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Chapter 3: Equations and how to manipulate them

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More time on the basics Chapter 3: Equations and how to manipulate them Factorization Multiplying out Rearranging quadratics Chapter 4: More advanced equation ... – PowerPoint PPT presentation

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Title: Chapter 3: Equations and how to manipulate them


1
More time on the basics
Chapter 3 Equations and how to manipulate
them Factorization Multiplying out Rearranging
quadratics Chapter 4 More advanced equation
manipulation More logs and exponentials Simultaneo
us equations Quality assurance
2
Isostacy
  • Length of a degree of latitude
  • Mass deficiency in the Andes Mountains
  • Everest
  • The Archdeacon and the Knight
  • Mass deficiency mass of mountains
  • Archimedes - a floating body displaces its own
    weight of water
  • Crust and mantle

3
The Airy Hypothesis
Airys idea is based on Archimedes principle of
hydrostatic equilibrium. Archimedes principle
states that a floating body displaces its own
weight of water. Airy applies Archimedes
principle to the flotation of crustal mountain
belts in denser mantle rocks.
4
Archimedes Principle
A floating body displaces its own weight of
water. Mathematical Statement
or
5
Make our geometry as simple as possible
Ice Cube
6
.r
r
rdepth ice extends beneath the surface of the
water
Apply definition
7
Divide both sides of equation by ?wxy
(substitution)
The depth of displaced water
since ?water1
How high does the surface of the ice cube rest
above the water ?
Let e equal the elevation of the top of the ice
cube above the surface of the water.
8
Specify mathematical relationship
substitute
Distributive axiom in reverse
skipped
Most of us would go through the foregoing
manipulations without thinking much about them
but those manipulations follow basic rules that
we learned long ago. An underlying rule we have
been following is the Golden Rule - as Waltham
refers to it. That rule is that whatever you do
(to an equation), the left and right hand sides
must remain equal to each other. So if we add
(multiply subtract ..) something to one side we
must do the same to the other side.
9
The operations of addition, subtraction,
multiplication and division follow these basic
axioms (which we may have forgotten long ago) -
the associative, commutative and distributive
axioms. Also - no matter what kind of math you
encounter in geological applications - however
simple it may be - you must honor the golden rule
and properly apply the basic axioms for
manipulating numbers and symbols.
Geological Application...
Ice Cubes to Mountain Belts
10
We can extend the simple concepts of equilibrium
operating in a glass of water and ice to large
scale geologic problems.
11
From Ice Cubes and Water to Crust and Mantle
The relationship between surface elevation and
depth of mountain root follows the same
relationship developed for ice floating in water.
12
r
Lets look more carefully at the equation we
derived earlier
13
Given -
. show that
The constant 0.1 is related to the density
contrast
or ...
14
Which, in terms of our mountain belt applications
becomes
or
Where ?m represents the density of the mantle and
?? ?m - ?c (where ?c is the density of the
crust.
15
We can extend the simple concepts of equilibrium
operating in a glass of water and ice to large
scale geologic problems.
16
From Ice Cubes and Water to Crust and Mantle
The relationship between surface elevation and
depth of mountain root follows the same
relationship developed for ice floating in water.
17
Back to isostacy- The ideas weve been playing
around with must have occurred to Airy. You can
see the analogy between ice and water in his
conceptualization of mountain highlands being
compensated by deep mountain roots shown below.
The Airy Hypothesis
18
Lets take Mount Everest as an example, and
determine the extent of the crustal root required
to compensate for the mountain mass that extends
above sea level. Given- ?c2.8gm/cm3, ?m
3,35gm/cm3, eE 9km
Thus Mount Everest must have a root which extends
46 kilometers below the normal thickness of the
continent at sea level.
19
Physical Evidence for Isostacy
Japan Archipelago
20
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21
The Earths gravitational field
In the red areas you weigh more and in the blue
areas you weigh less.
22
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23
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24
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25
The gravity anomaly map shown here indicates that
the mountainous region is associated with an
extensive negative gravity anomaly (deep blue
colors). This large regional scale gravity
anomaly is believed to be associated with
thickening of the crust beneath the area. The low
density crustal root compensates for the mass of
extensive mountain ranges that cover this region.
Isostatic equilibrium is achieved through
thickening of the low-density mountain root.
26
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27
The importance of Isostacy in geological problems
is not restricted to equilibrium processes
involving large mountain-belt-scale masses.
Isostacy also affects basin evolution because the
weight of sediment deposited in a basin disrupts
its equilibrium and causes additional subsidence
to occur. Isostacy is a dynamic geologic process
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