The thickness of a plate is reduced from $$30$$ $$mm$$ to $$10$$ $$mm$$ by successive cold rolling passes using identical rolls of diameter $$600$$ $$mm.$$ Assume that there is no change in width. If the coefficient of friction between the rolls and the work piece is $$0.1,$$ the minimum number of passes required is

In multi-pass operation, a round bar of $$10$$ $$mm$$ diameter and $$100$$ $$mm$$ length is reduced in cross-section by drawing it successively through a series of seven dies of decreasing exit diameter. During each of theses drawing operations, the reduction in cross-sectional area is $$35\% $$. The yield strength of the material is $$200$$ $$MPa.$$ Ignore strain hardening.

The total true strain applied and the final length (in $$mm$$), respectively, are

In multi-pass operation, a round bar of $$10$$ $$mm$$ diameter and $$100$$ $$mm$$ length is reduced in cross-section by drawing it successively through a series of seven dies of decreasing exit diameter. During each of theses drawing operations, the reduction in cross-sectional area is $$35\% $$. The yield strength of the material is $$200$$ $$MPa.$$ Ignore strain hardening.

Neglecting friction and redundant work, the force (in $$kN$$) required for drawing the bar through the first die, is

To measure the effective diameter of an external metric thread (included angle is $$60$$ degrees) of $$3.5$$ $$mm$$ pitch, a cylindrical standard of $$30.5$$ $$mm$$ diameter and two wires of $$2$$ $$mm$$ diameter each are used. The micrometer reading over the standard and over the wires are $$16.532$$ $$mm$$ and $$15.398$$ $$mm$$ respectively. The effective diameter (in $$mm$$) of the thread is