Question

If a rubber ball is taken down to a 100 m deep lake, its volume decreases by 0.1%. If \[g = 10m/{s^2}\] then the bulk modulus of elasticity for rubber, in \[N/{m^2}\], is
A.${10^8}$
B.${10^9}$
C.${10^{11}}$
D.${10^{10}}$

Answer 1

Hint: In this question we have to find the bulk modulus of elasticity for rubber. For this we will use the formula of bulk modulus. For this first we will find the difference between pressure and volume strain to use in the formula of bulk modulus. We will also be careful that all the variables are in the same unit system.

Complete step by step answer:
Given,
h=100 m
\[g = 10m/{s^2}\]
The volume decreases by 0.1%,
Formula used:
\[Bulk{\text{ }}modulus{\text{ }} = \dfrac{{\Delta P}}{{\dfrac{{\Delta V}}{V}}}\]
Where,
\[\Delta P\]is change in pressure
\[\dfrac{{\Delta V}}{V}\]is volume strain
Pressure on rubber ball, $P = {P_0} + \rho gh$
Where,
${P_0}$is the atmospheric pressure which is equal to $1.03 \times {10^5}{\text{ m}}$
$\rho $is density of water which is equal to \[\;1000{\text{ }}kg/{m^3}\]
g is acceleration due to gravity
$h$ is the distance deep in the lake.
Now, we will put the values of variables in the above equation of pressure.
 $P = {P_0} + \rho gh$
$P = 1.03 \times {10^5} + 1000 \times 10 \times 100$
$P = 1.03 \times {10^5} + 10 \times {10^5}$
$P = 11.03 \times {10^5}{\text{ N/}}{{\text{m}}^2}$
It is given that there is a decrease of 0.1 % in volume,
$\dfrac{{\Delta V}}{V} = \dfrac{{0.05}}{{100}}$
$\dfrac{{\Delta V}}{V} = 5 \times {10^{ - 4}}$
Change in pressure $\Delta P = P - {P_0}$
Putting the values of $P$and ${P_0}$in above equation,
$\Delta P = 11.03 \times {10^5} - 1.03 \times {10^5}$.
$\Delta P = 10.00 \times {10^5}N/{m^2}$
$\Delta P = {10^6}N/{m^2}$
\[Bulk{\text{ }}modulus{\text{ }} = \dfrac{{\Delta P}}{{\dfrac{{\Delta V}}{V}}}\]
Putting the values of \[\Delta P\]and \[\dfrac{{\Delta V}}{V}\]in above formula,
\[Bulk{\text{ }}modulus{\text{ }} = \dfrac{{{{10}^6}}}{{5 \times {{10}^{ - 4}}}}\]
\[Bulk{\text{ }}modulus{\text{ }} = 0.2 \times {10^{10}}N/{m^2}\]
Result- Hence, from the above calculation we have found the value of\[Bulk{\text{ }}modulus{\text{ }} = 0.2 \times {10^{10}}N/{m^2}\].

Note:
Hence, it is clear from the above explanation that in such types of questions we have to be aware of the formulae and we must know the concept to solve these questions. The units of all the variables must be in the same system. We should be careful while doing the calculation.