Phys 231
3
4
Lecture 7
2013 1
Wed. 9/18
Lab
Fri. 9/20
3.6-.10 Elect & Strong Force; planetary orbit simulation Quiz 2
L3: Predicting Motion under Non-Constant Forces
3.11 –.13 Conservation of Momentum & Multiple Particles
RE 3.b bring laptop, smartphone, pad,…
bring laptop and headphones if you have them
RE3.c
Mon. 9/23
Tues 9/24
Wed. 9/25
4.1-.5 Atomic nature of matter / springs
RE 4.a
EP 3, HW3: Ch 3 Pr’s 42, 46, 58, 65, 72 & CP
RE 4.b
4.6-.7, .9-.10 Stress, Strain, Young’s Modulus, Compression, Sound
Science Poster Session: Hedco7pm~9pm
Equipment
Look over WebAssign questions
Static electric stuff
Van der Graff
Three body.py
Double pendulum.py
Lab 3
3.6 Electric Interaction
Demo: Charge up rubber rod and push around pith ball or make electrometer deflect
electric (a subset of electro-weak) – everything else: tension, normal force,
friction, spring force, push, pull,… For unexplainable reasons (that’s why we call it
fundamental), almost all fundamental particles have another fundamental property, in
addition to mass, and just as how much mass a particle has effects its gravitational
interactions, how much of this property it has effects a new kind of interaction.


r
F

1
2
1 2
qq
F1 2 k c 12 2 rˆ
Coulombs Law
r1 2
q1
q2
kc = 8.99 × 109 N m2/C2
Charge has its own fundamental units, C for “Coulombs” (or you can
imagine C for Charge)
First off, it is quite remarkable that our new fundamental relation looks almost just like
our old one. To put some names to things, we honor Coulomb with this relation and we
call this type of force “electric”. The fundamental property that governs the interaction is
charge, symbolized by the letter “q”. Just as Newton’s Gravitational Law has a force
constant, G, this has one, kc. Saying the Law in English – the force between two particles
with charges is proportional to their charges and inversely proportional to their
separation. Unlike mass, charge causes particles to either attract or repel. Using this
equation as somewhat definitive of charge, that means that some charges must be + while
others are -. Two charges of like sign, give a positive force: a Push - repulsion. Two
charges of opposite sign give a negative force: a Pull - attraction. Note: charge is just as
fundamental, mysterious, and omnipresent property as is mass; however, its effects are
slightly more subtle because there are the two flavors, since most macroscopic objects are
built of nearly equal positive and negative charged particles, they have nearly no net
charge and thus hardly respond to the force.
Phys 231
Lecture 7
2013 2
Sub-Atomic Particles
o Zooming in to the atomic level, and even further, to the sub-atomic level,
almost all individual particles have charge. A particle physicist could
name off a whole zoo’s worth of particles, but as far as everyday
experience is concerned, electrons, protons, and neutrons are enough for
now. As the names suggest, a Neutron is charge neutral – it does not
interact by the Coulomb force. The Proton has positive charge and the
electron has negative charge (which gets called + and which – was an
arbitrary decision made long ago).
 Electron:
Mass: 9.11 × 10-31kg
Charge: - 1.60× 10-19C, denoted –e
 Neutron:
Mass: 1.675 × 10-27kg
Charge: 0 (“neutral”)
o Made of one up quark with +2/3 e charge and two
down quarks with -1/3 e charge, for 0 net charge:
udd
 Proton:
Mass: 1.673 × 10-27kg
Charge: +1.60× 10-19C, denoted +e
 Note: The charges of the Proton and the Electron are exactly the
same!
Properties and Interactions.
o Mass (and energy) and Gravitation. You’ll notice that all particles have
mass and all particles interact gravitationally according to their mass. In
fact, even light, which has no mass, but has energy, interacts
gravitationally.
o Charge and Electrical. The electron has charge and interacts electrically
according to that. So do the quarks within the proton and neutron. In the
proton’s case, they add up to a charge exactly opposite to that of the
electron, so a proton interacts just as strongly as does an electron. The
charges of the quarks in the neutron on the other hand, cancel out, leaving
the neutron electrically neutral and only subtly influenced by electric
interactions.
o Strong and Weak. There are two more types of interaction that are too
complicated for us to deal with in much detail, but are worth knowing
about.
 “Color” and Strong. Aside from mass and charge, the quarks
have another fundamental property which is, somewhat fancifully,
referred to as “color.” Folks chose that name because, unlike
charge which has two distinct flavors (+ and -) and equal quantities
of each make something electrically inert, there are three ‘flavors’
of this property and it takes one of each to make things ‘strongly’
inert. Analagously, a “white” spot of your computer screen is,
under a magnifying class, a red dot by a green dot by a blue dot.
This is, as the name suggests, the strongest of the forces; indeed,
quarks are inseparable because of it.
Lecture 7
2013 3
 Short range – in fact, the strong would actually be very strong
even at great distances; however, it’s so strong, that you always
have net-neutral objects; only right up close do you notice any
color distribution at all – kind of like the three color pixels on your
screen.
Nuclei – Strong vs. Electric
o Stable. Neutrons and protons are “strongly” attracted to each other, but
only to nearest neighbors. Meanwhile, protons are electrically repelled by
protons. So a nucleus is stable only if it’s got the right balance neutrons
and protons.
o Unstable. If the balance isn’t right, it can
 split apart.
 Loose bits
 Or even have a quark in a proton/neutron transform to change it
into a neutron/proton.
Atoms
o Alright, we have a new fundamental interaction: The Coulomb or Electric
force. We also have new particles that are affected by it according to their
new fundamental property: Charge.
o Put an Electron and a Proton in a room together and what motion results?

