A 0.55-m-long guitar string emits 220 Hz as its fundamental. The mass of the str

Question

A 0.55-m-long guitar string emits 220 Hz as its fundamental. The mass of the string that is vibrating is 1 gram. What is the period of this oscillation? What is the wavelength of this oscillation? What is the wave speed along the string? What is the linear density mu of the string? What is the tension in the string? What is the string's second harmonic (2^nd Mode or 1^st Overtone)? What is the string's third harmonic (3rd mode or 2^nd Overtone)? If you pluck the string when your finger is on the 5th fret (so 3/4 of the original length now vibrates), what fundamental frequency is emitted?

Explanation / Answer

The period of oscillation would be

a) T = 1 / f = 1 / 220 Hz = 0.0045 s

b) Wavelength of the oscillation will be

lambda = 2L

= 2(0.55) = 1.11m

c) V = f(lambda) = 220(1.11) = 242m/s

d) Linear density of the string = mu = m/L = 1g / o.55m = 0.00181 kg / m3

e) V = (T / mu)1/2

T = mu V2 = 0.00181(242)2 = 106.48 N

f) second harmonic will be = 2f1 = 2(220) = 440 Hz , third harmonic = 3f1= 3(220) - 660 Hz

g) lamda =( 3/4) L

f = 4V / 3L = 4(242) / 3(0.55) = 177.46 Hz

  

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