A helical$-$coil spring with plain ends, ground,

is to be used as a return spring on the cam$-$driven

valve mechanism shown The $1\mathrm{.}50-$inch$-$diameter

rod must pass freely through the spring. The cam

eccentricity is $0\mathrm{.}75$ inch $($i$.$e$.,$ the total stroke is

$1\mathrm{.}50$ inches$).$ The spring must exert a force of $300$

lb when at the HEDC position shown in the

sketch, and must exert a force of $150$ lb when the

crank$-$end$-$dead$-$ center $($CEDC$)$ at the bottom of

the stroke. That is$,$ the spring is preloaded into the

machine. The spring is to be made of a patented

spring steel wire that has a $200,000-$psi ultimate

tensile strength, $190,000-$psi tension yield

strength, and $90,000-$psi tensile fatigue endurance

limit$.$ G$=11,500,000$ psi. It has a mean diameter

of $2$ inches, and a wire diameter of $0\mathrm{.}28$ inch.

Determine the following:

$($a$)$ Factor of safety based on $$infinite$$ fatigue life

$($b$)$ Factor of safety based on yielding at the maximum load

$($c$)$ Number of active coils, N

$($d$)$ Spring rate, k

$($e$)$ Recommended free length of the spring, L$0($Assume a need for robust linearity$).$