$A$ force of $10^3 \ N$ stretches the length of a hanging wire by $1 \ mm$. The force required to stretch a wire of the same material and length,but having four times the diameter,by $1 \ mm$ is:

The area of cross-section of a steel wire $(Y = 2.0 \times 10^{11} \ N/m^2)$ is $0.1 \ cm^2$. The force required to double its length will be

The cross-sectional area of each wire is $10^{-4} \, m^2$. What is the displacement of point $B$?

In the determination of Young's modulus $\left(Y=\frac{4 MLg}{\pi \ell d^2}\right)$ by using Searle's method,a wire of length $L=2 \ m$ and diameter $d=0.5 \ mm$ is used. For a load $M=2.5 \ kg$,an extension $\ell=0.25 \ mm$ in the length of the wire is observed. Quantities $d$ and $\ell$ are measured using a screw gauge and a micrometer,respectively. They have the same pitch of $0.5 \ mm$. The number of divisions on their circular scale is $100$. The contributions to the maximum probable error of the $Y$ measurement:

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 ...... $mm$.

$A$ uniform cylindrical rod of length $L$ and radius $r$ is made from a material whose Young's modulus of elasticity equals $Y$. When this rod is heated by temperature $T$ and simultaneously subjected to a net longitudinal compressional force $F$,its length remains unchanged. The coefficient of volume expansion of the material of the rod is (nearly) equal to: