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Chapter 9: Problem 22

The Richter scale, developed in \(1935,\) has been used for years to measure earthquake magnitude. The Richter magnitude \(m\) of an earthquake is given by $$ m=\log \frac{A}{A_{0}} $$ where \(A\) is the maximum amplitude of the earthquake and \(A_{0}\) is a constant. What is the magnitude on the Richter scale of an earthquake with an amplitude that is a million times \(A_{0} ?\)

Short Answer

Expert verified
The magnitude of the earthquake on the Richter scale is 6.

Step by step solution

01

Understand the Given Formula

The Richter magnitude formula is given as \[ m = \log \left( \frac{A}{A_0} \right) \] where - \(m\) is the Richter magnitude - \(A\) is the maximum amplitude of the earthquake - \(A_0\) is a constant
02

Identify the Given Values

According to the problem, the amplitude of the earthquake, \(A\), is a million times the reference amplitude, \(A_0\). Thus, \(A = 1,000,000 \cdot A_0\).
03

Substitute the Given Values into the Formula

Replace \(A\) in the formula: \[ m = \log \left( \frac{1,000,000 \cdot A_0}{A_0} \right) \]
04

Simplify the Expression Inside the Logarithm

\(A_0\) will cancel out, and we are left with: \[ \frac{1,000,000 \cdot A_0}{A_0} = 1,000,000 \]
05

Calculate the Logarithm

Since \(1,000,000\) can be written as \(10^6\), we get: \[ m = \log(10^6) \] Using the properties of logarithms, we have \( \log(10^6) = 6\).
06

Conclude the Magnitude

Thus, the magnitude of the earthquake on the Richter scale is \(m = 6\).

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Key Concepts

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

Logarithms
Logarithms are a fundamental concept in mathematics. They help us understand and work with very large or very small numbers by transforming multiplicative relationships into additive ones. A logarithm is the inverse operation to exponentiation. For example, if we have the equation \(10^6 = 1,000,000\), the logarithm tells us how many times we need to multiply 10 to get 1,000,000. This can be written as \( \text{log}_{10} (1,000,000) = 6 \). This property is crucial for interpreting the Richter scale, where earthquake magnitude is calculated using a logarithmic formula.
So, when we need to find the magnitude of an earthquake with amplitude 1,000,000 times greater than \(A_0\), we use: \ m = \text{log}(1,000,000 \times A_0/A_0) = \text{log}(1,000,000) = 6 \.
Amplitude
Amplitude refers to the maximum height of a wave, whether it's sound, light, or in this case, an earthquake wave. Greater amplitude means a stronger, more intense wave. In the context of earthquakes, amplitude is crucial for determining the amount of energy released.
The formula for the Richter scale magnitude \(m = \text{log}(A/A_0) \) involves amplitude \(A\), indicating how it scales exponentially. A million times greater amplitude than the base amplitude \(A_0\) translates to \(A = 1,000,000 \times A_0\). When substituted back into the Richter formula, it simplifies to \(m = \text{log}(1,000,000)\), giving a magnitude of 6. This showcases the importance of understanding amplitude in measuring earthquake energy.
Earthquake Measurement
Measuring earthquakes accurately is vital for safety and preparedness. The Richter scale is a logarithmic scale, meaning each whole number increase on the scale represents a tenfold increase in measured amplitude and roughly 31.6 times more energy release.
For instance, an earthquake with a Richter magnitude of 6 releases significantly more energy than one with a magnitude of 5. This system helps scientists and engineers understand the potential damage and required safety measures.
The formula \( m = \text{log}(A / A_0) \) plays a crucial role in this measurement. By comparing the measured amplitude \(A\) to a reference amplitude \(A_0\), the Richter scale provides a standardized way to communicate the earthquake's strength.
Understanding how to use and interpret this formula is essential for anyone studying seismology or related fields.

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