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Transcript of Maths geology
Page 2
Index Index ........................................................................................................... 1
Mathematics vs Geology ............................................................................. 3
How can Mathematics applied to Geology be important for us? .............. 4
Radiometric dating ...................................................................................... 5
What is Radiometric Dating? ................................................................... 5
How is Radiometric Dating done? ........................................................... 5
Why is Radiometric Dating important for us? ........................................... 8
Earth’s Internal Structure ............................................................................ 8
How can we possibly know what the Earth’s internal structure is like? ...... 10
Mohorovicic discontinuity: crust vs mantle ............................................. 11
Gutenberg discontinuity : Mantle vs Liquid Outer Core .......................... 12
Earthquakes and logarithms .................................................................. 13
Scientific Notation ..................................................................................... 17
Huge and microscopic dimensions ........................................................ 17
Astronomical unit and light-years........................................................... 17
What is Scientific Notation? ................................................................... 19
How Scientific Notation is done? ........................................................... 19
Possible exercises of scientific notation ................................................ 20
Why is Scientific Notation important for us? .......................................... 22
A specific case: Maths and Dams (Hydroelectric powerplant) .................. 24
Conclusion ................................................................................................ 25
Links ......................................................................................................... 26
Students Involved on this work: ................................................................ 26
Teachers Involved on this work: ................................................................ 26
Page 3
Mathematics vs Geology
Mathematics has no generally accepted definition, it’s usually referred has
the abstract study of topics encompassing quantity, structure, space,
calculus and other properties.
Benjamin Peirce called mathematics "the science that draws necessary
conclusions."
Geology is the science that studies the solid features of the Earth, like the
rocks which it is made of, and the processes by which they change.
Geology also helps us understand the evolution and drift of the continents.
The evidences preserved on rocks like fossils, the evolution of past
climates, sustainable development, and the study of the features of
celestial bodies are all different fields of Geology.
Geologists study the way the planet has formed, changed and developed.
They predict future trends so that people can understand potential
problems and find ways to reverse, avoid or prepare for them. And
Mathematics is an essential tool to do that.
Let’s see how.
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How can Mathematics applied to Geology be important for us?
We can find Maths applied to Geology on different fields such as:
Radiometric Dating
Determination of the age of rocks
Earth’s Internal Structure
Variation of the Earth’s temperature with the depth
increase
Seismology
Velocity of seismic waves
Scientific Notation
Distances in the Universe
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Radiometric dating
What is Radiometric Dating?
Radiometric dating (often called radioactive dating) is a technique used to
date materials such as rocks, usually based on a comparison between the
observed abundance of a naturally occurring radioactive isotope and its
decay products, using known decay rates.
How is Radiometric Dating done?
As stated before, Radiometric Dating dates materials comparing the
observed abundance of a naturally occurring radioactive isotope and its
decay products.
An isotope is any of two or more forms of a chemical element having the
same number of protons in the nucleus, but having different numbers of
neutrons in the nucleus, or different atomic weights.
When the magma solidifies and forms a magmatic rock, it will acquire
radioactive elements; some particular isotopes are inherently unstable.
That is, at some point in time, an atom will spontaneously transform in
order to gain stability. This disintegration is irreversible, and the initial
isotope is called the parent isotope and the resulting isotope is called the
daughter isotope.
A particular isotope of a radioactive element decays into another element at
a distinctive rate. This constant rate is given in terms of a "half-life”.
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This table describes the radioactive decay of Carbon-14 and its
transformation into Nitrogen-14. The blue line represents the initial element,
Carbon-14 and the red line represents the result of the radioactive decay,
Nitrogen-14.
By the time of 5730 years, one half-life time has passed - this means that
50% of the carbon-14 atoms have transformed into Nitrogen-14. This is
shown on the graphic when the two lines intercept.
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By looking at the table we can see that, for example, when half of the
parent isotope potassium-40 has turned into argon-40, 1.25 billion years
will have passed.
In this way, if we find a rock that has 50% carbon-14 isotopes and
50% nitrogen-14 isotopes we can say that is 5370 years old or so.
The exact age of the oldest rocks exposed on the
surface of the Earth is difficult to determine, as they
are aggregates of minerals of possibly different
ages.
