In computer aided drafting practice, an arc is defined by

Two castings, a cube and a slab of the same material solidify under identical conditions. The volumes of the castings are equal but the slab dimensions are in the ratio of $$1:2:4$$, find the ratio of the solidification times of the cube to that of the slab.

A cylindrical billet of $$100$$ $$mm$$ diameter is forged from $$50$$ $$mm$$ height to $$40$$ $$mm$$ at $${1000^0}$$$$C$$. The material has constant flow stress of $$80$$ $$MPa.$$ Find the work of deformation. If a $$10$$ $$kN$$ drop hamper is used to complete the reduction in one blow. What will be the height of fall.

Two different pairs of sheets of same material are welded by resistance-spot welding. In one pair, the average radius $$(r)$$ of each spherical bridge is $$0.2$$ $$mm$$ and the number of bridges per $$c{m^2}\,\,\left( n \right)$$ is $$25.$$ In another pair, the number of bridges per $$c{m^2}$$ is $$50$$ with the same $$'r'$$ of bridge. The contact resistance $${R_c}$$ per unit area is given by $${R_c} = 0.85$$ $$\left( {\rho /n\pi r} \right)$$ where $$'\,\rho \,'$$ is the resistivity of metal. If the voltage applied is $$5$$ volts and the resistivity of metal is $$2 \times {10^{ - 5}}$$ ohm-cm. The rate of heat generated per $$c{m^2}$$ in each case