$A$ 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.

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.

$A$ steel rod of radius $20 \ mm$ and length $2 \ m$ is acted upon by a force of $400 \ kN$ along its length. The values of stress and strain are respectively $(Y_{\text{steel}} = 2 \times 10^{11} \ N \ m^{-2})$.

When the temperature of a wire of length $L_0$ is increased by $T$,what is the energy density? The volume expansion coefficient of the wire is $\gamma$ and the Young's modulus is $Y$.

$A$ uniform metallic wire is elongated by $0.04\, m$ when subjected to a linear force $F$. The elongation,if its length and diameter are doubled and subjected to the same force,will be ..... $cm$.

The Young's modulus of a rubber string $8\, cm$ long and density $1.5\, kg/m^3$ is $5 \times 10^8\, N/m^2$. If it is suspended from the ceiling in a room,the increase in length due to its own weight will be: