Hello students we know that according to Hooke’s law stress is directly proportional to strain (within elastic limit). This indicates that the ratio of stress and strain is constant, which is termed as modulus of elasticity. The modulus of elasticity in which study of change in shear modulus or modulus of rigidity
Let’s define the concept and derive its formula….!
The modulus of rigidity (n) is the modulus of elasticity corresponding to a change in the shape of a body.
Consider a metallic cube as shown in fig. below. Let the tangential force ‘F’ is applied to its face ABEF, so that the shape of cube is changed as parallelepiped. The face ABEF is displaced to face A’B’EF by distance ‘x’. The face ABEF is at distance ‘h’ from the fixed layer CDGH.
Then the shearing stress produced here is,
∴ shearing stress = force/area
∴ shearing stress = F/A
Also the shearing strain produced here is,
∴ shearing strain = lateral displacement of layer/distance of the the layer from fixed layer
∴ shearing strain = x/h
∴ shearing strain = tanθ
The ratio of the sharing stress to the shearing strain is called Modulus of Rigidity.
∴ Modulus of rigidity = shearing stress/shearing strain
∴ Modulus of Rigidity = Shearing stress/Shearing Strain
∴ η = F/A/tanθ
∴ η = F/A tanθ
Some important points to note……!
- As only solids have definite shape, Modulus of Rigidity is possessed by solids only.
- It depends on temperature and a material.
Let’s go in more detail with some numerical….!
Ex:1) The lateral displacement of layer of cube of side 10 cm is 0.2 cm when the tangential force of 2500000 dyne is applied on its surface. Find the modulus of rigidity of the material of cube.
Here, l =10 cm = 0.1 m, A=100 cm2 = 10-2 m2, x = 0.2 cm = 2 × 10-3 m, h =10 cm = 0.1 m, F = 25 N
we have,