In a resonance column the first resonant Iength is 0.16 in using a tuning fork of frequency 512 Hz. If the diameter of the tube is 0.025 m, calculate the velocity of sound ?

Answer

To calculate the velocity of sound in a resonance column, we will follow these steps:

### Step 1: Identify the given values
- First resonant length (L) = 0.16 m
- Frequency (f) = 512 Hz
- Diameter of the tube (D) = 0.025 m

### Step 2: Use the formula for the fundamental frequency in a closed tube
The formula for the fundamental frequency when one end of the tube is closed is given by:

\[ f = \frac{V}{4L + 0.4D} \]

Where:
- \( V \) = velocity of sound
- \( L \) = length of the resonance column
- \( D \) = diameter of the tube

### Step 3: Rearrange the formula to solve for V
To find the velocity of sound \( V \), we can rearrange the formula:

\[ V = f \times (4L + 0.4D) \]

### Step 4: Substitute the known values into the equation
Now, substitute the values of \( f \), \( L \), and \( D \) into the rearranged formula:

\[ V = 512 \times (4 \times 0.16 + 0.4 \times 0.025) \]

### Step 5: Calculate the terms inside the parentheses
First, calculate \( 4L \):

\[ 4L = 4 \times 0.16 = 0.64 \]

Next, calculate \( 0.4D \):

\[ 0.4D = 0.4 \times 0.025 = 0.01 \]

Now, add these two results:

\[ 4L + 0.4D = 0.64 + 0.01 = 0.65 \]

### Step 6: Calculate the velocity of sound
Now, substitute this back into the equation for \( V \):

\[ V = 512 \times 0.65 \]

Calculating this gives:

\[ V = 332.8 \text{ m/s} \]

### Step 7: Final calculation
Now, let's calculate the final value:

\[ V = 512 \times 0.65 = 332.8 \text{ m/s} \]

### Conclusion
The velocity of sound in the resonance column is approximately **332.8 m/s**.

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