A tuning fork produces a wave of wavelength 110 cm in air at `O^(0)C`. The wavelength at `25^(@)C` would be

Options

A. 110 cm
B. 115 cm
C. 120 cm
D. 130 cm

Answer

To find the wavelength of the wave produced by a tuning fork at 25°C, given that the wavelength at 0°C is 110 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between wavelength and temperature**:
The speed of sound in air is affected by temperature. The wavelength of a sound wave is related to its speed and frequency by the formula:
\[ v = \lambda \cdot f \]
where \( v \) is the speed of sound, \( \lambda \) is the wavelength, and \( f \) is the frequency.

2. **Recognize that speed is proportional to temperature**:
The speed of sound in air increases with temperature. Specifically, the speed of sound \( v \) is proportional to the square root of the absolute temperature \( T \) (in Kelvin):
\[ v \propto \sqrt{T} \]

3. **Set up the relationship between the two temperatures**:
Let \( \lambda_1 \) be the wavelength at 0°C (which is 110 cm) and \( \lambda_2 \) be the wavelength at 25°C. The corresponding temperatures in Kelvin are:
- \( T_1 = 0°C = 273 \, K \)
- \( T_2 = 25°C = 298 \, K \)

4. **Use the proportionality to relate the wavelengths**:
Since \( \lambda \) is proportional to \( \sqrt{T} \), we can write:
\[ \frac{\lambda_1}{\lambda_2} = \frac{\sqrt{T_1}}{\sqrt{T_2}} \]
Rearranging gives:
\[ \lambda_2 = \lambda_1 \cdot \frac{\sqrt{T_2}}{\sqrt{T_1}} \]

5. **Substitute the known values**:
Plugging in the values:
\[ \lambda_2 = 110 \, \text{cm} \cdot \frac{\sqrt{298}}{\sqrt{273}} \]

6. **Calculate the square roots**:
- \( \sqrt{298} \approx 17.26 \)
- \( \sqrt{273} \approx 16.52 \)

7. **Perform the division**:
\[ \frac{\sqrt{298}}{\sqrt{273}} \approx \frac{17.26}{16.52} \approx 1.04 \]

8. **Calculate \( \lambda_2 \)**:
\[ \lambda_2 \approx 110 \, \text{cm} \cdot 1.04 \approx 114.4 \, \text{cm} \]

9. **Final result**:
Rounding to a reasonable precision, we find:
\[ \lambda_2 \approx 115 \, \text{cm} \]

### Final Answer:
The wavelength at 25°C would be approximately **115 cm**.