$4.[25$ Pts$.]$ A $1040($Annealed$)$ steel shaft is to transmit $20$ hp while rotating at $1750$ rpm $.$ Determine the minimum required diameter for the shaft, if the tensile yield strength is $\backslash ($ S$\_\{$y$\}=75\backslash$mathrm$\{$ksi$\}\backslash )$ and the safety factor is $3$ based on initial yielding:

a$)$ According to the maximum$-$normal$-$stress theory.

b$)$ According to the maximum$-$shear$-$stress theory.

c$)$ According to the maximum$-$distortion$-$energy theory.