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Question: The bulk modulus of rubber is \(9.8 \times {10^8}{\text{ }}\dfrac{N}{{{m^2}}}\). To what depth shoul...

The bulk modulus of rubber is 9.8×108 Nm29.8 \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%0.1\% ?
A. 1 m1{\text{ }}m
B. 25 m{\text{25 }}m
C. 100 m100{\text{ }}m
D. 200 m{\text{200 }}m

Explanation

Solution

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 Nm2\dfrac{N}{{{m^2}}}. Let the initial volume of the rubber ball be VV.As, the volume decreases by 0.1%0.1\% , then the final volume V′V' corresponds to,
V′=V−V1000=9991000VV' = V - \dfrac{V}{{1000}} = \dfrac{{999}}{{1000}}V
The relation between bulk modulus and change in volume is,
k=P−ΔVV−−−−−(1)k = \dfrac{P}{{ - \dfrac{{\Delta V}}{V}}} - - - - - \left( 1 \right)
The variables are referred to,
k=k = bulk modulus
P=P = Pressure
ΔV=\Delta V = Change in volume== Final volume−- Initial volume
V=V = Initial volume

Pressure is given as the formula,
P=ρghP = \rho gh
where P=P = Pressure, ρ=\rho = density of water, g=g = acceleration due to gravity and h=h = height.
Density of water is 103 kgm3{10^3}{\text{ }}\dfrac{{kg}}{{{m^3}}} and acceleration due to gravity is 9.8 ms29.8{\text{ }}\dfrac{m}{{{s^2}}}.
The values given in the questions are,
k=9.8×108 Nm2k = 9.8 \times {10^8}{\text{ }}\dfrac{N}{{{m^2}}}
⇒P=103×9.8×h\Rightarrow P = {10^3} \times 9.8 \times h
⇒ΔV=9991000V−V=−V1000\Rightarrow \Delta V = \dfrac{{999}}{{1000}}V - V = - \dfrac{V}{{1000}}
Substituting all the values in equation (1)\left( 1 \right) we get,
9.8×108=−103×9.8×h−V1000V9.8 \times {10^8} = - \dfrac{{{{10}^3} \times 9.8 \times h}}{{\dfrac{{ - \dfrac{V}{{1000}}}}{V}}}
After cancelling the term 9.8 and VV we get,
h=108106h= \dfrac{10^8}{10^6}
∴h=100\therefore h = 100
If the ball is taken at a depth of 100 m100{\text{ }}m then the volume of the rubber ball will be decreased by 0.1%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.