In the experiment for the determination of the speed of sound in air using the resonance column method, the length of the air column that resonates in the fundamental mode, with a tuning fork is `0.1m`. When this length is changed to `0.35m`, the same tuning fork resonates with the first overtone. Calculate the end correction.

Explanation

To solve the problem of determining the end correction in the resonance column method for the speed of sound in air, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Length of the air column in the fundamental mode, \( L_1 = 0.1 \, \text{m} \)
- Length of the air column in the first overtone, \( L_2 = 0.35 \, \text{m} \)

2. **Understand the Resonance Conditions:**
- For the fundamental mode, the relationship between the speed of sound \( V \), the length of the air column \( L_1 \), and the end correction \( e \) is given by:
\[ f = \frac{V}{4(L_1 + e)} \]
- For the first overtone, the relationship is:
\[ f = \frac{3V}{4(L_2 + e)} \]

3. **Set Up the Equations:**
- From the fundamental mode:
\[ f = \frac{V}{4(0.1 + e)} \]
- From the first overtone:
\[ f = \frac{3V}{4(0.35 + e)} \]

4. **Equate the Two Frequencies:**
- Since both expressions represent the same frequency \( f \), we can set them equal to each other:
\[ \frac{V}{4(0.1 + e)} = \frac{3V}{4(0.35 + e)} \]

5. **Eliminate \( V \) from the Equation:**
- Cancel \( V \) from both sides (assuming \( V \neq 0 \)):
\[ \frac{1}{4(0.1 + e)} = \frac{3}{4(0.35 + e)} \]

6. **Cross-Multiply to Solve for \( e \):**
- Cross-multiplying gives:
\[ 1 \cdot (0.35 + e) = 3 \cdot (0.1 + e) \]
- This simplifies to:
\[ 0.35 + e = 0.3 + 3e \]

7. **Rearrange the Equation:**
- Rearranging gives:
\[ 0.35 - 0.3 = 3e - e \]
- Simplifying further:
\[ 0.05 = 2e \]

8. **Solve for \( e \):**
- Dividing both sides by 2:
\[ e = \frac{0.05}{2} = 0.025 \, \text{m} \]

### Final Answer:
The end correction \( e \) is \( 0.025 \, \text{m} \).