Solve this
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.

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

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

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

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


Correct Option: 1

Solution:

(1)

$\Rightarrow y=3 k(1-2 \sigma)$

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

$\Rightarrow y=2 \eta(1+\sigma)$

$\Rightarrow \sigma=\frac{y}{2 \eta}-1 \quad \ldots(2)$

by comparing equation (1) and (2), we get

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

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

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

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

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

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