An extension spring cycles between $4$ N and $8$ N $,$ with a deflection difference of $15$ mm between the two loads. Given: $D=12$

mm $,$ and the wire diameter is $1\mathrm{.}0$ mm $.$ bend radii are $r_1=\frac{D}{2},$ and $r_2=3d.$ No preload is applied.

a$.$ At the maximum load value of $8$ N $,$ check for static yielding of the coil and the hooks.

b$.$ Using the Zimmerli data for peened springs check for coil and hook fatigue using the Modified Goodman criterion for fatigue failure.