A metallic strip having a thickness of $$12$$ $$mm$$ is to be rolled using two steel rolls, each of $$800$$ $$mm$$ diameter. It is assumed that there is no change in width of the strip during rolling. In order to achieve $$10\% $$ reduction in cross-sectional area of the strip after rolling, the angle subtended (in degrees) by the deformation zone at the center of the roll is

In a single-pass rolling operation, a $$200$$ $$mm$$ wide metalic strip is rolled from a thickness $$10$$ $$mm$$ to a thickness $$6$$ $$mm.$$ The roll radius is $$100$$ $$mm$$ and it rotates at $$200$$ $$rpm$$. The roll-strip contact length is a function of roll radius and, initial and final thickness of the strip. If the average flow stress in plane strain of the strip material in the roll gap is $$500$$ $$MPa$$, the roll separating force (in $$kN$$) is _________________

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 __________________.