But the oldest such minerals analyzed to date –
small crystals of zircon from the Jack Hills of
Western Australia – are at least 4.404 billion years
old.
Crystal of Zircon(1)
Page 8
We can find more information about Radiometric Dating, on Isabel Doutor,
Eduardo Fernandes, Sofia Serra and Beatriz Lopes’ prezi. Just click on the
link bellow.
http://prezi.com/uvhnbzdbn3nf/untitled-prezi/
Why is Radiometric Dating important for us?
Nowadays we know that the Earth is about 4.6 billion years old. This
average age is based on evidence from radiometric age dating of meteorite
material and it is consistent with the ages of the oldest-known terrestrial
and lunar samples. Not only Radiometric Dating gives us information about
the solid features that surround us but also it allows us to broaden our
horizons about the Earth’s past, present and future.
Earth’s Internal Structure
The internal constitution of the Earth primarily relates to the structural and
compositional aspects of the layered earth.
The internal structure of the Earth deals primarily with the concentric
layering of the earth based on their physical characteristics, which
distinctively vary in their densities and seismic wave characteristics. And
also, the compositional layers of the earth are characterized by their
chemistry composition.
The image, on the right side of the
sheet, shows the Earth’s internal
structure arranged in layers
according to the Chemical Model.
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And the table below shows the Earth’s internal structure arranged in layers
according the Physical Model.
Depth
Earth Internal Structures(2)
Layer Kilometres Miles
0–60 0–37 Lithosphere (locally varies between 5 and
200 km)
0–35 0–22 … Crust (locally varies between 5 and 70 km)
35–60 22–37 … Uppermost part of mantle
35–2,890 22–1,790 Mantle
100–200 62–125 … Asthenosphere
35–660 22–410 … Upper mesosphere (upper mantle)
660–2,890 410–1,790 … Lower mesosphere (lower mantle)
2,890–
5,150
1,790–
3,160 Outer core
5,150–
6,360
3,160–
3,954 Inner core
Page 10
How can we possibly know what the Earth’s internal structure
is like?
The study of seismic waves’ behavior allowed geologists to know the
Earth’s internal structure.
Seismic waves are waves of energy that travel through the Earth's layers,
and are a result of an earthquake, explosion, or a volcano which erupts.
There are different types of seismic waves according to their
characteristics; we have P waves, S waves and Surface waves.
Primary waves (P-waves) are compressive waves that travel faster
than other waves through the earth, and arrive first at seismograph
stations.
Secondary waves (S-waves) are shear waves that are transverse in
nature and can only travel through solids, since fluids (liquids and gases)
do not support shear stresses.
Surface waves have low frequency, long duration, and large
amplitude, so they are the most destructive.
The propagation velocity of seismic waves depends on the density and the
elasticity of the medium, and is given by the complex following equation:
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From this formula we can know that the velocity of seismic waves depends
on the stiffness of the medium that they cross, and that, the more rigid the
medium is, the faster the seismic waves will be.
In fact, the velocity of seismic waves is not constant throughout the Earth.
That variation is the evidence used to confirm the physical model of the
internal structure of the Earth, as well as the existence of discontinuities
between the layers and an outer core which is in the liquid state.
Mohorovicic discontinuity: crust vs mantle
The crust is separated from the mantle by the Mohorovicic discontinuity,
usually referred to as the Moho.
The study of seismic waves was crucial to prove Moho existence.
Immediately above the Moho the velocities of primary seismic waves (P-
waves) are similar to those through basalt (6.7 – 7.2 km/s), and below the
velocities are similar to those through peridotitic (7.6 – 8.6 km/s) tested on
laboratory simulations.That suggests the Moho marks a change in
composition.
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Gutenberg discontinuity : Mantle vs Liquid Outer Core
When an earthquake occurs, seismographs near the epicenter are able to
record both P and S waves, but those at a greater distance no longer
detect the high frequencies of the first S wave. As was mentioned before,
the velocity of the seismic waves depends on the stiffness of the medium
that they cross; liquid means have zero rigidity, consequently, the S waves
have zero speed. Since S waves cannot pass through liquids, this
phenomenon was original evidence for the now well-established
observation that the Earth has a liquid outer core, as demonstrated
by Richard Dixon Oldham.
