At `20^(@)`C, an open organ pipe produces a note of frequency 256 Hz. What will be the length of pipe if velocity of sound at `0^(@)C` is 340 m/s ?

Answer

To solve the problem of finding the length of an open organ pipe that produces a note of frequency 256 Hz at a temperature of 20°C, we can follow these steps:

### Step 1: Determine the velocity of sound at 20°C
The velocity of sound in air increases with temperature. The formula to calculate the velocity of sound at a certain temperature is:

\[ v(T) = v(0) + 0.61 \times T \]

Where:
- \( v(0) \) is the velocity of sound at 0°C (given as 340 m/s).
- \( T \) is the temperature in degrees Celsius (20°C in this case).

Substituting the values:

\[ v(20) = 340 + 0.61 \times 20 \]

Calculating this gives:

\[ v(20) = 340 + 12.2 = 352.2 \, \text{m/s} \]

### Step 2: Use the formula for the fundamental frequency of an open organ pipe
The fundamental frequency \( f \) of an open organ pipe is given by the formula:

\[ f = \frac{v}{L} \]

Where:
- \( f \) is the frequency (256 Hz).
- \( v \) is the velocity of sound (calculated as 352.2 m/s).
- \( L \) is the length of the pipe.

### Step 3: Rearrange the formula to find the length of the pipe
We can rearrange the formula to solve for \( L \):

\[ L = \frac{v}{f} \]

### Step 4: Substitute the known values into the equation
Now, substitute the values of \( v \) and \( f \):

\[ L = \frac{352.2}{256} \]

Calculating this gives:

\[ L \approx 1.375 \, \text{m} \]

### Step 5: Finalize the answer
Thus, the length of the open organ pipe is approximately:

\[ L \approx 1.375 \, \text{m} \]

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