Food for thought

# How far did the earth move? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

## Solution

The magnitude is defined as a function of $A$, the maximum amplitude of horizontal displacement of the earth’s surface at a distance of $\quantity{100}{km}$ from the epicentre, $M_L=\log_{10}\left(\frac{A}{A_0}\right)$ where $A_0=\quantity{1}{\mu m}=\quantity{10^{-6}}{m}$

What is the magnitude of an earthquake causing a displacement, $A$, equal to

• $\quantity{1.0}{\mu m}$?

In this case, $A=A_0$, so $M_L=\log_{10}1=0 .$

Charles Richter chose this as the reference because no equipment existed at the time that could detect displacements this small.

Modern seismometers can detect movements smaller than this. How would they appear on the magnitude scale?

• $\quantity{1.0}{cm}$?

$\quantity{1.0}{cm}=\quantity{10^{-2}}{m}=\quantity{10^{4}}{\mu m}$, so $M_L=\log_{10}{10^4}=4.0 .$

This would be described as a minor or light earthquake, easily felt but causing minimal damage.

• $\quantity{30}{cm}$?

$\quantity{30}{cm}=\quantity{0.3}{m}=\quantity{3\times10^{5}}{\mu m}$, so $M_L=\log_{10}{(3\times10^5)}\approx5.5 .$

This would be a moderate earthquake, causing significant damage to poorly constructed buildings.

What was the surface displacement for the following earthquakes? What might that have looked or felt like?

• New Zealand, 2011, $M_L=6.1$
To find $A$ we can rearrange the formula and substitute in the $M_L$ value. \begin{align*} M_L &= \log_{10}\left(\frac{A}{A_0}\right)\\ \frac{A}{A_0} &= 10^{M_L}\\ A &= \quantity{10^{6.1}\times 1.0}{\mu m}\\ &\approx \quantity{1.3}{m} \end{align*} If the earth’s surface moved $\quantity{1.3}{m}$ you would expect it to cause severe damage to buildings, which is what happened in Christchurch in 2011.
• Lincolnshire, 2008, $M_L=5.2$
\begin{align*} \frac{A}{A_0} &= 10^{M_L}\\ A &= \quantity{10^{5.2}}{\mu m}\\ &\approx \quantity{16}{cm} \end{align*}

This caused minor damage to buildings.

• Chile, 1960, $M_L=9.5$
This was the strongest earthquake ever recorded on earth. \begin{align*} A&=\quantity{10^{9.5}}{\mu m}\\ &\approx\quantity{3.2}{km} \end{align*}

Do you find this surprising? How realistic do you think it is?

What factors may effect the reliability of this result?