Question

A pipe of length $l_1$ closed at one end is kept in a chamber of gas of density $p_1$. A second pipe open at both ends is placed in the second chamber of gas of density $p_2$. The compressibility of both the gases is equal. Calculate the length of the second pipe if the frequency of the first overtone in both the cases is equal.

• A. $\frac{4}{3}{l}_1\sqrt{\frac{p_2}{p_1}}$
• B. $\frac{4}{3}{l}_1\sqrt{\frac{p_1}{p_2}}$
• C. $l_1\sqrt{\frac{p_2}{p_1}}$
• D. $l_1\sqrt{\frac{p_1}{p_2}}$

Answer 1

Answer: (B)

$\frac{4}{3}{l}_1\sqrt{\frac{p_1}{p_2}}$

Frequency of $I^{st}$ of over tone of closed pipe = Frequency of $I^{st}$ of over tone of open pipe

$\Rightarrow \frac{3v}{4l_1}=\frac{v}{l_2}$

$\Rightarrow \frac{3}{4l_1}\sqrt{\frac{\gamma P}{P_1}}=\frac{1}{l_2}\sqrt{\frac{\gamma P}{P_2}}$

$\therefore l_2=\frac{4}{3}{l}_1\sqrt{\frac{P_1}{P_2}}$