How far from the end of the string should you place your finger to play the note C (523 Hz)?

A violin string is 28 cm long. It sounds the musical note A (440 Hz) when played without fingering.

How far from the end of the string should you place your finger to play the note C (523 Hz)?
There are two equations you'll need.





First, when you play a note on a string, the fundamental frequency causes a standing wave such that the length of the string equals 1/2 of a wavelength. So L = 1/2 wavelength.





Given the legnth of the string above (.28 m), you can figure out the wavelength.


.28 m = 1/2 wavelength


.56 m = wavelength





Second, velocity = frequency x wavelength.


Using the frequency given (440 Hz) and the wavelength just computed, you can find the velocity.





velocity = 440 Hz x .56 m = 246.4 m/s





This speed won't change even as you hold your finger on the string.





Starting with a velocity of 246.4 m/s and a desired frequency of 523 Hz, we can find the wavelength.





velocity = frequency x wavelength


246.4 m/s = 523 Hz x wavelength


.471 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 (.471 m)


Length of string = .236 m





Since the question asks how far from the end you should place your finger, take the original string length and subtract the answer above.





distance from end = .28 m - .236 m = .044 m or 4.4 cm.