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.

• A. $Y =\frac{9 K \eta}{3 K -\eta} N / m ^{2}$
• B. $\eta=\frac{3 Y K}{9 K+Y} N / m^{2}$
• C. $Y =\frac{9 K\eta }{2 \eta+3 K } N / m ^{2}$
• D. $K =\frac{ Y\eta }{9 \eta-3 Y } N / m ^{2}$

Answer 1

The Correct Option is D

To find the correct relation between Young's modulus $Y$, bulk modulus $K$, and modulus of rigidity $\eta$, we need to use fundamental relationships between these elastic constants.

The relationship between Young's modulus $Y$, bulk modulus $K$, and modulus of rigidity $\eta$ is given by the equation:

$Y = \frac{9K\eta}{3K + \eta}$

However, our task is to find the expression for $K$ in terms of $Y$ and $\eta$.

1. Starting from the known relationship:

   $Y = \frac{9K\eta}{3K + \eta}$

2. Cross-multiply to arrive at:

   $Y (3K + \eta) = 9K\eta$

3. Expand and rearrange terms to isolate the term involving $K$ on one side:

   $3YK + Y\eta = 9K\eta$

   $3YK - 9K\eta = -Y\eta$

   $K(3Y - 9\eta) = -Y\eta$

4. Solve for $K$:

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

Hence, the correct relationship is:

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

This matches the provided correct option.