Question

An open pipe 30 cm long and a closed pipe 23 cm long, both of the same diameter, are each sounding its first overtone and these are in unison. The end correction of these pipes is:

• A. 0.5 cm
• B. 0.3 cm
• C. 1 cm
• D. 1.2 cm

Answer 1

The correct option is C 1 cm

$2{\nu }_o=\frac{2v}{2L_1}$ ( In $I^{st}$ overtone open pipe )
$3{\nu }_c=\frac{3v}{4L_2}$ ( In $I^{st}$ overtone closed pipe )

By problem,
$2{\nu }_o=3{\nu }_c$
$\Rightarrow \frac{2\nu }{2L_1}=\frac{3\nu }{4L_2}$
$\frac{L_1}{L_2}=\frac{4}{3}$

Here $L_1=30+0.6\times 2\times r$ and $L_2=23+0.6\times r$
( $\because$ both have same diameter, radius will be same )
$\Rightarrow \frac{30+0.6\times 2\times r}{23+0.6\times r}=\frac{4}{3}\Rightarrow r=\frac{2}{1.2}$

End correction is $0.6r=0.6\times \frac{2}{1.2}=1cm$.