In a cutting operation, which is performed with a cutting tool with a normal rake angle of $20$ and a clearance angle of $7,$ a machining power of $1850$ J$/$s is used to machine a part at a cutting velocity of $100$ m$/$min$.$ Under these cutting conditions, the thrust force is measured as $550$ N$.$ The undeformed chip thickness is set to $0\mathrm{.}2$ mm while cutting a width of $3$ mm$.$ The chip thickness is measured as $1\mathrm{.}2$ mm$,$ and the length of contact between the chip and tool is measured as $1\mathrm{.}25$ mm$.($a$)$ Calculate the proportion ofshear zone heat conducted into the workpiece $($b$)$ Calculate the maximum temperature along the tool rake face. Room temperature is $22$C$,$ density of the material is $5000$ Kg$/$m$3,$ thermal conductivity is $40$ J$/$smK$,$ and specific heat is $500$ J$/$kgK$.$ The width of the secondary deformation zone divided by the chip thickness w$0$ is assumed to be $0\mathrm{.}15$ for the above$-$mentioned cutting conditions.