<Text-field style="Heading 1" layout="Heading 1">Earthquakes</Text-field>

Logarithmic Scales - Section 8D

Mth 111 Mathematics as Human Pursuit Lab

Earthquakes

The magnitude scale for earthquakes is defined so that each magnitude represents about 32 times as much energy as the prior magnitude. The formulas relating released energy (E) in joules to magnitude (M) are given by

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or equivalently

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See the U.S. Geological Survey Earthquake Hazards Program Webster:

http://earthquake.usgs.gov/

for lots of information about earthquakes.

Example

The energy released by a magnitude 6 earthquake is computed as:

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Exercise

a) On April 16th, 2008 there was an earquake nearVincinnes, IN of magnitude 5.1 How much energy in joules is released by an earthquake of magnitude 5.1?

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b) "At 4:53 pm on January 12 2010, the worst earthquake to hit the Caribbean in 200 years struck the impoverished nation of Haiti. The epicentre of the 7.0-magnitude earthquake was only 25 kilometres from the country\342\200\231s capital, Port-au-Prince; the devastation was massive and immediate." (From the website Inside the Haiti Earthquake (Note: At this website you can simulate experiencing the earthquake from the viewpoint of a survivor, aid worker, or journalist -- well worth your time.)

Compute the energy released in joules by this earthquake:

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From the computations above we can conclude that this magnitude 7.0 is almost ______ times as strong as the magnitude 5.1.

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Exercise: Go to the US Geological Survey web site on earthquakes: http://earthquake.usgs.gov/ -- Use the Historical links

http://earthquake.usgs.gov/earthquakes/world/historical.php to find the magnitudes of the May 12, 2008 earthquake in Eastern Sichuan, China lo\011ok up the magnitude of the largest earthquake recorded since 1900 in Chile, May 22, 1960. Compare as before their relative instensities.

JSFH

<Text-field style="Heading 1" layout="Heading 1">pH</Text-field>

Finding pH. The pH Scale is defined by

[H+ ] = 10-pH.

where H+ is the hydrogen ion concentration in moles per liter.

Equivalently

pH = -log 10[H+]

The lower the pH, the more acidic the solution.

Practice. How much more acidic (in terms of ratios) is acid rain with a pH equal to 3.5 than ordinary rain with a pH equal to 6?

Step 1: Compute Hydrogen Ion concentration for a pH equal to 6.0:

JSFH

Step 2: Compute Hydrogen Ion concentration for a pH equal to 3.5:

JSFH

Step 3: Take the ratio of your answers:

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This is the same as (in just one step)

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Or Short Cut Method: Just take 10 raised to the amount by which the pH decreases: 2.5 = -3.5 - (-6.0)

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JSFH

<Text-field style="Heading 1" layout="Heading 1">Sound </Text-field>

Sound -- we use dB to denote loudness in number of decibels, and IR to denote the ratio

(intensity of the sound ) / (intensity of softest audible sound).

The relationship between decibels and intensity are given by the formula

\134011loudness in decibels = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIzEwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnRjUvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGPi8lKnN5bW1ldHJpY0dGPi8lKGxhcmdlb3BHRj4vJS5tb3ZhYmxlbGltaXRzR0Y+LyUnYWNjZW50R0Y+LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTS1JJW1zdWJHRiQ2JS1GLDYlUSRsb2dGJy8lJ2l0YWxpY0dGPkY1LUYjNiRGMUY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNUYrRjU=(Intensity ratio of sound to softest audible sound)

or using our abbreviations:

\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

Alternatively we have an equivalent formula:

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For example: If we have a sound that has an intensity ratio IR of 100 (Example 3 page 502), we would use the first form to compute its loudness in decibels. 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 =10(2) = 20 decibels.

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On the other hand if we are given a loudness in decibels of 20, we use the second form to compute its intensity ratio:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEjMTBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Iy1JJm1mcmFjR0YkNigtRi82JFEjMjBGJ0YyLUYjNiNGLi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRC8lKWJldmVsbGVkR1EmZmFsc2VGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRic=

Example: A sound of what loudness in decibels is 63 times as intense as a 13 dB sound?

Step 1: Compute the intensity ratio of the 13 dB sound.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEjMTBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Iy1JJm1mcmFjR0YkNigtRi82JFEjMTNGJ0YyLUYjNiMtRi82JFElMTAuMEYnRjIvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkcvJSliZXZlbGxlZEdRJmZhbHNlRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYn

Step 2: Compute an intensity ratio 63 times you intensity ratio computed in Step 1.

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Step 3: Compute the corresponding decibels for this intensity ratio.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSNtbkdGJDYkUSMxMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJyZzZG90O0YnRjMvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjwvJSlzdHJldGNoeUdGPC8lKnN5bW1ldHJpY0dGPC8lKGxhcmdlb3BHRjwvJS5tb3ZhYmxlbGltaXRzR0Y8LyUnYWNjZW50R0Y8LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSy1GLDYlUSZsb2cxMEYnLyUnaXRhbGljR0Y8RjMtSShtZmVuY2VkR0YkNiQtRiM2Iy1GMDYkUSwxMjU3LjAxNTI1OEYnRjNGMw==

Try Exercise 19 page 507: (Compute the IR of each sound and then take their ratio.)

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Inverse Square Law for Sound:

Intensity of sound is inversely proportional to the square of the distance from the source of the sound.

This gives the Inverse Square Law:

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(Here K is a constant)

For example if you move from 5 feet from a band performing to 15 feet away, you have moved 3 times further away and reduced the intensity by a factor 32 = 9. The sound is 9 times less intense. If you moved to 50 feet away, you have moved 10 times further away. Thus the sound intensity has decreased by a factor of 102 = 100 times less intense.

Try Exercise 23 page 507.

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