Question

An organ pipe closed at one end produces \(5^{\text{th}}\) overtone of an air column. It is found that air in the pipe is vibrating with \(x\) nodes and \(y\) antinodes. The value of \(x\) and \(y\) is

• A. \(x=5\) and \(y=6\)
• B. \(x=6\) and \(y=6\)
• C. \(x=6\) and \(y=5\)
• D. \(x=5\) and \(y=5\)

Hint:

A closed pipe has a node at the closed end; each higher overtone adds one node and one antinode.

Answer 1

The Correct Option is B

Step 1: Harmonic number
Overtones in a closed pipe are the harmonics $3,5,7,9,11,\dots$. The 5th overtone is the 11th harmonic.

Step 2: Formula
For the $(2n+1)^{th}$ harmonic, there are $n+1$ nodes and $n+1$ antinodes. Here $2n+1=11$, so $n=5$.

Step 3: Result
Nodes $=6$ and antinodes $=6$.

Final Answer:
The 11th harmonic has equal numbers of nodes and antinodes, 6 each. \[ \boxed{\text{(B) }x=6\text{ and }y=6} \]