$A$ metal rod of Young's modulus $Y$ and coefficient of thermal expansion $\alpha$ is held at its two ends such that its length remains invariant. If its temperature is raised by $t^{\circ} C$,the linear stress developed in it is

Young's modulus depends upon

The two wires $A$ and $B$ of equal cross-section but of different materials are joined together. The ratio of Young's modulus of wire $A$ and wire $B$ is $20/11$. When the joined wire is kept under certain tension the elongations in the wires $A$ and $B$ are equal. If the length of wire $A$ is $2.2\text{ m}$, then the length of wire $B$ is . . . . . . m.

The speed of a transverse wave passing through a string of length $50 \; cm$ and mass $10 \; g$ is $60 \; ms^{-1}$. The area of cross-section of the wire is $2.0 \; mm^2$ and its Young's modulus is $1.2 \times 10^{11} \; Nm^{-2}$. The extension of the wire over its natural length due to its tension will be $x \times 10^{-5} \; m$. The value of $x$ is $...$

Four uniform wires of the same material are stretched by the same force. The dimensions of the wires are given below. Which one has the minimum elongation?

$A$ steel ring of radius $r$ and cross-sectional area $A$ is fitted onto a wooden disc of radius $R$ $(R > r)$. If Young's modulus is $Y$,then the force with which the steel ring is expanded is