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Chapter 17: Problem 74

The average density of Earth's crust \(10 \mathrm{~km}\) beneath the continents is \(2.7 \mathrm{~g} / \mathrm{cm}^{3}\). The speed of longitudinal seismic waves at that depth, found by timing their arrival from distant earthquakes, is \(5.4 \mathrm{~km} / \mathrm{s}\). Find the bulk modulus of Earth's crust at that depth. For comparison, the bulk modulus of steel is about \(16 \times 10^{10} \mathrm{~Pa}\).

Short Answer

Expert verified
The bulk modulus of Earth's crust at 10 km depth is \( 7.87 \times 10^{10} \mathrm{~Pa} \).

Step by step solution

01

Understand the Given Variables

We are given the density \( \rho \) of the Earth's crust as \( 2.7 \mathrm{~g/cm^3} \), which we need to convert to \( \mathrm{kg/m^3} \) knowing there are 1000 \( \mathrm{g} \) in a \( \mathrm{kg} \) and \( 10^6 \mathrm{~cm^3} \) in a \( \mathrm{m^3} \). This results in \( \rho = 2700 \; \mathrm{kg/m^3} \). The speed of the\ longitudinal seismic waves \( v \) is given as \( 5.4 \; \mathrm{km/s} \), which is \( 5400 \; \mathrm{m/s} \).
02

Understand the Relation Between Variables

The formula relating the density, speed of seismic waves, and bulk modulus \( K \) is given by \( v = \sqrt{\frac{K}{\rho}} \). We need to solve for the bulk modulus \( K \).
03

Solve the Equation for Bulk Modulus

Using the formula \( v = \sqrt{\frac{K}{\rho}} \), we square both sides to get \( v^2 = \frac{K}{\rho} \), which implies \( K = v^2 \cdot \rho \). Substitute the given values: \( K = (5400 \; \mathrm{m/s})^2 \times 2700 \; \mathrm{kg/m^3} \).
04

Calculate the Bulk Modulus

Calculate \( K = 5400^2 \times 2700 = 78732000000 \; \mathrm{Pa} \). To express this in \( 10^{10} \mathrm{~Pa} \), we write \( K = 7.87 \times 10^{10} \mathrm{~Pa} \).
05

Compare with Bulk Modulus of Steel

The bulk modulus of the Earth's crust at this depth is \( 7.87 \times 10^{10} \mathrm{~Pa} \), compared to the bulk modulus of steel, which is \( 16 \times 10^{10} \mathrm{~Pa} \). The Earth's crust is less stiff than steel.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Seismic Waves
Seismic waves are vibrations that travel through the Earth, typically as a result of natural events like earthquakes or volcanic activity. They are crucial in helping us understand the internal structure of the Earth.
There are two main types of seismic waves: body waves and surface waves. Body waves travel through the Earth's interior, whereas surface waves travel across the surface.
  • Body waves include Primary (P) waves and Secondary (S) waves. P waves are longitudinal waves that compress and expand the material they move through. They are faster than S waves and can move through both solids and liquids.
  • Surface waves include Love and Rayleigh waves. These are slower and tend to cause more destruction during an earthquake due to their motion along the Earth's surface.
Understanding these waves helps seismologists locate the epicenter of earthquakes and study the Earth's internal characteristics, such as density and composition.
Earth's Crust Density
The density of the Earth's crust is a significant factor in understanding the behavior of seismic waves. Density indicates how much mass is contained in a specific volume of a material. For the Earth's crust, the average density beneath the continents is approximately 2.7 g/cm鲁, which can be converted to 2700 kg/m鲁 for scientific calculations.
This density affects the speed at which seismic waves travel through the crust. Higher density typically means more closely packed particles, causing seismic waves to travel faster.
When comparing the crust's density to other materials, such as steel, we see that the crust is much less dense. Steel, for instance, has a density around 7850 kg/m鲁. This difference in density results in different physical properties, like speed of sound and stiffness, which are essential for understanding geological phenomena.
Longitudinal Seismic Wave Speed
The speed at which longitudinal seismic waves travel through a material is an important measure of its elastic properties. These waves, also known as P waves, are compressional waves that change the volume of the material as they pass through.
The speed of longitudinal seismic waves in the Earth's crust is approximately 5.4 km/s, or 5400 m/s. This speed is affected by the medium's density and bulk modulus.
To find the speed, you can use the formula:
  • \[ v = \sqrt{\frac{K}{\rho}} \]
Here v is the wave speed, K is the bulk modulus, and 蚁 is the density. Solving problems with this formula allows us to determine how various factors like density affect wave propagation.
For instance, in the original exercise, using the given density and wave speed through these calculations helped estimate the bulk modulus of the Earth's crust, serving as a comparison to materials like steel.

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Most popular questions from this chapter

A sound source \(A\) and a reflecting surface \(B\) move directly toward each other. Relative to the air, the speed of source \(A\) is \(29.9 \mathrm{~m} / \mathrm{s}\), the speed of surface \(B\) is \(65.8 \mathrm{~m} / \mathrm{s}\), and the speed of sound is \(329 \mathrm{~m} / \mathrm{s}\). The source emits waves at frequency \(1200 \mathrm{~Hz}\) as measured in the source frame. In the reflector frame, what are the (a) frequency and (b) wavelength of the arriving sound waves? In the source frame, what are the (c) frequency and (d) wavelength of the sound waves reflected back to the source?

You can estimate your distance from a lightning stroke by counting the seconds between the flash you see and the thunder you later hear. By what integer should you divide the number of seconds to get the distance in kilometers?

Ultrasound, which consists ultrasour of sound waves with frequencies above the human audible range, can Artery be used to produce an image of the interior of a human body. Moreover, ultrasound can be used to measure Figure 17-47 Problem \(83 .\) the speed of the blood in the body; it does so by comparing the frequency of the ultrasound sent into the body with the frequency of the ultrasound reflected back to the body's surface by the blood. As the blood pulses, this detected frequency varies. Suppose that an ultrasound image of the arm of a patient shows an artery that is angled at \(\theta=20^{\circ}\) to the ultrasound's line of travel (Fig. 17-47). Suppose also that the frequency of the ultrasound reflected by the blood in the artery is increased by a maximum of \(5495 \mathrm{~Hz}\) from the original ultrasound frequency of \(5.000000 \mathrm{MHz} .\) (a) In Fig. \(17-47\), is the direction of the blood flow rightward of leftward? (b) The speed of sound in the human arm is \(1540 \mathrm{~m} / \mathrm{s}\). What is the maximum speed of the blood? (Hint: The Doppler effect is caused by the component of the blood's velocity along the ultrasound's direction of travel.) (c) If angle \(\theta\) were greater, would the reflected frequency be greater or less?

The source of a sound wave has a power of \(1.00 \mu \mathrm{W}\). If it is a point source, (a) what is the intensity \(3.00 \mathrm{~m}\) away and (b) what is the sound level in decibels at that distance?

A tube \(1.20 \mathrm{~m}\) long is closed at one end. A stretched wire is placed near the open end. The wire is \(0.330 \mathrm{~m}\) long and has a mass of \(9.60 \mathrm{~g}\). It is fixed at both ends and oscillates in its fundamental mode. By resonance, it sets the air column in the tube into oscillation at that column's fundamental frequency. Find (a) that frequency and (b) the tension in the wire.
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