Question

An open organ pipe containing air resonates in fundamental mode due to a tuning fork. The measured values of length $\ell $ (in $cm$ ) of the pipe and radius $r$ (in $cm$ ) of the pipe are $\ell =94\pm0.1,r=5\pm0.05.$ The velocity of the sound in air is accurately known. The maximum percentage error in the measurement of the frequency of that tuning fork by this experiment is given by $x\%$ . Find the value of $100x$ .

Answer 1

$f=\frac{v}{2 \left(\right. \ell + 2 e \left.\right)}$ where e $=$ end correction $=0.6r$
$\therefore f=\frac{v}{2 \left(\right. \ell + 2 \times 0 . 6 r \left.\right)}=\frac{v}{2 \left(\right. \ell + 1 . 2 r \left.\right)}$
$\therefore \frac{\Delta f}{f}=\frac{\Delta v}{v}-\frac{\Delta \left(\right. \ell + 1 . 2 r \left.\right)}{\ell + 1 . 2 r}$
$=\frac{\Delta v}{v}-\frac{\Delta \ell + 1 . 2 \Delta r}{\ell + 1 . 2 r}$
here $\frac{\Delta v}{v}=0$ (given) $\frac{\Delta f}{f}\times 100$
$=-\frac{\Delta \ell + 1 . 2 \Delta r}{\ell + 1 . 2 r}\times 100$
for maximum $\%$ error: $\Delta \ell =0.1,\Delta r=0.05$
$\left(\frac{\Delta f}{f} \times 100\right)_{max}=\frac{0 . 1 + 1 . 2 \times 0 . 05}{94 + 1 . 2 \times 5}\times 100$
$=0.16\%$