$A$ wire of length $L$ is hanging from a fixed support. The length changes to $L_{1}$ and $L_{2}$ when masses $1 \, kg$ and $2 \, kg$ are suspended respectively from its free end. Then the value of $L$ is equal to:

$A$ $14.5\; kg$ mass,fastened to the end of a steel wire of unstretched length $1.0\; m$,is whirled in a vertical circle with an angular velocity of $2\; rev/s$ at the bottom of the circle. The cross-sectional area of the wire is $0.065\; cm^2$. Calculate the elongation of the wire when the mass is at the lowest point of its path.

The Young's modulus of steel is twice that of brass. Two wires of the same length and of the same area of cross-section,one of steel and another of brass,are suspended from the same roof. If we want the lower ends of the wires to be at the same level,then the weights added to the steel and brass wires must be in the ratio of

$A$ metal rod of cross-sectional area $3 \times 10^{-6} \,m^{2}$ is suspended vertically from one end and has a length of $0.4 \,m$ at $100^{\circ} C$. The rod is cooled to $0^{\circ} C$, but prevented from contracting by attaching a mass '$m$' at the lower end. Find the value of '$m$'. (Given: $Y = 10^{11} \,N/m^{2}$, coefficient of linear expansion $\alpha = 10^{-5} /K$, $g = 10 \,m/s^{2}$) (in $\,kg$)

$A$ uniform dense rod with non-uniform Young's modulus is hanging from the ceiling under gravity. If the elastic energy density at every point is the same,then how will Young's modulus change with $x$ as shown in the graphs?

$A$ wire is stretched by $0.01 \ m$ by a certain force $F$. Another wire of the same material whose diameter and length are double that of the original wire is stretched by the same force. Then its elongation will be (in $m$)