Two rods of different materials having coefficients of linear expansion $\alpha_1, \alpha_2$ and Young's moduli $Y_1$ and $Y_2$ respectively are fixed between two rigid massive walls. The rods are heated such that they undergo the same increase in temperature. There is no bending of rods. If $\alpha_1 : \alpha_2 = 2 : 3$,and the thermal stresses developed in the two rods are equal,then the ratio $Y_1 : Y_2$ is equal to:

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

$A$ steel wire is $1 \,m$ long and $1 \,mm^2$ in area of cross-section. If it takes $200 \,N$ to stretch this wire by $1 \,mm$,how much force will be required to stretch a wire of the same material as well as diameter from its normal length of $10 \,m$ to a length of $1002 \,cm$?

$A$ wire elongates by $l \ mm$ when a load $W$ is hanged from it. If the wire goes over a pulley and two weights $W$ each are hung at the two ends,the elongation of the wire will be (in $mm$)

$(a)$ $A$ 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?

If $\rho$ is the density of the material of a wire and $\sigma$ is the breaking stress, what is the greatest length of the wire that can hang freely without breaking?