Copper of fixed volume $V$ is drawn into a wire of length $l$. When this wire is subjected to a constant force $F$,the extension produced in the wire is $\Delta l$. Which of the following graphs is a straight line?

$A$ steel ring of radius $r$ and cross-section area $A$ is fitted onto a wooden disc of radius $R$ $(R > r)$. If Young's modulus is $E$,then the force with which the steel ring is expanded is

$A$ wire of cross-sectional area $10^{-6} \, m^2$ is elongated by $0.1 \%$ when the tension in it is $1000 \, N$. The Young's modulus of the material of the wire is (Assume radius of the wire is constant).

$(a)$ $A$ 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 cross-sectional area of all wires is $10^{-4} \ m^2$. What is the displacement of point $D$?

Consider the situation shown in the figure. The force $F$ is equal to $m_2g/2$. If the area of cross-section of the string is $A$ and its Young's modulus is $Y$,find the strain developed in it. The string is light and there is no friction anywhere.