A broken$-$in multiple$-$disk dry clutch is used to drive a commercial air compressor through a gear reducer $($ie gearbox$)$ as shown in the following figure. For this application, the air compressor requires $375$ hp at $1200$ rpm$,$ the gearbox has a gear ratio of Ngb $=1\mathrm{.}5,$ and there is a $12\%$ power loss through the gearbox. Based on space considerations, the outside diameter of the clutch is $8$ inches. The clutch plates are made of high$-$carbon steel and the clutch friction disks are made of sintered bronze. Design values for the high$-$carbon steel plates and sintered bronze disks are pmax $=125$ psi and mu $=0\mathrm{.}1.$ Using a factor of safety of $2\mathrm{.}0$ with respect to torque capacity, determine appropriate values for:

a$.)$ The clutch disk inside diameter for maximum torque transfer through the clutch.

b$.)$ The number of friction interface surfaces $($N$).$

c$.)$ The number of sintered bronze friction disks.

d$.)$ The number of high$-$carbon steel plates.

e$.)$ The smallest clamping force that will provide the desired torque capacity.