A rigid horizontal rod of length $$2L$$ is fixed to a circular cylinder of radius $$R$$ as shown in the figure. Vertical forces of magnitude $$P$$ are applied at the two ends as shown in the figure. The shear modulus for the cylinder is $$G$$ and the Young’s modulus is $$E.$$

The vertical deflection at point $$A$$ is

A thin cylindrical pressure vessel with closed-ends is subjected to internal pressure. The ratio of circumferential (hoop) stress to the longitudinal stress is

A circular metallic rod of length 250 mm is placed between two rigid immovable walls as shown in the figure. The rod is in perfect contact with the wall on the left side and there is a gap of 0.2 mm between the rod and the wall on the right side. If the temperature of the rod is increased by 200^oC, the axial stress developed in the rod is __________ MPa.

Young’s modulus of the material of the rod is 200 GPa and the coefficient of thermal expansion is 10⁻⁵ per ^oC.

A single degree of freedom mass-spring - viscous damper system with mass $$m,$$ spring constant $$k$$ and viscous damping coefficient $$q$$ is critically damped. The correct relation among $$m, k$$ and $$q$$ is