$A$ wire is loaded by $6 \ kg$ at its one end,the increase in length is $12 \ mm$. If the radius of the wire is doubled and all other magnitudes are unchanged,then the increase in length will be ......... $mm$.

$A$ steel rod has a radius of $20\,mm$ and a length of $2.0\,m$. $A$ force of $62.8\,kN$ stretches it along its length. Young's modulus of steel is $2.0 \times 10^{11}\,N/m^2$. The longitudinal strain produced in the wire is $..........\times 10^{-5}$.

Overall changes in volume and radii of a uniform cylindrical steel wire are $0.2 \%$ and $0.002 \%$ respectively when subjected to some suitable force. Calculate the longitudinal tensile stress acting on the wire. (Given: Young's modulus $Y = 2.0 \times 10^{11} \ N/m^2$)

$A$ steel rod of length $1\,m$ and area of cross-section $1\,cm^2$ is heated from $0\,^{\circ}C$ to $200\,^{\circ}C$ without being allowed to extend or bend. Find the tension produced in the rod $(Y = 2.0 \times 10^{11}\,N/m^2, \alpha = 10^{-5} \,^{\circ}C^{-1})$.

Two metallic wires $P$ and $Q$ have the same volume and are made up of the same material. If their areas of cross-section are in the ratio $4:1$ and a force $F_1$ is applied to $P$,an extension of $\Delta l$ is produced. The force required to produce the same extension in $Q$ is $F_2$. The value of $\frac{F_1}{F_2}$ is . . . . . . .

The proportional limit of steel is $8 \times 10^8 \, N/m^2$ and its Young's modulus is $2 \times 10^{11} \, N/m^2$. The maximum elongation,a one metre long steel wire can be given without exceeding the elastic limit is ...... $mm$.