A violin string is 30 cm long. It sounds the musical note A...

A violin string is $30 \mathrm{cm}$ long. It sounds the musical note
A $(440 \mathrm{Hz})$ when played without fingering. How far from the end of the string should you place your finger to play the note $\mathbf{C}(523 \mathrm{Hz}) ?$

Key Concepts

Fundamental Frequency of a Vibrating String

The fundamental frequency is the lowest frequency at which a string vibrates and is determined by its length, tension, and mass per unit length. In the simplest mode of vibration, the entire string forms a single antinode between fixed endpoints, and this frequency is given by an expression that includes the factor 1/(2L), where L is the effective vibrating length of the string.

Inverse Relationship Between String Length and Frequency

For a string under constant tension with unchanged linear density, the frequency is inversely proportional to the length of the vibrating part of the string. This means that when the effective length is reduced by placing a finger on the string, the vibrating frequency increases, which is why fretting changes the pitch produced by the instrument.

Proportional Reasoning in Adjusting Musical Pitch

Using the direct proportionality between frequency and the reciprocal of the effective length, one can determine the new vibrating length needed for a desired frequency. By comparing the open string frequency with the target frequency, a ratio is established that allows calculation of the effective length required for the new pitch without recalculating the wave speed explicitly.

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