Question

The bulk modulus of rubber is $9.8 \times {10^8}{\text{ }}\dfrac{N}{{{m^2}}}$. To what depth should a rubber ball be taken in a lake so that its volume is decreased by $0.1\% $ ?
A. $1{\text{ }}m$
B. ${\text{25 }}m$
C. $100{\text{ }}m$
D. ${\text{200 }}m$

Answer 1

Hint: First of all, we have to consider the volume of the rubber ball that it initially remains. Then by using the values here, we will find the new volume. After that by using the formula of bulk modulus we will find the answer to the solution.

Complete step by step answer:
Bulk Modulus also known as incompressibility refers to the measure of the ability of a substance to overcome changes in its volume when external pressure acts on it. The SI unit of bulk modulus is Pascal or $\dfrac{N}{{{m^2}}}$. Let the initial volume of the rubber ball be $V$.As, the volume decreases by $0.1\% $, then the final volume $V'$ corresponds to,
$V' = V - \dfrac{V}{{1000}} = \dfrac{{999}}{{1000}}V$
The relation between bulk modulus and change in volume is,
$k = \dfrac{P}{{ - \dfrac{{\Delta V}}{V}}} - - - - - \left( 1 \right)$
The variables are referred to,
$k = $ bulk modulus
$P = $ Pressure
$\Delta V = $ Change in volume$ = $ Final volume$ - $ Initial volume
$V = $ Initial volume

Pressure is given as the formula,
$P = \rho gh$
where $P = $ Pressure, $\rho = $ density of water, $g = $ acceleration due to gravity and $h = $ height.
Density of water is ${10^3}{\text{ }}\dfrac{{kg}}{{{m^3}}}$ and acceleration due to gravity is $9.8{\text{ }}\dfrac{m}{{{s^2}}}$.
The values given in the questions are,
$k = 9.8 \times {10^8}{\text{ }}\dfrac{N}{{{m^2}}}$
$\Rightarrow P = {10^3} \times 9.8 \times h$
$\Rightarrow \Delta V = \dfrac{{999}}{{1000}}V - V = - \dfrac{V}{{1000}}$
Substituting all the values in equation $\left( 1 \right)$ we get,
$9.8 \times {10^8} = - \dfrac{{{{10}^3} \times 9.8 \times h}}{{\dfrac{{ - \dfrac{V}{{1000}}}}{V}}}$
After cancelling the term 9.8 and $V$ we get,
$h= \dfrac{10^8}{10^6}$
$\therefore h = 100$
If the ball is taken at a depth of $100{\text{ }}m$ then the volume of the rubber ball will be decreased by $0.1\% $.

Hence, the correct answer is option C.

Note: It must be noted that the bulk modulus or incompressibility is a negative value because the volume of the material gets decreased than its initial volume. It is actually the ratio of amount of bulk stress to bulk strain. Bulk stress is pressure as pressure is the force that is applied upon a particular area. Bulk strain is changed in volume to original volume.