$A$ rod of length $l$ and area of cross-section $A$ is heated from $0^{\circ}C$ to $100^{\circ}C$. The rod is so placed that it is not allowed to increase in length,then the force developed is proportional to

The ratio of Young's modulus of three wires is $2 : 2 : 1$ and the ratio of their cross-sectional areas is $1 : 2 : 3$. If the same force is applied to each,what is the ratio of the increase in their lengths?

$A$ 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).

$A$ 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]

In an experiment to determine the Young's modulus of a wire of length exactly $1\;m$,the extension in the length of the wire is measured as $0.4\;mm$ with an uncertainty of $\pm 0.02\;mm$ when a load of $1\;kg$ is applied. The diameter of the wire is measured as $0.4\;mm$ with an uncertainty of $\pm 0.01\;mm$. The error in the measurement of Young's modulus $(\Delta Y)$ is found to be $x \times 10^{10}\;N/m^2$. The value of $x$ is (Take $g = 10\;m/s^2$)

$A$ steel rail of length $5\,m$ and area of cross-section $40\,cm^2$ is prevented from expanding along its length while the temperature rises by $10\,^{\circ}C$. If the coefficient of linear expansion and Young's modulus of steel are $1.2\times10^{-5}\,K^{-1}$ and $2\times10^{11}\,N/m^2$ respectively,the force developed in the rail is approximately: