Problem 97 - Applied Calculus

Question

\(97-98 .\) GENERAL: Earthquakes The sizes of major earthquakes are measured on the Moment Magnitude Scale, or MMS, although the media often still refer to the outdated Richter scale. The MMS measures the total energy relensed by an earthquake, in units denoted \(M_{W}\) (W for the work accomplished). An increase of \(1 M_{W}\) means the energy increased by a factor of \(32,\) so an increase from \(A\) to \(B\) means the energy increased by a factor of \(32^{B-A}\). Use this formula to find the increase in energy between the following earthquakes: The 1994 Northridge, California, earthquake that measured \(6.7 \mathrm{M}_{\mathrm{W}}\) and the 1906 San Francisco earthquake that measured \(7.8 \mathrm{M}_{W}\). (The San Francisco earthquake resulted in 3000 deaths and a 3 -day fire that destroyed 4 square miles of San Francisco.)

Step-by-Step Solution

Answer

The San Francisco earthquake released approximately 38.56 times more energy than the Northridge earthquake.

Step 1: Identify given magnitudes

The magnitudes given are \( A = 6.7 \) for the Northridge earthquake and \( B = 7.8 \) for the San Francisco earthquake.

Step 2: Calculate magnitude difference

Calculate the difference between the magnitudes of the two earthquakes: \[ B - A = 7.8 - 6.7 = 1.1 \]

Step 3: Use the formula for energy increase factor

Use the formula \(32^{B-A}\) to determine how much more energy the San Francisco earthquake released compared to the Northridge earthquake. \[ 32^{1.1} \]

Step 4: Compute the energy factor

Calculate \( 32^{1.1} \) using a calculator:\[ 32^{1.1} \approx 38.56 \]

Step 5: Interpret the result

The energy released by the San Francisco earthquake was approximately 38.56 times greater than the energy released by the Northridge earthquake.

Key Concepts

Understanding Earthquake Energy Calculation

Earthquakes are powerful natural events that release immense energy. To quantify this energy, scientists use the Moment Magnitude Scale (MMS). Unlike other scales, such as the Richter scale, MMS provides a more accurate representation of the energy released during an earthquake.

When calculating the energy difference between two earthquakes, we use the given formula related to the MMS. This formula uses differences in magnitude to determine how much more or less energy is released compared to another event.

The MMS is logarithmic; each whole number increase in magnitude reflects a significant increase in energy - precisely, a thirty-twofold increase. So, when calculating the energy differential between two earthquakes with magnitudes of, say, 6.7 and 7.8, a crucial step involves finding the magnitude difference (in this case, 1.1) and computing the energy factor as shown by the formula:

• Magnitude difference = 7.8 - 6.7 = 1.1
• Energy factor = 32^1.1

This reveals that the San Francisco earthquake released approximately 38.56 times the energy of the Northridge earthquake.

Richter Scale - The Predecessor

While the Moment Magnitude Scale is the modern approach, the Richter scale is historically significant and was commonly used before MMS. Developed in 1935 by Charles F. Richter, it aimed to quantify the size of seismic waves produced by earthquakes.

The Richter scale, just like the MMS, is a logarithmic scale. It meant that each unit increase in Richter magnitude equated to a tenfold increase in measured amplitude of the seismic waves. However, the scale was less effective for measuring large or distant earthquakes.

This limitation led to more advanced scales, with MMS being the preferred model for detailed energy readings. It's essential to recognize that although the Richter scale is a historical marker in seismology, MMS allows for a more comprehensive analysis.

Exploring the Logarithmic Scale

A logarithmic scale is a type of scale used for measurement that displays information in a way that helps visualize large quantities by reducing them to more manageable figures. Many sciences, including seismology, use logarithmic scales due to the wide range of values that they can represent.

In the context of earthquakes, both the Richter and Moment Magnitude Scales are logarithmic. This means that each increase by one unit represents a fixed multiplication of the previous value:

• For the Richter scale, each whole number increase means a tenfold increase in wave amplitude.
• For the moment magnitude, an increase by one leads to a thirty-twofold increase in energy release.

This kind of scaling is crucial for simplifying and interpreting the impact and energy involved in natural phenomena, allowing both scientists and the public to better understand the magnitude of such events.