An air column, closed at one end and open at the other resonates with a tuning fork when the smallest length of the column is 50 cm. The next larger length of the column resonating with the same tuning fork is

Options

A. `66.7 cm `
B. `100 cm`
C. `150 cm`
D. `200 cm`

Explanation

To solve the problem, we need to understand the behavior of an air column that is closed at one end and open at the other. The fundamental frequency (first harmonic) and the higher harmonics (overtones) can be determined based on the length of the air column.

### Step-by-Step Solution:

1. **Identify the Fundamental Frequency:**
- For an air column that is closed at one end and open at the other, the fundamental frequency (first harmonic) corresponds to a length of \( L_1 = \frac{\lambda}{4} \), where \( \lambda \) is the wavelength of the sound wave.
- Given that the smallest length of the column is 50 cm, we have:
\[ L_1 = 50 \, \text{cm} = \frac{\lambda}{4} \]

2. **Calculate the Wavelength:**
- To find the wavelength \( \lambda \), we rearrange the equation:
\[ \lambda = 4L_1 = 4 \times 50 \, \text{cm} = 200 \, \text{cm} \]

3. **Determine the Next Larger Length:**
- The next larger length that resonates with the same tuning fork corresponds to the third harmonic (or first overtone) for a closed-end air column. The length for the third harmonic is given by:
\[ L_3 = \frac{3\lambda}{4} \]
- Substituting the value of \( \lambda \):
\[ L_3 = \frac{3 \times 200 \, \text{cm}}{4} = \frac{600 \, \text{cm}}{4} = 150 \, \text{cm} \]

4. **Conclusion:**
- The next larger length of the column that resonates with the same tuning fork is 150 cm.

### Final Answer:
The next larger length of the column resonating with the same tuning fork is **150 cm**.