An air column in pipe, which is closed at one end, will be in resonance with a vibrating tuning fork of frequency `264 Hz` if the length of the column in `cm` is :

Options

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

Answer

To solve the problem of finding the length of an air column in a pipe that is closed at one end and resonates with a tuning fork of frequency 264 Hz, we can follow these steps:

### Step 1: Understand the Resonance Condition
For a pipe closed at one end, the fundamental frequency (first harmonic) is given by the formula:
\[ f = \frac{n \cdot v}{4L} \]
where:
- \( f \) = frequency of the tuning fork (in Hz)
- \( n \) = harmonic number (for closed pipes, \( n \) can be 1, 3, 5, ...)
- \( v \) = speed of sound in air (approximately 330 m/s)
- \( L \) = length of the air column (in meters)

### Step 2: Rearrange the Formula
We need to rearrange the formula to solve for \( L \):
\[ L = \frac{n \cdot v}{4f} \]

### Step 3: Substitute Known Values
Substituting the known values into the formula:
- \( f = 264 \) Hz
- \( v = 330 \) m/s

Thus, we have:
\[ L = \frac{n \cdot 330}{4 \cdot 264} \]

### Step 4: Calculate the Length for \( n = 1 \)
First, let's calculate \( L \) for the fundamental frequency (\( n = 1 \)):
\[ L = \frac{1 \cdot 330}{4 \cdot 264} = \frac{330}{1056} \approx 0.3125 \text{ m} \]
To convert this to centimeters:
\[ L \approx 0.3125 \times 100 = 31.25 \text{ cm} \]

### Step 5: Calculate Length for Higher Odd Harmonics
Next, we can calculate the lengths for higher odd harmonics:
1. For \( n = 3 \):
\[ L = \frac{3 \cdot 330}{4 \cdot 264} = 3 \cdot 0.3125 \approx 0.9375 \text{ m} = 93.75 \text{ cm} \]
2. For \( n = 5 \):
\[ L = \frac{5 \cdot 330}{4 \cdot 264} = 5 \cdot 0.3125 \approx 1.5625 \text{ m} = 156.25 \text{ cm} \]

### Step 6: Final Result
The lengths of the air column that will resonate with the tuning fork are:
- For \( n = 1 \): \( 31.25 \) cm
- For \( n = 3 \): \( 93.75 \) cm
- For \( n = 5 \): \( 156.25 \) cm

Thus, the possible lengths of the column in resonance with the tuning fork of frequency 264 Hz are \( 31.25 \) cm and \( 93.75 \) cm.