When a uniform wire of radius $r$ is stretched by a $2kg$ weight, the increase in its length is $2.00\, mm$. If the radius of the wire is $r/2$ and other conditions remain the same, the increase in its length is ………. $mm$

A wire of cross sectional area $A$, modulus of elasticity $2 \times 10^{11} \mathrm{Nm}^{-2}$ and length $2 \mathrm{~m}$ is stretched between two vertical rigid supports. When a mass of $2 \mathrm{~kg}$ is suspended at the middle it sags lower from its original position making angle $\theta=\frac{1}{100}$ radian on the points of support. The value of $A$ is. . . . . .  $\times 10^{-4} \mathrm{~m}^2$ (consider $\mathrm{x}<\mathrm{L}$ ).

(given: $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2$ )

A steel wire $1.5\,m$ long and of radius $1\,mm$ is attached with a load $3\,kg$ at one end the other end of the wire is fixed it is whirled in a vertical circle with a frequency $2\,Hz$ . Find the elongation of the wire when the weight is at the lowest position $(Y = 2 \times 10^{11}\,N/m^2$ and $g = 10\,m/s^2)$

Two wires of diameter $0.25 \;cm ,$ one made of steel and the other made of brass are loaded as shown in Figure. The unloaded length of steel wire is $1.5 \;m$ and that of brass wire is $1.0 \;m .$ Compute the elongations of the steel and the brass wires.

What should be the shape of the pillars or column in building and bridge ?