For a certain stretched string, three consecutive resonance frequencies are observed as 105, 175 and 245 Hz respectively. Then, the fundamental frequency is

Options

A. 30 Hz
B. 45 Hz
C. 35 Hz
D. None of these

Explanation

To find the fundamental frequency of the stretched string based on the given resonance frequencies, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Frequencies**:
The three consecutive resonance frequencies are:
- \( f_1 = 105 \, \text{Hz} \)
- \( f_2 = 175 \, \text{Hz} \)
- \( f_3 = 245 \, \text{Hz} \)

2. **Determine the Differences Between Frequencies**:
Calculate the differences between the consecutive frequencies:
- \( f_2 - f_1 = 175 \, \text{Hz} - 105 \, \text{Hz} = 70 \, \text{Hz} \)
- \( f_3 - f_2 = 245 \, \text{Hz} - 175 \, \text{Hz} = 70 \, \text{Hz} \)

This shows that the difference between consecutive resonance frequencies is constant.

3. **Identify the Pattern**:
Since the difference between the consecutive frequencies is constant (70 Hz), we can conclude that these frequencies are harmonics of the fundamental frequency.

4. **Find the Fundamental Frequency**:
The fundamental frequency \( f_0 \) is the lowest frequency, and it can be determined by finding the greatest common divisor (GCD) of the differences:
- The difference (70 Hz) can be expressed as a multiple of the fundamental frequency.
- We can also express the resonance frequencies in terms of the fundamental frequency:
- \( f_1 = 3f_0 \)
- \( f_2 = 5f_0 \)
- \( f_3 = 7f_0 \)

The difference between consecutive harmonics gives us:
- \( f_2 - f_1 = 2f_0 = 70 \, \text{Hz} \)
- Therefore, \( f_0 = \frac{70}{2} = 35 \, \text{Hz} \)

5. **Conclusion**:
The fundamental frequency of the stretched string is \( f_0 = 35 \, \text{Hz} \).

### Final Answer:
The fundamental frequency is \( 35 \, \text{Hz} \).