Question

Define three moduli of elasticity. State their formulae.

Answer 1

(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.
$\gamma =\frac{\text{longitudinal stress}}{\text{longitudinal strain}}$
$=\frac{F/A}{△L/L}$
$=\frac{Mg/\pi r^2}{△L/L}$
(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.
$K=\frac{\text{Volume stress}}{\text{Volume strain}}$
$=\frac{△p}{△V/V}$
(c) Shear modulus or modulus of rigidity : For small strain the ratio of the shearing stress to the shearing strain is called the modulus of rigidity of the material of the body.
$\eta =\frac{\text{shear stress}}{\text{shear strain}}$

$=\frac{F/A}{△x/y}\cong \frac{F/A}{\theta }$