Let a wire be suspended from the ceiling (rigid support) and stretched by a weight $W$ attached at its free end. The longitudinal stress at any point of cross-sectional area $A$of the wire is:

The two blocks $P$ and $Q$ are connected by a metal wire with breaking stress ${10}^9 \mathrm{N} {\mathrm{m}}^{-2}$ through a frictionless pulley. The minimum radius of the wire for it not to break is (consider $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

Wires $W_1$ and $W_2$ are made of same material having the breaking stress of $1.25\times {10}^9 \mathrm{N} {\mathrm{m}}^{-2}. {\mathrm{W}}_1$ and ${\mathrm{W}}_2$ have cross-sectional area of $8\times {10}^{-7}{ \mathrm{m}}^2$ and $4\times {10}^{-7}{ \mathrm{m}}^2$, respectively. Masses of $20 \mathrm{kg}$ and $10 \mathrm{kg}$ hang from them as shown in the figure. The maximum mass that can be placed in the pan without breaking the wires is _____$\mathrm{kg}$ (Use $\left.\mathrm{g}=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$

Two blocks of masses $3 \mathrm{kg}$ and $5 \mathrm{kg}$ are connected by a metal wire going over a smooth pulley. The breaking stress of the metal is $\frac{24}{\pi }\times {10}^2 \mathrm{N} {\mathrm{m}}^{-2}$. What is the minimum radius of the wire ?
( take $\left.g=10{ \mathrm{m} \mathrm{s}}^{-2}\right)$