A closed organ pipe and an open organ pipe are tuned to the same fundamental frequency. The ratio of their lengths is

Options

A. `1 : 2`
B. `2 : 1`
C. `1 : 4`
D. `4 : 1`

Explanation

To solve the problem of finding the ratio of the lengths of a closed organ pipe and an open organ pipe that are tuned to the same fundamental frequency, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Fundamental Frequency Formulas**:
- The fundamental frequency \( f_C \) of a closed organ pipe is given by:
\[ f_C = \frac{V}{4L_C} \]
where \( V \) is the speed of sound in air and \( L_C \) is the length of the closed organ pipe.
- The fundamental frequency \( f_O \) of an open organ pipe is given by:
\[ f_O = \frac{V}{2L_O} \]
where \( L_O \) is the length of the open organ pipe.

2. **Set the Frequencies Equal**:
- Since both pipes are tuned to the same fundamental frequency, we can set the two frequency equations equal to each other:
\[ f_C = f_O \]
This gives us:
\[ \frac{V}{4L_C} = \frac{V}{2L_O} \]

3. **Cancel the Speed of Sound**:
- The speed of sound \( V \) is common in both equations, so we can cancel it out:
\[ \frac{1}{4L_C} = \frac{1}{2L_O} \]

4. **Cross-Multiply to Solve for the Ratio**:
- Cross-multiplying gives us:
\[ 2L_O = 4L_C \]
- Rearranging this equation leads to:
\[ \frac{L_C}{L_O} = \frac{1}{2} \]

5. **Conclusion**:
- The ratio of the lengths of the closed organ pipe to the open organ pipe is:
\[ \frac{L_C}{L_O} = \frac{1}{2} \]

### Final Answer:
The ratio of the lengths of the closed organ pipe to the open organ pipe is \( \frac{1}{2} \).

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