Michael Fowler, U.Va.

**Discovering Gravity: Galileo, Newton,
Kepler PDF**

*Galileo analyzes a cannonball’s trajectory, Newton imagines the cannon
on a very high mountain shooting the cannonball into orbit, and sees the
analogy to the Moon’s motion, which leads him to conjecture that the
gravitational force extends to the moon and beyond, with strength proportional
to the inverse-square of the distance. Analyzing Kepler's Laws of planetary
motion indicates that a similar gravitational force keeps the planets in their
orbit, suggesting a Universal Law of Gravitation. We give an (optional)
calculus-based proof that the planets’ orbits **are in fact ellipses. *

**Visualizing Gravity: the Gravitational Field PDF**

*Finding the gravitational attraction form a single mass, a pair of masses,
a ring, a hollow sphere and finally a solid sphere, both inside and out. How does gravity change on going down a mine?*

**Working
with Gravity: Potential Energy PDF**

*How potential energy relates to the
gravitational field, near the earth’s surface and far away. Potential energy and escape velocity. Potential and kinetic energies in circular
orbits.*

**Elliptic
Orbits: Paths to the Planets PDF**

*The interesting orbits are ellipses,
or sequences of pieces of ellipses. Some
simple properties of the ellipse make it possible to understand these orbits
well. We briefly discuss other (hyperbolic) orbits, and also the important role
of the slingshot in actually reaching the outer planets.*

*More gravitational phenomena: pairs of
stars orbiting a common center; how a close gravitational source can distort a
planet. *

The Principle of Equivalence: a uniformly accelerating frame of reference is equivalent to a gravitational field. How it necessarily follows that a gravitational field deflects light, and that a clock on the surface of a big planet runs slow.

Create your own planetary orbit with the click of a mouse, and see Kepler’s Laws in action.

*How good is your aim at getting a spaceship to Mars?*

This Excel spreadsheet will calculate planetary orbits over a wide range of initial conditions, and will work for gravitational forces that are not inverse square, producing some strange looking orbits. Convenient numbers, such as GM = 8, correspond to mini solar systems with one-kilogram planets orbiting stars weighing only a hundred million tons or so, but the geometry of the orbits doesn’t depend on the scale, so we can gain intuition about real planetary systems.

*A rather extensive historical lecture, originally for my Galileo and
Einstein class, but you might find it interesting. I will not, however, expect
you to know historical facts on the exams—just understand the physics!*

*Some quotes from Boyle describing his discovery of the famous Law,
followed by a derivation of the Law of Atmospheres—just how the air thins out
at high altitudes. *

*What is viscosity? Exploring fluid friction at the molecular level,
and explaining some surprising results—for example, why the viscosity of a gas
is independent of its density. *

**Calculating Viscous Flow: Velocity Profiles
in Wide Rivers and Circular Pipes PDF**

*M, L and T: all physics equations must have the same dimensions on both
sides. How to make interesting physics predictions without doing much math.*

*Dropping a ball through very viscous fluid. A dimensional prediction of
the dependence of speed on radius. An experiment with glycerin. *

**Inertial Drag Force and the Reynolds Number
PDF**

*Another experiment, this time dropping coffee filters through air, with a
very different result. The Reynolds Number: the dimensionless ratio of
inertial to viscous drag. *

*Definition of the number e. The series expansion of e ^{x}.
Solving the differential equation dy/dx = ay. The Natural
Logarithm.*

*Real numbers as vectors. The square root of *-1*.
Polar Coordinates in the plane. The significance of the formula e ^{i}*

**Some Exponential Integrals PDF**

*An elementary discussion of various exponential integrals that arise in
understanding Maxwell’s velocity distribution in the kinetic theory of gases.*

**From Complex Numbers to the Simple
Harmonic Oscillator PDF**

**SHM as the Shadow of Circular Motion**

*A very simple spreadsheet animating the process. Press and hold the end
of thescrollbar.*

**Crazy Way to Demonstrate Simple Harmonic
Motion**

*From Vladimir Vasc, in the *

**Damped and/or Driven Oscillators
PDF**

*You can safely skip the * sections: they treat the differential equations
in more detail than we need at this point.*

**Spreadsheet for Damped Oscillator**

*This spreadsheet plots the motion of a damped oscillator: you can specify
the initial conditions and degree of damping, and find critical damping. *

**Spreadsheet for Damped Driven Oscillator**

*An external driving force is added to the previous spreadsheet: see how
damping and resonance compete.*

**Introducing Waves: Strings
and Springs PDF **

*Kinds of waves in one dimension: transverse, longitudinal, traveling,
standing.*

*How applying F = ma to a little piece of string leads to an equation that
describes many different waves.*

**Flashlet of Tension
Forces on a Bit of a Waving String**

Finding these forces and applying Newton’s Law to a small piece of string is the way to the Wave Equation. Flashlet by Patrick LeDuc.

**Interference
from Two Sources: Young’s Two Slit Experiment**

**Flashlet
for The Doppler Effect**

**Great Website for Sound
and Musical Instruments**

**Animated Spreadsheet for Adding Harmonic Traveling
Waves to Get Standing Waves**

**Animated Spreadsheet for Two Traveling Waves**

**Building a Triangular Wave: Fourier Series**

**Spreadsheet for Beats, Group Velocity and
Phase Velocity**

*Development of the concepts, some early applications, the Zeroth Law of
Thermodynamics, calorimetry.*

**Thermal Expansion and the Gas Law PDF**

*Linear and volume expansion coefficients, pressure versus temperature for
a gas, the Kelvin scale, the gas law. *

**Early Attempts to Understand Heat: The
Caloric Fluid Theory, and Rumford’s Experiment PDF**

*Lavoisier suggests heat flow is a conserved fluid, like
electricity. Carnot analyzes the steam engine using this idea, and
reaches many of the right conclusions. Rumford manufactures cannons, and
pours cold water on the caloric theory.*

**The Discovery of Energy Conservation: Mayer and Joule PDF**

**The Kinetic Theory of Gases PDF**

**Ideal Gas Thermodynamics: Isotherms, Adiabats,
Specific Heats PDF**

**Kinetic Theory Applets: One Atom Gas,
Many Atom Gas,
Brownian Motion,
Maxwell Distribution.**

**Spreadsheet for Isotherms and Adiabats**

**Carnot
Cycle Flashlet**