Physics 152: Gravity
Michael Fowler, UVa.
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.
Galileo's own diagram showing how "natural" steady horizontal motion and "natural" constant downward acceleration when put together create a parabolic path: we just animated it.
Animation of Newton's Classic Thought Experiment: shooting a cannonball horizontally from an imaginary mountaintop above the atmosphere,with sufficient speed, the ball goes into orbit.
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?
Find closed orbits, precessing orbits, unstable orbits, etc. This can also be done with the spreadsheet below.
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.