Two metallic wires $P$ and $Q$ have the same volume and are made up of the same material. If their areas of cross-section are in the ratio $4:1$ and a force $F_1$ is applied to $P$,an extension of $\Delta l$ is produced. The force required to produce the same extension in $Q$ is $F_2$. The value of $\frac{F_1}{F_2}$ is . . . . . . .

Which of the following substances has the highest elasticity?

Two wires of diameter $0.25 \; cm,$ one made of steel and the other made of brass,are loaded as shown in the figure. The unloaded length of the steel wire is $1.5 \; m$ and that of the brass wire is $1.0 \; m.$ Compute the elongations of the steel and the brass wires. (Given: Young's modulus of steel $Y_s = 2.0 \times 10^{11} \; Pa,$ Young's modulus of brass $Y_b = 0.91 \times 10^{11} \; Pa$)

$A$ one metre steel wire of negligible mass and area of cross-section $0.01 \,cm^2$ is kept on a smooth horizontal table with one end fixed. $A$ ball of mass $1 \,kg$ is attached to the other end. The ball and the wire are rotating with an angular velocity of $\omega$. If the elongation of the wire is $2 \,mm$, then $\omega$ is (Young's modulus of steel $= 2 \times 10^{11} \,N/m^2$)

$A$ brass rod of length $2\,m$ and cross-sectional area $2.0\,cm^2$ is attached end to end to a steel rod of length $L$ and cross-sectional area $1.0\,cm^2$. The compound rod is subjected to equal and opposite pulls of magnitude $5 \times 10^4\,N$ at its ends. If the elongations of the two rods are equal,then the length of the steel rod $(L)$ is ........... $m$ $(Y_{Brass}=1.0\times 10^{11}\,N/m^2$ and $Y_{Steel} = 2.0 \times 10^{11}\,N/m^2)$.

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