An open pipe of length `24cm` is in resonance with a frequency `660Hz` in fundamental mode. The radius of pipe is `(V=330ms^(-1))`

Options

A. `3cm`
B. `0.83cm`
C. `3.5cm`
D. `2cm`

Explanation

To solve the problem, we need to find the radius of an open pipe that is in resonance with a given frequency in its fundamental mode. Here’s a step-by-step solution:

### Step 1: Understand the relationship between frequency, wavelength, and speed of sound.
The fundamental frequency of an open pipe can be expressed using the formula:
\[ f = \frac{V}{\lambda} \]
where:
- \( f \) is the frequency (660 Hz),
- \( V \) is the speed of sound in air (330 m/s),
- \( \lambda \) is the wavelength.

### Step 2: Calculate the wavelength.
Rearranging the formula gives us:
\[ \lambda = \frac{V}{f} \]
Substituting the values:
\[ \lambda = \frac{330 \, \text{m/s}}{660 \, \text{Hz}} = 0.5 \, \text{m} \]

### Step 3: Relate the wavelength to the length of the pipe.
For an open pipe, the relationship between the wavelength and the length of the pipe in the fundamental mode is:
\[ \lambda = 2L \]
where \( L \) is the length of the pipe. Given that the length of the pipe is 24 cm (or 0.24 m), we can express this as:
\[ \lambda = 2 \times 0.24 \, \text{m} + 2e \]
where \( e \) is the correction for the end effects.

### Step 4: Express the end correction.
The end correction \( e \) is typically approximated as:
\[ e = 0.6R \]
where \( R \) is the radius of the pipe. Therefore, the equation becomes:
\[ 0.5 = 2 \times 0.24 + 2(0.6R) \]

### Step 5: Simplify the equation.
Substituting the values gives:
\[ 0.5 = 0.48 + 1.2R \]
Now, rearranging the equation to solve for \( R \):
\[ 1.2R = 0.5 - 0.48 \]
\[ 1.2R = 0.02 \]
\[ R = \frac{0.02}{1.2} \]
\[ R = 0.01667 \, \text{m} \]

### Step 6: Convert the radius to centimeters.
To convert meters to centimeters, multiply by 100:
\[ R = 0.01667 \times 100 = 1.667 \, \text{cm} \]

### Step 7: Final answer.
The radius of the pipe is approximately:
\[ R \approx 1.67 \, \text{cm} \]