Question

A string under a tension of 170 N has a frequency of 300 Hz.What will its frequency become if the tension is increased to 340 N?

Answer 1

The speed of the wave on a string is given by Taylor's formula:

$v=\sqrt[]{\frac{F}{\mu}}$

where

F = tension force

μ = linear density = mass per unit length

But also we can say the speed of any wave is given by:

$v=\lambda\times f$

where:

λ = wave length

f = frequency

Plug the second equation in the first one. We get:

$\lambda\times f=\sqrt[]{\frac{F}{\mu}}$

Now solve for f:

$f=\frac{1}{\lambda}\times\sqrt[]{\frac{F}{\mu}}$

Lets say wave length is the same on the second case. Since it's the same string μ will also be the same.

See that 340 N = 2 x 170, so we can write:

$\begin{gathered} f_{new}=\sqrt[]{2}\times\frac{1}{\lambda}\sqrt[]{\frac{F}{\mu}} \\ f_{new}=\sqrt[]{2}\times f_{old} \\ f_{new}=\sqrt[]{2}\times300 \\ f_{new}\approx424Hz \end{gathered}$