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Environmental Physics
Chapter 1:
Forces of Nature
Copyright © 2007 by DBS
Concepts
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Many natural systems contain motion, forces and momentum that can be
described by Newton’s laws
Frictional forces are important in any real-world motion, dissipating energy
Gravity acts between any two bodies, dependent on their mass and
distance apart
Rotational motion can be described by laws and equations analogous to
those for straight line motion, particuarly relevent to climate and orbits
Different types of wave observed in environmental systems have certain
properties in common
Electricity and magnetism and inextricably linked, as one induces the other
The Earth’s magnetic field provides a tool to investigate the geological
history of the Earth and a subsurface surveying technique
Question
What comes next?
1, 2, …
1, 2, 3…
Predictable?
1, 2, 3, 2, 1, 2, 3…
Predictable?
Physical events are predictable and quantifiable
More complex the pattern, the longer one must observe
and the greater the need for accurate record keeping
Newtonian Mechanics
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Why are blue whales bigger
than elephants
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Why don’t we just shoot CO2
into space?
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Why are skyscapers taller than
trees?
Buoyancy of the
surrounding ocean water
supports the weight of the
whale's body tissues
Use more energy
disposing of the gas
than is gained from
producing it
Forces acting on a
building are reduced by
use of appropriate
materials and design
Newton’s Laws
3 universal laws act in the universe
The First Law is just a special case of the Second
Law for which the net external force is zero
Newtonian Mechanics
1st law
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Law of inertia:
– An object that is not moving will not move until a net force acts
upon it.
– An object that is in motion will not change its velocity (accelerate)
until a net force acts upon it.
e.g. hockey puck…should continue to move forever why does it stop?
There are no perfect
demonstrations!
2-16
Newtonian Mechanics
Momentum and Inertia
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1st law is a statement of principle of ‘conservation of momentum’,
Momentum is the tendancy to continue moving once doing so
u = mv
Where u = momentum (kg ms-1), m = mass (kg), v = velocity (m s-1)
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Heavier something is, or faster it is moving, the more momemtum it has
Objects tend to "keep on doing
what they're doing"
(unless acted upon by a force)
Newtonian Mechanics
Momentum and Inertia
• Total momentum after collision is conserved
u1 = u2
m1v1 = m2v2
– Elastic – bounce off each other - kinetic energy remains the
same
– Inelastic – objects coalesce - Kinetic energy is converted
into heat or sound
e.g. sticky reaction: m1v1 + m2v2 = (m1+m2)vf
Question
Calculate the recoil speed of a 8 lb Winchester .308 rifle which
launches a bullet of mass 10 g with a speed of 2820 ft s-1.
[Conversions: 1 ft = 30.5 cm = 0.305 m]
Momentum is concerved: mgvg = -mbvb
mbvb
= ub = 0.01 kg x 2820 ft s-1 x 0.305 m / ft
= 8.6 kg m s-1
This gun has a mass = 3.8 kg
Momentum is concerved: mgvg = -mbvb
vg = -mbvb = - ub/mg = - 8.6 kg m s-1 / 3.8 kg = -2.3 m s-1
Answer: 2.3 m s-1
Newtonian Mechanics
Forces
• 2nd Law: rate of change of momentum (u) is proportional to the
applied force
• Heavier something is and the faster it is moving the more
difficult to stop it, cf. horse vs. cheetah
f = Δu / Δt
f = Δ(mv) / Δt = m Δv / Δt = m dv/dt = ma
• Acceleration produced by a force is proportional to the
magnitude of the force and inversely proportional to the mass of
the object
f = m a (units kg m s-2 or N)
Effects of an identical force F acting on two different
masses…
Quote
Quote
1-15
Law 3: Action and reaction are equal and
Quote
opposite
2-22
Newtonian Mechanics
Motion
• For constant velocity v,
Velocity = distance / time
v=s/t ,
v= u+ v
2
• For uniform motion in a straight line at constant acceleration s,
v, and a are linked by 3 simple equations
v = u + at
s = ut + ½ at2
v2 = u2 + 2as
Since a = dv/dt
a = (v – u)/t
v = u + at
Question
Derive s = ut + ½ at2 from the first 3 equations.
s = v t = [(u + v)/2] x t
Substitute: v = u + at
s = [(u + u + at)/2] x t = ](2u + at)/2] x t
s = ut + ½ at2
Question
A car is travelling at 30 m s-1 and takes 10 seconds to acceleration
to a new speed of 35 m s-1. What is its acceleration?
