Problem 2 - College Physics

Question

Earthquakes at fault lines in Earth's crust create seismic waves, which are longitudinal (P-waves) or transverse (S-waves). The P-waves have a speed of about \(7 \mathrm{~km} / \mathrm{s}\). Estimate the average bulk modulus of Earth's crust given that the density of rock is about \(2500 \mathrm{~kg} / \mathrm{m}^{3}\).

Step-by-Step Solution

Answer

The estimated average bulk modulus of Earth's crust is approximately \(1.225 * 10^{11} \mathrm{~Pa}\).

Step 1: Identify the Knowns and Unknowns

We're given that the speed of the P-waves (\(v\)) is \(7 \mathrm{~km/s}\), which we'll need to convert to \(m/s\). Also, we're given that the density of the rock (\(p\)) is \(2500 \mathrm{~kg/m^3}\). We're asked to find the bulk modulus (\(B\)).

Step 2: Convert Units

The speed of P-waves is given in \(km/s\). Convert this to \(m/s\) by multiplying by \(1000\). So, \(v\) becomes \(7000 \mathrm{~m/s}\).

Step 3: Rearrange the Formula to Solve for the Bulk Modulus

The formula is \(v = \sqrt{B/p}\). We need to isolate, \(B\), the bulk modulus. Squaring both sides gives \(v^2 = B/p\). Rearranging to solve for \(B\) gives \(B = v^2 * p\).

Step 4: Substitute the Given Values into the Formula

Substitute \(v = 7000 \mathrm{~m/s}\) and \(p = 2500 \mathrm{~kg/m^3}\) into the formula. \(B = (7000 \mathrm{~m/s})^2 * 2500 \mathrm{~kg/m^3}\).

Step 5: Perform the Calculation

Multiply out the values to calculate the bulk modulus. \(B\) = \(1.225 * 10^{11} \mathrm{~kg/(m \cdot s^2)}\) or \(1.225 * 10^{11} \mathrm{~Pa}\), since \(1 \mathrm{~Pa} = 1 \mathrm{~kg/(m \cdot s^2)}\).

Key Concepts

Seismic Waves

Seismic waves are vibrations that travel through the Earth's interior or along its surface. They are typically generated by the sudden release of energy in the Earth's crust, such as during an earthquake or by volcanic activity. These waves can tell us a lot about the internal structure of Earth because they travel at different speeds depending on the materials they move through. There are several types of seismic waves, including primary waves (P-waves), secondary waves (S-waves), and surface waves. Each type has its own distinct properties and behaviors as it interacts with various rock layers and compositions within the Earth.

P-waves

P-waves, or primary waves, are a type of seismic wave that can travel through solids, liquids, and gases, making them the fastest seismic waves. When P-waves travel through the Earth, they compress and expand the material in the same direction that the wave is moving. These waves are the first to be detected by seismographs after an earthquake occurs. Due to their ability to move through all types of matter, they provide essential information about the Earth's core and various boundaries within the Earth's interior.

Density of Rock

The density of rock is an important physical property that affects the propagation of seismic waves. It is defined as the mass per unit volume of the rock and typically measured in kilograms per cubic meter (\( \text{kg/m}^3 \)). Density can vary significantly between different rock types and conditions. For example, dense igneous rocks will have higher densities compared to less dense sedimentary rocks. Understanding the density of the Earth's crust is crucial for solving many geophysical problems, including the estimation of the bulk modulus that determines how much the material will compress under pressure.

Speed of P-waves

The speed of P-waves, or their velocity, is directly related to the properties of the material they are traveling through. The wave speed is a function of both the elasticity and density of the material, quantified through formulas that involve the modulus of elasticity, such as the bulk modulus, and material density. In the context of the Earth's crust, the average speed of P-waves is around 7 kilometers per second, but this number can vary depending on the specific composition and condition of the crust at different locations.

Physics Problem Solving

Physics problem solving often involves understanding and applying various formulas to find unknown quantities. In the context of the given exercise, the formula relating the speed of P-waves (v), density of rock (\rho), and bulk modulus (B) is a crucial element. This formula, often presented as \(v = \sqrt{B/\rho}\), serves as the foundation for calculating the bulk modulus of a material given its density and the speed at which P-waves travel through it. Physics problems such as this one not only require numerical calculations but also an understanding of the physical principles underlying the formulas.