Calculate the wavelength of sound emitted by a tuning fork which makes 240 vibrations per second at `27^@C.` Velocity of sound in air at `0^@"C is 330 ms"^(-1)`.

Answer

To calculate the wavelength of sound emitted by a tuning fork that makes 240 vibrations per second at 27°C, we can follow these steps:

### Step 1: Identify the given values
- Frequency (f) = 240 Hz
- Temperature (T1) = 27°C
- Velocity of sound at 0°C (V2) = 330 m/s

### Step 2: Convert the temperature to Kelvin
To convert Celsius to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273 \]
So,
\[ T1 = 27 + 273 = 300 \, K \]

### Step 3: Find the temperature at 0°C in Kelvin
For 0°C:
\[ T2 = 0 + 273 = 273 \, K \]

### Step 4: Use the relationship between the velocity of sound and temperature
The velocity of sound in air is directly proportional to the square root of the absolute temperature. The formula is:
\[ \frac{V1}{V2} = \sqrt{\frac{T1}{T2}} \]
Where:
- V1 = velocity of sound at 27°C
- V2 = velocity of sound at 0°C

### Step 5: Rearrange the equation to solve for V1
\[ V1 = V2 \times \sqrt{\frac{T1}{T2}} \]

### Step 6: Substitute the known values
Substituting the values we have:
\[ V1 = 330 \times \sqrt{\frac{300}{273}} \]

### Step 7: Calculate the square root
First, calculate the square root:
\[ \sqrt{\frac{300}{273}} \approx 1.048 \]

### Step 8: Calculate V1
Now, substituting back:
\[ V1 = 330 \times 1.048 \approx 346 \, m/s \]

### Step 9: Calculate the wavelength (λ)
The wavelength (λ) can be calculated using the formula:
\[ \lambda = \frac{V1}{f} \]
Substituting the values:
\[ \lambda = \frac{346}{240} \]

### Step 10: Perform the division
Calculating the wavelength:
\[ \lambda \approx 1.44 \, m \]

### Final Answer
The wavelength of sound emitted by the tuning fork is approximately **1.44 meters**.

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