The ratio of fundamental frequencies of an open organ pipe and a cloed organ pipe of same length at same temperature is

Options

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

Answer

To find the ratio of fundamental frequencies of an open organ pipe and a closed organ pipe of the same length, we can follow these steps:

### Step 1: Understand the characteristics of the pipes
- An open organ pipe has antinodes at both ends and a node in the middle.
- A closed organ pipe has a node at the closed end and an antinode at the open end.

### Step 2: Determine the wavelength for each pipe
- For the open organ pipe, the fundamental frequency corresponds to a wavelength (λ) that fits two halves of a wave in the length of the pipe:
\[ L = \frac{\lambda_1}{2} \implies \lambda_1 = 2L \]

- For the closed organ pipe, the fundamental frequency corresponds to a wavelength that fits one quarter of a wave in the length of the pipe:
\[ L = \frac{\lambda_2}{4} \implies \lambda_2 = 4L \]

### Step 3: Use the formula for frequency
The frequency (f) of a wave is given by the formula:
\[ f = \frac{v}{\lambda} \]
where \( v \) is the speed of sound.

### Step 4: Calculate the fundamental frequencies
- For the open organ pipe (frequency \( f_1 \)):
\[ f_1 = \frac{v}{\lambda_1} = \frac{v}{2L} \]

- For the closed organ pipe (frequency \( f_2 \)):
\[ f_2 = \frac{v}{\lambda_2} = \frac{v}{4L} \]

### Step 5: Find the ratio of the frequencies
Now, we can find the ratio \( \frac{f_1}{f_2} \):
\[ \frac{f_1}{f_2} = \frac{\frac{v}{2L}}{\frac{v}{4L}} = \frac{v}{2L} \cdot \frac{4L}{v} \]
The \( v \) and \( L \) cancel out:
\[ \frac{f_1}{f_2} = \frac{4}{2} = 2 \]

### Conclusion
The ratio of the fundamental frequencies of the open organ pipe to the closed organ pipe is:
\[ \frac{f_1}{f_2} = 2:1 \]