Michael Fowler, U.Va.
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.
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?
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.
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.
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.
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 ex. 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 eiq = cosq + isinq.
An elementary discussion of various exponential integrals that arise in understanding Maxwell’s velocity distribution in the kinetic theory of gases.
A very simple spreadsheet animating the process. Press and hold the end of thescrollbar.
From Vladimir Vasc, in the
You can safely skip the * sections: they treat the differential equations in more detail than we need at this point.
This spreadsheet plots the motion of a damped oscillator: you can specify the initial conditions and degree of damping, and find critical damping.
An external driving force is added to the previous spreadsheet: see how damping and resonance compete.
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.
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.
Development of the concepts, some early applications, the Zeroth Law of Thermodynamics, calorimetry.
Linear and volume expansion coefficients, pressure versus temperature for a gas, the Kelvin scale, the gas law.
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.
Carnot Cycle Flashlet