# 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...

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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### 1 Answer

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`

= `sqrt(0.5)*sqrt(T/m)`

= `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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