Question

A rubber ball is taken to depth $1$ km inside water so that its volume reduces by $0.05\% .$ what is the bulk modulus for rubber?
A. ${\text{2}} \times {\text{1}}{{\text{0}}^8}N/{m^2}$
B. ${\text{2}} \times {\text{1}}{{\text{0}}^9}N/{m^2}$
C. ${\text{2}} \times {\text{1}}{{\text{0}}^{10}}N/{m^2}$
D. ${\text{2}} \times {\text{1}}{{\text{0}}^{11}}N/{m^2}$

Answer 1

Bulk modulus of any substance can be defined as the measure of how resistant the substance is with respect to the compression. It can be defined as the ration as the increase in pressure with the relative decrease in volume. Use formula - $\beta = \dfrac{{ - PV}}{{\Delta V}}$. Before substituting values check and convert all the values in the same system of units.

Complete step by step solution:
Given that the pressure is at $1\,{\text{km}}$ depth inside the water.
$h = 1\,{\text{km}}$
Convert the given units in MKS (Meter Kilogram Second) system
$h = 1000\,{\text{m}}$
Now, the pressure, $p = \rho gh$
Where, $\rho$ is the density of the water$= 1000kg/{m^3}$
Gravitational acceleration, $g = 10m/{s^2}$
Height, $h = 1000m$
Place the values in the above equation –
p = 1000 \times 10 \times 1000 \\\ \Rightarrow p = {10^7}N{m^2} \\\
Reduction in volume be $= \Delta V$

Given that volume is reduced by $0.05\%$.
$\Delta V = V \times \dfrac{{0.05}}{{100}}$
By using the formula for the bulk modulus,
$\beta = \dfrac{{ - PV}}{{\Delta V}}$
Place the known values in the above equations –
$\beta = \dfrac{{ - {{10}^7} \times V}}{{\dfrac{{ - 0.05V}}{{100}}}}$
Common factors from the numerator and the denominator cancel each other. Therefore, remove “V” from both the denominator and the numerator.
$\beta = \dfrac{{ - {{10}^7}}}{{\dfrac{{ - 0.05}}{{100}}}}$
Negative sign from both the denominator and the numerator cancel each other and also the

Denominator’s denominator goes to the numerator part of the fraction –
$\beta = \dfrac{{{{10}^7}}}{{0.05}} \times 100$
Simplify the above equation
$\beta = \dfrac{{{{10}^7} \times 100 \times 100}}{5}$
Dividing implies
$\beta = {10^7} \times 100 \times 20 \\\ \Rightarrow\beta = 20 \times {10^9} \\\$
Re-writing in the form of the given multiple choices –
$\therefore\beta = 2 \times {10^{10}}N{m^2}$
Hence, from the given multiple choices – the third option is the correct answer.

Hence, option C is the correct answer.

Note: Sometimes, the bulk modulus is referred to as the incompressibility of the substance. Also, refer to the basic difference between the young’s modulus, bulk modulus and the shear modulus. Know the difference and apply its law accordingly.