Transcript Review 1 units v12 x

Structural Geology Basics
• We need to review fundamental
information about physical properties
and their units.
Exponents
Suppose m and n are rational numbers
• aman = am+n
• (am)n = amn
• (a/b)m = am/bm
am/an = am-n
(ab)m = ambm
a-n = 1/an
Logarithms
• Logarithms (Logs) are just exponents
• if by = x then y = logb x
http://www.engineeringtoolbox.com/average-velocity-d_1392.html
Scalars and Vectors
• A scalar is a quantity with a size, for
example mass or length
• A vector has a size (magnitude) and a
direction.
http://www.engineeringtoolbox.com/average-velocity-d_1392.html
Velocity
• Velocity is the rate and direction of
change in position of an object.
• For example, at the beginning of the
Winter Break, our car had an average
speed of 61.39 miles per hour, and a
direction, South. The combination of these
two properties, speed and direction, forms
the vector quantity Velocity
Vector Components
• Vectors can be broken down into
components
• For example in two dimensions, we can
define two mutually perpendicular axes in
convenient directions, and then calculate
the magnitude in each direction
• Vectors can be added
• The brown vector plus
the blue vector equals
the green vector
Vectors 2: Acceleration.
• Acceleration is the change in Velocity
during some small time interval. Notice
that either speed or direction, or both, may
change.
• For example, falling objects are
accelerated by gravitational attraction, g.
In English units, the speed of falling
objects increases by about
g = 32.2 feet/second every second, written
g = 32.2 ft/sec2
SI Units: Kilogram, meter, second
• Most scientists and engineers try to avoid
English units, preferring instead SI units.
For example, in SI units, the speed of
falling objects increases by about 9.81
meters/second every second, written
g = 9.81 m/sec2
• In geology, both english and SI units are
used. We must learn to use both.
Système international d'unités
pron dooneetay
http://en.wikipedia.org/wiki/International_System_of_Units
What’s in it for me?
• Petroleum Geologists trained in Structural
Geology and Stratigraphy make more money
than other Geology job centers.
• http://jobs.bhpbilliton.com/jobDetails.asp
• The mining industry also employs many
Structural Geologists, e.g.
• http://www.na.srk.com/en/page/na-geology-resourcejobs
Data and Conversion Factors
• In your work as a geologist, you will be
scrounging for data from many sources. It
won’t always be in the units you want. We
convert from one unit to another by using
conversion factors.
• Conversion Factors involve multiplication
by one, so nothing changes.
• 1 foot = 12 inches so 1 foot = 1
12 “
Example
• Rock is flowing at a velocity of 3 x 10-14 meters
per second at a depth of 35km. What is this
speed in feet per second?
• Steps: (1) write down the value you have, then
(2) select a conversion factor and write it as a
fraction so the unit you want to get rid of is on
the opposite side, and cancel. Then calculate.
• (1)
(2)
• 3 x 10-14 meters x 3.281 feet
second
meter
= 9.843 x 10-14 feet
second
Momentum (plural: momenta)
• Momentum (p) is the product of velocity
and mass, p = mv
• In a collision between two particles, for
example, the total momentum is
conserved.
• Ex: two particles collide and m1 = m2, one
with initial speed v1 ,
the other at rest v2 = 0,
• m1v1 + m2v2 = constant
Force
• Force is the change in momentum with
respect to time.
• A normal speeds, Force is the product of
Mass (kilograms) and Acceleration
(meters/sec2), so Force F = ma
• So Force must have SI units of kg . m
sec2
• 1 kg . m
sec2
is called a Newton (N)
Statics
• If all forces and Torques are balanced, an
object doesn’t move, and is said to be
static
• Discussion Torques, See-saw
F=2
• Reference frames
F=1
-1
0
F=3
+2
Pressure = Stress
• Pressure is Force per unit Area
• So Pressure must have units of kg . m
sec2 m2
• 1 kg . m is called a Pascal (Pa)
Pressure (stress) = density x gravity x depth
sec2 m2
s = ρgz
For solid-solid systems, Pressure is called “stress”
Density
• Density is the mass contained in a unit
volume
• Thus density must have SI units kg/m3
• The symbol for density is r, pronounced
“rho”
• Very important r is not a p, it is an r
• It is NOT the same as pressure
Chaining Conversion Factors
Suppose you need the density of water in
kg/m3. You may recall that 1 cubic centimeter
(cm3) of water has a mass of 1 gram.
1 gram water x (100 cm)3 x 1 kilogram = 1000 kg / m3
(centimeter)3
(1 meter)3
1000 grams
r water = 1000 kg / m3
Don’t forget to cube the 100
Energy
• Energy is the ability to do work, and work and
energy have the same units
• Work is the product of Force times distance,
• W = Fd
• 1 kg . m2 is called a N.m or Joule (J)
sec2
•
•
Energy in an isolated system is conserved
KE + PE + P/v + Heat = constant
N.m is pronounced Newton meter, Joule sounds like Jewel.
KE is Kinetic Energy, PE is Potential Energy, P/v is Pressure, v is unit
volume
An isolated system, as contrasted with an open system, is a physical
system that does not interact with its surroundings.
Kinetic Energy
• Kinetic Energy (KE) is the energy of
motion
• KE = 1/2 mass . Velocity 2 = 1/2 mV2
• SI units for KE are 1/2 . kg . m . m
•
sec2
Note the use of m both for meters and for mass. The context will tell you which.
That’s the reason we study units.
Note that the first two units make a Newton (force) and the remaining unit is meters,
so the units of KE are indeed Energy
Potential Energy
• Potential energy (PE) is the energy
possible if an object is released within an
acceleration field, for example above a
solid surface in a gravitational field.
• The PE of an object at height h is
PE = mgh Units are kg . m . m
sec2
Note that the first two units make a Newton (force) and the
remaining unit is meters, so the units of PE are indeed Energy
Note also, these are the same units as for KE
KE and PE exchange
• An object falling under gravity loses
Potential Energy and gains Kinetic Energy.
• A pendulum in a vacuum has potential
energy PE = mgh at the highest points,
and no kinetic energy because it stops
• A pendulum in a vacuum has kinetic
energy KE = 1/2 mass.V2 at the lowest
point h = 0, and no potential energy.
• The two energy extremes are equal
Stops v=0 at high point, fastest but h = 0 at low point.
Without friction, the kinetic energy at the lowest spot (1) equals
the potential energy at the highest spot, and the pendulum will
run forever.
Overburden
The upper crust has an average density of 2.7 g/cm3
• Stress caused by gravity is called overburden.
• Pressure (stress) = density x gravity x depth
we need to convert units for density
• Stress s = ρgz
s = 2.7 g/cm3 x 9.81 m/sec2 x 1000 m
ρ = 2.7 g x 1003 cm3 x 1 kg
= 2700 kg/m3
cm3
1 m3
1000 g
 s = 2700 kg/m3 x 9.81 m/sec2 x 1000 m
• The normal stress (pressure) at 1 km depth is
about 26487000 Pa ~25 MPa in the upper crust.