A concentric spring is used as a valve spring in a heavy$-$duty diesel engine. It

consists of two helical compression springs having the same free length and same

solid length. The composite spring is subjected to a maximum force of $6000$ N and

the corresponding deflection is $50$ mm$.$ The maximum torsional shear stress

induced in each spring is $800$ N$/$mm$2.$ The spring index of each spring is $6.$ Assume

same material for two springs and the modulus of rigidity of spring material is

$81370$ N$/$mm$2.$ The diametral clearance between the coils is equal to the difference

between their wire diameters. Calculate:

$($i$)$ the axial force transmitted by each spring;

$($ii$)$ wire and mean coil diameters of each spring;

and

$($iii$)$ number of active coils in each spring.