$A$ steel wire and a copper wire are joined end to end having equal cross-sections. The elongation of the two wires is found to be equal under tension. What is the ratio of the length of the steel wire to the length of the copper wire? (Young's modulus of steel $= 2.0 \times 10^{11} \ N \ m^{-2}$ and Young's modulus of copper $= 1.1 \times 10^{11} \ N \ m^{-2}$)

Three bars having lengths $l, 2l$,and $3l$ and areas of cross-section $A, 2A$,and $3A$ are joined rigidly end to end. The compound rod is subjected to a stretching force $F$. The total increase in the length of the rod is (Young's modulus of the material is $Y$ and the bars are massless).

One end of a steel wire of radius $r$ is fixed to a ceiling and a load of $3 \ kg$ is attached to the free end of the wire. Another wire made of copper of radius $2r$ is attached to the bottom of the $3 \ kg$ load and a $2 \ kg$ load is attached to the free end of the copper wire. The ratio of longitudinal strains produced in copper and steel wires is (Young modulus of steel $= 20 \times 10^{10} \ Nm^{-2}$,Young modulus of copper $= 12 \times 10^{10} \ Nm^{-2}$)

$A$ force is applied to a steel wire '$A$',rigidly clamped at one end. As a result,the elongation in the wire is $0.2\,mm$. If the same force is applied to another steel wire '$B$' of double the length and a diameter $2.4$ times that of the wire '$A$',the elongation in the wire '$B$' will be $............\times 10^{-2}\,mm$ (wires having uniform circular cross sections).

$A$ metal string $A$ is suspended from a rigid support and its free end is attached to a block of mass $M$. $A$ second block having mass $2M$ is suspended at the bottom of the first block using a string $B$. The area of cross-sections of strings $A$ and $B$ are the same. The ratio of lengths of strings $A$ to $B$ is $2$ and the ratio of their Young's moduli $(Y_A/Y_B)$ is $0.5$. The ratio of elongations in $A$ to $B$ is . . . . . . .

The speed of a transverse wave passing through a string of length $50 \; cm$ and mass $10 \; g$ is $60 \; ms^{-1}$. The area of cross-section of the wire is $2.0 \; mm^2$ and its Young's modulus is $1.2 \times 10^{11} \; Nm^{-2}$. The extension of the wire over its natural length due to its tension will be $x \times 10^{-5} \; m$. The value of $x$ is $...$