Please Solve this Machine Design Question and make sure to provide all the steps and final values.

PROBLEM $1(50$ points$)$

A $57$ tooth spur gear is in contact with a $23$ tooth pinion. The diametral pitch is $4$in$^(-1)$ and the

pressure angle is $20^($@$).$ The pinion transmits $60$ Hp at $1500$ rpm $.$ The tooth width of the pinion is

$2\mathrm{.}85$ in$.$ Find the bending safety factor of the pinion for $10^(7)$ cycles assuming steady$-$torque, full$-$

depth teeth, HPSTC loading, Q$\_($v$)=9,$ an AISI $4340$ steel pinion $($uncorrected endurance strength

of $40$ ksi $),$ class $40$ cast iron gear $($uncorrected endurance strength of $13$ ksi $),87\%$ reliability and

machined surface.