Problem $1(6$ points, consult Lecture $30)$: In this problem, you are asked to calculate bending fatigue life for spur gears

A $20$ spur pinion with $18$ teeth and a module of $2\mathrm{.}5$ mm transmits $1500$ W to a $36-$tooth gear. The pinion speed is $200$ rpm $,$ and the gears are grade $1$ induction$-$hardened with type A pattern, $18-$mm face width, uncrowned, manufactured at a No$.6$ quality standard, and considered to be of open gearing quality installation. The shock from the prime mover is normal, the reaction load torque shock is also normal. Pinion life is ${10}^8$cycles and reliability is $0\mathrm{.}98.$ Note: flame or induction hardening is a process of hardening gear teeth by heating them via conducting coils. Type A hardening refers to the entire contour of the tooth undergoing hardening whereas type B is a localized hardening, resulting in slightly weaker bending strength.

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

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

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

d$.$ Obtain the bending strength value $S_t$ as well as the various factors for the AGMA bending strength calculation.

e$.$ Obtain $S_F,$ the factor of safety in bending fatigue