The force required to stretch a steel wire of $1\,cm^2$ cross-section to $1.1$ times its original length is $(Y = 2 \times 10^{11}\,N/m^2)$.

Two blocks of mass $2 \ kg$ and $4 \ kg$ are connected by a metal wire going over a smooth pulley as shown in the figure. The radius of the wire is $4.0 \times 10^{-5} \ m$ and Young's modulus of the metal is $2.0 \times 10^{11} \ N/m^2$. The longitudinal strain developed in the wire is $\frac{1}{\alpha \pi}$. The value of $\alpha$ is [Use $g = 10 \ m/s^2$].

One end of the steel rod is clamped to the roof and the other end is attached to a mass of $1000 \,kg$ as shown in the figure. The length of the rod is $50 \,cm$ and its cross-sectional area is $1000 \,mm^2$. The change in the length of the rod due to the weight of the mass is (Young's modulus of steel $= 2 \times 10^{11} \,Nm^{-2}$ and acceleration due to gravity $= 10 \,ms^{-2}$) (in $\,mm$)

An aluminum rod (Young's modulus $Y = 7 \times 10^9 \, N/m^2$) has a breaking strain of $0.2\%$. The minimum cross-sectional area of the rod required to support a load of $10^4 \, N$ is:

$A$ wire of length $2 \ m$ and cross-sectional area $10^{-2} \ cm^2$ is fixed at one end. $A$ force of $200 \ N$ is applied at the other end. The coefficient of linear expansion of the wire is $1.1 \times 10^{-5} \ ^oC^{-1}$ and the Young's modulus is $1.2 \times 10^{11} \ N/m^2$. If the temperature is increased by $10^oC$,what will be the thermal stress developed in the wire?

$A$ metal rod of cross-sectional area $3 \times 10^{-6} \,m^{2}$ is suspended vertically from one end and has a length of $0.4 \,m$ at $100^{\circ} C$. The rod is cooled to $0^{\circ} C$, but prevented from contracting by attaching a mass '$m$' at the lower end. Find the value of '$m$'. (Given: $Y = 10^{11} \,N/m^{2}$, coefficient of linear expansion $\alpha = 10^{-5} /K$, $g = 10 \,m/s^{2}$) (in $\,kg$)