A uniformly loaded propped cantilever beam and its free body diagram are shown below. The reactions are

A stepped steel shaft shown below is subjected to $$10$$ N-m torque. If the modulus of rigidity is $$80$$ GPa, the strain energy in the shaft in N-mm 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

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