A guitar string is 1 meter long and is plucked. It vibrates in its first resonance pattern at a frequency of 256hz. The musician then places a finger on the fret board at 0.5m, and plucks the same string causing it to vibrate in its first resonance pattern. What is the frequency now.
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The first resonance frequency of a string is given by `F = sqrt(T/(m/L))/(2L)` where T is the tension in the string, m refers to the string mass and L is the string length.
When the guitar string that is 1 m long is plucked, the fundamental resonance frequency is 256 Hz. The musician then places a finger on the fret board at 0.5 m , this reduces the string length to 0.5 m.
`256 = sqrt(T/(m/L))/(2L)`
=> `sqrt 1*sqrt(T/m)/(2*1) = 256`
= `sqrt(T/m) = 256*2`
= `sqrt (T/m) = 512`
When the string length is reduced to 0.5 m, the frequency is F = `sqrt(T/(m/0.5))/1`
= `512*sqrt 0.5`
= 362.03 Hz
The frequency of the string when a finger is placed on the fret board is 362.03 Hz
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