Question

If Y, K and η 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. $Y = \frac{9K\eta}{2\eta + 3K} \text{ N/m}^2$
• B. $Y = \frac{9K\eta}{3K - \eta} \text{ N/m}^2$
• C. $K = \frac{Y\eta}{9\eta - 3Y} \text{ N/m}^2$
• D. $\eta = \frac{3YK}{9K + Y} \text{ N/m}^2$

Hint:

To remember the relation easily, use the "9, 3, 1" rule: $\frac{9}{Y} = \frac{3}{\eta} + \frac{1}{K}$.

Answer 1

The Correct Option is C

To find the correct relation between Young's modulus \(Y\), bulk modulus \(K\), and modulus of rigidity \(\eta\), we start by understanding the interrelations between these elastic constants. The following relationships are fundamental:

• Young's Modulus \(Y\) is related to Bulk Modulus \(K\) and Poisson's Ratio \(\nu\) by the formula: Y = 3K(1 - 2\nu)
• Young's Modulus \(Y\) is related to Modulus of Rigidity (or Shear Modulus) \(\eta\) and Poisson's Ratio \(\nu\) by: Y = 2\eta(1 + \nu)

Our goal is to express \(K\) in terms of \(Y\) and \(\eta\). Begin by using the above relationships to express \(\nu\) and then substitute to obtain the desired formula.

First, express \(\nu\) from the second relationship:

\nu = \frac{Y}{2\eta} - 1

Substitute \(\nu\) in the first relationship:

Y = 3K \left(1 - 2\left(\frac{Y}{2\eta} - 1\right)\right)

Simplifying:

Y = 3K \left(1 - \frac{2Y}{2\eta} + 2\right)

This leads to:

Y = 3K \left(\frac{9\eta - Y}{2\eta}\right)

Simplifying we arrive at:

K = \frac{Y\eta}{9\eta - 3Y}

This confirms the correct option is K = \frac{Y\eta}{9\eta - 3Y} \text{ N/m}^2.

Thus:

• The correct relation is mathematically derived, curing the confusion with other options.
• The option K = \frac{Y\eta}{9\eta - 3Y} \text{ N/m}^2 is the accurate relation.