If the length of a piano wire (of given density) is increased by 5%, what approximate change in tension?

Question

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) Increase of 10%
B) Increase of 5%
C) Decrease of 5%
D) Decrease of 10%
E) Decrease of 20%

Answer 1

To keep the pitch the same, the speed of propagation for a wave in the wire must be increased by 5% while maintaining the same density (mass per unit length) for the wire. Because the velocity of propagation v = √(T / ρ), the tension T = ρv² and if v increases by 5%, v² will increase by about 10%: more precisely, by the fraction 1.05² - 1.