Question

Define three moduli of elasticity. State their formulae.

Answer 1

Hint: In order to solve this question we should know about three modulus of elastic constants which are given. After this we can find the solution for each.

Complete step by step answer:
(a) Young's modulus: When strain is small, the ratio of the longitudinal stress to the corresponding longitudinal strain is called the Young's modulus of the material of the body.
$\gamma $ = longitudinal stress/​longitudinal strain

$\implies \gamma = \dfrac{{\dfrac{F}{A}}}{{\dfrac{{\Delta L}}{L}}} \\$
$\implies \gamma = \dfrac{{\dfrac{{Mg}}{{\pi {r^2}}}}}{{\dfrac{{\Delta L}}{L}}} \\ $

(b) Bulk modulus of elasticity: When strain is small the ratio of the normal stress to the volume strain is called the bulk modulus of the material of the body.
K = Volumetric stress/Volume strain
$\implies K = \dfrac{{\Delta p}}{{\dfrac{{\Delta V}}{V}}}$

(c) Shear modulus or modulus of rigidity: For small strains the ratio of the shearing stress to the shearing strain is called the modulus of rigidity of the material of the body.
$\eta$ = shear stress​/shear strain
$\implies \eta = \dfrac{{\dfrac{F}{A}}}{{\dfrac{{\Delta x}}{y}}} \\$
$\implies \eta = \dfrac{{\dfrac{F}{A}}}{\theta } \\$

Note:
The modulus of elasticity is that quantity which measures the amount of elasticity of an object or substance which resists from deformation which means not permanently when we apply stress. The modulus of elasticity which gives the information about the slope of stress–strain curve in the elastic deformation region.