The shaft shown in the figure is machined from AISI $1040$ CD steel. The shaft rotates at $1600rpm$ and is supported i rolling bearings at A and $B.$ The applied forces are $F_1=11kN$ and $F_2=4\mathrm{.}5kN.$

$($a$)$ Determine the minimum fatigue factor of safety based on achieving infinite life. If infinite life is not predicted, estimate the number of cycles to failure. Also, check for yielding. $S_y=490$MPa and $S_{ut}=590$MPa and take $f=0\mathrm{.}867.$

$($b$)$ Determine the critical speed of the shaft using Rayleigh's equation considering same diameters for each section assuming $d=40mm.$