$A$ beam of metal supported at the two ends is loaded at the centre. The depression at the centre is proportional to

$A$ metallic ring of radius $r$ and cross-sectional area $A$ is fitted into a wooden circular disc of radius $R$ $(R > r)$. If the Young's modulus of the material of the ring is $Y$, the force with which the metal ring expands is

Let $L$ be the length and $d$ be the diameter of the cross-section of a wire. Wires of the same material with different $L$ and $d$ are subjected to the same tension along the length of the wire. In which of the following cases will the extension of the wire be the maximum?

$A$ meter scale of mass $m$,Young's modulus $Y$,and cross-sectional area $A$ is hung vertically from the ceiling at the zero mark. The separation between the $30\ cm$ and $70\ cm$ marks will be: (Assume $\frac{mg}{AY}$ is dimensionless)

$A$ metal rod is fixed rigidly at two ends so as to prevent its thermal expansion. If $L$, $\alpha$, and $Y$ denote the length of the rod, the coefficient of linear thermal expansion, and the Young's modulus of its material respectively, then for an increase in temperature of the rod by $\Delta T$, the longitudinal stress developed in the rod is

$A$ steel rod has a radius of $50 \ mm$ and a length of $2 \ m$. It is stretched along its length with a force of $400 \ kN$. This causes an elongation of $0.5 \ mm$. Find the (approximate) Young's modulus of steel from this information.