Physics
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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.
Animation of
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?
Using Jupiter’s gravity to get way out there: read the instructions, then
go to full
screen.
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.