Question

If the young’s modulus of the material is 3 times its modulus of rigidity, then its volume elasticity will be:
A) Zero
B) Infinity
C) $2 \times {10^{10}}N{m^{ - 2}}$
D) $3 \times {10^{10}}N{m^{ - 2}}$

Answer 1

Hint: First find the relation between the young modulus and modulus of rigidity, then find the relation between the Poisson’s ratio and modulus of rigidity, from that relation we can find the value of volume elasticity.

Complete step by step solution:
Here, we are asked to find the value of volume elasticity or the volume elasticity of the material.
Let’s discuss the terms volume modulus.
The modulus of volume is also called bulk modulus. It is a numerical constant that helps to analyze the properties of elasticity of fluid or solid when pressure is acting on all of its surfaces. The volume reduces due to the applied pressure.
It is given that the young’s modulus is equal to three times its modulus of rigidity.
That is, $Y = 3\eta $
It is known that the value of Young’s modulus is given by,
$Y = 2\eta (1 + \sigma )$
Where, $\sigma $ is the Poisson’s ratio.
Applying the value of young’s modulus we get,
$3\eta = 2\eta (1 + \sigma )$
$ \Rightarrow \sigma = \dfrac{1}{2}$
Now we are using another formula for young’s modulus to find the value of bulk modulus.
That is, $Y = 3K(1 - 2\sigma )$
Where, $K$ is the bulk modulus.
We are applying the value of Poisson’s ratio to this equation, we get,
\[Y = 3K(1 - 2\sigma )\]
$ \Rightarrow K = \dfrac{Y}{{3(1 - 2\sigma )}}$
$ \Rightarrow K = \dfrac{Y}{{3\left\{ {1 - \left( {2 \times \dfrac{1}{2}} \right)} \right\}}}$
$ \Rightarrow K = \dfrac{Y}{{3 \times 0}}$
$ \Rightarrow K = \dfrac{Y}{0}$
We know if anything is divided by zero, the answer will be infinity,
That is $K = \infty $

Final answer is option (B): infinity.

Note: Young’s modulus is a numerical constant that describes the capacity of a material to undergo a lengthwise compression or tension. It is also called the modulus of elasticity, and given by the longitudinal stress divided by the stain. Modulus of rigidity is also called the shear modulus. The value modulus of rigidity is equal to the shear stress divided by shear strain.