Consider a rod of length $1.0 \ m$ with a cross-sectional area of $0.50 \ cm^2$. The rod supports a $500 \ kg$ platform that hangs attached to the rod's lower end. What is the elongation of the rod under the stress,ignoring the weight of the rod (in $mm$)? Consider Young's modulus to be $10^{11} \ Pa$ and $g = 10 \ m \ s^{-2}$.

At $40\,^oC$,a brass wire of radius $0.5\, mm$ (diameter $1\, mm$) is hung from the ceiling. $A$ small mass $M$ is hung from the free end of the wire. When the wire is cooled down from $40\,^oC$ to $20\,^oC$,it regains its original length of $0.2\, m$. The value of $M$ is close to ........$kg$. (Coefficient of linear expansion $\alpha = 10^{-5}/^oC$ and Young's modulus $Y = 10^{11}\, N/m^2$; $g = 10\, ms^{-2}$)

$A$ horizontal steel railroad track has a length of $100 \, m$ when the temperature is $25^{\circ} C$. The track is constrained from expanding or bending. The stress on the track on a hot summer day,when the temperature is $40^{\circ} C$,is ............. $\times 10^7 \, Pa$. (Note: The linear coefficient of thermal expansion for steel is $1.1 \times 10^{-5} /^{\circ} C$ and the Young's modulus of steel is $2 \times 10^{11} \, Pa$.)

$A$ $5\, m$ long aluminium wire $(Y = 7 \times 10^{10}\, N/m^2)$ of diameter $3\, mm$ supports a $40\, kg$ mass. In order to have the same elongation in a copper wire $(Y = 12 \times 10^{10}\, N/m^2)$ of the same length under the same weight,the diameter should now be,in $mm$.

$A$ load $W$ produces an extension of $1 \ mm$ in a thread of radius $r$. If the load is increased to $4W$ and the radius is increased to $2r$,while all other factors remain the same,what will be the new extension in $mm$?

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 $Y = 2.0 \times 10^{11} \ N/m^2$)