Problem $2(50$ points$).$ A single$-$tooth right$-$hand worm transmits $1$ hp at $1000re\frac{v}{m}in$ to a $15-$tooth

worm gear. The gear has a transverse diametral pitch of $6$ teeth$/$in and a face width of $1$ in $.$ The worm

has a pitch diameter of $2$ in and a face width of $2\mathrm{.}5$ in $.$ The normal pressure angle is $14\mathrm{.}5.$ Consider that

worms and gears are made of high$-$quality materials.

$($a$)$ Find the axial pitch, the center distance, the lead, and the lead angle.

$($b$)$ As shown in the following figure, the gear is supported by bearings A and $B.$ Find the forces exerted

against the worm$-$gear shaft and the output torque.