A cylindrical tube (L = 129 cm) is in resonance with in tuning fork of frequency 330 Hz. If it is filling by water then to get resonance again, minimum length of water column is `(V_("air") = 330 m//s)` :-

Options

A. 54 cm
B. 60 cm
C. 25 cm
D. 20 cm

Explanation

To solve the problem, we need to determine the minimum length of the water column in a cylindrical tube that is required to achieve resonance with a tuning fork of frequency 330 Hz, given that the tube has a total length of 129 cm and the speed of sound in air is 330 m/s.

### Step-by-Step Solution:

1. **Identify the Type of Pipe**:
The cylindrical tube is closed at one end, which means it behaves like a closed organ pipe. In such pipes, the resonant frequencies occur at odd multiples of the fundamental frequency.

2. **Calculate the Wavelength (λ)**:
The relationship between the speed of sound (v), frequency (f), and wavelength (λ) is given by the formula:
\[ v = f \cdot \lambda \]
Rearranging this gives:
\[ \lambda = \frac{v}{f} \]
Substituting the values:
\[ v = 330 \, \text{m/s}, \quad f = 330 \, \text{Hz} \]
\[ \lambda = \frac{330}{330} = 1 \, \text{m} \]

3. **Determine the Length of the Tube for Resonance**:
For a closed pipe, the first resonance occurs at a length of:
\[ L_1 = \frac{\lambda}{4} = \frac{1}{4} \, \text{m} = 25 \, \text{cm} \]
The second resonance occurs at:
\[ L_2 = \frac{3\lambda}{4} = \frac{3}{4} \, \text{m} = 75 \, \text{cm} \]

4. **Calculate the Minimum Length of Water Column (L')**:
The total length of the tube is given as 129 cm. To find the length of the water column (L'), we subtract the length corresponding to the second resonance from the total length of the tube:
\[ L' = \text{Total Length} - L_2 \]
\[ L' = 129 \, \text{cm} - 75 \, \text{cm} = 54 \, \text{cm} \]

5. **Final Answer**:
The minimum length of the water column required to achieve resonance again is:
\[ L' = 54 \, \text{cm} \]