Question

A wire of length 80 cm has a frequency of 256 Hz. Calculate the length of a similar wire under similar tension, which will have frequency 1024 Hz.

Answer 1

We are talking about fundamental frequency of a standing wave on the wire fixed at both ends.
v = speed $=\frac{\sqrt{T}}{\mu }$, $\mu$ = mass per unit length
T = tension force
L = length of wire $=\frac{\lambda }{2}$
Frequency $=\frac{v}{\lambda }=\frac{v}{2L}=\frac{1}{2L}\times \frac{1}{\sqrt{\mu }}\times \sqrt{T}$
$\frac{L_2}{L_1}=\frac{f_1}{f_2}{L}_2=80\times \frac{256}{1024}=20cm$