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

Frequency of the transverse waves produced in the string $\nu =\frac{1}{2L}\sqrt{\frac{T}{\mu }}$

where $L$ is the length of string, $T$ is the tension in the wire and $\mu$ is the mass per unit length of wire.

Since the second wire is of same material, so $\mu$ remains the same. Also, tension in the wire remains the same (given).

$⟹\nu \propto \frac{1}{L}$

SO, we get $\frac{{\nu }_1}{{\nu }_2}=\frac{L_2}{L_1}$

Given : $L_1=80cm$ and ${\nu }_1=256Hz$

Let the length of second wire be $L_2$.

Frequency of wave in second wire ${\nu }_2=1024Hz$

$\therefore$ $\frac{256}{1024}=\frac{L_2}{80}$

$⟹L_2=20cm$