Question

The P-waves have a speed of about \[6\,{km}/{s}\;\]. How do you estimate the average bulk modulus of Earth’s crust given that the density of rock is about \[2400\,{kg}/{{{m}^{3}}}\;\]. Answer the question in Pa?

Answer 1

Hint: The formula that defines the parameters such as, velocity, bulk modulus and the density should be used to solve this problem. The formula relates the energy of a photon and the wavelength through Plank’s constant and the speed of light.
 Formula used:
\[v=\sqrt{\dfrac{B}{\rho }}\]

Complete step by step answer:
Bulk modulus – when a uniform element is subjected to the equal stresses from all the three mutually perpendicular directions, then, the ratio of direct stress to the volumetric strain is called the bulk modulus.
From the given information, we have the data as follows.
The P-waves have a speed of about \[6\,{km}/{s}\;\].
The density of rock is about \[2400\,{kg}/{{{m}^{3}}}\;\].
Convert the units of the given parameters into SI units. The unit of the speed is given as km per sec, we need to convert it into metre per sec.
\[6\,{km}/{s}\;=6000\,{m}/{s}\;\]
Consider the formula that relates the parameters such as, velocity, bulk modulus and the density is given as follows.
\[v=\sqrt{\dfrac{B}{\rho }}\]
Where \[v\]is the speed of the wave, B is the bulk modulus and \[\rho \]is the density of the material.
Square the above equation on both the sides.
\[{{v}^{2}}=\dfrac{B}{\rho }\]
Represent the above equation in terms of Bulk modulus.
\[B=\rho {{v}^{2}}\]
Substitute the values in the above equation.
\[B=2400\times {{6000}^{2}}\]
Continue further computation.
\[B=8.64\times {{10}^{10}}Pa\]
Thus the value of the bulk modulus is,
\[B=8.64\times {{10}^{4}}MPa\]

\[\therefore \] The average bulk modulus of Earth’s crust in Pa is \[8.64\times {{10}^{4}}MPa\].

Note: After converting the units of the parameters, we have to continue the computation. The bulk modulus represents the change in volume. The bulk modulus is the volume of the object times the change in pressure exerted on the material by the change in the volume of the object.