Problem 89 - Brief Applied Calculus

Question

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of major earthquakes are measured on the Moment Magnitude Scale, or MMS, although the media often still refer to the outdated Richter scale.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 released 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 \(6.7 M_{W}\) and the 1906 San Francisco earthquake that measured \(7.8 M_{W}\). (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 energy increased by approximately 42.8 times between the two earthquakes.

Step 1: Identify Variables

Identify the values of the moment magnitude measurements for the two earthquakes. Here, earthquake A (Northridge, 1994) has a magnitude of \( M_W = 6.7 \) and earthquake B (San Francisco, 1906) has a magnitude of \( M_W = 7.8 \).

Step 2: Calculate Increase in Magnitude

Determine the difference in magnitude between the two earthquakes. Calculate: \( B - A = 7.8 - 6.7 = 1.1 \).

Step 3: Use the Energy Formula

Use the formula \( 32^{B-A} \) to calculate the increase in energy. Plug the values into the formula: \( 32^{1.1} \).

Step 4: Evaluate the Formula

Calculate the value of \( 32^{1.1} \) using a calculator. This gives approximately 42.8.

Key Concepts

Earthquake Energy Measurement

The Moment Magnitude Scale (MMS) is a standard tool used by seismologists to measure the energy released by an earthquake. Unlike the older Richter scale, MMS provides a more accurate depiction of an earthquake's size and potential impact. When people talk about an earthquake's magnitude, they usually refer to this scale.

The MMS works by evaluating several factors, including:

• Seismic wave amplitude
• Fault size
• Average slip along the fault

Each of these factors helps to give a complete picture of the energy released. This approach allows MMS to remain effective across different quake sizes, from everyday seismic activity to the most devastating natural events. By focusing on energy rather than just movement, seismologists can provide more precise data about the earthquake's true impact.

Mathematical Calculation in Seismology

Mathematical calculations are vital in understanding and measuring seismic events. The Moment Magnitude Scale relies heavily on mathematical formulas to determine the energy release of an earthquake. In our exercise, we look at two notable earthquakes to understand these calculations better.

To measure the difference in energy using MMS, we calculate the difference between the magnitudes. For instance, the Northridge earthquake (1994) was noted at 6.7 on the MMS, and the San Francisco earthquake (1906) was larger at 7.8. This difference of 1.1 in magnitude gives us a clear indication of the relative energy increase between these two events.

Using the formula \(32^{B-A}\), where \(B\) and \(A\) are the magnitudes of the two earthquakes, seismologists can calculate the factor by which energy increases. This formula utilizes the base of 32 to represent the exponential growth in energy released as an earthquake's magnitude incrementally increases.

Exponential Growth in Energy Release

Earthquakes are phenomenally powerful events, and the increase in energy is not linear but exponential. This means small increases in magnitude can result in massive increases in energy release.

In the formula \(32^{B-A}\), an increase of just 1 unit in the MMS results in 32 times more energy release. If we consider the specific case from the exercise, moving from a magnitude of 6.7 to 7.8 seems small on the surface. However, mathematically, it shows a considerable spike in energy.

To compute, consider 7.8 - 6.7, which results in 1.1. This exponentiation gives us \(32^{1.1}\), which approximately equals 42.8. This means the San Francisco earthquake released just over 42 times the energy of the Northridge earthquake. Such exponential growth patterns highlight why even minor differences in MMS readings can indicate significantly different catastrophic potential.