$4.$ Machining

In a single point turning operation, the cylinder shown below is being machined. The rough workpiece has an initial outer diameter of $150$ mm $,$ inner diameter of $100$ mm $,$ and is $250$ mm long. The required finish outer diameter is $142$ mm $.$ The radial depth of cut for each pass is $1$ mm $.$ The axial feed rate is $\backslash (200\backslash$mathrm$\{$~mm$\}/\backslash$mathrm$\{$min$\}\backslash ),$ and the spindle speed is $2000$ rpm $.$ The specific cutting energy of the workpiece material is $\backslash (\backslash$mathrm$\{$K$\}\_\{\backslash$mathrm$\{$s$\}\}=1300\backslash$mathrm$\{$~N$\}/\backslash$mathrm$\{$mm$\}^\{2\}\backslash ).$

a$.)$ What is the volume of material to be removed from the rough workpiece?

$\backslash [$

$\backslash$mathrm$\{$V$\}=\backslash$quad $\backslash$mathrm$\{$mm$\}^\{3\}$

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b$.)$ What is the tangential component $($perpendicular to feed direction and tangent to the cylinder$)$ of the cutting force?

c$.)$ What is the material removal rate during the first pass?

d$.)$ What is the minimum time required to machine the part? $($Ignore the time needed to reposition the tool for multiple passes.$)$