Question

If the Poisson’s ratio of a solid is $\dfrac{2}{5}$, then the ratio of its young’s modulus to the rigidity modulus is,
(A) $\dfrac{5}{4}$
(B) $\dfrac{7}{{15}}$
(C) $\dfrac{{14}}{9}$
(D) $\dfrac{{14}}{5}$

Answer 1

Hint: The solution for this question is determined by using the relation between the Young’ modulus and the rigidity modulus. By this equation, the ratio of the young’s modulus and the rigidity modulus can be determined by using the given Poisson’s ratio.

Formula used:
The relation between Young’s modulus and the rigidity modulus is given by,
$Y = 2G\left( {1 + \sigma } \right)$
Where, $Y$ is the young’s modulus, $G$ is the rigidity modulus, and $\sigma $ is the Poisson’s ratio of a solid.

Complete step by step answer:
Given that,
The Poisson’s ratio of the given solid is, $\sigma = \dfrac{2}{5}$
The relation between the Young’s modulus and the rigidity modulus is given by,
$Y = 2G\left( {1 + \sigma } \right)\,.................\left( 1 \right)$
To find the ratio between the Young’s modulus to the rigidity modulus, then the equation (1) is written as,
$\Rightarrow \dfrac{Y}{G} = 2\left( {1 + \sigma } \right)$
Now, the above equation shows the equation of the ratio of the young’s modulus to the rigidity modulus.
By substituting the Poisson’s ratio values given in the question in the above equation, then the above equation is given as,
$\Rightarrow \dfrac{Y}{G} = 2\left( {1 + \dfrac{2}{5}} \right)$
Now cross multiplying the terms inside the bracket, then the above equation is written as,
$\Rightarrow \dfrac{Y}{G} = 2\left( {\dfrac{{5 + 2}}{5}} \right)$
On adding the terms in the numerator in RHS, then the above equation is written as,
$\Rightarrow \dfrac{Y}{G} = 2\left( {\dfrac{7}{5}} \right)$
Now multiplying the term $2$ inside the bracket, then the above equation is written as,
$\Rightarrow \dfrac{Y}{G} = \dfrac{{14}}{5}$

Thus, the above equation shows the ratio of Young’s modulus to the rigid modulus for the given Poisson ratio of $\dfrac{2}{5}$. Hence, the option (D) is the correct answer.

Note:
The reciprocal of the final answer gives the ratio of the Rigidity modulus to Young’s modulus. By using the same formula, if the ratio of the young’s modulus to the rigid modulus is given, the Poisson’s ratio is determined.