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 $l=94 \pm 0.1, r=5 \pm 0.05 .$ 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)

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