Question

The Bulk modulus for an incompressible liquid is:
A. Zero
B. Unity
C. Infinity
D. Between 0 to 1

Answer 1

As, the question is referring about an incompressible liquid, the property of an incompressible fluid is that when the compressive force is applied on it, the change in volume is zero because of this volumetric strain to becomes zero of the liquid. The bulk modulus is calculated by $K = \dfrac{{\dfrac{F}{A}}}{{\dfrac{{\Delta V}}{V}}}$ or $K = \dfrac{{\Delta P}}{{\dfrac{{\Delta V}}{V}}}$.

Complete step by step solution:
It is given the liquid is incompressible, for any incompressible liquid, their volume does not change.
The bulk modulus is basically the measure of the resistance that any body provides to compressive force on that particular body.
Mathematically, bulk modulus is the ratio of normal stress to volumetric strain or in simplified form it is also the ratio of infinitesimal change in pressure to the relative decrease of the volume.
Bulk modulus is represented by K, and is given by $K = \dfrac{{\dfrac{F}{A}}}{{\dfrac{{\Delta V}}{V}}}$ or $K = \dfrac{{\Delta P}}{{\dfrac{{\Delta V}}{V}}}$.
As, for any incompressible liquid the $\Delta V$ will be approximately equal to zero.
So, if we substitute $\Delta V$ value in the formula $K = \dfrac{{\Delta P}}{{\dfrac{{\Delta V}}{V}}}$, then,
$K = \dfrac{{\Delta P}}{{\dfrac{0}{V}}} \\\ K = \dfrac{{\Delta P}}{0} \\\$
So, anything divided by zero gives the result as infinity.
Therefore, $K = \infty$, that means the incompressible fluid will totally resist the compressive force on it.

So, option (C) is correct.

Note: Here, there is a key point that is bulk modulus, whenever any liquid and gas is subjected to uniform pressure its volume gets decreased but for any ideal gas or liquid it does not happen as they are considered as an incompressible body, i.e., they will not have any volumetric strain, which results in the infinite value of bulk modulus, bulk modulus is the inverse of compressibility, so if bulk modulus is infinite, compressibility is zero.