Question

The fundamental frequency of a string of length \( l \) is \( n \). The string is cut into 3 parts \( l_1, l_2, \) and \( l_3 \), each having fundamental frequency \( n_1, n_2, n_3 \). Then, what is the relationship between \( n_1, n_2, n_3 \)?

• A. \( n_1 = n_2 = n_3 \)
• B. \( n_1<n_2<n_3 \)
• C. \( n_1>n_2>n_3 \)
• D. \( n_1 = 2n_2 = 3n_3 \)

Hint:

When a string is cut into equal parts, the frequency for each part remains the same because the tension and mass density do not change.

Answer 1

The Correct Option is A

When a string is cut into parts, the fundamental frequency for each part is determined by the length of the string and its tension. For a string of length \( l \), the fundamental frequency \( n \) is related to the length by the equation: \[ n = \frac{1}{2l} \sqrt{\frac{T}{\mu}} \] Where \( T \) is the tension in the string, and \( \mu \) is the linear mass density of the string. Since the tension and mass density do not change when the string is cut into smaller parts, the frequency is inversely proportional to the length of each part. For each part \( l_1, l_2, l_3 \), the fundamental frequency is given by: \[ n_1 \propto \frac{1}{l_1}, \quad n_2 \propto \frac{1}{l_2}, \quad n_3 \propto \frac{1}{l_3} \] If \( l_1 = l_2 = l_3 \), then \( n_1 = n_2 = n_3 \). Thus, the relationship is \( n_1 = n_2 = n_3 \).