q1 q 2
ee
e2
 F1 2 k c 2 rˆ k c 2 rˆ
k c 2 rˆ . The proton and electron
r1 2
r1 2
r1 2
+e
are attracted to each other. They fly to each other. Much like the
Earth and the sun, they form an orbiting system: an atom:
Hydrogen.
 Say we put another electron in the room. From far away, it feels
-e
equal attraction to the proton and repulsion from the electron, so it
doesn’t get pulled in, or pushed away.
+e
o Moral: Atoms are formed by equal numbers of protons and electrons.
(Neutrons slip in there to, via another interaction we haven’t spoken of
yet). The combination of equal and opposite charges yields a net neutral
atom.
o To understand atoms better, we’ll have to wait until we get a taste of
quantum mechanics (that’s what it was developed for).
Weak. The weak interaction is also associated with charged particles; however, it
is more an interaction of transformation than of attraction or repulsion. For
example, via the weak interaction, a neutron can transform into an electron + a
proton, or a proton can transform into a positron + a neutron. Not that neutrons
are composed of electrons and protons, or that protons are composed of positrons
and neutrons – they’re not; via the weak interaction they can transform into each
other.
Strong. In fact, a proton is composed of two up quarks with +2/3 e charge and
one down quark with -1/3 e charge, for +e net charge: uud; while a neutron is
composed of two down quarks and one up quark: udd. Aside from charge and
mass, quarks have a third fundamental property which by some poetic license is
known as ‘color.’ The strong interaction is the quite strong attraction of different
‘colored’ quarks, and this holds neutrons and protons together and to eachother.
Phys 231
-e
-e
Phys 231
Lecture 7
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3.8 Forces as Mental Tools
Newton introduced to us a conceptual/mathematical tool for considering interactions:
Forces. He also applied that tool quite successfully to the Gravitational Interaction. It’s
worth pointing out that while interactions are real facts of nature that exist all around us,
forces are intellectual tools that exist within our minds. As is the case with household
projects, sometimes different tools can be used, with varying degrees of ease and success,
for the same job. Rather than Forces, we could (and will later) use the tool of Work.
Lagrangians, Hamiltonians, and Action are all names of alternative tools for relating
motion and interaction.
General Relativity In the specific case of the gravitational force, Einstein came up with
a more successful tool than Newton’s – a way of looking at the geometry of space and
time. He was motivated by the unique duality of mass.
Consider color – it is the property by which objects interact strongly.
Consider charge – it is the property by which objects interact electrically and
weakly.
Consider mass – it is the property by which objects interact gravitationally AND
the property that quantifies inertia (an object’s tendency to resist deviating from
constant velocity: straight line / constant speed motion.)
Einstein found a way of uniting mass’s two roles as one, of seeing gravity as inertia.
Rather than thinking of a gravitational force, he said we could think of a gravitational
warping of space and time, much like a bowling ball warps a mattress it is set upon.
Then other masses (say a ping pong ball) will move through space and time (across
the mattress) according to that warp. In the absence of the bowling ball, the mattress
is flat, and the ping pong ball travels in a straight line, as its inertia would have it do.
Plop down the bowling ball and the mattress is warped, now the ping pong ball
follows what is known as a geodes of the curved surface (like the curved paths of
trans-Atlantic flights). Einstein proposed that following geodesses through warped
space-time was the more general rule (rather than just following straight lines through
flat space and time).
Now this is a much more complicated tool to use than Newton’s, and Newton’s
Gravitational Force tool is ‘good enough’ for most of our purposes. Still it’s worth
knowing that there is a more accurate tool out there and it is sometimes called for (for
example, GPS systems would be worthless without it).
It’s also interesting to note that Einstein went on to experiment with applying such a
geometric model to the electric interaction, but it wasn’t a good fit.
General Relativity & the tool of forces. Newton introduced to us a
conceptual/mathematical tool for considering interactions: Forces. He also applied that
tool quite successfully to the Gravitational Interaction. It’s worth pointing out that while
interactions are real facts of nature that exist all around us, forces are intellectual tools
that exist within our minds. As is the case with household projects, sometimes different
tools can be used, with varying degrees of ease and success, for the same job. Rather
than Forces, we could (and will later) use the tool of Work. In the specific case of the
gravitational force, Einstein came up with a more successful tool than Newton’s – a way
of looking at the geometry of space and time. Rather than thinking of a gravitational
force, he said we could think of a gravitational warping of space and time, much like a
bowling ball warps a mattress it is set upon. Then other masses (say a ping pong ball)
will move through space and time (across the mattress) according to that warp. Now this
is a much more complicated tool to use than Newton’s, and Newton’s Gravitational Force
tool is ‘good enough’ for most of our purposes. Still it’s worth knowing that there is a
Phys 231
Lecture 7
2013 5
more accurate tool out there and it is sometimes called for (for example, GPS systems
would be worthless without it).
3.9
M1

