$A$ brass wire $1.8\; m$ long at $27\;^{\circ}C$ is held taut with little tension between two rigid supports. If the wire is cooled to a temperature of $-39\;^{\circ}C$,what is the tension developed in the wire,if its diameter is $2.0\; mm$? (Coefficient of linear expansion of brass $\alpha = 2.0 \times 10^{-5}\; K^{-1}$; Young's modulus of brass $Y = 0.91 \times 10^{11}\; Pa$)

Same tension is applied to the following four wires made of same material. The elongation is longest in

$A$ copper wire of length $2.2 \; m$ and a steel wire of length $1.6 \; m$,both of diameter $3.0 \; mm$,are connected end to end. When stretched by a load,the net elongation is found to be $0.70 \; mm$. Obtain the load applied in $N$.

Two wires $A$ and $B$ of the same length are made of the same material. The load $(F)$ vs. elongation $(x)$ graph for these two wires is shown in the figure. Which of the following statement$(s)$ is/are true?

In nature,the failure of structural members usually results from large torque because of twisting or bending rather than due to tensile or compressive strains. This process of structural breakdown is called buckling. In cases of tall cylindrical structures like trees,the torque is caused by their own weight bending the structure,such that the vertical line through the centre of gravity does not fall within the base. The elastic torque caused by this bending about the central axis of the tree is given by $\frac{Y\pi r^4}{4R}$,where $Y$ is the Young's modulus,$r$ is the radius of the trunk,and $R$ is the radius of curvature of the bent surface along the height of the tree containing the centre of gravity (the neutral surface). Estimate the critical height of a tree for a given radius of the trunk.

Steel and copper wires of the same length are stretched by the same weight one after the other. The Young's modulus of steel and copper are $2 \times 10^{11} \, N/m^2$ and $1.2 \times 10^{11} \, N/m^2$,respectively. What is the ratio of the increase in their lengths?