One tooth of gear having $$4$$ module and $$32$$ teeth is shown in the figure. Assume that the gear tooth and the corresponding tooth space make equal intercepts on the pitch circumference.

The dimensions $$' a '$$ and $$' b '$$ respectively, are closest to

A clutch has outer and inner diameters $$100mm$$ and $$40mm$$ respectively. Assuming a uniform pressure of $$2MPa$$ and coefficient of friction of inner material $$0.4,$$ the torque carrying capacity of the clutch is

While cooling, a cubical casting of side $$40mm$$ undergoes $$3\% ,\,\,4\% $$ and $$5\% $$ volume shrinkage during the liquid state, phase transition and solid state respectively. The volume of metal compensated from the riser is

In a single pass rolling operation, a $$20mm$$ thick plate with plate width of $$100mm$$, is reduced to $$18mm$$ thick. The roller radius is $$250mm$$ and rotational speed is $$10$$ $$rpm.$$ The average flow stress for the plate material is $$300$$ $$MPa.$$ The power required for the rolling operation in $$kW$$ is closer to