The velocity of seismic waves (P and
S) tends to increase with the depth,
and ranges from approximately 2 to
8 km/s in the Earth's crust, up to
13 km/s in the deep mantle.
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Earthquakes and logarithms
An earthquake is a movement of the ground,
generated from tectonic force. These issue a
motion in earth's crust and release this energy
in a point of the underground called "focus".
From here the energy is emitted in a point
exactly above it called "epicentre". The
waves generated from an earthquake are
known as "seismic waves", and are elastic
and longitudinal.
The unit of measurement of a seism, is the
"magnitude", with who we can estimate the
power of an earthquake.
The term "magnitude was coined by
Richter and Gutemberg (that were
seismologists), in 1935 when they created
the Richter scale.
In the Richter scale they posed at
"magnitude 0" an earthquake that, on a
seismograph posed at 100 km from the
epicentre, drew a seismogram of 0.001 mm, and used as referenced model
the torsion Wood-Anderson seismograph. But to gauge slight and very
powerful seisms, they posed the Richter scale on a base-10 logarithm; so
every increase of one unit in the scale is an increase of 10 time of the
amplitude and of 30 times of the energy emitted.
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Magnitude What people see 0 – 1.9 Can be detected only by seismographs
2 – 2.9 Hanging objects might swing
3 – 3.9 Comparable to the vibrations of a passing truck
4 – 4.9 May break windows and cause small or unstable objects to fall down
5 – 5.9 Furniture moves, chunks of plaster may fall
6 – 6.9 Damages to well-built structures, severe damages to poorly-built ones
7 – 7.9 Buildings displaced by foundations, cracks in the earth, underground pipes broken
8 – 8.9 Bridges destroyed, few structures left standing
9 and over Near total destruction, waves moving through the earth visible with naked eyes
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But not all of the seismographs are positioned at 100 km from the
epicentre, and to estimate the earthquakes that happened between 20 and
600km from the seismograph, and at more than 600km...they invented the
"laws of attenuation of seismic waves" the laws of attenuation of seismic
waves.
Richter's original magnitude scale was then extended to observations of
earthquakes of any distance and of focal depths ranging between 0 and
700 km.
Two magnitude scales evolved - the mb and MSscales – as earthquakes
excite both body waves, which travel into and through the Earth, and
surface waves, which are constrained to follow the natural wave guide of
the Earth's uppermost layers:
✦ the standard body-wave magnitude formula is
mb = log10(A/T) + Q(D,h)
where A = amplitude of ground motion (in microns)
T = corresponding period (in seconds)
Q(D,h) = correction factor that is a function of distance
D = degrees between epicenter and station and focal depth
h = distance (in kilometers) of the earthquake
✦ The standard surface-wave formula is:
MS = log10 (A/T) + 1.66 log10 (D) + 3.30
These laws were designed to measure the earthquakes on a large scale.
We can observe that, using logarithms, the graph of the magnitude/energy
is not an exponential curve but a line: on the x axis we have the magnitude
and on the y axis we have the logarithm of the energy associated to the
seism.
However since the middle of 20th century the Richter scale was replaced
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by the "Moment magnitude scale" that measures the magnitude at the
exact moment in who the earthquake takes place.
But these two scales are not the only ones: there's also the Mercalli scale.
The most important difference between Mercalli and Richter scales is the
geographic connotation, that is only kept in account by the Mercalli scale;
for example in 1976 there was a big earthquake: that took place in Friuli
Venezia-Giulia, Italy; with a magnitude of 6.6 and it killed 1000 of people.
Nine years later, in 1985, an earthquake with the same magnitude of that in
Friuli, took place at Los Angeles, California, but this one killed only 6
people. This difference of deads is explainable for the tipe of ground
present in the two places. But there's another thing to say yet: Richter scale
are opened at any value because it haven't a maximum or minimum; for
example the most powerful earthquake registred in the history was the
Great Chilean Earthquake, happend in 1960 at Valdivia and with a
magnitude of 9.5. His power changed definitely the geologic appearance of
the place, he killed only 1000 people, but it shifted the earth's axis of 30cm
The Mercalli scale, differently from the previous, estimates the damage that
an earthquake makes, not is power; because a level in the Richter scale
isn't equivalent of a level on that of Mercalli, in fact this last one considers
Page 17
also the geographic structure of the hurted zone. Originally it included only
ten levels, but the same Mercalli added the 11th after the earthquake that
took place in Messina in 1908, and the 12th level was added in 1956. So
today this scale is called "Mercalli scale changed".
