An air column in a pipe, when is closed at one end, is in resonance with a vibrating tuning fork of frequency `264 H_(Z)`. If `upsilon = 330 m//s`, the length of the column in cm is (are) a) 31.25 b) 62.50 c) 93.75 d) 125

Options

A. `31.25`
B. `62.50`
C. `93.75`
D. `125`

Explanation

To solve the problem, we need to determine the length of the air column in a pipe that is closed at one end, which is in resonance with a tuning fork of frequency 264 Hz. The speed of sound in air is given as 330 m/s.

### Step-by-Step Solution:

1. **Understand the Resonance Condition**:
For a pipe closed at one end, the fundamental frequency (first harmonic) occurs when there is one quarter of a wavelength in the pipe. The relationship between frequency (f), speed of sound (v), and wavelength (λ) is given by:
\[ f = \frac{v}{\lambda} \]
For the fundamental mode in a closed pipe:
\[ L = \frac{\lambda}{4} \]

2. **Relate Wavelength to Frequency**:
Rearranging the formula for frequency gives us:
\[ \lambda = \frac{v}{f} \]

3. **Substituting for Wavelength**:
Substitute the expression for wavelength into the length formula:
\[ L = \frac{1}{4} \cdot \frac{v}{f} \]

4. **Plug in the Values**:
Now we can substitute the known values into the equation:
- \( v = 330 \, \text{m/s} \)
- \( f = 264 \, \text{Hz} \)
\[ L = \frac{1}{4} \cdot \frac{330}{264} \]

5. **Calculate the Length**:
First, calculate \( \frac{330}{264} \):
\[ \frac{330}{264} = 1.25 \]
Now, calculate \( L \):
\[ L = \frac{1.25}{4} = 0.3125 \, \text{m} \]

6. **Convert to Centimeters**:
Since the question asks for the length in centimeters, convert meters to centimeters:
\[ L = 0.3125 \, \text{m} \times 100 = 31.25 \, \text{cm} \]

7. **Consider Higher Harmonics**:
For higher harmonics (n = 3), the length can be calculated as:
\[ L = \frac{3 \cdot v}{4f} = \frac{3 \cdot 330}{4 \cdot 264} \]
Calculate:
\[ L = \frac{990}{1056} = 0.9375 \, \text{m} = 93.75 \, \text{cm} \]

### Final Answer:
The lengths of the column in resonance with the tuning fork are:
- **31.25 cm** (for n = 1)
- **93.75 cm** (for n = 3)

Thus, the correct options are:
- a) 31.25 cm
- c) 93.75 cm