Two solid cylinders of equal diameter have different heights. They are compressed plastically by a pair of rigid dies to create the same percentage reduction in their respective heights. Consider that the die-work-piece interface friction is negligible. The ratio of the final diameter of the shorter cylinder to that of the longer cylinder is _________________.

Open die forging of a cylinder made of a rigid perfectly plastic material with yield strength of $$200$$ $$MPa$$ having a height of $$25$$ $$mm$$ and diameter of $$25$$ $$mm$$ is being carried out. The cylinder is subjected to a true compressive strain of $$3.6$$ during the process. Assuming frictionless and homogeneous deformation, the energy expended, in $$kJ$$ is __________________.

A $$80$$ $$mm$$ thick steel plate with $$400$$ $$mm$$ width is rolled to $$40mm$$ thickness in $$4$$ passes with equal reduction in each pass, by using rolls of $$800mm$$ diameter. Assuming the plane-strain deformation, what is the minimum coefficient of friction required for unaided rolling to be possible?

In a single pass rolling process using $$410mm$$ diameter steel rollers, a strip of width $$140mm$$ and thickness $$8mm$$ undergoes $$10\% $$ reduction of thickness. The angle of bite in radians is