Question

The lowest string of a certain guitar is 64 cm long and has a mass density of 6.3 g/m. The string is fixed at its ends by the bridge and the nut of the guitar.

(a) What tension in the string is required to tune it so that its fundamental frequency matches the E2 musical note at 82.41 Hz? Give an answer in both N and lbs.

(b) The effective string length can be shortened by pressing your finger down on a fret on the neck of the guitar. At what string length should a fret be placed to play the F2 note at 87.31 Hz?

(c) When you pluck the string, it vibrates at the fundamental frequency and many harmonics. These harmonics determine the character of the resulting sound. In an electric guitar the vibrations are detected via magnetic induction in a small pickup coil. The pickup will miss a frequency if the corresponding wave has a node at the pickup location. If the pickup is 8.0 cm from the bridge, will it detect all the harmonics when an E2 note is played? If not, which harmonics between n = 1 and n = 20 will it miss?

Answer 1

fundamental frequency = sqrt(tension / (mass per unit length)) / 2 * length

82.41 = sqrt(tension / 0.006) / (2 * 0.64)

(82.41 * (2 * 0.64))^2 = tension / 0.006

(82.41 * (2 * 0.64))^2 * 0.006 = tension

tension = 66.7622 N

87.31 = sqrt(66.7622/ 0.006) / (2 * length)

length = 0.604 m

n * 82.41 = sqrt(52.158 / 0.006) / (2 * 0.08)

n = 7

it'll miss 7th harmonic