The Merchant circle diagram showing various forces associated with a cutting process using a wedge shaped tool is given in the Figure

The coefficient of friction can be estimated from the ratio

In a machining operation with turning tool, the tool life $$(T)$$ is related to cutting speed $$v$$ $$(m/s),$$ feed $$f(mm)$$ and depth of cut $$d(mm)$$ as $$$T = C\,v{\,^{ - 2.5}}\,{f^{ - 0.9}}\,{d^{ - 0.15}}$$$
where, $$C$$ is a constant. The suggested values for the cutting parameters are: $$V=1.5$$ $$m/s,$$ $$f=0.25$$ $$mm$$ and $$d=3$$ $$mm$$ for normal rough turning. If the operation is performed at twice the cutting speed and the other parameters remain unchanged, the corresponding percentage change in tool life is ______________.

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

A steel wire of $$2$$ $$mm$$ diameter is to be drawn from a wire of $$5mm$$ diameter. The value of true strain developed is _____________ (up to three decimal places)