Two rods of different materials having coefficients of linear expansion ${\alpha _1},\,{\alpha _2}$ and Young's moduli ${Y_1}$ and ${Y_2}$ respectively are fixed between two rigid massive walls. The rods are heated such that they undergo the same increase in temperature. There is no bending of rods. If ${\alpha _1}:{\alpha _2} = 2:3$, the thermal stresses developed in the two rods are equally provided ${Y_1}:{Y_2}$ is equal to

A steel uniform rod of length $2L$ cross sectional area $A$ and mass $M$ is set rotating in a horizontal plane about an axis passing through the centre. If $Y$ is the Young’s modulus for steel, find the extension in the length of the rod.

A wire of length $L$ and radius $r$ is clamped rigidly at one end. When the other end of the wire is pulled by a force $F$, its length increases by $5\,cm$. Another wire of the same material of length $4 L$ and radius $4\,r$ is pulled by a force $4\,F$ under same conditions. The increase in length of this wire is $....cm$.

Four identical hollow cylindrical columns of mild steel support a big structure of mass $50,000 \;kg$. The inner and outer radii of each column are $30$ and $60\; cm$ respectively. Assuming the load distribution to be uniform, calculate the compressional strain of each column.

Young’s moduli of two wires $A$ and $B$ are in the ratio $7 : 4$. Wire $A$ is $2\, m$ long and has radius $R$. Wire $A$ is $2\, m$ long and has radius $R$. Wire $B$ is $1.5\, m$ long and has radius $2\, mm$. If the two wires stretch by the same length for a given load, then the value of $R$ is close to ......... $mm$

Young's modulus depends upon