A double$-$thread steel worm rotates at $1800$ rev$/$min$,$ meshing with a $24-$tooth

worm gear transmitting $2$ hp to the output shaft. The worm pitch diameter is $3$ in and the tangential

diametral pitch of the gear is $4$ teeth$/$in$.$ The normal pressure angle is $20\backslash$deg $.$ The ambient temperature is

$60\backslash$deg F$.$ The application factor is $1\mathrm{.}25$ and the design factor is $1\mathrm{.}2$; the gear face width is $2$ in$,$ the lateral

case area is $600$ in$2,$ and the gear is centrifugally cast bronze. $($a$)$ Find the gear geometry. $($b$)$ Find the

transmitted gear forces and the mesh efficiency. $($c$)$ Is the mesh sufficient to handle the loading? $($d$)$

Estimate the lubricant sump temperature.