A tuning fork of frequency `340 Hz` is excited and held above a cylindrical tube of length `120 cm`. It is slowly filled with water. The minimum height of water column required for resonance to be first heard( Velocity of sound `=340 ms^-1`) is.

Options

A. `(a)25 cm`
B. `(b)95 cm`
C. `(c)75 cm`
D. `(d)45 cm`

Explanation

To solve the problem, we need to find the minimum height of the water column required for resonance to be first heard in a cylindrical tube when a tuning fork of frequency 340 Hz is used. The speed of sound is given as 340 m/s.

### Step-by-Step Solution:

1. **Calculate the Wavelength (λ)**:
The wavelength (λ) can be calculated using the formula:
\[ \lambda = \frac{V}{f} \]
where \( V \) is the velocity of sound and \( f \) is the frequency of the tuning fork.
\[ V = 340 \, \text{m/s}, \quad f = 340 \, \text{Hz} \]
\[ \lambda = \frac{340 \, \text{m/s}}{340 \, \text{Hz}} = 1 \, \text{m} = 100 \, \text{cm} \]

2. **Determine the Position of First Resonance**:
The first resonance in a tube closed at one end occurs at a length of \( \frac{\lambda}{4} \) from the closed end (the water surface).
\[ \text{First resonance position} = \frac{\lambda}{4} = \frac{100 \, \text{cm}}{4} = 25 \, \text{cm} \]

3. **Calculate the Height of Water Column**:
The total length of the tube is 120 cm. To find the height of the water column needed to achieve this resonance, we subtract the resonance position from the total length of the tube:
\[ \text{Height of water column} = \text{Total length of tube} - \text{First resonance position} \]
\[ \text{Height of water column} = 120 \, \text{cm} - 25 \, \text{cm} = 95 \, \text{cm} \]

4. **Conclusion**:
The minimum height of the water column required for the first resonance to be heard is **95 cm**.

### Final Answer:
The minimum height of the water column required for resonance to be first heard is **95 cm**.