Question

A metal wire under tension T0 vibrates at its fundamental

frequency. For what tension will the second-harmonic frequency

be the same as the fundamental frequency at tension $T_0$?

Short answer

The tension will be$T=\frac{T_0}{4}$

Step-by-step solution

Frenquency and Tension 

Frequency is proportional to the square root of its tension.

Using formula of frequency  

Tension or T has two nodes on the end of the string that produces the same frequency. It got produced under the frequency $T_0$. the formula of the frequency is

$fm=m\frac{v}{2L}=m\frac{\sqrt{{\displaystyle \frac{T}{\mu }}}}{2L}=m\frac{\sqrt{{\displaystyle T}}}{2L\sqrt{\mu }}$

Step 3:Simplification of the formula 

The substitution of the two cases by equating them is

$1\frac{\sqrt{T_0}}{2L\sqrt{\mu }}=f=2\frac{\sqrt{T}}{2L\sqrt{\mu }}$

The obtained simplification is