The modulus of rigidity, also known as the shear modulus or the torsion modulus, is a measure of a material's ability to resist shear stress. It is defined as the ratio of shear stress to shear strain.

Definition of Modulus of Rigidity:
The modulus of rigidity, denoted by G, is a material property that relates the shear stress, denoted by τ, to the shear strain, denoted by γ. It is given by the equation:

G = τ / γ

Where:
G = Modulus of rigidity
τ = Shear stress
γ = Shear strain

Explanation:
Shear stress is the force acting parallel to the cross-sectional area of a material, divided by the cross-sectional area. It is a measure of the internal forces within a material that cause it to deform under shear. Shear strain, on the other hand, is the deformation per unit length in the plane perpendicular to the applied shear stress.

The modulus of rigidity represents the material's resistance to shear deformation. It quantifies how much shear stress is required to produce a given amount of shear strain in a material.

Example:
For example, if a shear stress of 100 N/m^2 is applied to a material, and it produces a shear strain of 0.05, then the modulus of rigidity can be calculated as follows:

G = 100 N/m^2 / 0.05 = 2000 N/m^2

This means that the material has a modulus of rigidity of 2000 N/m^2, indicating that it requires a shear stress of 2000 N/m^2 to produce a shear strain of 0.05.

Importance:
The modulus of rigidity is an important material property for applications involving shear stress and strain. It is commonly used in the design and analysis of structures and components subjected to torsional or shearing loads, such as shafts, springs, and beams.

It is worth noting that the modulus of rigidity is a specific measure of a material's response to shear stress, and should not be confused with other material properties such as Young's modulus, which relates linear stress to linear strain, or Poisson's ratio, which relates lateral strain to linear strain.