$12\mathrm{.}29$ A helical compression spring used for essentially static loading has $d=0\mathrm{.}100in.,D=0\mathrm{.}625in.,N=8,$ and squared and ground ends. It is made of ASTM A$227$ cold$-$drawn steel wire.

$($a$)$ Compute the spring rate and the solid height.

$($b$)$ Estimate the greatest load that can be applied without causing long$-$term permanent set in excess of $2$ percent.

$($c$)$ At what spring free length will the load determined in part $($b$)$ cause the spring to become solid?