It is required to design a helical compression spring subjected to a force of $500$ N$.$ The deflection of the spring corresponding to this force is approximately $20$ mm$.$ The spring index should be $6.$ The spring is made of cold$-$drawn steel wire with ultimate tensile strength of $1000$ N$/$mm$2.$ The permissible shear stress for the spring wire can be taken as $50\%$ of the ultimate tensile strength $($G $=81370$ N$/$mm$2).$