Use v = u + at
35 m s-1 = 30 m s-1 + a x 10 s
10 a = 35 - 30 = 5 m s-1
 a = 0.5 m s-2
Note About Homework
• Mistake in question 1, units of acceleration are m s-2
• Use 10 for g (m s-2)
• Note for HW you must show full working in problems!!!
End
• Review
Newtonian Mechanics
SI Units
• Units must be consistent!
• Answers must have units!
• What is the unit of force?
kg ms-2 = N (Newtons)
Newtonian Mechanics
SI Units
• The Reynolds number is used to characterize fluid flow (laminar
or turbulent) it is the ratio of inertial (ρv) to viscous (μ/l) forces
Re = ρvl
μ
Where ρ = density (kg m-3), v = velocity (m s-1), l = length (m)
μ = viscosity (kg m-1 s-1)
• Show that Re has dimensionless units
Newtonian Mechanics
Scalars and Vectors
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Vectors: direction and magnitude
– Force, velocity and acceleration
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Scalar: magnitude but no direction
– Mass, time, length, charge
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Scalars can be added, vectors cannot…why is the wind speed not 35 km h-1?
15 km h-1
20 km h-1
Newtonian Mechanics
Scalars and Vectors
15 km h-1
(i) Find apparent wind velocity for cyclilst
20 km h-1
v a2 = v h2 + v s 2
(152 + 202)1/2 = 25 km h-1
(ii) Resolve forces on kite to find wind
speed, v
v = fr x cosθ
θ
Newtonian Mechanics
Friction and Air Resistance
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Friction – force between two objects due to roughness of their
touching surfaces
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Static or dynamic
– Depends on surface roughness
– Affected by lubricants (oil, ball bearings, polymers etc.)
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Friction between solid objects and fluids = drag
– Depends on shape, size, surface characteristics and speed
– Important for wind pollination, wind stress resistance and erosion
What is friction in an electrical circuit called?
Newtonian Mechanics
Friction and Air Resistance
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Skin drag
– Important for small particles (e.g. PM)
or objects in viscous fluid
– Fd ~ radius x viscosity x velocity
Fd = 6πrμv
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Form drag
– Important for larger objects in air
– Fd ~ radius2 x velocity2 x air density x drag coefficient
(dependant on objects shape)
Fd = ½ ρaC πr2v2
Where ρa = air density (1.2 kg m-3), C = drag coefficient
Form drag >>> Skin drag
at high velocity (v2)
Freefall Demo
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Which of two objects will strike
the floor first if dropped
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All objects fall at the same rate
when air resistance can be
ignored
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If the mass is too low compared
to surface area (as in a feather)
air resistance becomes
important
http://nssdc.gsfc.nasa.gov/planetary/image/featherdrop_sound.mov
Apollo 15 Movie (1971)
1-14
Newtonian Mechanics
Gravity
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Newtonian Gravity
– Anything falling under gravity falls at
same rate
g = 9.8 m s-2
f = mg
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Calculate time of fall and velocity
s = ut + ½at2
v = u + at
Let a = g = 9.8 m s-2
t = 0.9 s, v = 8.9 m s-1
Newtonian Mechanics
Mass, Weight and Density
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Mass
– Measurement of the amount of matter something contains
Weight
– Measurement of the pull of gravity on an object, w = mg
Mass of an object
– doesn't change with location
– Weight does
e.g. A 10 kg mass weights 10 x 9.8 = 98 N
Moon gravity is 1/6 Earth,
gm = 9.8 / 6 = 1.6 m s-2
10 kg mass is now, w = mgm = 16 N
Question
The moon’s gravitational pull is 1/6th as great as the
Out of 168 people taking a
Earth’s attraction
quiz, 48 missed the question.
