$4-$ A rectangular keyway is to be broached in a cylindrical steel shaft with a hardness of $200$ HB using a broaching machine. The details of the operation are as follows: Length of the keyway: $L=150mm,$ Width of the keyway: w$=20$ mm$,$ Depth of the keyway: $d=6mm,$ Cutting speed: $v_c=10\frac{m}{m}in,$ Material removal rate per tooth: $MRR=0\mathrm{.}2m\frac{m^3}{?}$ tooth, Total number of cutting teeth on the broach: $n_t=30,$ Distance between successive teeth: $p=4mm,$ Cutting force per tooth: $F_t=200N,$ Efficiency of the broaching machine: $=85\%,$ Determine:

a$)$ Total time to complete the broaching operation, considering the full stroke length includes the keyway length L plus an additional $20$ mm for setup and exit.

b$)$ Material removal rate $($MRR$)$ for the operation.

c$)$ Power required to perform the broaching operation.

d$)$ The force acting on the broach during cutting, assuming all teeth engage simultaneously after the first few passes.

e$)$ The total energy consumption for the operation $[25$ Pts$].$

Please solve all parts with steps