A string is fixed at both ends and is vibrating at 130 Hz,...

A string is fixed at both ends and is vibrating at 130 Hz, which is its third harmonic frequency. The linear density of the string is $5.6 \times 10^{-3} kg/m,$ and it is under a tension of 3.3 N. Determine the length of the string.

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Key Concepts

Harmonics in Vibrating Strings

When a string is fixed at both ends, it supports standing waves at discrete frequencies known as harmonics. The nth harmonic has a frequency that is n times the fundamental frequency. This concept is crucial because it relates the observed vibration frequency to the length of the string, with the condition that the string length accommodates an integral number of half-wavelengths.

Wave Speed on a String

The speed at which waves travel along a string depends on the physical properties of the string, specifically its tension and linear mass density. This relationship is described by the equation v = ?(T/?), where T is the tension and ? (mu) is the linear mass density. This concept allows one to determine the propagation speed of the wave given the string's characteristics.

Frequency, Wavelength, and Wave Speed Relationship

The fundamental relationship between frequency (f), wavelength (?), and wave speed (v) is given by v = f?. In the context of harmonics on a string, the wavelength is determined by the boundary conditions imposed by the fixed ends. For the nth harmonic, the wavelength can be expressed in terms of the string length, leading to a direct relation between the harmonic frequency, the wave speed, and the physical dimensions of the string.

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