r1
M2
2
Predicting the future of Complex
The limits of the analytical and computational.
o As we’ve seen, an exact, analytical solution can be found for a single mass
moving in a gravitational field. In Advanced Classical Mechanics, you’ll
find the other solutions for the single mass. Analytical solutions can also
be found for the case when both masses are free to move. There are even
some 3-body analytical solutions, but there are many other trajectories that
3 bodies can take and that aren’t expressible as analytic functions.
o We can however ask a computer to calculate the motions of three, four,
five… several bodies interacting with each other gravitationally.
o Now, our solar system is made of 9 planets, several moons, a number of
comets, and uncounted asteroids, all gravitationally interacting with each
other. In theory, one could model all of that. Aside from taking an awful
long time to program everything in, all those calculations would bring
even the fastest of computers to it knees – there is a limit to even what can
be done computationally (all be it an every moving limit.)
o Why bother trying?
 Interception. If we can predict where an asteroid will be, we can
plan out how to send a probe to encounter it and learn about it.
 Occultation of the Moons of Jupiter.
 One of my colleagues studies the volcanoes on Io, a moon of
Jupiter. Whenever another moon crosses our view of Io, she can
learn about the relative strengths of the volcanoes by watching Io
get dimmer and dimmer and then brighter and brighter again as
each volcano is blocked out and then revealed again.
 To know which volcanoes are getting blocked and revealed when,
she has to know quite precisely how the two moons cross in our
view – thus the motion of the Earth, the motion of Jupiter, and the
motions of the two moons about Jupiter.
2.9.1 The Three-body problem
o We saw last time the range of kinds of trajectories we can have with one
body moving in a gravitational field. Two planets is of course a bit more
complicated. You know how to model both of these, and you’ll do so in
lab Monday.
Demo: Three Body.py two planets about stationary center of mass (essentially keeping
the camera fixed on the center.)

o But what about three bodies? Perhaps a planet, a sun, and a moon; or
r1 3
perhaps three bodies of more comparable masses (so no one can be
ignored when considering the motions of the other two.)
M3
o Say we have three bodies interacting, we’ve got their masses, their initial
positions, and for that matter, their momenta. What series of calculations
do we go through to trace their trajectories?
 So we can calculate the forces on each, use those to update the
momenta, use those to update the positions, and then do it all over
again.
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2013 6
o So what’s does the net force on mass 1 look like, in terms of the positions
and masses?



