ell_2}{\ell_1}-1=\frac{\mathrm{V}-\mathrm{V}_{\mathrm{T}}}{\mathrm{~V}}-1 \\ & \frac{\ell_2-\ell_1}{\ell_1}=\frac{-\mathrm{V}_{\mathrm{T}}}{\mathrm{~V}} \\ & \frac{\ell_2-\ell_1}{\ell_1} \times 100=\frac{-2}{320} \times 100=-0.625 \end{aligned} $$ Therefore smallest value of percentage change required in the length of pipe is 0.625