Question

A pipe closed at end vibrating in the fourth overtone is in unison with the open pipe vibrating in its fourth overtone. The ratio of \(l_{c}:l_{o}\) is {\(l_{c}\) = vibrating length of closed pipe, \(l_{o}\) = vibrating length of open pipe}

• A. \(5:6\)
• B. \(7:8\)
• C. \(9:10\)
• D. \(11:12\)

Hint:

The fourth overtone is the 9th harmonic in a closed pipe and the 5th harmonic in an open pipe.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
A closed pipe gives only odd harmonics, so its 4th overtone is the 9th harmonic: \(f_c=\dfrac{9v}{4l_c}\).

Step 2: Open pipe
An open pipe gives all harmonics, so its 4th overtone is the 5th harmonic: \(f_o=\dfrac{5v}{2l_o}\).

Step 3: Calculation
Unison means \(f_c=f_o\): \(\dfrac{9}{4l_c}=\dfrac{5}{2l_o}\).

Step 4: Result
\(\dfrac{l_c}{l_o}=\dfrac{9\times2}{4\times5}=\dfrac{9}{10}\).

Final Answer:
The ratio of lc to lo is 9 : 10. \[ \boxed{\text{(C) }9:10} \]