Question

-. A metal wire is 0.400 m long and is under a Tension force of 75.0N. a) What is the mass per unit length if it has a fundamental frequency of 440 Hz? b) How fast does a sound wave travel in the wire? c) What are the frequencies of the 2nd, 3rd, and 7th harmonics? d) What are the wavelengths for the 1st, 2nd, and 7th harmonics? e) What new tension must you exert on the string for the frequency of the 2nd harmonic at 75.0N to become the fundamental frequency? f) Draw a diagram of a standing wave on a string at the third harmonic with all nodes and antinodes clearly labeled.

Answer 1

Length of the wire, L = 0.4 m

Tension in the wire, T = 75 N

a)

fundamental frequency, f1 = 440 Hz

fundamental frequency, f1 = 1/2L x {T/m}^1/2

440 = 1/ 2 x 0.4 x {75/m}^1/2

Mass per unit length of the wire, m = 6.05 x 10^-4 kg/m

b)

the wavelength in fundamental mode, λ = 2L = 2 x 0.4 = 0.8 m

speed of the wave, V = f λ = 440 x 0.8 = 352 m/s

c)

frequency of 2^nd harmonic, f2 = 2f1 = 2 x 440 = 880 Hz

frequency of 3^rd harmonic, f3 = 3f1 = 3 x 440 =1320Hz

frequency of 7^th harmonic, f7 = 7f1 = 7 x 440 = 3038 Hz

d)

wavelength for 1^st harmonic, λ1 = V/f1 = 352/440 = 0.8 m

wavelength for 2^nd harmonic, λ2 = V/f2 = 352/880 = 0.4 m

wavelength for 7^th harmonic, λ7 = V/f7 = 352/3038 = 0.116 m

e)

new fundamental frequency, f1 = 880 Hz

let T1 be the new tension.

fundamental frequency, f1 = 1/2L x {T1/m}^1/2

880 = 1/ 2 x 0.4 x {T1/6.05 x 10^-4 }^1/2

New tension, T1 = 300 N