Question

If $Y$, $K$ and $\eta$ are the values of Young’s modulus, bulk modulus and modulus of rigidity of any material respectively. Choose the correct relation for these parameters.

• A. $K=\frac{Y\eta }{9\eta -3Y}{\text{N/m}}^2$
• B. $\eta =\frac{3YK}{9K+Y}{\text{N/m}}^2$
• C. $Y=\frac{9K\eta }{3K-\eta }{\text{N/m}}^2$
• D. $Y=\frac{9K\eta }{2\eta +3K}{\text{N/m}}^2$

Answer 1

The correct option is A $K=\frac{Y\eta }{9\eta -3Y}{\text{N/m}}^2$

$\Rightarrow Y=3K(1-2\sigma )$
$\Rightarrow \sigma =\frac{1}{2}(1-\frac{Y}{3K})....\left(1\right)$

$\Rightarrow Y=2\eta (1+\sigma )$
$\Rightarrow \sigma =\frac{Y}{2\eta }-1....\left(2\right)$

On comparing equation $\left(1\right)$ and $\left(2\right)$ , we get :

$\frac{Y}{2\eta }-1=\frac{1}{2}(1-\frac{Y}{3K})$

$\Rightarrow \frac{Y}{\eta }-2=1-\frac{Y}{3K}$

$\Rightarrow \frac{Y}{\eta }=1+2-\frac{Y}{3K}$

$\Rightarrow \frac{Y}{\eta }=3-\frac{Y}{3K}$

$\Rightarrow \frac{Y}{3K}=3-\frac{Y}{\eta }$

$\Rightarrow \frac{Y}{3K}=\frac{3\eta -Y}{\eta }$

$\Rightarrow K=\frac{Y\eta }{9\eta -3Y}{\text{N/m}}^2$