A machine with uniform loading is driven uniformly by a $3$ kW electric motor running at $1440\frac{r}{m}in$ through a $2\mathrm{.}5$:$1$ spur gear reduction gearbox. The spur gears are machine cut with a $20$ degree normal pressure angle, and a gear module of $2\mathrm{.}5$ mm $.$ The $En24(Y)Su=1240$MPa,$Se=413,3$MPa,$J=0\mathrm{.}32,$ alloy steel pinion has $18$ teeth and is keyed directly to the motor shaft as shown below. The gear is made from $En9(T)Su=850$MPa,$Se=283,3$MPa,$J=0,39$ carbon steel and the endurance limit in both cases may be taken as $\frac{1}{3}Su.$ The operating temperature of the gearbox is not expected to exceed $80$ degrees C $,$ and $50\%$ reliability is required. Hint : $Sd=S\frac{e}{R}F,Se=S\frac{u}{3}.$

a$)$ Which is weaker, the gear or the pinion? Determine the reserve factor for the gearbox based on the weaker part, considering the bending fatigue failure of the teeth.

b$)$ Calculate the life expectancy of the gearbox based on the surface endurance strength and surface fatigue stress of the pinion.