Problem $3(8$ points$)$: The Figure below shows a drive system in which a $20-$hp electric motor drives three separate output shafts. Gear $A$ is mounted on the motor shaft that has a rotational speed of $1750$ rpm clockwise. Gear A drives a gear train consisting of gears B$,$ C$,$ and D that deliver power through the shafts on which they are mounted. All gears have a diametral pitch of $P_d=8.$ The following data are given for the gear system:

Power delivered by gears $B,C,$ and $D$:$P_B=8hp,P_C=7hp,P_D=5$

Numbers of teeth for all gears: $N_A=24,N_B=48,N_C=96,N_D=24$

Determine the following of the drive system:

A$)$ The pitch diameter of each gear.

B$)$ The center distance of each gear mesh.

C$)$ The rotational speed of each output shaft.

D$)$ The torque through each output shaft.

E$)$ The tangential force on the teeth of each gear