Discussion
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 \(n\) is given by
Answer: B
The relation is \(\frac{1}{n} = \frac{1}{n_{1}} + \frac{1}{n_{2}} + \frac{1}{n_{3}}\).
For a string under the same tension and with the same mass per unit length, the fundamental frequency is \(n = \frac{1}{2l}\sqrt{\frac{T}{\mu}}\), so \(l = \frac{k}{n}\) with the same constant \(k\) for every segment. Since the lengths add, \(l = l_{1} + l_{2} + l_{3}\):
\[ \frac{k}{n} = \frac{k}{n_1} + \frac{k}{n_2} + \frac{k}{n_3} \]
Dividing by \(k\) gives the answer. Adding the frequencies themselves would make the long string's note higher than each piece's, the reverse of the truth; the square-root forms would apply to a quantity varying as \(\frac{1}{\sqrt{l}}\) or \(\sqrt{l}\).
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