Solution:

Given data:
- Outer diameter of the shaft (d) = 25 cm
- Inner diameter of the shaft (d₁) = 17.5 cm
- Maximum shearing stress (𝜏) = 75 Mega Newton per meter square

We need to find the torque (T) required to produce the given maximum shearing stress in the hollow circular shaft.

Formula:
- The maximum shearing stress (𝜏) in a hollow circular shaft is given by:

𝜏 = (T×r) / J
where,
T = torque applied
r = radius of the shaft
J = polar moment of inertia of the shaft

- The polar moment of inertia of a hollow circular shaft is given by:

J = (π/2)×(d⁴ - d₁⁴)

where,
d = outer diameter of the shaft
d₁ = inner diameter of the shaft

Calculation:
- The radius of the shaft (r) = (d/2) = 12.5 cm
- The polar moment of inertia (J) = (π/2)×[(0.25m)⁴ - (0.175m)⁴] = 1.083×10⁻³ m⁴
- The maximum shearing stress (𝜏) = 75 Mega Newton per meter square = 75×10⁶ N/m²

Using the formula, we can find the torque (T) as:

T = (𝜏×J) / r

Substituting the values, we get:

T = [(75×10⁶)×(1.083×10⁻³)] / 0.125
T = 650.625 Nm

Therefore, the torque required to produce a maximum shearing stress of 75 Mega Newton per meter square in the material of a hollow circular shaft of 25 cm outer diameter and 17.5 cm inside diameter is 650.625 Nm.

Explanation:
- The given problem is solved using the formula for maximum shearing stress in a hollow circular shaft.
- The radius of the shaft and the polar moment of inertia of the shaft are calculated using the given dimensions of the shaft.
- The torque required to produce the given maximum shearing stress is calculated using the formula for maximum shearing stress and substituting the values.
- The final answer is obtained in Newton-meter (Nm) unit which represents the torque required to produce the given maximum shearing stress in the hollow circular shaft.