$7.$ A spring$-$loaded relief valve is shown in Fig. It consists of a plunger, which is mounted in the main body covering the outlet opening. It is held against the inlet opening by means of a helical compression spring, with adjustable compression. When the force due to hydraulic pressure acting on the plunger exceeds the initial setting of the spring, the plunger is lifted up$,$ permitting the oil to relieve into the outlet opening. Since the valve is occasionally used, the spring is to be designed for static load. The diameter of the plunger is $25$ mm and the outer diameter of the spring should not exceed $20$ mm because of space limitations$.$ The normal working pressure is $\backslash (0\mathrm{.}25\backslash$mathrm$\{$~N$\}/\backslash$mathrm$\{$mm$\}2\backslash )$ and the valve should open at a pressure of $\backslash (1\backslash$mathrm$\{$~N$\}/\backslash$mathrm$\{$mm$\}2\backslash ),$ with a valve$-$lift of $6$ mm from the normal position. The spring is made of oil hardened and tempered valve spring wire of GradeVW $(\backslash (\backslash$mathrm$\{$G$\}=81370\backslash$mathrm$\{$~N$\}/\backslash$mathrm$\{$mm$\}2\backslash )).$ The permissible shear stress in the wire can be taken as $\backslash (50\backslash \%\backslash )$ of the ultimate tensile strength. Design the spring and determine: $($i$)$ the wire diameter; $($ii$)$ the mean coil diameter; $($iii$)$ the stiffness of the spring; and $($iv$)$ the number of active coils. $[($i$)3\mathrm{.}6$ mm $($ii$)15$ mm $($iii$)\backslash (61\mathrm{.}36\backslash$mathrm$\{$~N$\}/\backslash$mathrm$\{$mm$\}\backslash )($iv$)9]$