The ratio of the lengths of two wires of the same material is $1:2$ and the ratio of their radii is $1:\sqrt{2}$. If they are stretched by the same force,what is the ratio of the increase in their lengths?

Young's modulus of a perfectly rigid body material is

$A$ steel rod has a radius of $50 \ mm$ and a length of $2 \ m$. It is stretched along its length with a force of $400 \ kN$. This causes an elongation of $0.5 \ mm$. Find the (approximate) Young's modulus of steel from this information.

$A$ 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 $Y_{Steel} = 2.0 \times 10^{11}\,N/m^2)$.

$A$ force of $200\, N$ is applied at one end of a wire of length $2\, m$ and having area of cross-section $10^{-2}\, cm^2$. The other end of the wire is rigidly fixed. If the coefficient of linear expansion of the wire is $\alpha = 8 \times 10^{-6}\, ^\circ C^{-1}$,Young's modulus is $Y = 2.2 \times 10^{11}\, N/m^2$,and its temperature is increased by $5\, ^\circ C$,then the increase in the tension of the wire will be ........ $N$.

As shown in the figure,a light uniform rod $PQ$ of length $150 \ cm$ is suspended from the ceiling horizontally using two metal wires $A$ and $B$ tied to the ends of the rod. The ratios of the radii and the Young's moduli of the materials of the two wires $A$ and $B$ are respectively $2:3$ and $3:2$. The position at which a weight should be suspended from the rod such that the elongations of the two wires become equal is