$1.$ It is required to design a helical compression spring subjected to a maximum force of $1250$ N$.$ The deflection of the spring corresponding to the maximum force should be approximately $30$ mm$.$ The spring index can be taken as $6.$ The spring is made of patented and cold$-$drawn steel wire. The ultimate tensile strength and modulus of rigidity of the spring material are $1090$ and $81370$ N$/$mm$2$ respectively. The permissible shear stress for the spring wire should be taken as $50\%$ of the ultimate tensile strength. Design the spring and calculate: $($i$)$ wire diameter; $($ii$)$ mean coil diameter; $($iii$)$ number of active coils; $($iv$)$ total number of coils; $($v$)$ free length of the spring; and $($vi$)$ pitch of the coil. Draw a neat sketch of the spring showing various dimensions.