Which of the following formulas gives the shaft diameter d in terms of torque T and allowable shear stress τ_allow?

• A. d = [8 T /(π τ_allow)]^(1/3)
• B. d = [16 T /(π τ_allow)]^(1/3)
• C. d = [16 T /(π τ_allow)]^(1/2)
• D. d = [32 T /(π τ_allow)]^(1/3)

Answer: (C)

When a solid circular shaft is subjected to pure torsion, the shear stress at the outer surface relates to torque and diameter through the geometry of the shaft. The polar moment of inertia for a solid circle is J = π d^4 / 32, and the shear stress at the surface is τ = T c / J with c = d/2. Substituting gives τ = (16 T) / (π d^3).

To find the diameter in terms of torque and allowable shear stress, solve for d: d^3 = 16 T / (π τ_allow), so d = [16 T / (π τ_allow)]^(1/3).

This cube-root relationship reflects how the diameter must grow with torque and shrink as allowable shear stress increases. A square-root form would imply a different, incorrect dependence for torsion in a solid shaft.