A steel rod of length $$L$$ and diameter $$D$$, fixed at both ends, is uniformly heated to a temperature rise of $$\Delta T.$$ The Young's modulus is $$E$$ and the coefficient of linear expansion is $$'\alpha '\,.$$ The thermal stress in the rod is

In a simply-supported beam loaded as shown below, the maximum bending moment in N-m is

A disc of radius' r 'has a hole of radius 'r/2' cut-out as shown. The centroid of the remaining disc (shaded portion) at a radial distance from the centre ''O'' is

A quick return mechanism is shown in below. The crank OS is drive at $$2$$ rev/s in counter clockwise direction

If the quick return ratio is $$1:2$$, then the length of the crank in mm is