If a pen is dropped
on afloat
moon,
willbecause
it:
e.g. "It will
away
A) Float away
the gravitational force is less than
B) Float where
is the Earth where it would
hereit on
C) Fall to thefall.
surface of the moon
I think it will float away because
of what I have seen of the space
rooms NASA uses to get
astronauts ready for flight."
Newtonian Mechanics
Landslides
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Wind, water and glaciers move
large amounts of material
So does gravity
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Landslide when component of
gravity fs > frictional forces
supporting it
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Steeper slope has > fs
Newton decided that g was somehow related to orbital motion…
Newtonian Mechanics
The Universal Force of Gravity
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Any two objects in the Universe exert gravitational attraction on
each other, with the force having a universal form:
f = GMm
r2
Force of gravity ~ M x m and ~ 1 / r2
Where M and m are two masses,
r is the distance between them
and G is Universal gravitational
constant,
The more massive two objects
are the greater the force between them
G = 6.67 x 10-11 Nm2 kg-2
The farther apart they are, the less the
force will be
Newtonian Mechanics
Relationship Between Big G and Little g
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Now notice also that at the Earth’s surface it is true by Newton's 2nd
Law that:
f = -GMEm1 = m1a = m1g
rE 2
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Let ME = 6.02 x 1024 kg, rE = 6.40 x 106 m and solve for g:
g = -GME
rE 2
g = 9.8 m s-2
This says that the acceleration you feel due to the planet is
independent of your mass. That's just what Galileo showed in his
famous freefall experiments
Newtonian Mechanics
Terminal Velocity and Settling Velocity
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Objects accelerate on falling until:
drag force = force of gravity
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Higher terminal velocity (vt) means large particles settle more
quickly
For a small sphere (PM) when viscous forces dominate
(skin drag)
Fg = mg = Fd
mg = 6πrμv
4/3 πr3ρsg = 6πrμv
vt = 2g ρs r2
9μ
(m = ρs x 4/3 π r3)
Where ρs is the density of the sphere.
Stokes’ law: settling velocity is highest for a
larger, dense object in a less viscous medium
Question
Derive an equation for vt for larger objects in less viscous
fluids.
Drag force predominates:
mg = Fd
4/3 πr2ρsg = ½ ρaC πr 2 v 2
vt =
8 ρsrg
3 ρaC
Question
Find vt for a human being…
vt = (8 ρsrg / 3 ρaC)1/2
=
8 x 1000 kg m-3 x 0.5 m x 10 m s-2
3 x 1.2 kg m-3 x 1.0
= 105 m s-1
Newtonian Mechanics
Settling Chambers
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For a chamber of height h and
length l and gas velocity u, gas
takes l /u seconds to traverse
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Particle settles in h/vs seconds
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Particles with large enough
diameters such that
h< l
vs u
settle out
vs < hu
l
l
h
Newtonian Mechanics
Settling Chambers
vs < hu
l
Substitute into Stoke’s law (for small particles in air) and find the
size of particle removed, r
vt = 2g ρs r2
9μ
Question
Substitute into Stoke’s law and find r.
r=
9μuh
2ρsgl
Cheap method of removing large particles, not so good for fine!
Important Note
• Note: since the force due to bouyancy is not included these
calculations assume large particulate density, ρs >>> ρa
• If bouyancy were included:
Fg = Fd + Fb
mg = 6πrμv + Fb
4/3 πr3ρsg - 4/3 πr3ρag = 6πrμv
4/3 πr3g (ρs – ρa) = 6πrμv
vt = 2r2g (ρs - ρa)
9μ
End
• Review
Rotational Dynamics
Moments of Inertia
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Angular momentum: the tendency for something that is spinning to continue
to spin
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Conserved – contunes to spin unless external forces act
Linear motion and forces:
Rotational motion and forces:
distance, s
velocity, v
mass, m
momentum, u = mv
angle,
angular velocity, ω
moment of inertia, I
angular momentum, L = I ω
Rotational Dynamics
Moments of Inertia
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Angular momentum: depends on speed of spin (ω) and mass distribution (I)
L=Iω
L = angular momentum (torque), I = moment of inertia, ω = angular velocity
large I
To conserve L
ω increases
Smaller I
Rotational Dynamics
Central Forces
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Central Forces
– Objects velocity is at a tangent to
spin direction (ω = v / r)
– Change in direction = change in
velocity = acceleration
fc = mv2 = mr ω2
r
v2 / r = r ω2 is accn.
