If $n_1, n_2 and n_3$ are the fundamental frequencies of three segments into which a string is divided, then the original fundamental frequency 'n' of the string is given by following :

Question

Accepted Answer

To understand why Option 2 is the correct answer, we need to delve into the physics of vibrating strings and how their fundamental frequencies relate when a string is divided into segments.

### Explanation for Option 2:

The fundamental frequency of a string is determined by its length, tension, and mass per unit length. When a string is divided into segments, each segment can be considered as an independent vibrating string with its own fundamental frequency. The relationship between the original string and its segments can be understood through the concept of harmonic frequencies.

1. **Fundamental Frequency and Length**: The fundamental frequency $ n $ of a string is inversely proportional to its length $ L $. When a string is divided into segments, the length of each segment determines its fundamental frequency. If the original string is divided into three segments with fundamental frequencies $ n_1, n_2, $ and $ n_3 $, the relationship between these frequencies and the original frequency $ n $ is given by the harmonic series.

2. **Harmonic Series**: The harmonic series for a string implies that the reciprocal of the fundamental frequency of the entire string is the sum of the reciprocals of the fundamental frequencies of its segments. This is because the segments are effectively in series with respect to their vibrational modes.

Therefore, the correct relationship is:
$$
\frac{1}{n} = \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3}
$$
This is precisely what Option 2 states, making it the correct answer.

### Explanation for Why Other Options are Incorrect:

1. **Option 1: $ n = n_1 + n_2 + n_3 $**

This option suggests that the original frequency is simply the sum of the frequencies of the segments. This is incorrect because frequencies do not add linearly when segments are in series. The fundamental frequency of a string is not a direct sum of the frequencies of its parts.

2. **Option 3: $ \sqrt{n} = \sqrt{n_1} + \sqrt{n_2} + \sqrt{n_3} $**

This option implies a square root relationship between the frequencies, which does not align with the physics of wave propagation in strings. The square root relationship is not applicable in this context, as it does not reflect the harmonic nature of the frequencies.

3. **Option 4: $ \frac{1}{\sqrt{n}} = \frac{1}{\sqrt{n_1}} + \frac{1}{\sqrt{n_2}} + \frac{1}{\sqrt{n_3}} $**

This option suggests a reciprocal square root relationship, which is not supported by the principles of wave mechanics in strings. The correct relationship involves the reciprocals of the frequencies themselves, not their square roots.

In conclusion, Option 2 correctly represents the relationship between the fundamental frequency of the original string and the frequencies of its segments, based on the harmonic series and the physics of vibrating strings.