Introduction:
In this problem, we are given two steel wires of equal lengths and same diameter. The wires are made of two springs, s1 and s2, with mean diameters of 75mm and 60mm respectively. We need to find the ratio of their stiffness.

Explanation:
To find the ratio of stiffness, we need to understand the concept of stiffness and how it is related to the diameter of the wire.

Stiffness:
Stiffness is a measure of how much a spring resists deformation when a force is applied to it. It is defined as the force required to produce a unit deflection. Stiffness is directly proportional to the cross-sectional area of the wire and inversely proportional to its length.

Formula:
The stiffness of a spring can be calculated using the formula:

Stiffness (k) = (G * A) / L

Where:
k = Stiffness
G = Shear modulus of the material
A = Cross-sectional area of the wire
L = Length of the wire

Analysis:
From the given information, we know that the wires have the same length and diameter. Let's assume the length of the wires is L and the diameter of the wires is D.

Wire s1:
- Mean diameter (D) = 75mm
- Radius (r) = D/2 = 75/2 = 37.5mm = 0.0375m

Wire s2:
- Mean diameter (D) = 60mm
- Radius (r) = D/2 = 60/2 = 30mm = 0.03m

Calculating the Stiffness:
To calculate the stiffness of the wires, we need to know the shear modulus of the material. Let's assume it is constant and denoted by G.

Wire s1:
Stiffness (k1) = (G * A1) / L

Wire s2:
Stiffness (k2) = (G * A2) / L

Since the wires have the same length, the ratio of their stiffness can be calculated as:

Ratio of Stiffness (k1/k2) = (G * A1) / (G * A2) = A1 / A2

Calculating the Cross-sectional Area:
The cross-sectional area of a wire can be calculated using the formula:

Area (A) = π * r^2

Wire s1:
Area (A1) = π * (0.0375)^2

Wire s2:
Area (A2) = π * (0.03)^2

Calculating the Ratio of Stiffness:
Using the values of A1 and A2, we can calculate the ratio of stiffness:

Ratio of Stiffness (k1/k2) = (π * (0.0375)^2) / (π * (0.03)^2)

Simplifying the equation:

Ratio of Stiffness (k1/k2) = (0.0375)^2 / (0.03)^2

Ratio of Stiffness (k1/k2) = (0.00140625

Y