Lecture Notes
Lecture Notes
From Dr. Fowler's Galileo
& Einstein site.
- Counting
in Babylon
Spanish
Version
Babylon
had in all probability the earliest written language. At the same time,
an
elegant system of weights and measures kept the peace in the
marketplace. Their
method of counting was in some ways better than our present one! We
look at
some ancient math tables, and ideas about Pythagoras' theorem 1,000
years
before Pythagoras.
- Early
Greek Science: Thales to Plato Spanish
Version
In the ancient port city of
Miletus,
there took place a "discovery of nature": philosophers tried to
understand natural phenomena without invoking the supernatural. The
Greeks
imported basic geometric ideas from Egypt, and developed them
further.
Members of the Pythagorean cult announced the famous theorem, and the
(to them)
alarming discovery of irrational numbers! The Greeks had some
ideas
about elements and atoms. Hippocrates looked for non-supernatural
causes of
disease. Plato formulated a rationale for higher education, and thought
about
atoms.
- Motion
in the Heavens: Stars, Sun, Moon, Planets
A brief review for moderns of facts
familiar to
almost everybody in the ancient world: how the sun, moon and planets
move
through the sky over the course of time.
- Aristotle
A brief look at the beginnings of
science and
philosophy in Athens:
Plato's Academy and Aristotle's Lyceum. On to Aristotle's science:
"causes" in living things and inanimate matter, Aristotle's elements,
and laws of motion.
- Measuring
the Solar System
We look at some startlingly good
measurements by
the Greeks of the size of the earth and the distance to the moon, and a
less
successful (but correct in principle) attempt to find the distance to
the sun.
- Greek
Science after Aristotle
Strato understood that falling
bodies pick up speed
(contrary to Aristotle's assertions). Aristarchus gave a completely
correct
view of the solar system, anticipating Copernicus by 2,000 years or so.
Science
flourished for centuries in Alexandria,
Egypt:
Euclid,
Apollonius, Hypatia and others lived there, Archimedes studied there.
Archimedes understood leverage and buoyancy, developed military
applications,
approximated Pi very closely, and almost invented calculus! (See also
the next
lecture.)
- Basic
Ideas in Greek Mathematics
Nailing down the square root of 2.
Zeno's
paradoxes: Achilles and the tortoise. Proving an arrow can never move -
analyzing motion, the beginning of calculus. How Archimedes calculated
Pi to
impressive accuracy, squared the circle, and did an integral to find
the area
of a sphere.
- How
the Greeks used Geometry to Understand the Stars
The universe is like an onion of
crystal spheres:
Plato, Eudoxus, Aristotle. More earthly ideas: Eudoxus and Aristarchus.
Understanding planetary motion in terms of cycles and epicycles:
Hipparchus and
Ptolemy. These methods were refined to the point where they gave
accurate
predictions of planetary positions for centuries (even though Ptolemy
believed
the earth was at rest at the center of the universe).
- Galileo
and the Telescope
Copernicus challenged Ptolemy's
worldview.
Evolution of the telescope. Galileo saw mountains on the moon, and
estimated
their height - the first indication that the moon was earthlike, not a
perfect
ethereal sphere at all.
- Life
of Galileo
A few facts and anecdotes to try to
give
something of the flavor of Galileo's life and times, plus references to
books
for those who would like a more complete picture.
- Scaling:
why giants don't exist
One of Galileo's most important
contributions to
science (and engineering): the realization that since areas and volumes
scale
differently when the size of an object is increased keeping all
proportions the
same, physical properties of large objects may be dramatically
different from
similar small objects, not just scaled up versions of the same thing.
We
explore some of the consequences.
- Galileo's
Acceleration Experiment
Galileo argued against Aristotle's
assertions
that falling bodies fall at steady speeds, with heavier objects falling
proportionately faster. Galileo argued that falling bodies pick up
speed at
a steady rate (until they move so fast that air resistance becomes
important). He constructed an experiment to prove his point (and we
reproduced
it).
- Naturally
Accelerated Motion
This lecture presents the core of
Galileo's
analysis of motion in free fall, which he referred to as "naturally
accelerated motion". This is challenging material if you're new to it,
but
crucial in progressing from an Aristotelian or medieval
worldview to
that of Galileo and Newton,
the basis of our modern understanding of nature. Galileo used his
new-found
understanding of falling motion to prove that a projectile follows a
parabolic
path, if air resistance can be ignored.
- Describing
Motion
A simple introduction to the modern
way of
describing motion using arrows - "vectors" - to indicate speed and
direction. Galileo (and, later, Newton)
made heavy use of Greek geometry in analyzing motion. It's much easier,
and
just as valid, to use vectors.
