$A$ 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

$A$ uniform heavy rod of mass $20\,kg$,cross-sectional area $0.4\,m^{2}$,and length $20\,m$ is hanging from a fixed support. Neglecting the lateral contraction,the elongation in the rod due to its own weight is $x \times 10^{-9}\,m$. The value of $x$ is (Given: Young's modulus $Y = 2 \times 10^{11}\,N/m^{2}$ and $g = 10\,m/s^{2}$)

An Indian rubber cord $L$ metre long and area of cross-section $A$ $m^2$ is suspended vertically. The density of rubber is $D$ $kg/m^3$ and Young's modulus of rubber is $E$ $N/m^2$. If the wire extends by $l$ metre under its own weight,then the extension $l$ is:

$A$ block of weight $100 \ N$ is suspended by copper and steel wires of same cross-sectional area $0.5 \ cm^2$ and lengths $\sqrt{3} \ m$ and $1 \ m$,respectively. Their other ends are fixed on a ceiling as shown in the figure. The angles subtended by the copper and steel wires with the ceiling are $30^{\circ}$ and $60^{\circ}$,respectively. If the elongation in the copper wire is $\Delta \ell_C$ and the elongation in the steel wire is $\Delta \ell_S$,then the ratio $\frac{\Delta \ell_C}{\Delta \ell_S}$ is. . . . . .
[Young's modulus for copper and steel are $1 \times 10^{11} \ N/m^2$ and $2 \times 10^{11} \ N/m^2$ respectively]

$A$ rigid bar of mass $15 \, kg$ is supported symmetrically by three wires,each $2 \, m$ long. The wires at each end are made of copper,and the middle one is made of steel. The Young's modulus of elasticity for copper and steel are $110 \times 10^9 \, N/m^2$ and $190 \times 10^9 \, N/m^2$ respectively. If each wire is to have the same tension,the ratio of their diameters (diameter of copper wire to diameter of steel wire) will be ............

$A$ steel rod with $Y = 2.0 \times 10^{11} \, N/m^2$ and $\alpha = 10^{-5} \, ^\circ C^{-1}$ of length $4 \, m$ and area of cross-section $10 \, cm^2$ is heated from $0^\circ C$ to $400^\circ C$ without being allowed to extend. The tension produced in the rod is $x \times 10^5 \, N$ where the value of $x$ is ....... .