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Consider a string in a guitar whose leng...

Consider a string in a guitar whose length is 80 cm and a mass of 0.32 g with tension 80 N is plucked . Compute the first four lowest frequencies produced when it is plucked .

Text Solution

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The velocity of the wave , `v = sqrt((T)/(mu))`
The length of the string , L = 80 cm = 0.8 m . The mass of the string , `m = 0.32` g = `0.32 xx 10^(-3)` kg
Therefore , the linear mass density , `mu = (0.32 xx 10^(-3))/(0.8) = 0.4 xx 10^(-3) kg m^(-1)`
The tension in the string , T = 80 N
`v = sqrt((80)/(0.4 xx 10^(-3))) = 447.2 ms^(-1)`
The wavelength corresponding to the fundamental frequency `f_(1)` is `lambda_(1) = 2 L = 2 xx 0.8 = 1.6` m
The fundamental frequency `f_(1)` corresponding to the wavelength `lambda_(1)`
`f_(1) = (v)/(lambda_(1)) = (447.2)/(1.6) = 279.5` Hz
Similarly , the frequency corresponding to the second harmonics , third harmonics and fourth harmonics are
`f_(2) = 2f_(1) = 559` Hz
`f_(3) = 3f_(1) = 838.5`Hz
`f_(4) = 4f_(1) = 1118` Hz
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