- Tycho
Brahe and Johannes Kepler
These two colorful characters made
crucial
contributions to our understanding of the universe: Tycho's
observations were
accurate enough for Kepler to discover that the planets moved in elliptic
orbits, and find some simple rules about how fast they moved. These
became
known as Kepler's Laws, and gave Newton
the clues he needed to establish universal inverse-square gravitation.
This lecture links to more detailed
lectures I
gave previously.
- Isaac
Newton
A brief account of Newton's life,
followed by a discussion of
perhaps his most important insight: that a cannonball shot
horizontally, and
fast enough, from an imagined mountaintop above the atmosphere might
orbit the
earth. This tied together Galileo's understanding of projectiles with
the
motion of the moon, and was the first direct understanding (as opposed
to
description) of motion in the heavens.
- How
Newton Built on Galileo's Ideas
Newton's
famous Laws of Motion generalized and extended Galileo's discussion of
falling
objects and projectiles. Putting these laws together with his Law of
Universal
Gravitation, Newton
was able to account for the observed motions of all the planets. This
lecture
gives a careful development of the basic concepts underlying Newton's Laws,
in particular the tricky
concept of acceleration in a moving body that is changing direction -
essential
to really understanding planetary motion.
- The
Speed of Light
Aristotle thought it was infinite,
Galileo tried
unsuccessfully to measure it with lanterns on hilltops, a Danish
astronomer
found it first by observing Jupiter's moons. Rival Frenchmen found it
quite
accurately about 1850, but a far more precise experiment was carried
out in
1879 in Annapolis,
Maryland by Albert
Abraham Michelson.
- The
Michelson-Morley Experiment Spanish
Version
By the late 1800's, it had been
established that
light was wavelike, and in fact consisted of waving electric and
magnetic
fields. These fields were thought somehow to be oscillations in a
material
aether, a transparent, light yet hard substance that filled the
universe (since
we see light from far away). Michelson devised an experiment to detect
the earth's
motion through this aether, and the result contributed to the
development of
special relativity.
- Special
Relativity
Galileo had long ago observed that
in a closed
windowless room below decks in a smoothly moving ship, it was
impossible to do
an experiment to tell if the ship really was moving. Physicists call
this
"Galilean relativity" - the laws of motion are the same in a smoothly
moving room (that is to say, one that isn't accelerating)as in a room
"at
rest". Einstein generalized the notion to include the more recently
discovered laws concerning electric and magnetic fields, and hence
light. He
deduced some surprising consequences, recounted below.
- Special
Relativity: What Time is it?
The first amazing consequence of
Einstein's
seemingly innocuous generalization of Galileo's observation is that
time must
pass differently for observers moving relative to one another - moving
clocks
run slow. We show how this comes about, and review the experimental
evidence
that it really happens. We also show that if times pass differently for
different observers, lengths must look different too.
- Special
relativity: Synchronizing Clocks
Another essential ingredient in the
relativistic
brew is that if I synchronize two clocks at opposite ends of a train
I'm on,
say, they will not appear to be synchronized to someone on the
ground
watching the train go by. (Of course, the discrepancy is tiny at
ordinary
speeds, but becomes important for speeds comparable to that of light).
- Time
Dilation: A Worked Example
At first sight, it seems impossible
that each of
two observers can claim the other one's clock runs slow. Surely one of
them must
be wrong? We give a detailed analysis to demonstrate that this is a
perfectly
logically consistent situation, when one remembers also to include
effects of
length contraction and of lack of synchronization - special relativity
makes
perfect sense!
- More
Relativity: the Train and the Twins
Some famous paradoxes raised in
attempts to show
that special relativity was self-contradictory. We show how they were
resolved.
- Momentum,
Work and Energy
An elementary review of these basic
concepts in
physics, placed here for the convenience of nonscience majors who may
be a
little rusty on these things, and will need them to appreciate
something of
what relativity has to say about dynamics - the science of motion.
- Adding
Velocities: A Walk on the Train
A straight forward application of
the new
relativistic concepts of time dilation, length contraction etc.,
reveals that
if you walk at exactly 3 m.p.h. towards the front of a train that's
going
exactly 60 m.p.h., your speed relative to the ground is not 63 m.p.h.
but a
very tiny bit less! Again, this difference from common sense is only
detectable
if one of the speeds is comparable with that of light, but then it
becomes very
important.
- Conserving
Momentum: the Relativistic Mass Increase
How the very general physical
principle of
momentum conservation in collisions, when put together with special
relativity,
predicts that an object's mass increases with its speed, and how this
startling
prediction has been verified experimentally many times over. The
increase in
mass is related to the increase in kinetic energy by E = mc2.
This formula turns out to be more general: any kind of energy, not just
kinetic
energy, is associated with a mass increase in this way. In particular,
the
tight binding energies of nuclei, corresponding to the energy released
in
nuclear weapons, can be measured simply by weighing nuclei of the
elements
involved.
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