Question

The velocity of sound in water is $1400m/s$. The density of the water is $1000kg/{m^3}$. The bulk modulus of elasticity is:
A. $5 \times {10^9}N/{m^2}$
B. $1.96 \times {10^9}N/{m^2}$
C. $4 \times {10^9}N/{m^2}$
D. None of these

Answer 1

In the question, the velocity of sound in water and density of water are given. Therefore to find the bulk modulus of elasticity, we need to use the relation among the velocity of sound, the density of the medium and the bulk modulus of elasticity.

Formula used:
$v = \sqrt {\dfrac{B}{\rho }}$,
Where, $v$ is the velocity of the sound in a given medium, $B$ is the bulk modulus of elasticity and $\rho$ is the density of the medium.

Complete step by step answer:
Here, velocity of sound is given as $v = 1400m/s$ and the medium is water which density is $\rho = 1000kg/{m^3}$.
For finding the bulk modulus we will now use the formula $v = \sqrt {\dfrac{B}{\rho }}$
Taking square on both sides, we get
{v^2} = \dfrac{B}{\rho } \\\ \Rightarrow B = {v^2}\rho \\\
Now, put the given values of , velocity of sound and density of water
B = {\left( {1400} \right)^2} \times 1000 \\\ \Rightarrow B = 196 \times {10^7}N/{m^2} \\\ \therefore B = 1.96 \times {10^9}N/{m^2} \\\
Thus, the value of the bulk modulus for the given condition is $1.96 \times {10^9}N/{m^2}$

Hence, option B is the right answer.

Note: Here, we have determined the value of bulk modulus with the help of velocity of sound in its medium which is water. We know that the bulk modulus is nothing but a numerical constant which is used to measure and describe the elastic properties of a solid or fluid when pressure is applied on all the surfaces. More accurately, it is defined as the proportion of volumetric stress related to the volumetric strain of specified material, while the material deformation is within the elastic limit. Also, bulk modulus is used to measure how incompressible a solid is. Therefore, we can say that the more the value of B for a material, the higher it is likely to be incompressible.