$A$ wire of cross-sectional area $A$,modulus of elasticity $2 \times 10^{11} \text{ N m}^{-2}$,and length $2L = 2 \text{ m}$ is stretched between two vertical rigid supports. When a mass of $2 \text{ kg}$ is suspended at the middle,it sags from its original position,making an angle $\theta = \frac{1}{100} \text{ radian}$ with the horizontal at the points of support. The value of $A$ is . . . . . . $\times 10^{-4} \text{ m}^2$. (Given: $g = 10 \text{ m/s}^2$)

Wires $A$ and $B$ are connected with blocks $P$ and $Q$ as shown. The ratio of lengths,radii,and Young's modulus of wires $A$ and $B$ are $r, 2r$,and $3r$ respectively ($r$ is a constant). Find the mass of block $P$ if the ratio of the increase in their corresponding lengths is $1/(6r^2)$. The mass of block $Q$ is $3M$.

$A$ 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 $N$.

$A$ cylindrical rod made of aluminum has a length of $1 \,m$ and a diameter of $10 \,cm$. The rod is subjected to a tensile force of $100 \,kN$. Calculate the elongation in the rod. (Young's modulus of aluminum $= 70 \,GPa$)

There are two wires of the same material and same length. The diameter of the second wire is two times the diameter of the first wire. What is the ratio of the extensions produced in the wires by applying the same load?

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.