Question

The formula for the Richter magnitude, \(M\) of an earthquake is \(M=\log \frac{A}{A_{0}},\) where \(A\) is the amplitude of the ground motion and \(A_{0}\) is the amplitude of a standard earthquake. In \(1985,\) an earthquake with magnitude 6.9 on the Richter scale was recorded in the Nahanni region of the Northwest Territories. The largest recorded earthquake in Saskatchewan occurred in 1982 near the town of Big Beaver. It had a magnitude of 3.9 on the Richter scale. How many times as great as the seismic shaking of the Saskatchewan earthquake was that of the Nahanni earthquake?

Short answer

The shaking in Nahanni was 1000 times greater than in Big Beaver

Step-by-step solution

- Understand the problem

The problem asks to find out how many times greater the seismic shaking (ground motion amplitude) of the earthquake in Nahanni was compared to the one in Big Beaver.

- Write down the given magnitudes

Nahanni earthquake magnitude: - - 6.9 -Big Beaver magnitude: 3.9

- Recall the formula

The Richter magnitude formula is given by: R

- Solving the equations for each earthquake

We will use the magnitude formula each calculate the amplitudes (A) for earthqueakes to for the be erals amplitude ao the st nder quake am

- Calculate ground motion amplitude

Nahanni's earthquake: 6.9_ _arithmeticoe amplit

- Find the ratio of amplitudes

We use the ratio of amplitudes to find how many times greater the shaking in Nahanni was compared to Big Beaver. This is calculated as: By dividing through exponential exponents By we've we get r difference magnitude Richter solution the step

- Simplify and calculate the final answer

Simplify the division step to get the final ratio of the amplitudes: dealt simply exponent ratio with over amplitudes

Key concepts

logarithmic scales

Logarithmic scales are a unique way of displaying numerical data. A key feature of logarithmic scales is that each unit increase on the scale represents a tenfold increase in the quantity being measured. For example, moving from 1 to 2 on a logarithmic scale implies a tenfold increase, and moving from 2 to 3 implies another tenfold increase.
This type of scale is commonly used for quantities that can span many orders of magnitude.
It’s particularly useful when dealing with natural phenomena like earthquakes because the difference in intensity between weak and strong earthquakes is enormously vast.
Thus, a logarithmic scale like the Richter scale allows for a more manageable representation.
Rather than presenting extremely large numbers, it condenses them into a scale where each step represents a proportional increase in intensity.

ground motion amplitude

Ground motion amplitude is a measurement of the ground displacement or shaking caused by seismic waves during an earthquake.
It is essentially how much the ground moves back and forth as the seismic waves pass through.
This movement can be captured by instruments called seismographs.
The greater the amplitude, the stronger the shaking during the earthquake.
When comparing earthquakes, the amplitude of ground motion is critical because it directly relates to the level of damage and energy released.
The Richter scale formula, which incorporates amplitude, gives us a way to quantify these differences accurately.

earthquake magnitude calculation

The magnitude of an earthquake represents the energy released at its source, and it's calculated using the Richter magnitude formula. The formula is:
\[M = \text{log} \frac{A}{A_{0}}\],
Where:

• \(M\): The magnitude of the earthquake.
• \(A\): The amplitude of the ground motion recorded by a seismograph.
• \(A_{0}\): The amplitude of a standard earthquake.

To find the magnitude, you take the logarithm (base 10) of the ratio between the amplitude of the current earthquake and a reference amplitude.
This equation allows us to standardize the measurement of earthquakes, making it simple to compare the magnitudes of different events.
For example, if one earthquake has a magnitude of 6.9 and another a magnitude of 3.9, the difference in shaking can be found by calculating the ratio of their amplitudes.

Richter scale

The Richter scale, introduced by Charles F. Richter in 1935, is a system for measuring the magnitude of earthquakes.
It assigns a single number to quantify the energy that's released by an earthquake.
It's important to know that the Richter scale is logarithmic.
This means that each whole number increase on the scale corresponds to a tenfold increase in measured amplitude and roughly 31.6 times more energy release.
For instance, an earthquake that measures 6.0 on the Richter scale has 10 times the ground motion amplitude of one that measures 5.0, and about 31.6 times more energy release.
This makes the Richter scale very effective in conveying the power and potential destructiveness of an earthquake in a simple to understand, numeric form.