Question

The average depth of Indian Ocean is about $3000 \mathrm{m}$. Calculate the fractional compression, $\dfrac{\Delta V}{V}$ of water at the bottom of the ocean, given that the bulk modulus of water is $2.2 \times 10^{9} \mathrm{Nm}^{-2}$ (consider $g=10m{{s}^{-2}}$)
(A) 0.82%
(B) 0.91%
(C) 1.24%
(D) 1.36%

Answer 1

Hint: We know that elasticity, ability of a deformed material body to return to its original shape and size when the forces causing the deformation are removed. A body with this ability is said to behave (or respond) elastically. The bulk elastic properties of a material determine how much it will compress under a given amount of external pressure. The ratio of the change in pressure to the fractional volume compression is called the bulk modulus of the material.

Complete step by step answer
We know one thing $P=P_{0}+\rho g h$
Where $\mathrm{Po}$ is the atmospheric pressure, $\mathrm{g}$ is acceleration due to gravity, $\mathrm{h}$ is the height from the Earth surface and $\rho$ is density of water
Here, $\mathrm{P}_{0}=10^{5} \mathrm{N} / \mathrm{m}^{2}, \mathrm{g}=10 \mathrm{m} / \mathrm{s}^{2}, \mathrm{h}=3000 \mathrm{m}$ and $\rho=10^{3} \mathrm{Kg} / \mathrm{m}^{3}$
Now, $\mathrm{P}=10^{5}+10^{3} \times 10 \times 3000=3.01 \times 10^{7} \mathrm{N} / \mathrm{m}^{2}$
Again, we have to use formula, $\mathrm{B}=\mathrm{P} /\{-\Delta \mathrm{V} / \mathrm{V}\}$
Here, $\mathrm{B}$ is bulk modulus and $\{-\Delta \mathrm{V} / \mathrm{V}\}$ is the fractional compression $\mathrm{So},-\Delta \mathrm{V} / \mathrm{V}=\mathrm{P} / \mathrm{B}$
Put, $P=3.01 \times 10^{7} \mathrm{N} / \mathrm{m}^{2}$ and $\mathrm{B}=2.2 \times 10^{9} \mathrm{N} / \mathrm{m}^{2}$
$\therefore$ fractional compression $=3.01 \times 10^{7} / 2.2 \times 10^{9}=1.368 \times 10^{-2}$

Hence the correct answer is option D

Note: We know that the basic difference between young's modulus, bulk modulus, and shear modulus is that Young's modulus is the ratio of tensile stress to tensile strain, the bulk modulus is the ratio of volumetric stress to volumetric strain and shear modulus is the ratio of shear stress to shear strain. Higher the value of Bulk Modulus indicates that it is difficult to compress the fluid. Take water for example. Its Bulk Modulus is. meaning that it requires an enormous pressure to change the volume of water by a small amount. Out of solids, liquids and gases, which one has all the three types of modulus of elasticity and why gases have only bulk modulus of elasticity.