Question $1$

Achieving optimal cutting speed involves two objectives, one of which is the maximization of the production rate. For maximum production rate, the speed that minimizes machining time per workpiece is determined. Also, minimizing cutting time per unit is equivalent to maximizing production rate. Regarding turning operation, three elements contribute to the total production cycle time, $T_c$ for a part. These are part handling time, $T_c,$ machining time, $T_m$ and tool change time, $T_t.$ These can be mathematically expressed as $T_c=T_h+T_m+\frac{T_t}{n_p}.$

If the cycle time per piece is minimum at a certain cutting speed such that $\frac{d{T}_c}{dv}=0,$ show that $T_{\text{m}\text{a}\text{x}}=(n^{-1}-1)T_t$

Note: All the steps must be clearly shown.

Question $2$

Apart from the removal of heat and reduction of friction from the tool$-$chips interface, cutting fluids provide additional benefits, such as flushing away chips $($especially in grinding and milling$),$ reducing the temperature of the workpart for easier handling, reducing cutting forces and power requirements, improving dimensional stability of the workpart, and improving surface finish. Cutting fluid also increases the value of $C$ in the Taylor tool life equation. In a certain machining situation using HSS tooling, the $C$ value was increased from $C=1\mathrm{.}016$ to $C=1\mathrm{.}143$ as a result of the application of the cutting fluid. The $n$ value is the same with or without fluid at $n=0\mathrm{.}125.$ Cutting speed used in the operation is $v=0\mathrm{.}635\frac{m}{s}.$ Feed $=0\mathrm{.}254m\frac{m}{r}ev$ and depth $=2\mathrm{.}54mm.$ The effect of the cutting fluid can be to either increase cutting speed $($at the same tool life$)$ or increase tool life $($at the same cutting speed$).($a$)$ What is the cutting speed that would result from using the cutting fluid if tool life remains the same as when no cutting fluid was applied? $($b$)$What will be the tool life if the cutting speed remained at $0\mathrm{.}635\frac{m}{s}?($c$)$ Economically, which effect is better, given that tooling cost $=$ R$34$ per cutting edge, tool change time is $2\mathrm{.}5$ min $,$ and operator and machine rate is R$522/$hr$?$ Justify your answer with calculations, using cost per $m^3$ of metal machined as the criterion of comparison. Ignore the effects of workpart handling time.