A tuning fork of frequency 340 Hz is vibrated just above the tube of 120 cm height. Water is poured slowly in the tube. What is the minimum height of water necessary for the resonance? (speed of sound in the air = 340m/s)

Options

A. 15 cm
B. 25 cm
C. 30 cm
D. 45 cm

Answer

To solve the problem, we need to determine the minimum height of water necessary for resonance in a tube that is open at one end and closed at the other.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a tube of height 120 cm (1.2 m).
- The tuning fork has a frequency of 340 Hz.
- The speed of sound in air is given as 340 m/s.

2. **Determine the Wavelength**:
- The relationship between speed (v), frequency (f), and wavelength (λ) is given by the formula:
\[ v = f \cdot \lambda \]
- Rearranging this gives us:
\[ \lambda = \frac{v}{f} = \frac{340 \, \text{m/s}}{340 \, \text{Hz}} = 1 \, \text{m} \]

3. **Identify the Resonance Condition**:
- For a tube that is open at one end and closed at the other, the fundamental frequency (first harmonic) has a node at the closed end and an antinode at the open end.
- The length of the tube (L) for the fundamental frequency is given by:
\[ L = \frac{\lambda}{4} \]
- Substituting the value of λ:
\[ L = \frac{1 \, \text{m}}{4} = 0.25 \, \text{m} = 25 \, \text{cm} \]

4. **Calculate the Length for Higher Harmonics**:
- The first overtone (third harmonic) has a length given by:
\[ L = \frac{3\lambda}{4} \]
- Substituting the value of λ:
\[ L = \frac{3 \cdot 1 \, \text{m}}{4} = 0.75 \, \text{m} = 75 \, \text{cm} \]

5. **Finding the Height of Water**:
- The total height of the tube is 120 cm. For resonance to occur at the first overtone (75 cm), the height of the water must be:
\[ \text{Height of water} = \text{Total height} - \text{Length of air column} \]
- Thus:
\[ \text{Height of water} = 120 \, \text{cm} - 75 \, \text{cm} = 45 \, \text{cm} \]

### Final Answer:
The minimum height of water necessary for resonance is **45 cm**.

---