A wire of length L and radius r is clamped at one end. If its other...

The formula to calculate the Young's modulus of the material of the wire, in the first case, can be written as

$Y = \dfrac{\dfrac{F}{\pi{r}^{2}}}{\dfrac{l}{L}} . . . \left(1\right)$

From equation (1), it follows that

$F = Y\pi{r}^{2} \times \dfrac{l}{L} . . . \left(2\right)$

For the second case, the Young's modulus is given by

$Y = \dfrac{\dfrac{F / 2}{\pi{r}^{2} / 4}}{\dfrac{l '}{L}} . . . \left(3\right)$

From equation (3), it follows that

$F = Y\dfrac{l '}{L} \times 2 \times \dfrac{\pi{r}^{2}}{4} . . . \left(4\right)$

From equations (2) and (4), it implies that

$Y\pi{r}^{2} \times \dfrac{l}{L} = Y\dfrac{l '}{L} \times 2 \times \dfrac{\pi{r}^{2}}{4} ⇒ l ' = 2l$