Q$-13$ The extension spring shown in the figure has fulltwisted loop ends. With $OD=37\mathrm{.}5mm,$ wire diameter $d=4mm,r_1=16\mathrm{.}75mm,r_2=12\mathrm{.}5mm.$ The material is AISI $1065$ OQ&T wire. The spring's body coils $N_b=84$ coils and is closewound with a preload of $F_i=70N.$

$($a$)$ Find the closed length $L_{\text{@}}$ of the spring.

$($b$)$ Find the torsional stress ${}_i$ in the spring corresponding to the preload.

$($c$)$ Estimate the spring rate, k in $\frac{N}{m}m.$

$($d$)$ What load would cause permanent deformation