$A$ mild steel wire of length $2l$ meter and cross-sectional area $A \; m^2$ is fixed horizontally between two pillars. $A$ small mass $m \; kg$ is suspended from the midpoint of the wire. If the extension in the wire is within the elastic limit,then the depression $x$ at the midpoint of the wire will be:

$(a)$ $A$ steel wire of mass $\mu$ per unit length with a circular cross-section has a radius of $0.1\,cm$. The wire is of length $10\,m$ when measured lying horizontal and hangs from a hook on the wall. $A$ mass of $25\,kg$ is hung from the free end of the wire. Assuming the wire to be uniform and lateral strains $\ll$ longitudinal strains,find the extension in the length of the wire. The density of steel is $7860\,kg/m^3$ and Young's modulus $Y = 2 \times 10^{11}\,N/m^2$.
$(b)$ If the yield strength of steel is $2.5 \times 10^8\,N/m^2$,what is the maximum weight that can be hung at the lower end of the wire?

The speed of a transverse wave passing through a string of length $50 \; cm$ and mass $10 \; g$ is $60 \; ms^{-1}$. The area of cross-section of the wire is $2.0 \; mm^2$ and its Young's modulus is $1.2 \times 10^{11} \; Nm^{-2}$. The extension of the wire over its natural length due to its tension will be $x \times 10^{-5} \; m$. The value of $x$ is $...$

One end of a steel rod of radius $10.0 \ mm$ and length $50.0 \ cm$ is clamped on a horizontal table. The other end of the rod is pulled with a force of magnitude $10.0 \times \pi \ kN$. This force is uniform across the flat surface of the rod and is perpendicular to it. The change in the length of the rod due to this applied force is (Use Young's modulus $= 2.0 \times 10^{11} \ N/m^2$) (in $mm$)

$A$ $3 \ m$ long wire of radius $3 \ mm$ shows an extension of $0.1 \ mm$ when loaded vertically by a mass of $50 \ kg$ in an experiment to determine Young's modulus. The value of Young's modulus of the wire as per this experiment is $P \times 10^{11} \ Nm^{-2}$,where the value of $P$ is: (Take $g = 3 \pi \ m/s^2$)

$A$ force is applied to a steel wire '$A$',rigidly clamped at one end. As a result,the elongation in the wire is $0.2\,mm$. If the same force is applied to another steel wire '$B$' of double the length and a diameter $2.4$ times that of the wire '$A$',the elongation in the wire '$B$' will be $............\times 10^{-2}\,mm$ (wires having uniform circular cross sections).