Question

A spherical ball contracts in volume by 0.01 % when subjected to a normal uniform pressure of 100 atm. The Bulk modulus of its material is
(A). $1.01 \times {10^{11}}N{m^{ - 2}}$
(B). $1.01 \times {10^{12}}N{m^{ - 2}}$
(C). $1.01 \times {10^{10}}N{m^{ - 2}}$
(D). $1.01 \times {10^{13}}N{m^{ - 2}}$

Answer 1

Hint: Bulk modulus is a numerical constant that defines solid or fluid elastic properties when it is under pressure on all surfaces. The pressure applied increases the volume of a substance and when the pressure is removed returns to its original volume.
Formula used: ${\text{B = }}\dfrac{{\vartriangle {\text{P}}}}{{(\dfrac{{\vartriangle {\text{v}}}}{{\text{v}}}{\text{)}}}}$Where,
B = Bulk modulus
$\vartriangle P$ = Change of the pressure or force applied on the material per unit area
$\vartriangle V$= Change of material volume due to compression
V = Initial material volume in English system units, and $N/{m^2}$in metric scale.

Complete step-by-step solution -
Since the number of spherical contracts is 0.01%
$\Rightarrow \dfrac{{\vartriangle v}}{v} = \dfrac{{0.01}}{{100}}$
So Bulk modulus ‘B’ is given by
${\text{B = }}\dfrac{{\vartriangle {\text{P}}}}{{(\dfrac{{\vartriangle {\text{v}}}}{{\text{v}}}{\text{)}}}}$
$\Rightarrow {\text{B = }}\dfrac{{100 \times 1.01 \times {{10}^5}}}{{\dfrac{{0.01}}{{100}}}}$
$\because \;{\text{1 atm = 1}}{\text{.01 }} \times {\text{ 1}}{{\text{0}}^5}N/{m^2}$
$\Rightarrow B = 1.01 \times {10^{11}}N{m^{ - 2}}$
Hence option A is the correct answer.

Additional information-
Bulk modulus is used to calculate the incompressibility of a solid. Besides, the higher value of K for a substance, the more incompressible its existence is. For example, for steel, the value of K is $1.6 \times {10^{11}}N/{m^2}$and the value of K for glass is $4 \times {10^{10}}N/{m^2}$. K for steel, here, is more than three times the glass value of K. Which means more compressible glass than steel.
The standard atmosphere (symbol: atm) is a pressure unit specified as 101325 Pa (1.01325 bar). It is commonly used as a reference or standard pressure. At sea level, it is nearly equal to the air pressure.

Note: Bulk elasticity modulus is the one measure of solids' mechanical properties. Certain elastic modules include module Young and module Shear. In any case, a material's bulk elastic properties are used to determine how much it can compress under a specified amount of outer pressure. Here the ratio of the change in pressure to the compression of the fractional volume is important to find and remember.