If you increase the length of a vibrating string while keeping mass per unit length and tension constant, what happens to the fundamental frequency?

If you increase the length of a vibrating string while keeping mass per unit length and tension constant, what happens to the fundamental frequency?

• A. It lowers the frequency.
• B. It raises the frequency.
• C. It stays the same.
• D. It changes randomly.

Answer: (A)

The basic idea is that the fundamental frequency of a stretched string depends on its length, tension, and mass per unit length. The wave speed on the string is v = sqrt(T/μ), and the fundamental wavelength is λ1 = 2L, so f1 = v/λ1 = (1/2L) sqrt(T/μ). If you lengthen the string while keeping T and μ constant, v stays the same but λ1 increases, which makes f1 decrease. In short, a longer string with the same tension and density vibrates more slowly, producing a lower fundamental frequency.

The basic idea is that the fundamental frequency of a stretched string depends on its length, tension, and mass per unit length. The wave speed on the string is v = sqrt(T/μ), and the fundamental wavelength is λ1 = 2L, so f1 = v/λ1 = (1/2L) sqrt(T/μ). If you lengthen the string while keeping T and μ constant, v stays the same but λ1 increases, which makes f1 decrease. In short, a longer string with the same tension and density vibrates more slowly, producing a lower fundamental frequency.