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

Answer 1

Answer: (D)

Y- Younge modulus, K- Bulk modulus,
$\eta$ - modulus of rigidity
We know that
$y=3k(1-2\sigma )$
$\sigma =\frac{1}{2}\left(1-\frac{y}{3k}\right)$...(i)
$y=2\eta (1+\sigma )$
$\sigma =\frac{y}{2\eta }-1$...(ii)
From Eq.(i) and Eq. (ii)
$\frac{1}{2}\left(1-\frac{Y}{3k}\right)=\frac{y}{2\eta }-1$
$1-\frac{y}{3k}=\frac{y}{\eta }-2$
$\frac{y}{3k}=3-\frac{y}{\eta }$
$\frac{y}{3k}=\frac{3\eta -y}{\eta }$
$\frac{\eta y}{3k}=3\eta -y$
$k=\frac{\eta y}{9\eta -3y}$