A violin string has a length of 0.350 m and is tuned to concert G, with fC=392 Hz . Where must t

step 1 We have a string of a violin that gives off a certain frequency with the given length of 0 .35 m

A violin string has a length of 0.350 m and is tuned to...

A violin string has a length of 0.350 $\mathrm{m}$ and is tuned to concert $\mathrm{G},$ with $f_{\mathrm{C}}=392 \mathrm{Hz}$ . Where must the violinist place her finger to play concert $\mathrm{A},$ with $f_{\mathrm{A}}=440 \mathrm{Hz}$ ? If this position is to remain correct to one-half the width of a finger (that is, to within 0.600 $\mathrm{cm}$ ), what is the maximum allowable percentage change in the string tension?

Key Concepts

Frequency-Tension Relationship

Because the frequency of a vibrating string is proportional to the square root of the tension, small changes in frequency correspond to smaller relative changes in tension. Mathematically, if the frequency changes by a small amount, the percentage change in tension can be approximated by twice the percentage change in frequency. This concept is crucial when assessing how precisely the string tension must be controlled in order to maintain the correct pitch.

Vibrating String Fundamentals

This concept refers to the fundamental physics of a vibrating string, where the frequency of vibration is determined by its length, tension, and mass per unit length, typically expressed by the equation f = (1/2L)?(T/?). This relationship is central to understanding how changes in any of these parameters, such as shortening the effective length by finger placement, alter the pitch of the sound produced.

Effective Length Adjustments Through Finger Placement

Instruments such as violins alter pitch by changing the effective length of the string via finger placement. This concept highlights the practical application of the vibrating string equations, where moving a finger on the string changes its length and therefore the frequency of vibration. Additionally, precise tolerance in finger placement is important because even small changes in the effective length can lead to noticeable changes in pitch, underlining the need for accuracy in both physical placement and corresponding tension adjustments.

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