Question

If the longitudinal strain in a cubical body's three times the lateral strain then the bulk modulus $K$, young’s modulus $Y$, and rigidity $\eta$ are related by:
A) $K=Y$
B) $\eta ={\displaystyle \frac{3Y}{8}}$
C) $Y={\displaystyle \frac{3\eta }{8}}$
D) $Y=\eta$

Answer 1

Hint: The Poisson’s ratio which is the ratio of lateral strain to the longitudinal strain is to be found. The relation between the young’s modulus, rigidity modulus, bulk modulus, and the Poisson’s ratio was to be analyzed for a new relation.

Complete step by step answer:
The strain is the ratio of change in dimension to the original dimension. A rubber band tends to be thinner when it is stretched. This is because, when the material is subjected to stretching, the compression will happen in the direction perpendicular to the force we applied. This can be measured using the Poisson’s ratio.
Suppose the lateral strain is $x$, then the longitudinal strain is $3x$.
The ratio of lateral strain to longitudinal strain is termed as Poisson’s ratio.
Therefore the Poisson’s ratio is given as,
$$\begin{gathered} \sigma ={\displaystyle \frac{\text{lateral}\text{strain}}{\text{longitudinal}\text{strain}}} \\ \Rightarrow \sigma ={\displaystyle \frac{x}{3x}} \\ \Rightarrow \sigma ={\displaystyle \frac{1}{3}} \end{gathered}$$
The relation is connected as young’s modulus $Y$, bulk modulus $K$, rigidity modulus $\eta$, and Poisson’s ratio $\sigma$.
$Y=3K\left(1-2\sigma \right)$
Substitute for the Poisson’s ratio.
$\Rightarrow Y=3K\left(1-2\times {\displaystyle \frac{1}{3}}\right)$
$\Rightarrow Y=3K\times {\displaystyle \frac{1}{3}}$
$\Rightarrow Y=K$
Another relation connecting the Poisson’s ratio is given as,
$Y=2\eta \left(1+\sigma \right)$
Substitute for the Poisson’s ratio.
$$\begin{gathered} \Rightarrow Y=2\eta \left(1+{\displaystyle \frac{1}{3}}\right) \\ \Rightarrow Y={\displaystyle \frac{8\eta }{3}} \\ \Rightarrow \eta ={\displaystyle \frac{3Y}{8}} \end{gathered}$$

Therefore, $Y=K$ and $\eta ={\displaystyle \frac{3Y}{8}}$. The correct options are, option A and B.

Note:
We have to note that the lateral strain is transverse contraction and longitudinal strain is a longitudinal extension. And both are along the direction of the force we apply. And both strains are the change in dimension to the original dimension.