Tension of a guitar string

Question

A 0.500 m guitar string is placed under a tension of 270 N and the fundamental mode of vibration is at 150 Hz.

a) What is the weight of the string per unit length (µ in kg m^-1)?

b) What tension would need to be applied to the string such that its fundamental mode of vibration is a middle C (440 Hz)?

Answer 1

T/µ = (f₀)² * (2L)²

Where T is tension in Newtons, µ is mass per unit length in kg/m, f₀ is the fundamental frequency in Hz, and L is the length of the string in m.

µ = T / ((f₀)² * (2L)²) = (270 N) / ((150 Hz)² * (2*0.500m)²) = 0.012 kg/m

Given a new f₀, we can use the same equation to find T, now that we know µ.

T = µ[(f₀)² * (2L)²] = (0.012 kg/m) * [(440 Hz)² * (2*0.500m)²] = 2323.2 N

(A quick unrelated note: 440 Hz is typically the 'A' above middle 'C', not middle 'C' itself, which is usually closer to 260 Hz)