Question

A string of length \( l \) is divided into three segments of lengths \( l_1, l_2 \) and \( l_3 \) with the fundamental frequencies \( n_1, n_2 \) and \( n_3 \) respectively. The original fundamental frequency of the string is given by

• A. \( n = n_1 + n_2 + n_3 \)
• B. \( \frac{1}{n} = \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3} \)
• C. \( \sqrt{n} = \sqrt{n_1} + \sqrt{n_2} + \sqrt{n_3} \)
• D. \( \frac{1}{\sqrt{n}} = \frac{1}{\sqrt{n_1}} + \frac{1}{\sqrt{n_2}} + \frac{1}{\sqrt{n_3}} \)
• E. \( n = n_1 n_2 n_3 \)

Hint:

Frequency inversely proportional to length in strings.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The fundamental frequency (`n`) of a vibrating string is determined by its length (`l`), tension (`T`), and linear mass density (\(\mu\)). The relationship shows that frequency is inversely proportional to length, assuming tension and density are constant.
Step 2: Key Formula or Approach:
The formula for the fundamental frequency of a string is: \[ n = \frac{1}{2l}\sqrt{\frac{T}{\mu}} \] Since the tension `T` and linear mass density \(\mu\) are the same for the original string and its segments, we can say that the product of frequency and length is a constant: \[ nl = \text{constant} = k \] We are given that the total length \(l\) is the sum of the lengths of the segments: \[ l = l_1 + l_2 + l_3 \] Step 3: Detailed Explanation:
From the relationship \(nl = k\), we can express the length of each segment in terms of its frequency:
For the original string: \(l = \frac{k}{n}\)
For the first segment: \(l_1 = \frac{k}{n_1}\)
For the second segment: \(l_2 = \frac{k}{n_2}\)
For the third segment: \(l_3 = \frac{k}{n_3}\)
Now, substitute these expressions back into the length summation equation: \[ l = l_1 + l_2 + l_3 \] \[ \frac{k}{n} = \frac{k}{n_1} + \frac{k}{n_2} + \frac{k}{n_3} \] Since `k` is a non-zero constant, we can divide the entire equation by `k`: \[ \frac{1}{n} = \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3} \] This is the required relationship.
Step 4: Final Answer:
The relationship between the frequencies is \(\frac{1}{n} = \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3}\).