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 $\ell =94\pm 0.1,r=5\pm \mathrm{0.05.}$ 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

Answer: 16

$f=\frac{v}{2(\ell +2e)}$ where e $=$ end correction $=0.6r$
$\therefore f=\frac{v}{2(\ell +2\times 0.6r)}=\frac{v}{2(\ell +1.2r)}$
$\therefore \frac{\Delta f}{f}=\frac{\Delta v}{v}-\frac{\Delta (\ell +1.2r)}{\ell +1.2r}$
$=\frac{\Delta v}{v}-\frac{\Delta \ell +1.2\Delta r}{\ell +1.2r}$
here $\frac{\Delta v}{v}=0$ (given) $\frac{\Delta f}{f}\times 100$
$=-\frac{\Delta \ell +1.2\Delta r}{\ell +1.2r}\times 100$
for maximum $\%$ error: $\Delta \ell =0.1,\Delta r=0.05$
${\left(\frac{\Delta f}{f}\times 100\right)}_{max}=\frac{0.1+1.2\times 0.05}{94+1.2\times 5}\times 100$
$=0.16\%$