A source of frequency 10kHz when viberted over than mouth of a closed organ is in unison at 300K. The beats produced when temperature rises by 1K

Options

A. 30Hz
B. 13.33Hz
C. 16.67Hz
D. 40Hz

Answer

To solve the problem of finding the beat frequency produced when the temperature rises by 1K, we will follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between temperature and frequency:**
The frequency of sound in a closed organ pipe is affected by the temperature of the medium. The speed of sound in air increases with temperature, which in turn affects the frequency.

2. **Use the formula for the speed of sound:**
The speed of sound (v) in air can be approximated by the formula:
\[ v = v_0 \sqrt{\frac{T}{T_0}} \]
where \( v_0 \) is the speed of sound at a reference temperature \( T_0 \) (usually 273 K), and \( T \) is the temperature in Kelvin.

3. **Relate speed, frequency, and wavelength:**
The relationship between speed (v), frequency (f), and wavelength (λ) is given by:
\[ v = f \cdot \lambda \]
Since the wavelength (λ) remains constant in this case (as the medium does not change), any change in speed will result in a change in frequency.

4. **Differentiate the frequency with respect to temperature:**
From the relationship of frequency with temperature, we can derive:
\[ \frac{dF}{dT} = \frac{F}{2T} \]
where \( F \) is the initial frequency and \( T \) is the initial temperature.

5. **Calculate the change in frequency (ΔF) for a temperature change (ΔT):**
Given:
- Initial frequency \( F = 10 \, \text{kHz} = 10 \times 10^3 \, \text{Hz} \)
- Initial temperature \( T = 300 \, \text{K} \)
- Change in temperature \( ΔT = 1 \, \text{K} \)

Substitute these values into the differentiated equation:
\[ \Delta F = \frac{F}{2T} \cdot \Delta T = \frac{10 \times 10^3}{2 \times 300} \cdot 1 \]

6. **Perform the calculation:**
\[ \Delta F = \frac{10 \times 10^3}{600} = \frac{10000}{600} = \frac{50}{3} \approx 16.67 \, \text{Hz} \]

7. **Conclusion:**
The beat frequency produced when the temperature rises by 1K is approximately 16.67 Hz.

### Final Answer:
The beat frequency produced when the temperature rises by 1K is approximately **16.67 Hz**.