Power transmission design: The maximum twisting moment a circular shaft can safely resist equals the permissible shear stress multiplied by which geometric property?

Introduction / Context:
For shafts under torsion, the torsional strength depends on material strength (allowable shear stress) and a geometry-dependent property. Recognizing the correct property is essential for sizing shafts in mechanical and civil applications (e.g., hoists, drives, and rotating machinery).

Given Data / Assumptions:

• Elastic torsion theory applies.
• Allowable shear stress is τ_allow.
• Geometric properties include J (polar moment) and Zp (polar section modulus).

Concept / Approach:
Torsion formula: τ_max = T * r / J. Rearranging gives T = τ_max * (J / r). The quotient (J / r) is the polar section modulus Zp. Hence the safe torque equals τ_allow * Zp.

Step-by-Step Solution:
Start with τ = T * r / J.At allowable stress: τ_allow = T_allow * r / J.Therefore T_allow = τ_allow * (J / r) = τ_allow * Zp.

Verification / Alternative check:
For a solid circular shaft, J = (π d^4) / 32 and r = d/2, so Zp = J/r = (π d^3) / 16. Using this in T_allow = τ_allow * (π d^3) / 16 matches standard design tables.

Why Other Options Are Wrong:
Area moment of inertia and polar moment J alone do not directly give torque capacity without dividing by r.Modulus of rigidity G affects angle of twist, not ultimate torque.Section area is not the governing torsion property.

Common Pitfalls:
Confusing J (polar moment) with Zp (polar section modulus). Zp is the correct multiplier with τ_allow for maximum safe torque.

Final Answer:
Polar section modulus (polar modulus)