$A$ one metre steel wire of negligible mass and area of cross-section $0.01 \,cm^2$ is kept on a smooth horizontal table with one end fixed. $A$ ball of mass $1 \,kg$ is attached to the other end. The ball and the wire are rotating with an angular velocity of $\omega$. If the elongation of the wire is $2 \,mm$, then $\omega$ is (Young's modulus of steel $= 2 \times 10^{11} \,N/m^2$)

Four identical hollow cylindrical columns of mild steel support a big structure of mass $50 \times 10^{3} \; \text{kg}$. The inner and outer radii of each column are $50 \; \text{cm}$ and $100 \; \text{cm}$ respectively. Assuming uniform load distribution,calculate the compressive strain of each column. [Use $Y = 2.0 \times 10^{11} \; \text{Pa}$,$g = 9.8 \; \text{m/s}^2$]

$A$ steel wire of length $3 \ m$ and a copper wire of length $2.2 \ m$ are connected end to end. When the combination is stretched by a force, the net elongation is $1.05 \ mm$. If the area of cross-section of each wire is $6 \ mm^2$, then the load applied is (Young's modulus of steel and copper are respectively $2 \times 10^{11} \ N/m^2$ and $1.1 \times 10^{11} \ N/m^2$) (in $N$)

$A$ steel wire of length $20 \text{ cm}$ and area of cross-section $1 \text{ mm}^2$ is tied rigidly at both the ends. When the temperature of the wire is changed from $40^{\circ} \text{C}$ to $20^{\circ} \text{C}$, find the change in its tension. Given, the coefficient of linear expansion for steel is $\alpha = 1.1 \times 10^{-5} {}^{\circ} \text{C}^{-1}$ and Young's modulus of steel is $Y = 2.0 \times 10^{11} \text{ N/m}^2$. (in $\text{ N}$)

$A$ $100\,m$ long wire having cross-sectional area $6.25 \times 10^{-4}\,m^2$ and Young's modulus $10^{10}\,N/m^2$ is subjected to a load of $250\,N$. The elongation in the wire will be:

$A$ structural steel rod has a radius of $10 \;mm$ and a length of $1.0 \;m$. $A$ $100 \;kN$ force stretches it along its length. Calculate $(a)$ stress,$(b)$ elongation,and $(c)$ strain on the rod. Young's modulus of structural steel is $2.0 \times 10^{11} \;N \;m^{-2}$.