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

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

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

The input link Q₂P of a four bar linkage is rotated at $$2$$ rad/s counter clockwise direction as shown below. The angular velocity of the coupler PQ in rad/s, at an instant when $$\angle {O_4}{O_2}P = {180^ \circ },$$ is