A helical compression spring will be used in a pressure relief valve. When the valve is closed, the spring length is $50$ mm $,$ and the spring force is to be $7$ N $.$ The spring uses a wire diameter of $d=1\mathrm{.}5mm$ and a mean spring diameter of $D=10mm,$ and has a solid length of $L_s=30mm.$ Use hard$-$drawn ASTM A$227$ wire, squared and ground ends, and ignore buckling since the spring is in a cage. Use $G=79\mathrm{.}3$GPa.

Perform the following:

a$)$ Find the force needed to compress the spring to its solid length,

b$)$ Find the number of active coils and spring index,

c$)$ Find the spring rate,

d$)$ Find the spring's free length, and

e$)$ Find the safety factor against torsional yielding