component
These are wrong way
around in text!
Demo
http://www.ac.wwu.edu/~vawter/PhysicsNet/QTMovies/Rotations/Centrif
ugalH2OMain.html
Rotational Dynamics
Coriolis Force
• Force felt by any body moving relative to something rotating
fc = 2mωv
• Anything moving on the Earth’s surface is subjected to CF
• Movement of air towards axis reduces inertia (I) so by law of
conservation of momentum angular velocity (ω) increases
– CF increases towards poles
• CF = 0 at equator
Rotational Dynamics
Cyclone Separators
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Air pollution control
– Centrifugal force amplifies gravitational settling
– Outer and inner vortex
Rotational Dynamics
The Vortex
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Vorticity is a measure of the
rotation of a fluid about an axis
Fluid speeds up as it is drawn
towards center
(conservation of angular
momentum)
e.g. Tornadoes, cyclonic winds
and hurricanes
Vortices form in turbulent flow
http://en.wikipedia.org/wiki/Vortex
Rotational Dynamics
Orbits
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Remote Sensing: geostationary or polar orbits/Low Earth Orbits
(LEO)
For LEO (< 2000 km) centripetal force = gravitational force
mv2 = mg
R
Where m = mass of satellite, v = velocity of satellite, R = Earth’s
radius (6400 km) and g = 10 m s-2
v = √ (gR) = 8000 m s-1
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Time period T = 2πR/v = 83 mins
Low orbit satellite travels at
8 km s-1 or 28,800 km hr-1
> 8 km s-1 escape velocity
Question
For geostationary orbit need to find r.
Given T = 2π/ω equate the formulas for
gravitational force and centripetal
force and solve for r.
r
fc = fg
msat rω2 = GmE msat / r2
rω2 = GmE / r2
r = (GmE / ω2)1/3
T = 2π/ω
ω = 2π/86164 s
Orbital radius r = 42164 km from center of earth.
Subtracting Earth’s radius gives an altitude of 36,000 km
Use v = ωr to find velocity
Waves
Wave Characteristics
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Wavelength and frequency
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v = fλ
Where v = speed (m s-1), f = frequency (cyces s-1), λ = wavelength (m)
Speed varies according to medium waves travel in
Waves
Transverse and Longitudinal Waves
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Transverse: wave travels perpendicular to vibration
e.g. light, EM waves
Longitudinal: vibrations travel in same direction as wave
e.g. sound
Shape: natural waves often sine waves
I = Imax sin 2π/λ (vt-x)
Where I = intensity, t = time, x = distance
Maybe progressive (moving) or standing (stationary)
Waves
Wave Properties
(a) Incident wave
(b) Reflection
(c) Transmission and absorption
(d) Refraction
(a) Diffraction
(b) Scattering
(diffraction+reflection)
(c) Interference
• Polarization
Waves
Seismic Waves
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Body waves consist of ‘P’
and ‘S’ waves
– P travel in solid and
liquid
– S in solid only
Surface waves cause more
damage
Resonance
P waves are refracted
End
• Review
Question
How does the electrical force differ from the gravitational
force?
