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.