Torsion of circular shafts — choose the correct unified torsion equation below (valid for homogeneous, isotropic shafts under elastic twist). State the standard relationship among applied torque, polar moment of inertia, maximum shear stress at the outer radius, shear modulus, angle of twist, and shaft length.

Introduction:
In strength of materials, the elastic torsion of circular shafts is governed by one compact relation that links torque, section geometry, shear stress, material stiffness in shear, and twist. Identifying the correct form tests understanding of symbols and the assumptions of linear elasticity.

Given Data / Assumptions:

• Prismatic, circular shaft; homogeneous and isotropic.
• Small deformations; linear elastic behavior.
• Symbols: T (torque), J (polar moment of inertia), τ (shear stress at radius R), R (outer radius), G (shear modulus), θ (angle of twist in radians), L (gauge length).

Concept / Approach:
The torsion formula for circular shafts simultaneously expresses stress distribution and angular twist. J is for torsion (not I), and G controls shear deformation (not E).

Step-by-Step Solution:

Verification / Alternative check:
Dimensional check confirms consistency: [T]/[L^4] = [stress]/[length] = [modulus][rotation]/[length].

Why Other Options Are Wrong:

• Uses σ or E: bending/axial symbols (incorrect for torsion).
• Using I or y belongs to bending (T/I or σ/y are bending analogs).
• Replacing G with E confuses shear and normal stiffness.

Common Pitfalls:
Mixing I with J; swapping E and G; omitting the outer radius R when relating τ and T.

Final Answer: