The maximum twisting moment a shaft can resist is determined by the product of the permissible shear stress and the polar modulus.

Explanation:
A shaft is a cylindrical component that is used to transmit power or torque from one part of a machine to another. When a shaft is subjected to a twisting force, it experiences a twisting moment, also known as torque. The ability of a shaft to resist this twisting moment is crucial to ensure its structural integrity and prevent failure.

The permissible shear stress is the maximum shear stress that a material can withstand without experiencing permanent deformation or failure. It is an important design parameter that depends on the material properties and the safety factor considered.

The polar modulus, also known as the torsion modulus or the modulus of rigidity, is a material property that relates the shear stress to the shear strain in a shaft. It quantifies the stiffness of a material to resist shear deformation under torsional loading.

The maximum twisting moment that a shaft can resist is given by the product of the permissible shear stress and the polar modulus. This relationship is derived from the torsion equation, which relates the twisting moment, the polar moment of inertia, and the shear stress.

The polar modulus is a fundamental property of a material and is determined experimentally. It represents the ratio of shear stress to shear strain in a material under torsional loading. A higher polar modulus indicates a stiffer material that can resist larger twisting moments.

Therefore, the correct answer is option 'D' - polar modulus. The maximum twisting moment a shaft can resist is determined by the product of the permissible shear stress and the polar modulus. This relationship ensures the design of a shaft that can withstand the expected torsional loading without failure.