$A$ wire of length $L$ and radius $r$ is clamped rigidly at one end. When the other end of the wire is pulled by a force $F$,its length increases by $5\,cm$. Another wire of the same material of length $4L$ and radius $4r$ is pulled by a force $4F$ under the same conditions. The increase in length of this wire is $....cm$.

$A$ steel wire of length $L$ at $40^{\circ}C$ is suspended from the ceiling and then a mass $m$ is hung from its free end. The wire is cooled down from $40^{\circ}C$ to $30^{\circ}C$ to regain its original length $L$. The coefficient of linear thermal expansion of the steel is $10^{-5} /^{\circ}C$,Young's modulus of steel is $10^{11} N/m^2$,and the radius of the wire is $1 mm$. Assume that $L \gg$ diameter of the wire. Then the value of $m$ in $kg$ is nearly:

If the ratio of length,radii,and Young's modulus of steel and aluminium wire are $a, b, c$ respectively,then the corresponding ratio of increase in their length would be:

The length of a light string is $1.4 \ m$ when the tension on it is $5 \ N$. If the tension increases to $7 \ N$,the length of the string is $1.56 \ m$. The original length of the string is . . . . . . $m$.

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.

$A$ copper wire of length $2.4 \ m$ and an aluminum wire of length $0.7 \ m$, both having diameter $2 \ mm$, are connected end to end. When stretched by a load, the total elongation is found to be $0.6 \ mm$. The applied load is (Young's modulus of copper $= 1.2 \times 10^{11} \ N/m^2$ and Young's modulus of aluminum $= 0.7 \times 10^{11} \ N/m^2$). (in $\pi \ N$)