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.
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