A $$200$$ $$mm$$ long down sprue has an area of cross section of $$650$$ $$m{m^2}$$ where the pouring basin meets the down sprue (i.e. at the beginning of the down sprue). A constant head of molten metal is maintaining by the pouring basin. The molten metal flow rate is $$6.5 \times {10^5}\,\,\,m{m^3}/s.$$ considering the end of down sprue to be open to atmosphere and an acceleration due to gravity of $${10^4}mm/{s^2},$$ the area of the down sprue in $$m{m^2}$$ at its end (avoiding aspiration effects) should be

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

A machine frame shown in the figure below is subjected to a horizontal force of $$600$$ $$N$$ parallel to $$Z$$-direction.

The maximum principal stresses in $$MP$$a and the orientation of the corresponding principal plane in degrees are respectively