A string in a guitar is made of steel (density `7962 kg//m^(3)`). It is `63.5` cm long, and has diameter of `0.4` mm. The fundamental frequency is `f = 247 Hz`.
(a) Find the string tension (F).
(b) If the tension F is changed by a small amount `DeltaF`, the frequency `f` changes by a small amount `Deltaf`. Show that `(Deltaf)/(f)=1/2 (DeltaF)/(F)`
(c) The string is tuned with tension equal to that calculated in part (a) when its temperature is `18^(@)C`. Continuous playing causes the temperature of the string to rise, changing its vibration frequency. Find `Deltaf` if the temperature of the string rises to `29^(@)C`. The steel string has a Young’s modulus of `2.00 xx 10^(11) Pa` and a coefficient of linear expansion of `1.20 xx 10^(–5) (.^(@)C)^( –1)`. Assume that the temperature of the body of the guitar remains constant. Will the vibration frequency rise or fall?

Answer

(a) 98.4 N
(c) 4.2 Hz decrease.