Imagine a guitar strung with Copper (Cu) strings. (Information about Cu can be f
ID: 1624180 • Letter: I
Question
Imagine a guitar strung with Copper (Cu) strings. (Information about Cu can be found on page 1.) (a) A (longitudinal) sound wave in a copper string of radius 0.5 mm travels at 2590 m/s. A transverse wave on the same string travels at 230 m/s. What is the tension in the string? Answer should be a number. (b) Manufacturers of guitar strings usually make strings of different radii to allow players to tunic their instruments with different string tensions. A company offers copper strings of radius 0.16 mm and radius 0.19 mm, both of which are intended to produce a high-E note. When these strings are properly tuned, what is the ratio of the tension in the thinner string to the tension in the thicker string? Answer should be a number.Explanation / Answer
a) vl = sqrt(Y/rho) = 2590 m/s
Y of Cu = 117 GPa
=> rho = Y/vl^2 = 117 * 10^9 / 2590^2 = 17441.6 kg/m^3
A = pi * (0.5*10^-3)^2 = 7.85 * 10^-7 m^2
vt = sqrt (T/m) = 230 m/s where m = M/L = rho * A
=> T = vt^2 * rho * A
=> T = 230^2 * 7.85 * 10^-7 * 17441.6 = 724.3 N
b) Since both strings are tuned to same frequency and have same length
L/v1 = L/v2 where v = sqrt (T/m) and m = M/L = rho*A
=> v1 = v2
=> T1/(rho*A1) = T2/(rho*A2)
T1/T2 = A1/A2 = 0.16^2/0.19^2 = 0.71