If the length of a piano wire (of given density) is increased by 5%, what approximate change in tension is necessary to keep its fundamental frequency unchanged?
A) decrease of 10%
B) decrease of 5%
C) increase of 5%
D) increase of 10%
E) increase of 20%

Question

Quizplus · Accepted Answer

The fundamental frequency of a stretched string is given by
f = (1/2L) * sqrt(T/μ)
where L is the length of the string, T is the tension, and μ is the linear density (mass per unit length) of the string.

If we increase the length of the string by 5%, the new length is 1.05L. To keep the frequency unchanged, we need to solve for the new tension T', such that:
f = (1/2)(1.05L)*sqrt(T'/μ)

Since the frequency is unchanged, we can set these two expressions for f equal to each other:
(1/2L) * sqrt(T/μ) = (1/2)(1.05L)*sqrt(T'/μ)
Simplifying this equation gives:
T' = T/1.1025

Therefore, the necessary change in tension is an increase of 5%: (T' - T)/T = (T/1.1025 - T)/T = 0.0479 = 4.79% (approximately 5%).

If the Length of a Piano Wire (Of Given Density)