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.

Updated On: Dec 30, 2025
  • $Y =\frac{9 K \eta}{3 K -\eta} N / m ^{2}$
  • $\eta=\frac{3 Y K}{9 K+Y} N / m^{2}$
  • $Y =\frac{9 K\eta }{2 \eta+3 K } N / m ^{2}$
  • $K =\frac{ Y\eta }{9 \eta-3 Y } N / m ^{2}$
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The Correct Option is D

Solution and Explanation

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.

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