A rod of length $1.05\; m$ having negligible mass is supported at its ends by two wires of steel (wire $A$) and aluminium (wire $B$) of equal lengths as shown in Figure. The cross-sectional areas of wires $A$ and $B$ are $1.0\; mm ^{2}$ and $2.0\; mm ^{2}$. respectively. At what point along the rod should a mass $m$ be suspended in order to produce $(a)$ equal stresses and $(b)$ equal strains in both steel and alumintum wires.

If the length of a wire is made double and radius is halved of its respective values. Then, the Young's modules of the material of the wire will :

A steel wire is $1 \,m$ long and $1 \,mm ^2$ in area of cross-section. If it takes $200 \,N$ to stretch this wire by $1 \,mm$, how much force will be required to stretch a wire of the same material as well as diameter from its normal length of $10 \,m$ to a length of $1002 \,cm$ is …….. $N$

A copper wire of length $4.0m$ and area of cross-section $1.2\,c{m^2}$ is stretched with a force of $4.8 \times {10^3}$ $N.$ If Young’s modulus for copper is $1.2 \times {10^{11}}\,N/{m^2},$ the increase in the length of the wire will be

If the ratio of lengths, radii and Young's moduli of steel and brass wires in the figure are $a, b$ and $c$ respectively, then the corresponding ratio of increase in their lengths is