$A$ compressive force, $F$ is applied at the two ends of a long thin steel rod. It is heated, simultaneously, such that its temperature increases by $\Delta T$. The net change in its length is zero. Let $l$ be the length of the rod, $A$ its area of cross-section, $Y$ its Young's modulus, and $\alpha$ its coefficient of linear expansion. Then, $F$ is equal to

At $40\,^oC$,a brass wire of radius $0.5\, mm$ (diameter $1\, mm$) is hung from the ceiling. $A$ small mass $M$ is hung from the free end of the wire. When the wire is cooled down from $40\,^oC$ to $20\,^oC$,it regains its original length of $0.2\, m$. The value of $M$ is close to ........$kg$. (Coefficient of linear expansion $\alpha = 10^{-5}/^oC$ and Young's modulus $Y = 10^{11}\, N/m^2$; $g = 10\, ms^{-2}$)

$A$ fixed volume of iron is drawn into a wire of length $l$. The extension produced in this wire by a constant force $F$ is proportional to

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.

$A$ wire of length $2 \ m$ and cross-sectional area $10^{-2} \ cm^2$ is fixed at one end. $A$ force of $200 \ N$ is applied at the other end. The coefficient of linear expansion of the wire is $1.1 \times 10^{-5} \ ^oC^{-1}$ and the Young's modulus is $1.2 \times 10^{11} \ N/m^2$. If the temperature is increased by $10^oC$,what will be the thermal stress developed in the wire?

Two wires $A$ and $B$ made of different materials of length $6.0 \ cm$ and $5.4 \ cm$,respectively,and area of cross-sections $3.0 \times 10^{-5} \ m^2$ and $4.5 \times 10^{-5} \ m^2$,respectively,are stretched by the same magnitude under a given load. The ratio of the Young's modulus of $A$ to that of $B$ is $x : 3$. The value of $x$ is . . . . . . . . . . .