A force $F$ is applied on the wire of radius $r$ and length $L$ and change in the length of wire is $l.$ If the same force $F$ is applied on the wire of the same material and radius $2r$ and length $2L,$ Then the change in length of the other wire is

$(a)$ A steel wire of mass $\mu $ per unit length with a circular cross section has a radius of $0.1\,cm$. The wire is of length $10\,m$ when measured lying horizontal and hangs from a hook on the wall. A mass of $25\, kg$ is hung from the free end of the wire. Assuming the wire to be uniform an lateral strains $< \,<$ longitudinal strains find the extension in the length of the wire. The density of steel is $7860\, kgm^{-3}$ and Young’s modulus $=2 \times 10^{11}\,Nm^{-2}$.

$(b)$ If the yield strength of steel is $2.5 \times 10^8\,Nm^{-2}$, what is the maximum weight that can be hung at the lower end of the wire ?

Two wires $A$ and $B$ are of same materials. Their lengths are in the ratio $1 : 2$ and diameters are in the ratio $2 : 1$ when stretched by force ${F_A}$ and ${F_B}$ respectively they get equal increase in their lengths. Then the ratio ${F_A}/{F_B}$ should be

Stress required in a wire to produce $0.1\%$ strain is $4 \times10^8\, N/m^2$. Its yound modulus is $Y_1$. If stress required in other wire to produce $0.3\%$ strain is $6 \times 10^8\, N/m^2$. Its young modulus is $Y_2$. Which relation is correct

According to Hook’s law of elasticity, if stress is increased, the ratio of stress to strain

The proportional limit of steel is $8 \times 10^8 \,N / m ^2$ and its Young's modulus is $2 \times 10^{11} \,N / m ^2$. The maximum elongation, a one metre long steel wire can be given without exceeding the elastic limit is ...... $mm$