Question

In an organ pipe closed at one end produces a fundamental note of frequency '\(ν\)'. The pipe is cut into two pipes of equal length. The fundamental frequencies produced in the two pipes are

• A. \(ν\) , \(2ν\)
• B. \(\frac{ν}{2}\) , \(ν\)
• C. \(2ν\) , \(4ν\)
• D. \(\frac{ν}{2}\) , \(2ν\)

Hint:

A closed pipe has fundamental v/4L. Cut in half, one half is still closed and the other is open at both ends.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
A pipe closed at one end has a node at the closed end and an antinode at the open end. Its fundamental frequency is \(\nu = \dfrac{v}{4L}\). An open pipe of length \(l\) has a fundamental frequency \(\dfrac{v}{2l}\).

Step 2: Key Formula or Approach:
Cut the pipe into two halves, each of length \(\dfrac L2\).

Step 3: Detailed Explanation:
The first half still has the original closed end, so it is a closed pipe of length \(\dfrac L2\):
\[ \nu_1 = \frac{v}{4(L/2)} = \frac{v}{2L} = 2\nu \]
The second half has two open ends, so it is an open pipe of length \(\dfrac L2\):
\[ \nu_2 = \frac{v}{2(L/2)} = \frac vL = 4\nu \]
The fundamental frequencies are \(2\nu\) and \(4\nu\). Options (A), (B) and (D) all contain a frequency of \(\nu\), or \(\nu/2\), which could only be right if the length had not been shortened.

Final Answer:
The two pipes give \(2\nu\) and \(4\nu\), option (C). \[ \boxed{2\nu,\ 4\nu \text{ (C)}} \]