Question

A wire under tension vibrates with a frequency of 550Hz; what would be the
fundamental frequency if the wire were half as long, twice as thick and under
vibration in pipes.

Answer 1

Answer on Question #72316, Physics / Mechanics — Relativity

Question A wire under tension vibrates with a frequency of 550Hz; what would be the fundamental frequency if the wire were half as long, twice as thick and under vibration in pipes.

Solution The formula for fundamental frequency is

$f_{1}=\frac{\sqrt{TL/m}}{2L}=\frac{1}{2}\sqrt{T/mL}$

where $L$ is length of string $M$ is its mass and $T$ is tension. With constant $T$ we have 2 times smaller $L$ and 2 times bigger mass, as it increases in 4 times with increasing thickness in 2 times and decreasing 2 times with decreasing length in 2 times. Hence, we have

$f^{\prime}_{1}=\frac{1}{2}\sqrt{\frac{T}{2mL/2}}=f_{1}$

As we can see, fundamental frequency will not change.