Question

For an elastic material
A. $Y>\eta $
B. $Y<\eta $
C. $Y\eta =1$
D. $Y=\eta $

Answer 1

Hint: Define stress and types of stress. Understand the physical meaning of the symbols in the question. Define them in terms of stress and strain. Since, for both the quantities the type of stress and strain is different, you can compare them to find the relation between the quantities.

Complete step-by-step answer:
Elasticity of a material is a property which resists the change in shape due to expansion and compression of the body.
A body which can regain its original shape When we remove the deformation force is called an elastic body. Now, stress can be defined as the restoring force per unit area of a deformed body.
$stress=\dfrac{force}{area}$
Strain can be defined as the deformation produced in a body with respect to the original shape of the body. Strain is of three kinds – longitudinal strain, shearing strain and bulk strain.
Stress is directly proportional to the strain with a constant of proportionality called the modulus of elasticity.
Young’s modulus can be defined as within the elastic limit the ratio of longitudinal stress to the longitudinal strain.
Where, Young’s modulus,
\[\begin{align}
  & Y=\dfrac{\text{longitudinal stress}}{\text{longitudinal strain}} \\
 & Y=\dfrac{\dfrac{F}{A}}{\dfrac{\Delta l}{l}} \\
\end{align}\]
The rigidity modulus can be explained as within the elastic limit the ratio of the shearing stress to the shearing strain.
Where, rigidity modulus,
$\begin{align}
  & \eta =\dfrac{\text{tangential stress}}{\text{shearing strain}} \\
 & \eta =\dfrac{\dfrac{F}{A}}{\dfrac{\Delta l}{l}}=\dfrac{\dfrac{F}{A}}{\theta } \\
\end{align}$
Now, since longitudinal strain is less than the shearing strain from the above two equations, we will always get young’s modulus will be greater than the rigidity modulus.
So, $Y>\eta $
The correct option is (A)

Note: A perfectly elastic body can be defined as a body which will regain its original shape and size completely after removing the deforming force.
A plastic body is defined as an object which cannot regain its original shape even after we remove the deforming force on the body.