A wire of length L and radius r is clamped rigidly at one end. When...

Young's modulus is same for both the wires since the material is same.

For the first wire,

$Y = \dfrac{fL}{\pi{r}^{2}l} . . . ( i )$

For the second wire,

$Y = \dfrac{2f ( 2L )}{4\pi{r}^{2}l '} . . . ( ii )$

Dividing (i) by (ii)

$\dfrac{fL}{\pi{r}^{2}l} \times \dfrac{4\pi{r}^{2}l '}{4fL} = 1 ⇒ l ' = l$