Problem $2(6$ points, consult Lecture $31)$: In this problem, you are asked to calculate contact fatigue life for spur gears A $20$ spur pinion with $20$ teeth and a module of $3$ mm transmits $1000$ W to a $38-$tooth gear. The pinion speed is $400$ rpm $,$ and the gears are grade $2,$ nitrided $(2\mathrm{.}5\%$ chrome, $87\mathrm{.}5$ HR$15$N$)($side note: what is the difference between nitriding and carburizing?$),20-$mm face width, uncrowned, manufactured at a No$.8$ quality standard, and considered to be a commercial enclosed gearing unit. The shock from the prime mover is normal, reaction load torque shock is also normal. Pinion life is ${10}^9$cycles and reliability is $0\mathrm{.}99.$

a$.$ Obtain the magnitude of the transmitted load and the pinion pitch$-$line velocity.

b$.$ Obtain all the modification factors in the AGMA contact stress equation.

c$.$ If the problem were in the US customary units, what are the modification factor values for AGMA contact stress?

d$.$ Obtain the contact strength value $S_c$ as well as the various factors for the AGMA contact strength calculation.

e$.$ Obtain $S_H,$ the factor of safety in contact fatigue.