Question

If the bulk modulus of water is 2100 MPa, what is the speed of sound in water?
(A) $ 1450m/s $
(B) $ 2100m/s $
(C) $ 1400m/s $
(D) $ 1200m/s $

Answer 1

Hint
From the formula for the calculation of the speed of sound in water in form of the bulk modulus and the density of water given by $ $ $ v = \sqrt {\dfrac{K}{\rho }} $ , we can calculate the value of the speed of sound by putting the value of the bulk modulus from the question and the known density of water.
Formula used: In this question, we can calculate the speed of sound by,
 $ v = \sqrt {\dfrac{K}{\rho }} $
where $ v $ is the speed of sound in water,
 $ K $ is the bulk modulus of water,
and $ \rho $ is the density of water.

Complete step by step answer
We know that for any substance, for any substance, the bulk modulus is a measure of how resistant to compression that substance is. Bulk modulus is formally defined by the equation,
 $ K = - V\dfrac{{dP}}{{dV}} $ where $ P $ is the pressure and $ V $ is the initial volume of the substance.
In a fluid, by using the Newton-Laplace formula, using the bulk modulus and the density of the fluid, we can measure the speed of sound in that fluid.
 $ \therefore v = \sqrt {\dfrac{K}{\rho }} $
So in the question, we are given the bulk modulus as,
 $ K = 2100MPa $ , we can write this as,
 $ K = 2100 \times {10^6}Pa $
And we know that the density of water is given by,
 $ \rho = 1000kg/{m^3} $
Hence, by substituting these values in the bulk modulus we can calculate,
 $ v = \sqrt {\dfrac{{2100 \times {{10}^6}}}{{1000}}} m/s $
So by cancelling $ {10^3} $ from numerator and denominator, we get
 $ \therefore v = \sqrt {2100 \times {{10}^3}} m/s $
On doing square root on the value, we get
 $ \Rightarrow v = \sqrt {2100 \times {{10}^3}} = 1449.13m/s $
So approximating this value of $ v $ we get,
 $ v \simeq 1450m/s $ .
So, the correct option will be (A).

Additional Information
For solid, there are two other important terms for stress, the shear modulus, and Young’s modulus. But in the case of fluid, only the bulk modulus is important.

Note
To calculate the speed of sound in a liquid, we can use the Newton-Laplace equation to determine it, when the bulk modulus is given. The speed of sound in any fluid is predictable from their density and the elastic property of the media.