An open organ pipe produces a note of frequency 512 Hz at `15^@C.` Calculate the length of the pipe. Velocity of sound at `0^@C" is 335 ms"^(-1).`

Answer

To solve the problem of finding the length of an open organ pipe that produces a note of frequency 512 Hz at a temperature of 15°C, we will follow these steps:

### Step 1: Calculate the velocity of sound at 15°C
The formula for the velocity of sound at a temperature \( t \) is given by:
\[ v_t = v_{0} + 0.61 \times t \]
where:
- \( v_{0} \) is the velocity of sound at 0°C (335 m/s),
- \( t \) is the temperature in degrees Celsius (15°C).

Substituting the values:
\[ v_{15} = 335 + 0.61 \times 15 \]
Calculating:
\[ v_{15} = 335 + 9.15 = 344.15 \, \text{m/s} \]

### Step 2: Use the frequency to find the wavelength
For an open organ pipe, the relationship between the frequency \( f \), velocity \( v \), and wavelength \( \lambda \) is given by:
\[ f = \frac{v}{\lambda} \]
Rearranging this gives:
\[ \lambda = \frac{v}{f} \]
Substituting the values:
\[ \lambda = \frac{344.15}{512} \]
Calculating:
\[ \lambda \approx 0.672 \, \text{m} \]

### Step 3: Relate the wavelength to the length of the pipe
For an open organ pipe, the length \( L \) of the pipe is related to the wavelength by:
\[ L = \frac{\lambda}{2} \]
Substituting the value of \( \lambda \):
\[ L = \frac{0.672}{2} \]
Calculating:
\[ L \approx 0.336 \, \text{m} \]

### Final Answer
The length of the open organ pipe is approximately \( 0.336 \, \text{m} \).

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