Question

A uniform cube is subjected to volume compression. If each side is decreased by 1% Then what is bulk strain?
(A) 0.01
(B) 0.03
(C) 0.06
(D) 0.09

Answer 1

Hint: To solve this problem students should know what bulk strain definition and formula and the formula for density are. With the general equation of bulk strain, we can find the solution to this problem. Volumetric strain or bulk strain It is defined as the change in volume per unit original volume, when the body is deformed by external forces.

Formula used:
${\text{volumetric strain = }}\dfrac{{{\text{change in volume}}}}{{{\text{original volume}}}} = \dfrac{{\Delta V}}{V}$

Complete step by step solution:
A uniform cube is subjected to volume compression there let the initial volume be $V$ and the final volume or change in volume be $\Delta V$
Therefore, we know the volume $\left( V \right){\text{ = }}{{\text{a}}^3}$
Now, change in volume = change in length
We have given that length is decreased by 1%
Using, percentage error formula we get
$\% {\text{ change in volume = 3 }} \times {\text{ % change in length}}$
$\dfrac{{\Delta V}}{V}{\text{ = 3}} \times \dfrac{{\Delta a}}{a} - - - (1)$
It is given that each side is decreased by 1% therefore $\dfrac{{\Delta a}}{a} = 1\% $
Substituting the above relation in equation (1) we get
$ \Rightarrow \dfrac{{\Delta V}}{V}{\text{ = 3 }} \times {\text{ 1% }}$
$ \Rightarrow \dfrac{{\Delta V}}{V}{\text{ = 3 }} \times {\text{ }}\dfrac{1}{{100}}$
Now simplifying, we get
$\therefore \dfrac{{\Delta V}}{V}{\text{ = 0}}{\text{.03}}$
Therefore, change in bulk strain is equal to 0.03


Additional information:
Tangential or shear stress: It is defined as the restoring force acting per unit area tangential to the surface of the body.
Bulk stress or volume stress: When the force is acting all along the surface normal to the area, then force acting per unit area is known as volume stress. The effect of pressure is to produce change in volume. The shape of the body may or may not change depending upon the homogeneity of the body.
 Longitudinal stress: When an object is one dimensional then force acting per unit area is called longitudinal stress. It is of two types: tensile stress and compressive stress.
Longitudinal strain: It is defined as the increase in the length per unit original length, when the body is deformed by external force.
Shear strain: It is defined as the angle (in radian), through which a face originally perpendicular to the fixed face gets turned on applying tangential deforming force.

Note:
We have taken 3 times the change in length from the equation of percentage error. That is if $Z{\text{ = }}{{\text{a}}^n}$ then we can calculate the error by $\dfrac{{\Delta Z}}{Z}{\text{ = n}} \times \dfrac{{\Delta a}}{a}$.