Problem $3$

A $20$ spur pinion with $20$ teeth and a module of $2\mathrm{.}5mm$ transmits $120W$ to a $36-$tooth gear. The pinion speed is $100re\frac{v}{m}in,$ and the gears are grade $1,18mm$ face width, through$-$hardened steel at $200$ Brinell, uncrowned, manufactured to a No$.6$ quality standard, and considered to be of open gearing quality installation. Find the AGMA bending and contact stresses and the corresponding factors of safety for a pinion life of ${10}^8$ cycles and a reliability of $0\mathrm{.}95.$