A closed organ pipe of length 1.2 m vibrates in its first overtone mode . The pressue variation is maximum at

Options

A. 0.8m from the open end
B. 0.4m from the open end
C. closed end
D. 1.0m from the open end

Answer

To solve the problem of where the pressure variation is maximum in a closed organ pipe vibrating in its first overtone mode, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Closed Organ Pipe**:
- A closed organ pipe has one end closed and one end open. In such pipes, the closed end is a node (minimum pressure variation) and the open end is an antinode (maximum pressure variation).

2. **Identify the Mode of Vibration**:
- The first overtone mode (also known as the second harmonic) in a closed organ pipe consists of one node at the closed end and one antinode at the open end, with an additional node and antinode in between.

3. **Determine the Length of the Pipe**:
- Given that the length \( L \) of the closed organ pipe is 1.2 m.

4. **Relate the Length to Wavelength**:
- For a closed organ pipe, the relationship between the length of the pipe and the wavelength \( \lambda \) in the first overtone mode is given by:
\[ L = \frac{3\lambda}{4} \]
- This is because in the first overtone mode, there are three-quarters of a wavelength fitting into the length of the pipe.

5. **Calculate the Wavelength**:
- Rearranging the equation gives:
\[ \lambda = \frac{4L}{3} \]
- Substituting \( L = 1.2 \) m:
\[ \lambda = \frac{4 \times 1.2}{3} = \frac{4.8}{3} = 1.6 \text{ m} \]

6. **Determine the Position of Maximum Pressure Variation**:
- In the first overtone mode, the pressure variation is maximum at the antinode, which is located at a distance of \( \frac{1}{4} \lambda \) from the closed end.
- Calculate \( \frac{1}{4} \lambda \):
\[ \frac{1}{4} \lambda = \frac{1.6}{4} = 0.4 \text{ m} \]

7. **Final Answer**:
- Therefore, the pressure variation is maximum at a distance of 0.4 m from the closed end of the pipe.

### Summary:
The pressure variation is maximum at 0.4 m from the closed end of the organ pipe.

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