maths geology

Maths and

Geology

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

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.

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

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”.

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.

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)

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.

http://prezi.com/uvhnbzdbn3nf/untitled-prezi/

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

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:

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.

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.

http://en.wikipedia.org/wiki/Crust_(geology)

http://en.wikipedia.org/wiki/Mantle_(geology)

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.

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

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

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

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)

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.

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

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.

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.

All Rights Reserved

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

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/

http://htwins.net/scale2/?bordercolor=white

http://htwins.net/scale2/?bordercolor=white

http://prezi.com/r35t87iz3t55/untitled-prezi/

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

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

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.”

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

http://www.minerals.net/

http://www.wikipedia.com/

http://www.livescience.com/

maths geology

Education

geology page

scientific notation

radiometric dating important

different fields of

crust vs mantle

earths internal structure

index index

mantle vs liquid outer