Question

A cable stretches by an amount d when it supports a crate of mass M. The
cable is replaced by another cable of the same material having the same
length and twice the diameter. If the same crate is supported by the
thicker cable, by how much will the cable stretch?

Answer 1

$$\frac{A'}{A}=\frac{r'^2}{r^2}=\frac{(2r)^2}{r^2}=4$$

$$\frac{\sigma'}{\sigma}=\frac{A}{A'}=0.25$$

But

$$\frac{d'}{d}=\frac{\sigma'}{\sigma}=0.25$$

$$d'=0.25d$$