Question

A solid cube is subjected to a pressure of $5\times {{10}^{5}}\,N/{{m}^{2}}$. Each side of the cube is shortened by $1\%$. Then what is the magnitude of volumetric strain and Bulk modulus of the cube?
A. $0.03,\,\,5\times {{10}^{5}}\,N/{{m}^{2}}$
B. $0.03,\,\,1.67\times {{10}^{7}}\,N/{{m}^{2}}$
C. $3,\,\,1.67\times {{10}^{-7}}\,N/{{m}^{2}}$
D. $0.01,\,\,1.67\times {{10}^{7}}\,N/{{m}^{2}}$

Answer 1

Hint: Volumetric strain is defined as change in volume per unit volume of the solid. Bulk modulus of the substance is defined as the relative change in volume of the body when a unit pressure is applied uniformly over its surface.
The change in side of the cube is given. Obtain change in volume by using this value. Then obtain volumetric strain i.e. the ratio of change in volume to actual volume of cube.
Formula used:
Volumetric strain =$\dfrac{\delta V}{V}$; Bulk modulus, $B=\dfrac{V\delta P}{\delta V}$

Complete step by step answer:
We assume that the initial side of the cube before pressure is applied to be $a$.
Then initial volume of the cube is
$V={{a}^{3}}$
When pressure is applied, each side of the cube is shortened by $1\%$. We can now calculate the change in volume of the cube with respect to change in its length as
$\dfrac{\delta V}{\delta a}=\dfrac{\delta {{a}^{3}}}{\delta a}=3{{a}^{2}}\Rightarrow \delta V=3{{a}^{2}}\delta a$
Volumetric strain is defined as change in volume per unit volume of the solid. Mathematically,
Volumetric strain =$\dfrac{\Delta V}{V}=\dfrac{3{{a}^{2}}\delta a}{{{a}^{3}}}=3\times (0.01)=0.03$
Since, the change in length of the cube is in the direction of applied force, volumetric strain is positive. Therefore, volumetric strain is 0.03
Bulk modulus of the substance is defined as the relative change in volume of the body when a unit pressure is applied uniformly over its surface. It is given as
$B=\dfrac{V\delta P}{\delta V}$
This can also be written as
$B=\dfrac{\delta P}{\text{Volumetric Strain}}=\dfrac{5\times {{10}^{5}}}{0.03}=1.67\times {{10}^{7}}\,N/{{m}^{2}}$

Hence, C is the correct option.

Note:
Volumetric strain is the ratio of change in volume of solid to its initial volume. Thus, it is a unit less quantity.
Bulk modulus of elasticity is related to pressure applied and it has units of pressure. That is, its SI unit is $N/{{m}^{2}}$.