$7.$ A $2-$tooth right hand worm transmits $1$ hp at $1200$ rpm to a $30-$tooth worm gear. The gear has a transverse diametral pitch of $6$ teeth$/$in and a face width of $\backslash (1\backslash$mathrm$\{$in$\}\backslash ).$ The worm has a pitch diameter of $2$ in and a face width of $2\mathrm{.}5$ in $.$ The normal pressure angle is $\backslash (14\mathrm{.}5^\{\backslash$circ$\}\backslash ).$ The arrows indicate the direction of the three forces acting on the gear. The gear is supported by bearings $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash ).$ The axial resistance force on the shaft can be assumed at position $\backslash ($ B $\backslash ).(15\text{'}$ in total$)$

$(1)$ Find the axial pitch, the pitch diameter of the gear, the lead, and the lead angle.

$(2)$ Find the pitch line speed of the worm and gear.

$(3)$ Find the forces on the gear.

$(4)$ Find the forces exerted by the bearings $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash )$ against the worm$-$gear shaft.

$(5)$ Find the output torque by the gear shaft. Schematic Figure for Problem $\backslash$#$7$