Physics - Elasticity Question with Solution | TestHub

$Y-$Young's modulus, $K-$Bulk modulus, $\eta -$modulus of rigidity
We know that, 

$Y=3k\left(1-2\sigma \right)$

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

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

From the equation $\left(1\right)$ and the equation $\left(2\right)$

$\frac{1}{2}\left(1-\frac{Y}{3k}\right)=\frac{Y}{2\eta }-1$

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

$\Rightarrow \frac{\eta Y}{3k}=3\eta -Y$$\Rightarrow k=\frac{\eta Y}{9\eta -3Y} \mathrm{N} {\mathrm{m}}^{-2}$.