Torsional Rigidity: Definition and Relation to Torque

Torsional rigidity is a crucial mechanical property that describes a shaft's resistance to twisting deformation when subjected to a torque. It is fundamentally derived from the material's shear stiffness and the cross-sectional geometry of the component.

For a circular shaft subjected to torsion, the relationship between the applied torque (\( T \)), the angle of twist (\( \theta \)), the shaft's length (\( L \)), the material's modulus of rigidity (\( G \)), and the cross-section's polar moment of inertia (\( J \)) is given by the torsion equation:

\[ T = \frac{G J \theta}{L} \]

In this fundamental equation, each term plays a specific role:

• \( T \) represents the applied torque, which is the twisting force (typically measured in N-m or lb-in).
• \( G \) is the modulus of rigidity (also known as shear modulus), an intrinsic material property that measures its resistance to shear deformation (measured in Pa or psi).
• \( J \) is the polar moment of inertia, a geometric property of the cross-section that indicates its resistance to twisting (measured in m⁴ or in⁴). For a solid circular section with diameter \( d \), \( J = \frac{\pi d^4}{32} \).
• \( \theta \) is the angle of twist, representing the angular deformation of the shaft along its length (measured in radians).
• \( L \) is the length of the shaft over which the twist occurs (measured in m or in).

Understanding Torsional Rigidity for Unit Conditions

The term torsional rigidity itself is commonly defined as the product of the modulus of rigidity (\( G \)) and the polar moment of inertia (\( J \)), often denoted as \( G J \). It signifies the resistance of a shaft to twist.

The question specifies conditions of "unit twist" and "unit length". Let's interpret these conditions within the torsion formula:

• Unit twist implies that the angle of twist \( \theta \) is equal to 1 radian (\( \theta = 1 \)).
• Unit length means that the length of the shaft \( L \) is equal to 1 unit (e.g., 1 meter or 1 inch, so \( L = 1 \)).

Deriving the Relationship Under Unit Twist and Unit Length

By substituting \( \theta = 1 \) and \( L = 1 \) into the torsion equation \( T = \frac{G J \theta}{L} \), we can observe the resulting relationship:

\[ T = \frac{G J (1)}{1} \]

\[ T = G J \]

This derivation clearly shows that when a circular section is subjected to a unit twist over a unit length, the applied torque (\( T \)) becomes numerically equal to the torsional rigidity (\( G J \)) of that section. Therefore, torsional rigidity can be conceptualized as the amount of torque required to produce a unit angle of twist over a unit length of the shaft.

Analysis of the Given Options

Let's consider each option in light of our understanding:

• modulus of rigidity (\( G \)): While an essential component, the modulus of rigidity is a material property that, when combined with the polar moment of inertia, forms torsional rigidity. It is not the torsional rigidity itself, nor the torque under the specified conditions.
• torque (\( T \)): As demonstrated by the derivation, for a unit twist and unit length, the torque applied is precisely equal to the torsional rigidity. Thus, under these specific conditions, the torsional rigidity is represented by the torque.
• radius of the section (\( R \)): The radius is a geometric parameter of the cross-section that influences the polar moment of inertia. It is not directly equivalent to torsional rigidity or the torque under the given conditions.
• polar moment of inertia (\( J \)): This is a geometric property of the cross-section that dictates its shape's resistance to torsion. It is a part of the torsional rigidity term (\( G J \)), but not torsional rigidity itself, nor the torque under the specified conditions.

Based on this analysis, for a circular section undergoing a unit twist over a unit length, the value of its torsional rigidity is equivalent to the torque required to achieve this specific deformation.