$A$ 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 . . . . . . .

$A$ metal wire of length $L_1$ and area of cross-section $A$ is attached to a rigid support. Another metal wire of length $L_2$ and of the same cross-sectional area is attached to the free end of the first wire. $A$ body of mass $M$ is then suspended from the free end of the second wire. If $Y_1$ and $Y_2$ are the Young's moduli of the wires respectively,the effective force constant of the system of the two wires is:

$A$ wire is stretched by $0.01 \ m$ by a certain force $F$. Another wire of the same material whose diameter and length are double that of the original wire is stretched by the same force. Then its elongation will be (in $m$)

$A$ copper wire and a steel wire of the same diameter and length are connected end to end and a force is applied,which stretches their combined length by $1 \ cm$. The two wires will have

$A$ 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]

Under the same load,wire $A$ having length $5.0 \, m$ and cross-section $2.5 \times 10^{-5} \, m^2$ stretches uniformly by the same amount as another wire $B$ of length $6.0 \, m$ and a cross-section of $3.0 \times 10^{-5} \, m^2$ stretches. The ratio of the Young's modulus of wire $A$ to that of wire $B$ will be