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[]{(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* f`

where:

λ = wave length

f = frequency

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

`\lambda* f=\sqrt[]{(F)/(\mu)}`

Now solve for f:

`f=(1)/(\lambda)*\sqrt[]{(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}*(1)/(\lambda)\sqrt[]{(F)/(\mu)} \\ f_(new)=\sqrt[]{2}* f_(old) \\ f_(new)=\sqrt[]{2}*300 \\ f_(new)\approx424Hz \end{gathered}`