Scientific Notation
Huge and microscopic dimensions
As time goes by, Mankind continues to broaden his knowledge as far as
everything which surrounds it is concerned. Although in the dawn of times,
Man only studied what was closer to him and possible to be studied by his
own senses, it faster became clear that the Universe was much larger than
it appeared.
Therefore, with the advent of the technology and the development of new
gadgets, which would allow us to study in more
detail our surroundings, it became necessary to
measure not only really huge distances but also
tiny ones. Thus, there was the creation of new
units and ways of representing numbers.
Astronomical unit and light-years
In order to measure the distances in the Universe, we can nowadays use
three particular units of measurement: the astronomical unit, the light-year
and the parsec. On the one hand, an astronomical unit (abbreviated as AU)
Page 18
corresponds to the distance between the Earth and the Sun and it is
approximately 149,597,870,700 meters.
As the distance between the Earth and the Sun varies while our planet
describes its orbit, an astronomical unit can be more precisely defined as
the length of the semi-major axis of the Earth’s elliptical orbit around the
Sun.
On the other hand, a light-year (abbreviated as ly) is defined as the
distance that light travels in vacuum in one Julian year, which means
365,25 days. Although it can seem a measure of time, this unit is indeed a
measure of space and one light-year corresponds to nearly 10 trillion
kilometers.
Finally, parsec is the biggest unit of distance and it is the most used in
scientific works for astronomy. The name parsec is an abbreviated form of
“a distance corresponding to a parallax of one second”. It was created in
1913 due to the suggestion of a British astronomer named Herbert Hall
Turner. A parsec is the distance from the Sun to an astronomical
object which has a parallax angle of
one arcsecond (1⁄3,600 of a degree). In
other words, if a straight line were drawn
from the object to the Earth, another line
were drawn from the object to the Sun
and the angle formed between the two
lines were exactly one arcsecond, then
the object's distance
would be exactly
one parsec. One
parsec is equivalent
to 3,26 light-years.
Page 19
What is Scientific Notation?
Despite the creation of new units to measure space, it was also necessary
to create a new way of representing numbers due to their order of
magnitude, which was too big or even too small. Scientific notation (also
called Standard Form in Britain) is a way of writing numbers that are too big
or too small to be conveniently written in decimal form. Scientific notation
has a number of useful properties and is commonly used in calculators and
by scientists, mathematicians and engineers; therefore it can be applied in
a wide range of areas, including, for example, Physics and Geology.
In scientific notation all numbers are written in the form of .
This means, “a times ten raised to the power of b”, where the coefficient a
is any real number, and the exponent b is an integer .The set of integers is
a subset of the real numbers, and consists of the natural numbers (0, 1, 2,
3, ...) and the negatives natural numbers (−1, −2, −3, ...).
How Scientific Notation is done?
In order to convert a number from decimal notation to scientific notation,
first, move the decimal separator point the required amount, n, to make the
number's value within a desired range, between 1 and 10 for normalized
notation. If the decimal was moved to the left, append x 10n; otherwise, if it
was moved to the right, append x 10-n. To represent the number 1,230,400
in normalized scientific notation, the decimal separator would be moved 6
Page 20
digits to the left and x 106 appended, resulting in 1.2304×106. The number -
0.004 0321 would have its decimal separator shifted 3 digits to the right
instead of the left and yield −4.0321×10−3 as a result. On the other hand, in
order to covert a number from scientific notation to decimal notation, we
just need to do the inverse process.
Possible exercises of scientific notation
Write 124 in scientific notation.
This is not a very large number, but it will work nicely for an example. To
convert this to scientific notation, we first write "1.24". This is not the same
number, but (1.24)(100) = 124 is, and 100 = 102. Then, in scientific
notation, 124 is written as 1.24 × 102.
