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.

Options

A. 0.012 m
B. 0.025 m
C. 0.05 m
D. 0.024 m

Answer

To solve the problem, we will use the resonance column method and the relationship between the lengths of the air column that resonate in the fundamental mode and the first overtone.

### Step-by-Step Solution:

1. **Identify the lengths of the air column:**
- Fundamental mode length, \( L_1 = 0.1 \, \text{m} \)
- First overtone length, \( L_2 = 0.35 \, \text{m} \)

2. **Understand the resonance conditions:**
- In the fundamental mode, the length of the air column is given by:
\[ L_1 + e = \frac{v}{4f} \]
where \( e \) is the end correction, \( v \) is the speed of sound, and \( f \) is the frequency of the tuning fork.

- In the first overtone, the length of the air column is given by:
\[ L_2 + e = \frac{3v}{4f} \]

3. **Set up the equations:**
- From the fundamental mode:
\[ 0.1 + e = \frac{v}{4f} \quad \text{(1)} \]
- From the first overtone:
\[ 0.35 + e = \frac{3v}{4f} \quad \text{(2)} \]

4. **Subtract equation (1) from equation (2):**
\[ (0.35 + e) - (0.1 + e) = \frac{3v}{4f} - \frac{v}{4f} \]
This simplifies to:
\[ 0.35 - 0.1 = \frac{3v - v}{4f} \]
\[ 0.25 = \frac{2v}{4f} \]
\[ 0.25 = \frac{v}{2f} \quad \text{(3)} \]

5. **Substitute \( v \) from equation (3) back into equation (1):**
From equation (3):
\[ v = 0.5 \times 2f \times 0.25 = 0.5f \]
Substitute \( v \) into equation (1):
\[ 0.1 + e = \frac{0.5}{4f} \quad \text{(4)} \]

6. **Now substitute \( v \) into equation (2):**
\[ 0.35 + e = \frac{3 \times 0.5}{4f} \]
\[ 0.35 + e = \frac{1.5}{4f} \quad \text{(5)} \]

7. **Now we have two equations (4) and (5) to solve for \( e \):**
From equation (4):
\[ e = \frac{0.5}{4f} - 0.1 \]
From equation (5):
\[ e = \frac{1.5}{4f} - 0.35 \]

8. **Set the two expressions for \( e \) equal to each other:**
\[ \frac{0.5}{4f} - 0.1 = \frac{1.5}{4f} - 0.35 \]
Rearranging gives:
\[ 0.35 - 0.1 = \frac{1.5 - 0.5}{4f} \]
\[ 0.25 = \frac{1}{4f} \]
\[ 4f = 4 \quad \Rightarrow \quad f = 1 \, \text{Hz} \]

9. **Substituting \( f \) back to find \( e \):**
Substitute \( f = 1 \) into either equation for \( e \):
\[ e = \frac{0.5}{4 \times 1} - 0.1 = 0.125 - 0.1 = 0.025 \, \text{m} \]

10. **Convert to centimeters:**
\[ e = 0.025 \, \text{m} = 2.5 \, \text{cm} \]

### Final Answer:
The end correction \( e \) is \( 0.025 \, \text{m} \) or \( 2.5 \, \text{cm} \).