In strength of materials, identify the incorrect proportionality in the torsional resistance of a circular shaft (choose the statement that is NOT directly proportional to torsional resistance).

Introduction / Context:
Torsion of circular shafts is a core topic in strength of materials. The torque–twist relation for a prismatic circular shaft is T / J = G * θ / L, where T is torque, J is the polar moment of inertia, G is modulus of rigidity, θ is angle of twist, and L is length. This question asks which proportionality statement is incorrect when discussing what torsional resistance (torque for a given twist) depends upon.

Given Data / Assumptions:

• Torsion of a homogeneous, circular shaft.
• Elastic (Hookean) behavior, Saint-Venant torsion.
• Symbols: G (shear modulus), θ (angle of twist), L (length), J (polar moment of inertia).

Concept / Approach:
From T = (G * J / L) * θ, torsional resistance grows with G, J, and θ, and decreases with L. Importantly, the relevant geometric property is the polar moment of inertia J, not the area (second) moment of inertia I used for bending about an axis. Confusing I (bending stiffness) with J (torsional stiffness) is a common mistake.

Step-by-Step Solution:
Start with T = (G * J / L) * θ.Identify proportionalities: T ∝ G, T ∝ J, T ∝ θ, and T ∝ (1 / L).Note that “moment of inertia of the shaft section (I)” refers to bending, not torsion.Therefore, the statement linking torsional resistance directly to I is incorrect; it should be J.

Verification / Alternative check:
Check limiting cases: doubling G or J doubles T for the same θ and L; doubling L halves T; using I in place of J fails dimensional and physical consistency for torsion.

Why Other Options Are Wrong:

• Modulus of rigidity (G): Correct proportionality—higher G increases torsional resistance.
• Angle of twist (θ): For a given shaft, larger θ requires proportionally larger T.
• Reciprocal of length (1 / L): Longer shafts twist more easily; resistance varies inversely with L.
• Polar moment of inertia (J): Correct geometric measure for torsion.

Common Pitfalls:
Mixing up I (bending) and J (torsion); assuming length does not influence torsional stiffness; forgetting that G—not E—controls torsional response.

Final Answer:
Moment of inertia of the shaft section (second moment, I)