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

Answer 1

The correct option is A

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

Step 1: Given :

Young's modulus is $Y$.

Bulk modulus is $K$.

Modulus of rigidity is $\eta$.

Step 2: Formulas

The formula of Young's modulus in terms of the bulk modulus is $Y=3K(1-2\mu )$ where $\mu$ is the Poisson's ratio

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

The formula of Young's modulus in terms of the rigidity of the material is $Y=2\eta (1+\mu )$

⇒ $\mu =\frac{Y}{2\eta }-1$ ------- (2)

Step 3: Comparing the equations

Equating the equations (1) and (2)

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

⇒ $\frac{1}{2}-\frac{Y}{6K}=\frac{Y}{2\eta }-1$

⇒ $\frac{3}{2}-\frac{Y}{2\eta }=\frac{Y}{6K}$

⇒ $\frac{6K}{Y}=\frac{2\eta }{3\eta -Y}$

⇒ $K=\frac{Y\eta }{9\eta -3Y}$

⇒ The correct relation between Young's modulus, Bulk modulus, and Rigidity of the material is $K=\frac{Y\eta }{9\eta -3Y}N/m^2$

Hence, the correct option is A.