A single$-$bodied helical$-$coil torsion spring has a wire diameter of $1$ mm and an outside coil diameter of $9$ mm $.$ The tightly coiled

spring has $15\mathrm{.}5$ coils, with an end extension of $l=12mm$ from the coil center to the point of load application at each end. The spring material is a

steel alloy with $S_{ut}=2030$MPa. Let $E=200$GPa and $S_y=0\mathrm{.}78S_{ut}$

a$.$ Calculate the torsional spring rate for this spring. Neglect contribution of end extensions.

b$.$ If a torque $0\mathrm{.}10$ N $-$m were applied, what angular deflection $($in degrees$)$ would be expected? Neglect contribution of end extensions.

c$.$ What is the value of the maximum stress in the spring wire under the $0\mathrm{.}10N-m$ static torque? Obtain the static factor of safety.

d$.$ If a design safety factor of $1\mathrm{.}5,$ based on minimum ultimate tensile strength, is desired, would the spring design be acceptable? Compare

with your answers from part $($c$).$

e$.($optional part, no extra credit, fair game for exam$)$ If the applied moment fluctuates between $0\mathrm{.}1$ Nm and $0\mathrm{.}2$ Nm $,$ determine the factor

of safety against fatigue. Use a fatigue life of one million cycles, and non$-$shot peened ASTM A$228$ steel for fatigue date. The Goodman

fatigue criterion is to be used for this part, with endurance limits defined from a repeated bending test.