A pipe 1 m length is closed at one end. Taking the speed of sound in air as `320ms ^(-1)` .the air column in the pipe cannot resonate for the frequency (in Hz )

Options

A. 80
B. 160
C. 240
D. 560

Answer

To solve the problem, we need to find the frequencies at which the air column in a closed pipe of length 1 meter cannot resonate.

### Step-by-Step Solution:

1. **Understanding Resonance in Closed Pipes**:
- A pipe closed at one end resonates at odd harmonics. The fundamental frequency and its harmonics can be represented as:
\[ L = \frac{(2n + 1) \lambda}{4} \]
where \( L \) is the length of the pipe, \( \lambda \) is the wavelength, and \( n \) is a non-negative integer (0, 1, 2, ...).

2. **Finding the Wavelength**:
- Rearranging the formula for wavelength, we get:
\[ \lambda = \frac{4L}{2n + 1} \]

3. **Relating Frequency to Wavelength**:
- The frequency \( f \) is related to the speed of sound \( c \) and wavelength \( \lambda \) by the equation:
\[ f = \frac{c}{\lambda} \]
- Substituting the expression for \( \lambda \):
\[ f = \frac{c}{\frac{4L}{2n + 1}} = \frac{c(2n + 1)}{4L} \]

4. **Substituting Known Values**:
- Given that the speed of sound \( c = 320 \, \text{m/s} \) and the length of the pipe \( L = 1 \, \text{m} \):
\[ f = \frac{320(2n + 1)}{4 \times 1} = 80(2n + 1) \]

5. **Calculating Frequencies**:
- For \( n = 0 \):
\[ f = 80(2 \times 0 + 1) = 80 \, \text{Hz} \]
- For \( n = 1 \):
\[ f = 80(2 \times 1 + 1) = 240 \, \text{Hz} \]
- For \( n = 2 \):
\[ f = 80(2 \times 2 + 1) = 400 \, \text{Hz} \]
- For \( n = 3 \):
\[ f = 80(2 \times 3 + 1) = 560 \, \text{Hz} \]

6. **Identifying Non-Resonant Frequencies**:
- The air column in the pipe cannot resonate at frequencies that are not of the form \( 80(2n + 1) \). Therefore, any frequency that is not equal to 80 Hz, 240 Hz, 400 Hz, or 560 Hz will not resonate.

### Conclusion:
The air column in the pipe cannot resonate at any frequency that is not of the form \( 80(2n + 1) \). Thus, the frequencies that the air column cannot resonate at are all other frequencies not equal to 80 Hz, 240 Hz, 400 Hz, or 560 Hz.