If Y, K and $η$ are the value of young' s modulus, bulk and modulus of rigidly of any material respectively. Choose the correct relation for those parameters

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$η = \frac{3 YK}{9 K + Y} N / m^{2}$

$K = \frac{Yη}{9 η + 3 Y} N / m^{2}$

$Y = \frac{9 Kη}{3 K - η} N / m^{2}$

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

answer is C.

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Detailed Solution

We know that,

Y - Young modulus
K - Bulk modulus,
η - modulus of rigidity

$Y = 3 K (1 - 2 σ)$

$σ = \frac{1}{2} (1 - \frac{Y}{3 K})______(1)$

$Y = 2 η (1 + σ)$

$σ = \frac{Y}{2 η} - 1_______(2)$

From equations 1 and 2,

$\frac{1}{2} (1 - \frac{Y}{3 K}) = \frac{Y}{2 η} - 1$

$1 - \frac{Y}{3 K} = \frac{Y}{η} - 2$

$\frac{Y}{3 K} = 3 - \frac{Y}{η}$

$\frac{Y}{3 K} = \frac{3 η - Y}{η}$

$\frac{ηY}{3 K} = 3 η - Y$

$K = \frac{ηY}{9 η - 3 Y} N / m^{2}$

Hence the correct answer is $\frac{ηY}{9 η - 3 Y} N / m^{2} .$

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