Question

For a given material, the Young's modulus is $2.4$ times its modulus of rigidity. Its Poisson's ratio is
A) $2.4$
B) $1.2$
C) $0.4$
D) $0.2$

Answer 1

Young’s modulus gives us the ratio of stress to strain for a material while Poisson’s ratio is the negative of the ratio of change in length in the transverse to the longitudinal direction of a material under stress. Use the relation of Young’s modulus with the modulus of rigidity and Poisson’s ratio to solve for Poisson’s ratio.

Formula used:
$\Rightarrow Y = 2\eta (1 + \sigma )$, where $Y\,$ is the young’s modulus, $\eta$ is the modulus of rigidity and $\sigma$ is the Poisson’s ratio.

Complete step by step solution:
The relation between Young’s modulus, the modulus in the rigidity and Poisson’s ratio for a given material is:
$\Rightarrow Y = 2\eta (1 + \sigma )$ ......(1)
But, we’ve been given that for the material in question, the Young's modulus is $2.4$ times its modulus of rigidity. So,
$\Rightarrow Y = 2.4\eta$
Substituting the value of $Y$ in equation (1), we get
$\Rightarrow 2.4\eta = 2\eta (1 + \sigma )$
Dividing both side by $2\eta$, we get,
$\Rightarrow (1 + \sigma ) = 1.2$
$\therefore \sigma = 0.2$ which corresponds to option (D).
Hence, the correct option is option (D).

Additional Information:
The Poisson’s ratio tells us how the dimensions of a material will change relatively under the application of stress. For e.g., if our material was rectangular in shape and we were applying stress on the breadth of the rectangle, for a Poisson’s ratio of $0.2$, the breadth would change by $0.2$ times the change in length due to the stress. Typically, materials have Poisson ratios between $- 1$ to $1/2$.

Note:
The relation between various mechanical quantities must be known to solve this question. The relation used here between young’s modulus, modulus of rigidity and Poisson’s ratio is only applicable to isotropic solids i.e. those materials whose mechanical properties are independent of the direction in which we measure them.