GRAVITATION

Film Loop: Orbiting bodies in force fields: Part I: Positive power laws.

Length: 2:35 min., Black and White, No sound

Using computer generated animation these demonstrations show the behavior of two bodies attracted to each other by a force varying as the positive power of R: the force gets larger as the bodies move farther apart. The behavior is shown for three forces; F proportional to: R, (Rsquared)and (R cubed).

NOTES: The conventions used in the film are all shown in the figure (deleted): the size of the circle shows the mass (2:1 ratio), the direction of the force is along the arrow and the magnitude of the force is proportional to the length of the arrow, the center of mass of the two bodies is marked by a cross, and a line traces the path of each body. The initial velocities are chosen so that the center of mass does not move.

In general, if the period of the X and Y or the radial (r) and the angular (q) components of motion are equal or have an integer ratio, then the orbits will be reentrant - they will close. This is the case in the first sequence which shows two-dimensional simple harmonic motion: F proportional to R. In an inertial frame of reference the bob of a spherical pendulum, of small amplitude, would trace out a pattern similar to either of the two bodies. In other cases of positive power laws shown, the orbits do not close; the figure shows the orbit for F proportional to (Rcubed). The limiting case of the force proportional to large positive powers of R is shown in the last example to be similar to a ball bouncing inside a ring: the source of the force - center of the ring - is not shown.