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Write in decimal notation: 3.6 × 1012
Since the exponent on 10 is positive, we know we are looking for a large
number, so we’ll need to move the decimal point to the right, in order to
make the number larger. Since the exponent on 10 is "12", I'll need to move
the decimal point twelve places to the right.
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Thus, the number is 3,600,000,000,000, or 3.6 trillion
Write 0.000 000 000 043 6 in scientific notation.
In scientific notation, the number part (as opposed to the ten-to-a-power
part) will be "4.36". So we will count how many places the decimal point
has to move to get from where it is now to where it needs to be:
Therefore the power on 10 has to be –11. "eleven", because that's how
many places the decimal point needs to be moved, and "negative",
because we’re dealing with a small number and so the decimal point is
moved to the right. So, in scientific notation, the number is written as 4.36 ×
10–11
Page 22
Why is Scientific Notation important for us?
First of all scientific notation give us another concept
of space, allowing us to compare distances and
lengths. And also, scientific notation is applied to
numbers with useful properties, often required to
scientists, mathematicians and engineers, who work
with such numbers and have the necessity to simplify
its writing.
As it is not always easy to have these spatial notions,
this site - whose link appears after the text - allows the
reader to draw a comparison between values which
are familiar with the abstract values of scientific
notation.
http://htwins.net/scale2/?bordercolor=whi
te
Do you want to know more about Scientific Notation? Check Ana Silva’s
prezi.
http://prezi.com/r35t87iz3t55/untitled-prezi/
Page 23
Where we can find Scientific Notation Applied to Geology?
We can find Scientific Notation Applied to Geology in the Study of the
Universe:
Distances between planets;
Temperature at the surface and inside of the planets;
Density and mass of the celestial bodies;
Energy (Radiation) emitted and received;
Source: http://www.physicsoftheuniverse.com/numbers.html
Scientific Notation
Number (conventional form)
Mass (in kilograms) of the Earth.
6 × 1024 6,000,000,000,000,000,000,000,0
00
Distance (in metres) to the Andromeda Galaxy, the nearest galaxy to our own (2.36 million light years).
2.23 × 1022
22,300,000,000,000,000,000,000
Total solar radiation (in Joules) received from the Sun by one square meter of the Earth's surface per second.
1.366 × 103
1,366
Temperature (in ° Kelvin) at the core of the Sun.
1.56 × 107 15,600,000
Page 24
A specific case: Maths and Dams (Hydroelectric powerplant)
A hydroelectric power plant has a height of 40 meters and flow rate of 20
m3/s. Knowing that an agglomeration with x homes, each has a
consumption of 400 kWh / month. Determine the number of homes
supplied by the central, knowing that the central works 5 hours
consecutively.
Electrical Power = 8xQxh
Q = flow rate
h = height
Elec. Power = 8x20x40
Elec. Power = 6400 Kw
W Elec. = Elec. Power x t
W Elec. = 6400 x 5 hours
W Elec. = 3200 Kwh
Number of homes supplied by the central
http://turismoenportugal.blogspot.com.es/2011/04/tierras-de-alqueva.html
Page 25
Conclusion
We may not notice how complex the Earth can be and there are a lot of
gaps in the timeline of our planet´s evolution. However, it is the task of
Geology to fill those gaps by trying to understand, simplify and explain
them.
Geology alone cannot answer all our questions and this is where
Mathematics steps in. Mathematical logic guides the geologist’s reasoning
in the way to a scientific answer. The numbers above are evidence to this.
It is easy to forget how much of what we know today about the Earth is due
to Mathematics applied to Geology.
But like Galileo Galilei once said "The universe cannot be read until we
have learned the language and become familiar with the characters in
which it is written. It is written in mathematical language, and the letters are
triangles, circles and other geometrical figures, without which means it is
humanly impossible to comprehend a single word.”
Page 26
Links 1) Crystal of Zircon –
a. www.minerals.net
2) Earth’s Internal Structure –
a. www.wikipedia.com
b. www.livescience.com
Students Involved on this work:
Ana Silva
António
Beatriz Lopes
Cláudia Sintra
Eduardo Fernandes
Isabel Doutor
Leonel Queimada
Maria Dias
Sofia Serra
Matteo Barchi
Erika Carbone
Pietro Polimeni
Teachers Involved on this work:
Ana Antunes
Ana Castro