The G string on an instrument is 48 cm long. When played without fingering, it v

Question

The G string on an instrument is 48 cm long. When played without fingering, it vibrates at a frequency of 196 Hz. The next higher notes on the C-major scale are A (220 Hz), B (247 Hz), C (262 Hz), and D (294 Hz). What distance x from the end of the string must a finger be placed to play each of these notes?

The G string on an instrument is 48 cm long. When played without fingering, it vibrates at a frequency of 196 Hz. The next higher notes on the C-major scale are A (220 Hz), B (247 Hz), C (262 Hz), and D (294 Hz). What distance x from the end of the string must a finger be placed to play each of these notes? XA= cm cm cm xc=1 xD = cm

Explanation / Answer

Given the legnth of the string above (0.48 m), you can figure out the wavelength.
0.48 m = 1/2 wavelength
0.96 m = wavelength
velocity = frequency x wavelength.
Using the frequency given (196 Hz) and the wavelength just computed, you can find the velocity.
velocity = v = 196 Hz x 0.96 m = 188.2 m/s
This speed won't change even as you hold your finger on the string.

Starting with a velocity of 188.2 m/s and a desired frequency, we can find the wavelength.

a) velocity = frequency x wavelength
188.2 m/s = 220 Hz x wavelength
0.855 m = wavelength

Using this wavelength and the fact that the length of the string is 1/2 of a wavelength long, you get:

Length of string = 1/2 wavelength
Length of string = 1/2 (0.855 m)
Length of string = 0.4275 m

distance from end = 0.48 m – 0.4275 m = 0.0525 m or 5.25 cm.

b) velocity = frequency x wavelength
188.2 m/s = 247 Hz x wavelength
0.762 m = wavelength

Using this wavelength and the fact that the length of the string is 1/2 of a wavelength long, you get:

Length of string = 1/2 wavelength
Length of string = 1/2 (0.762 m)
Length of string = 0.381 m

distance from end = 0.48 m – 0.381 m = 0.099 m or 9.9 cm.

c) velocity = frequency x wavelength
188.2 m/s = 262 Hz x wavelength
0.718 m = wavelength

Using this wavelength and the fact that the length of the string is 1/2 of a wavelength long, you get:

Length of string = 1/2 wavelength
Length of string = 1/2 (0.718 m)
Length of string = 0.359 m

distance from end = 0.48 m – 0.359 m = 0.121 m or 12.1 cm.

d) velocity = frequency x wavelength
188.2 m/s = 294 Hz x wavelength
0.64 m = wavelength

Using this wavelength and the fact that the length of the string is 1/2 of a wavelength long, you get:

Length of string = 1/2 wavelength
Length of string = 1/2 (0.64 m)
Length of string = 0.32 m

distance from end = 0.48 m – 0.32 m = 0.16 m or 16.0 cm.

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