Two blocks of masses 1 ~kg and 2 ~kg are connected by a metal wire...

The stress in the wire $=\frac{\text { tension }}{\text { area of cross }- \text { section }}$ To avoid breaking, this stress should not exceed the breaking stress. Let the tension in the wire be $T$. The equations of motion of the two blocks are, $T-10=1 a$ and $\quad 20 \mathrm{~N}-T=2 a$ Eliminating $a$ from these equations, $T=\left(\frac{40}{3}\right) \mathrm{N}$ Stress $\quad T=\frac{\left(\frac{40}{3}\right)}{\pi r^2}$ If the minimum radius needed to avoid breaking is $r$, $2 \times 10^9=\frac{\left(\frac{40}{3}\right)}{\pi r^2}$ Solving this, $r=4.6 \times 10^{-5} \mathrm{~m}$