If $\mathrm{Y}, \mathrm{K}$ 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.
Correct Option: , 4
Y- Younge modulus, $\mathrm{K}$ - Bulk modulus,
$\eta$ - modulus of rigidity
We know that
$y=3 k(1-2 \sigma)$
$\sigma=\frac{1}{2}\left(1-\frac{\mathrm{y}}{3 \mathrm{k}}\right)$...(1)
$y=2 \eta(1+\sigma)$
$\sigma=\frac{\mathrm{y}}{2 \eta}-1$...(2)
From Eq.(i) and Eq. (ii)
$\frac{1}{2}\left(1-\frac{\mathrm{Y}}{3 \mathrm{k}}\right)=\frac{\mathrm{y}}{2 \eta}-1$
$1-\frac{\mathrm{y}}{3 \mathrm{k}}=\frac{\mathrm{y}}{\eta}-2$
$\frac{\mathrm{y}}{3 \mathrm{k}}=3-\frac{\mathrm{y}}{\eta}$
$\frac{\mathrm{y}}{3 \mathrm{k}}=\frac{3 \eta-\mathrm{y}}{\eta}$
$\frac{\eta \mathrm{y}}{3 \mathrm{k}}=3 \eta-\mathrm{y}$
$\mathrm{k}=\frac{\eta \mathrm{y}}{9 \eta-3 \mathrm{y}}$