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

Answer 1

Answer: (D)

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