Question

The relationship between $Y,\eta $ and $\sigma $ is
${\text{A}}{\text{.}}$ $Y = 2\eta (1 + \sigma )$
${\text{B}}{\text{.}}$ $\sigma = \dfrac{{2T}}{{(1 + \eta )}}$
${\text{C}}{\text{.}}$ $\eta = 2Y(1 + \sigma )$
${\text{D}}{\text{.}}$ $Y = \eta (1 + \sigma )$

Answer 1

Hint: We can find the relationship between young’s modulus, modulus is rigidity, and Poisson radio. Also, by using stress relation on the unit solid element
Finally we get the required answer.

Complete step-by-step solution:
The given the relation between $Y$, $\eta $ and $\sigma $
Young’s modulus (Y)
It is a measure of a solid’s stiffness or resistance to elastic deformation under load. It relates stress and strain along an axis or line.
When Stress – force per unit area
Strain – proportional deformation
${\text{Y = }}\dfrac{{{\text{longitudinal stress}}}}{{{\text{longitudinal strain}}}}$
$Y = \dfrac{{FL}}{{A.\Delta L}}$
Modulus of rigidity ($\eta $)
Shear modulus is also known as modulus of rigidity is the measure of the rigidity of the body, given by the ratio of shear stress to shear strain. Often denoted by G sometimes by $S$ or $\eta $.
\[{{\eta = }}\dfrac{{{\text{shear stress}}}}{{{\text{shear strain}}}}\]
$\eta = \dfrac{\tau }{r}$
Poisson radio ($\sigma $)
Ratio of lateral contraction to linear elongation is called Poisson’s radio. It also can be defined because the ratio of unit transverse strain to unit longitudinal strain. The formula for Poisson’s ratio $\sigma $ is given as, Poisson’s ratio =$\sigma = $ lateral contraction/ linear elongation.
${{\sigma = }}\dfrac{{{\text{Lateral strain}}}}{{{\text{Longitudinal strain}}}}$
By using stress relation on the unit solid element, these relations are often derived:
$\eta = \dfrac{Y}{{2(1 + \sigma )}}$
Hence, the young’s modulus of rigidity and Poisson radio is given by
$Y = 2\eta (1 + \sigma )$

So, option A is the correct answer.

Note: Young’s modulus calculation,
The young’s modulus equation is ${\text{E = }}\dfrac{{{\text{tensile stress}}}}{{{\text{tensile strain}}}}$ where ${\text{F}}$ that is the applied force ${\text{L}}$ that is the initial length, ${\text{A}}$ is that the square area, and ${\text{E}}$ is young’s modulus in Pascal (Pa).
Using a graph, you'll determine whether a cloth shows elasticity.