A wire of length $L,$ area of cross section $A$ is hanging from a fixed support. The length of the wire changes to $L_{1}$ when mass $M$ is suspended from its free end. The expression for Young's modulus is 

A thick rope of density $\rho$ and length $L$ is hung from a rigid support. The Young's modulus of the material of rope is $Y$. The increase in length of the rope due to its own weight is

Two wires are made of the same material and have the same volume. However wire $1$ has crosssectional area $A$ and wire $2$ has cross-section area $3A$. If the length of wire $1$ increases by $\Delta x$ on applying force $F$, how much force is needed to stretch wire $2$ by the same amount?

A wire of length $L$ and radius $r$ is rigidly fixed at one end. On stretching the other end of the wire with a force $F$, the increase in its length is $l$. If another wire of same material but of length $2L$ and radius $2r$ is stretched with a force of $2F$, the increase in its length will be

An elastic material of Young's modulus $Y$ is subjected to a stress $S$. The elastic energy stored per unit volume of the material is

Young's moduli of the material of wires $A$ and $B$ are in the ratio of $1: 4$, while its area of cross sections are in the ratio of $1: 3$. If the same amount of load is applied to both the wires, the amount of elongation produced in the wires $A$ and $B$ will be in the ratio of

[Assume length of wires $A$ and $B$ are same]