**Physics
152 Home Page Next topic: Fluids**

**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.*