State the torsion formula for a circular shaft and identify each variable.

State the torsion formula for a circular shaft and identify each variable.

• A. tau = J/(T*r); tau is shear stress, T = torque, r = outer radius, J = polar moment.
• B. tau = T/(r*J); tau equals shear modulus, T = torque, r = inner radius, J = polar moment.
• C. tau = T*r/J; tau = shear stress, T = torque, r = outer radius, J = polar moment of inertia.
• D. tau = J*T*r; tau is shear strain, T = torque, r = outer radius, J = polar moment.

Answer: (C)

Under torsion, the shear stress in a circular shaft at a radius r is proportional to r and inversely proportional to the polar moment of inertia J. The formula is tau(r) = T r / J, where T is the applied torque, r is the distance from the center (the outer radius gives the maximum stress), and J is the polar moment of inertia of the cross-section. For a solid circular shaft, J = π R^4 / 2, so the maximum shear stress occurs at the outer surface: tau_max = T R / J. This matches the statement that tau equals shear stress, T is torque, r is the outer radius, and J is the polar moment of inertia. The other expressions misplace variables, mix up units or physical meanings (such as shear modulus or shear strain), or use the inner radius, which does not describe the standard torsion relation.

Under torsion, the shear stress in a circular shaft at a radius r is proportional to r and inversely proportional to the polar moment of inertia J. The formula is tau(r) = T r / J, where T is the applied torque, r is the distance from the center (the outer radius gives the maximum stress), and J is the polar moment of inertia of the cross-section. For a solid circular shaft, J = π R^4 / 2, so the maximum shear stress occurs at the outer surface: tau_max = T R / J. This matches the statement that tau equals shear stress, T is torque, r is the outer radius, and J is the polar moment of inertia. The other expressions misplace variables, mix up units or physical meanings (such as shear modulus or shear strain), or use the inner radius, which does not describe the standard torsion relation.