A glass tube of `1.0 m` length is filled with water . The water can be drained out slowly at the bottom of the tube . If a vibrating tuning fork of frequency `500 c//s` is brought at the upper end of the tube and the velocity of sound is `330 m//s`, then the total number of resonances obtained will be

Options

A. `4`
B. `3`
C. `2`
D. `1`

Explanation

To find the total number of resonances obtained in a glass tube of length 1.0 m filled with water, we can follow these steps:

### Step 1: Calculate the wavelength (λ) of the sound.
We know the relationship between the velocity of sound (V), frequency (f), and wavelength (λ):
\[ V = f \cdot \lambda \]
Given:
- Velocity of sound, \( V = 330 \, \text{m/s} \)
- Frequency, \( f = 500 \, \text{c/s} \)

Rearranging the formula to solve for λ:
\[ \lambda = \frac{V}{f} = \frac{330 \, \text{m/s}}{500 \, \text{c/s}} = 0.66 \, \text{m} \]

### Step 2: Determine the positions of the resonances.
In a tube closed at one end (the bottom end is closed due to water), the resonances occur at odd multiples of \( \frac{\lambda}{4} \). The positions of the resonances can be calculated as follows:

1. **First resonance (n=1)**:
\[ L_1 = \frac{1}{4} \lambda = \frac{1}{4} \times 0.66 = 0.165 \, \text{m} \]

2. **Second resonance (n=3)**:
\[ L_2 = \frac{3}{4} \lambda = \frac{3}{4} \times 0.66 = 0.495 \, \text{m} \]

3. **Third resonance (n=5)**:
\[ L_3 = \frac{5}{4} \lambda = \frac{5}{4} \times 0.66 = 0.825 \, \text{m} \]

4. **Fourth resonance (n=7)**:
\[ L_4 = \frac{7}{4} \lambda = \frac{7}{4} \times 0.66 = 1.155 \, \text{m} \]

### Step 3: Count the number of resonances within the length of the tube.
The tube length is 1.0 m. We will check how many of the calculated resonance positions fit within this length:

- \( L_1 = 0.165 \, \text{m} \) (fits)
- \( L_2 = 0.495 \, \text{m} \) (fits)
- \( L_3 = 0.825 \, \text{m} \) (fits)
- \( L_4 = 1.155 \, \text{m} \) (does not fit, exceeds 1.0 m)

Thus, the total number of resonances that can occur in the tube is **3**.

### Final Answer:
The total number of resonances obtained is **3**.

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