A source of frequency 10kHz when vibrted 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 step by step, we will follow the concepts of wave motion and the relationship between frequency, temperature, and the speed of sound.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Frequency of the source (f) = 10 kHz = 10,000 Hz
- Initial temperature (T1) = 300 K
- Change in temperature (ΔT) = 1 K

2. **Understand the Relationship between Frequency and Temperature:**
- The frequency of sound in a closed organ pipe is directly proportional to the square root of the temperature:
\[ f \propto \sqrt{T} \]
- Therefore, we can express the relationship as:
\[ \frac{f'}{f} = \sqrt{\frac{T'}{T}} \]
- Where \(f'\) is the new frequency at temperature \(T'\) and \(f\) is the frequency at temperature \(T\).

3. **Calculate the New Temperature:**
- The new temperature after the increase is:
\[ T' = T1 + \Delta T = 300 K + 1 K = 301 K \]

4. **Calculate the New Frequency (f'):**
- Using the ratio of frequencies:
\[ \frac{f'}{10,000} = \sqrt{\frac{301}{300}} \]
- Squaring both sides gives:
\[ f'^2 = 10,000^2 \cdot \frac{301}{300} \]
- Taking the square root:
\[ f' = 10,000 \cdot \sqrt{\frac{301}{300}} \]

5. **Approximate the Square Root:**
- For small changes, we can use the binomial approximation:
\[ \sqrt{1 + x} \approx 1 + \frac{x}{2} \text{ for small } x \]
- Here, \(x = \frac{1}{300}\):
\[ \sqrt{\frac{301}{300}} \approx 1 + \frac{1}{2 \cdot 300} = 1 + \frac{1}{600} \]
- Therefore:
\[ f' \approx 10,000 \left(1 + \frac{1}{600}\right) = 10,000 + \frac{10,000}{600} = 10,000 + 16.67 \approx 10,016.67 \text{ Hz} \]

6. **Calculate the Beat Frequency:**
- The beat frequency (Δf) is given by the difference between the new frequency and the original frequency:
\[ Δf = f' - f = 10,016.67 - 10,000 = 16.67 \text{ Hz} \]

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