Q$3$: $(25$ pts$)$

The external loads acting on the shaft in a gear system are given as $M_a=412\mathrm{.}5kN*mm,T_m=366,0kN*mm,M_m=T_a=$

$0.$ For the shaft the material $1020CD$ steel is selected wchic has $S_{ut}=470$MPa. Other knowns for the design are: $K_t=$

$1\mathrm{.}7,K_{ts}=1\mathrm{.}5,K_f=K_t,K_{fs}=K_{ts},a=3,04,b=-0,217,k_b=0\mathrm{.}9,k_c=k_d=k_e=1,n=1,5.$

a$)$ Calculate the schaft dimameter

b$)$ Calculate the safety factor according to the Goodmann ciriterion.