Gravity always pulls together, electrical force acts both
ways (attracts and repels)
Electrical force is many times more powerful than gravity
(action of a comb on paper)
Electromagnetism
Electric Charge and Current
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Electric charge (Coulomb = 1 amp / s)
Current (A) is how much charge is
moving
Voltage (V) is difference in electrical
potential energy (provides push)
Ohm’s Law
V=IxR
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Where R = resistance (ohms)
AC and DC current
Electromagnetism
Electric Fields
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Objects carrying a static charge emit an
electric field in all directions that
becomes weaker at greater distances
Like chartges repel, unlike attract
Field strength:
E=V/d
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Where V = voltage (v), d = distance (m)
Electromagnetism
Electrostatic Precipitators
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Used to control dust and particulate matter (aerosols)
High voltage
Flow of e- charges particles
Electromagnetism
Magnets
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A magnet is a metallic object (Fe, Co, Ni, alloys) that attracts another metallic
object
– Every magnet has at least two poles, N and S (dipole)
– Like magnetic poles repel each other, while unlike poles attract
– Lines of force extend from N to S create a field
– Loss of magnetic field at Curie point
– Presence of iron in the earth creates a field
Demo
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Strength is measured in Teslas (T)
Flux density 1 T = 1 N A-1 m-1
Electromagnetism
The Earth’s Magnetic Field
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Core is above Curie point, induced by
fluid and electrical currents in outer
liquid iron core
- Geomagnetic dynamo
Influenced by solar flux
Reverses periodically –
300x during last 200 million yrs
Mid-ocean ridge shows magnetic
reversals
Electromagnetism
The Earth’s Magnetic Field
Computer
model
Source: NASA
http://science.nasa.gov/headlines/y2003/29dec_magneticfield.htm
Electricity and Magnetism
Induction
• Electrical current creates
magnetism, changing
magnetic fields produce
current
– The Electromagnet
– Electromagnetic
induction
• Electricity and magnetism
combine to produce force
and motion
20-12
Electromagnet
Electromagnetic
Induction
Demo
20-12
Electromagnetism
In Animals and Plants
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First noticed in Robins,
seen in all Migratory birds
Organisms as diverse as hamsters,
salamanders, sparrows, rainbow
trout, spiny lobsters, and bacteria
Whale beachings
Bacteria contain magnetite crystals
so they can tell ‘up’ from ’down’
Electromagnetism
Transmission Lines
• Transmission Lines – harmful to humans?
• Hypothesized EMF link to childhood leukemia
• Proposed:
– Direct effect of electric field disrupting electrical activity in
the body
– Indirect effect of polarized water vapor which dissolves
ionized gaseous pollutants from the air allowing them to
be more readily absorbed
• No conclusive evidence
• Magnetic fields still an issue
Demos
19-01
19-02
19-06
19-19
19-20
19-24
Further Reading
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http://www.phys.unsw.edu.au/~jw/demo/projectiles.html
The monkey and the hunter
http://physics.bu.edu/~duffy/semester1/c04_monkeyhunter.html
Monkey-hunter
Journals
• Fews, A.P., Henshaw, D.L., Wilding, R.J., and Keitch, P.A., (1999) Corona ions
from powerlines and increased exposure to pollutant aerosols. International
Journal of Radiation Biology, Vol. 75, No. 12, pp. 1523-1531.
• Fews, AP, Henshaw, D.L., Keitch, P.A., Close, J.J. and Wilding, R.J. (1999)
Increased exposure to pollutant aerosols under high voltage powerlines.
International Journal of Radiation Biology, Vol. 75 No. 12, pp. 1505-1521.
• Gailitis, A., Lielausis, O., Dement’ev, S. et al. (1999) Detection of a flow induced
magnetic field eigenmode in the Riga dynamo facility. Physical Review Letters,
Vol. 84, No. 19, pp. 4365-4368.
• UK Childhood Cancer Study Investigators (1999) Exposure to power-frequency
magnetic fields and the risk of childhood cancer. The Lancet, Vol. 354, No. 9194,
pp. 1925-1931.
Books
• Cutnell, J.D., and Johnson, K.W. (2000) Physics (5th Edition).
John Wiley, New York.
• Warren, P. (1988) Physics for Life. John Murray, London.
Movies
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Magnetic Storm
http://www.pbs.org/wgbh/nova/magnetic/
Movies
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Harmful Effects of
Electromagnetism