Question

An open organ pipe containing air resonates in fundamental mode due to a tuning fork. The measured values of length $l$ (in $cm$ ) of the pipe and radius $r$ (in $cm$ ) of the pipe are $l=94\pm 0.1,r=5\pm \mathrm{0.05.}$ 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, will be

• A. 0.16
• B. 0.64
• C. 1.2
• D. 1.6

Answer 1

Answer: (A)

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