Standing waves on a string: Two violinists are trying to tune their instruments in an orchestra. One is producing the desired frequency of 440.0 Hz. The other is producing a frequency of 448.4 Hz. By what percentage should the out-of-tune musician change the tension in his string to bring his instrument into tune at 440.0 Hz?
A) +1.9%
B) -1.9%
C) +3.7%
D) -3.7%
E) +8.4%

Question

Quizplus · Accepted Answer

The frequency produced by a vibrating string is directly proportional to the tension in the string. Therefore, the out-of-tune musician needs to adjust the tension in the string. The frequency ratio between the desired frequency of 440.0 Hz and the current frequency of 448.4 Hz is:

$$\frac{448.4}{440.0} = 1.019$$

To bring the frequency down to 440.0 Hz, the tension in the string needs to be decreased by the same factor of 1.019. The change in tension is therefore:

$$(1-1/1.019)	imes100\% \approx -3.7\%$$

Therefore, the musician needs to decrease the tension in his string by about 3.7% to bring his instrument into tune at 440.0 Hz. The correct answer is (D).

Standing Waves on a String: Two Violinists Are Trying to Tune