# If Y, K and

Question:

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.

1. $\mathrm{Y}=\frac{9 \mathrm{~K} \eta}{3 \mathrm{~K}-\eta} \mathrm{N} / \mathrm{m}^{2}$

2. $\eta=\frac{3 Y K}{9 K+Y} N / m^{2}$

3. $\mathrm{Y}=\frac{9 \mathrm{~K} \eta}{2 \eta+3 \mathrm{~K}} \mathrm{~N} / \mathrm{m}^{2}$

4. $\mathrm{K}=\frac{\mathrm{Y} \eta}{9 \eta-3 \mathrm{Y}} \mathrm{N} / \mathrm{m}^{2}$

Correct Option: , 4

Solution:

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)

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

$1-\frac{\mathrm{y}}{3 \mathrm{k}}=\frac{\mathrm{y}}{\eta}-2$

$\frac{\mathrm{y}}{3 \mathrm{k}}=3-\frac{\mathrm{y}}{\eta}$

$\frac{\mathrm{y}}{3 \mathrm{k}}=\frac{3 \eta-\mathrm{y}}{\eta}$

$\frac{\eta \mathrm{y}}{3 \mathrm{k}}=3 \eta-\mathrm{y}$

$\mathrm{k}=\frac{\eta \mathrm{y}}{9 \eta-3 \mathrm{y}}$