A down sprue of $$180mm$$ length has diameter of $$20mm$$ at its top end. The liquid metal in the pouring cup is main-tained upto $$60$$ $$mm$$ height. What should be the theoretical diameter of down sprue at its lowest end to ensure that there is no aspiration effects.

Riser is designed so as to

A 6 metre long supported wooden beam of rectangular section 10 cm $$ \times $$ 20 cm deep is strengthened by mild steel plates 0.5 cm $$ \times $$ 10 cm wide at the top and bottom fibre over the entire length. Find the minimum supportable uniformly distributed load considering failures in steel and wood due to flexure. Weakening of wood due to screws and weakening of the steel plates due to drilled holes may be ignored.
Permissible tensile stress for steel = 156.8 N/mm²
Permissible tensile stress for steel = 14.89 N/mm²
Young's modulus of mild steel = 1.96 $$ \times $$ 10⁵ N/mm²
Young's modulus of wood = 0.117 $$ \times $$ 10⁵ N/mm²

The figure shows arrangements of springs. They have stiff nesses $${{K_1}}$$ and $${{K_2}}$$ as marked. Which of the following arrangements offers a stiffness $$ = \,{{2{K_1}{K_2}} \over {{K_1} + 2{K_2}}}$$