A compression spring is needed to fit within a $25\mathrm{.}400-mm$ diameter hole. To allow for some clearance, the outside diameter of the spring is to be no larger than $22\mathrm{.}860$ mm $.$ To ensure a reasonable coil, use a spring index of $8.$ The spring is to be used in a machine by compressing it from a free length of $76\mathrm{.}200$ mm to a solid length of $25\mathrm{.}400$ m $.$ The spring should have squared ends, and is unpeened, and is to be made from music wire.

$($a$)$ Determine a suitable wire diameter.

$($b$)$ Determine a suitable total number of coils.

$($c$)$ Determine the spring constant.

$($d$)$ Determine the static factor of safety when compressed to solid length.

$($e$)$ Determine the fatigue factor of safety when repeatedly cycled from free length to solid length. Use the Gerber$-$Zimmerli fatigue$-$failure criterion.$($Please give the detail of caculation and the dates table sources$)$