A piano wire of length 1.2m and linear density 6.0×10−3kgm−1 is initially under a tension that makes it sound the musical note A (440Hz). The pianist wishes to raise the pitch to 466Hz (one semitone higher) by tightening the wire. The length and mass per unit length remain unchanged.
(i) Derive a general relation showing how the fundamental frequency of a stretched string depends on its tension when the length and mass per unit length are fixed.
(ii) Using the relation obtained in part (i), calculate the percentage increase in tension required to change the frequency from 440Hz to 466Hz.
(iii) If the original tension was 600N, compute the new tension that must be applied to achieve the desired higher frequency.