Two identical flutes produce fundamental notes of frequency `300Hz` at `27^@C`. If the temperature of air in one flute is increased to `31^@C`, the number of the beats heard per second will be

Options

A. 3
B. 2
C. 1
D. 4

Answer

To solve the problem, we need to find the number of beats heard per second when the temperature of the air in one flute is increased from \(27^\circ C\) to \(31^\circ C\).

### Step-by-Step Solution:

1. **Understanding the relationship between frequency and temperature**:
The speed of sound in air increases with temperature. The frequency of a sound wave is given by the formula:
\[ v = n \cdot \lambda \]
where \(v\) is the speed of sound, \(n\) is the frequency, and \(\lambda\) is the wavelength.

2. **Finding the speed of sound at different temperatures**:
The speed of sound in air can be approximated by the formula:
\[ v = \sqrt{\frac{\gamma R T}{M}} \]
where \(T\) is the absolute temperature in Kelvin. The speed of sound is directly proportional to the square root of the temperature:
\[ v \propto \sqrt{T} \]

3. **Converting temperatures to Kelvin**:
Convert the given temperatures from Celsius to Kelvin:
- \(T_1 = 27^\circ C = 27 + 273 = 300 \, K\)
- \(T_2 = 31^\circ C = 31 + 273 = 304 \, K\)

4. **Calculating the change in frequency**:
Since the frequency is related to the speed of sound, we can express the ratio of the speeds at the two temperatures:
\[ \frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}} = \sqrt{\frac{304}{300}} \]

5. **Calculating the new frequency**:
The initial frequency \(n_1\) is given as \(300 \, Hz\). Therefore, the new frequency \(n_2\) can be calculated as:
\[ n_2 = n_1 \cdot \frac{v_2}{v_1} = 300 \cdot \sqrt{\frac{304}{300}} \]

6. **Calculating the approximate value**:
To simplify the calculation:
\[ \sqrt{\frac{304}{300}} \approx \sqrt{1.0133} \approx 1.00665 \]
Thus,
\[ n_2 \approx 300 \cdot 1.00665 \approx 301.995 \, Hz \approx 302 \, Hz \]

7. **Finding the number of beats**:
The number of beats per second is given by the absolute difference in frequencies of the two flutes:
\[ \text{Beats} = |n_2 - n_1| = |302 - 300| = 2 \, Hz \]

### Final Answer:
The number of beats heard per second will be \(2 \, Hz\).