Two springs, S1 and S2, with mean diameters of 90 mm and 75 mm, respectively, are made up of two equal lengths of hardened steel wires of the same diameter. The ratio of the stiffness of S1 to that of S2 is: 

Concept:-

Stiffness or Spring rate of the spring - 

⇒ \(k = \frac{Gd^4}{8D^3N}\;\)      ...(1)

Where,

N = Number of active coils

D = Mean diameter of the coil

G = Modulus of rigidity  

d = Diameter of spring wire

k = Spring stiffness or spring rate  

Given:-

D₁ = 90 mm, D₂ = 75 mm, G₁ = G₂, d₁ = d₂ , N₁ = N₂

Calculation:-

From equation (1) and the given conditions, it can be deduced that Stiffness depends only on Mean coil diameters. 

So, \(k \propto \frac{1}{D^3}\;\)

Now the ratio of the stiffness of S1 to that of S2 is, 

⇒ \(\frac{k_1}{k_2}=\left ( \frac{D_2}{D_1} \right )^3\;\;\)

⇒ \(\frac{k_1}{k_2}=\left ( \frac{75}{90} \right )^3 = \frac{125}{216}\;\;\)