M M
M M
o Fnet 1 F2 1 F3 1 G  1 22 rˆ1 2 G  1 23 rˆ1 3
r1 2
r1 3



o This figures into p1. f p1.i Fnet 1 t and that in turn goes into



r1. f r1.i p1 t to update the position.
o Similar force, momentum, and position relations give us new positions for
the other two bodies as well.
o So, what do the trajectories look like?
Demo: Three body.py Run some of the ordered patterns and some of the more messy.
Also do the three dimensional ones.
o A system with 4 bodies is infinitely more complex. The whole solar
system…
2.9.2 Sensitivity to initial conditions
o This complexity brings to light an oft important characteristic of physical
systems: extreme sensitivity to initial conditions.
o Simple system, not too sensitive. Now, in a very simple system, like a
single coin tossed in the air, if you change the initial velocity, it’s pretty
easy to predict how that will change its trajectory. In fact, you don’t even
need to mathematically model it to predict the qualitative change in
behavior: Throw more upward, and the path is a taller & narrower arc;
throw faster and the arc is taller and longer…
o Modest system, moderately sensitive. Now in the case of a two body
system, like the earth going around the sun, or even a pair of stars, the
behavior changes smoothly between qualitatively different kinds of
behavior, but still quite smooth ones.
o Complex system, surprising sensitivity. But for our three-body system,
some of those trajectories were quite complicated, nothing that could have
easily been foreseen. For that matter, the difference between two
qualitatively different outcomes lies in subtlety different initial conditions.
Demo: Three body.py, X=-2.76 vs. X = -2.765
Those two terribly different trajectories came from initial conditions that
differed by only 0.2%!
Phys 231
Demo: Alan’s Chaos magnetic pendulum demo.
o You can imagine how sensitive a system of even more particles is!
o Initial Conditions
 On the one hand, this means that tiny errors in specifying the initial
conditions can lead to drastically erroneous predictions.
o Neglected influences
 Similarly, the smallest of interactions, ones that seemingly should
be negligible, can, when thrown into the mix with all the other
complicated ones, spell the difference between two drastically
different outcomes – the old ‘straw that broke the camels back.’
Phys 231
Lecture 7
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2.10 Determinism
Given initial conditions, in theory, the tools that we’ve developed spell out the
fate of the universe and each and every one of us – it’s all predetermined. This
school of thought is known as determinism. There are two fundamental problems
with this perpective.
Practical limitations. Be this as it may, based on what we’ve seen for something
as ‘simple’ as a three-body system, what are our prospects of predicting the future
in detail?
o As we’ve just seen, for relatively simple systems, in practice, we can
never know the initial conditions to absolute accuracy, and we can never
take into consideration all interactions, and the universe is a complex place
– so even if it is all predetermined, there is no way that we could predict
the future in any detail.
2.10.2 Chaos
Systems like these display what we call “chaotic” behavior. That is, their
behavior depends on the infinitesimal level of their initial conditions.
They aren’t completely undeterministic, each step may follow very simple
rules, and from one step to the next, the behavior can be predicted. But
with each step the finest details are amplified so that it’s impossible to
predict the out come several moves down the line.
This disorder itself has some patterns and can be characterized and
quantified.
Often the systems behave almost periodically.
Chaotic system include
o the magnetic pendulum – the small difference between
approaching a magnet just left of center or just right of center has a
huge effect.
o Water dripping from a tap – exactly when the next drop will break
loose depends on such fine detail, it’s unpredictable.
o The weather – one of the first places chaotic behavior was
recognized and studied was a simple model of convection cells in
the atmosphere.
o Population – somewhere being hunted out by predators and being
so fruitful that they overwhelm their resources is a happy medium,
however, the smallest growth or decline can send the population on
an unpredictable roller coaster ride of boom and bust.
Demo: Double Pendulum
o A double hinged pendulum is a chaotic system. Each move
follows naturally from the previous one, but over the long term,
it’s impossible to predict the pattern. This can be seen by running
it again from, seemingly, the same initial conditions, yet
eventually, very different outcome.
Demo: Double Pendulum.py
Here you see two double pendula simulated; starting with virtually the
same initial conditions, but eventually their paths become quite
different.
Phys 231
Lecture 7
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2.10.3 Breakdown of Newton’s laws
2.10.4 Probability and uncertainty
Now think of things on the scale of atoms and electrons. If the
slightest perturbation can set off a simple double pendulum, imagine
what it can do to an electron! Imagine tossing an electron in a box,
like you would a marble. Even if you knew pretty well where you
tossed it and in what direction it was going. Not too much later, you’d
have little idea where it was any more, you could only speak of the
probability of it’s being in one place or another.
Similarly for an electron just zipping through space, we can’t say with
absolute certainty where it is (or for that matter whether an electron is
fundamentally a thing that has a finite location.) So we describe it
more as a traveling cloud of probability.