A 20\" long steel guitar string (linear density = 4.8 x 10 -3 kg/m 3 ) is to be

Question

A 20" long steel guitar string (linear density = 4.8 x 10-3 kg/m3) is to be tuned to 409 Hz.

a) What must the tension be?

b) If the string is not in tune at 409 Hz, and one hears a beat frequency 2 Hz using a tuning fork, what are the possible frequencies the string is currently tuned to?

c) To play the the harmonic f3 of 409 Hz, at what distance from the bridge must the string be fingered?

d) What is the frequency of the sound wave when the tuned string is played at full length?

e) What is the wavelength of the sound wave striking the ear at 22 deg C from the tuned frequency at 409 Hz?

Explanation / Answer

speed v = sqrt(T/u)

a) f = 3V/2L = sqrt(T/u)/2L

T = f^2*4*L^2*u/9 = 409*409*4*0.5*0.5*4.8e-3/9 = 89.21 N

b) f = 409+2 = 411 Hz     or 409-2 - 407 Hz

c) x = L/3 = 0.5/3 = 0.167 m

d) f1 = f3/3 = 136.3 Hz

e) wavelength = 2x = 0.334 m

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