Question

A 2.0 g string, 0.67 m long, is under tension. The string produces a 700 Hz tone when it vibrates in the third harmonic. The speed of sound in air is 344 m/s. The wavelength of the tone in air, in SI units, is closest to ______.
0.57
0.55
0.61
0.49
0.45

Answer 1

Answer:

0.49

Explanation:

The fundamental frequency in string is expressed as Fo = V/2L where;

Fo is the fundamental frequency

V is the speed of sound in air

L is the length of the string

Since harmonics are integral multiples of the fundamental frequency, the first harmonic F1 = 2fo, second harmonic F2 = 3fo and third harmonics f3 = 4fo

Since the string produces a 700 Hz tone when it vibrates in the third harmonic, this means that F3 = 700Hz

From F3 = 4fo... 1

Substituting Fo = V/2L in equation 1, this will give;

F3 = 4(V/2L)

F3 = 4V/2L

Also note that the length of the string in terms of wavelength in the third harmonics is L = 2¶ where ¶ is the wavelength.

F3 = 4V/2(2¶)

F3 = 4V/4¶

F3 = V/¶

¶ = V/F3

Given V = 344m/s and F3 = 700Hz

¶ = 344/700

¶ = 0.49

The wavelength of the tone in air is closest to 0.49