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

Options

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

Answer

To solve the problem, we need to determine the next larger length of the air column that resonates with the same tuning fork after the smallest length of 50 cm.

### Step-by-Step Solution:

1. **Understanding the Resonance in a Closed Air Column:**
- An air column that is closed at one end and open at the other supports standing waves. The closed end is a node (minimum displacement), and the open end is an antinode (maximum displacement).
- The fundamental frequency (first harmonic) corresponds to a quarter of the wavelength fitting into the length of the column.

2. **Finding the Wavelength:**
- Given that the smallest length (L) of the column is 50 cm, we can express this in terms of the wavelength (λ).
- For the fundamental frequency, the relationship is:
\[ L = \frac{\lambda}{4} \]
- Rearranging this gives:
\[ \lambda = 4L \]
- Substituting the value of L:
\[ \lambda = 4 \times 50 \text{ cm} = 200 \text{ cm} \]

3. **Finding the Next Larger Length:**
- The next resonance occurs at the third harmonic (first overtone) for a closed-end air column, which corresponds to three-quarters of a wavelength fitting into the length of the column.
- The relationship for the next larger length (L') is:
\[ L' = \frac{3\lambda}{4} \]
- Substituting the value of λ:
\[ L' = \frac{3 \times 200 \text{ cm}}{4} = \frac{600 \text{ cm}}{4} = 150 \text{ cm} \]

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**.