Question

A wire of length L and radius r is elongated by l in length when subjected to extension force F. What will be the radius of the wire of same material of length 2L and extension force 2F so to have the same elongation?

Answer 1

Let Y be the young modulus of material.
Young modulus remains same if force 2F is applied on wire of length 2l of same material.

$$\begin{gathered} Y=\frac{F/{\mathrm{\pi r}}^2}{l/L}=\frac{FL}{{\mathrm{\pi r}}^2\mathrm{l}}--\left(1\right). \\ Y=\frac{2F/{\mathrm{\pi R}}^2}{l/2L}=\frac{4FL}{{\mathrm{\pi R}}^2\mathrm{l}}---\left(2\right) \\ PuttingvalueofYfrom\left(1\right)in\left(2\right) \\ \frac{FL}{\pi r2}=\frac{4FL}{{\mathrm{\pi R}}^2\mathrm{l}} \\ {R}^2=4r^2,R=2r \end{gathered}$$

Radius of wire will be 2r.