In nature the failure of structural members usually result from large torque because of twisting or bending rather than due to tensile or compressive strains. This process of structural breakdown is called buckling and in cases of tall cylindrical structures like trees, the torque is caused by its own weight bending the structure. Thus, the vertical through the centre of gravity does not fall withinthe  base. The elastic torque caused because of this bending about the central axis of the tree is given by $\frac{{Y\pi {r^4}}}{{4R}}$ $Y$ 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.

A uniform wire (Young's modulus $2 \times 10^{11}\, Nm^{-2}$ ) is subjected to longitudinal tensile stress of $5 \times 10^7\,Nm^{-2}$ . If the over all volume change in the wire is $0.02\%,$ the fractional decrease in the radius of the wire is close to

The following four wires are made of the same material. Which of these will have the largest extension when the same tension is applied ?

The length of metallic wire is $l$. The tension in the wire is $T_1$ for length $l_1$ and tension in the wire is $T_2$ for length $l_2$. Find the original length.

The length of a wire is $1.0\, m$ and the area of cross-section is $1.0 \times {10^{ – 2}}\,c{m^2}$. If the work done for increase in length by $0.2\, cm$ is $0.4\, joule$, then Young's modulus of the material of the wire is