If the ratio of length,radii,and Young's modulus of steel and aluminium wire are $a, b, c$ respectively,then the corresponding ratio of increase in their length would be:

Explain the experimental determination of Young's modulus.

$A$ metal rod of length $L$ and cross-sectional area $A$ is heated through $T^{\circ} C$. What is the force required to prevent the expansion of the rod lengthwise? ($Y=$ Young's modulus of the material of the rod,$\alpha=$ coefficient of linear expansion of the rod.)

The length of a wire is $l_1$ when a mass $M_1$ is hung from it,and it is $l_2$ when both masses $M_1$ and $M_2$ are hung from it. The natural length of the wire is ........

Two exactly similar wires of steel and copper are stretched by equal forces. If the difference in their elongations is $0.5 \ cm$,find the elongation $(l)$ of each wire. Given: ${Y_s} = 2.0 \times {10^{11}} \ N/m^2$ and ${Y_c} = 1.2 \times {10^{11}} \ N/m^2$.

$A$ compressive force, $F$ is applied at the two ends of a long thin steel rod. It is heated, simultaneously, such that its temperature increases by $\Delta T$. The net change in its length is zero. Let $l$ be the length of the rod, $A$ its area of cross-section, $Y$ its Young's modulus, and $\alpha$ its coefficient of linear expansion. Then, $F$ is equal to