On a cello, like most other stringed instruments, the positioning of the fingers
ID: 1707717 • Letter: O
Question
On a cello, like most other stringed instruments, the positioning of the fingers by the player determines the fundamental frequencies of the strings. Suppose that one of the strings on a cello is tuned to play a middle C (262 Hz) when played at its full length. By what fraction must that string be shortened in order to play a note that is the interval of a third higher (namely, an E (330 Hz))?How about a fifth higher or a G (392 Hz)?
note:The speed of a wave on a string is a function of its wavelength and frequency. In this problem, the standing waves are at the fundamental frequency; that is, the only nodes are at the ends of the strings. The speed of a wave on a string is determined by the tension in the string and its mass density. Pressing the string against the neck of the cello does not change the tension in the string appreciably and so you can ignore this very small increase in tension in your solution of the problem.
Explanation / Answer
The fundamental frequency produced by a string be f
Then
f = 1/2L (T/)
Where L is the elngth of the vibrating segment
T is the applied tension
is the linear density of the string
For a given string under the constant tension,
f1/f2 = L2/L1
Where L1 and L2 are the lengths of the vibrating segments for corresponding frequencies
Here the length of the string produciong frequency of 262 Hz is L (say)
Then the length of the string that produces frequency of 330 Hz is
262/330 = L'/L
L' = 0.7939L
Therefore in order to produce a higher frequency E the string must be shortened by a fraction of 0.7939
The length of the string that produces frequency of 392 Hz is
262/392 = L'/L
L' = 0.6683L
Therefore in order to produce a higher frequency G the string must be shortened by a fraction of 0.6683
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